uuid int64 541B 3,299B | dataset stringclasses 1
value | text stringlengths 1 4.29M |
|---|---|---|
2,877,628,090,154 | arxiv | \section*{Main text}
\section{Introduction}
Photonics has deep utility in a broad range of scientific and technological domains. Integrated photonic systems enable novel light sources~\cite{Watts2013, Lipson2018}, classical communication platforms \cite{SiliconPhotonics2016, Bergman2018}, and quantum optical processors \cite{Vuckovic2009}. Subwavelength-scale metal and dielectric composites can scatter and localize electromagnetic waves in customizable and extreme ways, producing new modalities in sensing \cite{Tittl1105} and light-matter interactions \cite{Baumberg2019}. Emergent concepts of metamaterials \cite{Urbas2016} and metasurfaces \cite{Capasso2014}, in which materials are structured to perform wavefront engineering tasks, are transforming the field of optical engineering. Much of the versatility of photonics can be attributed to the strong relationship between device geometry and optical response, which allows a wide diversity of optical properties and functions to be tailored from structured dielectric and metallic materials.
Photonic systems are typically analyzed through the framework of two problems. The first is the forward problem: given a structure, what is the electromagnetic response? This is the easier of the two problems and can be solved using one of a number of well-established numerical electromagnetic simulators \cite{FDTD2011, FEM2014}. While these simulators can accurately evaluate Maxwell’s equations, it remains a challenge to manage computational resources when evaluating large simulation domains and large batches of simulations. The second is the inverse problem: given a desired electromagnetic response, how does one design a suitable photonic structure? The solution to the inverse problem cannot be directly evaluated and is challenging to solve because the solution space is non-convex, meaning there exists many local optima. Approaches to this problem are often framed as an exercise in optimization \cite{Campbell2019} and include simulated annealing\cite{SimulatedAnnealing2004, SimulatedAnnealing2016}, evolutionary \cite{lipson2005,jerry2017ACSPhotonics}, objective-first \cite{vuckovic2015}, and adjoint variables algorithms \cite{yablonovitch2013,Sigmund2011}. While great strides have been made in solving the inverse problem, it remains a challenge to identify the best overall device given a desired objective and constraints.
Deep neural networks are a versatile class of machine learning algorithms that utilize the serial stacking of nonlinear processing layers to enable the capture and modeling of highly non-linear data relationships. They offer a fresh perspective on the forward and inverse problems due to the ability of neural networks to mimic non-linear physics-based relationships, such as those between photonic system geometries and their electromagnetic responses. In this Review, we will discuss how deep neural networks can facilitate solutions for both the forward and inverse problem in photonics. We will examine how trained neural networks can function as high speed surrogate Maxwell solvers and how they can be configured to serve as high performance device optimizers. We note that there have been a number of recent reviews on machine learning and photonics \cite{Yuebing2019Review, Rho2020Review, Hegde2020Review} that cover complementary topics to this Review, including reinforcement learning \cite{Rho2020Review, Shirakawa2019RL, Rho2019RL, Rho2019RL2} and the application of photonics hardware to machine learning computation \cite{Yuebing2019Review, Shen2017Hardware}. This Review is unique in that it serves as both a tutorial introducing basic machine learning concepts and a comprehensive guide for current state-of-the-art research developments. Additionally, it covers electromagnetic technologies spanning microwave to optical frequencies, providing a broader conceptual overview of the topic.
It is undeniable that machine learning is a fashionable area of research today, making it difficult to separate the hype from true utility. In spite of the hype, deep learning has the potential to strongly impact the simulation and design process for photonic technologies for a number of reasons. First, deep learning is a proven method for the capture, interpolation, and optimization of highly complex phenomena in a multitude of fields, ranging from robotic controls \cite{abbeel2010autonomous} and drug discovery \cite{DrugDiscovery2015} to image classification \cite{AlexNet2012} and language translation \cite{MachineTranslation2016}. These algorithms will only be getting more powerful, particularly given recent explosive growth of the data sciences field.
Second, deep learning is broadly accessible. Software, ranging from TensorFlow \cite{tensorflow2015} to PyTorch \cite{2017pytorch}, are open source and free to use, meaning that anyone can immediately start implementing and training neural networks. Furthermore, researchers in the machine learning community practice a culture of openness and sharing, making many state-of-the-art algorithms openly available and easy to access. There also are tremendous educational resources, including curricula at universities and online courses, to help researchers get up to speed with the theory and implementation of neural networks.
Third, photonic structures can be readily evaluated with a broad range of electromagnetic simulation tools. These widely available tools enable the quantification of the near- and far-field electromagnetic response by a structure, which is important to facilitate the solving of inverse problems. In addition, they can be used to calculate analytic and numerical gradients, such as the impact of dielectric perturbations to a desired figure of merit. As we will see later in this Review, the computation of such gradient terms can combine with deep learning to yield entirely new and effective modalities of inverse design, such as global topology optimization. The implementation of electromagnetic simulation software tools in conjunction with deep learning programming packages is streamlined using application programming interfaces that come standard with many mainstream computational software \cite{COMSOL-Python, Lumerical-Python, MATLAB-Python}.
Fourth, there exist broadly accessible computational resources that enable large quantities of electromagnetic simulations to be performed, which plays well into the strengths of deep learning approaches. Distributed computing, including cloud-based platforms, allows anyone with an internet connection to parallelize large quantities of simulations \cite{CloudComputing}. Additionally, the advent and advancement of new computing hardware platforms, such as those based on graphical \cite{GPU2010} and tensor \cite{TensorUnit2017} processing units, will also promise to push the computational efficiency and capacity for both electromagnetic simulations and neural network training.
An outline of this Review is as follows. First, we will provide an overview of discriminative and generative neural networks, how artificial neural networks are formulated, and how different electromagnetics phenomena can be modeled through the processing of different data structure types. Second, we will discuss how deep discriminative networks can serve as surrogate models for electromagnetic solvers and be used to expedite the solving of forward and inverse problems. Third, we will show how generative networks are a natural framework for population-based inverse design and can be configured to perform the global optimization of nanophotonic devices. Finally, we will compare and contrast deep learning methods with classical modeling tools for electromagnetics problems, discuss a pathway for future research that pushes the capabilities and speed of neural networks for photonics inverse design, and suggest effective research practices that can accelerate progress in this field.
\section{Principles of deep neural networks}
Deep neural networks consist of multiple layers of neurons connected in series. A neuron is a mathematical function that takes one or more values as its input, and it performs a non-linear operation on a weighted sum of those input values, yielding a single output value (See Box 1). With layer-by-layer processing of data inputted to the network, data features with higher levels of abstraction are captured from lower level features, and complex network input-output relations can be fitted. To train a neural network, a large training data set that the network is to model is first generated using electromagnetic simulations. The training data aids in the iterative adjustment of the neuron weights until the network correctly captures the data distributions in the training set. Modifications to the network weights are performed using a process termed backpropagation that minimizes the network loss function, which specifies the deviation of the network output from the ground truth training set. These terms and concepts are discussed in more detail in Box 2.
The photonic devices modeled by neural networks are described by two types of labels (FIG. \ref{fig:fig1}a). The first type is physical variables describing the device, and it includes the device geometry, material, and electromagnetic excitation source. These labels are delineated by the variable $\textbf{\emph{x}}$. The second type is physical responses describing a range of spectral and performance characteristics. These labels are delineated by the variable $\textbf{\emph{y}}$. In electromagnetics, physical responses can be described as a function of physical variables that is single valued, so that a given input $\textbf{\emph{x}}$ maps to a single $\textbf{\emph{y}}$. For example, a thin-film stack with a fixed geometric and material configuration will produce a single transmission spectrum. However, the opposite is not true: for most problems in electromagnetics, a given physical response $\textbf{\emph{y}}$ does not map to a single $\textbf{\emph{x}}$ but instead maps onto multiple $\textbf{\emph{x}}$'s. For example, a single transmission spectrum can be produced using different thin-film stack configurations. As a result, different classes of neural networks need to be considered depending on the type of device labels being processed by the network. For electromagnetics, the two most commonly used network classes are discriminative and generative networks.
\subsection{Discriminative and generative deep neural networks}
Discriminative networks are capable of regression and classification tasks and can specify complex, nonlinear mapping relations between inputs and outputs \cite{lecun2015deep}. For regression tasks, discriminative networks can interpolate relationships within training data, and the relationship between input and output mappings is that of a single valued function that can support one-to-one or many-to-one mappings. As such, discriminative neural networks can capture the relationship $\textbf{\emph{y}} = f(\textbf{\emph{x}})$ and serve as surrogate physical models that solve the forward problem (FIG. \ref{fig:fig1}b). Compared to a numerical electromagnetic solver, a trained discriminative network can evaluate the forward problem in order-of-magnitude faster time scales \cite{Padilla2019}. It is noted that in these models, the physical variable inputs and physical response outputs must be formulated as discretized data structures. Such a representation is contrary to the continuous form of many real world input-output types, such as freeform device layouts and time sequential events. Nonetheless, these forms can be readily discretized in numerical representation without loss of generality due to ability for Maxwell's equations to be accurately discretized.
\begin{figure}[ht!]
\centering
\includegraphics[width=400pt]{./Figures/Figure1.png}
\caption{
\textbf{Overview of deep learning for photonics.} \textbf{a}| Photonic devices are described by two types of labels, physical variables $\textbf{\emph{x}}$ and physical responses $\textbf{\emph{y}}$. \textbf{b}| Discriminative neural networks can serve as surrogate physical models. The trained network models the relationship $\textbf{\emph{y}} = f(\textbf{\emph{x}})$, which matches with the discrete values $(\textbf{\emph{x}},\hat{\textbf{\emph{y}}})$
in the training set. \textbf{c}| Generative neural networks map latent variables, $\textbf{\emph{z}}$, and conditioning labels, $\theta$, to a distribution of generated devices, $P(\textbf{\emph{x}}|\theta)$. Upon training, the network matches $P(\textbf{\emph{x}}|\theta)$ to the training set distribution, $\hat{P}(\textbf{\emph{x}}|\theta)$. FDFD: finite-difference frequency-domain; FDTD: finite-difference time-domain; RCWA: rigorous coupled-wave analysis; FEM: finite-element method.
}
\label{fig:fig1}
\end{figure}
Deep generative neural networks appear deceptively similar to discriminative neural networks, utilizing the same concepts in deep network architecture and neuron-based data processing. The key difference is that one of the inputs to the network is a latent variable, $\textbf{\emph{z}}$, which is a random variable internal to the network. The term `latent' refers to the fact that this variable does not have an explicit physical meaning. In generative neural networks, the latent variables are sampled from a standard probability distribution, such as a uniform or Gaussian distribution. A single instance of latent variable sampling maps to a single network output, while a continuum of latent variable samplings map to a distribution of network outputs. The neural network can therefore be regarded as a function that maps a standard probability distribution to a complex output distribution.
In photonics, generative networks are typically configured to output a distribution of device layouts. A schematic of such a network is shown in FIG. \ref{fig:fig1}c. The network is conditional and its inputs include the latent variable, $\textbf{\emph{z}}$, and a set of labels, $\theta$, which is a subset of all device labels and can include physical variables and physical responses. These networks are termed `conditional' because the outputted distributions can be considered as probability distributions conditioned on $\theta$. The network learns from a training set consisting of an ensemble of discrete labeled devices, which can be treated as samples from the distribution $\hat{P}(\textbf{\emph{x}}|\theta)$. A properly trained network outputs a device distribution $P(\textbf{\emph{x}}|\theta)$ that matches the training set distribution. Unconditional networks, for which the only input is the latent variable, can also be trained, and these networks generate devices that match an unlabeled training set distribution. There also exist schemes for performing inverse design with generative networks without the use of training sets \cite{GLOnets2019NanoLett, GLOnets2019Nanophot}, which will be discussed later in this Review.
The stochastic nature of generative neural networks distinguishes these networks from discriminative networks. While discriminative networks can capture the relationship between device layouts and optical response from a training set, generative networks focus on learning the properties of the device layout distributions themselves \cite{ma2019probabilistic, Cai2018GAN, Rho2019GAN, an2019multifunctional, Jiang2019ACSNano, an2019generative, liu2020topological}. Moreover, for a given input value of $\theta$, generative networks produce a distribution of outputs and therefore perform one-to-many mappings. We note that there are also classes of generative networks that do not utilize latent random variable inputs \cite{van2016conditional}, but these have limited capabilities and are not widely used to model photonic systems.
\begin{framed}
\noindent \textbf{Box 1 | Building blocks of artificial neural networks.}
\begin{center}
\includegraphics[width=400pt]{./Figures/FigureBox1new.png}
\label{fig:FigureBox1_1}
\end{center}
The fundamental unit in an artificial neural network is the neuron, which receives an input vector $\textbf{\emph{x}}$ and outputs a scalar value $\emph{y}$. The neuron performs two mathematical operations, a weighted sum followed by a non-linear mapping \cite{goodfellow2016deep}. The weighted sum $\emph{a}$ is calculated as: $a = \sum_{i}{w_i}{x_i}+\emph{b} = \textbf{\emph{w}}^{T}\textbf{\emph{x}}+\emph{b}$, where $\textbf{\emph{w}}$ is a trainable weight vector that possesses the same dimension as the input $\textbf{\emph{x}}$ and $\emph{b}$ is a trainable bias term. The non-linear mapping applied to $a$ is performed using a continuous, differentiable, non-linear function \emph{f}, known as the activation function, such that the output of the neuron is $y = f(a)$. Some of the most widely used activation functions are the sigmoid function, the rectified linear unit function, and the hyperbolic tangent function.
Neurons can be connected in different ways to form different modules within a deep neural network. The most commonly used modules in electromagnetics are the fully connected (FC) and convolutional layers. FC layers comprise a vector of neurons, and the inputs to each neuron in one layer are the output values from every neuron in the prior layer. The number of neurons in each layer can be arbitrary and differ between layers.
Upon stacking a large number of FC layers in sequence, the relationship between the input and output data can be specified to be increasingly more complex. The non-linear activation function $\emph{f}$ applied within each layer ensures that such stacking of FC layers leads to added computational complexity that cannot be captured from just a single layer.
Convolutional layers are designed to capture local spatial features within data. Convolutional layers are typically used to process image-based data structures, though they can be generally applied to vector, matrix, and tensor data structures. For a convolutional layer processing a two-dimensional image, a kernel filter is spatially displaced over the width and height of the input image with a constant displacement step. The kernel filter is a small matrix with trainable weights, and it processes small groups of neighboring image elements. At each kernel position, a single value in the output matrix is computed using the same operations as in a single neuron: a dot product between the image elements and kernel filter, followed by a non-linear activation, is performed. As each value in the output matrix derives from a small region of the input image, the output matrix is typically referred to as a feature map that highlights regions of the input image with kernel-like local features. For most convolutional layers, an input matrix is processed with multiple kernels, each producing a unique feature map. These maps are then stacked together to produce an output tensor.
\end{framed}
\clearpage
\begin{framed}
\noindent \textbf{Box 2 | Training of artificial neural networks.}
When a neural network is initialized, all neurons have randomly assigned weights. To properly specify the weights in a manner that captures a desired input-output relationship, the network weights are iteratively adjusted to push the network input-output relationship towards those specified in the training set. This training objective can be framed as the minimization of a loss function, which quantifies the difference between the outputs of the network and the ground truth values from the training set.
For discriminative networks performing regression, consider terms in the training set to be $(\textbf{\emph{x}}, \hat{y})$ and the outputs of the network, given network inputs of $\textbf{\emph{x}}$ from the training set, to be $y$. A common loss function for this problem is mean squared error:
\begin{equation}
L(y, \hat{y}) = \frac{1}{N}\sum_{n=1}^N(y^{(n)} - \hat{y}^{(n)})^2
\label{Eq1}
\end{equation}
$N$ is the batch size. If $N$ is equal to the training set size, the entire training set is used each iteration and the training process is termed batch gradient descent. If $N$ is equal to one, a single training set term is randomly sampled each iteration and the training process is termed stochastic gradient descent. If $N$ is less than the training set size but greater than one, a fraction of the training set is randomly sampled each iteration and the training process is termed mini-batch gradient descent. This training process is typically used in practice, as it provides a good approximation of the gradient calculated using the entire training set while balancing the computational cost of network training.
To understand the loss function form in generative networks, the training set and generated devices must be treated in the context of probability distributions. The training set devices $\{\textbf{\emph{x}}_i\}$ can be regarded as samples from the desired probability distribution spanning the design space, $S$, denoted as $\hat{P}(\textbf{\emph{x}})$. This distribution represents the probability that device $\textbf{\emph{x}}$ is chosen upon random sampling of a device from the design space. Similarly, the distribution of devices produced by the generative network can be treated as a probability distribution spanning the design space and is denoted as $P(\textbf{\emph{x}})$. This distribution represents the probability that device $\textbf{\emph{x}}$ is generated by the network upon sampling of the input latent random variable.
The goal of the training process is to match the distribution of outputted device layouts, $P(\textbf{\emph{x}})$, with the statistical distribution of structures within the training set, $\hat{P}(\textbf{\emph{x}})$. To accomplish this objective, the network loss function should quantify the dissimilarity between the two distributions. One such function is Kullback–Leibler (KL) divergence \cite{kullback1997information}, also known as relative entropy, which is a metric from information theory that quantifies how different one probability distribution is from another. It is defined as:
\begin{equation}
D_{KL}(\hat{P} || P) = \int_{S} \hat{P}(\textbf{\emph{x}}) \log \frac{\hat{P}(\textbf{\emph{x}})}{P(\textbf{\emph{x}})} d\textbf{\emph{x}}
\label{Eq2}
\end{equation}
Another related function is Jensen–Shannon (JS) divergence \cite{lin1991divergence}, which is a symmetric function defined as the average KL divergence of $\hat{P}(\textbf{\emph{x}})$ and $P(\textbf{\emph{x}})$ from their mixed distribution $(\hat{P}(\textbf{\emph{x}}) + P(\textbf{\emph{x}}))/2$:
\begin{equation}
D_{JS}(\hat{P}, P) = \frac{1}{2} D_{KL}(\hat{P} || \frac{\hat{P}+P}{2}) + \frac{1}{2} D_{KL}(P || \frac{\hat{P}+P}{2})
\label{Eq3}
\end{equation}
Both KL divergence and JS divergence are minimized when $P(\textbf{\emph{x}})$ and $\hat{P}(\textbf{\emph{x}})$ are the same. Therefore, specifying either term as the loss function fulfills the network training objective.
For both discriminative and generative networks, backpropagation is used to calculate adjustments to the network weights that lead to a reduction in the loss function, $\nabla_{\textbf{\emph{w}}}L$.
To visualize backpropagation in a simple example, consider a discriminative network comprising a single neuron (see Box 1) that is being trained using stochastic gradient descent. We start from $\nabla_{y}L$, which is the gradient of the loss function with respect to $y$, and we “propagate” the gradient back to $\textbf{\emph{w}}$ using the chain rule.
To compute $\frac{\partial L}{\partial \textbf{\emph{w}}}$, we first compute $\frac{\partial L}{\partial a} = \frac{\partial L}{\partial y} \cdot\frac{\partial y}{\partial a}$, which specifies how $a$ in $f(a)$ should be adjusted to reduce the loss function. We then compute $\frac{\partial L}{\partial \textbf{w}} = \frac{\partial L}{\partial a} \cdot\frac{\partial a}{\partial \textbf{w}}$, which specifies how $\textbf{\emph{w}}$ in $a(\textbf{\emph{w}})$ should be adjusted. Note that for the chain rule to work, all of the mathematical functions involved must be differentiable. Backpropagation readily generalizes to deep networks comprising many layers of connected neurons, in a manner where $\nabla_{\textbf{\emph{w}}}L$ can be calculated for every neuron. Once $\nabla_{\textbf{\emph{w}}}L$ is calculated for all neurons, all weight vectors are updated by gradient descent: ${\textbf{\emph{w}}} := {\textbf{\emph{w}}} - \alpha\nabla_{\textbf{\emph{w}}}L$, where $\alpha$ is the learning rate. For mini-batch gradient descent, $\nabla_{\textbf{\emph{w}}}L$ is calculated for each sampled training set term and these gradients are summed up at each neuron to produce a single term for gradient descent.
\end{framed}
\subsection{Data structures describing electromagnetics phenomena}
There exist distinct data structures in electromagnetics that describe a broad range of phenomena. In this section, we discuss four types of data structures: vectors that describe discrete parameters, images that describe freeform devices, graphs that describe interacting structures, and time sequences that describe time-dependent phenomena. Network architectures and layer configurations are subsequently tailored depending on data structure type.
\begin{figure}[ht!]
\centering
\includegraphics[width=400pt]{./Figures/Figure2.png}
\caption{
\textbf{Overview of data structures for photonics.} \textbf{a}| Image data structures are well suited to represent freeform device layouts, such as those of curvilinear nanostructures. Images are typically processed using convolutional layers in a convolutional neural network (CNN), which use small two- and three-dimensional kernels to transform images into feature map representations. \textbf{b}| Graph data structures are well suited to represent photonic systems with interacting parts, such as nanostructures coupled in the near field. Graphs are processed in graph neural networks (GNNs) that can generalize the optical response of a structure based on its node and edge attributes. \textbf{c}| Time-sequential phenomena can be captured using recurrent neural networks (RNNs). In the wave filter example shown here, the output of the network, $I_{out}$, is a function of the input intensity, $I_{in}$, and the internal state of the network, which learns $\mathbf{E}(t)$. The depiction of the RNN unrolled in time shows how the internal state of the network updates after each time step.
}
\label{fig:fig2}
\end{figure}
\textbf{Discrete data structures}. For relatively basic photonic structure layouts, geometric and optical properties can be described by a vector of discrete parameters. Some of these parameter types are summarized in FIG. \ref{fig:fig1}a and include the height, width, and period of a device geometry, permittivity and permeability of a material, and the wavelength and angle of an electromagnetic excitation source. Many optical properties can also be described as discrete parameters and include device efficiency, Q factor, bandgap, and spectral response sampled at discrete points. Discrete data structures naturally interface with neural network layers that are fully connected (See Box 1). If the objective of the network is to relate discrete input and output data structures, a deep fully connected network architecture will often suffice.
\textbf{Image data structures}. Many photonic devices have freeform geometries that cannot be parameterized by a few discrete variables but are best described as two- or three-dimensional images. An example is a freeform metagrating that diffracts incident light to specific orders (FIG. \ref{fig:fig2}a) and is described as a pixelated image with thousands of voxels. Image data types are effectively processed using a set of convolutional layers in series (see Box 1) that can extract and process spatial features\cite{krizhevsky2012imagenet, szegedy2017inception, simonyan2014very, szegedy2015going}. Neural networks that process image data structures using convolutional layers are termed convolutional neural networks (CNNs). If the objective of the network is to output an image data structure, such as the internal polarization or field distributions within a device, a CNN comprising all convolutional layers can be implemented \cite{Muskens2020}. If the objective of the network is to output a discrete data structure, such as the spectral response or efficiency of a device, high level feature maps from a series of convolutional layers can join with fully connected layers for processing and conversion to the proper data structure.
\textbf{Graph data structures}. For electromagnetic systems consisting of physically interacting discrete objects, graphs are ideal data structure representations. As an example, consider an on-chip photonics system consisting of ring resonators coupled in the near-field (FIG. \ref{fig:fig2}b). The physical attributes of each resonator are embedded in a node of the graph, and edges between two nodes in the graph describe the near-field interactions between two neighboring resonators. The graph structure can be irregular, meaning that different nodes can connect with different sets of neighbors. For the example shown here, neighboring nodes connect only when there is significant near-field coupling.
Graph data structures are suitably processed in graph neural networks (GNN) \cite{henaff2015deep, niepert2016learning, velivckovic2017graph}, which analyze and operate on aggregated information between neighboring nodes in each layer. GNNs can learn of physical interactions between nodes through the training process and are able to generalize the nature of these interactions to different configurations of neighboring nodes. As more layers in the network are stacked, the interactions between nodes that are more distant from each other, such as nearest-nearest neighbors, are accounted for and learned. The outputs of the GNN are abstracted representations of the node and edge properties and the graph structure, and they can be further processed using fully connected layers to output a desired discrete physical response. While GNNs have not yet been extensively studied in the context of photonics, they have been applied to a broad range of physical systems, including the modeling of phase transitions in glasses \cite{bapst2020unveiling}, molecular fingerprint analysis \cite{duvenaud2015convolutional}, and molecular drug discovery \cite{torng2019graph}. The network architectures are highly specialized depending on the application and include Graph Attention Networks \cite{velivckovic2017graph}, Graph Recurrent Networks \cite{li2015gated}, and Graph Generative Networks \cite{ma2018constrained}.
\textbf{Time sequence data structures}. For dynamical electromagnetic systems, the physical variables and responses can be described in terms of time sequences. These continuous time electromagnetics phenomena can be represented in terms of discrete time sequences without loss of generality as long as the discrete time steps are sufficiently small. Consider electromagnetic wave propagation in a waveguide modulated by a ring resonator (FIG. \ref{fig:fig2}c). Both the input and output port signals are time sequences, and the output at a given time not only depends on the input signal at that time, but also on the state of the device (i.e., its internal electric fields) at the previous time step.
Recurrent neural networks (RNN) feed the network outputs back into the input layer, thereby maintaining a memory that accounts for the past state of the system and making them ideally suited to model time sequential systems \cite{graves2013generating, sutskever2014sequence, luong2015effective, weston2014memory}. Unlike the CNN and GNN concepts above, RNNs are more general and can be adapted to all of the network architectures described earlier, allowing them to process discrete, image, and graph data structures. For the RNN unrolled in time in the example shown here (FIG. \ref{fig:fig1}e), we see at the $t_{k}$ time step that the current signal $I_{\text{in}}(t_{k})$ and previous electromagnetic field $\mathbf{E}(t_{k-1})$ are network inputs, and the RNN processes these inputs to update its state $\mathbf{E}(t_{k})$ and output the signal $I_{\text{out}}(t_{k})$. While the output of the network varies in time, the neural network itself is fixed and does not change in time due to the time-translation invariance of Maxwell's equations. It is noted that RNNs are particularly well-suited for electromagnetic wave phenomena modeling because the master equations describing the recurrence relations in RNNs have an exact correspondence to the equations describing wave propagation in the time domain \cite{hughes2019wave}.
\section{Surrogate modeling and inverse design with discriminative models}
\subsection{Overview of electromagnetic devices modeled by discriminative networks}
Initial demonstrations of the neural network modeling of electromagnetic devices date back to the early 1990’s in the microwave community (FIG. \ref{fig:fig3}a). Microwave circuits are an analogue to nanophotonic structures as they are based on components described by the subwavelength limits of Maxwell’s equations. Amongst the first published implementations was the use of a Hopfield neural network, which is a recurrent neural network, for microwave impedance matching \cite{Vai1993}. Through an iterative process, the network could specify how changes in stub position and length could improve network matching. Network weights were determined not through a training process, but by known relationships between stub position and network matching determined from simulations. While this demonstration did not entail a classical trainable discriminative network, it showcased the early potential of neural networks in electromagnetics problems.
\begin{figure}[ht!]
\centering
\includegraphics[width=140mm]{./Figures/Figure3.png}
\caption{
\textbf{Surrogate modeling with discriminative networks.} \textbf{a}| Initial research in the application of neural networks to electromagnetic devices started with microwave systems in the early 1990's and led to the modeling of lumped component devices, transmission lines, and structured surfaces. \textbf{b}| Neural networks were applied to model guided-wave photonic components and waveguides starting in the early 2010's. \textbf{c}| There has been immense recent interest to use neural networks to model various nanostructured optical media, from scatterers to metasurfaces. \textbf{d}| The scattering spectra of concentric nanoshell scatterers with shell thicknesses as inputs can be modeled using a deep, fully connected network. \textbf{e}| The electric polarization distribution within nanostructures under electromagnetic excitation can be modeled using a convolutional neural network. \textbf{f}| The relationship between coupled split ring resonators and the transmission spectrum of a microwave filter can be modeled using a graph neural network. Panel \textbf{a} adapted from REFs. \cite{Gupta1997, Zhang2003, Assuncao2010, Cui2019, katabi2019}. Panel \textbf{b} adapted from REFs. \cite{Figueroa2012, Figueroa2018_coupler, Parsons2019, da2018computing, Ye2019, Obayya2014, Soliman2016, xu2019, Noda2018, Camacho2019}. Panel \textbf{c} adapted from REFs. \cite{peurifoy2018nanophotonic, zongfu2018, Yongmin2018, Mosallaei2018, rho2019, Suchowski2018, conti2018, Zhang2019, hughes2019wave, Muskens2020}. Panel \textbf{d} adapted from REF. \cite{peurifoy2018nanophotonic}. Panel \textbf{e} adapted from REF. \cite{Muskens2020}. Panel \textbf{f} adapted from REF. \cite{katabi2019}.}
\label{fig:fig3}
\end{figure}
Shortly thereafter, deep fully connected discriminative networks with at least two layers were used to model more complex microwave circuit elements, including MESFETs \cite{Nakhla1994}, heterojunction bipolar transistor amplifiers \cite{Prasad1998}, coplanar waveguide components \cite{Gupta1997}, and lumped 3D integrated components such as capacitors and inductors \cite{Zhang2003}. As the complexity of the devices increased, accurate modeling required a concept termed space mapping, in which a deep network that learned spatially coarse device features paired with one that learned spatially fine features \cite{Zhang2003, Bandler2004}. Over the last decade, the scope of deep discriminative networks for microwave technologies has expanded to include frequency selective surfaces \cite{Assuncao2010, Assuncao2014}, metamaterials \cite{Cruz2013}, metasurfaces \cite{Cui2019}, and filters \cite{liu2018hybrid, katabi2019}. More details on deep learning developments in the microwave community can be found in a number of reviews \cite{Devabhaktuni2003, Sanchez2004, Bandler2008} and are worth noting here because of their potential to apply to photonic systems.
Researchers working on silicon photonics and optical fibers started exploring the neural network modeling of guided wave systems in the early 2010’s, around the time when deep learning as a field started undergoing tremendous growth (FIG. \ref{fig:fig3}b). Initial deep discriminative networks utilized two to three total neural layers and could learn the bandgap properties of simple photonic crystals \cite{Figueroa2012}, the dispersion properties of photonic crystal fibers \cite{Obayya2014}, and the propagation characteristics of plasmonic transmission lines \cite{Soliman2016}. More recent demonstrations focused on devices with additional geometric degrees of freedom and included different classes of photonic crystal fibers \cite{Rahman2019}, 3D photonic crystals \cite{da2018computing}, photonic crystal cavities \cite{Noda2018}, plasmonic waveguide filters\cite{xu2019}, in-plane mode couplers and splitters \cite{Figueroa2018_coupler, Parsons2019, Smy2019}, Bragg gratings \cite{Camacho2019}, and free space grating couplers \cite{Ye2019,Takenaka2020}.
Deep learning modeling of free space-based nanophotonic systems has only been recently researched in the last few years (FIG. \ref{fig:fig3}c). The first published example to the best of our knowledge is from 2017 and is the modeling of the scattering spectral response of concentric metallic and dielectric nanoshells \cite{peurifoy2018nanophotonic}. Discriminative networks have since modelled the spectral responses of other plasmonic systems including chiral nanostructures \cite{Yongmin2018,Fang2019,tao2020exploiting}, planar scatters \cite{Suchowski2018}, absorbers \cite{rho2019}, lattice structures with tailored coloration profiles \cite{ramunno2019}, phase change material-based smart windows \cite{Abdulhalim2019}, and nanoslit arrays supporting Fano resonances \cite{Martin2020}. Discriminative networks have also been used to model artificial photonic materials in the form of dielectric metagratings \cite{Mosallaei2018}, dielectric metasurfaces \cite{Zhang2019, Padilla2019}, graphene-based metamaterials \cite{QingLiu2019,Wakabayashi2020}, and scatterers for color design \cite{Zongfu2019}, and they have been applied to thin film dielectric stacks serving as color filters \cite{zongfu2018} and topological insulators \cite{conti2018,chen2019,singh2020mapping}.
The collection of surrogate models summarized above encompasses a wide range of deep learning strategies that span differing network architectures, training strategies, and modeling capabilities. To capture in more detail how deep discriminative networks are implemented in electromagnetics systems, we discuss a few representative examples.
\textbf{Scattering spectra modeling}. In REF. \cite{peurifoy2018nanophotonic}, a fully connected deep neural network reconstructed the scattering properties of nanoparticles consisting of eight concentric dielectric shells of alternating silica and titania dielectric material (FIG. \ref{fig:fig3}d). The input values to the network were the discrete nanoshell thicknesses, and the output was the scattering cross section between 400 and 800 nm, sampled over 200 points. The network itself contained four fully connected layers with 250 neurons each, and the spectra of 50,000 scatterers with random nanoshell thicknesses were generated for training using a transfer matrix formalism. Scattering spectra generated by the trained neural network from a random geometry produced accurate profiles that demonstrated the ability for the network to perform high level interpolation of the training data.
\textbf{Electric polarization modeling}. In REF. \cite{Muskens2020}, a CNN predicted the vectorial polarization distribution internal to a nanophotonic structure, given a fixed electromagnetic excitation (FIG. \ref{fig:fig3}e). The input to the network was a three-dimensional matrix that represented a nanophotonic structure discretized into small subwavelength-scale voxels. The output was matrices that represented the electric polarization components at each voxel. A fully convolutional CNN was a sensible choice for this task because the electric field distributions within a nanostructure are strongly spatially correlated with the detailed geometric features of the nanostructure. The network architecture utilized an "encoder-decoder" scheme consisting of a series of convolutional and deconvolutional layers, which is a dimensionality reduction scheme that enables high level features of the input matrix to be captured and used for data processing. The training data comprised approximately 30,000 random structures and their calculated field profiles. The trained network could predict the internal fields of a random structure with high accuracy, though approximately 5\% of random structure inputs produced predicted internal fields that strongly deviated from simulated values.
\textbf{Microwave filter modeling}. In REF. \cite{katabi2019}, a deep GNN predicted the $s_{21}$ characteristics, which are the microwave analogue to transmission spectra, of a microwave circuit comprising three to six split ring resonators. The input to the network was a graphical representation of the circuit, where the graph nodes contained information about individual ring resonator geometries and the edges contained information about the relative resonator positions (FIG. \ref{fig:fig3}f). Each GNN layer contained two sub-networks, an edge processor that captured the near-field coupling between neighboring resonators and a node processor that captured the electromagnetic properties of individual ring resonators based on their geometries and coupling with neighboring resonators. The $s_{21}$ spectra of 80,000 randomly generated circuits were generated as training data using a fullwave commercial simulator. The trained network was capable of accurately computing $s_{21}$ for many resonator configurations four orders of magnitude faster than a commercial solver.
These representative examples capture a number of general trends common to deep discriminative networks. First, trained neural networks serve as reasonably accurate surrogate models for nanophotonic systems with limited complexity (i.e., described by on the order of ten physical parameters). An exception is the CNN mapping of nanostructure layout with internal electric field, which is a special case due to the close correlation between nanostructure geometry and polarization profile. As the surrogate model represents a simplified approximation of the simulation space, there are always a fraction of cases where model accuracy is poor. Second, discriminative neural network training is computationally expensive. Most of this expense arises from the generation of training data, which involves the simulation of tens of thousands of device examples or more using full wave electromagnetic solvers. Third, a trained network can calculate an output response that is orders of magnitude faster than a full wave solver. As such, the decision to train a discriminative neural network requires an application in which the benefits of having a high speed surrogate solver outweighs the substantial one-time computational cost for network training.
\subsection{Inverse design with deep discriminative networks}
There are three general classes of inverse design methods based on trained discriminative networks. The first class, outlined in FIG. \ref{fig:fig4}a, is a gradient descent method based on backpropagation. Initially, a device consisting of a random geometry or educated guess is evaluated by the trained network. The error between the outputted and desired response is then evaluated by the loss function and is iteratively reduced using backpropagation. Unlike the network training process, in which backpropagation reduces the loss by adjusting the network weights, loss here is reduced by fixing the network weights and adjusting the input device geometry. It is noted that many optical design problems do not possess unique solutions: there exist multiple device layouts that can exhibit the same desired optical response. As such, for neural networks that capture this non-uniqueness, different initial device layouts located within distinct domains of the design space will produce different final device layouts after optimization.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\linewidth]{./Figures/Figure4.png}
\caption{
\textbf{Inverse design with discriminative networks.} \textbf{a}| Backpropagation can be used with a trained discriminative neural network to perform gradient-based inverse design. An initially random device geometry input gets iteratively perturbed in a manner that pushes its optical response closer to that of a desired value. \textbf{b}| Examples of microwave filters designed using backpropagation with a trained graph neural network model. The resulting designs often match (top) but do not always match (bottom) the desired spectral response. \textbf{c}| Classical optimization algorithms can utilize neural networks as high speed surrogate solvers to expedite the optimization process. In this algorithm, a surrogate solver is utilized in a genetic algorithm to coarsely search the design space, and the backpropagation method is then used to locally optimize the device. \textbf{d}| Schematic of the inverse design space of a toy model, where the independent variable is the optical response and the dependent variable is the device layout. Due to the presence of multiple devices that exhibit the same optical response, the design space has multiple branches. Discriminative models that attempt to directly capture this design space will not get properly trained. \textbf{e}| A multi-branched neural network can be trained to fit a multi-branched inverse design space. \textbf{f}| Inverse networks can be implemented by first training a forward surrogate model, which learns to model a simplified version of the design space, and then to train an inverse network in tandem with the forward network.
Panel \textbf{b} adapted from REF. \cite{katabi2019}. Panel \textbf{c} adapted from REF. \cite{liu2018hybrid}. Panels \textbf{d} and \textbf{e} adapted from REF. \cite{Zhang2018MultiBranch}. Panel \textbf{f} adapted from REF. \cite{Zhang2018MultiBranch, zongfu2018}.}
\label{fig:fig4}
\end{figure}
The backpropagation method was initially used in the 1990's for the inverse design of microwave circuits \cite{Nakhla1995, Prasad1998} and more recently to tailor the spectral properties of nanoparticle scatterers \cite{peurifoy2018nanophotonic}, microwave filters \cite{katabi2019}, and photonic crystals \cite{Noda2018}. As an inverse design tool, this method can produce devices that exhibit the desired optical response, as shown in the example of microwave filters (FIG. \ref{fig:fig4}b, upper). However, it does not always work well (FIG. \ref{fig:fig4}b, lower). One reason is that the neural network may not be accurately capturing the physical relationships between device geometry and response. This problem can be addressed by increasing the size of the training set. Another reason is that the device can get trapped in an undesired local optimum during the backpropagation process. This issue can be addressed by attempting optimizations with different initial device layouts, which leads to the exploration of different parts of the surrogate neural network design space. This issue can also be mitigated using alternatives to gradient descent, such as Adam optimization \cite{kingma2014adam}, which uses momentum terms during backpropagation.
The second class of inverse methods is hybrid optimization packages that utilize discriminative networks as solvers for conventional iterative optimization algorithms. These algorithms span a wide range of well-established concepts in the optimization community and include Newton's methods \cite{Camacho2019}, interior-point algorithms \cite{Mosallaei2018}, evolutionary algorithms \cite{Cruz2013, Figueroa2018_coupler, liu2018hybrid, Hegde2019}, iterative multivariable approaches \cite{ramunno2019}, trust-region methods \cite{Abdulhalim2019}, a fast forward dictionary search \cite{Padilla2019}, and particle swarm optimization \cite{Assuncao2010, Cui2019}. Compared to optimization with conventional fullwave solvers, these hybrid packages can perform optimization with orders-of-magnitude faster speeds, reducing the total optimization time from hours and days to minutes.
Unlike the backpropagation method discussed above, hybrid methods can support customized optimization strategies based on the needs of the designer. If a global search of the design space is required, global optimizers such as genetic algorithms can be used. Local optimizers such as Newton's methods are sufficient if good initial device layouts are known and the design space is sufficiently smooth. Optimizers can even be configured to combine both global and local searching of the design space. The flowchart of such an example is shown in FIG. \ref{fig:fig4}c for the optimization of microwave patch antennas and filters \cite{liu2018hybrid}: a genetic algorithm coarsely searches for good regions of the design space while gradient-based optimization locally refines the devices. The neural network accelerates the full optimization algorithm by serving both as a surrogate solver and as a gradient-based optimizer.
In the third class, an inverse discriminative network can be configured such that its input is the desired optical response and its output is the device geometry. While this problem setup nominally appears to be the most straightforward way to perform inverse design, it is difficult to execute in practice because of the non-uniqueness of optical design solutions for many problems (see Section 2.1) The issue is visualized in FIG. \ref{fig:fig4}d for a toy model and shows regions of the inverse design space in which individual inputs can take three possible outputs located along three distinct branches. During the training process, training data from a particular branch will attempt to push the surrogate model to that branch, and the net result is that the surrogate model will not converge to any branch and remain improperly trained.
One solution is to limit the design space through proper parameterization of the problem, to enforce a mostly unique mapping between device geometry and optical response. This concept was used to train an inverse network for plasmonic metasurfaces consisting of coupled metallic disks \cite{Li2020OptLett}. Another solution is to use a multi-branched neural network that can be configured to output multiple devices for a given input (FIG. \ref{fig:fig4}e). In these networks, a special loss function ensures that specific network branches produce outputs that map onto unique design space branches \cite{Zhang2018MultiBranch}. One other solution is to first train a forward discriminative network serving as a surrogate model, fix its weights, and then to use it in tandem with an inverse network to train the inverse network\cite{zongfu2018} (Figure \ref{fig:fig4}f). The forward surrogate model represents a simplified version of the full design space, reducing the quantity of non-unique solutions posed to the inverse network. Tandem networks have been effectively used for the inverse design of core-shell particles \cite{Rho2019ACS}, metasurface filters \cite{Zhang2019}, topological states in photonic crystals \cite{conti2018, chen2019, singh2020mapping}, planar plasmonic scatterers \cite{Suchowski2018}, and dielectric nanostructure arrays supporting high quality factors \cite{xu2019enhanced}.
\subsection{Dimensionality reduction with discriminative networks}
In order to train a discriminative network that adequately captures the correct mapping between the whole design and response space, a sufficient number of data points in the training set is required to adequately sample these spaces. For devices residing in a relatively low dimensional design space, a brute force sampling strategy is computationally costly but tractable: as we have seen, tens of thousands of training set devices can be simulated for device geometries described by a handful of geometric parameters. However, the problem becomes increasingly intractable as the dimensionality of the design space increases due to the curse of dimensionality \cite{zimek2012survey,marimont1979nearest}, which states that the number of points in the training set required to properly sample the design space increases exponentially as the dimensionality of the design space increases. This scaling behavior is plotted in FIG. \ref{fig:fig5}a together with data points from ten different studies, which follow the exponential trend line. For freeform devices described as images with hundreds to thousands of voxels, a brute force sampling strategy involving all of these voxels would require many billions of training set devices. Even with distributed computing resources, such an approach is not practical.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\linewidth]{./Figures/Figure5.png}
\caption{\textbf{Methods for dimensionality reduction.} \textbf{a}| Plot of training set size versus the geometric degrees of freedom of the modeled device. Points from ten different studies show exponential growth in training set size as a function of degrees of freedom, which is consistent with trends from the curse of dimensionality. \textbf{b}| Visualization of the method of principle components analysis, where a hyperplane in the full design space of grating couplers defines a low-dimensional subspace of high performance devices. \textbf{c}| Dimensionality reduction of images of devices is performed by encoding the image into a Fourier representation and then cropping that representation. \textbf{d}| Schematic of an autoencoder neural network. The encoder transforms input data to low dimensional latent vectors while the decoder attempts to reconstruct the input data from the latent vectors with minimal distortion. \textbf{e}| An autoencoder reduces the dimensionality of the design and optical response space of a reconfigurable metasurface, enabling an inverse discriminative network to be trained for the problem. \textbf{f}| A convex hull delineates suitable design space regions within a dimensionality-reduced data set representing differing dielectric meta-atom layouts. Panel \textbf{a} adapted from REFS. \cite{Zongfu2019,peurifoy2018nanophotonic,Yongmin2018,zongfu2018,Suchowski2018,Muskens2020,Mosallaei2018,Padilla2019,Smy2019,MetaNet,rho2019}. Panel \textbf{b} adapted from REF. \cite{melati2019mapping}. Panel \textbf{c} adapted from REF. \cite{liu2020topological}. Panel \textbf{e} adapted from REF. \cite{kiarashinejad2020deep}. Panel \textbf{f} adapted from REF. \cite{Adibi2020ConvexHull}}.
\label{fig:fig5}
\end{figure}
Fortunately, for many high dimensional design spaces, candidate devices actually reside in or can be well approximated by a subspace described by a reduced number of feature parameters. For these problems, preprocessing these device representations from a high to low dimensional space without critical information loss would make the discriminative modeling of these devices more tractable. In this section, we discuss three dimensionality reduction techniques that have been applied to photonic systems.
\textbf{Principal components analysis}. Principal components analysis (PCA) is a classical statistical technique in which a set of device parameters residing in a high dimensional space is projected to a low dimensional subspace. The first step to performing this projection is identifying a new orthogonal basis for the device parameters in the high dimensional space. This basis is determined through the sequential specification of basis vectors with maximal component scores, which refer to the amount of information preserved when the device parameters are projected to the corresponding basis vector. A subset of these basis vectors with the largest component scores is then selected to define the low dimensional subspace. High dimensional device data structures projected to such a subspace will retain the salient features of the original devices with minimal distortion.
PCA was used in REF. \cite{melati2019mapping} to expedite the inverse design of vertical fiber grating couplers (FIG. \ref{fig:fig5}b). Local optimization was first performed on a sparse distribution of devices within the full five-dimensional design space. PCA was then performed on this collection of locally optimized devices, which yielded a two-dimensional subspace that captured locally optimal devices. Finally, a brute force search within this low dimensional subspace was performed with modest computation resources to map out this subspace and identify even higher efficiency devices. While this study did not utilize a deep neural network, the low dimensional representation of devices based on PCA could be utilized in a discriminative network to perform device surrogate modeling in a particular design subspace.
\textbf{Fourier transformations}. Dimensionality reduction can be performed on device structures by eliminating high spatial frequency terms in image representations of the devices \cite{liu2020topological}. The concept is outlined in FIG. \ref{fig:fig5}c. First, level set functions of the device images are created, which ensures that the initial and dimensionality-reduced image device representations are binary. Next, the level set functions, which represent the shape boundaries, are Fourier transformed and high frequency components in the Fourier space are cropped, reducing image dimensionality from 64$\times$64 to 9$\times$9. Due to symmetry considerations, nine of the elements in this concatenated Fourier representation are unique and constitute the low dimensional device space. To reconstruct the images in real space, the low dimensional image representations in the Fourier domain are converted using an inverse Fourier transform and thresholding. As a demonstration, this dimensionality-reduced subspace was used to encode freeform grating elements that could diffract light to different channels. With this low dimensional representation, a deep discriminative network that could learn the relationship between the grating subspace representation and diffractive response could be properly trained using only 12,000 devices.
\textbf{Autoencoders}. Autoencoders, schematically shown in FIG. \ref{fig:fig5}d, are neural networks that comprise two parts \cite{kramer1991nonlinear, baldi2012autoencoders}. The first is an encoder network that maps input data from a high dimensional design space to a low dimensional latent vector. The second is a decoder network that takes the latent vector representation of the data and maps it back to the high dimensional space, in an attempt to reconstruct the original data. Dimensionality reduction is therefore achieved through the encoding process. The loss function, also termed reconstruction loss, attempts to minimize the difference between the decoded and original data and often has the form of least mean squares error. As such, the network attempts to learn the best encoding-decoding scheme to achieve a high level of dimensionality reduction while also maintaining a low level of information loss upon decoding. Compared to PCA, which is limited to linear transformations, autoencoders can learn more complex, low dimensional representations of the data due to the highly nonlinear nature of neural networks.
Autoencoders were utilized in REF. \cite{kiarashinejad2020deep} to design reconfigurable metasurfaces, based on phase change materials, which can perform amplitude modulation of a normally incident beam (FIG. \ref{fig:fig5}e). In this work, dimensionality reduction was performed to reduce the design space of device geometry parameters from 10 to 5 dimensions and the response space of discretized spectra from 200 to 10 dimensions. The resulting subspace supported an approximate one-to-one mapping between the reduced design and response spaces, from which an inverse discriminative network was trained with only 4000 devices. The trained inverse network, together with the encoder and decoder networks that mapped the data between high and low dimensions, could perform inverse design with high accuracy.
Autoencoders were also used in an inverse algorithm that could predict the layout of a digital metasurface, comprising a grid of metal or air pixels, given a desired spectral response \cite{Qu2020SVM}. To train this algorithm, an autoencoder was first used to perform a dimensionality reduction on spectra in the training set, from 1000 to 128 data points. The resulting latent space representation of these spectra and their corresponding device layouts were then used to train multiple support vector machines (SVMs), one for each device pixel. SVMs are classical machine learning algorithms that serve as binary classifiers, and each SVM learned to classify latent space inputs to be either metal or air, for the corresponding pixel. The final encoder-SVM scheme produced metasurface layouts with spectral properties that closely matched the desired inputted spectra.
For autoencoders trained on device geometries, additional analysis of the low dimensional latent space representation of these devices can be performed in an attempt to further delineate the full design space of all feasible devices. These include fitting training set devices to a convex geometric manifold, termed a convex hull, in the low dimensional space (FIG. \ref{fig:fig5}f), and using a SVM to classify devices as either feasible or unfeasible. These concepts have been applied to analyze digital plasmonic nanostructures and dielectric nanopillar arrays \cite{Adibi2020ConvexHull}.
\section{Generative networks}
\subsection{Adapting generative networks to photonic systems}
Deep generative networks have been an active topic of study in the computer science community for the last decade and have produced impressive, eye-catching results. In one famous example, a generative network was trained to produce photorealistic images of faces, using a training set of millions of face images collected from the internet \cite{karras2018progressive}. This example is representative of the conventional way generative networks are typically utilized in the computer science community: networks are trained to generate classes of images that intrinsically exhibit a wide range of diversity, such as faces, animals, and handwritten digits. Training methods exclusively rely on learning the statistical structure of the training set, as there are no analytic expressions to directly quantify the quality of generated images or to calculate gradients to directly improve the images. For example, there is no equation to score the quality of a face image or gradients to make an image more face-like \cite{salimans2016improved}.
In photonics inverse design, on the other hand, the goal is qualitatively different: it is to find just one or a small handful of devices that achieve a specific design objective. In addition, electromagnetic simulators can assist in the network training process by directly evaluating the electromagnetic fields, scattering profiles, and performance gradients of generated devices. Performance gradients refer to structural perturbations that can be made to a device to improve its performance, and they can be calculated using the adjoint variables method or auto-differentiation. With the adjoint variables method, performance gradients that specify perturbations to the dielectric constant value at every device voxel, in a manner that improves a figure of merit, are calculated using forward and adjoint simulations \cite{Sigmund2011,yablonovitch2013, Sell2017NanoLett, Phan2019LightSci, Fan2020MRSBull, hughes2018adjoint}. Auto-differentiation is mathematically equivalent to backpropagation \cite{Autograd, minkov2020inverse, hughes2019forward} and can directly evaluate performance gradients at every device voxel, pending the use of a differentiable electromagnetic simulator. Both methods can be performed iteratively to serve as a local freeform optimizer based on gradient descent.
\begin{figure}[ht!]
\centering
\includegraphics[width=400pt]{./Figures/Figure6.png}
\caption{
\textbf{Concepts for implementing generative neural networks for inverse design.} \textbf{a}| An unconditional network learns favorable regions of the design space from a limited training set of devices and can be sampled to find similar, better devices. \textbf{b}| A conditional network can generate distributions of devices for conditional parameter labels interpolated from the training set. In this schematic, a network learns the distributions of training set devices operating at 800 nm and 1000 nm and can interpolate the distribution of devices operating at 900 nm. \textbf{c}| Classical optimization methods can be used to search the latent space within a trained network for optimal devices. The search space is limited to a device distribution specified by the training set.
}
\label{fig:fig6}
\end{figure}
These considerations have led to new implementations of generative networks for photonics inverse design, and design schemes that utilize training sets for network training can be grouped into one of three strategies. The first is to train an unconditional network with a set of devices that samples a small, targeted subset of the design space (FIG. \ref{fig:fig6}a). These training sets can be ensembles of disparate shapes that collectively exhibit a set of desired optical responses, in which case a trained network would generate distributions of devices that more thoroughly fill out this design subspace. These training sets can also comprise variants of the same device type such that a trained network would generate even more geometric variations of that device type, some of which are higher performing than those in the training set.
The second is to train a conditional network with sets of high performance devices (FIG. \ref{fig:fig6}b). If the training set consists of devices operating with specific discrete values of these conditional labels, the trained network will be able to generalize and produce device layouts across the continuous spectrum of label values. This ability for the network to generate devices with conditional label values interpolated from those in the training set is analogous to regression with discriminative networks.
The third is to initially train either a conditional or unconditional generative network and then to use conventional optimization methods to search within the latent space for an input value that generates a structure with the desired optical properties (FIG. \ref{fig:fig6}c). This method is related to that discussed previously in which a discriminative model is used as a surrogate electromagnetic solver in conjunction with conventional optimization methods. A principle difference here is that generative networks enable more control over the search space of candidate devices: the distribution of generated devices is constrained by the training set, and furthermore, the generative network uses a proper, low-dimensional latent space to represent the training set.
\subsection{Generative model types}
Deep generative networks are a relatively new innovation, and much of the foundation of the field has been set in the last decade. Unlike discriminative networks, which are described by relatively generic deep network architectures, a range of generative models have been developed that assume different statistical properties about the training set and generated data distributions. Amongst the first deep generative models developed were autoregressive models \cite{uria2013rnade, oord2016wavenet}, which were applied to image generation in 2011 \cite{larochelle2011neural}. Images are generated pixel-by-pixel with the assumption that the $i^{th}$ generated pixel depends only on the value of all previous pixels \cite{oord2016pixel, van2016conditional}. These models do not use a latent variable but determine the value of each generated pixel by sampling an explicit conditional probability distribution. By explicit, we mean that these statistical distributions are described by analytic expressions.
Variational autoencoders (VAEs) were introduced in 2013 \cite{kingma2013autoencoding} and are able to learn salient geometric features in a training set and statistically reconstruct variations of these features through sampling an explicit latent variable distribution. While these explicit statistical forms may not perfectly capture variations within the training set, they have explicit fitting parameters that help simplify the training process. The most advanced and best performing generative network to date is the generative adversarial network (GAN), which was introduced in 2014 \cite{goodfellow2014generative} and learns the implicit statistical distributions of training sets. By implicit, we mean that the distributions have no predefined form. These models have the potential to better capture highly complex statistical trends within training sets but are less straight forward to train. GANs have very quickly evolved to support a high degree of sophistication, leading to demonstrations such as photorealistic face generation described earlier. In this section, we will discuss in more detail the architecture of VAEs and GANs, which are two of the more mainstream models used in photonics.
\begin{figure}[ht!]
\centering
\includegraphics[width= 0.8\linewidth]{./Figures/Figure7.png}
\caption{
\textbf{Overview of variational autoencoders (VAEs).} \textbf{a}| Schematic of a VAE. Like an autoencoder, a VAE possesses an encoder-decoder network structure and attempts to reconstruct inputted data with minimal distortion. However, instead of encoding input data to fixed latent vectors, the encoder maps input data to multivariate Gaussian distributions in the latent space, which is then sampled and decoded. \textbf{b}| A conditional VAE can train a decoder serving as an inverse network for meta-atoms. The inputs include a desired spectrum and latent variables, and the output is a distribution of devices. \textbf{c}| The decoder of a trained VAE can be used in the inverse design of meta-atoms by optimizing the latent space with evolutionary optimization algorithms. E: encoder; D: decoder.
Panel \textbf{b} adapted from REF. \cite{ma2019probabilistic}. Panel \textbf{c} adapted from REF. \cite{liu2020hybrid}. }
\label{fig:fig7}
\end{figure}
\textbf{Variational autoencoders}.
To understand how a VAE works, we first revisit the autoencoder, which uses an encoder to map high-dimensional data to low-dimensional latent vectors and a decoder to map the latent vectors back to the high dimensional space. The latent vectors capture principle features common to the training set. Given the objective that we want to generate variants of structures similar to the training set, our goal is to frame the decoder as a generative neural network: by treating the latent vectors as latent variables and sampling within this latent space, the hope is that variations of different principle features get sampled and decoded into viable structures. Unfortunately, autoencoders do not properly interpolate the training set, and random sampling of the latent vectors followed by decoding produces devices with no relation to the training set.
VAEs are regularized versions of autoencoders that overcome these limitations by better managing the latent space \cite{kingma2013autoencoding, CVAE2015, makhzani2015adversarial, kingma2014semi}, as shown in FIG. \ref{fig:fig7}a. We summarize two key features of VAEs with the caveat that there is a lot of probabilistic modeling required to fully understand VAEs. First, the encoder maps input data points not to discrete points in the latent space, but to distributions over the latent space. These latent space distributions vary as a function of the input data and are typically set to be multivariate Gaussian distributions, each of which are described by a mean and covariance matrix. The encoder therefore outputs mean and covariance matrix values that are a function of the input data. The decoder now has the form of a generative network and uses these latent space distributions as the latent variable input to generate distributions of data. Second, the loss function includes both reconstruction loss, which is the same term used for autoencoders, and regularization loss, which is the KL divergence between the multivariate Gaussian distribution returned by the encoder and a standard multivariate Gaussian distribution. The regularization loss helps to ensure that the latent space is not irregular and that samplings of the latent variable correspond to principle features learned from the training set.
Two strategies for using VAEs in the inverse design of freeform subwavelength-scale meta-atoms are summarized as follows. In the first, the training set consisted of images of meta-atom patterns and their spectra, which were both encoded into the low-dimensional latent space\cite{ma2019probabilistic} (FIG. \ref{fig:fig7}b). For the decoder, the inputs were the latent variable and the desired spectra, so that the decoder was able to generate a distribution of device patterns given the spectra input. The trained decoder was able to decode the desired spectra, together with latent variables sampled from a standard Gaussian distribution, into device patterns. In the second strategy, a VAE was combined with evolutionary optimization \cite{liu2020hybrid} (FIG. \ref{fig:fig7}c). The training set comprised meta-atom patterns of various shapes, such as circles, crosses, and polygons, and a VAE was trained to map this subset of design patterns to the low-dimensional latent space. This latent space was then used as the basis for genetic algorithms, where a batch of latent vectors was decoded as device patterns, evaluated using electromagnetic simulations or a surrogate network, and evolved until an optimized latent vector corresponding to a suitable device pattern was identified.
\textbf{Generative adversarial networks}. GANs are actually a pair of neural networks that train together (FIG. \ref{fig:fig8}a): a generative network that generates distributions of device images from a latent variable input, and a discriminative network that serves as a classifier and attempts to determine if an image is from the training set or generator \cite{goodfellow2014generative, radford2015unsupervised, gulrajani2017improved, brock2018large}. In the original GAN concept \cite{goodfellow2014generative}, the training process is framed as a two player game in which the generator attempts to fool the discriminator by generating structures that mimic the training set, while the discriminator attempts to catch the generator by better differentiating real from fake structures. This training process is captured in the loss functions specified for each network. For the generator, the loss function minimizes JS divergence between the training set and generated device distributions, in an attempt to get these device distributions to converge. For the discriminator, the loss function maximizes JS divergence, in an attempt to differentiate these two distributions as best as possible. Upon the completion of training, the fully trained discriminator will be unable to differentiate generated images from those in the training set, indicating that the generator is producing device distributions that approximate the training set. With this training method, the generator learns the implicit form of the training set distribution without any explicit assumptions of its statistical properties. Since the inception of the GAN concept, alternative loss functions including Wasserstein distance \cite{WGAN2017} and Wasserstein distance with gradient penalties \cite{WGAN2} have been implemented in so called WGANs. These loss functions help stabilize the network training process and prevent the network from generating an overly narrow distribution of output values.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\linewidth]{./Figures/Figure8.png}
\caption{
\textbf{Generative adversarial networks (GANs) for freeform device modeling.} \textbf{a}| Schematic of a GAN. The discriminator attempts to differentiate whether an inputted device is from the training set or generator, while the generative network attempts to fool the discriminator by generating devices mimicking the training set. The trained generator generates device distributions that match the training set distribution. \textbf{b}| Images of generated devices and their spectral properties from a GAN conditioned on spectral response. The generator trains with two discriminators, one that ensures that the generated devices match those in the training set and one that ensures that the generated devices have spectra matching the desired conditional input. \textbf{c}| Device layouts and performance histogram of thermal emission structures generated by an unconditional GAN. \textbf{d}| Progressive growing GAN (PGGAN) that trains using progressive growth of the network architecture and training set over multiple cycles. The trained network could generate robust metagratings for a range of wavelengths and deflections with efficiencies comparable to or exceeding those of the best devices designed using adjoint-based topology optimization. The black squares in the bottom left plot indicate wavelength-deflection angle pairs for devices in the original training set. PGGAN: Progressive-growing generative adversarial network.
Panel \textbf{b} adapted from REF. \cite{Cai2018GAN}. Panel \textbf{c} adapted from REF. \cite{Boltasseva2019GAN}. Panel \textbf{d} adapted from REF. \cite{wen2019progressivegrowing}.}
\label{fig:fig8}
\end{figure}
In an early example of GANs applied to photonics inverse design, a conditional generator was trained that could generate images of plasmonic nanostructures when the desired transmission spectrum was inputted (FIG. \ref{fig:fig8}b) \cite{Cai2018GAN}. The training set comprised a variety of freeform shapes, ranging from disks to crosses, and their corresponding transmission spectra. The training process involved the use of two discriminative networks to improve the generator. The first was the adversarial discriminator, which learned to differentiate generated from training set patterns and ensured that the generated patterns mimicked the training set. The second was a pretrained surrogate simulator, which evaluated the spectral response of generated nanostructures and ensured that the generated patterns exhibited the desired transmission spectrum. The trained generator produced distributions of shapes that mimicked those in the training set and served as an inverse network within this design space. A similar type of network scheme that utilized two discriminators was used to generate single-layer \cite{hodge1} and multi-layer \cite{hodge2} radio frequency metasurfaces, where individual metasurface layers were generally represented as matrix elements stacked in a tensor. GANs were also used to generate plasmonic nanostructures, given the desired reflection spectrum as the conditional input \cite{Rho2019GAN}, and freeform dielectric meta-atom structures as a function of amplitude and phase \cite{an2019multifunctional}. These GANs did not use a surrogate simulator during training. Instead, the former study added terms to the generator loss function to facilitate matching of the generated shapes with the desired input spectrum, while the latter used a simulator to evaluate and filter for high quality devices produced from the trained generator.
GANs have also been implemented using training data consisting of topology-optimized nanostructures. By restricting the design space of the training data to only high-performance freeform devices, the generative network exclusively focuses on learning the geometric features of freeform structures and does not expend resources exploring any other extraneous part of the design space. In one demonstration, an unconditional GAN was applied to plasmonic structures serving as thermal emitters for thermophotovoltaic systems \cite{Boltasseva2019GAN}. The training set comprised images of different topology-optimized structures, each locally optimized from random initial dielectric distributions. Upon learning the statistical distribution of these training set devices, the network could generate many topologically-complex devices within this distribution, some of which exhibited better performance than the training set devices. The result is summarized in FIG. \ref{fig:fig8}c. The same group also analyzed thermal emitters using an adversarial autoencoder (AAE), which is a variation of the VAE model except that an adversarial discriminative network is used instead of KL divergence to match the encoded latent space with a standard Gaussian distribution\cite{makhzani2015adversarial}. For their network implementation schemes, AAEs generated better devices than GANs \cite{kudyshev2019machine}.
GANs have also been configured to learn features from topology-optimized metagratings, which are periodic metasurfaces that diffract incident light to the +1 diffraction order. Metagratings are good model systems for metasurfaces because they capture the essential light-matter interactions in diffraction processes \cite{Jianji2017OptLett, Yang2018Annalen, Yang2017OptExp, Sell2017OptMat, Sell2018ACSPhot, Wang2019OptExp}. In the first demonstration, a conventional GAN architecture, conditioned on operating wavelength and deflection angle, was trained with images of silicon-based metagratings operating for select wavelength-deflection angle pairs \cite{Jiang2019ACSNano}. The final network could generate topologically-complex devices for a continuous range of wavelength and deflection angle values, showing the ability for the network to learn and interpolate device layouts within this parameter space. However, the best generated devices were not robust or highly efficient, and they required additional optimization refinement to match the performance of training set devices. In a following demonstration, the GAN training process was modified to incorporate progressive growth of the network architecture and training set over the course of multiple training cycles. Self-attention network layers that could capture long range spatial correlations within images were also added to the network architecture \cite{wen2019progressivegrowing} (FIG. \ref{fig:fig8}d). When fully trained, this progressive-growing GAN (PGGAN) could generate robust devices with efficiencies comparable to the best topology-optimized devices, showing the potential of generative networks to learn highly complex and intricate geometric trends in freeform photonic structures.
\subsection{Global topology optimization networks} \label{GLOnets}
A long standing challenge in inverse photonics design is the global optimization of freeform devices. Existing methods to perform inverse freeform design, ranging from heuristic to gradient-based topology optimization, are not able to effectively solve for the global optimum because the design space for photonic devices is vast and non-convex. In all neural network-based inverse design methods discussed thus far, which rely on a training set, global optimization is only possible if devices near or at the global optimum are included in the training set. While discriminative and generative neural networks can be effective at fitting training data, they cannot perform meaningful extrapolation tasks beyond the training set.
Global topology optimization networks (GLOnets), outlined in FIG. \ref{fig:fig9}a, are a newly developed class of generative network that are capable of effectively searching the design space for the global optimum \cite{GLOnets2019NanoLett, GLOnets2019Nanophot}. Unlike conventional implementations of generative networks, which are trained to fit a training set distribution, GLOnets attempt to fit a narrowly-peaked function centered around the global optimum and do so without a training set (FIG. \ref{fig:fig9}b). In this manner, GLOnets reframe the the topology optimization process through the dataless training of a neural network.
\begin{figure}[]
\centering
\includegraphics[width=\linewidth]{./Figures/Figure9.png}
\caption{
\textbf{Global topology optimization networks (GLOnets).} \textbf{a}| Schematic of the GLOnet optimizer. A generative network is trained to output a narrow distribution of devices centered around the global optimum. The training process involves evaluating generated devices with an electromagnetic solver and using those results in backpropagation to improve the mapping of the latent variable to device distribution. \textbf{b}| Schematic of the design space, where devices $\textbf{\emph{x}}$ have efficiencies $\mbox{Eff}(\textbf{\emph{x}})$. The design space has many local optima and a single globally optimal device, $\textbf{\emph{x}}^*$, which has an efficiency of $\mbox{Eff}(\textbf{\emph{x}}^*)$. Prior to training, the generative neural network is randomly initialized with weights $w$ and generates a uniform distribution of devices spanning the entire design space, $P_w (\textbf{\emph{x}})$. Upon training completion, the final network has optimal weights $w^*$ and produces the narrow distribution $P_{w^*} (\textbf{\emph{x}})$. \textbf{c}| Histogram of silicon metagrating efficiencies for devices designed using the adjoint variables method and GLOnet. The adjoint variables-optimized devices have a broad distribution and the best device operates with 93\% efficiency. The GLOnet devices have a relatively narrow distribution with high efficiencies, and the best device operates with 98\% efficiency. \textbf{d}| Plot of metagrating deflection efficiency as a function of minimum feature size for globally optimiazed devices comprising four silicon bars, solved using a reparameterized GLOnet. \textbf{e}| Plot of emissivity enhancement as a function of layers for thin film stacks serving as an incandescent light bulb filter. Multi-objective GLOnets outperforms the genetic algorithm reference and can produce high performing devices with relatively few layers. Inset: Schematic of filters, based on stacks of thin films, that transmit visible light and reflect infrared light.
Panels \textbf{a} and \textbf{c} adapted from REF. \cite{GLOnets2019Nanophot}. Panel \textbf{d} adapted from REF. \cite{mingkun2020reparam}. Panel \textbf{e} adapted from REF. \cite{jiang2020global}. }
\label{fig:fig9}
\end{figure}
The basic GLOnet architecture shown in FIG. \ref{fig:fig9}a is a deep generative CNN with conditional device labels and a latent variable as inputs, and it outputs a distribution of devices. During each training step, a batch of devices is generated and the performance metric and performance gradient of each device is evaluated using a Maxwell solver. The latter can be calculated using either the adjoint variables method or auto-differentiation. These performance metric and gradient terms are then incorporated into the loss function and backpropagated to adjust the network weights. The loss function is engineered to push the distribution of generated devices towards the global optimum and is:
\begin{equation}
L(\textbf{\emph{x}}, \textbf{\emph{g}}, \mbox{Met}) = -\frac{1}{N}\sum_{n=1}^{N} \frac{1}{\sigma} \exp{\left(\frac{\mbox{Met}^{(n)}}{\sigma}\right)}\ \textbf{\emph{x}}^{(n)}\cdot \textbf{\emph{g}}^{(n)}
\label{Eq5}
\end{equation}
$N$ is the batch size, ${\sigma}$ is a tunable hyperparameter, and $\mbox{Met}^{(n)}$, $\textbf{\emph{x}}^{(n)}$, and $\textbf{\emph{g}}^{(n)}$ represent the performance metric, device layout, and performance gradient of the $\emph{n}^{th}$ device, respectively. The biasing of the network towards the global optimum is captured by the exponential weighing of the performance metric in the loss function. Interestingly, the value of the globally optimal performance metric does not need to be known.
In an initial demonstration, GLOnets were used to globally optimize the efficiencies of metagratings consisting of silicon ridges \cite{GLOnets2019Nanophot}. Sixty-three unconditional GLOnets were trained, each searching for optimal devices with distinct combinations of operating wavelength and deflection angle, and each network was benchmarked with five hundred locally optimized devices designed using the adjoint variables method. For fifty-seven of these networks, the best GLOnets device had the same or higher efficiency compared to the best locally optimized device. Histograms of device efficiencies from these two methods show that the distributions of generated GLOnet devices are relatively narrow and biased towards high efficiencies, which is consistent with the training goal of GLOnets (FIG. \ref{fig:fig9}c). The stability of the GLOnets method is demonstrated with the training of eight different randomly initialized networks, each with the same design objective: six of the eight trained networks produce the same device possessing an efficiency of 97\%.
Conditional GLOnets for metagratings, which could simultaneously optimize devices with a range of wavelengths and deflection angles, were also examined \cite{GLOnets2019NanoLett}. For this demonstration, network training with conditional labels worked well because the design space and optimal devices for different conditional labels were strongly correlated. A comparison of the best devices generated from a single GLOnet and the best devices locally optimized using the adjoint variables method showed that 75\% of the devices from the conditional GLOnet had higher efficiencies than those based on the local optimization. The computational resources required to train the GLOnet were 10$\times$ less than those used for the local optimization of the benchmark devices. Such computational efficiency arises because the GLOnet does not expend computational resources in unpromising parts of the design space. It instead is constantly shifting the generated device distribution towards the globally optimal device during training.
\textbf{Incorporation of constraints with reparameterization}.
An important consideration with all inverse design methods is the incorporation of practical experimental constraints, such as the specification of a minimum feature size or robustness to fabrication imperfections. A typical method to incorporating these constraints in inverse design is to add terms in the figure of merit that penalize violations to these constraints \cite{vercruysse2019analytical, huang2019implementation, vercruysse2019dispersion}. While this concept will generally push devices towards regions of the design space that satisfy the desired constraints, it does not guarantee the enforcement of constraints. An alternative method that can impose hard constraints in optimization is to reparameterize the problem, in a manner where the optimizer processes devices in a latent space that can span an unconstrained range of values \cite{mingkun2020reparam}. Mathematical transformations are then used to transform the latent space representation to the real device space, and constraints imposed within the real device space are defined by the transformation itself. Finally, the constrained device in the real device space is evaluated. For gradient-based optimizers, such as the adjoint variables method or GLOnets, performance gradients are calculated for devices in the real device space and are backpropagated to the latent device space representation for the optimizer to process. Backpropagation is generally possible as long as the mathematical transformations linking the two spaces are differentiable.
Reparameterization was applied to GLOnet algorithms for silicon metagratings that deflect incident monochromatic light to a 65 degree angle. In this example, the topology was fixed to contain four silicon nanoridges, and the unconstrained latent space variables mathematically transformed to ridge width and ridge separation values with a hard minimum feature size constraint. The silicon and air regions of the device were still defined to possess gray scale values with spatial profiles defined by analytic functions, which allowed gradients from the adjoint variables method to be directly applied to this shape optimization problem. The resulting globally-optimized device efficiencies as a function of minimum feature size are summarized in FIG. \ref{fig:fig9}d. The unconstrained globally optimal device has a minimum feature size of 20 nm, such that reparameterized GLOnets with minimum feature sizes equal to or smaller than 20 nm generated the same optimal device. As the minimum feature size constraint increased, the efficiency of the globally optimized devices decreased. Images of the device layouts show that the globally optimal devices each possess at least one feature with the minimum feature size posed by the constraint, indicating the utility of small features to enhance light diffraction efficiency in these devices.
\textbf{Multi-objective GLOnets}.
The metagrating topology optimization problem above is single objective: all parameters in the problem are fixed except the refractive indices of each voxel, which are specified to be either silicon or air. Multi-objective problems are more complex and require more than just binary decisions to be made, but they can be readily handled with the GLOnets formalism without loss of generality. As an example, consider the design of multi-layer stacks in which each layer can be one of $M$ distinct material types. With GLOnets, these multi-layer stacks are represented as matrices, where each row is a $1\times M$ dimensional vector and each term corresponds to a particular material type in a given layer. Each of these vectors is computed into a probability distribution using the softmax function, which specifies the likelihood that a particular material in a given layer is optimal. The expected refractive index of each layer given by this likelihood matrix is calculated, evaluated with an electromagnetic solver, and used to evaluate the loss function and perform backpropagation. As the training process progresses, each row of the likelihood matrix converges to have one predominant term, which is the optimal material.
Multi-objective GLOnets have been applied to a number of thin film stack systems \cite{jiang2020global}. One is anti-reflection coatings intended for broadband and broad angle usage on silicon solar cells, where a continuum of dielectric values was selected for each layer. Existing benchmarks for a three layer system included a brute force search of the global optimum, which took over 19 days of CPU computation\cite{azunre2019guaranteed}, and a multi-start gradient optimizer\cite{azunre2019guaranteed}, which took 15 minutes to find the global optimum. GLOnets solved for the global optimum in seven seconds with a single GPU, demonstrating its efficiency and efficacy. GLOnets were also applied to thermal filters that could transmit visible light and reflect infrared light. The results are summarized in FIG. \ref{fig:fig9}e for thin film stacks comprising seven different dielectric material types: magnesium fluoride, silicon dioxide, silicon carbide, silicon mononitride, aluminum oxide, hafnium dioxide, and titanium dioxide. The broadband reflection characteristics of a 45 layer GLOnet-optimized device showed that the device operates with nearly ideal transmission at 500--700 nm and nearly ideal reflection at near-infrared wavelengths, for both normal incidence and for incidence angles averaged over all possible solid angles. The application of GLOnets to different layer numbers and a comparison with a genetic algorithm benchmark \cite{shi2017optimization} showed that for optimized devices with 45 layers, GLOnets clearly outperformed the genetic algorithm reference. Furthermore, GLOnets could produce devices with the same performance as the genetic algorithm reference but with approximately two thirds the number of layers, which is important for translating these designs to experiment.
\section{Future research directions and practices}
Deep neural networks are poised to be a disruptive force in the solving of forward and inverse design problems in photonics. In just the last few years, discriminative networks have been shown to serve as effective surrogate models of Maxwell solvers, learning and generalizing the complex relationship between nanoscale layouts and their optical properties. Generative models have proven to serve as a new framework for the inverse design of freeform devices, through the learning of geometric features within device datasets and by dataless network training using Maxwell solvers.
Neural network-based models are not a general replacement tool for conventional electromagnetic simulators, which will continue to be a workhorse tool for most problems, but they have complementary strengths and weaknesses. The main drawback of neural networks is they require large training sets of thousands to millions of devices, which is a significant one-time computational cost. If conventional simulation and optimization methods can solve a problem with an equivalent or smaller computational budget, it is more judicious and straightforward to stick with conventional approaches. Another issue is that even the best trained networks cannot guarantee accuracy and should not be used in lieu of an electromagnetic simulator when an exact physics calculation is required. It is also noted that low-dimensional electromagnetic systems described by a small number of design parameters can often be modeled and optimized using a number of classical statistical, machine learning, and optimization packages \cite{Gerstner1998, Rockstuhl2019, Genevet2019}, many of which are available as standard numerical toolboxes in scientific computing software. Compared to the training of deep networks, these methods can work as effectively and do not require extensive hyperparameter tuning.
Neural network-based models also have a number of strengths that make them uniquely suited for a number of problems. First, a trained neural network operates with orders-of-magnitude faster speeds than a conventional simulator and is ideal in situations where simulation time is a critical factor. Second, the regression capabilities of neural networks surpass those of classical data fitting methods and can extend to complex, high-dimensional systems, due to the scalability of neural networks to accommodate thousands of neurons with tunable parameters. Third, neural networks are particularly computationally efficient at simulating and designing many device variants that utilize related underlying physics. These devices range from grating couplers that require different input mode conditions to metasurface sections that require different amplitude and phase properties, and these device variants can be readily co-designed by training a single conditional neural network. Fourth, neural network approaches to inverse design can produce electromagnetic devices with better overall performance. Global topology optimization based on GLOnets has already been shown to supersede conventional gradient-based optimizers, and continued advancements in neural network-based optimization promise even better and more computationally efficient design algorithms.
Looking ahead, multiple innovations will be required to push the capabilities of deep learning algorithms towards the inverse design of complex, technologically-relevant devices. First, while generic machine learning algorithms will continue to play a role in solving photonics problems, new concepts that intimately combine the underlying physical structure of Maxwell's equations with machine learning need to be developed. GLOnets, which combine machine learning with physics-based solvers, is one such example showing how new hybrid algorithms can enhance the capabilities of neural networks. There have also been recent demonstrations showing that neural networks can be trained to solve differential equations \cite{2019DIffEqn}. To integrate physics with neural networks, we anticipate new innovations in network architectures, training procedures, and loss function engineering, as well as entirely new ways of using discriminative and generative networks both independently and together. We predict that dataless training, in which physics-based calculations are used to train neural networks, will serve as particularly effective and computationally efficient means to harnessing machine learning for photonics problems.
Second, new electromagnetic simulators need to be developed that can operate at significantly faster time scales than what is currently offered with conventional full wave solvers. Fast solvers are needed because as device complexity increases, significantly larger training sets for supervised learning and larger simulation batches for dataless training methods are required. We anticipate that application-specific electromagnetic solvers will play a major role as ultra-fast solvers in deep learning photonics problems. One path forward is the augmentation of existing Maxwell solvers with neural network-enhanced preconditioners \cite{Vuckovic2019PhotAccel}. With a neural network that can predict an approximate solution to the electromagnetics problem on hand, that solution can be used as a starting point for the solver and dramatically speed up the calculation. Specialized algorithms that can evaluate the scattering properties of structures with high computational efficiency, such as integral-equation solvers \cite{Chew2008IntegralMethods} and T-matrix approaches \cite{tmatrix1996}, are also worth revisiting.
Third, the training and refinement of neural networks for solving photonics problems need to be better streamlined, both from a data usage and user interface point of view. Currently, every time a new problem is proposed, a data scientist needs to train and fine tune a neural network from scratch. One avenue that can help address data usage is transfer learning, in which a subset of network weights from an trained network solving an initial problem is applied to a network intending to solve a related problem \cite{TransferLearning2010}. The initial problem can be that of a related physical system for which a trained network already exists, or it can be a simplified version of the desired problem, from which an initial neural network can be trained with computationally 'cheap' data. In a recent demonstration, network weights from a trained network that could predict the scattering spectra of concentric shell scatterers were transferred to a network intended to predict the spectral properties of dielectric stacks, leading to improved training accuracy of the latter \cite{transferlearning2019}. For the user interface problem, we anticipate that meta-learning, in which neural networks learn to learn \cite{NIPS2016Metalearning,ICML2017MetaLearning}, will help automate the setup and training process for photonics-based machine learning algorithms. Meta-learning is currently an active area of research in the computer sciences community, and while it is a highly data intensive proposition, it promises to simplify the interface between the algorithms and user.
Also looking ahead, it would be judicious for our community to take inspiration from the computer sciences community and engage in a more open culture of sharing. In the computer sciences community, extreme progress and proliferation of the data sciences can be attributed in part to the willingness of computer scientists to openly share algorithms and benchmark their approaches by solving common problems. For example, it is typical in computer vision research for groups to use established databases of labeled images, such as ImageNet \cite{ImageNet} and CIFAR-10 \cite{CIFAR10}, as training data for algorithm benchmarking. The computer vision community even engages in regular contests, such as the ImageNet Large Scale Visual Recognition Challenge, in which researchers attempt to solve the same image classification task with the same training data. This community-centric design-of-experiments approach allows researchers to rapidly prototype, compare, and evolve their algorithms at a rapid rate, to the benefit of the whole community.
In this spirit, an online repository for device designs and inverse design codes for nanophotonic systems, termed MetaNet, has been developed \cite{MetaNet}. As of this paper publication, MetaNet contains design files of over 100,000 freeform metagrating structures, as well as codes for local and global topology optimization. We hope that with continued dialogue within the photonics community, we can agree on important design problems to tackle and to open source training sets and basic code formulations so that we may build on each other's algorithmic approaches. At the very least, with inverse design strategies that produce freeform device layouts, we need standardized methods share device layouts so that we can benchmark and openly evaluate the capabilities of these structures, not just in terms of device performance but also other metrics such as robustness to geometric imperfections. By working together, we can effectively push optical and photonics engineering to the next and possibly final frontier.
\noindent\textbf{Author contributions}\\
All authors researched data for the article, discussed the content, and contributed to the writing and revising of the manuscript.\\
\noindent\textbf{Competing interests}\\
The authors declare no competing interests. \\
|
2,877,628,090,155 | arxiv | \section{Introduction}
\label{sec:intro}
We consider the large time behavior for the nonlinear Schr\"odinger
equation
\begin{equation}
\label{eq:nls}
i{\partial}_t u +\frac{1}{2}\Delta u = V(x)u + |u|^{2{\sigma}}u,
\end{equation}
where $u:(t,x,y)\in \R\times \R\times \R^{d-1}\to \C $, with $d\geqslant 2$,
$\Delta$ is the Laplacian in $(x,y)$,
and $0<{\sigma}<\frac{2}{(d-2)_+}$ (where $1/a_+$ stands for $+\infty$ if $a\leqslant 0$, and for $1/a$ if $a>0$): the nonlinearity is energy-subcritical
in terms of the whole space dimension $d$. The external potential $V$
depends only on $x$. More precisely, we suppose:
\begin{hyp}
\label{hyp:V}
The potential $V\in L^2_{\rm loc}(\R)$ is real-valued and bounded from
below:
\begin{equation*}
\exists C_0,\quad V(x)+C_0\geqslant 0,\quad \forall x\in \R.
\end{equation*}
\end{hyp}
It follows from \cite[Theorem X.28]{ReedSimon2} that
\begin{equation*}
H = -\frac{1}{2}\Delta +V(x)
\end{equation*}
is essentially self-adjoint on $C_0^\infty(\R^d)$, with domain
(\cite[Theorem X.32]{ReedSimon2})
\begin{equation*}
D(H) = \{f\in L^2(\R^d),\quad -\frac{1}{2}\Delta f+Vf\in L^2(\R^d)\}.
\end{equation*}
The goal of this paper is to understand the large time dynamics in
\eqref{eq:nls}. This framework is to be compared with the analysis in
\cite{TzVi-p}, where there is no external potential ($V=0$), but where
the $x$ variable belongs to the torus $\T$ (which is the only
one-dimensional compact manifold without boundary). It is proven
there that if a short range scattering theory is available for the
nonlinearity $|u|^{2{\sigma}}u$ in $H^1(\R^{d-1})$, that is if
$\frac{2}{d-1}<{\sigma}<\frac{2}{(d-2)_+}$, then the solution of the Cauchy
problem for $(x,y)\in
\T\times \R^{d-1}$ (is global and) is asymptotically linear as $t\to
\infty$.
\smallbreak
In this paper, we prove the analogous result in the case of
\eqref{eq:nls}, as well as the existence of wave operators (Cauchy
problem with behavior prescribed at infinite time). This extends some
of the results from \cite{AnCaSi-p} where the special case of an
harmonic potential $V$ is considered. The properties related
to the harmonic potentials are exploited to prove the existence of
wave operators in the case of a multidimensional confinement
($V(x)=|x|^2$, $x\in \R^n$, $n\geqslant 1$), a case that we do not consider
in the present paper (see Remark~\ref{rem:higherD}): essentially, if the nonlinearity is
short range on $\R^{d-n}$, then it remains short range on $\R^d$ with
$n$ confined directions. Long range effects are described in
\cite{HaTh-p}, in the case $n=d-1$ and ${\sigma}=1$ (cubic nonlinearity,
which is exactly the threshold to have long range scattering in one
dimension). A technical difference with \cite{TzVi-p} is that for the
Cauchy problem, we do
not make use of inhomogeneous Strichartz for non-admissible pairs like
established in \cite{CW92,FoschiStri,Vil07}, and for scattering
theory, such estimates are not needed when $d\leqslant 4$.
\smallbreak
We emphasize that here, the potential $V$
can have essentially any behavior, provided that it remains
bounded from below. It can be bounded (in which case the term
``confinement'' is inadequate), or grow arbitrarily fast as $x\to \pm
\infty$. This is in sharp contrast with
e.g. \cite{Miz14,YajZha01,YajZha04}, where Strichartz estimates (with
loss) are established in the presence of super-quadratic potentials,
or with \cite{BCM08}, where a functional calculus adapted to confining
potentials is developed: in all these cases, typically, an exponential growth of
the potential is ruled out, since in this case, no pseudo-differential
calculus is available.
\smallbreak
Introduce the notation
\begin{equation*}
M_x =-\frac{1}{2}{\partial}_x^2+V(x)+C_0.
\end{equation*}
We define the spaces
\begin{align*}
& B_x=\left\{u\in L^2(\R),M_x^{1/2}u\in L^2(\R)\right\},\quad
\Sigma_y=\left\{u\in H^1(\R^{d-1}),yu\in L^2(\R^{d-1})\right\}, \\
& Z = L^2_yB_x\cap L^2_xH^1_y,\quad \tilde Z=L^2_yB_x\cap L^2_x\Sigma_y,
\end{align*}
endowed with the norms
$$\|u\|_{B_x}^2=\|u\|_{L^2_x(\R)}^2+\|M_x^{1/2}u\|_{L^2_x(\R)}^2=
\|u\|_{L^2_x(\R)}^2+\<M_x u,u\>,$$
$$\|u\|_{\Sigma_y}^2=\|u\|_{L^2_y(\R^{d-1})}^2+\|\nabla_y
u\|_{L^2_y(\R^{d-1})}^2+\|yu\|_{L^2_y(\R^{d-1})}^2,$$
and
$$\|u\|_Z^2=\|u\|_{L^2_{xy}(\R^d)}^2+\|M_x^{1/2}u\|_{L^2_{xy}(\R^d)}^2+
\|\nabla_y u\|_{L^2_{xy}(\R^d)}^2 ,\quad \|u\|_{\tilde Z}^2=
\|u\|_Z^2+ \|y u\|_{L^2_{xy}(\R^d)}^2.$$
The group $e^{-itH}$ is unitary on $Z$, but not on $\tilde Z$, a
property which is discussed in the proof of
Lemma~\ref{lem:opA}.
\begin{remark}
Note that $B_x$ is the domain of the operator $M_x^{1/2}$, which is
defined as a fractional power of the self-adjoint operator $M_x$
acting on $L^2(\R)$: for $u\in B_x$, $M_x^{1/2}u$ is defined by
$$M_x^{1/2}u=\int_0^\infty\lambda^{1/2}dE_\lambda(u),$$
where $M_x=\int_0^\infty\lambda dE_\lambda$ is the spectral
decomposition of $M_x$.
\end{remark}
\begin{theorem}[Cauchy problem]\label{theo:cauchy}
Let $d\geqslant 2$, $V$ satisfying Assumption~\ref{hyp:V} and
$0<{\sigma}<\frac{2}{(d-2)_+}$. Let $t_0\in \R$ and $u_0\in Z$. There
exists a unique solution
$ u\in C(\R;Z) $ to \eqref{eq:nls} such that $u_{\mid
t=t_0}=u_0$. The following two quantities are independent of time:
\begin{align*}
&\text{Mass: }\|u(t)\|_{L^2_{xy}(\R^d)}^2,\\
&\text{Energy: }\frac{1}{2}\|\nabla_{xy}u(t)\|_{L^2_{xy}(\R^d)}^2 +
\frac{1}{{\sigma}+1}\|u(t)\|_{L_{xy}^{2{\sigma}+2}(\R^d)}^{2{\sigma}+2} +\int_{\R^d} V(x) |u(t,x,y)|^2dxdy.
\end{align*}
If in addition $u_0\in \tilde Z$, then $u\in
C(\R;\tilde Z)$.\end{theorem}
\begin{theorem}[Existence of wave operators]\label{theo:waveop}
Let $d\geqslant 2$, and $V$ satisfying Assumption~\ref{hyp:V}. \\
$1.$ If $u_-\in Z$ and $\frac{2}{d-1}\leqslant {\sigma}< \frac{2}{(d-2)_+}$,
there exists $u\in C(\R;Z)$
solution to \eqref{eq:nls} such that
\begin{equation*}
\| u(t)-e^{-itH}u_-\|_Z=\|e^{itH} u(t)-u_-\|_Z\Tend t {-\infty}0.
\end{equation*}
This solution is such that
\begin{equation*}
u\in L^\infty(\R;Z) \cap L^p((-\infty,0];L^k_yL^2_x)
\end{equation*}
for some pair $(p,k)$ given in the proof, and it is unique in this class. \\
$2.$ If $u_-\in \tilde Z$ and
$\frac{2}{d}<{\sigma}<\frac{2}{(d-2)_+}$,
there exists a unique $u\in C(\R;\tilde Z)$
solution to \eqref{eq:nls} such that
\begin{equation*}
e^{itH} u\in L^\infty(]-\infty,0];\tilde Z) \quad\text{and}\quad
\|e^{itH} u(t)-u_-\|_{\tilde Z}\Tend t {-\infty}0.
\end{equation*}
\end{theorem}
In the second case, the lower bound ${\sigma}>\frac{2}{d}$ is weaker than
in the first case, so there is some gain in working in the smaller
space $\tilde Z$ rather than in $Z$. However, this lower bound is larger than in the
corresponding
result from \cite{AnCaSi-p} where only the case $V(x)=x^2$ is
considered. Indeed in \cite{AnCaSi-p}, the general lower bound is
${\sigma}>\frac{2d}{d+2}\frac{1}{d-1}$, which is smaller than the present
one as soon as $d\geqslant 3$. The main technical reason is that specific
properties of the harmonic oscillator (typically, the fact that it
generates a flow which is periodic in time) makes it possible to
establish a larger set of Strichartz estimates than the one which we
use in the present paper. In all cases, the expected borderline
between short range and long range scattering is
${\sigma}_c=\frac{1}{d-1}$ ($d-1$ is the ``scattering dimension''), so
our result is sharp in the case $d=2$, and
most likely only in this case.
\begin{theorem}[Asymptotic completeness]\label{theo:AC}
Let $d\geqslant 2$, $V$ satisfying Assumption~\ref{hyp:V}, and
$\frac{2}{d-1}<{\sigma}<\frac{2}{(d-2)+}$. For any $u_0\in Z$, there
exists a unique $u_+\in Z$ such that the solution to \eqref{eq:nls}
with $u_{\mid t=0}=u_0$ satisfies
\begin{equation*}
\|u(t)-e^{-itH}u_+\|_Z =\|e^{itH}u(t)-u_+\|_Z\Tend t {+\infty}
0.
\end{equation*}
\end{theorem}
\begin{remark}\label{rem:higherD}
When a confinement is present (due either to a harmonic potential,
or to a bounded geometry) in $n$ directions, for a total space
dimension $d$, it is expected that the ``scattering dimension'' is
$d-n$. This was proven systematically in the case of a harmonic
confinement in \cite{AnCaSi-p}, complemented by
\cite{HaTh-p}; see also \cite{HPTV-p,TzVi12}. Therefore, to prove asymptotic
completeness thanks to Morawetz estimates, it is natural to assume
${\sigma}>\frac{2}{d-n}$ (essentially because it is not known how to take
advantage of these estimates otherwise, except in the $L^2$-critical
case, where many other tools are used). On the other hand, for the Cauchy
problem to be locally well-posed at the $H^1$-level, it is necessary
to assume ${\sigma} \leqslant \frac{2}{d-2}$ if $d\geqslant 3$. For the above two
conditions to be consistent in the energy-subcritical case
${\sigma}<\frac{2}{d-2}$, we readily see that the only possibility is
$n=1$, as in \cite{TzVi-p} and the present paper. To treat the case
$n=2$,the analysis of a doubly critical case would be required: $L^2$-critical
in $\R^{d-n}$ with ${\sigma}=\frac{2}{d-n}$, and energy-critical in $\R^d$
with ${\sigma}=\frac{2}{d-2}$.
\end{remark}
\section{Technical preliminaries}
\label{sec:prelim}
\subsection{Sobolev embeddings}
\begin{lemma}\label{lem-emb-bm-hm}
$B_x$ is continuously embedded into $H^{1}_x(\R)$.
\end{lemma}
\begin{proof}
Since $V $ is bounded from below, we have
\begin{align*}
\|u\|_{H^1_x(\R)}^2&\leqslant\|u\|_{L^2_x(\R)}^2+\|\partial_xu\|_{L^2_x(\R)}^2+2\int_\R
\(V(x)+C_0\)|u(x)|^2dx\\
&\leqslant \|u\|_{L^2_x(\R)}^2+2\<M_x u,u\>\lesssim\|u\|_{B_x}^2,
\end{align*}
hence the result.
\end{proof}
Introduce, for $\gamma,s\geqslant 0$, the anisotropic Sobolev space
$$H^\gamma_yH^s_x=(1-\Delta_y)^{-\gamma/2}(1-\partial_x^2)^{-s/2}L^2_{x,y},$$
endowed with the norm
$$\|u\|_{H^\gamma_yH^s_x}^2=\int_{\R\times\R^{d-1}}\<\xi\>^{2s}
\<\eta\>^{2\gamma}|\widehat{u}(\xi,\eta)|^2d\xi
d\eta,$$
where $\hat{u}$ denotes the Fourier transform of $u$ in both $x$ and
$y$ variables. $\dot{H}^\gamma_yH^s_x$ denotes the corresponding
homogeneous space, endowed with the norm
$$\|u\|_{\dot{H}^\gamma_yH^s_x}^2=\int_{\R\times\R^{d-1}}\<\xi\>^{2s}|\eta|^{2\gamma}|\widehat{u}(\xi,\eta)|^2d\xi
d\eta.$$
\begin{lemma}\label{lem2}
If $\varepsilon\in (0,1/2)$, $s=\frac{1}{2}+\varepsilon$ and
$\gamma=\frac{1}{2}-\varepsilon$, then
$$\|u\|_{\dot{H}^{\gamma}_yH^{s}_x}\leqslant
\|u\|_{{H}^{\gamma}_yH^{s}_x}\lesssim\|u\|_{Z},\quad \forall
u\in Z.$$
\end{lemma}
\begin{proof}
From Young inequality and Lemma~\ref{lem-emb-bm-hm},
\begin{align*}
\|u\|_{{H}^{\gamma}_yH^{s}_x}^2&=\int_{\R\times\R^{d-1}}\<\xi\>^{2\gamma}
\<\eta\>^{2s}|\widehat{u}(\xi,\eta)|^2d\xi
d\eta\\
& \lesssim
\int_{\R\times\R^{d-1}}\left[(1+\xi^2)+(1+|\eta|^2)\right]|\widehat{u}(\xi,\eta)|^2d\xi
d\eta\ \lesssim\ \|u\|_{L^2_yH^1_x}^2+\|u\|_{L^2_x\dot{H}^1_y}^2,
\end{align*}
hence the result.
\end{proof}
\subsection{Anisotropic Gagliardo-Nirenberg inequality}
\begin{proposition}\label{sob-anis}
Let $k,s,\gamma>0$ such that
\begin{equation}\label{ass-ksg}
s>1/2\quad \text{and} \quad \frac{1}{2}>
\frac{1}{k}>\frac{1}{2}-\frac{\gamma}{d-1}>0.
\end{equation}
Then $H^\gamma_yH^s_x\subset L^k_yL^\infty_x$, and there exists $C>0$
such that for every
$u\in H^\gamma_yH^s_x$,
\begin{equation*}
\|u\|_{L^k_yL^\infty_x}\leqslant C
\|u\|_{L^2_yH^{s}_x}^{1-\delta}\|u\|_{\dot{H}^{\gamma}_yH^{s}_x}^{\delta},
\quad\text{where }\delta=\frac{d-1}{\gamma
}\left(\frac{1}{2}-\frac{1}{k}\right).
\end{equation*}
\end{proposition}
\begin{proof}
We first use the Sobolev inequality in the $x$ variable and Minkowski
inequality (which is possible because $k>2$). We get
\begin{equation}\label{23}
\|u\|_{L^k_yL^\infty_x}\lesssim
\|u\|_{L^k_yH^{s}_x}=\|\<\xi\>^{s}\mathcal{F}_xu(\xi,y)\|_{L^k_yL^2_\xi}
\lesssim\|\<\xi\>^s\mathcal{F}_xu(\xi,y)\|_{L^2_\xi L^k_y},
\end{equation}
where $\mathcal{F}_x$ denotes the
Fourier transform in the $x$ variable. Similarly, we denote by $\mathcal{F}_y$ the
Fourier transform in $y$ and
$\widehat{u}(\xi,\eta)=(\mathcal{F}_x\mathcal{F}_yu)(\xi,\eta)$. Then
for a fixed value of $\xi\in\R$, Hausdorff-Young
inequality yields
\begin{equation}\label{24}
\|\mathcal{F}_xu(\xi,y)\|_{L^k_y}\lesssim \|\widehat{u}(\xi,\eta)\|_{L^{k'}_\eta}.
\end{equation}
Omitting the dependence of the right hand side in $\xi$, let us denote by
$v(\eta)=\widehat{u}(\xi,\eta)$. It follows from
the triangle and H\"older inequality that for any $R>0$,
\begin{align}
\|v\|_{L^{k'}_\eta} &\leqslant
\|v\|_{L^{k'}(|\eta|<R)}+\|v\|_{L^{k'}(|\eta|>R)}\nonumber\\
&\lesssim
\|\mathbf{1}_{\{|\eta|<R\}}\|_{L^p(|\eta|<R)}\|v\|_{L^2_\eta}+\||\eta|^{-\gamma}\|_{L^p(|\eta|>R)}\||\eta|^{\gamma}v\|_{L^2_\eta}\nonumber\\
&\lesssim
R^{(d-1)/p}\|v\|_{L^2_\eta}+R^{(d-1)/p-\gamma}\||\eta|^{\gamma}v\|_{L^2_\eta},\label{25}
\end{align}
where $p$ is given by $1/p=1/2-1/k$.. Note that (\ref{ass-ksg}) implies
that $\gamma p>d-1$, and
therefore $|\eta|^{-\gamma}\in L^p(|\eta|>R)$. Optimizing in $R$ in the right hand side of \eqref{25}, we get
\begin{equation}\label{26}
\|v\|_{L^{k'}_\eta} \lesssim \|v\|_{L^2_\eta}^{1-\delta}\||\eta|^{\gamma}v\|_{L^2_\eta}^\delta,
\end{equation}
where $\delta=\frac{d-1}{\gamma p}\in (0,1)$. Combining \eqref{23},
\eqref{24} and \eqref{26}, H\"older inequality yields
\begin{align*}
\|u\|_{L^k_yL^\infty_x}&\lesssim\left(
\int \<\xi\>^{2s(1-\delta)}\|\widehat{u}\|_{L^2_\eta}^{2(1-\delta)}
\<\xi\>^{2s\delta}\||\eta|^{\gamma}\widehat{u}\|_{L^2_\eta}^{2\delta}d\xi\right)^{1/2}\\
&\lesssim \left(
\int
\<\xi\>^{2s}\|\widehat{u}\|_{L^2_\eta}^{2}d\xi\right)^{(1-\delta)/2}
\left(\int \<\xi\>^{2s}\||\eta|^{\gamma}\widehat{u}\|_{L^2_\eta}^{2}
d\xi\right)^{\delta/2}\\
&=\|u\|_{L^2_yH^{s}_x}^{1-\delta}\|u\|_{\dot{H}^{\gamma}_yH^{s}_x}^{\delta}.
\end{align*}
\end{proof}
\begin{corollary}\label{coro}
Let $2<k<\frac{2(d-1)}{(d-2)_+}$. Then $Z$ is continuously embedded in $L^k_yL^\infty_x$.
\end{corollary}
\begin{proof}
Pick $\varepsilon>0$ small enough such that
$$\frac{1}{2}-\frac{1/2-\varepsilon}{d-1}=\frac{d-2}{2(d-1)}+\frac{\varepsilon}{d-1}<\frac{1}{k}.$$
Then $(s,\gamma)=(1/2+\varepsilon,1/2-\varepsilon)$ satisfy the assumptions of Proposition \ref{sob-anis} and Lemma \ref{lem2}. Thus, using also Lemma \ref{lem-emb-bm-hm},
\begin{equation}
\|u\|_{L^k_y L^\infty_x}\lesssim
\|u\|_{L^2_yB_x}^{1-\delta}
\|u\|_Z^{\delta}\lesssim\|u\|_Z.\nonumber
\end{equation}
\end{proof}
\subsection{Strichartz estimates}
Following the idea from \cite{TzVi12}, with the generalization from
\cite{AnCaSi-p} (noticing that the spectral decomposition from the
proof in \cite{TzVi12} is not needed), we have, since $M_x$ commutes
with $H$:
\begin{proposition}\label{prop:strichartz}
Let $d\geqslant 2$. We have
\begin{equation*}
\|e^{-itH}u_0\|_{L^q_tL^r_yL^2_x}+
\left\|\int_0^te^{-i(t-s)H}F(s)ds\right\|_{L^{q_1}_tL^{r_1}_yL^2_x}\lesssim
\|u_0\|_{L^2_{y}L^2_x}+
\|F\|_{L^{q_2'}_tL^{r_2'}_yL^2_x},
\end{equation*}
provided that the pairs are $(d-1)$-admissible, that is
\begin{equation*}
\frac{2}{q}+\frac{d-1}{r}= \frac{2}{q_1}+\frac{d-1}{r_1}=
\frac{2}{q_2}+\frac{d-1}{r_2}=\frac{d-1}{2},
\end{equation*}
with $(q,r)\neq (2,\infty)$
if $d=3$.
\end{proposition}
\subsection{Vectorfields}
We introduce the notation
\begin{align*}
&A_0(t)=A_0={\rm Id}, \quad A_1(t)=A_1=M^{1/2}_x, \quad A_2(t)=A_2=\nabla_y,\\
&
A_3(t)=y+it\nabla_y=it e^{i|y|^2/(2t)}\nabla_y\(\cdot \ e^{-i|y|^2/(2t)}\)=e^{-itH}ye^{itH}.
\end{align*}
The operator $A_3$ is the standard Galilean operator on $\R^{d-1}$,
see e.g. \cite{CazCourant}, so the last identity stems from the fact
that $e^{-itM_x}$ commutes with both $e^{i\frac{t}{2}\Delta_y}$ and
$y$. We readily have:
\begin{lemma}\label{lem:opA}
The operators $A_j$ satisfy the following properties:
\begin{itemize}
\item Commutation: for $j\in \{0,\dots,3\}$, $[i\partial_t-H,A_j]=0$.
\item Action on the nonlinearity: for all $j\in \{0,\dots,3\}$,
\begin{equation*}
\left\|A_j\(|u|^{2{\sigma}}u\)\right\|_{L^2_x}\lesssim
\|u\|_{L^\infty_x}^{2{\sigma}}\|A_ju\|_{L^2_x}.
\end{equation*}
\item Equivalence of norms: for all $u\in C_0^\infty(\R^d)$, we have,
uniformly in $t\in \R$,
\begin{equation}\label{eq:equiv-norm}
\| e^{itH} u\|_{Z}= \|u\|_Z\approx
\sum_{j=0}^2\|A_ju\|_{L^2_{xy}},\quad \| e^{itH} u\|_{\tilde Z}\approx
\sum_{j=0}^3\|A_j(t)u\|_{L^2_{xy}} .
\end{equation}
\item Gagliardo-Nirenberg inequalities: for all $g\in
\Sigma_y$, $2\leqslant p< \frac{2}{(d-3)_+}$,
\begin{align*}
&\| g\|_{L^p_y}\leqslant C \|g\|_{L^2}^{1-\delta}\|A_2
g\|_{L^2_y}^\delta,\\
&\| g\|_{L^p_y}\leqslant \frac{C}{|t|^{\delta}} \|g\|_{L^2}^{1-\delta}\|A_3(t)
g\|_{L^2_y}^\delta,\quad t\not =0,
\end{align*}
where $C$ is independent of $t$, and
$\delta=(d-1)\(\frac{1}{2}-\frac{1}{p}\)$.
\end{itemize}
\end{lemma}
\begin{proof}
The commutation property is straightforward. For the action on the
nonlinearity, it is trivial in the case of $A_0$ and $A_2$. For
$A_3$, it stems classically from the fact that $A_3$ is the gradient in $y$
conjugated by an exponential of modulus one and that the
nonlinearity we consider is gauge invariant. Concerning $A_1$, we
compute
\begin{align*}
\|M_x^{1/2}\left(|u|^{2\sigma}u\right)\|_{L^2_x}^2 & = \<M_x
\(|u|^{2{\sigma}}u\),|u|^{2{\sigma}}u\>\\
&=\frac{1}{2}\|\partial_x\left(|u|^{2\sigma}u\right)\|_{L^2_x}^2+\int_{-\infty}^{+\infty}\(V(x)+C_0\)|u|^{4\sigma+2}dx\\
&\leqslant
(2\sigma+1)^2\|u\|_{L^\infty_x}^{4\sigma}\left(\frac{1}{2}\|\partial_x
u\|_{L^2_x}^2+\int_{-\infty}^{+\infty}\(V(x)+C_0\)|u|^{2}dx\right)\\
& = (2\sigma+1)^2\|u\|_{L^\infty_x}^{4\sigma}\|M_x^{1/2}u\|_{L^2_x}^2.
\end{align*}
Recall that $A_0,A_1$ and $A_2$
commute with $e^{itH}$, which is unitary on
$L^2(\R^d)$, hence the first equivalence of norms. The identity
$A_3(t) = e^{-itH}y e^{itH}$ yields the second equivalence of norms,
uniformly in time: note that $\|e^{itH}u\|_{\tilde Z}$ is equivalent
to $\|u\|_{\tilde Z}$ only locally in time, due to the factor $t$ in
the identity $A_3(t) =y+it\nabla_y$.
\smallbreak
Finally, the Gagliardo-Nirenberg inequalities stated in the lemma are
the classical ones, using once more the factorization of $A_3$.
\end{proof}
\section{Cauchy problem}
\label{sec:cauchy}
In this section, we prove Theorem~\ref{theo:cauchy}. The existence
part relies on a a standard fixed point argument, adapted to the
present framework. Since the problem is invariant by translation in
time, we may assume $t_0=0$. Duhamel's formula reads
\begin{equation*}
u(t)=e^{-itH}u_0-i\int_{0}^te^{-i(t-s)H}\left(|u|^{2\sigma}u\right)(s)ds=:\Phi(u)(t).
\end{equation*}
This Cauchy problem will be solved thanks to a fixed point argument in
a ball of the Banach space
\begin{equation*}
Z_T=\{u\in L^\infty([0,T];Z),\quad A_ju\in
L^q\([0,T];L^r_yL^2_x\),
\forall j\in \{0,1,2\}\},
\end{equation*}
where $(q,r)$ is a $(d-1)$-admissible
pair that will be fixed later. The space $Z_T$ is naturally equipped
with the norm
\begin{align*}
\|u\|_{Z_T}= \sum_{j=0}^2 \(\|A_ju\|_{L^\infty_T
L^2_{xy}}+\|A_ju\|_{L^q_TL^r_yL^2_x}\) .
\end{align*}
Denote
$L^a_TX=L^a([0,T];X)$. Proposition~\ref{prop:strichartz} and
the first point of Lemma~\ref{lem:opA} imply, for $ j\in \{0,1,2\}$:
\begin{equation*}
\|A_j\Phi(u)\|_{L^\infty_TL^2_{xy}} + \|A_j\Phi(u)\|_{L^q_TL^r_yL^2_x}\lesssim \|A_j
u_0\|_{L^2_{xy}}+\|A_j(|u|^{2\sigma}u)\|_{L^{q'}_TL^{r'}_yL^2_x}.
\end{equation*}
The second point of Lemma~\ref{lem:opA} and H\"older inequality yield
\begin{equation*}
\|A_j(|u|^{2\sigma}u)\|_{L^{q'}_TL^{r'}_yL^2_x} \lesssim
\|u\|_{L^\theta_TL^k_yL^\infty_x}^{2\sigma}\|A_j
u\|_{L^{q}_TL^{r}_yL^2_x},
\end{equation*}
where $\theta$ and $k$ are given by
\begin{equation}\label{def-theta-k}
\frac{1}{q'}=\frac{2\sigma}{\theta}+\frac{1}{q},\quad
\frac{1}{r'}=\frac{2\sigma}{k}+\frac{1}{r}.
\end{equation}
We infer
\begin{equation}
\label{phi0ZT}
\|\Phi(u)\|_{Z_T}\lesssim
\|u_0\|_Z+\|u\|_{L^\theta_TL^k_yL^\infty_x}^{2\sigma}\|u\|_{Z_T}.
\end{equation}
Let us now explain how the parameters $q,r,\theta,k$ are
chosen.
\smallbreak
\noindent {\bf Case $d=2$.} We choose $r\in(2,\infty)$ if $\sigma\geqslant
1$, $2<r<\frac{2}{1-{\sigma}} $ if $0<\sigma<1$, and $(q,r)$ the corresponding
$1$-admissible pair. Then, \eqref{def-theta-k} defines a number $k$
that
belongs to $(2,\infty)
\smallbreak
\noindent {\bf Case $d=3$.} $(q,r)$ is a
$2$-admissible pair with $r\in (2,\infty)$ such
that
$$\frac{1}{4}<\frac{1}{2\sigma}\left(1-\frac{2}{r}\right)=:\frac{1}{k}<\frac{1}{2}.$$
Note that this is made possible thanks to the assumption
$\sigma<2$.
\smallbreak
\noindent {\bf Case $d\geqslant 4$.} As $(q,r)$ describes the set of all
$(d-1)$-admissible pairs,
$r$ varies between the two extremal values $2$ and $\frac{2(d-1)}{d-3}$, and
therefore $\frac{1}{2\sigma}(1-\frac{2}{r})$ varies between $0$ and
$\frac{1}{\sigma(d-1)}$, where the latter number is larger than
$\frac{d-2}{2(d-1)}$ thanks to the assumption
$\sigma<2/(d-2)$. Thus, one can choose $2<r<\frac{2(d-1)}{d-3}$ such
that if $k$ is defined by
\eqref{def-theta-k},
$$\frac{d-2}{2(d-1)}<\frac{1}{k}<\frac{1}{2}.$$
For these choices of the
parameters, Corollary \ref{coro} and
H\"older inequality in time imply
\begin{equation}\label{l-theta-k}
\|u\|_{L^\theta_TL^k_yL^\infty_x}\lesssim\|u\|_{L^\theta_TZ}
\lesssim T^{1/\theta}\|u\|_{Z_T}.
\end{equation}
Note that we have chosen admissible pairs such that $q>2$. Thus, since
$\theta$ is defined by \eqref{def-theta-k}, $1/\theta>0$. From the
combination of \eqref{phi0ZT} and \eqref{l-theta-k}, we deduce that if
$u$ belongs to the ball $B(R,Z_T)$ of $Z_T$ with radius $R>0$ centered at the
origin, we have
\begin{equation}\label{stab_ball}
\|\Phi(u)\|_{Z_T}\leqslant
C_1\|u_0\|_Z+CT^{2\sigma/\theta}R^{2\sigma+1}.
\end{equation}
Chosing $R=2C_1\|u_0\|_Z$ and $T=T(\|u_0\|_Z)>0$ sufficiently
small, $B(R,Z_T)$ is
stable by $\Phi$. Then, we note that $B(R,Z_T)$ endowed with the
norm
$$\|u\|_{B(R,Z_T)}=\|u\|_{L^\infty_TL^2_{xy}}+
\|u\|_{L^q_TL^r_yL^2_x}$$
is a complete metric space (Kato's method, see
e.g. \cite{CazCourant}).
For $u_2,u_1\in B(R,Z_T)$, the same estimates as above yield
\begin{align*}
\|\Phi(u_2) -\Phi( u_1)\|_{L^\infty_T L^2_{xy}}+&\|\Phi(u_2) - \Phi(u_1)\|_{L^q_T
L^r_yL^2_x}\\
&\lesssim
\(\|u_2\|_{L^\theta_T L^k_yL^\infty_x}^{2{\sigma}}+
\|u_1\|_{L^\theta_TL^k_yL^\infty_x}^{2{\sigma}}\)\|u_2-u_1\|_{L^q_T
L^r_yL^2_x}\\
&\lesssim T^{2{\sigma}/\theta} \( \|u_2\|_{Z_T}^{2{\sigma}} +
\|u_1\|_{Z_T}^{2{\sigma}} \)
\|u_2-u_1\|_{L^q_T
L^r_yL^2_x} \\
&\lesssim T^{2{\sigma}/\theta} R^{2{\sigma}}
\|u_2-u_1\|_{L^q_T
L^r_yL^2_x} .
\end{align*}
Therefore, $\Phi$ is a
contraction on $B(R,Z_T)$ endowed with the above norm, provided that
$T=T(\|u_0\|_Z)$ is
sufficiently small, hence the existence of a local solution in $Z$.
\smallbreak
The conservation of mass and energy follows from standard arguments
(see e.g. \cite{CazCourant}). Under Assumption~\ref{hyp:V}, this
implies an a priori bound for $\|u(t)\|_Z$, and so the solution $u$ is
global in time, $u\in L^\infty(\R;Z)$.
\smallbreak
Unconditional uniqueness as stated in Theorem~\ref{theo:cauchy}
follows from the same approach as in \cite{TzVi-p}. If $u_1,u_2\in
C([0,T];Z)$ are two solutions of \eqref{eq:nls} with the same initial
datum, then
\begin{equation*}
u_2(t)-u_1(t) = -i\int_0^t e^{-i(t-s)H}
\(|u_2|^{2{\sigma}}u_2-|u_1|^{2{\sigma}}u_1\)(s)ds.
\end{equation*}
Resuming the same estimates as above, we now have, for
$0<\tau\leqslant T$:
\begin{align*}
\|u_2 - u_1\|_{L^q_\tau L^r_yL^2_x}&\lesssim
\(\|u_2\|_{L^\theta_\tau L^k_yL^\infty_x}^{2{\sigma}}+
\|u_1\|_{L^\theta_\tau
L^k_yL^\infty_x}^{2{\sigma}}\)\|u_2-u_1\|_{L^q_\tau L^r_yL^2_x}\\
&\lesssim \tau^{2{\sigma}/\theta} \( \|u_2\|_{Z_T}^{2{\sigma}} +
\|u_1\|_{Z_T}^{2{\sigma}} \)
\|u_2-u_1\|_{L^q_\tau
L^r_yL^2_x},
\end{align*}
and uniqueness follows by taking $\tau>0$ sufficiently small.
\smallbreak
To complete the proof of Theorem~\ref{theo:cauchy}, we just have to
check that the extra regularity $u_0\in \tilde Z$ is propagated by the
flow. To do so, it suffices to replace the space $Z_T$
with
\begin{equation*}
\tilde Z_T=\{u\in L^\infty((0,T),Z),\quad A_j(t)u\in
L^q\((0,T);L^r_yL^2_x\),
\forall j\in \{0,1,2,3\}\},
\end{equation*}
that is, to add the field $A_3$. The second point of
Lemma~\ref{lem:opA}, and the above computations then yield
\begin{align*}
\|A_3\Phi(u)\|_{L^\infty_TL^2_{xy}}+
\|A_3\Phi(u)\|_{L^q_TL^r_yL^2_{x}}&\lesssim \|yu_0\|_{L^2_{xy}}+
\|u\|_{L^\theta_T
L^k_yL^\infty_x}^{2{\sigma}}\|A_3u\|_{L^q_TL^r_yL^2_{x}}\\
&\lesssim \|yu_0\|_{L^2_{xy}}+
T^{2{\sigma}/\theta}\|u\|_{Z_T}^{2{\sigma}}\|A_3u\|_{L^q_TL^r_yL^2_{x}}.
\end{align*}
The above fixed point argument can then be resumed: we construct a
local solution in $\tilde Z$, $u\in C([-T,T];\tilde Z)\cap L^\infty(\R;Z)$. The latest
property and the previous estimate show that $A_3u\in C(\R;L^2_{xy})$
is global in time.
\section{Existence of wave operators}
\label{sec:wave}
To prove the existence of wave operators, we construct a fixed point
for the related Duhamel's formula,
\begin{equation}\label{eq:duhamel-}
u(t)=e^{-itH}u_--i\int_{-\infty}^t e^{-i(t-s)H}\(|u|^{2{\sigma}}u\)(s)ds
=:\Phi_-(u)(t),
\end{equation}
on some time interval $(-\infty,-T]$ for $T$ possibly very large but
finite. According to the regularity assumption on $u_-$, we construct
a solution in $Z$ or in $\tilde Z$. This solution is actually global
in time from either case of Theorem~\ref{theo:cauchy}. We therefore
focus on the construction of a fixed point for $\Phi_-$, as well as on
uniqueness. In a similar fashion as in Section~\ref{sec:cauchy}, we
denote $L_T^aX = L^a((-\infty,-T];X)$.
\subsection{Wave operators in $Z$}
Resume the $(d-1)$-admissible pair $(q,r)$ used in
Section~\ref{sec:cauchy}, and
$(\theta,k)$ given by \eqref{def-theta-k}.
For $(q_1,r_1)$ a $(d-1)$-admissible pair, and $j\in
\{0,1,2\}$, Strichartz estimates and H\"older
inequality yield:
\begin{align*}
\left\| A_j\Phi_-(u)\right\|_{L^{q_1}_T L^{r_1}_yL^2_x}
&\lesssim\|A_ju_-\|_{L^2_{xy}} +
\left\| A_j\(|u|^{2{\sigma}}u\)\right\|_{L^{q'}_TL^{r'}_y L^2_x}
\\
&\lesssim\|A_ju_-\|_{L^2_{xy}} +
\|u\|_{L^{\theta}_TL^{k}_y L^\infty_x}^{2{\sigma}}\| A_j u\|_{L^{q}_TL^{r}_y L^2_x}.
\end{align*}
By construction,
\begin{equation*}
2\leqslant k<\frac{2(d-1)}{(d-2)_+}< \frac{2(d-1)}{(d-3)_+},
\end{equation*}
so we can find $p$ such that $(p,k)$ is $(d-1)$-admissible. Putting
the definition of admissible pairs and \eqref{def-theta-k} together,
we get
\begin{equation*}
1-\frac{2{\sigma}}{\theta} =\frac{2}{q}= (d-1)\(\frac{1}{2}-\frac{1}{r}\)
= \frac{(d-1){\sigma}}{k} = {\sigma}\(\frac{d-1}{2}-\frac{2}{p}\).
\end{equation*}
By assumption, ${\sigma}\geqslant \frac{2}{d-1}$, so $p\leqslant \theta$, and there
exists $\beta\in (0,1]$ such that
\begin{equation*}
\|u\|_{L^{\theta}_TL^{k}_y L^\infty_x}\leqslant \|u\|_{L^{p}_TL^{k}_y
L^\infty_x}^\beta
\|u\|_{L^{\infty}_TL^{k}_y L^\infty_x}^{1-\beta}.
\end{equation*}
Corollary~\ref{coro} implies
\begin{equation*}
\left\| A_j\Phi_-(u)\right\|_{L^{q_1}_T L^{r_1}_yL^2_x}
\lesssim\|A_ju_-\|_{L^2_{xy}} +
\|u\|_{L^{p}_TL^{k}_y
L^\infty_x}^{2{\sigma}\beta}
\|u\|_{L^{\infty}_TZ}^{2{\sigma}(1-\beta)}\| A_j u\|_{L^{q}_TL^{r}_y L^2_x}.
\end{equation*}
Now the one-dimensional Gagliardo-Nirenberg inequality
\begin{equation*}
\|f\|_{L^\infty_x}\leqslant \sqrt 2 \|f\|_{L^2_x}^{1/2}\|{\partial}_x f\|_{L^2_x}^{1/2}
\end{equation*}
and according to the proof of Lemma~\ref{lem-emb-bm-hm}, we have
\begin{equation}\label{eq:waveH1}
\begin{aligned}
\left\| A_j\Phi_-(u)\right\|_{L^{q_1}_T L^{r_1}_yL^2_x}
&\leqslant C\|A_ju_-\|_{L^2_{xy}} \\
&+
C\|u\|_{L^{p}_TL^{k}_y
L^2_x}^{{\sigma}\beta}\|A_1u\|_{L^{p}_TL^{k}_y
L^2_x}^{{\sigma}\beta}
\|u\|_{L^{\infty}_TZ}^{2{\sigma}(1-\beta)}\| A_j u\|_{L^{q}_TL^{r}_y L^2_x}.
\end{aligned}
\end{equation}
for $C$ sufficiently large. We can now define
\begin{align*}
B_T:=\Big\{ & u\in C(]-\infty,-T];Z),\\
&\left\|
A_j u\right\|_{L^q_TL^r_yL^2_x} + \left\|
A_j u\right\|_{L^\infty_TL^2_{xy}} \leqslant 4C \|A_ju_-\|_{L^2_{xy}},
\quad j\in \{0,1,2\},\\
& \left\| A_j u\right\|_{L^p_T L^k_yL^2_x} \leqslant 2 \left\|
A_j e^{-itH}u_-\right\|_{L^p_T L^k_yL^2_x} ,\quad j\in \{0,1\}\Big\}.
\end{align*}
From Strichartz estimates, we know that
for $j\in \{0,1\}$,
\begin{equation*}
A_j e^{-itH}u_- \in L^p(\R;L^k_yL^2_x),\quad \text{so}\quad
\left\|A_j e^{-itH}u_-\right\|_{L^p_TL^k_TL^2_x} \to 0\quad \text{as }T\to +\infty.
\end{equation*}
Since $\beta>0$, we infer that $\Phi_-$ maps $B_T$ to itself, for
$T$ sufficiently large, by \eqref{eq:waveH1}, and since the same
estimates yield,
for $j\in \{0,1\}$,
\begin{align*}
\left\| A_j\Phi_-(u)\right\|_{L^{p}_T L^{k}_yL^2_x}
&\leqslant\|A_je^{-itH}u_-\|_{L^{p}_T L^{k}_yL^2_x} \\
&\quad+
C\|u\|_{L^{p}_TL^{k}_y
L^2_x}^{{\sigma}\beta}\|A_1u\|_{L^{p}_TL^{k}_y
L^2_x}^{{\sigma}\beta}
\|u\|_{L^{\infty}_TZ}^{2{\sigma}(1-\beta)}\| A_j u\|_{L^{q}_TL^{r}_y L^2_x}.
\end{align*}
We have also, for $u_2,u_1\in B_T$, and typically $(q_1,r_1)\in \{(q,r),(\infty,2)\}$:
\begin{align*}
\left\| \Phi_-(u_2)-\Phi_-(u_1)\right\|_{L^{q_1}_T L^{r_1}_yL^2_x}&\lesssim
\max_{j=1,2}\| u_j\|_{L^\theta_TL^k_yL^\infty_x}^{2{\sigma}} \left\|
u_2-u_1\right\|_{L^q_T L^r_yL^2_x}\\
\lesssim \left\|e^{-itH}u_-\right\|_{L^p_TL^k_yL^2_x}^{{\sigma}\beta}&
\left\|A_1 e^{-itH}u_-\right\|_{L^p_TL^k_yL^2_x}^{{\sigma}\beta} \|u_-\|_Z^{2{\sigma}(1-\beta)} \left\|
u_2-u_1\right\|_{L^q_T L^r_yL^2_x}.
\end{align*}
Up to choosing $T$ larger, $\Phi_-$ is a contraction on $B_T$, so $\Phi_-$
has a unique fixed point in $B_T$, which solves
\eqref{eq:duhamel-}. Uniqueness as stated in Theorem~\ref{theo:waveop}
is an easy consequence of the above estimates.
\subsection{Wave operators in $\tilde Z$}
In the case $u_-\in \tilde Z$, we consider the whole set of
vector fields, $(A_j)_{0\leqslant j\leqslant 3}$. For $(q,r)$ a $(d-1)$-admissible
pair to be chosen later, we define
\begin{equation*}
\tilde Z_T=\{u\in C((-\infty,-T];\tilde Z),\quad A_j(t)u\in
L^q_TL^r_yL^2_x\cap L^\infty_TL^2_{xy},
\forall j\in \{0,1,2,3\}\}.
\end{equation*}
We have, for all $(d-1)$-admissible pairs $(q_1,r_1)$, and all
$j\in \{0,1,2,3\}$,
\begin{equation}\label{phi-YT}
\|A_j\Phi_-(u)\|_{L^{q_1}_TL^{r_1}_yL^2_x}\lesssim \|u_-\|_{\tilde
Z}+\|u\|_{L^\theta_TL^k_yL^\infty_x}^{2\sigma}\|A_j u\|_{L^q_TL^r_yL^2_x},
\end{equation}
where $\theta$ and $k$ are again given by \eqref{def-theta-k}. If
\begin{equation}\label{eq:sob-emb}
H^{1/2-}(\R^{d-1}_y)\hookrightarrow L^k(\R^{d-1}_y),\text{ that is},\quad
2\leqslant k<\frac{2(d-1)}{d-2},
\end{equation}
we can find $s$ and $\gamma$ satisfying \eqref{ass-ksg} and $s+\gamma=1$.
To obtain explicit time decay, apply Proposition~\ref{sob-anis} to
$v=e^{-i|y|^2/(2t)}u$. This yields
\begin{equation*}
\|u\|_{L^k_yL^\infty_x}=\|v\|_{L^k_yL^\infty_x}\lesssim
\|v\|_{L^k_yH^{s}_x}\lesssim\|v\|_{L^2_yH^{s}_x}^{1-\delta}\|v\|_{\dot{H}^\gamma_yH^{s}_x}^\delta,
\end{equation*}
where $\delta$ is defined by
$$\delta \gamma=(d-1)\left(\frac{1}{2}-\frac{1}{k}\right).$$
Then, since $\gamma+s=1$, it follows from the Young inequality as
in Lemma~\ref{lem2} that
\begin{align}\label{t-g}
\|v\|_{\dot{H}^\gamma_yH^{s}_x}&=|t|^{-\gamma}\left(\int|t\eta|^{2\gamma}(1+\xi^2)^s|\widehat{v}(\xi,\eta)|^2d\xi
d\eta\right)^{1/2}\\
&\lesssim|t|^{-\gamma}\left(\int\left(|t\eta|^{2}+(1+\xi^2)\right)|\widehat{v}(\xi,\eta)|^2d\xi
d\eta\right)^{1/2} \nonumber \\
&\lesssim
|t|^{-\gamma}\left(\|A_3(t)u\|_{L^2_xL^2_y}+\|u\|_{L^2_yH^1_x}\right),\nonumber
\end{align}
where in the last line, we have used Plancherel formula and
$$A_3(t)u=it e^{i|y|^2/(2t)}\nabla_y e^{-i|y|^2/(2t)}u=it e^{i|y|^2/(2t)}\nabla_y v.$$
Then, we deduce from \eqref{t-g} and Lemma~\ref{lem2} that for any
$u\in\tilde Z_T$ and $t\leqslant -T$, we have
\begin{equation}\label{pre-scat}
\|u(t)\|_{L^k_yL^\infty_x}
\lesssim
\frac{1}{|t|^{(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)}}\sum_{j=0}^3
\|A_j u\|_{L^\infty_TL^2_{xy}}
\lesssim \frac{1}{|t|^{(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)}}\|u\|_{\tilde
Z_T}.
\end{equation}
Then, provided $t\mapsto |t|^{-(d-1)(1/2-1/k)}$
belongs to $L^\theta(-\infty,-1)$, \eqref{phi-YT} and \eqref{pre-scat}
imply that for every
$u\in \tilde Z_T$,
\begin{align}\label{phi-YT-bis}
\|A_j\Phi_-(u)\|_{\tilde Z_T}\lesssim\|u_-\|_{\tilde
Z}+T^{2\sigma\(\frac{1}{\theta}-(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)\)}\|u\|_{\tilde
Z_T}^{2\sigma+1}.
\end{align}
Let us now explain how the parameters $\theta,k,q,r$ are chosen. Since
$\sigma>1/(d-1)$, one can choose $q>2$ large enough such that
\begin{equation}\label{cond-scat-0}
(d-1)\sigma>\frac{2}{q}+1.
\end{equation}
Then, $r$ is chosen such that $(q,r)$ is a $(d-1)$-admissible pair, in
such a way that \eqref{cond-scat-0} becomes
\begin{equation*}\label{cond-scat-1}
(d-1)\left(\sigma+\frac{1}{r}-\frac{1}{2}\right)>1,
\end{equation*}
which is equivalent to
\begin{equation*}\label{cond-scat-2}
(d-1)\left(\sigma-\frac{2\sigma}{k}\right)=(d-1)\left(\sigma-1+\frac{2}{r}\right)>1-(d-1)\left(\frac{1}{2}-\frac{1}{r}\right)=1-\frac{2}{q}=\frac{2\sigma}{\theta},
\end{equation*}
where $\theta$ and $k$ are defined by \eqref{def-theta-k}. This is
precisely the condition $\theta(d-1)(\frac{1}{2}-\frac{1}{k})>1$ which
ensures that the right hand side of \eqref{pre-scat} belongs to
$L^\theta$. In terms of $k$, \eqref{cond-scat-0} is equivalent to
\begin{equation*}
\frac{1}{k}<1-\frac{1}{(d-1){\sigma}}.
\end{equation*}
This condition is consistent with \eqref{eq:sob-emb} if and only if
\begin{equation*}
\frac{d-2}{2(d-1)}<1-\frac{1}{(d-1){\sigma}},
\end{equation*}
which is equivalent to ${\sigma}>\frac{2}{d}$.
\smallbreak
The rest of the proof is similar to the proof of local
well-posedness of the Cauchy problem: we take
$R$ and $T$ sufficiently large so that the ball of radius $R$ in
$\tilde Z_T$ is stable under the action of
$\Phi_-$, and so that $\Phi_-$ is a contraction on this ball, equipped
with the distance $\|u\|_{L^\infty_TL^2_{xy}}+
\|u\|_{L^q_TL^r_yL^2_x}$, in view of the previous estimates and
\begin{equation*}
\|\Phi_-(u_2)-\Phi_-(u_1)\|_{L^{q_1}_TL^{r_1}_yL^2_x} \lesssim
\max_{j=1,2}\|u_j\|^{2{\sigma}}_{L^\theta_TL^k_yL^\infty_x}
\|u_2-u_1\|_{L^q_TL^r_yL^2_x}.
\end{equation*}
In view of \eqref{eq:equiv-norm}, the solution that we have constructed satisfies
\begin{equation*}
e^{itH} u\in L^\infty((-\infty,-T];\tilde Z).
\end{equation*}
Uniqueness in this class follows from \eqref{eq:equiv-norm} and the
same approach as for the Cauchy problem. If $u_1$ and $u_2$ are two
solutions of \eqref{eq:nls} satisfying
\begin{equation*}
e^{itH} u_j\in L^\infty((-\infty,-T];\tilde Z),\quad
\|e^{itH}u_j(t)-u_-\|_{\tilde Z}\Tend t {-\infty}0,\quad j=1,2,
\end{equation*}
then for $\tau>T$,
\begin{equation*}
\|u_2-u_1\|_{L^{q}_\tau L^{r}_yL^2_x} \lesssim
\max_{j=1,2}\|u_j\|^{2{\sigma}}_{L^\theta_\tau L^k_yL^\infty_x}
\|u_2-u_1\|_{L^q_\tau L^r_yL^2_x},
\end{equation*}
and \eqref{pre-scat} implies
\begin{equation*}
\|u_2-u_1\|_{L^{q}_\tau L^{r}_yL^2_x} \lesssim
\tau^{2\sigma\(\frac{1}{\theta}-(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)\)}
\|u_2-u_1\|_{L^q_\tau L^r_yL^2_x}.
\end{equation*}
Choosing $\tau$ sufficiently large, we have $u_2=u_1$ for $t\leqslant -\tau$,
and Theorem~\ref{theo:cauchy} yields $u_2\equiv u_1$.
\section{Asymptotic completeness}
\label{sec:ac}
In this section, we prove Theorem~\ref{theo:AC}. Three approaches are
available to prove asymptotic completeness for
nonlinear Schr\"odinger equations (without potential). The initial
approach (\cite{GV79Scatt}) consists in working with a
$\Sigma$ regularity. This
makes it possible to use the operator $x+it\nabla$, whose main
properties are essentially those stated in Lemma~\ref{lem:opA}, and to
which an important evolution law (the
pseudo-conformal conservation law) is associated. This law provides
important a priori estimates, from which asymptotic completeness
follows very easily in the case ${\sigma}\geqslant 2/d$, and less easily for
some range of ${\sigma}$ below $2/d$; see
e.g. \cite{CazCourant}. Unfortunately, this conservation law seems to
be bound to isotropic frameworks: an analogous identity is available
in the presence on an isotropic quadratic potential (\cite{CaSIMA}),
but in our present framework, anisotropy seems to rule out a similar
algebraic miracle.
\smallbreak
The second historical approach relaxes the localization assumption on
the data, and allows
to work in $H^1(\R^d)$, provided that ${\sigma}>2/d$. It is based on
Morawetz inequalities: asymptotic completeness is then established in
\cite{LiSt78,GV85} for the case $d\geqslant 3$, and in \cite{NakanishiJFA} for the low
dimension cases $d=1,2$, by introducing more intricate Morawetz
estimates.
\smallbreak
The most recent approach to prove asymptotic completeness in $H^1$
relies on the introduction of interaction Morawetz estimates in \cite{CKSTTAnnals},
an approach which has been revisited since, in particular in
\cite{PlVe09} and \cite{GiVe10}. In the anisotropic case, interaction
Morawetz have been used in \cite{AnCaSi-p} and \cite{TzVi-p} with two
different angles: in both cases, it starts with the choice of an
anisotropic weight in the virial computation from
\cite{GiVe10,PlVe09}, but the interpretations of this computation are
then different. We start by presenting a unified statement of this
aproach in the next paragraph.
\subsection{Morawetz estimates}
\label{sec:morawetz}
For $(x,y)\in \R^d$ and $\mu>0$, we denote by $Q(x,y,\mu)$ a dilation of
the unit cube
centered in $(x,y)$,
\begin{equation*}
Q(x,y,\mu) = (x,y)+[-\mu,\mu]^d.
\end{equation*}
\begin{proposition}\label{prop:morawetz}
Let $u\in C(\R;Z)$ be as in Theorem~\ref{theo:cauchy}. For every
$\mu>0$, there exists $C_\mu>0$ such that
\begin{align*}
\left\| |\nabla_y|^{\frac{4-d}{2}}R\right\|_{L^2_{ty}(\R\times
\R^{d-1})}^2 + \int_{\R}& \(\sup_{(x_0,y_0)\in
\R^d}\iint_{Q(x_0,y_0,\mu)}|u(t,x,y)|^2dxdy \)^{{\sigma}+2} dt \\
&\leqslant
C_\mu\sup_{t\in \R}\|u(t)\|_{H^1_{xy}}^4\lesssim \|u_0\|_Z^4,
\end{align*}
where
\begin{equation*}
R(t,y) = \int_{-\infty}^{+\infty} |u(t,x,y)|^2dx
\end{equation*}
is the marginal of the mass density.
\end{proposition}
\begin{proof}
We resume the computations from \cite[Section~5]{AnCaSi-p}, and
simply recall the main steps.
\smallbreak
To shorten the notations, we set $z=(x, y)$. Following
\cite{GiVe10}, we write that if $u$ is a solution
to \eqref{eq:nls}, then we have
\begin{equation}\label{eq:cons_laws}
\left\{\begin{aligned}
&{\partial}_t\rho+\diver J=0\\
&{\partial}_tJ+\diver\(\RE(\nabla\bar u\otimes\nabla u)\)+\frac{\sigma}{\sigma+1}\nabla\rho^{\sigma+1}
+\rho\nabla V=\frac14\nabla\Delta\rho,
\end{aligned}\right.
\end{equation}
where $\rho(t, z):=|u(t, z)|^2$ and $ J(t, z):=\IM(\bar u\nabla u)(t, z)$.
Let us define the virial potential
\begin{equation*}
I(t):=\frac12\iint_{\R^d\times\R^d}\rho(t, z)a(z-z')\rho(t, z')\,dzdz'=\frac12\langle\rho, a\ast\rho\rangle,
\end{equation*}
where $a$ is a sufficiently smooth even weight function which will be
be eventually a function of $y$ only. Here $\langle\cdot,\cdot\rangle$
denotes the scalar
product in $L^2(\R^d)$. By using \eqref{eq:cons_laws}, we see that the time
derivative of $I(t)$ reads
\begin{equation}\label{eq:mor_act}
\frac{d}{dt}I(t)=-\langle\rho,\nabla a\ast J\rangle=\iint\rho(t, z')\nabla a(z-z')\cdot J(t, z)\,dz'dz=:M(t),
\end{equation}
where $M(t)$ is the Morawetz action. By using again the balance laws
\eqref{eq:cons_laws} we have
\begin{equation}\label{eq:Moraw}
\begin{aligned}
\frac{d}{dt}M(t)=&-\langle J,\nabla^2a\ast J\rangle+\langle\rho,\nabla^2a\ast\RE(\nabla\bar u\otimes\nabla u)\rangle
+\frac{\sigma}{\sigma+1}\langle\rho,\Delta a\ast\rho^{\sigma+1}\rangle
\\
&-\langle\rho,\nabla a\ast(\rho\nabla V)\rangle-\frac14\langle\rho,\Delta a\ast\Delta\rho\rangle\\
=&-\langle\IM(\bar u\nabla u), \nabla^2a\ast\IM(\bar u\nabla u)\rangle
+\langle\rho,\nabla^2a\ast(\nabla\bar u\otimes\nabla u)\rangle\\
&+\frac{\sigma}{\sigma+1}\langle\rho,\Delta a\ast\rho^{\sigma+1}\rangle
-\langle\rho,\nabla a\ast(\rho\nabla V)\rangle
-\frac14\langle\rho,\Delta a\ast\Delta\rho\rangle,
\end{aligned}
\end{equation}
where in the second term we dropped the real part because of the
symmetry of $\nabla^2a$ (here, the notation $\nabla^2a\ast\RE(\nabla\bar
u\otimes\nabla u)$ stands for
$\sum_{j,k}\partial^2_{jk}a\ast\RE(\partial_k\bar u\partial_j u)$). Leaving out the details presented in
\cite{AnCaSi-p} and \cite{TzVi-p}, the computation shows that if
$\nabla^2 a$ is non-negative and if $a$ depends on $y$ only (so we
have $\nabla a(z_1)\cdot \nabla V(z_2)=0$ for all $z_1,z_2\in \R^d$),
then we have:
\begin{equation}\label{eq:dtm2}
\frac{d}{dt}M(t)\geqslant \frac12\langle\nabla_y\rho,
\Delta_ya\ast\nabla_y\rho\rangle
+\frac{\sigma}{\sigma+1}\langle\rho, \Delta_ya\ast\rho^{\sigma+1}\rangle.
\end{equation}
Now we consider two choices for the weight $a$. First, for $a(y)=|y|$,
we have indeed $\nabla^2 a\geqslant 0$ as a symmetric matrix, and for $d\geqslant
3$,
$\Delta_ya(y)=\frac{d-2}{|y|}$: it is, up to a multiplicative
constant, the integral kernel of the operator
$(-\Delta_y)^{-\frac{d-2}{2}}$, that is,
\begin{equation*}
\((-\Delta_y)^{-\frac{d-2}{2}}f\)(y)=\int_{\R^{d-1}}\frac{c}{|y-y'|}f(y')\,dy'.
\end{equation*}
Thus, by recalling $z=(x, y)$, we obtain
\begin{multline*}
\iint_{\R^d\times\R^d}\frac{1}{|y-y'|}\nabla_y\rho(t, z')\cdot\nabla_y\rho(t, z)\,dz'dz\\
=\iiint_{\R\times\R\times\R^{d-1}}\nabla_y\rho(t, x, y)
\cdot\nabla_y(-\Delta_y)^{-\frac{d-2}{2}}\rho(t, x', y)\,dxdx'dy.
\end{multline*}
Hence, if we define the marginal of the mass density
\begin{equation*}
R(t, y):=\int_{\R}\rho(t, x, y)\,dx,
\end{equation*}
the last integral also reads
\begin{equation*}
\int_{\R^{d-1}}\left||\nabla_y|^{\frac{4-d}{2}}R(t, y)\right|^2\,dy.
\end{equation*}
We now plug this expression into \eqref{eq:dtm2} and we integrate in
time. Furthermore, the second term in the right hand side in
\eqref{eq:dtm2} is positive. We then infer
\begin{equation}\label{eq:mor_est}
\int_{-T}^T\int_{\R^{d-1}}\left||\nabla_y|^{\frac{4-d}{2}}R(t,
y)\right|^2\,dydt\leqslant C\sup_{t\in[-T, T]}|M(t)|.
\end{equation}
Furthermore, with our choice of the weight $a$, we have
\begin{equation*}
|M(t)|=\left|\iint\rho(t, z')\frac{y-y'}{|y-y'|}\cdot\IM(\bar u\nabla_yu)(t, z)\,dz'dz\right|
\leqslant\|u_0\|_{L^2(\R^d)}^3\|\nabla_yu(t)\|_{L^2(\R^d)},
\end{equation*}
hence the first part of Proposition~\ref{prop:morawetz} in the case
$d\geqslant 3$. In the case $d=2$, the choice $a(y)=|y|$ leads to $a''(y) =
2\delta_0$, and the conclusion remains the same.
\smallbreak
Now, as in \cite{TzVi-p}, consider the weight $a(y)=\<y\>$: we still
have $\nabla^2 a\geqslant 0$. Resume \eqref{eq:Moraw}: the computations from
\cite{TzVi-p,PlVe09}
yield a rearrangement of the terms so that instead of
\eqref{eq:dtm2}, we now have
\begin{equation*}
\frac{d}{dt}M(t)\geqslant \frac{\sigma}{\sigma+1}\langle\rho,
\Delta_ya\ast\rho^{\sigma+1}\rangle.
\end{equation*}
The right hand side is equal to
\begin{equation*}
\frac{{\sigma}}{{\sigma}+1}\iint\iint |u(t,x_1,y_1)|^2 \Delta a (y_1-y_2)
|u(t,x_2,y_2)|^{2{\sigma}+2}dx_1dy_1dx_2dy_2.
\end{equation*}
Following \cite{TzVi-p}, we note that
\begin{equation*}
\inf_{Q(0,0,2\mu)}\Delta_y\(\<y\>\)>0,
\end{equation*}
so the above term is bounded from below by constant times
\begin{equation*}
\sup_{(x_0,y_0)\in \R^d}\iint_{Q(x_0,y_0,\mu)}
\iint_{Q(x_0,y_0,\mu)} |u(t,x_1,y_1)|^2 |u(t,x_2,y_2)|^{2{\sigma}+2}dx_1dy_1dx_2dy_2.
\end{equation*}
H\"older inequality yields
\begin{equation*}
\iint_{Q(x_0,y_0,\mu)} |u(t,x_2,y_2)|^{2{\sigma}+2}dx_2dy_2\gtrsim
\(\iint_{Q(x_0,y_0,\mu)} |u(t,x_2,y_2)|^{2}dx_2dy_2\)^{{\sigma}+1}.
\end{equation*}
Finally, with this second choice for $a$, we still have
\begin{equation*}
|M(t)|\leqslant \|u_0\|_{L^2_{xy}}^3\|\nabla_y u(t)\|_{L^2_{xy}},
\end{equation*}
hence the result by integrating in time.
\end{proof}
\subsection{End of the argument}
To prove Theorem~\ref{theo:AC} in the case $d\leqslant 4$, one can resume
the approach followed in \cite[Section~6]{AnCaSi-p} which is readily adapted to
our framework, the only difference being that the function space and
the related set of vectorfields are not the same here.
\smallbreak
However, as
pointed out in \cite{TzVi-p}, the fact that negative order derivatives
are involved in the first term in Proposition~\ref{prop:morawetz}
makes it delicate to use this term when $d\geqslant 5$, and requires fine
harmonic analysis estimates in the case $V=0$; it is not clear whether
or not
these tools can be adapted to the present setting. This is why the second term in
Proposition~\ref{prop:morawetz}, which corresponds to the one
considered in \cite{TzVi-p}, is more efficient then, and allows to
prove Theorem~\ref{theo:AC} for all $d\geqslant 2$.
\smallbreak
The first step stems from \cite{Vi09}:
Theorem~\ref{theo:cauchy} and Proposition~\ref{prop:morawetz} imply that
\begin{equation*}
\|u(t)\|_{L^r_{xy}}\Tend t {+\infty} 0,\quad \forall 2<r<\frac{2d}{(d-2)_+}.
\end{equation*}
The end of the proof is presented in \cite{TzVi-p}, and is readily
adapted to our framework: it consists in choosing suitable Lebesgue
exponents and applying inhomogeneous Strichartz estimates for
non-admissible pairs, which follow in our case from
\cite{AnCaSi-p,FoschiStri}. Since the proof is then absolutely the
same as in \cite{TzVi-p}, we choose not to reproduce it here.
\bibliographystyle{siam}
|
2,877,628,090,156 | arxiv | \section{Introduction}
Observations with the Atacama Large Millimeter/submillimeter Array (ALMA) have revealed that many circumstellar disks contain intricate radial substructures \citep{2015ApJ...808L...3A,2016ApJ...820L..40A,2016ApJ...818L..16Z,2016ApJ...819L...7N,2016Sci...353.1519P,2016PhRvL.117y1101I,2016Natur.535..258C,2017A&A...597A..32V,2017A&A...600A..72F}, opening the door for previously inaccessible studies of the physical nature of disks. A number of physical processes have been proposed to explain the formation of rings and gaps in disks, including gap clearing by planets \citep{2015ApJ...809...93D,2015MNRAS.453L..73D,2017ApJ...843..127D,2017ApJ...850..201B}, rapid pebble growth at the condensation fronts of abundant volatile species \citep{2015ApJ...806L...7Z}, the pileup of volatile ices in sintering zones just outside snow lines \citep{2016ApJ...821...82O}, sharp changes in the disk viscosity at the boundaries of non-turbulent `dead zones' \citep{2015A&A...574A..68F,2016A&A...590A..17R}, magnetic self-organization through zonal flows \citep{2017A&A...600A..75B}, and the secular gravitational instability \citep{2014ApJ...794...55T}.
We presented, in \citet{2017MNRAS.468.3850S} (herein, \citetalias{2017MNRAS.468.3850S}), a novel mechanism for forming rings and gaps in magnetically coupled disk-wind systems in the presence of Ohmic resistivity, which is the dominant non-ideal magnetohydrodynamic (MHD) effect in the inner (sub-au) part of the disk \citep{2007Ap&SS.311...35W,2014prpl.conf..411T}. It relies on a magnetically driven disk wind \citep{1982MNRAS.199..883B} to remove angular momentum from the disk at a rate that varies strongly with radius, leading to a large spatial variation in accretion rate and thus the disk surface density. Observationally, there is now growing evidence for rotating winds removing angular momentum from disks \citep{2016ApJ...831..169S,2017NatAs...1E.146H,2017A&A...607L...6T,2018arXiv180203668L}. Theoretically, a picture of wind-driven disk evolution is also beginning to emerge, with non-ideal MHD effects (Ohmic resistivity, ambipolar diffusion, and the Hall effect) suppressing MHD turbulence from the magnetorotational instability (MRI; \citealt{1991ApJ...376..214B}) over a wide range of radii, which leaves MHD disk winds as the primary driver of disk accretion in these regions \citep{2000ApJ...530..464F,2003ApJ...585..908F,2011ApJ...736..144B,2013ApJ...769...76B,2013MNRAS.434.2295K,2015ApJ...801...84G}.
In this follow-up work, we focus on the intermediate radii of young star disks (a few to tens of au) where ambipolar diffusion (AD) starts to become the most important non-ideal MHD effect, especially in the upper layers of the disk \citep{2007Ap&SS.311...35W,2014prpl.conf..411T}. We find that rings and gaps are naturally produced in the presence of a significant poloidal magnetic field, just as in the resistive case studied in \citetalias{2017MNRAS.468.3850S}. We show that a relatively laminar disk-wind system develops in the presence of a relatively strong ambipolar diffusion, which makes it easier to analyse the simulation results and identify a new mechanism for ring and gap formation. The mechanism is driven by reconnection of the highly pinched poloidal magnetic field in a midplane current sheet steepened by ambipolar diffusion \citep{1994ApJ...427L..91B}, in a manner that is reminiscent of the tearing mode \citep{1963PhFl....6..459F} or the pinch-tearing mode \citep{2009MNRAS.394..715L}. We show that the reconnection leads to weakening of the poloidal field in some regions, which accrete more slowly and form rings, and field concentration in others, where accretion is more efficient creating gaps.
The rest of the paper is organized as follows. In Section~\ref{sec:setup} we describe the simulation setup, including the equations solved, the initial disk model, and the boundary conditions. Section~\ref{sec:ref} analyses the results of a reference simulation in detail and explains how rings and gaps are formed in the coupled disk-wind system in the presence of relatively strong ambipolar diffusion. In Section~\ref{sec:param} we explore how changes in the magnetic field and ambipolar diffusion strength modify the picture of the reference run. In Section~\ref{sec:discuss} we compare to other similar works in the field and discuss the implications of our work on dust settling, growth, and trapping that are important to the formation of planetesimals and planets. Finally, Section~\ref{sec:conc} concludes with the main results of this study.
\section{Problem setup}\label{sec:setup}
\subsection{MHD equations}
We use the ZeusTW code \citep{2010ApJ...716.1541K} to solve the time-dependent magnetohydrodynamic (MHD) equations in axisymmetric spherical coordinates ($r,\theta,\phi$). The ZeusTW code is based on the ideal MHD code, ZEUS-3D (version 3.4; \citealt{1996ApJ...457..291C,2010ApJS..187..119C}), which is itself developed from ZEUS-2D \citep{1992ApJS...80..753S,1992ApJS...80..791S}. In the ZeusTW code, Ohmic resistivity is treated using the algorithm described in \citet{2000ApJ...530..464F} and AD is implemented using the fully explicit method of \citet{1995ApJ...442..726M} (see also \citealt{2011ApJ...738..180L}). The equations solved are
\begin{equation}
\frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \bm{v} \right) = 0,
\end{equation}
\begin{equation}
\rho\frac{\partial\bm{v}}{\partial t} + \rho\left(\bm{v}\cdot\nabla\right)\bm{v} = -\nabla P + \bm{J}\times\bm{B}/c - \rho\nabla\Phi_g,
\end{equation}
\begin{equation}
\frac{\partial\bm{B}}{\partial t} = \nabla\times\left(\bm{v}\times\bm{B}\right) - \frac{4\pi}{c}\nabla\times\left(\eta_O\bm{J} + \eta_A\bm{J_\perp}\right),\label{eq:induction}
\end{equation}
\begin{equation}
\frac{\partial e}{\partial t} + \nabla \cdot \left(e \bm{v} \right) = -P \nabla \cdot \bm{v},
\end{equation}
where the internal energy is $e=P/(\Gamma-1)$ and $\Gamma$ is the adiabatic index. The current density is $\bm{J}=(c/4\pi)\nabla\times\bm{B}$ and the current density perpendicular to the magnetic field is $\bm{J}_\perp=-\bm{J}\times\bm{B}\times\bm{B}/B^2$. The Ohmic resistivity is $\eta_O$ and the effective ambipolar diffusivity $\eta_A$ is defined as
\begin{equation}
\eta_A=\frac{B^2}{4\pi\gamma\rho\rho_i},
\label{ADdiffusion}
\end{equation}
where $\rho_i$ is the ion density and $\gamma=\langle\sigma v\rangle_i/(m+m_i)$ is the frictional drag coefficient with units of $\rm{cm}^3~\rm{g}^{-1}~\rm{s}^{-1}$. The remaining parameters have their usual definitions. When referring to cylindrical coordinates, we will use the notation ($R,\phi,z$) such that $R=r\sin{\theta}$ and $z=r\cos{\theta}$.
\subsection{Initial conditions}
The initial conditions are similar to those in \citetalias{2017MNRAS.468.3850S}. We describe them here in detail for completeness. Specifically, the simulation domain is separated into two regions: a thin, cold, rotating disk orbiting a 1~\text{M$_\odot$}~central source at the grid origin and an initially non-rotating, hot corona above the disk that is quickly replaced by a magnetic wind driven from the disk. We choose the adiabatic index to be $\Gamma=1.01$ so that the material in the simulation domain is locally isothermal in the sense that any parcel of disk material nearly retains its initial temperature no matter where it moves. The initial temperature distribution is assumed to decrease with radius as a power-law $T\propto r^{-1}$, so that the sound speed is proportional to the local Keplerian speed.
\subsubsection{Disk}
The geometrically thin disk is characterized by the dimensionless parameter $\epsilon = h/r = c_s/v_K \ll 1$, where $h$ is the disk scale height, $c_s$ is the isothermal sound speed, and $v_K$ is the Keplerian speed. The initial value of $\epsilon$ is set to 0.05 for all simulations in this work. The disk is limited to the equatorial region where the polar angle $\theta \in {[\pi/2 - \theta_0,\pi/2 + \theta_0]}$, with disk (half) opening angle set to $\theta_0=\arctan(2\epsilon)$, i.e., the initial disk half-thickness is set to twice the scale height. This choice is somewhat arbitrary, but a more elaborate treatment of the initial disk surface is not warranted because the structure of the disk surface is quickly modified by a magnetic wind. The disk density takes the form of a radial power law multiplied by a Gaussian function of $z/r=\cos\theta$,
\begin{equation}
\rho_d(r,\theta) = \rho_{0} \left(\frac{r}{r_0}\right)^{-\alpha} \exp \left(-\frac{\cos^2\theta}{2 \epsilon^2}\right),
\end{equation}
as determined by hydrostatic balance. The subscript `0' refers to values on the disk midplane at the inner radial boundary. For all simulations shown in this paper, we use $\alpha=3/2$. The disk pressure is set as
\begin{equation}
P_d(r,\theta)=\rho_d c_s^2,
\end{equation}
with $c_s = \epsilon v_K$. The radial pressure gradient causes the equilibrium rotational velocity $v_\phi$ to be slightly sub-Keplerian,
\begin{equation}
v_\phi = v_K \sqrt{1-(1+\alpha)\epsilon^2}.
\end{equation}
\subsubsection{Corona}
We require that the hydrostatic corona is initially in pressure balance with the disk surface. This constraint sets the density drop from the disk surface to the corona by $1/[(1+\alpha)\epsilon^2]=160$, and a corresponding increase in temperature from the disk surface to the corona by the same factor. Therefore, the coronal density and pressure are
\begin{equation}
\rho_{c}(r)=\rho_{0} \epsilon^2(1+\alpha)\exp\left[-\frac{\cos^2\theta_0}{2 \epsilon^2}\right] \left(\frac{r}{r_0}\right)^{-\alpha}\equiv\rho_{c,0}\left(\frac{r}{r_0}\right)^{-\alpha},
\end{equation}
\begin{equation}
P_c(r)= \rho_c v_K^2/(1+\alpha).
\end{equation}
It is, however, important to note that the initial hot coronal material is quickly replaced by the colder disk material that remains nearly isothermal as it is launched from the disk surface into a wind.
\subsubsection{Magnetic field}
To ensure that the magnetic field is divergence-free initially, we set the magnetic field components using the magnetic flux function $\Psi$ as in \citet{2007A&A...469..811Z},
\begin{equation}
\Psi(r,\theta) = \frac{4}{3}r_0^2 B_{\mathrm{p},0}\left(\frac{r\sin\theta}{r_0}\right)^{3/4} \frac{m^{5/4}}{\left(m^{2}+\cot^2\theta\right)^{5/8}}\label{eq:psi},
\end{equation}
where $B_{\mathrm{p},0}$ sets the scale for the poloidal field strength and the parameter $m$ controls the bending of the field. The value of $B_{\mathrm{p},0}$ is set by the initial plasma-$\beta$, the ratio of the thermal to magnetic pressure, on the disk midplane, which is approximately $10^3$ ($0.922\times 10^3$ to be more exact) for most of the simulations. Since varying $m$ from 0.1 to 1 has little effect on the long-term disk or wind magnetic field structure \citep{2014ApJ...793...31S}, we use $m=0.5$ for all simulations presented in this work. The initial magnetic field components are then calculated as
\begin{equation}
B_r=\frac{1}{r^2\sin{\theta}}\frac{\partial\Psi}{\partial\theta},
\end{equation}
\begin{equation}
B_\theta=-\frac{1}{r\sin\theta}\frac{\partial\Psi}{\partial r}.
\end{equation}
\subsection{Ambipolar diffusion}\label{sec:AD}
The magnetic diffusivities associated with non-ideal MHD effects, including ambipolar diffusion, depend on the densities of charged particles, which can in principle be computed through detailed chemical networks (e.g., \citealt{2009ApJ...701..737B}). Here, as a first step toward a comprehensive model, we will simply parametrize the density of ions as
\begin{equation}
\rho_i = \rho_{i,0} f(\theta) \left(\frac{\rho}{\rho_0}\right)^{\alpha_{AD}},
\end{equation}
where
\begin{equation}\label{eq:ftheta}
f(\theta) =
\begin{cases}
\exp\left(\frac{\cos^2(\theta+\theta_0)}{2 \epsilon^2}\right) & \theta<\pi/2-\theta_0 \\
1 & \pi/2-\theta_0\leq\theta\leq\pi/2+\theta_0 \\
\exp\left(\frac{\cos^2(\theta-\theta_0)}{2 \epsilon^2}\right) & \theta>\pi/2+\theta_0.
\end{cases}
\end{equation}
The angular dependence $f(\theta)$ is chosen such that, at a given radius, the ion density increases rapidly in the tenuous disk atmosphere, to mimic the ionization by high energy photons (UV and X-rays) from the central young star in addition to cosmic rays (e.g., \citealt{1981PASJ...33..617U,2011ApJ...735....8P,2017MNRAS.472.2447G}). In the simulations presented in this work, we take $\alpha_{AD}=0.5$. This power-law dependence for the ion density is roughly what is expected when the volumetric cosmic ray ionization rate is balanced by the recombination rate of ions and electrons, under the constraint of charge neutrality (i.e., $\zeta n\propto n_e n_i \propto n_i^2$, where $\zeta$ is the cosmic ray ionization rate per hydrogen nucleus; see page 362 of \citealt{1992phas.book.....S}).
The magnitude of the ion density, and therefore the ion-neutral drag force, $\bm{F_d}=\gamma\rho\rho_i(\bm{v_i}-\bm{v})$, is sometimes quantified through the dimensionless ambipolar Elsasser number \citep{2007NatPh...3..604C,2011ApJ...727....2P,2011ApJ...736..144B},
\begin{equation}
\Lambda=\frac{\gamma\rho_i}{\Omega}=\frac{v_A^2}{\eta_A\Omega},
\end{equation}
where $\gamma$ is the frictional drag coefficient. Physically, the Elsasser number is the collision frequency of a neutral particle in a sea of ions of density $\rho_i$, normalized to the Keplerian orbital frequency. For example, the Elsasser number will be unity when the neutral particle collides $2\pi$ times in one orbital period. As the neutral-ion collision frequency increases to infinity, so does the Elsasser number, and the bulk neutral medium becomes perfectly coupled to the ions/magnetic field (i.e., the ideal MHD limit). Similarly, as the Elsasser number drops to zero, the neutrals and ions no longer collide; the neutrals are entirely decoupled from the magnetic field. For our reference simulation, we choose the Elsasser number to be $\Lambda_0=0.25$ at the inner boundary on the disk midplane, but will vary this parameter to gauge its effects on the coupled disk-wind system. The choice of $\alpha_{AD}=0.5$, assuming that the drag coefficient $\gamma$ is constant, implies that the Elsasser number is proportional to $r^{3/4}$, thus larger radii are better coupled than smaller radii in the reference simulation.
In some cases, we have considered an explicit Ohmic resistivity $\eta_O$ in addition to ambipolar diffusion. In these cases, it is useful to compute the effective ambipolar diffusivity $\eta_A$ according to equation~(\ref{ADdiffusion}), to facilitate the comparison of the two effects.
\subsection{Grid}\label{sec:grid}
The equations are solved for $r\in{[1,100]}$~au and $\theta\in{[0,\pi]}$. The radial grid limits are chosen such that they encompass the anticipated disk region where ambipolar diffusion is the most important non-ideal effect, especially in the upper layers of the disk \citep{2014prpl.conf..411T}. A `ratioed' grid is used in the radial direction such that $dr_{i+1}/dr_i=1.012$ is constant and $r_{i+1}=r_i+dr_i$. The grid spacing at the inner edge is set as $dr_0=2.3r_0d\theta$. The grid is uniform in $\theta$ with a resolution of $n_r\times n_\theta = 400\times720$. This results in 24 grid cells from the disk midplane to surface (at two disk scale heights) in the reference run.
\subsection{Boundary conditions}\label{sec:bc}
Both the inner and outer radial boundaries use the standard outflow condition in Zeus codes, where the flow quantities in the first active zone are copied into the ghost zones except for the radial component of the velocity, $v_r$, which is set to zero in the ghost zones if it points into the computation domain in the first active zone (i.e., if $v_r > 0$ in the first active zone at the inner radial boundary or $v_r < 0$ in the first active zone at outer radial boundary). The standard axial reflection boundary condition is used on the polar axis ($\theta=0$ and $\pi$), where the density and radial components of the velocity and magnetic field ($v_r$ and $B_r$) in the ghost zones take their values in the corresponding active zones while the polar and azimuthal components ($v_\theta$, $B_\theta$, and $v_\phi$) take the negative of their values in the corresponding active zones. We set $B_\phi$ to vanish on the polar axis and on the inner radial boundary, since it is taken to be non-rotating.
\section{Reference model and the formation of rings and gaps through reconnection}\label{sec:ref}
We will first discuss in detail a reference simulation to which other simulations with different parameters can be compared. The parameters used in all simulations are listed in Table~\ref{tab:sims}. The initial density at $r_0$ on the disk midplane is $\rho_0=1.3\times 10^{-10}~\text{g~cm$^{-3}$}$. Of the parameters to be changed in later simulations, this `reference' simulation uses $\beta_0\sim10^3$ and $\Lambda_0=0.25$. Note that the radial power-law dependence of the ambipolar Elsasser number is $\Lambda\propto r^{3/4}$, whereas the initial distribution of plasma-$\beta$ is constant with radius.
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig1.png}
\caption{A representative (`reference') axisymmetric simulation. Shown is the mass volume density (logarithmically spaced colour contours in units of \text{g~cm$^{-3}$}), the poloidal magnetic field lines (magenta), and the poloidal velocity unit vectors (black). Panels (a)-(d) corresponding to simulation times of 0, 200, 1000, and 2500 inner orbital periods, respectively. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:global}
\end{figure*}
\subsection{Global picture}
The overall global evolution of the disk is seen in Fig.~\ref{fig:global}. The frames show the simulation at times of 0, 200, 1000, and 2500$t_0$ (left to right), where $t_0=1$~yr is the orbital period at the inner edge of the simulation domain ($r_0=1$~au). The disk wind launching proceeds in an inside-out fashion, i.e, the wind is launched from larger disk radii as the simulation progresses. By 1000$t_0$ (one orbit at the outer radius $r=100$~au), the simulation appears to have no memory of the initial simulation setup and the magnetic field geometry is conducive to launching an MHD disk wind at all radii. From initial inspection, the disk wind that is launched is very steady in time (see the supplementary material online for an animated version of Fig.~\ref{fig:global}). The magnetic flux is also very much fixed in time following an initial adjustment period of approximately $t/t_0=100$. From this point on, the poloidal magnetic field lines stay in close proximity to their equilibrium footpoint in the disk (magenta lines in Fig.~\ref{fig:global}). However, there is persistent laminar accretion radially inward through the disk and across magnetic field lines.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig2.pdf}
\caption{The mass outflow rates (\text{M$_\odot$~yr$^{-1}$}) as a function of time in the reference simulation through a sphere of radius $r=10$~au. The total mass outflow rate both above and below the disk ($\vert \pi/2-\theta \vert>2\epsilon$) is shown in black. It is separated into two velocity components, with the fast component ($v_r>10$~\text{km~s$^{-1}$}) shown in red and the slow component ($v_r\leq10$~\text{km~s$^{-1}$}) in blue. The green line shows the mass accretion rate through the disk ($\vert \pi/2-\theta \vert<2\epsilon$).}
\label{fig:mdot}
\end{figure}
A relatively massive and steady wind is launched from the disk surface for almost the entire simulation. Figure~\ref{fig:mdot} plots the mass outflow rate for the wind through a sphere of radius $r=10$~au, excluding the disk region ($\vert\pi/2-\theta\vert<2\epsilon$; black line). The mass outflow rate slowly decreases for times $t/t_0\gtrsim300$ from a value of $\dot{M}_\mathrm{w}\approx7\times10^{-7}~\text{M$_\odot$~yr$^{-1}$}$ down to approximately $3\times10^{-7}~\text{M$_\odot$~yr$^{-1}$}$ by $t/t_0=5000$. Most of the mass lost in the wind moves rather slowly, with a radial expansion speed less than the local Keplerian rotation speed at $r=10$~au for a one solar-mass star ($v_r<10$~\text{km~s$^{-1}$}; blue line). This picture is reminiscent of \citetalias{2017MNRAS.468.3850S}, where a weak initial magnetic field (again, $\beta\sim10^3$) is able to drive slow and massive outflow. There is, however, a substantial mass loss from a relatively fast outflow ($v_r > 10$~\text{km~s$^{-1}$}) as well (red line). Note that the mass infall rate in the low-density polar region is of the order $10^{-13}~\text{M$_\odot$~yr$^{-1}$}$, which is much smaller than the total mass outflow rate. The mass accretion rate through the disk at this same radius ($r=10$~au) is of the same order as the total outflow rate (green line). It stays relatively constant at $\dot{M}_\mathrm{acc}\approx 8\times10^{-7}~\text{M$_\odot$~yr$^{-1}$}$, even as the outflow rate decreases slowly at later times.
\subsection{Disk-wind connection}
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig3.png}
\caption{The reference simulation at time $t/t_0=2500$. The colour contours show (a) the logarithm of poloidal velocity (\text{cm~s$^{-1}$}), (b) the logarithm of plasma-$\beta$, (c) the ratio of the toroidal to the poloidal magnetic field components, $B_\phi/B_\mathrm{p}$, (d) the logarithm of density (\text{g~cm$^{-3}$}), (e) an axisymmetric, face-on view of the disk surface density distribution normalized to the initial power-law distribution, $\Sigma_i=\Sigma_0(r/r_0)^{-1/2}$, and (f) the mass flux per unit polar angle, $d\dot{M}/d\theta=2\pi r^2\rho v_r\sin\theta$, normalized to $\dot{M}_0=r_0^2\rho_0 c_{s,0}$. Poloidal magnetic field lines (i.e., magnetic flux contours) are shown in grey in panel (c). Panel (d) shows two specific poloidal magnetic field lines with midplane footpoints at $r=8$~au (magneta; see Fig.~\ref{fig:path_8au}) and $r=7$~au (black; see Fig.~\ref{fig:path_7au}). Poloidal velocity unit vectors are plotted in black in panels (a), (c), (d), and (f). (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:beta}
\end{figure*}
Figure~\ref{fig:beta} plots various physical quantities of the reference simulation at $t/t_0=2500$ up to a radius of $r=20$~au. Panel (a) shows the poloidal velocity as the accreting disk material is peeled off the disk surface and launched in a wind. The velocity of the disk accretion is of the order 1~m~s$^{-1}$, while in the wind, material is accelerated up to velocities of $v_\mathrm{p}\gtrsim 10$~\text{km~s$^{-1}$}. The disk and wind regions are easily distinguishable in this plot. They are also quite distinct in panel (b), where the ratio of the thermal pressure to the magnetic pressure, i.e., plasma-$\beta$, is plotted. The initial $\beta$ at the disk midplane is $\sim 10^3$ and it is constant as a function of radius. At the time shown, the thermal and magnetic pressures become approximately equal at the base of the wind ($\beta\sim1$), while $\beta$ decreases to $10^{-2}$ or less in the wind zone. The bulk of the disk has plasma-$\beta$ of approximately 10, although there remains a thin layer where the plasma-$\beta$ peaks at a value slightly larger than the initial $\beta_0$. This thin layer is where the toroidal magnetic field goes to zero as it reverses direction; it is therefore a current layer (see also the simulations of \citealt{2013ApJ...769...76B,2013ApJ...772...96B,2015ApJ...801...84G,2017ApJ...836...46B,2017A&A...600A..75B,2017ApJ...845...75B}). This can be seen in panel (c), which plots the ratio of the toroidal magnetic field to the poloidal magnetic field, $B_\phi/B_\mathrm{p}$. The white regions are where $B_\phi>0$ in the disk and the black regions have $B_\phi<0$. The poloidal magnetic field lines (i.e., constant magnetic flux contours) are shown in grey. Note that the toroidal field greatly dominates the poloidal field in distinct radial locations, while the poloidal field is stronger (with a smaller $\vert B_\phi/B_\mathrm{p}\vert$) in adjacent regions. These regions with less twisted field lines correspond to regions with lower densities in the disk, as shown in panel (d). They will be referred to as `gaps.' The neighbouring regions, where the field lines are more twisted, have higher densities; they will be referred to as `rings.' The rings and gaps are shown more clearly in the face-on view of the disk in panel (e), where the distribution of the surface density (normalized to the initial distribution) is plotted. How such rings and gaps form will be discussed in detail in Section~\ref{sec:ring}. Here, we would like to point out that there is vigorous accretion (and some expansion) in both types of regions, as shown in panel (f), which plots the spatial distribution of the radial mass flux per unit polar angle, $d\dot{M}/d\theta=2\pi r^2\rho v_r\sin\theta$. The blue cells show negative mass flux or inward accretion and the red cells represent positive mass flux or outward expansion. We see that most of the accretion in the disk occurs along the thin current sheet previously mentioned. In disk regions above and below this layer, there is a strong variation in mass flux, with larger outward mass flux from rings compared to those from the adjacent gaps, although this strong variation does not appear to extend into the (faster) wind zone.
\begin{figure*}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig4a.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig4b.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig4c.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig4d.pdf}
\caption{Physical quantities plotted along a poloidal magnetic field line as a function of the vertical height $z$. This representative field line passes through a low-density gap at $r=8$~au. The panels show (a) the density distribution, (b) plasma-$\beta$ for the total magnetic field strength (black) and for the poloidal magnetic field strength (red), (c) the poloidal components of the neutral (solid lines) and ion velocities (dashed lines), and the adiabatic sound speed (dash-dotted line), and (d) the magnetic field components. The yellow circles show the sonic point (where the poloidal velocity is equal to the adiabatic sound speed) and the vertical dashed lines show the initial disk height of $z=\pm2h_0$.}
\label{fig:path_8au}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig5a.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig5b.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig5c.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig5d.pdf}
\caption{Physical quantities plotted along a poloidal magnetic field line as a function of the vertical height $z$. This representative field line passes through a dense ring at $r=7$~au. The panels show (a) the density distribution, (b) plasma-$\beta$ for the total magnetic field strength (black) and for the poloidal magnetic field strength (red), (c) the poloidal components of the neutral (solid lines) and ion velocities (dashed lines), and the adiabatic sound speed (dash-dotted line), and (d) the magnetic field components. The yellow circles show the sonic point (where the poloidal velocity is equal to the adiabatic sound speed) and the vertical dashed lines show the initial disk height of $z=\pm2h_0$.}
\label{fig:path_7au}
\end{figure*}
To understand the connection between the disk region and wind region more quantitatively, we plot various physical quantities along two representative field lines. In Fig.~\ref{fig:path_8au}, we show plots along a magnetic field line that passes through a gap at a radius of 8~au on the midplane (see the magenta magnetic field line plotted in Fig.~\ref{fig:beta}d). Panel (a) of Fig.~\ref{fig:path_8au} shows that the density peaks near the midplane, at a value of approximately $10^{-12}$~\text{g~cm$^{-3}$}. It drops off quick inside the disk (the vertical dashed lines mark the initial disk surface at two scale heights). Beyond the sonic point (marked by a yellow circle on the curve), the decrease in density becomes less steep, transitioning from an exponential drop-off to a power-law decline, as expected as the approximately hydrostatic disk transitions to a supersonic wind. Even in this low density gap region, the disk is still dominated by the thermal pressure, with a plasma-$\beta$ of order 10 near the midplane (panel b). The plasma-$\beta$ decreases rapidly away from the midplane, reaching a value of order 0.1 at the sonic point; beyond the sonic point, the supersonic wind is completely magnetically dominated. This magnetic field, specifically its pressure gradient, is responsible for accelerating the outermost layer of the disk through the sonic point to produce the wind. The wind acceleration along this particular field line is illustrated in panel (c), which shows clearly the transition from slow inward motion near the disk midplane ($v_r < 0$, i.e., accretion) to fast outflow through the sonic point at approximately three disk scale heights. The sound speed remains nearly constant along the field line, which is indicative of a `cold' wind with a temperature comparable to that of the disk. Note that the outflow acceleration beyond the sonic point is rather gradual, with the velocity increasing over many disk scale heights. This is consistent with magnetically driven winds with heavy mass loading \citep{2005ApJ...630..945A}.
In panel (c), we have plotted ion speeds together with the speeds for the bulk neutral material. The largest difference is between the velocity component $v_\theta$ and $v_{i,\theta}$, especially in the wind zone. In particular, $v_{i,\theta}$ is more positive than $v_\theta$ below the disk (negative $z$), indicating that the ions are moving faster than the neutrals {\it away from} the disk. There must be a magnetic force pointing away from the disk which drives the ion-neutral drift through ambipolar diffusion; it is the same force that accelerates the wind in the first place. The situation is similar above the disk as the magnetic force that drives the wind also moves the ions away from the disk faster than the neutrals in the negative $\theta$ direction. This force comes mostly from the toroidal component of the magnetic field, which dominates the poloidal component in the wind zone, as shown in panel (d). The gradual decrease of the toroidal component, evident in panel (d), yields a magnetic pressure gradient along the poloidal field line that lifts up the material near the disk surface against gravity and slowly accelerates it to produce a wind. Closer to the midplane, the poloidal field component (particularly $B_\theta$) becomes more dominant, indicating that the field line passes through the gap region of the disk with relatively little bending in the $r$ direction or twisting in the $\phi$ direction.
The situation is quite different along the field line that passes through a dense ring at a radius of 7~au (Fig.~\ref{fig:path_7au}). Here, the density at the midplane is an order of magnitude higher than that of the neighbouring gap (see Fig.~\ref{fig:path_8au}a). It decreases rapidly away from the midplane and the decrease becomes slower beyond the sonic point, signalling the transition from the disk to a wind. The plasma-$\beta$ through the ring is more than $10^3$ near the disk midplane, more than two orders of magnitude larger than that in the gap. In other words, as measured by $\beta$, the ring is much less magnetized than the gap. The difference is even larger when only the poloidal field component is considered. From the red curve in Fig.~\ref{fig:path_7au}(b), which plots the ratio of the thermal pressure to the magnetic pressure due to the poloidal field only, it is clear that the poloidal field in the ring is not only weak (and much weaker than the toroidal component) but also highly variable as a function of $z$. Nevertheless, the basic disk-wind structure is preserved, as shown in panel (c), where the poloidal components of the ion and neutral velocities are plotted. Again, the transition from a slowly moving disk (in the poloidal plane) to a faster expanding supersonic wind is evident. Compared to the relatively smooth accretion in the gap, which has a single negative peak in $v_r(z)$ (see red line in Fig.~\ref{fig:path_8au}c), the radial flow in the ring is much more complex: it has six negative peaks and at least three positive peaks, indicating the coexistence of multiple channels of accretion and expansion in the ring. These channels are reflected in the magnetic structure (panel d), particularly in the vertical distribution of the radial component, $B_r(z)$, which has several sign reversals consistent with channel flows in the weakly magnetized ring. As alluded to earlier, the most striking difference in the magnetic field between the ring and the gap is the strength of the poloidal magnetic field, especially the polar component, $B_\theta$, which is much lower in the ring than in the gap (compare Fig.~\ref{fig:path_8au}d and \ref{fig:path_7au}d). In the ring, the magnetic field is completely dominated by the toroidal component, except near the midplane where $B_\phi$ changes direction.
\subsection{Formation of rings and gaps}\label{sec:ring}
In this subsection, we will demonstrate that the formation of rings and gaps in the reference simulation is closely related to the magnetic structure that develops in the disk, particularly the sharp pinching of the poloidal field line near the midplane that leads to magnetic reconnection. This field pinching is caused by the development of a current sheet near the midplane, as we show next.
\begin{figure*}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig6a.pdf}
\includegraphics[width=1.0\columnwidth]{figures/fig6b.pdf}
\caption{The toroidal magnetic field $B_\phi$ (left) and the radial current density $J_r$ (right) plotted versus $90^\circ-\theta$ (zero at the midplane and negative below it) at radius $r=20$~au. The legend labels are the simulation time in units of the inner orbital period $t_0$.}
\label{fig:jmid}
\end{figure*}
\subsubsection{Midplane current sheet}\label{sec:jmid}
We start by reminding the reader of the simulation setup, where the disk is rotating slightly sub-Keplerian for $\vert\pi/2-\theta\vert<2\epsilon$, the coronal regions above and below the disk are not rotating, and the magnetic field has no $\phi$ component initially. As the simulation begins to run, a toroidal magnetic field is quickly generated near the boundary between the rotating disk and the stationary corona due to differential rotation. This can been seen in the left panel of Fig.~\ref{fig:jmid}, which plots the toroidal component of the magnetic field as a function of $\theta$ at a representative disk radius $r=20$~au. At $t/t_0=10$, the solid black line shows that a toroidal magnetic field has already been generated near the disk surface, but has yet to penetrate into the bulk of the disk. Associated with the variation of $B_\phi$ with polar angle $\theta$ is a radial current,
\begin{equation}
J_r = \frac{c}{4\pi}~\frac{1}{r\sin\theta}\frac{\partial(B_\phi\sin\theta)}{\partial\theta}\approx\frac{c}{4\pi}\frac{dB_\phi}{r d\theta},
\end{equation}
where $\sin\theta\approx 1$ in the thin disk. This current is plotted in the right panel of Fig.~\ref{fig:jmid}, which shows two positive peaks near $5^\circ$ above and below the disk midplane at time $t/t_0=10$, corresponding to the sharp drop of $\vert B_\phi\vert$ going from the corona into the disk. At later times, the region of high toroidal field, $\vert B_\phi\vert$, above and below the midplane expands both outward into the corona and, more importantly, toward the disk midplane. The latter drives the two current layers (one on each side of the midplane) towards the midplane, until they merge together into a single, thin, current sheet, as shown in Fig.~\ref{fig:jmid} (right).
Ambipolar diffusion plays a key role producing the thin midplane current sheet. First, were it not for the presence of AD (or some other magnetic diffusivity, such as resistivity), prominent avalanche streams would have developed near the disk surface, which would drive the entire disk and its envelope into an unsteady state and make the formation of a laminar midplane current sheet impossible (see \citetalias{2017MNRAS.468.3850S} and Section~\ref{sec:els} below)\footnote{In this reference run, AD does not appear to suppress the development of the MRI completely. Channel-like features are evident at large radii where the disk is better coupled to the magnetic field as measured by the Elsasser number (see Fig.~\ref{fig:global}c).}. Second, as first stressed by \citet{1994ApJ...427L..91B}, AD tends to steepen the magnetic gradient near a magnetic null, i.e., where the magnetic field changes polarity. The reason is that the Lorentz force associated with the magnetic pressure gradient drives the ions (relative to the neutrals) toward the null from both sides. Since the field lines (of opposite polarity across the null) are tied to the ions, they are dragged towards the null as well, leading to a steepening of the magnetic gradient, which in turn yields a stronger magnetic force that drives the ions and the field lines even closer to the null. Since the ambipolar magnetic diffusivity, $\eta_{A}$, is proportional to the field strength (see equation~\ref{ADdiffusion}) and thus vanishes at the null, this steepening would result in an infinitely thin, singular, current sheet in principle. In practice, the thickness of the current sheet is limited by the grid resolution.
Sharp magnetic field reversals that give rise to thin current sheets are prone to reconnection. However, this is not the case for the midplane current sheet shown in Fig.~\ref{fig:jmid}, because it is produced by the reversal of the toroidal field component $B_\phi$ and it is impossible to reconnect oppositely directing toroidal fields under the adopted axisymmetry\footnote{Reconnection of the highly pinched toroidal field is expected in 3D, and will be explored in a future investigation.}. Nevertheless, this primary current sheet leads to another current component that does allow for reconnection.
\subsubsection{Reconnection of pinched radial magnetic field}
The secondary component of the midplane current sheet develops as a result of mass accretion in the disk, which is concentrated in the primary radial current ($J_r$) sheet near the midplane (see Fig.~\ref{fig:jmid}). The mass accretion is driven by the removal of angular momentum due to the Lorentz force ($\propto J_r B_\theta$) in the azimuthal direction, which peaks in the radial current sheet where the toroidal magnetic field changes sign. Pictorially, as the $\phi$ component of the magnetic field changes from the $+\phi$ direction below the disk to the $-\phi$ direction above the disk in a thin midplane layer, the magnetic field lines become severely kinked in the azimuthal direction. The sharp field kink generates a magnetic tension force in the $-\phi$ direction that exerts a braking torque and drives the disk accretion in the current sheet.
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig7.png}
\caption{Poloidal magnetic field lines at four different times are shown in grey, with a reconnecting field line highlighted in black. The colours show the ratio of the toroidal to poloidal magnetic field, $B_\phi/B_\mathrm{p}$.}
\label{fig:reconnect}
\end{figure*}
The accretion through the midplane current sheet now drags the poloidal magnetic field lines inward with the flow. This results in a pronounced radial pinch of the poloidal magnetic field, which transports the poloidal magnetic flux inward and yields another current component in the azimuthal direction, $J_\phi$. Eventually, the radial pinch becomes so severe that the magnetic field reconnects, forming poloidal magnetic field loops that are reminiscent of the tearing mode instability (\citealt{1963PhFl....6..459F}; see also \citealt{2009ARA&A..47..291Z}). An example of the reconnection process is shown in Fig.~\ref{fig:reconnect}, which plots lines of constant poloidal magnetic flux, i.e. poloidal magnetic field lines, along with the ratio of the toroidal to poloidal magnetic field, $B_\phi/B_\mathrm{p}$ (colour contours), at a radius centred on $r=6.5$~au from time $t/t_0=200$ to 230. At the first frame shown, the poloidal magnetic field highlighted in bold already has a kink across the midplane. The pinch grows with time until the field line is stretched almost parallel to the midplane over approximately four radial grid cells at $t/t_0=210$. By the next frame shown at $t/t_0=220$, the field has reconnected, forming a loop near $r=6.4$~au. In the last frame the loop has disappeared, however, the process of its formation has left a lasting mark on the magnetic field structure. The region now has a much larger toroidal magnetic field component compared to its poloidal component. The now weaker poloidal field strength can be seen clearly in the last panel from the lack of field lines in the post-reconnection region.
\subsubsection{Reconnection-driven ring and gap formation}\label{sec:formation}
The general picture for the reconnection-driven ring and gap formation is as follows. A poloidal magnetic field line that initially threads the disk rather smoothly is dragged by preferential accretion near the midplane into a highly pinched radial configuration (see upper-right panel of Fig.~\ref{fig:reconnect} for an illustration). Reconnection of the highly pinched field line produces a poloidal magnetic loop next to a poloidal field line that still threads the disk more smoothly (see lower-left panel of Fig.~\ref{fig:reconnect}). After reconnection, the material trapped on the magnetic loop is detached from the original (pre-reconnection) poloidal field line, which results in the separation of matter and (poloidal) magnetic flux. Specifically, there is no net poloidal magnetic flux passing through the matter on the loop and there is less matter remaining on the original poloidal field line (since some of the matter on the original field line is now on the detached magnetic loop). The net effect is a redistribution of the poloidal magnetic flux away from the loop-forming region into an adjacent region where the poloidal flux accumulates. Since mass accretion tends to be faster in regions with stronger poloidal magnetic fields, it is not surprising that the reconnection-induced variation of the poloidal field strength with radius would lead to a spatial variation in the mass accretion rate that would in turn lead to spatial variation in the mass (surface) density, i.e., the formation of rings and gaps. \footnote{The stronger poloidal field in a gap can also lead to a faster depletion of the local disk material via a stronger magnetically driven wind (e.g., \citealt{2000A&A...353.1115C}). However, this effect is less important compared to the faster accretion in the gap in general.}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig8.png}
\caption{The density, magnetic field strength, and velocities plotted up to $r=15$~au at $t/t_0=2500$. The panels show (a) the logarithm of the density (\text{g~cm$^{-3}$}) in colour and the poloidal magnetic field lines (i.e., magnetic flux contours) in black, (b) the surface density normalized to the initial radial distribution, $\Sigma_i\propto r^{-1/2}$, and the vertical magnetic field strength at the midplane normalized to its initial distribution, $B_{z,i}\propto r^{-5/4}$, (c) the radial velocity (\text{km~s$^{-1}$}) of neutrals (black) and ions (red) at the midplane, and (d) the density-weighted vertical average of the radial velocity (\text{km~s$^{-1}$}) of neutrals (black) and ions (red) over $z=\pm2h$. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:bunch}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig9.png}
\caption{The surface density of the disk (normalized to its initial radial distribution) as a function of radius and time, showing that most of rings and gaps created in the reference run remain stable for thousands of inner orbital periods.}
\label{fig:surfden_v_time}
\end{figure}
The preferential concentration of poloidal field lines inside the low-density gaps of the disk is illustrated pictorially in Fig.~\ref{fig:bunch}(a), which is a snapshot of the inner part of the reference simulation (up to 15~au) at a representative time when the simulation has reached a quasi-steady state, $t/t_0=2500$. The field concentration is quantified in panel (b), which plots the vertical component of the magnetic field ($B_z$, in red) at the midplane and surface density of the disk (in black), both relative to their initial values at $t=0$. Two features stand out: (1) there is a strong anti-correlation between the surface density and vertical field strength, as expected in the reconnection-induced scenario of ring and gap formation, and (2) the poloidal magnetic field is typically much weaker in the high-density ring regions (where $B_z$ is close to zero) than in the low-density gap regions (where $B_z$ is increased over its initial value by a factor of $2-4$). This drastic segregation of poloidal magnetic flux relative to matter appears to be quite stable with time (persisting for thousands of inner orbits), despite the fact that there is continued mass accretion through both the rings and gaps.
For our reference run, we believe that the key to maintaining the concentration of poloidal magnetic flux in the gap regions is ambipolar diffusion: it allows the bulk neutral material to accrete radially inward through the gaps without dragging the ions (and the magnetic field lines tied to them) with it. This is illustrated in panel (c) of Fig.~\ref{fig:bunch}, which plots the radial component of the ion (red line) and neutral (black line) velocity at the midplane. As expected, the neutrals accrete faster in the low-density gaps than in the high-density rings. One may naively expect the same trend for the ions but, in the presence of significant ambipolar diffusion, this is not necessarily the case. Indeed, at the time shown in Fig.~\ref{fig:bunch}, the ions are moving {\it outward} in several of the gaps, especially the two near $r=5$ and 6.5~au. The ions are forced to expand against the infalling neutrals by an outward Lorentz force due to either a temporary poloidal flux concentration near the inner edge of the gap (in a manner that is reminiscent of the forced ion-neutral separation in the AD-shock in magnetized accretion onto low-mass protostars first described in \citealt{1996ApJ...464..373L}) or an outward magnetic pressure gradient from the toroidal field component. In any case, the radially outward Lorentz force in the gap appears strong enough to keep the ions (and the poloidal field lines) in a state of dynamic equilibrium against the rapid infall of neutrals, at least under the assumption of (2D) axisymmetry. The dynamic equilibrium of the ions (and the field lines attached to them) is shown more clearly in Fig.~\ref{fig:bunch}(d), which plots the vertically averaged radial velocity weighted by density. The average ion speed fluctuates around zero as the neutrals accrete inward, especially in the gap regions. Whether this remains true in full 3D simulations is unclear and will be explored in future investigations.
The rings and gaps, once fully developed, remain remarkably stable over time. This is illustrated in Fig.~\ref{fig:surfden_v_time}, where the surface density (relative to its initial distribution) is plotted at each radius as a function of time, as done in \citet{2017A&A...600A..75B} for plasma-$\beta$ (see their Fig.~30). Note that most of the rings and gaps are stable for at least 4000 inner orbital periods. There are a few exceptions. For example, the two rings near 10~au appear to merge together around $t/t_0=5000$, whereas the ring at 6~au starts to fade away at later times. It would be interesting to determine whether these rings and gaps remain stable for long periods of time in full 3D simulations.
\begin{table*}
\centering
\caption{Model parameters for all simulation runs.}
\label{tab:sims}
\begin{tabular}{l c c c c c c c c}
\hline
\hfill & $\beta/10^3$ & $\gamma/10^{-3}$ & $\Lambda_0=\gamma\rho_{i,0}/\Omega_0$ & $\eta_{A,0}/10^{14}$ & $\eta_O/10^{14}$ \\
\hfill & & $[\rm{cm}^3~\rm{g}^{-1}~\rm{s}^{-1}]$ & & $[\rm{cm}^2~\rm{s}^{-1}]$ & $[\rm{cm}^2~\rm{s}^{-1}]$ \\
\hline
ad-els0.01 & 0.922 & 0.1763 & 0.01 & 243 & -- \\
ad-els0.05 & 0.922 & 0.8816 & 0.05 & 48.6 & -- \\
ad-els0.1 & 0.922 & 1.763 & 0.1 & 24.3 & -- \\
ad-els0.25~(ref)& 0.922 & 4.408 & 0.25 & 9.71 & -- \\
ad-els0.5 & 0.922 & 8.816 & 0.5 & 4.86 & -- \\
ad-els1.0 & 0.922 & 17.63 & 1.0 & 2.43 & -- \\
ad-els2.0 & 0.922 & 35.26 & 2.0 & 1.21 & -- \\
ideal & 0.922 & -- & -- & -- & -- \\
oh0.26 & 0.922 & 4.408 & 0.25 & 9.71 & 2.5 \\
oh2.6 & 0.922 & 4.408 & 0.25 & 9.71 & 25 \\
oh26 & 0.922 & 4.408 & 0.25 & 9.71 & 250 \\
beta1e2 & 0.0922& 4.408 & 0.25 & 97.1 & -- \\
beta1e4 & 9.22 & 4.408 & 0.25 & 0.971 & -- \\
\hline
\end{tabular}
\end{table*}
\section{Effects of magnetic coupling and field strength on ring and gap formation}\label{sec:param}
The most important features in the reference simulation are the rings and gaps that develop spontaneously in the disk. In this section, we will explore how their formation is affected by how well the ions (and therefore the magnetic field) are coupled to the bulk neutral fluid. The magnetic field coupling is changed by varying the AD Elsasser number at $r_0$ and $\theta=\pi/2$, $\Lambda_0$, which sets the scale for the Elsasser number everywhere; the ion density profile is unchanged. The AD Elsasser number controls the coupling between the ions and the bulk neutral fluid, as $\Lambda\propto\gamma\rho_i$, where $\gamma\rho_i$ is the collision frequency between the neutrals and ions. When $\Lambda$ is small, so is the collision frequency, and, therefore, the ions/magnetic field are poorly coupled to the neutral fluid. As $\Lambda$ increases, the magnetic field becomes increasingly coupled to the motions of the bulk neutral fluid. The ideal MHD limit is reached as the AD Elsasser number approaches infinity, $\Lambda\rightarrow\infty$. In Table~\ref{tab:sims}, we list the simulation runs performed to examine the effect that the magnetic coupling strength has on the ring and gap formation mechanism described in the previous section. First, we will present the results of the simulations as the Elsasser number increases from 0.01 to 2 (in the simulations named ad-els0.01, ad-els0.05, ad-els0.1, ad-els0.5, ad-els1.0, and ad-els2.0), as well as the ideal MHD case (Section~\ref{sec:els}). Next, we show the effects of introducing an explicit Ohmic resistivity into the reference simulation, where the Ohmic resistivity, $\eta_O$, is constant everywhere and is equal to 0.26, 2.6, and 26 times the initial effective ambipolar resistivity, $\eta_{A,0}$, at $r_0$ on the disk midplane ($\theta=\pi/2$). These simulations are named oh0.26, oh2.6, and oh26 respectively. They are discussed in Section~\ref{sec:beta} together with simulations that have different initial magnetic field strengths, with the midplane plasma-$\beta$ approximately ten times higher (model beta1e4) and lower (beta1e2) than that of the reference run. We conclude this section with an analysis of the magnetic stresses in the disk (Section~\ref{sec:stress}).
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig10.png}
\caption{Snapshots at $t/t_0=2000$ of the eight simulations where the AD Elsasser number is varied. Shown is the mass volume density (logarithmically spaced colour contours in units of \text{g~cm$^{-3}$}), the poloidal magnetic field lines (magenta), and the poloidal velocity unit vectors (black). The AD Elsasser number increases sequentially from panels (a)-(h). The reference simulation (ad-els0.25) is shown in panel (d). The simulation panels in alphabetical order are: (a) ad-els0.01; (b) ad-els0.05; (c) ad-els0.1; (d) ad-els0.25; (e) ad-els0.5; (f) ad-els1.0; (g) ad-els2.0; (h) ideal. See Table~\ref{tab:sims} for details. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:ad_panels}
\end{figure*}
\subsection{AD Elsasser number}\label{sec:els}
Before describing the results of the simulations, we will briefly describe our expectations as the AD Elsasser number is varied in the disk. In the reference run, we see a rather steady disk wind launched as disk material concentrates into rings and poloidal magnetic flux concentrates into gaps. The region demagnetized of poloidal field (where the density will grow to form a ring) develops as the radial magnetic field is stretched towards the $-r$ direction due to rapid accretion in the primary midplane current layer ($J_r$). As discussed in Section~\ref{sec:jmid}, the development of a strong midplane current layer where $B_\phi=0$ is a direct result of ambipolar diffusion because it is formed as the ions and toroidal magnetic field lines drift towards the magnetic null ($B_\phi=0$) relative to the bulk neutral material. In the limiting case that the Elsasser number goes to infinity, i.e., the ideal MHD case, the AD-enabled midplane current layer is not expected to develop and this ring and gap formation mechanism would be turned off. Instead, the so-called `avalanche' accretion streams are expected to develop near the disk surface, which may form rings and gaps through another mechanism (see \citetalias{2017MNRAS.468.3850S}). In the other limiting case where the Elsasser number approaches zero, the ions have no knowledge of the bulk fluid and the magnetic field will straighten out vertically without any effect on the disk at all. In particular, the field will neither launch a disk wind nor create any disk substructure. Therefore, there must be a minimum Elsasser number below which the formation of rings and gaps is expected to be suppressed.
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig11.png}
\caption{Face-on surface density profiles (up to a radius of 20~au) of the eight simulations where the AD Elsasser number is varied at $t/t_0=2000$. The AD Elsasser number increases sequentially from panels (a)-(h). The reference simulation (ad-els0.25) is shown in panel (d). The simulation panels in alphabetical order are: (a) ad-els0.01; (b) ad-els0.05; (c) ad-els0.1; (d) ad-els0.25; (e) ad-els0.5; (f) ad-els1.0; (g) ad-els2.0; (h) ideal. See Table~\ref{tab:sims} for details. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:face-on}
\end{figure*}
These expectations are borne out by the simulations. Figures~\ref{fig:ad_panels} and \ref{fig:face-on} show, respectively, the edge-on and face-on view of simulations at a common time $t/t_0=2000$ with increasing AD Elsasser numbers from panel (a) to (h). In the most magnetically diffusive case ($\Lambda_0=0.01$), a wind is launched steadily from the disk, removes angular momentum from the disk and drives disk accretion that is rather laminar (Fig.~\ref{fig:ad_panels}a), but there is little evidence for the development of rings and gaps (Fig.~\ref{fig:face-on}a). Specifically, there is little evidence for the sharp radial pinching of poloidal magnetic field lines near the midplane that is conducive to reconnection, which is the driver of the redistribution of the poloidal magnetic flux relative to matter and is crucial to the ring and gap formation in the scenario discussed in Section~\ref{sec:ring}. The lack of sharp radial pinching is to be expected, because it would be smoothed out quickly by the large magnetic diffusivity. As the diffusivity decreases (i.e., the magnetic field becomes better coupled to the bulk disk material), it becomes easier for the midplane mass accretion to drag the poloidal field lines into a radially pinched configuration that is prone to reconnection. Indeed, this occurs over at least one decade in the AD Elsasser number, from $\Lambda_0=0.05$ to 0.5, where the disk wind remains rather steady (see Fig.~\ref{fig:ad_panels}b-e), but repeated field pinching and reconnection have created multiple rings and gaps in the disk (Fig.~\ref{fig:face-on}b-e).
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig12.pdf}
\caption{Surface density profiles at time $t/t_0=2000$ for simulations with different AD Elsasser numbers. The surface density profiles are normalized to the initial surface density distribution, $\Sigma_i = \Sigma_0 (r/r_0)^{-1/2}$. The AD Elsasser number increases sequentially from the top panel to the bottom panel: (a) ad-els0.01; (b) ad-els0.05; (c) ad-els0.1; (d) ad-els0.25 (ref); (e) ad-els0.5; (f) ad-els1.0; (g) ad-els2.0; (h) ideal.}
\label{fig:surfden1D}
\end{figure}
In the intermediate parameter regime between AD Elsasser number $\Lambda_0=0.05$ and 0.5, rings and gaps are more prominent in the inner part of the disk than in the outer part. This is quantified in Fig.~\ref{fig:surfden1D}, where the surface density of the disk (normalized to its initial value) is plotted as a function of radius. It is clear from panels (b)-(e) that most of the rings and gaps of high contrast are confined to a radius of order 10~au. One reason may be that the inner part of the disk has had more time (relative to its orbital period) for the substructures to develop. Another is that mass accretion, especially during the initial adjustment before a quasi-steady state is reached, may have redistributed some poloidal magnetic flux from the outer part of the disk to the inner part, making it easier to form rings and gaps there (the effects of magnetic field strength will be discussed in the next subsection). As the disk becomes better magnetically coupled (going from panel b to e), the number of rings and gaps in the inner 10~au region appears to decrease somewhat and the contrast between adjacent rings and gaps tends to increase on average. The higher contrast may be related to the fact that a better magnetic coupling would allow more poloidal magnetic flux to be trapped in the inner disk region.
As the AD Elsasser number increases further (to $\Lambda_0=1.0$ and larger), another feature starts to become important. It is the development of the classical `channel flows' of the magnetorotational instability. The channels flows are already present in the intermediate regime for $\Lambda_0$, especially in the outer part of the disk (see panels c and d of Fig.~\ref{fig:global} for the reference run), where the magnetic field is better coupled to the disk compared to the inner part as measured by the Elsasser number, which scales with radius as $\Lambda\propto r^{3/4}$. Their growth was kept in check by ambipolar diffusion in relatively diffusive models with $\Lambda_0$ up to 0.5 (panel e in Fig.~\ref{fig:ad_panels}, \ref{fig:face-on}, and \ref{fig:surfden1D}). For better magnetically coupled disks, these channel flows run away, especially near the disk surface, forming the so-called avalanche accretion streams (see \citealt{1996ApJ...461..115M,1998ApJ...508..186K} and \citetalias{2017MNRAS.468.3850S}). When fully developed, they dominate the dynamics of both the disk and the wind, driving both to an unsteady state (see panels f-h of Fig.~\ref{fig:ad_panels}).
Despite the transition to a more chaotic dynamical state, prominent rings and gaps are still formed, especially in the outer part of the disk (see panels f-h of Fig.~\ref{fig:face-on}). Part of the reason is that the strong variability of the clumpy wind is able to create variation in the disk surface density. Another, perhaps more important, reason is that the distribution of the poloidal magnetic flux is highly inhomogeneous in the disk and regions with concentrated magnetic flux tend to accrete faster leading to lower surface densities (i.e., gaps), similar to the more magnetically diffusive cases (e.g., the reference run and Fig.~\ref{fig:bunch}). Since the poloidal field bunching is present even in the ideal MHD case,\footnote{In the ideal MHD case, the wind is significantly stronger in the upper hemisphere than in the lower hemisphere (see Fig.~\ref{fig:ad_panels}h). Such an asymmetry has been observed in the non-ideal shearing box simulations of \citet{2014A&A...566A..56L} and \citet{2015ApJ...798...84B}, and in the global non-ideal MHD simulations of \citet{2017A&A...600A..75B}. The fact that it shows up in global ideal MHD simulations as well indicates that it may be a general feature of magnetically coupled disk-wind systems that should be examined more closely.} its formation does not require ambipolar diffusion, which is formally different from the more diffusive reference case\footnote{Poloidal field bunching in the ideal MHD limit has been observed in the shearing box simulations of \citet{2012A&A...548A..76M} in the case of strong disk magnetization corresponding to plasma-$\beta$ of order unity, however, artificial injection of matter onto the field lines (to prevent rapid depletion of disk material) complicates the interpretation of the result. Current global 3D ideal MHD simulations of weak field cases of $\beta\sim 10^3$ or larger (e.g., \citealt{2017arXiv170104627Z}) do not appear to show as prominent poloidal field bunching as our 2D (axisymmetric) case. Whether this difference is due to the difference in dimensionality of the simulations or some other aspects (e.g., initial and boundary conditions) remains to be determined.} (see discussion in Section~\ref{sec:formation}). Nevertheless, there is widespread reconnection in these better coupled cases as well (this is best seen in the movie version of Fig.~\ref{fig:ad_panels} available online). The reconnection is still driven by sharp radial pinching of the poloidal field lines. The difference is that here the pinching is caused by the non-linear development of unstable channel flows \citep{2014ApJ...796...31B} rather than the AD-driven midplane current sheet. As a result, the reconnection occurs more sporadically and is less confined to the midplane. The net result is the same: a redistribution of poloidal magnetic flux relative to the matter, creating regions of stronger (poloidal) magnetization that tend to form gaps and regions of weaker (poloidal) magnetization that tend to form rings. These considerations strengthen the case for reconnection as a key to ring and gap formation in a coupled, magnetized disk-wind system, either through an AD-driven midplane current sheet in relatively diffusive disks, the non-linear development of MRI channel flows in better coupled disks, or some other means.
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig13.png}
\caption{Snapshots of simulations where the explicit Ohmic resistivity and plasma-$\beta$ are varied at $t/t_0=2000$. Shown is the mass volume density (logarithmically spaced colour contours in units of \text{g~cm$^{-3}$}), the poloidal magnetic field lines (magenta), and the poloidal velocity unit vectors (black). In the top row, the explicit resistivity is decreased from panels (a)-(c). Plasma-$\beta$ varies from high to low across the bottom row in panels (d)-(f). The reference simulation (ad-els0.25) is shown in panel (e). The simulation panels in alphabetical order are: (a) oh26; (b) oh2.6; (c) oh0.26; (d) beta1e4; (e) ad-els0.25; (f) beta1e2. See Table~\ref{tab:sims} for details. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:beta_panels}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=2.0\columnwidth]{figures/fig14.png}
\caption{Face-on surface density profiles (up to a radius of 20~au) of the simulations where the explicit Ohmic resistivity and plasma-$\beta$ are varied at $t/t_0=2000$. In the top row, the explicit resistivity is decreased from panels (a)-(c). Plasma-$\beta$ varies from high to low across the bottom row in panels (d)-(f). The reference simulation (ad-els0.25) is shown in panel (e). The simulation panels in alphabetical order are: (a) oh26; (b) oh2.6; (c) oh0.26; (d) beta1e4; (e) ad-els0.25; (f) beta1e2. (See the supplementary material in the online journal for an animated version of this figure.)}
\label{fig:face-on_beta}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig15.pdf}
\caption{Surface density profiles at time $t/t_0=2000$ for simulations with different explicit Ohmic resistivities (top) and initial magnetic field strengths, i.e., $\beta_0$ (bottom). The surface density profiles are normalized to the initial surface density distribution, $\Sigma_i = \Sigma_0 (r/r_0)^{-1/2}$.}
\label{fig:surfden1D_beta}
\end{figure}
\subsection{Explicit resistivity and magnetic field strength}\label{sec:beta}
The introduction of explicit Ohmic resistivity into the reference simulation can give us some important insights on the ring and gap formation mechanism. We add an Ohmic resistivity, $\eta_O$, that is constant in both space and time (as in \citetalias{2017MNRAS.468.3850S}). Specifically, three values of $\eta_0$ are considered corresponding to 0.26, 2.6, and 26 times the effective ambipolar resistivity at the inner edge of the disk at the midplane, $\eta_{A,0}=9.71\times 10^{14}$~cm$^2$s$^{-1}$; they are named oh0.26, oh2.6, and oh26, respectively. These simulations are plotted in panels (a)-(c) of Fig.~\ref{fig:beta_panels} and \ref{fig:face-on_beta}. In the most diffusive case with $\eta_O=26~\eta_{A,0}$, there is some concentration of mass at small radii, indicating that there is still mass accretion. However, there is little evidence for rings and gaps with the formation mechanism apparently turned off by the addition of a large resistivity. This strengthens the case for reconnection-driven ring and gap formation, because the large resistivity erases the sharp magnetic field geometries needed for reconnection. As the resistivity decreases, rings and gaps start to appear. In particular, when the resistivity $\eta_O$ drops below the characteristic AD resistivity $\eta_{A,0}$ (model oh0.26), the simulation looks very similar to the reference run that does not have any explicit resistivity. Their similarity, particularly in the location and structure of the rings and gaps, is quantified in Fig.~\ref{fig:surfden1D_beta}(a).
Besides magnetic diffusivity, the magnetic field strength also strongly affects the ring and gap formation. The second column of Fig.~\ref{fig:beta_panels} and \ref{fig:face-on_beta} (panels d-f) shows the effects of varying the initial magnetic field strength, as characterized by the midplane plasma-$\beta$, keeping everything else the same as in the reference run (panel e). These simulations are quantitatively similar, in that a wind is launched from the disk and rings and gaps are formed in all three cases. However, it takes longer for the weaker magnetic field case to produce a well-developed wind. Specifically, in the most weakly magnetized case (model beta1e4), it takes approximately 700 inner orbital periods for the disk wind from the inner part of the disk to become fully developed. This is because it takes longer to generate a strong enough toroidal field out of the weaker initial poloidal field to push the outer layers of the disk to large distances. The weakest field case should be the most prone to the MRI, however, there is no evidence for accretion streams developing near the disk surface.
As in the reference run, the ambipolar diffusion is able to concentrate the radial current ($J_r$) into a thin sheet near the midplane, where preferential accretion leads to severe radial pinching of the poloidal field, eventually leading to reconnection-driven ring and gap formation. The rings and gaps formed in this simulation have a relatively low contrast, however. This is because, with a weak initial field, there is less poloidal magnetic flux concentrated in the gaps after reconnection making the accretion of disk material from the gaps into the neighbouring rings less efficient.
In the stronger magnetic field case (model beta1e2), a quasi-steady wind is quickly established (see panel f of Fig.~\ref{fig:beta_panels}). It drives fast disk accretion, especially near the midplane, where reconnection of the sharply pinched poloidal field leads to demagnetization in some regions (creating rings) and bunching of poloidal field lines in others (creating gaps). The stronger poloidal field drives a more complete depletion of disk material, creating wider gaps with lower column densities, as illustrated in Fig.~\ref{fig:face-on_beta}(f) and quantified in Fig.~\ref{fig:surfden1D_beta}(b). In addition, the stronger overall field allows more material to be moved from the outer part of the disk to smaller disk radii, where several rings have much higher surface densities than their counterparts in the weaker field cases. The most massive inner ring at $r=3$~au has a contrast ratio of $\sim 10^2$. In any case, in the presence of the reference level of ambipolar diffusion, the same reconnection-driven ring and gap formation mechanism appears to operate over a range of disk plasma-$\beta$ with more strongly magnetized disks forming rings and gaps with higher surface density contrast.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figures/fig16.pdf}
\caption{The $\alpha$ parameter from the vertical wind stress, $\alpha_{\theta\phi}$ (see equation~\ref{eq:alpha_theta}; solid blue line), and the radial shear stress, $\alpha_{r\phi}$ (see equation~\ref{eq:alpha_radial}; dashed red line). The vertical wind stress is calculated at the surface $\theta=\pi/2\pm2\epsilon$ and the radial shear stress is integrated between these surfaces. The dotted lines show where the effective $\alpha$ is negative. The AD Elsasser number increases sequentially from the top panel to the bottom panel: (a) ad-els0.01; (b) ad-els0.05; (c) ad-els0.1; (d) ad-els0.25 (ref); (e) ad-els0.5; (f) ad-els1.0; (g) ad-els2.0; (h) ideal.}
\label{fig:stress}
\end{figure}
\subsection{Magnetic stresses and two modes of accretion}\label{sec:stress}
We have shown in Section~\ref{sec:els} and \ref{sec:beta} that the level of magnetic diffusivity, particularly ambipolar diffusion, plays a key role in determining the structure of the magnetically coupled disk-wind system. Specifically, more magnetically diffusive systems tend to be more laminar, with a well-developed wind that is expected to play a dominant role in driving disk accretion. Better magnetically coupled systems are more prone to MRI channel flows, which drive the system to a chaotic state. Although a wind is still developed, it may not play as important a role in disk accretion. In this subsection, we will try to quantify this expectation.
We do this through the dimensionless $\alpha$ parameters \citep{1973A&A....24..337S}, corresponding to the $r\phi$ and $z\phi$ components of the Maxwell stress, $T_{r\phi}=-\frac{B_r B_\phi}{4\pi}$ and $T_{\theta\phi}=-\frac{B_\theta B_\phi}{4\pi}$, respectively, defined as:
\begin{equation}
\alpha_{r\phi}\equiv \frac{\int T_{r\phi} dz} {\int P dz},\label{eq:alpha_theta}
\end{equation}
\begin{equation}
\alpha_{\theta\phi}\equiv \frac{T_{\theta\phi}\Big\vert_{\pi/2-\theta_0}^{\pi/2+\theta_0}}{P_\mathrm{mid}}, \label{eq:alpha_radial}
\end{equation}
where $P$ is the thermal pressure and $P_\mathrm{mid}$ is the pressure on the disk midplane. In the first term, the integration is between the lower and upper surfaces of the initial disk, $\theta=\pi/2\pm\theta_0$. The second term is evaluated at these surfaces. Since the polar shear stress $T_{\theta\phi}$ is the magnetic stress that moves angular momentum across the disk surface (at a constant polar angle $\theta$), we will refer it to as the `wind stress.' The other component, $T_{r\phi}$, will be referred to as the `radial shear stress.'
Figure \ref{fig:stress} compares these two $\alpha$ parameters for the set of simulations where the AD Elsasser number is varied. In the least coupled simulation, the wind stress is larger than the radial shear stress by a factor of a few. As the Elsasser number increases, the radial shear stress begins to grow until it is approximately equal to the wind stress by the simulation where the AD Elsasser number is $\Lambda_0=0.5$ (see Fig.~\ref{fig:stress}e). In the intermediate parameter regime ($\Lambda_0$ between 0.05 and 0.5), where the wind remains relatively laminar and the rings and gaps are rather steady, the two stresses are strongly correlated with both peaking in low-density gaps where the poloidal magnetic field lines are concentrated (compare, for example, panel d of Fig.~\ref{fig:stress} to panel d of Fig.~\ref{fig:surfden1D}, which shows that both stresses peak in the low-density gaps for the reference run). As the Elsasser number increases further, the avalanche accretion streams become prevalent, driving the atmosphere of disk and the base of the wind to be chaotic. This transition to a more chaotic disk-wind system is already present in the outer part of the disk in the $\Lambda_0=0.5$ case beyond $\sim 50$~au (see panel e), where the avalanche accretion flows have reversed the direction of the polar shear stress $T_{\theta\phi}$ at the initial surface of the disk ($\theta=\pi/2\pm\theta_0$). For the cases with higher $\Lambda_0=1.0,~2.0,$ and $\infty$~(ideal MHD), the effective $\alpha$ parameter for $T_{\theta\phi}$ becomes highly variable in both space and time and is often negative. However, $T_{r\phi}$ stays mostly positive, indicating that angular momentum in the disk is more persistently transported radially outward by avalanche accretion streams rather than vertically across the initial disk surface.
\section{Discussion}\label{sec:discuss}
\subsection{Comparison to other works}
This work examines how radial substructure can be created in a circumstellar disk in the presence of ambipolar diffusion on the scale of a few to tens of au, as part of a magnetically coupled disk-wind system. In \citetalias{2017MNRAS.468.3850S}, a similar phenomenon was observed to operate near the innermost disk radii ($\sim0.1$~au), where Ohmic resistivity dominates. As in \citetalias{2017MNRAS.468.3850S}, we again find that rings and gaps are formed solely from MHD processes. Here, the effects of AD have a clear and physically motivated interpretation as to how radial substructure is formed in simulations where the ions and neutrals are moderately coupled. The mechanism, described in Section~\ref{sec:ring}, relies on mass accretion through an AD-induced, midplane ($J_r$) current layer, where the poloidal magnetic field is dragged radially inward until it reconnects. The reconnection creates regions with magnetic loops where the net poloidal flux is decreased and mass accretion is less efficient, allowing matter to pile up into rings. It also enables the post-reconnection poloidal field to bunch up in localized regions, where mass accretion is more efficient, creating gaps.
The formation of radial disk substructures in MHD simulations (besides those formed by planets) has been seen at the boundaries of dead zones \citep{2010A&A...515A..70D,2015A&A...574A..68F,2016A&A...590A..17R} and in the context of zonal flows \citep{2009ApJ...697.1269J,2013MNRAS.434.2295K,2013ApJ...763..117D,2014ApJ...784...15S,2014ApJ...796...31B,2015ApJ...798...84B,2016A&A...589A..87B,2017A&A...600A..75B}. The concentration of poloidal magnetic field lines specifically in the presence of AD was observed in the disk simulations of \citet{2014ApJ...796...31B} and \citet{2017A&A...600A..75B}. This has been interpreted through the following form of the induction equation:
\begin{equation}
\frac{d\Phi_B(R,z)}{dt}=-2\pi R\mathcal{E}_\phi,
\end{equation}
where $\Phi_B$ is the vertical magnetic flux, and $\mathcal{E}_\phi$ is the $\phi$ component of the electromotive force (EMF; equation 8 of \citealt{2017ApJ...836...46B}; see also equation 23 of \citealt{2017A&A...600A..75B}). The EMF induced by AD is equal to $\bm{\mathcal{E}}_A=\eta_A \bm{J}_\perp$, where $\bm{J}_\perp$ is the component of the current perpendicular to the magnetic field (as defined in equation \ref{eq:induction}). When the azimuthal component of this perpendicular current, $J_{\perp,\phi}$, has a sign opposite to that of $J_\phi$, the AD EMF becomes anti-diffusive in nature, which would lead to the concentration of poloidal magnetic field lines. We have examined $J_{\perp,\phi}$ and $J_\phi$ in our reference run (where there is large spatial variation of the poloidal magnetic field strength, see Fig.~\ref{fig:bunch}b), and found that they have opposite signs in some regions but the same sign in others, which makes it hard to establish unambiguously the extent to which this mechanism may be operating in our simulations. In any case, we find that our results can be explained by a more pictorial mechanism: reconnection of sharply pinched poloidal field lines (e.g., Fig.~\ref{fig:reconnect}) that drives the segregation of poloidal magnetic flux relative to matter, which in turn leads to the formation of radial substructure. We note that \citet{2014ApJ...796...31B} also considered the possibility of reconnection playing a role in concentrating magnetic flux in the zonal flows found in their shearing box simulations (see their Fig.~9 for a cartoon illustrating the possibility). The relatively laminar nature of the disk accretion in the presence of a moderately strong ambipolar diffusion allowed us to isolate the reconnection events more clearly in our (2D) global simulations (see Fig.~\ref{fig:reconnect}). Whether it has a deeper physical connection with the mechanism that relies on $J_{\perp,\phi}$ and $J_\phi$ having opposite signs remains to be ascertained.
Although rings and gaps are prominent in most of our simulations, they are not a common feature of previous simulations. For example, the recent global accretion disk simulations of \citet{2017ApJ...836...46B} and \citet{2017ApJ...845...75B} do not seem to show such radial substructure. This is likely due to the fact that a weaker initial poloidal field strength is used (however, see the shearing box simulations of \citealt{2015ApJ...798...84B} where zonal flows develop with $\beta=10^5$). Specifically, in the simulations of \citet{2017ApJ...845...75B} the initial magnetic field in the disk is characterized by $\beta=10^5$, which is higher than the largest initial value of plasma-$\beta$ used in our simulations. In our $\beta\sim10^4$ simulation, rings and gaps are still present. However, the surface density contrast is reduced compared to the reference run of $\beta\sim10^3$ (see Fig.~\ref{fig:surfden1D_beta}). Although the same magnetic field variations and midplane pinching still occur in the weaker magnetic field simulation of $\beta\sim10^4$ (see Fig.~\ref{fig:beta_panels}c), the magnetic field is less able to move matter around to form rings and gaps and the timescale for the magnetic field to dynamically influence the matter will be longer compared to the stronger field case. As such, we expect the formation of rings and gaps to become increasingly less efficient as the magnetic field strength is reduced towards the purely hydrodynamic limit.
The initial field strengths in the ideal simulations of \citet{2017arXiv170104627Z} are similar to those adopted here ($\beta_0\sim10^3$). They show that most of the accretion occurs in a vertically extended disk `envelope,' with radial (as opposed to vertical) transport of angular momentum playing a dominant role in driving disk accretion. As discussed extensively in \citetalias{2017MNRAS.468.3850S}, this is consistent with the development of avalanche accretions streams as the Ohmic resistivity is reduced. It is also in agreement with the simulations in this work as we move towards the ideal MHD regime of large Elsasser numbers (see Fig.~\ref{fig:stress}). This agreement strengthens the case for the transition from a laminar disk-wind system to a more chaotic system dominated by the rapid formation and break up of accretion streams as the magnetic diffusivity (either Ohmic or ambipolar) is reduced.
\subsection{Dust dynamics and grain growth}
Wind-driven laminar disk accretion is an important feature of the moderately well magnetically coupled disks studied both in this paper (from AD) and in \citetalias{2017MNRAS.468.3850S} (due to Ohmic dissipation). There is some evidence that such a laminar accretion may be required for the HL Tau disk. As stressed by \citet{2016ApJ...816...25P}, there is tension between the small scale height of (sub)millimeter-emitting dust grains (inferred from the lack of azimuthal variation in the gap widths for the inclined HL disk, indicating strong dust settling) and the substantial ongoing mass accretion observed in the system, which, if driven by turbulence, would require a turbulence too strong to allow for the inferred degree of dust settling. This tension can be removed if the accretion is driven by ordered magnetic stresses rather than MRI-induced turbulence \citep{2017ApJ...845...31H}, as in our simulations with high to moderate levels of AD (such as the reference run), since dust grains can still settle to the midplane even with strong accretion. Furthermore, rings and gaps are naturally produced in these laminarly accreting disk-wind systems through the AD-aided magnetic reconnection; this mechanism can in principle produce the rings and gaps observed in the HL Tau disk. In practice, our model parameters are chosen for the purposes of illustrating the basic principles of ring and gap formation in the presence of ambipolar diffusion rather than for comparison with any specific object. Taken at the face value, the typical mass accretion rate of $10^{-6}~\text{M$_\odot$~yr$^{-1}$}$ found in the reference simulation is at least an order of magnitude larger than that inferred for classical T Tauri stars \citep{2016ARA&A..54..135H}. However, it is more consistent with the accretion rates inferred for younger protostellar disks (e.g., \citealt{2017ApJ...834..178Y}), although, it is possible to reduce the mass accretion rates in these simulations through rescaling (e.g., by adopting a lower initial disk density, $\rho_0$; see Appendix of \citealt{2014ApJ...793...31S}).
The formation of rings and gaps in a relatively laminar disk has important implications for the dynamics of dust grains. Pressure maxima, such as those formed from dense gas rings, are known sites of dust trapping \citep{1972fpp..conf..211W,2010AREPS..38..493C}. Without such traps, large millimeter-sized grains would migrate inward quickly as they lose angular momentum to the more slowly rotating gas that is partially supported by the radial pressure gradient \citep{1977MNRAS.180...57W}. This rapid radial drift is particularly problematic for low-mass disks around brown dwarfs \citep{2013A&A...554A..95P}. For example, in the case of 2M0444, \citet{2017ApJ...846...19R} has shown explicitly that, without any dust trap, millimeter-sized grains would be quickly depleted from the outer part of this disk (on the scale of tens to a hundred au; see the upper-left panel of their Fig.~3), in direct contradiction to observations. They also demonstrated that this fundamental problem can be resolved if there are multiple pressure peaks in the outer disk (see the lower-right panel of their Fig.~3). Such pressure peaks are naturally produced in our simulations (see, e.g., Fig.~\ref{fig:bunch} of the reference run).
Our mechanism of producing rings has two strengths. First, it takes into account ambipolar diffusion, which is the dominant non-ideal MHD effect in the outer disk where dust trapping is needed to be consistent with dust continuum observations. Second, it can in principle operate not only in relatively evolved protoplanetary disks but also younger protostellar disks as long as such disks are significantly magnetized with a poloidal field. Indeed, our mechanism is likely to work more efficiently in the earlier phases of disk evolution where the disk is expected to be threaded by a strong poloidal magnetic field, perhaps inherited from the collapse of dense cores, which are known to be magnetized with rather ordered magnetic fields (e.g., \citealt{2008ApJ...680..457T,2014prpl.conf..173L,2014prpl.conf..101L}). Such ordered poloidal fields can drive fast disk accretion expected in the early phases without generating a high level of turbulence in the outer (AD-dominated) region, which should make it easier for the dust to settle vertically and grow near the midplane, even during the early, perhaps Class 0, phase of star formation. In other words, strong accretion does not necessarily mean strong turbulence. Even in the earliest, Class 0 phase of star formation, large grains (if they are present) can be trapped in principle by the pressure bumps that naturally develop in the magnetically coupled disk-wind systems. Observationally, whether rings and gaps are prevalent in Class 0 disks is unknown at the present time, because they are more difficult to observe in the presence of a massive protostellar envelope, however, there is some evidence that rings and gaps are already present in at least the Class I phase (see observations of IRS 63 in $\rho$ Oph by Segura-Cox et al., \textit{in prep.}).
Lastly, we note that the laminar disk wind in our reference and related simulations can preferentially remove gas from the disk, if the dust has settled to the midplane (or perhaps been trapped near the rings). As discussed by \citet{2010ApJ...718.1289S}, this could lead to an increase in the dust-to-gas mass ratio (see also \citealt{2015ApJ...804...29G,2016ApJ...818..152B}), conducive to the development of the streaming instability \citep{2005ApJ...620..459Y,2017arXiv171103975S}, which may facilitate the formation of planetesimals and eventually planets (e.g., \citealt{2010AREPS..38..493C}). This process of grain settling, growth, and trapping may be as efficient, in not more, in the early, protostellar phase of star formation compared to the later, protoplanetary phase. We will postpone a quantitative exploration of this interesting topic to a future investigation.
\section{Conclusion}\label{sec:conc}
We have carried out 2D (axisymmetric) simulations of magnetically coupled disk-wind systems in the presence of a poloidal magnetic field and ambipolar diffusion (AD). The field strength is characterized by the plasma-$\beta$ and AD by the dimensionless Elsasser number $\Lambda_0$. We focused on $\beta\sim10^3$ and explored a wide range of values for $\Lambda_0$, from 0.01 to $\infty$ (ideal MHD). Our main conclusions are as follows:
\begin{enumerate}
\item In moderately well coupled systems with $\Lambda_0$ between 0.05 and 0.5, including the reference simulation (ad-els0.25), we find that prominent rings and gaps are formed in the disk through a novel mechanism, AD-assisted reconnection. This mechanism starts with the twisting of the initial poloidal magnetic field into a toroidal field that reverses polarity across the disk midplane. Ambipolar diffusion enables the Lorentz force from the toroidal field pressure gradient to drive the ions (and the toroidal field lines tied to them) towards the magnetic null near the midplane, which steepens the radial ($J_r$) current sheet in a run-away process first described in \citet{1994ApJ...427L..91B}. The field kink generates a toroidal Lorentz force that removes angular momentum from the thin radial current sheet, forcing it to accrete preferentially relative to the rest of the disk. The preferential midplane accretion drags the poloidal field lines into a sharply pinched configuration, where the radial component of the magnetic field reverses polarity over a thin, secondary azimuthal ($J_\phi$) current sheet. Reconnection of the radial pinch produces two types of regions with distinct poloidal field topologies: one occupied by magnetic loops and another that remains threaded by ordered poloidal fields. The weakening of the net poloidal field in the former makes angular momentum removal less efficient, allowing disk material to accumulate to form dense rings. Conversely, those regions that gained poloidal flux after reconnection are magnetically braked more strongly, with a faster draining of disk material that leads to gap formation. In addition, AD allows for a quasi-steady state of the ring and gap structure, where the field lines can stay more or less fixed in place despite rapid mass accretion in gaps because of the ion-neutral drift. We find little evidence for the formation of prominent rings and gaps in the case of the highest ambipolar diffusion considered in this work ($\Lambda=0.01$) and cases with large, additional Ohmic diffusivities. This finding is consistent with the above scenario because the radial pinching of the poloidal field is smoothed out by the excessive magnetic diffusivity, suppressing the reconnection that lies at the heart of the mechanism.
\item In better magnetically coupled disk-wind systems with larger $\Lambda_0$, as well as the ideal MHD limit, we find that avalanche accretion streams develop spontaneously near the disk surface. The accretion streams lead to unsteady/chaotic disk accretion and outflow, as found previously in \citetalias{2017MNRAS.468.3850S} for cases of low or zero Ohmic resistivities (see also \citealt{2017arXiv170104627Z}). Prominent rings and gaps are still formed in the disk. Part of the reason is the large temporal and spatial variations induced by the constant formation and destruction of the streams will inevitably produce spatial variation in the mass accretion rate and thus the surface density. Perhaps more importantly, the poloidal field lines are concentrated in some regions and excluded from others, with the more strongly magnetized regions producing gaps and the less magnetized regions forming rings, just as in the more magnetically diffusive reference case. We suggest that this segregation of poloidal magnetic flux and matter is also due to reconnection of highly pinched poloidal fields. In this case, the pinching is caused by the avalanche accretion streams (a form of MRI channel flows) rather than the midplane current sheet steepened by AD. The fact that rings and gaps are formed in both laminar and chaotic disk-wind systems over a wide range of magnetic diffusivities suggests that they are a robust feature of such systems, at least when the initial poloidal magnetic field is relatively strong. For more weakly magnetized systems, reconnection may still occur but the resulting redistribution of poloidal magnetic flux would have less of a dynamical effect on the gas, making ring and gap formation less efficient.
\item If young star disks are threaded by a significant poloidal magnetic field, especially during the early phases of star formation, it may drive rapid disk accretion through a magnetic wind without necessarily generating strong turbulence in the disk, particularly in the outer parts of the disk that are only moderately well coupled to the magnetic field. The lack of a strong turbulence despite rapid accretion may allow dust to settle early in the process of star formation, facilitating early grain growth. Large grains may be trapped in the rings that are naturally produced in the system, which may promote the formation of planetesimals and eventually planets.
\end{enumerate}
\section*{Acknowledgements}
We thank Xuening Bai, Zhaohuan Zhu, and Takeru Suzuki for helpful discussions and the referee, William B\'{e}thune, for detailed constructive comments. This work is supported in part by National Science Foundation (NSF) grant AST-1313083 and AST-1716259 and National Aeronautics and Space Administration (NASA) grant NNX14AB38G.
|
2,877,628,090,157 | arxiv | \section{Introduction}
At high velocities space and time are merged together into Minkowski
spacetime, and even though both distances and durations depend on the
observers frame, then the 4-vector $(ct, \vec x)$ has invariant length
under Lorentz transformations. Similarly the energy-momentum
4-vector, $(\varepsilon/c , \vec p)$, is a proper 4-vector in
Minkowski space. The classical conservation laws, like energy and
momentum conservation arising from symmetries in time and space, thus
have related conservation laws in relativistic physics.
However, for other objects, such as the (polar vector) dynamic mass
moment, $\vec N = ct \vec p -\varepsilon \vec x/c $, or the (axial
vector) angular momentum, $\vec L = \vec x \times \vec p$, the
corresponding relativistic conservation laws are often not discussed
in detail. One reason for this is that there is no way of combining
$\vec N$ and $\vec L$ into a proper 4-vector. Instead, one must
combine $\vec N$ and $\vec L$ into an anti-symmetri rank-2 tensor,
$M^{\mu \nu}$ \citep{landau}.
A very similar issue is well-known from electromagnetisme, where the
(polar vector) electric field, $\vec E$, and the (axial vector)
magnetic field, $\vec B$, also combine into an antisymmetri rank-2
tensor, $F^{\mu \nu}$. When learning about electromagnetism this is
frustrating to some, since most of our intuition is based on the
fields $(\vec E, \vec B)$, however, when performing a Lorentz
transformation, one most often finds oneself performing the
calculations with the physically somewhat less transparant tensor
$F^{\mu \nu}$.
While contemplating the physical meaning of the mathematical space
where $F^{\mu \nu}$ and $M^{\mu \nu}$ live, one is naturally drawn to
spacetime algebra (STA) \citep{hestenes1966, 2003AmJPh..71..691H},
which provides a geometric explanation for the connection between
tensors like $F^{\mu \nu}$ and $M^{\mu \nu}$ and Minkowski space.
In the sufficiently mature scientific field of STA, it is well-known
how the electromagnetic bi-vector field, ${\bf F}$, naturally is used
to derive both Maxwell's equations and the Lorentz force. Given the
strong mathematical similarities between the ``electromagnetic''
bi-vector ${\bf F}$ and the ``mechanics'' bi-vector
${\bf M}$, it appears natural to derive the equations and forces
which are dictated by ${\bf M}$, in particular since the
structure of STA uniquely defines these equations and
forces.
Below the same methods are applied to the bi-vector field, ${\bf M}$,
as have previously been applied to ${\bf F}$ in electromagnetism, and
it is shown how new forces appear naturally from the bi-vector field
${\bf M}$. One of these inverse distance-squared force-terms depends on the
internal velocity dispersion of an object (which for instance could be
a distant galaxy). It is speculated to which degree this new force possibly may
be related to the force which was recently suggested as an explanation
for the observed acceleration of the
universe~\citep{2021ApJ...910...98L}.
\section{Spacetime algebra}
Spacetime algebra starts with Minkowski space, ${\cal M}_{1,3}$, with
the metric signature $(+,-,-,-)$, and a chosen basis $\{\gamma_\mu
\}_{\mu=0}^3$ of ${\cal M}_{1,3}$. These 4 orthonormal vectors are the
basis for 1-blades. The 2-blade elements are the 6 antisymmetric
products $\gamma _{\mu \nu} \equiv \gamma_{\mu} \gamma_{\nu} $. The
product is here given by the sum of the dot and wedge product: $a b =
a \cdot b + a \wedge b$ \citep{hestenes2015,doranlasenby}.
The wedge operator, $\wedge$, is the 4-dimensional generalization
of the 3-dimensional cross-product.
Continuing over 3-blades, $\gamma _{\mu
\nu \delta}$, one finally reaches the highest grade, the
pseudoscalar $I \equiv \gamma _0 \gamma _1 \gamma _2 \gamma _3$, which
represents the unit 4-volume in any basis. Interestingly one has $I^2
= -1$.
For the discussion below, the bi-vectors are important:
these are oriented plane segments, and examples include the
electromagnetic field $ {\bf F} = \vec E + \vec B \, I$, and the
angular momentum ${\bf M} = x \wedge p$, where $x$ and $p$ are proper
4-vectors~\citep{hestenes2015,doranlasenby}.
From a notational point of view vector-arrows are used above spatial 3-vectors
like $\vec E$ or $\vec p$, no-vector-arrows are used for proper 4-vectors like
$x$ and $w$, and boldface is used for bi-vectors like ${\bf F}$ and
${\bf M}$.
\section{Electromagnetism}
The case of electromagnetism in STA is well described in the
literature \citep{hestenes2015, dressel2015}, and serves as a starting
point here. The 4 Maxwells equations can be written
\begin{equation}
\nabla {\bf F} = j \,
\label{eq:maxwell}
\end{equation}
where the complex current may contain both eletric (vector) and
magnetic (trivector) parts, $j_e + j_m I$. The bi-vector is given by
${\bf F} = \vec E + \vec B I$, and the derivative $\nabla {\bf F} = \nabla
\cdot {\bf F} + \nabla \wedge {\bf F}$ produces both a vector and a
trivector field.
Using the time-direction, $\gamma_0$, one can decompose the derivative along
a direction parallel to and perpendicular to $\gamma_0$,
$\nabla = \left( \partial _0 - \vec \nabla \right) \gamma_0 $,
where $ \vec \nabla $ is the frame-dependent relative 3-vector derivative.
It is now straight forward to expand eq.~(\ref{eq:maxwell}) to the
4 Maxwells equation~\citep{hestenes2015, dressel2015}.
It is important to stress, that eq.~(\ref{eq:maxwell}) is not only a
matter of compact notation, it is indeed the only logical extension
beyond the most trivial equation in STA, $\nabla {\bf F} = 0$. The
only thing missing is to connect the bi-vector field to observables:
this is done through observations, which also establish the units of
$\vec E$ and $\vec B$.
\subsection{The Lorentz force}
The classical Lorentz force is given by
\begin{equation}
\frac{d\vec p}{d t} = q \left( c \vec E + \vec v \times \vec B \right) \, ,
\label{eq:lorentz}
\end{equation}
which effectively arose as a clever guess to explain observations.
The cross, $\times$, refers to the normal 3-dimensional cross-product.
In
the standard Euler-Lagrange formalism the Lorentz force appears when
adding a term to the Lagrangian, $q w_\mu A^\mu$, where $A$ is the
4-vector potential.
In STA the Lorentz force appears when one contract the bi-vector field
${\bf F}$ with a proper 4-vector velocity $w$. Using that the 4-vector
$w$ is connected with the para-vector, $w_0+\vec w$, via
$\gamma_0$~\citep{doranlasenby}, namely~\footnote{The
right-multipliation by the timelike vector $\gamma_0$ isolates the
relative quantities of that frame~\citep{dressel2015}, e.g. $x
\gamma_0 = \left( ct + \vec x \right)$.} $w = \left( w_0 + \vec w
\right) \gamma_0$, one gets
\begin{equation}
\left( {\bf F} \cdot w \right) q \frac{d\tau}{dt} \gamma_ 0=
q \vec E \cdot \vec v +
q \left( c \vec E + \vec v \times \vec B\right) \, ,
\label{eq:lorentzforce}
\end{equation}
where the first term on the r.h.s. is the rate of work,
$d\varepsilon/d(ct)$, we use $\vec w = \gamma \vec v$,
and the last parenthesis on the r.h.s is exactly
the Lorentz force in eq.~(\ref{eq:lorentz}).
If the current was complex
there could be another force term allowed~\citep{dressel2015}, namely
\begin{equation}
{\bf F} \cdot \left( w I \right) \gamma_0 = \left( \left( {\bf F} \wedge w \right) I \right) \gamma_0
= \vec B \cdot \vec v + \left( c \vec B - \vec v \times \vec E \right) \, .
\end{equation}
To summarize, the full 4 Maxwells equations appear naturally from the
geometric structure of STA, through the equation $\nabla {\bf F} =
j$. The Lorentz force also appears naturally in STA when contracting
the bi-vector field, ${\bf F}$ with the 4-vector current, $dp/d\tau =
{\bf F} \cdot (qw)$, where $p$ is the proper energy-momentum 4-vector.
\section{Angular momentum}
In order to generalize the 3-dimensional angular momentum, $\vec L =
\vec x \times \vec p$, one uses the proper 4-dimensional $x=(ct +\vec
x)\gamma_0$ and $p= (\varepsilon/c + \vec p) \gamma_0$, to create the
bi-vector ${\bf M}$~\citep{landau, dressel2015}
\begin{eqnarray}
{\bf M} &=& x \wedge p \nonumber \\
&=& \frac{\varepsilon \vec x}{c} - ct \vec p - \vec x \times \vec p I \nonumber \\
&=& - \vec N - \vec L I \, .
\end{eqnarray}
Only ${\bf M}$ is a proper geometric object, and the split into
dynamic mass moment and angular momentum requires that one specifies
$\gamma_0$, in exactly the same way that ${\bf F}$ is the proper
geometric object of electromagnetism, and the separation into $\vec E$
and $\vec B$ fields requires specification of a frame by the choice of
$\gamma_0$.
It is now clear how everything can be repeated from the case of
electromagnetism: where one had a bi-vector ${\bf F}$ and relative
3-vectors $\vec E$ (polar) and $\vec B$ (axial), then one now has a
bi-vector ${\bf M}$ and relative 3-vectors $- \vec N$ (polar) and
$-\vec L$ (axial). The signs could have been defined away, but are kept
to agree with the standard notation in the literature~\citep{landau}. When
deriving the Lorentz force for electromagnetism, by dotting the
bi-vector field ${\bf F}$ with a charge-current, $q w$, one needs
experimental data to get the units right ($\epsilon_0$ and $\mu_0$ for
the E- and B-fields, respectively)~\citep{hestenes2015}. In a similar
fashion experimental data is needed to get the units for a force
defined by dotting the field ${\bf M}$ with a ``mass-current'', $m_t w$.
The simplest possible equation describing the evolution of the
bi-vector field is specified by the structure of STA, namely
\begin{equation}
\nabla {\bf M} = j_{\bf M} \, .
\end{equation}
This Letter is not focusing on the details of the source on the r.h.s. (which could be zero),
however, for the sake of generality it is allowed to contain both a
vector and a trivector term $ j_{\bf M} = j_{\bf 1} + j_{\bf 3} I$.
The resulting equations split into two equations for the relative
scalars
\begin{eqnarray}
-\vec \nabla \cdot \vec N &=& \rho_1 \, , \\
\label{eq:relativescalars}
- \vec \nabla \cdot \vec L &=& \rho_3 \, ,
\label{eq:relativescalars2}
\end{eqnarray}
(where $\rho_i$ refer to the 0-component of the sources) and two
equations for the relative 3-vectors, just like Maxwell's equations
did.
\begin{eqnarray}
-\partial_0 \vec N + \vec \nabla \times \vec L &=& - \vec J_1 \, ,\\
\partial_0 \vec L + \vec \nabla \times \vec N &=& \vec J_3 \, ,
\label{eq:relvec2}
\end{eqnarray}
where $\vec J_i$ refer to the 3 spatial components of the sources.
The details of these 4 equations
will be discussed elsewhere~\citep{students}.
Instead, the force which appears from the
contraction with a mass-current, $m_t \omega$, where $w$ again is a
proper 4-velocity, and $m_t$ is the inertia of the test particle,
will now be calculated. From the term ${\bf M} \cdot w$ one gets
\begin{equation}
\left( {\bf M} \cdot w \right) \frac{d\tau}{dt} \gamma_0 = -\vec N \cdot \frac{\vec v}{c} +
\left( - \vec N - \frac{\vec v}{c} \times \vec L \right) \,.
\label{eq:newforce}
\end{equation}
The first term on the r.h.s. is similar to a rate of work. However,
the last parenthesis of eq.~(\ref{eq:newforce})
contains the new forces of interest here,
and
will be discussed in section \ref{sec:newforce}
below. One could also have considered a force arising from $\left(
{\bf M} \wedge w \right) I$, which looks like
\begin{equation}
\left( {\bf M} \wedge w \right) I \frac{d\tau}{dt} \gamma_0 =
- \vec L \cdot \frac{\vec v}{c}
+ \left( -\vec L + \frac{\vec v}{c} \times \vec N \right) \, ,
\label{eq:newforce2}
\end{equation}
however, it is left for a future analysis to study this.
\section{The new force terms}
\label{sec:newforce}
Let us consider a collection of particles at a large distance, $\vec
r_0$. The particles may have different inertia, $m_i$, but move
collectively with an average velocity, $\vec V$. If one considers
particles in a cosmological setting, then the velocity is a
combination of the Hubble expansion and peculiar velocity, $\vec V = H
\vec r_0 + \vec v_p$, and at large distances the peculiar velocity is
subdominant. The collection of particles may have internal motion,
which is simplified with an internal velocity dispersion,
$\sigma^2$. Practically when calculating the velocity
dispersion there will be terms including both the Hubble expansion,
$v_H = H r$, and also the background density of both matter and the
cosmological constant, however, these terms happen to exactly cancel
each other~\citep{2013MNRAS.431L...6F}, and one can therefore
calculate $\sigma^2$ as if the structure is alone in a non-expanding
universe.
The dynamic mass moment is given by
\begin{equation}
\vec N = \sum \left( ct \vec p_i - \frac{\varepsilon_i \vec r_i}{c} \right) \, ,
\end{equation}
where the sum is over all particles involved~\citep{landau}. If one
divides both terms by the total energy, $\varepsilon_{tot} = \sum
\varepsilon_i$, then one gets
\begin{equation}
\frac{\vec N }{\varepsilon_{tot}} = \left( \frac{ct \sum \vec
p_i}{\sum \varepsilon_i} - \frac{\sum \varepsilon_i \vec r_i}{c \sum
\varepsilon_i} \right) \, ,
\end{equation}
The first term is just $ct$ times the average velocity. At small velocities
one has $\varepsilon \approx m_ic^2$ and hence the last term describes
the relativistic center of inertia, $\vec R_{cm} = \sum (m_i \vec
r_i)/ \sum m_i$.
If the centre of inerti moves at constant velocity (now ignoring sums over
particles), then one has $\vec r = \vec r_0 + \vec V t$, and hence
\begin{equation}
\vec N = - m c \vec r_0 \, .
\label{eq:Nr}
\end{equation}
When considering the Lorentz force in eq.~(\ref{eq:lorentzforce}) one needs
to get the units right to get
$\vec E = q \vec r /(4 \pi \epsilon_0 |r|^3)$, which includes
the observable vacuum permittivity, $\epsilon_0 $, and also Coulombs
inverse distance-square law (resulting from Gauss' and Faradays' laws combined).
Effectively this means dividing by $\epsilon_0 |r|^2$.
The new force terms in the parenthesis of eq.~(\ref{eq:newforce}) will
now be considered. Since the mass-current is related to gravity, one
should multiply by Newtons gravitational constant, $G = 6.67 \times
10^{-11} {\rm m}^3/({\rm s}^2 \, {\rm kg})$. To get a well-behaved
field one divides by distance to power 3, which will lead to an inverse
distance-square force: this comes from the integral over
eq.~(\ref{eq:relativescalars}), using the sphericity from
eq.~(\ref{eq:relvec2}) with $\vec J_3=0$ and the expression in
eq.~(\ref{eq:Nr}). This last point is easily recognized by considering
the change of notation, $\vec N \rightarrow \vec g /(4 \pi G)$, and
$\rho_1 \rightarrow \rho_m$ which is the mass density, which means
that eq.~(\ref{eq:relativescalars}) is written as $\vec \nabla \cdot
\vec g = - 4 \pi G \rho_m$. This equation is clearly regognized as
leading to Newtons gravitational law. Finally to get the units right,
it is divided by $c$.
This implies that one has a force-term that looks like
\begin{equation}
- \kappa \, \frac{G m_t m \vec r_0} {| \vec r_0 |^3} \, .
\label{eq:newton}
\end{equation}
where $\kappa$ is
an unknown, dimensionless number, which must be
determined from observations,
and $m_t$ is from the
mass-current, $m_t w$.
In the case of $\kappa = 1$ this is
just Newtons gravitational force.
In the above picture, it thus appears that the Newtonian gravity may
be interpreted as a gravitational analogue to the Coulomb force from
electromagnetism. Since the masses are always positive, the
gravitational force is always attractive.
When the structure under consideration contains a dynamical term
proportional to the velocity dispersion, $\sigma^2$, which for
instance can arise in a dwarf galaxy where the stars and dark matter
particles are orbiting in the local gravitational potential, then the
potential will be minus 2 times the kinetic energy according to the
virial theorem~\citep{bt2}, $2T+U=0$, and hence one writes the
energy as
\begin{equation}
\varepsilon_i = m_ic^2 - \frac{1}{2} m_i \sigma_i^2 \, .
\end{equation}
In this case one ends up with a new force of the form
\begin{equation}
\frac{\tilde \kappa}{2} \, \frac{\sigma^2}{c^2} \,
\frac{m_tm_iG \vec r_0}{\| \vec r_0 \|^3} \, ,
\label{eq:sigma2}
\end{equation}
where the dispersion has been normalized to $c$. This force is always
repulsive. In the case of $\tilde \kappa =1$ this is just a minor
correction to the normal gravitational attraction, e.g. for galaxy
clusters with velocity dispersions of $1000$ km/sec this is a
$10^{-5}$ correction, and for dwarf galaxies much less. Possibly for
motion near very compact objects, this correction may eventually be
observable.
Velocity-dependent forces are well-known, including the Coriolis-force
and the Lorentz-force, however, this is, to our knowledge, the first
derived long-distance force depending on velocity squared. In an
atttempt to make kinetic energy depend on relative velocities (rather
than absolute velocities) similarly to how potential energy depends on
relative position, Schr\"odinger suggested a new gravitational force
proportional to velocity squared \citep{1925AnP...382..325S}. His
force has essentially the same form as the force derived above,
however, its existence was postulated on rather philosofical grounds.
A recent paper demonstrated that a universe which contains no dark
energy, but instead includes a new repulsive inverse distance-square force
proportional to internal velocity dispersion squared, just like
equation~(\ref{eq:sigma2}), could have an accelerated expansion which
fairly closely mimics the accelerated expansion induced by the
cosmological constant~\citep{2021ApJ...910...98L}. In that paper it
was implicitely suggested that such a force might conceivably exist amongst the
dark matter particles. What has been shown in this Letter is, that
such a force indeed may exist, and that it is not specifically related
to the dark matter particle, but instead related to gravity in general.
One of the concerns with the suggestion discussed in
\cite{2021ApJ...910...98L} is the potential instability of
cosmological structures, however, from the derivation above it is clear
that the new force derived here
comes from the internal dispersion (as opposed to relative velocities),
and hence there is no instability concern.
The magnitude of the dimensionless $\tilde \kappa$ in
eq.~(\ref{eq:sigma2}) is unknown in the present derivation, however,
it should logically be unity. The new force of the
paper~\citep{2021ApJ...910...98L} should have a numerical value of
$\tilde \kappa \sim 10^6-10^8$. From the derivation above there is no
indication where such a large factor should come from.
\section{Conclusion}
It was recently suggested that if a force which depends on velocity
squared exists in Nature, then it may induce an effect on cosmological
scales which mimics the accelerated expansion of the standard
cosmological constant~\citep{2021ApJ...910...98L}. The present Letter
demonstrates one such concrete possibility. The derivation here is
framed in the geo\-metric structure of spacetime
algebra~\citep{hestenes2015}, and takes as starting point the
relativistic generalization of angular momentum which includes the
dynamic mass moment, $\vec N = ct \vec p - \varepsilon \vec
r/c$. Since the energy in this term, $\varepsilon$, contains the
velocity dispersion of a distant cosmological object, then an inverse
distance-square force naturally appears, which is proportional to
$\sigma^2$.
Such a force
may lead to a slightly reduced
gravitational force for extremely compact objects.
The magnitude of the force derived here is significantly
smaller than needed to explain the present day accelerating
universe~\citep{2021ApJ...910...98L}.
\section*{Acknowledgement}
It is a pleasure to thank Max Emil K.S. Sondergaard, Magnus B. Lyngby
and Nicolai Asgreen for interesting discussions. I thank Mario
Pasquato for bringing the 1925 Schr\"odinger paper to my attention.
\section{Data availability}
No new data were generated or analysed in support of this research.
|
2,877,628,090,158 | arxiv | \section{Introduction}
The binary $\beta$ Lyr is a nearly edge-on, semi-detached interacting
system that has undergone mass reversal and remains in a phase of
large-scale mass transfer. The primary, mass-losing star (the ``Loser'')
is a B6-B8 IIp star. The mass-gaining star (the ``Gainer'') is embedded
in an optically thick accretion disk and is not directly visible.
Although the embedded source had been considered as a possible compact
object (Devinney 1971; Wilson 1971), it is probably a main sequence B0 star
(Hubeny \& Plavec 1991). The system is very complex, having bipolar
jet-like structures (Harmanec \hbox{et~al.\/}\ 1996; Hoffman \hbox{et~al.\/}\ 1998),
a circumbinary envelope (Batten \& Sahade 1973; Hack \hbox{et~al.\/}\ 1975),
and a substantial kilo-Gauss magnetic field (Leone \hbox{et~al.\/}\ 2003).
\begin{table}
\begin{center}
\caption{Properties of $\beta$ Lyrae \label{tab1}}
\begin{tabular}{lcc}
\hline\hline
Component$^a$ & Gainer & Loser \\
\hline
\rule[0mm]{0mm}{3.5mm}$T_{\rm eff}$[K] & 32,000 & 13,300 \\
\rule[0mm]{0mm}{4mm}Spectral Type & $\approx$B0 V & B6-8 IIp \\
\rule[0mm]{0mm}{4mm}$M/M_\odot$ & $\approx$13 & $\approx$3 \\
\rule[0mm]{0mm}{4mm}$\dot{M}^b[M_{\odot}~{\rm yr}^{-1}]$ & $4.7 \times 10^{-8}$ & $7.2 \times 10^{-7}$ \\
\rule[0mm]{0mm}{4mm}$ \varv_\infty^b$[km s$^-1$] & 1470 & 390 \\
\\\hline\hline
System Property$^a$ & \multicolumn{2}{c}{Value} \\ \hline
\rule[0mm]{0mm}{3.5mm}Orbital Period & \multicolumn{2}{c}{12.9 days}\\
\rule[0mm]{0mm}{3.5mm}Viewing Inclination & \multicolumn{2}{c}{$86^\circ$}\\
\rule[0mm]{0mm}{4mm}Binary Separation & \multicolumn{2}{c}{$55-60 R_\odot$}\\
\rule[0mm]{0mm}{4mm}Distance & \multicolumn{2}{c}{270 pc} \\
\rule[0mm]{0mm}{4mm}$\log N_H$ (cm$^{-2}$) & \multicolumn{2}{c}{20.76} \\
\rule[0mm]{0mm}{4mm}{\it ROSAT} PSPC$^c$: & \multicolumn{2}{c}{0.07 cps} \\
\rule[0mm]{0mm}{4mm}{\it Einstein} SSS$^d$: & \multicolumn{2}{c}{0.11 cps}\\
\hline
\end{tabular}
\parbox{2.5in}{\small $^a$ Component and system properties taken
from Harmanec (2002) unless otherwise noted. \\
$^b$ Mass-loss rate and terminal speed for the stellar
winds of the respective binary components (Mazzali 1987).\\
$^c$ Bergh\"{o}fer \&
Schmitt 1994)\\
$^d$ Waldron (private comm.)}
\end{center}
\end{table}
The optical light curve of the system features a primary minimum that
is $\approx 1$ magnitude deep and a secondary minimum $\approx 0.4$
magnitudes deep (see Fig.~\ref{fig1}); however, the secondary minimum
is deeper than the primary minimum at shorter wavelengths, and below
Ly$\alpha$ the eclipses no longer appear (Kondo \hbox{et~al.\/}\ 1994). A summary
of the system properties is given in Table~\ref{tab1}. The orbital period
is 12.9~days, and the mean light curves appear stable with epoch.
The UV spectrum of $\beta$ Lyr is dominated by an anomalous continuum
and emission lines with unusually strong P~Cygni profiles typical of
hot star winds (Hack \hbox{et~al.\/}\ 1975; Aydin \hbox{et~al.\/}\ 1988; Mazzali 1987).
Despite numerous and ongoing modeling attempts (e.g., Wilson 1974;
Linnell \& Hubeny 1996; Bisikalo \hbox{et~al.\/}\ 2000; Linnell 2002; Nazarenko \&
Glazunova 2003, 2006ab), no model is yet capable of matching the observed
light curves of $\beta$ Lyr from the IR through the UV. Strangely,
$\beta$~Lyr has been largely unstudied in the X-ray regime, despite
the strong X-ray flux detected by the {\it ROSAT} HRI (Bergh\"{o}fer \&
Schmitt 1994). An unpublished spectrum taken with the {\it EINSTEIN}/SSS
in 1979 reveals X-ray emission at relatively high energies, suggesting
that phase-dependent observations may provide new clues to resolving
the puzzle of the $\beta$ Lyr geometry and interactions. Exploiting
{\it Suzaku's} excellent sensitivity to hard X-rays, we conducted three
pointed observations of $\beta$~Lyr within the same orbit. In the
following section the observations and reduction of data are detailed.
Analyses of the spectra with phase are described in \S 3, and a discussion
of the results is presented in \S 4.
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{f1.eps}}
\caption{
Illustration of the Suzaku pointings with
respect to the binary phase. The solid line represents the normalized
V-band light curve (Fourier fit by Harmanec et al. 1996).
Primary eclipse (phases 0.0 and 1.0) occurs when
the mass-losing star is occulted by the disk.
The three intervals marked on the curve represent our Suzaku pointings,
which occurred during a single orbit in May 2006. Phases were calculated
from the quadratic ephemeris of Harmanec \& Scholz (1993).
\label{fig1}}
\end{figure}
\section{Observations and Data Reduction}
The joint Japan/US X-ray astronomy satellite {\em Suzaku} (Mitsuda
\hbox{et~al.\/}\ \cite{mit}) observed \mbox{$\beta\,$Lyr}\ in May 2006 on three occasions at
orbital phases as shown in Fig.~\ref{fig1}, with corresponding viewing
perspectives of the \mbox{$\beta\,$Lyr}\ system illustrated in Fig.~\ref{fig2}.
(Note that the primary minimum occurs when the loser star is eclipsed,
and the secondary minimum when the disk component is eclipsed.)
The pointings were spaced approximately 4.3~days apart to sample the
full 12.9~day orbit of the binary. Exposure times and count rates are
tabulated in Table~\ref{tab2}.
{\em Suzaku} carries four X-ray Imaging Spectrometers (XIS; Koyama
\hbox{et~al.\/}\ \cite{xis}) and a collimated Hard X-ray Detector (HXD; Takahashi
\hbox{et~al.\/}\ \cite{hxd}). The field-of-view (FOV) for the XIS detectors
is $17\arcmin\ \times\,17\arcmin$. One of the XIS detectors (XIS1) is
back-side illuminated (BI) and the other three (XIS0, XIS2, and XIS3) are
front-side illuminated (FI). The bandpasses are $\sim$\,0.4\,--\,12\,keV
for the FI detectors and $\sim$\,0.2\,--\,12\,keV for the BI detector.
The BI CCD has higher effective area at low energies, however its
background level across the entire bandpass is higher compared to the
FI CCDs.
The angular resolution of the X-ray telescope onboard {\em Suzaku} is
$\approx 2\arcmin$. Therefore in the XIS image, \mbox{$\beta\,$Lyr}\ is not resolved
from two nearby B-type stars: HD\,174664 (\mbox{$\beta\,$Lyr}\,B) and HD\,174639. While
the latter star was not detected by {\it ROSAT}, the former has a
{\it ROSAT} HRI count rate of $4\times 10^{-3}$ cps, as compared to the
\mbox{$\beta\,$Lyr}\ {\it ROSAT} HRI count rate $4\times 10^{-2}$ cps. We are confident
that the X-ray flux detected by {\em Suzaku} is at least 90\%
dominated by \mbox{$\beta\,$Lyr}.
The HXD consists of two non-imaging instruments (the PIN
and GSO; see Takahashi \hbox{et~al.\/}\ \cite{hxd}) with bandpasses of
$\sim$\,10\,--70\,keV (PIN) and $\sim40-600$~keV (GSO), and a FOV of
$34\arcmin\ \times\,34\arcmin$ (PIN). Both of the HXD instruments are
background-limited. The background subtraction for the HXD is performed by
modeling the background spectrum. Presently, the non-X-ray background
model (e.g., particle events) is known for the PIN detector with
$\sim$\,3-5\% accuracy (Kokubun \hbox{et~al.\/}\ \cite{hxd-o}). The background
modeling for GSO data is currently far less certain than for the PIN,
and so we do not report on the GSO measurements.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{f2.eps}}
\caption{Graphic topview representation of the \mbox{$\beta\,$Lyr}\ star-disk system,
following the model of Hoffman \hbox{et~al.\/}\ (1998). Components are illustrative
and not to scale. The three {\it Suzaku} pointings occurred at phases
$\varphi=0.24$, 0.55, and 0.91 all within the same orbit. Arrows indicate
the Earth's line-of-sight for these three phases.
\label{fig2}}
\end{figure}
HXD-PIN data reduction and extraction of spectra were performed using the
latest calibration sources and background models. We account for cosmic
X-ray background (CXB) in fitting the spectral models using the procedure
suggested by the {\em Suzaku} team based on a ``typical'' CXB spectrum
(see www.astro.isas.ac.jp/suzaku). We do not correct for contributions to
the HXD background due to the South Atlantic Anomaly (SAA; see Kokubun
\hbox{et~al.\/}\ \cite{hxd-o}) because our observations of \mbox{$\beta\,$Lyr}\ were performed
when the background count-rate was at its lowest.
\begin{table*}
\begin{center}
\caption{XIS Observations of $\beta$ Lyrae \label{tab2}}
\begin{tabular}[t]{ccccccc
\hline\hline
Date & $\varphi^a$ & Exposure &
\multicolumn{4}{c}{XIS Count rate ($10^2$\,cps)} \\
& & (ksec) & XIS0 & XIS1 & XIS2 & XIS3 \\
\hline
\rule[0mm]{0mm}{3.5mm}2006 May 7& 0.55 & 15.4 &
$6.35 \pm 0.24$ & $8.53\pm 0.36$ & $6.33 \pm 0.24$ &
$ 5.15 \pm 0.22$ \\
\rule[0mm]{0mm}{4mm} 2006 May 12 & 0.91 & 17.7 &
$5.76\pm 0.21$ & $8.14\pm 0.33$ & $5.56 \pm 0.21$ &
$4.86 \pm 0.20$ \\
\rule[0mm]{0mm}{4mm} 2006 May 16 & 0.24 & 15.7 &
$6.15\pm 0.23 $ & $9.58\pm 0.36$ & $6.23\pm 0.22$ &
$4.93 \pm 0.22$ \\
\hline
\end{tabular}
\parbox{5in}{\small $^a$ The orbital phases for \mbox{$\beta\,$Lyr}\ were computed at the midpoint of each
observation using the quadratic ephemeris of
Harmanec \& Scholz (1993). Phases 0.0 and 1.0 correspond to
primary eclipse.}
\end{center}
\end{table*}
Based on the Rosat All-Sky Survey (RASS), there are a few X-ray sources in
the $34\arcmin\ \times\,34\arcmin$ PIN's field of view (FOV);
however, our target \mbox{$\beta\,$Lyr}\ is by far the brightest. The stellar
coronal X-ray sources present in the FOV are expected to be faint in
the HXD's energy range. There is also an active galactic nucleus (AGN)
in the FOV about 15\arcmin\ away from $\beta$~Lyr. This AGN, QSO\,B1847+3330,
is cataloged in the RASS with a count-rate of $0.05$\,cps. To estimate
its potential contribution to the $10-70$~keV energy range, we adopted a
standard AGN power-law spectrum with $\Gamma\,=\,2$ and a low interstellar
absorption column of $N_{\rm H}=5\times 10^{20}\,{\rm cm}^{-2}$. The
predicted count-rate for the {\em Suzaku} PIN is $\approx 0.006$~cps. The
observed HXD-PIN count-rates are listed
in Table~\ref{tab3}. The contribution of QSO\,B1847+3330 is between
10--30\% of the detected flux.
\section{Analysis}
A comparison of our three XIS spectra below 10 keV with the XIS detectors
indicates very little variability with phase (see Fig.~\ref{fig3}).
Based on chi-square model fits to the data, there is no evidence
for statistically significant differences between the three spectra.
For illustration a model fit for the $\varphi=0.24$ pointing appears
as the solid line in all three panels of Fig.~\ref{fig3}; its
good agreement with the data at all phases demonstrates that the soft
X-ray spectrum of $\beta$~Lyr is nearly constant in shape and strength.
Independent fits for each pointing are very similar.
These XIS X-ray spectra are most probably thermal in
nature, since emission lines are detected at 1.35~keV (Mg {\sc xi})
and 1.86~keV (Si {\sc xiii}). Our fits indicate a two-component model
with the majority of the XIS X-ray emission arising from a temperature
of $\approx 7.2$~MK and a hotter but much weaker component of $\gtrsim
20$~MK. Solar abundances were adopted except for nitrogen that was
enhanced by more than 10 times to achieve an adequate fit. The derived
hydrogen column density from the model is $N_H = 6.5 \pm 0.24 \times
10^{20}$ cm$^{-2}$, largely consistent with the interstellar value.
Hard emission in the 10--60~keV band of the PIN is background dominated.
The observed total and the background count rates are given in
Table~\ref{tab3}. The source counts come from the difference of the
total and background values. Based on the total counts, there is a
significant source detection at phase $\varphi=0.55$, a non-detection
for $\varphi=0.91$, and a marginal detection at $\varphi=0.24$.
Emission at such hard energies is exceptionally unusual and unexpected
for a system like $\beta$~Lyr that consists of two early-type stars.
The implied hard X-ray luminosity above 10~keV at $\varphi\approx 0.55$
is about $10^{-3}~L_\odot$. If confirmed, this hard component would be
an important tracer of the flow geometry and shock structure since the
circumstellar material is quite transparent to such high energy X-rays.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{f3.eps}}
\caption{Spectra of $\beta$ Lyr from the XIS1 detector
for each pointing.
Phase within the orbit is indicated in each panel. The data are of
reasonably high signal-to-noise, and it is clear that the spectral distribution
shows little change with phase, being of nearly constant brightness and
spectral shape. The solid line is a model fit to the data of
$\varphi=0.24$ that is replotted in the other phases for comparison.
\label{fig3}}
\end{figure}
\begin{table}
\begin{center}
\caption{PIN Observations of $\beta$ Lyrae \label{tab3}}
\begin{tabular}[t]{cccc
\hline\hline
$\varphi^a$ & Exposure & Count Rate & Background \\
& (ksec) & (cps) & (cps) \\
\hline
\rule[0mm]{0mm}{3.5mm}0.55 & 14.7 & $0.623 \pm 0.007$ & $0.536 \pm 0.002$ \\
\rule[0mm]{0mm}{4mm}0.91 & 17.0 & $0.505 \pm 0.005$ & $0.509 \pm 0.002$ \\
\rule[0mm]{0mm}{4mm}0.24 & 14.1 & $0.573 \pm 0.006$ & $0.564 \pm 0.002$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Discussion}
The results of our {\it Suzaku} study are both revealing and perplexing.
We had expected to see an eclipse of soft X-rays and essentially no
hard emission. Instead, we found that the soft X-rays were nearly
constant, and there is an indication of a quite hard emission component
in the system. The near constancy of the soft spectrum suggests that
the X-ray source must be axially symmetric. This is quite surprising,
since the system is intrinsically non-axisymmetric.
It is worth recalling that the UV lightcurve below Ly$\alpha$ is
notable for lacking any eclipses as well (Kondo \hbox{et~al.\/}\ 1994). A somewhat
analogous source to $\beta$~Lyr is the near edge-on interacting binary
W~Ser. Weiland \hbox{et~al.\/}\ (1995) interpret UV spectra of W~Ser in terms
of an extended boundary layer between the star and the accretion disk.
Thus, it may be possible that the largely steady FUV and soft X-ray
emission could arise from a similar region in the $\beta$~Lyr system.
However, we do not currently favor this explanation since the most
recent modeling of the disk and star components of \mbox{$\beta\,$Lyr}\ (Linnell 2002)
indicate that gainer star is entirely obscured at all phases.
An alternative view is suggested by hydrodynamic simulations. Bisikalo
\hbox{et~al.\/}\ (2000) modeled the mass transfer of $\beta$~Lyr and found that
they could reproduce a disk-like structure but without a well-identified
``hot spot''. Instead, their simulations yielded a complex distribution
of extended shocks. They also were able to generate a bipolar flow
similar to the jet detected by Harmanec \hbox{et~al.\/}\ (1996) and Hoffman \hbox{et~al.\/}\
(1998). However, radiative cooling was not treated self-consistently
in those simulations, but approximated through the equation of state.
It is unclear whether this model can generate gas at sufficiently high
temperatures to produce the observed X-rays. Hydrodynamic modeling
of the $\beta$~Lyr system was also pursued by Nazarenko \& Glazunova
(2003; 2006ab), first in 2D simulations and then in 3D simulations.
These researchers included cooling curves in their simulations and
allowed for a stellar wind by the gainer. Their models also lead to
a disk, a bipolar flow, and an environment permeated with elongated
shocks at different azimuthal orientations about the gainer. ``Warm''
gas temperatures up to about 200,000~K are achieved, but such gas can not
contribute significant emission to the X-ray band. The gas dynamical
simulations are qualitatively promising in terms of predicting shocked
structures that are spatially distributed in radius and azimuth throughout
the disk. In such a model, the hot gas could potentially that hot gas
could be viewable at every phase. However, X-ray emissions were not the
focus of those studies, and it would be useful to have new simulations
that emphasize the hot plasma structures.
We prefer an interpretation in which the X-rays arise from distributed
shocks in the wind of the gainer star. At $kT \approx 0.6$~keV, the
spectral characteristics of the {\it Suzaku} spectra are compatible with
a typical early main sequence B star, and the XIS luminosity $L_X({\rm
XIS}) \approx 6.6\times 10^{30}$ erg s$^{-1}$ is commensurate with the
soft emission expected from an early B0-B1 star (Cohen, Cassinelli, \&
MacFarlane 1997). Although the favored model for the binary geometry may
preclude a direct view of the gainer star, it does permit a reasonably
deep view into its wind, even at secondary eclipse (with the giant star at
front). Moreover, we know from the polarimetric study of Hoffman \hbox{et~al.\/}\
(1998) that there is substantial scattering opacity above and below the
disk plane. The near constancy of the X-rays seen with the XIS may result
from an extended ``halo'' of scattered X-ray light, similar in spirit
to the scattered optical light observed in images of some Herbig-Haro
objects seen edge-on to their disks, such as HH~30 (Burrows \hbox{et~al.\/}\ 1996).
In this picture the soft X-rays ultimately originate in the shocked wind
of the early gainer star. Part of the emission is observed directly
from hot plasma at large radii in the wind, and part is scattered into
our line-of-sight from above and below the disk. Thus the 0.6~keV
temperature of the hot gas would represent an upper limit value owing to
the fact that photoabsorption of X-rays is more severe toward the soft
end of the spectrum. On the other hand, the observed X-ray luminosity
must be a lower limit. \\
We have considered three distinct origins for the observed X-ray emissions.
At this point we favor the gainer wind as the source of the soft X-rays,
but acknowledge that future hydrodynamic modeling may change this picture.
In addition, there seems to be an unusually hard component of emission in
the system, in excess of 10~keV. Such hard emission is more typically
associated with compact objects, and not the winds of hot stars (e.g.,
Bergh\"{o}fer \hbox{et~al.\/}\ 1997). However, the nature of the gainer is still
ambiguous; perhaps the X-ray properties of \mbox{$\beta\,$Lyr}\ could be understood
in relation to a central compact object (Devinney 1971; Wilson 1971).
Although this seems unlikely, the X-ray observations certainly do not
conform to our original expectations, and so at least a reconsideration of
the possibility seems in order. The X-ray pointings by {\it Suzaku} offer
the tantalizing promise of providing new and valuable information about
the \mbox{$\beta\,$Lyr}\ system, but it is clear that a far more rigorous sampling
of the X-ray lightcurve and source spectrum and new models for the
system will be needed to test the suppositions that we have put forth.
We have obtained time with the RXTE satellite to create a more complete
X-ray light curve of $\beta$~Lyr, which should provide new insight into
this complex system in the near future.
\begin{acknowledgements}
We wish to thank the referee Ed Devinney, an anonymous referee, and
Steve Shore for several valuable comments. We are grateful for advice
given by Hugh Hudson in relation to the background of hard X-rays and
by Georg Lamer in relation to X-ray spectra of QSOs. We also thank
Koji Mukai and Nick White for technical assistance regarding Suzaku.
RI received support from NASA grant award NNX06AI04G. LMO acknowledges
support by the Deutsche Forschungsgemeinschaft with grant Fe\,573/3.
WLW was supported in part by NASA contract NNG07EF47P. JLH acknowledges
the support of a NSF Astronomy \& Astrophysics Postdoctoral Fellowship
under award AST-0302123.
\end{acknowledgements}
|
2,877,628,090,159 | arxiv | \section{Introduction}
One of the outstanding unsolved problems in molecular biology is protein
folding\cite{bra} \cite{dag}. The principle through which the amino acid
sequence determines the native structure, as wells as the dynamics of the
process, remain open questions. Generally speaking, there have been two
types of approaches to the problem: bioinformatics\cite{kan} and molecular
dynamics (MD)\cite{MD}.
Bioinformatics is purely data analysis, and does not involve dynamics at
all. It massages the data base of known proteins in different ways, using
very sophisticated computer programs, in order to discover correlations
between sequence and structure. By its very nature, it cannot provide any
physical understanding.
On the other hand, MD solves the Newtonian equations of motion of all the
atoms in the protein on a computer, using appropriate inter-atomic
potentials. To describe the solvent, one inculdes thousands of water
molecules explicitly, treating all the atoms in the water on same footing as
those on the protein chain. Not surprisingly, such an extravagant use of
computing power is so inefficient that one can follow the folding process
only to about a microsecond, whereas the folding of a real protein takes
from one second to ten minutes.
We shall try an approach from the point of view of statistical mechanics\cite%
{hualec}. After all, the protein is a chain molecule immersed in water, and,
like all physical systems, will tend towards thermodynamic equilibrium with
the environment. Our goal is to design a model that embodies physical
principles, and at the same time amenable to computer simulation in
reasonable time.
We treat the protein as a chain performing Brownian motion in water,
regarded as a medium exerting random forces on the chain, with the
concomitant energy dissipation. In addition, we include regular (non-random)
interactions within the chain, as well as between the chain and the medium.
The unfolded chain is assumed to be a random coil described by SAW
(self-avoiding walk), as suggested by Flory\cite{flo} some time ago. That
is, each link in the chain corresponds to successive random walks, in which
the chain is prohibited from revisiting an occupied position. Two types of
interactions are included in our initial formulation:
\begin{itemize}
\item the hydrophobic action due to the medium, which causes the chain to
fold;
\item the hydrogen-bonding within the chain, which leads to helical
structure.
\end{itemize}
\noindent Other interactions can be added later.
We model the protein chain in 3D space, keeping only degrees of freedom
relevant to folding, which we take to be the torsional angles between
successive links. In the computer simulation, we first generate an ensemble
of SAW's, and then choose a subensemble through a Monte Carlo method, which
generates a canonical ensemble with respect to a Hamiltonian that specifies
the interactions. We call the model CSAW\cite{hua0} \cite{hua1} (conditioned
self-avoiding walk). Mathematically speaking, it is based on a Langevin
equation\cite{hualec} describing the Brownian motion of a chain with
interaction. There seems little doubt that such an equation does describe a
protein molecule in water, for It is just Newton's equation with the
environment treated as a stochastic medium.. The model can be implemented
efficiently on a computer, and is flexible enough to be used as a
theoretical laboratory.
Both CSAW and MD are based on Newtonian mechanics, and differ only in the
idealization of the system. In CSAW we replace the thousands of water
molecules used in MD by a stochastic medium --- the heat reservoir of
statistical mechanics. We ignore inessential degrees of freedom, such as
small fluctuations in the lengths and angles of the chemical bonds that link
the protein chain. The advantages of these idealizations are that
\begin{itemize}
\item we avoid squandering computer power on irrelevant calculations;
\item we gain a better physical understanding of the folding process.
\end{itemize}
One often hears a debate on whether the folding process is "thermodynamic"
or "kinetic". There is also an oft cited \textquotedblleft Levinthal
paradox", to the effect that the folding time should be much larger than the
age of the universe, since the protein (presumably) had to search through an
astronomically large number of states before finding the right one. From our
point of view, these are not real issues.
The question of thermal equilibrium merely hangs on whether the protein can
reach equilibrium in realistic time, instead being trapped in some
intermediate state. For any particular protein, simulation of the Langevin
equation will answer the question.
As to Levinthal's "paradox", the protein is blithely unaware of that. It
just follows pathways guided by the Langevin equation.
After a brief review of the basics of protein folding and stochastic
processes, we shall describe the model in more detail, and illustrate its
use through examples involving realistic protein fragments. We will
demonstrate folding pathways, elastic properties, helix formation, and
protein collective modes.
The results indicate that the model has been successful in describing
qualitative features of folding in simple proteins.
\section{Protein basics}
\subsection{ The protein chain}
The protein chain consists of a sequence of units or "residues", which are
amino acids chosen from a pool of 20. This sequence is called the \textit{%
primary structure.} The center of each amino acid is a carbon atom called $%
C_{\alpha }$. Along the protein chain, the $C_{\alpha }$'s are connected by
covalent chemical bonds in the shape of a \textquotedblleft crank" that lies
in one plane. Two cranks join at a $C_{\alpha }$ with a fixed angle between
them, the tetrahedral angle $\theta _{\text{tet}}=-\arccos \left( 1/3\right)
\approx 110^{\circ }$. The amino acids differ from each other only in the
side chains connect to the $C_{\alpha }$'s. There are 20 possible choices
for side chains.
The relative orientation of successive cranks is determined by two torsional
angles $\phi $ and $\psi $, as schematically illustrated in Fig.1. These
torsional angles are the only degrees of freedom relevant to protein
folding, and small oscillations in bond lengths and bond angles can be
ignored. For our purpose, therefore, a protein of $N$ residues has $2(N-1)$
degrees of freedom.\FRAME{ftbpFU}{3.2846in}{1.2808in}{0pt}{\Qcb{Schematic
representation of the protein chain. Centers of residues are carbon atoms
labeled $\protect\alpha $. They are connected by rigid chemical bonds in the
shape of a planar crank. The only degrees of freedom we consider are the
torsional angles $\protect\phi ,\protect\psi $ that specify the relative
orientations of successive cranks. Residues can differ only in the side
chains labeled $R_{i},$ chosen from a pool of twenty. Atoms connected to the
cranks are omitted for clarity.}}{}{schematicchain.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 3.2846in;height 1.2808in;depth
0pt;original-width 6.7101in;original-height 2.5996in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/SchematicChain.jpg';file-properties "XNPEU";}}
\subsection{Secondary and tertiary structures}
At high temperatures, or in an acidic solution, the protein exists in an
unfolded state that can be represented by a random coil\cite{flo}. When the
temperature is lowered, or when the solution becomes aqueous, it folds into
a "native state" of definite shape. Fig.2 shows the native state of
myoglobin with different levels of detail. Local structures, such as
helices, are called \textit{secondary structures}. When these are blurred
over, one sees a skeleton called the \textit{tertiary structure}.\FRAME{%
ftbpFU}{3.525in}{1.657in}{0pt}{\Qcb{Native state of Myoglobin showing
different degrees of detail.}}{}{myoglobin.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 3.525in;height 1.657in;depth
0pt;original-width 7.8205in;original-height 3.6599in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/Myoglobin.jpg';file-properties "XNPEU";}}
Secondary structures are of two main types, the alpha helix and the beta
sheet, as shown in Fig.3. The former is stabilized by hydrogens bonds that
connect \ residues 1 to 4, 2 to 5, \textit{etc}. The beta sheet is a global
mat sewn together by hydrogen bonds.\FRAME{ftbpFU}{3.2707in}{2.4094in}{0pt}{%
\Qcb{Secondary structures. Dotted lines in the alpha helix denote hydrogen
bonds. The beta sheet is composed of \textquotedblleft beta strands" matted
together by hydrogen bonds. two ajacent strands are connect by a
\textquotedblleft beta hairpin".}}{}{alphabeta.jpg}{\special{language
"Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width
3.2707in;height 2.4094in;depth 0pt;original-width 4.2194in;original-height
3.4402in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'figs/AlphaBeta.jpg';file-properties "XNPEU";}}
\subsection{Hydrophobic effect}
The molecules of liquid water form hydrogen bonds with each other, resulting
in a dense fluctuating network, in which bonding partners change on a time
scale of 10$^{-12}$s. A computer simulation of such a network is shown in
Fig.4a\cite{water}. A foreign molecule introduced into water disrupts the
network, unless it can participate in hydrogen bonding. If it can
hydrogen-bond with water, it is said to be \textquotedblleft soluble", or
\textquotedblleft hydrophilic", and will be received by water molecules as
one of their kind. Otherwise it is unwelcome, and said to be
\textquotedblleft insoluble", or "hydrophobic". Protein side chains can be
hydrophilic or hydrophobic.
When immerse in water, the protein chain folds in order to shield the
hydrophobic residues from water. In effect, the water network squeezes the
protein into shape. This is called the "hydrophobic effect". However, a
\textquotedblleft frustration" arises in this process, because the skeleton
is hydrophilic, and likes to be in contact with water, as indicated in
Fig.4b. The frustration is resolved by the formation of secondary
structures, which use up hydrogen bonds internally. The folded chain reverts
to a random coil when the temperature becomes too high, or when the pH of
the solution becomes acidic.\FRAME{ftbpFU}{4.3535in}{1.6717in}{0pt}{\Qcb{(a)
Computer simulation of network of hydrogen bonds in liquid water. (b) The
hydrophobic side chains $R_{1}$ and $R_{2}$ cannot form hydrogen bonds, and
prefer to be shielded from water. However, the atoms $O$ and $H$ on the main
chain need to form hydrogen bonds. A \textquotedblleft frustration" thereby
arises, and is resolved by formation of secondary structures that use up\
hydrogen bonds internally. }}{}{water.jpg}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 4.3535in;height 1.6717in;depth 0pt;original-width
5.6697in;original-height 2.1594in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'figs/water.jpg';file-properties "XNPEU";}}
\subsection{Folding stages}
As depicted schematically in Fig.5, a typical folding process consists of a
very rapid collapse into an intermediate state called the \textquotedblleft
molten globule". The latter takes a relatively long time to undergo fine
adjustments to reach the native state. The collapse time is generally less
than 200 $\mu $s, while the molten globule can last as long as 10 minutes.%
\FRAME{ftbpFU}{4.7106in}{1.8775in}{0pt}{\Qcb{Being squeezed by a water net,
the protein chain rapidly collapses into the molten globule state, which
slowly adjusts itself into the native state.}}{}{foldingstages.jpg}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 4.7106in;height 1.8775in;depth
0pt;original-width 7.8698in;original-height 3.1194in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/FoldingStages.jpg';file-properties "XNPEU";}}
\subsection{Statistical nature of the folding process}
We have to distinguish between protein assembly inside a living cell (%
\textit{in vivo}), and folding in a test tube (\textit{in vitro}). In the
former the process takes place within factory molecules called ribosomes,
and need the assistance of "chaperon" molecules to prevent premature
folding. In the latter, the molecules freely fold or unfold, reversibly,
depending on the pH and the temperature.
We deal only with folding \textit{in vitro}, in which ten of thousands of
protein molecules undergo the folding process independently, and they do not
fold in unison. We are thus dealing with an ensemble of protein molecules,
in which definite fractions exist in various stages of folding at any given
time. The Langevin equation naturally describes the time evolution of such
an ensemble. Behavior of individual molecules fluctuate from the average,
even after the ensemble has reached equilibrium. In macroscopic systems
containing the order of 10$^{23}$ atoms, such fluctuations are unobservably
small. For a protein with no more than a few thousand atoms, however, these
fluctuations are expected to be pronounced.
\section{Stochastic process}
\subsection{Stochastic variable}
A stochastic process is one involving random forces, and is described
through a so-called stochastic variable (or random variable), which does not
have a definite value, but is characterized instead by a probability
distribution of values. Practically everything we deal with in the
macroscopic world involve random variables, from the position of a billiard
ball to the value of a stock.
Einstein pointed out the essence of a stochastic variable in his theory of
Brownian motion. He emphasized that every Brownian step we can observe is
the result of a very large number of smaller random steps, which in turn are
the result of a very number of even smaller steps, and so on, until we reach
the cutoff imposed by atomic structure. This self-similarity leads to a
Gaussian distribution, regardless of the underlying mechanism --- a result
known as the \textit{central limit theorem}\cite{huastat}.
\subsection{Brownian motion}
The simplest stochastic process is the Brownian motion of a single particle
suspended in a medium. Its position $x(t)$ is a stochastic variable
described by the Langevin equation
\begin{equation}
m\ddot{x}=F(t)-\gamma \dot{x}
\end{equation}%
Here, the force exerted by the medium on the particle is split into two
parts: a randomly fluctuating force $F\left( t\right) $ and a friction $%
-\gamma \dot{x}$. The random force is a member of a statistical ensemble
with the properties%
\begin{align}
\left\langle F(t)\right\rangle & =0 \notag \\
\left\langle F(t)F(t^{\prime })\right\rangle & =c_{0}\delta \left(
t-t^{\prime }\right)
\end{align}%
where the brackets $\langle \rangle $ denote ensemble average. The two
forces are not independent, but related through the fluctuation-dissipation
theorem:%
\begin{equation}
\frac{c_{0}}{2\gamma }=k_{B}T
\end{equation}%
where $k_{B}$ is Boltzmann's constant and $T$ is the absolute temperature, a
property of the medium.
The Langevin equation can be solved exactly, and also be simulated by
\textit{random walk}. Both methods lead to diffusion, in which the position
has a Gaussian distribution with variance $\sqrt{2Dt}$, where $t$ is the
time, and $D=c_{0}/\left( 2\gamma ^{2}\right) $ is called the diffusion
constant. An equivalent expression is \textit{Einstein's relation}%
\begin{equation}
D=\frac{k_{B}T}{\gamma }
\end{equation}%
Thus, a random force must generate energy dissipation, and the dissipation
constant $\gamma $ can be deduced from the variance of the distribution of
positions.
\subsection{Monte Carlo}
If a particle undergoes Brownian motion in the presence of a regular
(non-random) external force $G(x)$, we may not be able to solve the Langevin
equation exactly, but we can still simulate it on a computer by \textit{%
conditioned random walk,} as follows\textit{. }We first generate a random
trial step, but accept it only according to the Monte Carlo algorithm. Let $%
E $ be the potential energy corresponding to the external force $G$. Let $%
\Delta E$ be the energy change in the proposed update. The algorithm is as
follows:
\begin{itemize}
\item if $\Delta E\leq 0,$ accept it;
\item if $\Delta E>0,$ accept it with probability $\exp \left( -\Delta
E/k_{B}T\right) .$
\end{itemize}
\noindent The last condition simulates thermal fluctuations, which may drive
the system to a higher energy. After a sufficiently large number of updates,
the sequence of state generated will yield a canonical ensemble with
temperature $T$ . That is, the Monte Carlo procedure tends to minimize not
the energy, but the free energy.
Mathematically speaking, conditioned random walk simulates a generalized
Langevin equation, as indicated in the following:
\begin{equation}
m\ddot{x}=\underset{\text{Treat via random walk}}{\left[ F(t)-\gamma \dot{x}%
\right] }+\underset{\text{Treat via Monte Carlo}}{G(x).}
\end{equation}%
Of course, we could integrate the whole equation as a stochastic
differential equation, as an alternative to Monte Carlo. The equivalence of
these two methods is illustrated by example in the appendix of Ref.\cite{lei}%
.
\section{CSAW}
In protein folding, we are dealing with the Brownian motion of a chain with
interactions. All we need to do, in principle, is to generalize conditioned
random walk to conditioned SAW (self-avoiding walk). The resulting model is
called CSAW (conditioned self-avoiding walk).
We can generate a SAW representing an unfolded protein chain by the \textit{%
pivot algorithm}\cite{li} \cite{ken}, as follows. Choose an initial chain in
3D continuous space, and hold one end of the chain fixed.
\begin{itemize}
\item Choose an arbitrary point on the chain as pivot.
\item Rotate the end portion of the chain rigidly about the pivot (by
changing the torsional angles at the pivot point).
\item If this does not result in any overlap, accept the configuration,
otherwise repeat the procedure.
\end{itemize}
\noindent By this method, we can generate a uniform ergodic ensemble of
SAW's, which simulates a Langevin equation of the form
\begin{equation}
m_{k}\mathbf{\ddot{x}}_{k}=\mathbf{F}_{k}(t)-\gamma _{k}\mathbf{\dot{x}+U}%
_{k}\text{,\ \ \ \ (}k=1,\cdots ,N)\text{\ \ \ \ \ \ \ \ \ \ \ }
\end{equation}%
where the subscripts $k$\ label the residues along the chain. The terms $%
\mathbf{U}_{k}$ denote the regular (non-random) forces that maintain the
rigid bonds between successive residues, and that prohibit the residues from
overlapping one another.
We now add other regular forces $G_{k},$ which include the hydrophobic
interaction and hydrogen-bonding. Treating this force via Monte Carlo
results in CSAW, which simulates a generalized Langevin equation as
indicated in the following:
\begin{equation}
m_{k}\mathbf{\ddot{x}}_{k}=\,\underset{\text{ \ \ Treat via SAW}}{\left(
\mathbf{F}_{k}-\gamma _{k}\mathbf{\dot{x}+U}_{k}\right) }+\underset{\text{%
Treat via Monte-Carlo}}{\mathbf{G}_{k}.}\text{ \ \ (}k=1\cdots N)
\end{equation}%
Now we shall specify the forces $G_{k}$ explicitly.
\section{Implementation of CSAW}
To reiterate, the system under consideration is a sequence of centers
corresponding to $C_{\alpha }$ atoms, connected by planar \textquotedblleft
cranks". The degrees of freedom of the system are the pairs of torsional
angles $\left\{ \phi _{i},\psi _{i}\right\} $ specifying the relative
orientation of two successive cranks. There are $O$ and $H$ atoms attached
to each crank, through rigid bonds lying in the same plane as the crank. The
residues can differ from one another only through the side chains attach to $%
C_{\alpha }$, and there are 20 of them to choose from. As indicated in
Fig.6, the center of the side chain is located at an apex of a tetrahedron
with $C_{\alpha }$ at the center.\FRAME{ftbpFU}{2.5685in}{2.9343in}{0pt}{%
\Qcb{ The side chain is at the apex of a tetrahedron with $C_{\protect\alpha %
}$ at the center.}}{}{chaindetails.jpg}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 2.5685in;height 2.9343in;depth 0pt;original-width
4.9in;original-height 5.5997in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'figs/ChainDetails.jpg';file-properties "XNPEU";}%
}
We can start with a chain of bare cranks, and then add other components one
by one, as desired. We can first represent the side chains by hard spheres,
and put in the atoms in a more elaborate version. In this manner, we can
tinker with different degrees of buildup, and investigate the relative
importance of each element.
For Monte Carlo, we take the energy $E$ to be
\begin{align}
E& =-g_{1}K_{1}-g_{2}K_{2} \label{energy} \\
K_{1}& =\text{Total contact number of all hydrophobic residues} \notag \\
K_{2}& =\text{Number of hydrogen bonds} \notag
\end{align}
The first term in $E$ expresses the hydrophobic effect. The contact number
of a residue is the number of atoms touching its side chain. In the simplest
version, in which we do not explicitly put in the side chain, the contact
number is simply the number of atoms in contact with $C_{\alpha }$, \textit{%
not counting the other }$C_{\alpha }$'s\textit{\ lying next to it along the
chain}. This is illustrated in Fig.7a.
\FRAME{ftbpFU}{3.6754in}{2.437in}{0pt}{\Qcb{(a) The shaded hydrophobic
residue illustrated here has four contact neighbors. The permanent neighbors
along the chain are not counted. (b) Hydrogen bonding occurs between $O$ and
$H$ on the main chain, from different residues. }}{}{interactions.jpg}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 3.6754in;height 2.437in;depth
0pt;original-width 6.5596in;original-height 4.3405in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/interactions.jpg';file-properties "XNPEU";}}
The contact number measures how well a residues is being shielded from the
medium. When two hydrophobic residues are in contact, the total contact
number increases by 2, an this induces an effective attraction between
hydrophobic residues. The unfolded chain corresponds to $g_{1}=0.$
The second term in $E$ describes hydrogen bonding. As illustrated in Fig.7b,
a hydrogen bond is deemed to have formed between $O$ and $H$ $\ $from
different cranks when
\begin{itemize}
\item the distance between $O$ and $H$ is 2.5 A, within given tolerance;
\item The bonds $C=O$ and $N-H$ are antiparallel, within given tolerance.
\end{itemize}
Only the combinations $g_{1}/k_{B}T$ and $g_{2}/k_{B}T$ appear in the Monte
Carlo procedure. They are treated as adjustable parameters.
Note that $E$ only includes the potential energy. We can leave out the
kinetic energy because it contributes only a constant factor to the
configurational probability of the ensemble.
\
\section{Exploratory runs}
It is instructive to run the program with minimal components, as described
in Refs.\cite{hua0} \cite{hua1}. For a chain of 30 residues, the main
findings are the following:
\begin{itemize}
\item Under hydrophobic forces alone, without hydrogen-bonding, the chain
folds into a reproducible shape. This shows that the hydrophobic effect
alone can produce tertiary structure. There is no secondary structure in
this case, and the chain rapidly collapses to the final structure without
passing through an intermediate state..
\item When there is no hydrophobic force and the interaction consists purely
of hydrogen-bonding, the chain rapidly folds into one long alpha helix.
\item When both hydrophobic force and hydrogen bonding are taken into
account, secondary structure emerges. The folding process exhibits two-stage
behavior, with a fast collapse followed by slow \textquotedblleft
annealing", in qualitative agreement with experiments.
\end{itemize}
We now recount some simulations of realistic protein fragments.
\section{Folding pathways and energy landscape}
Chignolin is a synthetic peptide of 10 residues \cite{chignolin}, in the
shape of a \textquotedblleft beta hairpin" -- a turn in a beta sheet as
depicted in Fig.3. Jinzhi Lei \cite{lei0} of Tsinghua University modeled it
in CSAW, with side chains modeled as hard spheres. The native state emerges
after about 70000 trial steps, as shown in Fig.8. The computation took less
than 5 minutes on a work station. In contrast, an MD simulation on the same
work station did not reach the native state in one month's computation. The
run was repeated 100 times independently, to obtain an ensemble of folding
paths.\FRAME{ftbpFU}{4.3846in}{2.6757in}{0pt}{\Qcb{Folding of Chignolin, a
beta hairpin with ten residues.}}{}{chig_folding.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 4.3846in;height 2.6757in;depth
0pt;original-width 7.4097in;original-height 4.51in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/chig_folding.jpg';file-properties "XNPEU";}}
To display the folding pathways, we project them onto a two-dimensional
subspace of the configuration space, chosen as follows. Define a $10\times
10 $ distance matrix $D_{ij}=|\mathbf{R}_{i}-\mathbf{R}_{j}|$, where $%
\mathbf{R}_{i}$ is the vector position of the $i$th $C_{\alpha }$. Let its
eignevalues be $\lambda _{1},\ldots ,\lambda _{10}$ in ascending order.
Through experimentation, we find that it is best to project the pathways
onto the $\lambda _{1}$-$\lambda _{10}$ plane, and we rotate the viewpoint
to obtain the clearest representation. This is achieved by using $\lambda
_{1}$ and $\lambda _{1}+\lambda _{10}$ as axes. Fig.9 shows the evolution of
100 folding paths. We can see that the ensemble of 100 points, identified by
given shading, migrates towards an attractor as time goes on. The energy
landscape is shown below the migration map. \FRAME{ftbpFU}{3.1929in}{4.721in%
}{0pt}{\Qcb{Evolution of 100 folding paths of Chignolin. The ensemble
evolves towards an attractor. Lower panel shows the energy landscape. See
text for explanation of the axes. }}{}{chig_landscape.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 3.1929in;height 4.721in;depth
0pt;original-width 6.5397in;original-height 9.6997in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/chig_landscape.jpg';file-properties "XNPEU";}}
In Fig.10 we show 4 individual paths. They get trapped in various local
pockets, and breakout after long searches for outlets. In this respect, the
paths are similar to Levy flights.\FRAME{ftbpFU}{2.9248in}{3.3849in}{0pt}{%
\Qcb{Invidual pathways in the folding of Chignolin. Starting point are
marked with an open circle, and endpoints are marked 1.}}{}{chig_paths.jpg}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9248in;height 3.3849in;depth
0pt;original-width 6.4403in;original-height 7.4599in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/chig_paths.jpg';file-properties "XNPEU";}}
Finally, in Fig.11, we exhibit the elastic property of the protein chain by
plotting the energy as a function of molecular radius, in a semilog plot.
The behavior is consistent with an exponential force law. The flat portion
in the middle corresponds to the breaking of hydrogen bonds that held the
beta hairpin together.
\FRAME{ftbpFU}{3.2448in}{2.1811in}{0pt}{\Qcb{Elastic property of Chignolin:
semilog plot of potential energy vs. radius, averaged over an ensemble of 50
samples. The flat part corresponding to the breaking of hydrogen bonds. The
general shape of the curve is consistent with an exponential force law.
Energy unit is not calibrated.}}{}{elastic1.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 3.2448in;height 2.1811in;depth
0pt;original-width 4.0603in;original-height 2.7198in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/Elastic1.jpg';file-properties "XNPEU";}}
\section{Nucleation and growth of an alpha helix}
Next we report on Polyalanine (Ala$_{20}$), a protein fragment of 20
identical amino acids alanine, which is hydrophobic\cite{lei}. The native
state is known to be a single alpha helix. We tune $g_{1}/k_{B}T$ and $%
g_{2}/k_{B}T$ \ to maximize helical content.
An ensemble of 100 folding paths was generated. Fig.12 shows the fractions
of unfolded, intermediate, and folded molecules, as functions of time. The
solid curves are fits made according to a specific model, in which the
molecular radius reaches equilibrium first, while the helical content
continues to grow. The helical growth is described by a set of rate
equations, while the relaxation of the radius is akin to that of an elastic
solid. This shows that the tertiary structure was established before the
secondary structure, and their evolutions are governed by different
mechanisms.\FRAME{ftbpFU}{3.3356in}{3.1038in}{0pt}{\Qcb{Fractions of
Polyalanine at various stages of folding, as functions of time. Picuture at
top shows the native state of the protein fragment.}}{}{ala_rates.jpg}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 3.3356in;height 3.1038in;depth
0pt;original-width 3.2897in;original-height 3.0597in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'figs/Ala_rates.jpg';file-properties "XNPEU";}}
Fig.13 shows a contour plot of the ensemble average of helicity, with time
on the horizontal axis, and residue number along the vertical. We can see
that the alpha helix grew from two specific nucleation points.\FRAME{ftbpFU}{%
3.0588in}{2.4725in}{0pt}{\Qcb{Contour map of ensemble average of helicity as
a function of time and residue sequence, in the folding of Polyalanine.
Nuclearion occured near the two positions marked by arrows.}}{}{%
ala_nucleate.jpg}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
3.0588in;height 2.4725in;depth 0pt;original-width 6.89in;original-height
5.5599in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'figs/Ala_nucleate.jpg';file-properties "XNPEU";}}
\section{All-atom model}
Finally we show some preliminary results of Weitao Sun \cite{sun} of
Tsinghua University on the histone 1A7W, which has 68 residues. This is a
test of an all-atom CSAW model, in which atoms on the side chains are
explicitly included. The model also includes the electrostatic interactions
among all atoms. Fig.14 compares the simulated shape of the protein with the
native state. It was found that inclusion of electrostatic interactions
makes a noticeable improvement.
\FRAME{ftbpFU}{2.7207in}{1.4339in}{0pt}{\Qcb{Folding the histone 1A7W (68
residues) with an all-atom CSAW model including electrostatic interactions.}%
}{}{histone.jpg}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.7207in;height 1.4339in;depth 0pt;original-width 4.1597in;original-height
2.1793in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'figs/Histone.jpg';file-properties "XNPEU";}}
\FRAME{ftbpF}{3.4644in}{4.7227in}{0in}{}{}{skw.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 3.4644in;height 4.7227in;depth
0in;original-width 4.5904in;original-height 6.2699in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'figs/Skw.jpg';file-properties
"XNPEU";}}The main purpose of this calculation is to study the evolution of
the dynamical structure function%
\begin{equation*}
S(k,\omega )=\left\langle \left\vert n(\mathbf{k},\omega )\,\right\vert
^{2}\right\rangle
\end{equation*}%
where $n(\mathbf{k},\omega )$ is the space-time Fourier transform of the
particle density, and $\langle \rangle $ denotes ensemble average. In
principle, this function can be experimentally measured via inelastic x-ray
scattering. A peak occurring at particular $k,\omega $ will indicate the
existence of an excitation mode. The integral of $S(k,\omega )$ over $k$
will yield the normal mode spectrum, and the integral over $\omega $ will
yield the static structure factor. Preliminary results are shown in Fig.15.
We see that at update=3000 the collective modes of the final structure have
not yet formed, but they emerge at update=5000. The lower panel of Fig.15
shows details of a sound-wave mode with constant velocity.
\section{Discussion and outlook}
In treating protein folding as a physical process, the CSAW model differs
from MD in two important aspects, namely
\begin{itemize}
\item irrelevant degrees of freedom are ignored;
\item the environment is treated as a stochastic medium.
\end{itemize}
\noindent These, together with simplifying treatment of interactions, enable
the model to produce qualitatively correct results with minimal demands on
computer time.
An important simplification is separating the hydrophobic effect and
hydrogen bonding, as expressed by the separate terms in the potential energy
(\ref{energy}). Since both effects arise physically from hydrogen bonding,
it is not obvious that we can make such a separation. The implicit
assumption is that hydrogen bonding with water involves only the side
chains, while internal hydrogen bonding involves only atoms along the main
chain. This property is supported by statistical data, but should be a
result rather an assumption of the model. We should try to remedy this in an
improved version of the model.
The successful examples discussed here deal either with the alpha helix or
the beta hairpin. Our next goal is to study the formation of a beta sheet.
This is a much more difficult problem, for it involves global instead of
local properties of the protein chain. Not knowing which elements are
crucial for the project, we have made the following enhancements to-date:
\begin{itemize}
\item All-atom side chains can now be installed, with fractional
hydrophobicity.
\item Electrostatic interactions among all atoms can be included.
\item Hard-sphere repulsions between atoms can be replaced by Lennard-Jones
potentials.
\item Hydrogen bonds can switch among qualifying partners, with given
probability.
\end{itemize}
We hope to make progress on this problem.
|
2,877,628,090,160 | arxiv | \section{Introduction}
The versatile technique of positron annihilation makes use of the fact that positrons ($e^+$) are trapped at free
volume-type defects which allows their detection by a specific variation of the positron-electron annihilation characteristics \cite{Hautojaervi79, Brossmann12, Krause-Rehberg99, Puska94}.
Whereas the kinetics of $e^+$ trapping at vacancy-type point defects can be well described by
rate theory (so-called simple trapping model), it is well known that for trapping at extended defects like
grain boundaries, interfaces, voids, clusters, or precipitates, diffusion limitation of the trapping process may be an issue.
Diffusion-limited positron trapping at interfaces and grain boundaries has been quantatively modeled by several groups,
ranging from entirely diffusion-controlled trapping \cite{Brandt72}, diffusion-reaction controlled trapping including detrapping
\cite{Dupasquier93, Wuerschum96, Koegel96, dryzek1999}, up to diffusion-reaction controlled trapping at grain boundaries and competetive transition-limited trapping at point defects in crystals
\cite{Cizek02, Oberdorfer09, dryzek1998, Koegel96}.
Compared to grain boundaries, diffusion-limited $e^+$ trapping at voids and clusters has not been studied in such detail
despite the undoubted relevance of positron annihilation for studying this important class of defects \cite{Nieminen79a, Bentzon90, Eldrup03}.
One approach to deal with diffusion-limited trapping is based on effective diffusion-trapping rates
which then allow an implementation in standard rate theory (e.g., \cite{Bentzon90}).
Diffusion-limited trapping at point-like defects was studied by Dryzek \cite{dryzek1998diffusion}
for the one-dimensional case. A full treatment of $e^+$ trapping and annihilation in voids
in the framework of diffusion-reaction theory was given by Nieminen et al. \cite{Nieminen79}.
This treatment of Nieminen et al. \cite{Nieminen79} is conceptionally analogous to the subsequent work of Dupasquier et al.
\cite{Dupasquier93} for diffusion-limited $e^+$ trapping at grain boundaries, both of which lead to solutions exclusively in terms of infinite series.
Another treatment of the diffusion-reaction problem of $e^+$ trapping at grain boundaries
was given by W\"urschum and Seeger \cite{Wuerschum96} which yields closed-form expressions
for the mean $e^+$ lifetime and the intensity of the annihilation component associated with
the trapped state. This approach is applied in the present work to the diffusion-reaction problem
of $e^+$ trapping and annihilation in spherical extended defects (voids, clusters, precipitates).\footnote{For the sake of
simplicity, representatively for all kinds of spherical extended defects (voids, clusters, or precipitates)
the term voids is used in the following.}
Following our earlier further work on grain boundaries
\cite{Oberdorfer09}, now in addition competitive reaction rate-limiting trapping at point defects is taken into account.
The present treatment yields closed-form expressions of the major $e^+$ annihilation parameters
for this application-relevant case of competitive
$e^{+}$ trapping in voids and point defects.
These closed-form expressions allow deeper insight in the physical details of
$e^+$ annihilation characteristics as well as an assessment of the so far often used approach based on
effective diffusion-trapping rates. Above all, the results can be conveniently applied for
the analysis of experimental data.
In a further part, the model presented here and the previous model on positron trapping at grain boundaries
are merged in order to study precipitates embedded in matrix.
Here, diffusion- and reaction limited trapping is considered for both the trapping
from the matrix into the precipitate$-$matrix interface and for
the trapping from inside the precipitates into the interfaces.
\section{The Model}
The model describes positron ($e^+$) trapping and annihilation in voids in the general case that both the $e^+$ diffusion and the transition reaction has to be taken into account (so called diffusion-reaction controlled trapping process).
In order to cover more complex cases, competitive transition-limited trapping at vacancy-type points defects is also considered
(see Fig. \ref{fig:1}).
This procedure follows our earlier study
where concomitant positron trapping at grain boundaries and at point defects in crystallites has been considered \cite{Oberdorfer09}.
The behavior of the positrons is described by their bulk (free) lifetime $\tau_f$,
by their lifetime ($\tau_t)$ in the voids,
by their lifetime ($\tau_v)$ in the vacancy-type point defects in the lattice (matrix),
and by their bulk diffusivity $D$.
Trapping at the point defects of the matrix is characterized by the specific $e^+$ trapping rate $\sigma_v$ (unit s$^{-1}$), as usual.
The voids are considered as spherical-shaped extended defects (radius $r_0$) with a specific trapping rate $\alpha$ (unit m\,s$^{-1}$) which is related to the surface area of the void. In units of s$^{-1}$ the specific trapping rate of voids reads
\begin{equation}
\label{eq:sigma_t}
\sigma_t= \frac{\alpha 4\pi r_0^2}{\Omega},
\end{equation}
where $\Omega$ denotes the atomic volume.
The temporal and spatial evolution of the density $\rho_l$ of free positrons in the lattice is governed by:
\begin{equation}
\label{differential_eq}
\frac{\partial \rho_l}{\partial t}=
D\nabla^{2}\rho_l-\rho_l\left(\frac{1}{\tau_f}+
\sigma_v C_v\right)
\end{equation}
where $C_v$ denotes the concentration of vacancy-type point defects in the matrix.
The positrons trapped in the voids are described in terms of their density $\rho_t$ obeying the rate equation
\begin{equation}
\label{eq:rho_t}
\frac{\mathrm{d}\rho_t}{\mathrm{d}t}=
\alpha \rho_l(r_{0},t)-\frac{1}{\tau_t}\rho_t.
\end{equation}
The temporal evolution of the number $N_v$ of $e^+$ trapped in the point defects in the lattice is given by
\begin{equation}
\frac{\mathrm{d}N_v}{\mathrm{d}t}=
-\frac{1}{\tau_v} N_v+\sigma_v C_v N_f \, ,
\end{equation}
where the number $N_f$ of positrons in the free state follows from integration of $\rho_l$:
\begin{equation}
N_f=\int \rho_l \mathrm{d}V.
\end{equation}
The continuity of the ${\rm e}^{+}$ flux at the boundary between the lattice and the void surface is expressed by\footnote{Note the negative sign in contrast to the model of $e^+$ trapping at grain boundaries (e.g., \cite{Oberdorfer09}) where the corresponding continuity equation refers to the outer boundary.}
\begin{equation}
D\nabla \rho_l\Big|_{r=r_{0}}-\alpha \rho_l(r_{0},t)=0.
\end{equation}
The outer radius $R$ of the diffusion sphere is related to the void concentration
\begin{equation}
\label{eq:C_t}
C_t = \frac{3 \Omega}{4 \pi R^3} \, .
\end{equation}
The outer boundary condition
\begin{equation}
\frac{\partial \rho_l}{\partial r}\Big|_{r=R}=0
\end{equation}
reflects the vanishing $e^+$ flux through the outer border ($r = R$) of the diffusion sphere.
This boundary condition is the same as applied earlier in a quite
different diffusion-reaction model of ortho-para conversion of positronium at reaction cerntres \cite{Wuerschum95}.
As initial condition we adopt the picture that at $t=0$ all thermalized positrons are in the free state and homogeneously distributed in the lattice, i.e., initial density $\rho_l = \rho_l (0)$, $\rho_t(0)=0$, $N_v (0) = 0$.
Under this initial condition the solution of Eq. (\ref{differential_eq}) exhibits spherical symmetry.
Up to this point the above formulated diffusion-reaction problem is identical to that of Nieminen et al. \cite{Nieminen79}
apart from the additional rate-limited trapping at vacancy-type point defects which is considered here.
However, compared to \cite{Nieminen79}, in the following part of the present work the time dependence is handled by means of Laplace transformation which will lead to the more convenient
closed-form solutions. Applying the Laplace transformation
\begin{eqnarray}
\nonumber
\tilde{\rho}_{l,t}(p)=\int\limits _{0}^{\infty}\exp(-pt)\rho_{l,t}(t)\mathrm{d}t,
\end{eqnarray}
\begin{equation}
\tilde{N}_{v,f}(p)=\intop_{0}^{\infty}\exp(-pt)N_{v,f}(t)\mathrm{d}t
\end{equation}
leads to the basic equations
\begin{equation}
\label{differential_eq_r}
\frac{\mathrm{d^{2}}\tilde{\rho}_l}{\mathrm{d}r^{2}}+\frac{2}{r}
\frac{\mathrm{d}\tilde{\rho}_l}{\mathrm{d}r}-\gamma^{2}\tilde{\rho}_l
=-\frac{\rho_l(0)}{D}
\end{equation}
with
\begin{eqnarray}
\label{eq:gamma}
\gamma^{2}=\gamma^{2}(p)=\frac{\tau_f^{-1}+\sigma_v C_v+p}{D} \, ,
\end{eqnarray}
and
\begin{equation}
\label{tilde_rho_t}
\tilde{\rho}_t=
\frac{\alpha \tilde{\rho}_l(r_{0},p)}{
\tau_{t}^{-1}+p} \: ,
\end{equation}
\begin{equation}
\label{N_v}
\tilde{N}_v=
\frac{\sigma_v C_v}{\tau_v^{-1}+p} \times \int\limits^{R}_{r_0} 4 \pi r^2 \tilde{\rho_l}(r,p) {\rm d} r\: ,
\end{equation}
with the boundary conditions
\begin{equation}
\label{boundary_condition1}
D\,\frac{\mathrm{d}\tilde{\rho}_l}{\mathrm{d}r}\Bigg|_{r=r_{0}}-
\alpha \tilde{\rho}_l(r_{0},p)=0
\end{equation}
and
\begin{equation}
\label{boundary_condition2}
\frac{\mathrm{d}\tilde{\rho}_l}{\mathrm{d}r}\Bigg|_{r=R}=0 \, .
\end{equation}
The solution of the differential equation (\ref{differential_eq_r}) satisfying equations
[Eq. (\ref{boundary_condition1})] and [Eq. (\ref{boundary_condition2})] can be written as
\begin{equation}
\label{solution_rho_tilde}
\tilde{\rho}_l(r,p)=A\, i_{0}^{(1)}(\gamma r)+ B\, i_{0}^{(2)}(\gamma r) +
\frac{\rho_l(0)}{\tau_f^{-1}+\sigma_v C_v + p}
\end{equation}
with
\begin{eqnarray}
\nonumber
A:=& \displaystyle{\alpha~\frac{\rho_l(0)}{\tau_f^{-1}+\sigma_v C_v+p}~\times~\frac{i_{1}^{(2)}(\gamma R)}{-D \gamma F_1 + \alpha F_2}} \, , \\
\label{AB}
B:= & \displaystyle{\alpha~\frac{\rho_l(0)}{\tau_f^{-1}+\sigma_v C_v+p}~\times~\frac{i_{1}^{(1)}(\gamma R)}{D \gamma F_1 - \alpha F_2}}
\end{eqnarray}
and
\begin{eqnarray}
\nonumber
F1= i_1^{(2)}(\gamma r_0) i_1^{(1)}(\gamma R) - i_1^{(1)}(\gamma r_0) i_1^{(2)}(\gamma R) \, , \\
F2= i_0^{(2)}(\gamma r_0) i_1^{(1)}(\gamma R) - i_0^{(1)}(\gamma r_0) i_1^{(2)}(\gamma R) \, .
\end{eqnarray}
$i_{n}^{(1)}$ and $i_{n}^{(2)}$ ($n = 0,1$) denote the modified spherical Bessel functions of order $n$
\cite{olver2010nist}
\begin{eqnarray}
\nonumber
i_{n}^{(1)}(z):=(\frac{\pi}{2z})^{1/2}I_{n+1/2}(z), \,
\end{eqnarray}
\begin{equation}
i_{0}^{(1)}=\frac{\sinh z}{z},\quad
i_{1}^{(1)}=\frac{\cosh z}{z}-\frac{\sinh z}{z^2}
\end{equation}
\begin{eqnarray}
\nonumber
i_{n}^{(2)}(z):=(\frac{\pi}{2z})^{1/2}I_{-n-1/2}(z), \,
\end{eqnarray}
\begin{equation}
i_{0}^{(2)}=\frac{\cosh z}{z},\quad
i_{1}^{(2)}=\frac{\sinh z}{z}-\frac{\cosh z}{z^2}
\end{equation}
where $I_{\pm n\pm 1/2}(z)$ represents the Bessel function.
Basis for analyzing positron annihilation experiments is the total probability $n(t)$ that a
$e^+$ implanted at $t=0$ has not yet been annihilated at time $t$.
Here $n(t)$ is given by the number density of $e^+$ per lattice sphere at time
$t$:
\begin{equation}
n(t)= \frac{1}{\frac{4}{3}\pi (R^3 - r_{0}^{3})\rho_l(0)} \times
\left\{
\int\limits _{r_{0}}^{R}4\pi r^{2}\rho_l(r,t)\mathrm{d}r
+4\pi r_{0}^{2}\rho_t (t) + N_v (t)
\right\} \:.
\end{equation}
The Laplace transform of $n(t)$ can be calculated taking into account the solution of $\tilde{N}_v$ [Eq. (\ref{N_v})]
and the solution of the differential equation (\ref{solution_rho_tilde}) which yields
\begin{equation}
\tilde{n}(p)=\frac{1}{\frac{4}{3}\pi (R^3-r_{0}^{3})\rho_l(0)}
\times
\left\{
\left(1+\frac{\sigma_v C_v}{\tau_v^{-1}+p}\right)
\int\limits^{R}_{r_0} 4\pi r^{2} \tilde{\rho}_l(r,p) {\rm d} r
+ 4\pi r^{2}_{0} \tilde{\rho}_t(p)
\right\} \:.
\end{equation}
Solving the integral after substituting $\tilde{\rho_t}(p)$ by Eq. (\ref{tilde_rho_t}),
insertion of $A$ and $B$ [Eq. (\ref{AB})],
yields after some algebra
\begin{equation}
\label{Laplace_n}
\tilde{n}(p)=\frac{1}{t_{fc}^2 t_{v} t_{t}} \Biggl\{t_{vc} t_{fc} t_{t} + \frac{K (t_{fc} t_{v} - t_{vc} t_{t})
\Bigl(\gamma \hat{R} - \tanh(\gamma \hat{R}) [1- \gamma^2 r_0 R]\Bigr)}
{\gamma \hat{R}- \tanh(\gamma \hat{R}) [1-\gamma^2 r_0 R] + \frac{\alpha r_0}{D}[\gamma R - \tanh(\gamma \hat{R})]} \Biggr\}
\end{equation}
with
\begin{equation}
\label{eq:K}
K=\frac{3\alpha r_0^2}{R^3-r_0^3} \, ,
\end{equation}
\begin{equation}
\label{eq:R}
\hat{R} = R-r_0 \, ,
\end{equation}
and the abbreviations
\begin{eqnarray}
\nonumber
t_{t}=\tau_t^{-1}+p \, ; & t_{v}=\tau_v^{-1}+p \,; \\
t_{vc}=\tau_v^{-1}+\sigma C+p \, ; & t_{fc}=\tau_f^{-1}+\sigma C+p \, .
\end{eqnarray}
The Laplace transform $\tilde{n}(p)$ [Eq. (\ref{Laplace_n})] represents the
solution of the present diffusion and trapping model from which
both the mean positron lifetime and the positron lifetime spectrum
can be deduced.
The mean positron lifetime $\overline{\tau}$ is obtained by
taking the Laplace transform at $p = 0$:
\begin{equation}
\label{tauq_n_tilde}
\overline{\tau} = \tilde{n}(p=0) = \int\limits^{\infty}_{0} n(t)
dt \, .
\end{equation}
The positron lifetime spectrum follows from $\tilde{n}(p)$ by means of Laplace inversion. The single poles $p = - \lambda_i$
of $\tilde{n}(p)$ in the complex $p$ plane define the decay rates $\lambda_i (i=0,1,2,\dots)$ of the positron lifetime spectrum:
\begin{equation}
\label{spectrum}
n(t) = \sum_{i=0}^{\infty}I_i \exp (-\lambda_i t) \, ,
\end{equation}
where $I_i$ denote the relative intensities.
\section{Analysis}
At first, we consider the most important case that $e^+$ trapping exclusively occurs at voids, i.e.,
we omit $e^+$ trapping at point defects in the lattice ($C_v=0$).
For this case, we present the solution of the general diffusion-reaction theory (Sect.~\ref{sec:general}) and compare
it with the limiting cases of entirely reaction-controlled trapping (Sect.~\ref{sec:rate_limit})
and entirely diffusion-controlled trapping (Sect.~\ref{sec:diffusion_limit}). Finally, the case of competitive
reaction-controlled trapping at lattice defects is considered (Sect.~\ref{sec:vacancies})
and an extension to larger precipitates is presented for describing precipitate$-$matrix composite structures
(Sect.~\ref{sec:extended}).
\subsection{\label{sec:general}
General case with trapping at voids, exclusively ($C_v=0$)}
For negligible trapping at vacancies within the lattice ($C_v=0$),
the diffusion-reaction model according to Eq.~(\ref{Laplace_n}) yields
for positron trapping in voids as the single type of trap:
\begin{equation}
\label{eq:n}
\tilde{n}(p) = \frac{1}{\tau_f^{-1}+p} \Biggl\{ 1+\frac{K(\tau_f^{-1}-\tau_t^{-1})}{(\tau_t^{-1}+p)(\tau_f^{-1}+p)} \times
\frac{\gamma \hat{R}-\tanh(\gamma \hat{R}) [1-\gamma^2 r_0 R]}{\gamma \hat{R}-\tanh(\gamma \hat{R}) [1-\gamma^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma R- \tanh(\gamma \hat{R})]}
\Biggr \}
\end{equation}
and, hence, for the mean positron lifetime
\begin{equation}
\label{eq:tauq}
\overline{\tau}=\tilde{n}(0)= \tau_f \Biggl\{1+ K (\tau_t-\tau_f) \times
\frac{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]}{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma_0 R- \tanh(\gamma_0 \hat{R})]}
\Biggr \} \, .
\end{equation}
The pole of Eq.~(\ref{eq:n}) for $p = - \tau_t^{-1}$ corresponds to the positron lifetime component $\tau_t$
of the void-trapped state for which the following intensity is obtained:
\begin{equation}
\label{eq:I_t}
I_t= \frac{K}{\tau_f^{-1}-\tau_t^{-1}}\times
\frac{\gamma_t \hat{R}-\tanh(\gamma_t \hat{R}) [1-\gamma_t^2 r_0 R]}{\gamma_t \hat{R}-\tanh(\gamma_t \hat{R}) [1-\gamma_t^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma_t R- \tanh(\gamma_t \hat{R})]} \,.
\end{equation}
In equations (\ref{eq:n}), (\ref{eq:tauq}), (\ref{eq:I_t}):
\begin{equation}
\label{eq:gamma_0_t}
\gamma^2=\frac{\tau _f^{-1}+p}{D}; \,
\gamma_0^2=\frac{\tau _f^{-1}}{D}; \,
\gamma_{t}^2=\frac{\tau _f^{-1}-\tau_{t}^{-1}}{D} \, .
\end{equation}
In addition to the annihilation component $\tau_t^{-1}$ of the void-trapped state,
$\tilde{n}(p)$ [Eq.~(\ref{eq:n})] comprises a sequence of first-order poles $p = - \lambda_{0,j}$ for
$\lambda_{0,j} > \tau_f^{-1}$. These components $\lambda_{0,j}$,
which define the fast decay rates ($\lambda_{0,j} > \tau_f^{-1}$) of the $e^+$ lifetime spectrum,
are given by the solutions of the transcendental equation
\begin{equation}
\tan(\gamma^{\star} \hat{R}) = \frac{\gamma^{\star} (\alpha r_0 R + D \hat{R})}{D(1+\gamma^{\star 2} r_0 R) + \alpha r_0}
\label{eq:transcendent}
\end{equation}
with
\begin{equation}
\gamma^{\star 2} = \frac{\lambda_{0,j}-\tau_f^{-1}}{D}
\label{eq:gamma'}
\end{equation}
in agreement with the aforementioned earlier work of Nieminen et al. \cite{Nieminen79}.\footnote{Eq. (\ref{eq:transcendent})
is identical to the corresponding eq. (15) in the work of Nieminen et al. when $\nu$ in \cite{Nieminen79}
is identified with $4\pi r_0^2 \alpha$.}$^,$\footnote{We note that
the same problem was treated in the framework of a more general theoretical approach by K\"ogel \cite{Koegel96}. The
quoted specific function in dependence of $\gamma \hat{R}$ [eq. (75) in \cite{Koegel96}], which
determines the mean $e^+$ lifetime and the intensity of the trap component,
however, is not readily applicable.
}
As usual for this kind of diffusion-reaction problem (see, e.g. \cite{Oberdorfer09}),
the intensities of these decay rates rapidly decrease.
Experimentally only a single fast decay rate can be resolved in addition to the decay rate $\tau_t^{-1}$ of the trapped state. An experimental two-component $e^+$ lifetime spectrum is practically entirely defined by
$\overline{\tau}$ [Eq.~(\ref{eq:tauq})] and by $\tau_t$ with the corresponding intensity $I_t$ [Eq.~(\ref{eq:I_t})].
The appearance of a second-order pole in Eq.~(\ref{eq:n}) at $p = - \tau_f^{-1}$ (i.e., $\gamma = 0$) is spurious. Closer inspection by applying Taylor expansion shows that the intensity associated with this pole cancels.
Following the consideration of Dryzek \cite{dryzek2002}, in analogy to the mean $e^+$ lifetime [Eq. (\ref{eq:tauq})]
a respective relation for the mean line shape parameter $\overline{S}$ of
Doppler broadening of the positron-electron annihilation
can be given:
\begin{equation}
\label{eq:Doppler}
\overline{S}= S_f \Biggl\{1+ K (S_t-S_f) \times
\frac{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]}{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma_0 R- \tanh(\gamma_0 \hat{R})]}
\Biggr \} \, ,
\end{equation}
where $S_f$ and $S_t$ denote the line shape parameters of the free and trapped state, respectively.
For the sake of completeness, we quote $\tilde{n}(p)$ without derivation for the case that
at time zero positrons are homogeneously distributed in the voids and the lattice,
i.e., for the initial condition $\rho_t (0) = r_0 \rho_l (0)/3$:
\begin{eqnarray}
\label{eq:n_homogen}
\nonumber
\tilde{n}(p) = \displaystyle{\frac{1}{\tau_f^{-1}+p} \Biggl\{ 1+ \frac{r_0^3}{R^3} \times
\frac{\tau_f^{-1}-\tau_t^{-1}}{\tau_t^{-1} + p} +}
\frac{3 \alpha r_0^2}{R^3} \times \frac{\tau_f^{-1}-\tau_t^{-1}}{(\tau_t^{-1}+p)(\tau_f^{-1}+p)}
\times \\
\displaystyle{\frac{\gamma \hat{R}-\tanh(\gamma \hat{R}) [1-\gamma^2 r_0 R]}{\gamma \hat{R}-\tanh(\gamma \hat{R}) [1-\gamma^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma R- \tanh(\gamma \hat{R})]}
\Biggr \}} \, .
\end{eqnarray}
Eq.~(\ref{eq:n_homogen}) includes in the limiting case of negligible trapping ($\alpha = 0$)
as mean $e^+$ lifetime $\overline{\tau} = \tilde{n}(0) =
[(R^3-r_0^3) \tau_f + r_0^3 \tau_t]/R^3$ the expected volume-averaged mean value of $\tau_f$ and $\tau_t$.
\subsection{\label{sec:rate_limit}
Limiting case of entirely reaction limited trapping ($C_v=0$)}
If the e$^+$ diffusivity is high ($\gamma \hat{R} \ll 1$), the hyperbolic tangent
in Eq.~(\ref{eq:n}) can be expanded.
Expansion up to the third order
\begin{equation}
\tanh(z)\approx z-\frac{z^3}{3}
\end{equation}
yields the mean $e^+$ lifetime
\begin{equation}
\label{eq:tauq_rate}
\overline{\tau}=\displaystyle{
\tau_f \frac{1+ K \tau_t}{1+K \tau_f}}
\end{equation}
and for the $e^+$ lifetime component $\tau_t$ the intensity
\begin{equation}
\label{eq:I_t_rate}
I_t=\displaystyle{
\frac{K }{\tau_f^{-1}+ K -\tau_t^{-1}}
}
\end{equation}
with $K$ according to Eq.~(\ref{eq:K}).
Equations (\ref{eq:tauq_rate}) and (\ref{eq:I_t_rate}) are the well-known solutions of the simple trapping model
when we identify $K$ for vanishing defect volume with the trapping rate $\sigma_t C_t$
[equations (\ref{eq:sigma_t}) and (\ref{eq:C_t})].
Note that the standard trapping model does not take into account the finite defect volume (here $4 \pi r_0^3/3$)
and, therefore, does not contain the subtrahend $r_0^3$ as in Eq. (\ref{eq:K}). With this subtrahend,
equations (\ref{eq:tauq_rate}) and (\ref{eq:I_t_rate}) correctly contain
the exact values $\overline{\tau} = \tau_t$ and $I_t = 1$
as limiting case for $R=r_0$.
\subsection{\label{sec:diffusion_limit}
Limiting case of entirely diffusion limited trapping ($C_v=0$)}
The present solution includes in the limiting special case $\alpha \rightarrow \infty$ the relationships
for an entirely diffusion-limited trapping, i.e., for Smoluchowski-type boundary condition
\begin{equation}
\label{eq:Smoluchowski}
\rho_l (r_0, t) = 0 \, .
\end{equation}
In this limit one obtains from the Laplace transform [Eq. (\ref{eq:n})]
the mean $e^+$ lifetime
\begin{equation}
\label{eq:tauq_diffusion}
\overline{\tau}=\tau_f \Biggl \{ 1+ \frac{3r_{\rm 0}D}{R^3-r_{\rm 0}^3}(\tau_t-\tau_f)
\frac{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R})[1-\gamma_0^2 r_0 R]}{\gamma_0 R-\tanh(\gamma_0 \hat{R})} \Biggr\}
\end{equation}
and for the trap component $\tau_t$ the intensity
\begin{equation}
\label{eq:I_t_diffusion}
I_t=\frac{3 r_0 D}{R^3-r_0^3} \times \frac{1}{\tau_f^{-1}-\tau_t^{-1}} \times \frac{\gamma_t \hat{R}-\tanh(\gamma_t \hat{R})[1-\gamma_t^{2}r_0 R]}{\gamma_t R-\tanh(\gamma_t \hat{R})}
\end{equation}
with $\gamma_0$, $\gamma_t$ according to eq. (\ref{eq:gamma_0_t}).
\subsection{\label{sec:vacancies}
General case with voids {\em and} lattice vacancies}
The positron annihilation characteristics of diffusion-reaction
controlled trapping at voids and concomitant transition-limited
trapping at point defects in the lattice is given by Eq.
(\ref{Laplace_n}) in combination with Eq. (\ref{tauq_n_tilde}) and
Eq. (\ref{spectrum}).
The mean positron lifetime [Eq. \ref{tauq_n_tilde}], obtained from
Eq. (\ref{Laplace_n}) for $p=0$, reads in the general case:
\begin{eqnarray}
\label{eq:tau_q_general}
\nonumber
\overline{\tau}= \displaystyle{
\frac{1}{(\tau_f^{-1}+\sigma_v C_v)^{2}} }
\Biggl\{(\tau_f^{-1}+\sigma_v C_v) (\tau_v^{-1}+\sigma_v C_v) \tau_v+ \\
\displaystyle{
\frac{K \Bigl((\tau_f^{-1}+\sigma_v C_v) \tau_{t} - (\tau_v^{-1}+\sigma_v C_v) \tau_{v} \Bigr)\Bigl(\gamma_0 \hat{R} - \tanh(\gamma_0 \hat{R})[1- \gamma^2 r_0 R]\Bigr)}
{\gamma_0 \hat{R}- \tanh(\gamma_0 \hat{R})[1-\gamma_0^2 r_0 R] + \frac{\alpha r_0}{D}[\gamma_0 R - \tanh(\gamma_0 \hat{R})]}\Biggr\}
}
\end{eqnarray}
with
\begin{equation}
\label{eq:gamma_v}
\gamma_0^2=\frac{\tau_f^{-1}+\sigma_v C_v}{D} \, .
\end{equation}
In addition to the pole $p = - \tau_t^{-1}$ which characterizes the void trapped state,
$\tilde{n}(p)$ [Eq. (\ref{Laplace_n})] contains the further defect-related pole
$p = - \tau_v^{-1}$ for the vacancy-type defect in the lattice.
From the residues of $\tilde{n}(p)$ [Eq. (\ref{Laplace_n})],
the corresponding relative intensities
\begin{equation}
\label{eq:I_t_general}
I_t=\frac{K}{\tau_f^{-1}+\sigma_v C_v-\tau_t^{-1}}\times\frac{\gamma_t \hat{R} - \tanh(\gamma_t \hat{R})[1- \gamma_t^2 r_0 R]}{\gamma_t \hat{R}- \tanh(\gamma_t \hat{R})[1-\gamma_t^2 r_0 R] +
\frac{\alpha r_0}{D}[\gamma_t R - \tanh(\gamma_t \hat{R})]}
\end{equation}
and
\begin{eqnarray}
\nonumber
I_v=
\frac{\sigma_v C_v}{\tau_f^{-1}+\sigma_v C_v-\tau_v^{-1}} \Biggl\{1- \frac{K}{\tau_f^{-1}+\sigma_v C_v-\tau_v^{-1}} \times \\
\label{eq:I_v}
\frac{\gamma_v \hat{R} - \tanh(\gamma_v \hat{R})[1- \gamma_v^2 r_0 R]}{\gamma_v \hat{R}- \tanh(\gamma_v \hat{R})[1-\gamma_v^2 r_0 R] + \frac{\alpha r_0}{D}(\gamma_v R - \tanh(\gamma_v \hat{R})]}\Biggr\}
\end{eqnarray}
are deduced with
\begin{equation}
\label{eq:gamma_t_v}
\gamma_{t,v}^2=\frac{\tau _f^{-1}+\sigma_v C_v-\tau_{t,v}^{-1}}{D} \, .
\end{equation}
\subsection{\label{sec:extended}
Extended model for larger preciptates with $e^+$-trapping from both sides of precipitate$-$matrix interface}
The model presented above describes $e^+$ annihilation from a trapped state ($\tau_t$)
in spherical defects. Particularly, for larger precipitate sizes a situation may prevail where
$e^+$ annihilation inside the precipitates occurs from a free state with a characteristic $e^+$ lifetime $\tau_p$
and where also from this free precipitate state positrons may get trapped into the spherical interfacial shell between the precipitate and the surrounding matrix. This means that the precipitates are characterized by two compoments, one corresponding to the precipitate volume ($\tau_p$) and one corresponding to the trapped state in the matrix$-$precipitate interface ($\tau_t$).
The present model can be extended in a straight forward manner to this case under the reasonable assumption that the $e^+$ trapping from inside the precipitates is entirely reaction controlled. This is pretty well fulfilled as long as
the precipitate diameter is remarkably lower than the $e^+$ diffusion length in the precipitate.\footnote{A further model extension avoiding this
constraint will be outlined below.}
In this case the extension can be described by an additional rate equation for the temporal evolution of the number $N_p$ of $e^+$
inside the precipitates
\begin{equation}
\label{eq:N_p}
\frac{\mathrm{d}N_p}{\mathrm{d}t}=
-\Bigl( \frac{1}{\tau_p} + \frac{3\beta}{r_0} \Bigr) N_p \, ,
\end{equation}
where $\beta$ denotes the specific trapping rate (in units of m/s) at the spherical interfacial shell.
This trapping from inside the precipitates, which occurs in addition to the diffusion- and reaction-limited trapping
into the interfacial shell from the surrounding matrix, has to be taken into account in the rate equation for
$\rho_t$ (Equation \ref{eq:rho_t})
by the additional summand $\beta \rho_p(t)$ with the number density
$\rho_p = 3 N_p / (4 \pi r_0^3)$ of $e^+$ in the precipitate.
Assuming a homogeneous distribution of $e^+$ at time zero in the matrix and the
precipitate ($\rho_l (0) = \rho_p (0)$) without $e^+$ in the trapped state ($\rho_t (0) = 0$) for $t=0$,
one obtains with the Laplace transform of eq.~(\ref{eq:N_p})
\begin{equation}
\label{eq:Laplace_N_p}
\tilde{N_p} = \displaystyle{\frac{N_p(0)}{\tau_p^{-1} + \frac{3 \beta}{r_0} + p}}
\end{equation}
the additional summand
\begin{equation}
\label{eq:n_extension}
\Bigl( \frac{r_0}{R} \Bigr)^3
\displaystyle{\Bigl(\frac{\frac{3 \beta}{r_0}}{\tau_t^{-1} +p} +1 \Bigr)
\frac{1}{\tau_p^{-1} + \frac{3 \beta}{r_0} + p}}
\end{equation}
in eq. (\ref{Laplace_n}) of $\tilde{n}(p)$.
Moreover, in the bracket of eq. (\ref{Laplace_n}) the first summand is extended by the weighting factor
$[ 1 - (r_0/R)^3]$ and the trapping rate $K$ (Equation \ref{eq:K}) in the second summand is replaced by
$3 \alpha r_0^2/R^3$.
For \underline{$C_v = 0$} this leads to the mean $e^+$ lifetime
\begin{eqnarray}
\label{eq:tauq_extended}
&\overline{\tau}= \tau_f \Biggl\{\displaystyle{ \Bigl[1 - \Bigl(\frac{r_0}{R} \Bigr)^3 \Bigr]} +
\nonumber \\
& \displaystyle{\frac{3 \alpha r_0^2}{R^3}
\times (\tau_t-\tau_f) \times
\frac{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]}{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma_0 R- \tanh(\gamma_0 \hat{R})]} \Biggr \} } + \nonumber \\
& \displaystyle{\Bigl(\frac{r_0}{R} \Bigr)^3 \times \tau_t \times \frac{\tau_t^{-1} + \frac{3 \beta}{r_0}}{\tau_p^{-1} + \frac{3 \beta}{r_0}}}
\, ,
\end{eqnarray}
as compared to eq. (\ref{eq:tauq}).
Eq.~(\ref{eq:tauq_extended}) includes in the limiting case of negligible trapping ($\alpha = \beta = 0$)
as mean $e^+$ lifetime $\overline{\tau} =
[(R^3-r_0^3) \tau_f + r_0^3 \tau_p]/R^3$ the expected volume-averaged mean value of $\tau_f$ and $\tau_p$.
The additional pole for $p = - (\tau_p^{-1} + 3 \beta /r_0)$ of $\tilde{n}(p)$ yields the intensity
of the $e^+$ lifetime component $\tau_p$ in the precipitate:
\begin{equation}
I_p = \Bigl(\frac{r_0}{R} \Bigr)^3 \times
\Biggl(1 - \displaystyle{\frac{\frac{3 \beta}{r_0}}{\tau_p^{-1} + \frac{3 \beta}{r_0} - \tau_t^{-1}} \Biggr)}
\, .
\label{eq:I_p}
\end{equation}
Apart from the weighting prefactor ($(r_0/R)^3$), $I_p$ corresponds to the solution of the simple trapping model.\footnote{
Note that $I_p$ characterizes the free state in the precipitate.} Without trapping ($\beta = 0$), $I_p$ simply takes the form of the weighting prefactor $(r_0/R)^3$.
Since $e^+$ trapping into the precipitate$-$matrix interface occurs both from inside the precipitate and from the surrounding matrix, the intensity
of the trap component $\tau_t$ is given by the sum
\begin{equation}
I_t = I_t^{precip} + I_t^{matrix} \text{ with }
I_t^{precip} = \Bigl(\frac{r_0}{R} \Bigr)^3 \times
\displaystyle{\frac{\frac{3 \beta}{r_0}}{\tau_p^{-1} + \frac{3 \beta}{r_0} - \tau_t^{-1}} }
\, ,
\label{eq:I_t_tot}
\end{equation}
where $I_t^{matrix}$ corresponds to the intensity $I_t$ according to eq.~(\ref{eq:I_t}) with
$K$ replaced by $3 \alpha r_0^2/R^3$.\footnote{The identical equation for $I_t$ (equation~\ref{eq:I_t_tot}) follows
from the root $p = - \tau_t$ of the Laplace transform $\tilde{n}(p)$ in which the above mentioned extensions
of eq. (\ref{Laplace_n}) are taken into consideration.}
We note that the two $e^+$ trapping processes into the precipitate$-$matrix interface, namely that from inside the precipitate and that from the surrounding matrix, are completely decoupled. The trapping process from inside the precipitate can, therefore, be treated independently.
This also means that the process has not to be restricted to the case of entirely reaction-controlled trapping as given above, but
that $e^+$ trapping at the precipitate$-$matrix interface from inside the spherical precipitates can also be treated
in the framework of diffusion-reaction theory. Hence, the available solutions for
diffusion- and reaction-limited trapping at grain boundaries (GBs) of spherical crystallites \cite{Wuerschum96, Oberdorfer09} can
be directly applied. For this purpose the solutions for the GB-model have simply to be weighted by the factor $(r_0/R)^3$ which denotes the volume fraction of the precipitates.\footnote{Given the above initial condition $\rho_l (0) = \rho_p (0)$ and $\rho_t (0) = 0$.}
For instance, for the mean $e^+$ lifetime, the last summand in Eq.~(\ref{eq:tauq_extended}), i.e., the rate-equation solution has
to be replaced by that calculated for diffusion- and reaction-limited trapping at GBs \cite{Wuerschum96, Oberdorfer09},
yielding:
\begin{eqnarray}
\label{eq:tauq_double_extended}
&\overline{\tau}= \tau_f \Biggl\{\displaystyle{ \Bigl[1 - \Bigl(\frac{r_0}{R} \Bigr)^3 \Bigr]} +
\nonumber \\
& \displaystyle{\frac{3 \alpha r_0^2}{R^3}
\times (\tau_t-\tau_f) \times
\frac{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]}{\gamma_0 \hat{R}-\tanh(\gamma_0 \hat{R}) [1-\gamma_0^2 r_0 R]+ \frac{\alpha r_0}{D}[\gamma_0 R- \tanh(\gamma_0 \hat{R})]} \Biggr \} } + \nonumber \\
& \displaystyle{\Bigl(\frac{r_0}{R} \Bigr)^3 \Biggl\{
\tau_p + (\tau_t-\tau_p) \times
\frac{3 \beta L(\gamma'_0 r_0)}{r_0 \gamma'_0
\Bigl( \beta + \gamma'_0 D L (\gamma'_0 r_0) \Bigr) }\Biggr\}}
\, ,
\end{eqnarray}
with $\gamma'_0 = (\tau_p D)^{-1/2}$, $\gamma_0 = (\tau_f D)^{-1/2}$
and the Langevin function
\begin{equation}
L (z) = \coth z - \frac{1}{z} \, .
\label{eq:langevin}
\end{equation}
Likewise the intensity component $I_t^{precip}$
of the rate-equation solution in Eq.~(\ref{eq:I_t_tot}) has to be replaced by \cite{Wuerschum96, Oberdorfer09}:
\begin{equation}
\label{eq:I_p_diff}
I_t^{precip} = \Bigl(\frac{r_0}{R} \Bigr)^3 \times \frac{3 \beta}{r_0(\tau_p^{-1}-\tau_t^{-1})} \times
\Biggl\{
\frac{\gamma'_t D L(\gamma'_t r_0)}{
\beta + \gamma'_t D L(\gamma'_t r_0) }
\Biggr\}
\,
\end{equation}
with
\begin{equation}
\gamma'^2_t = \frac{\tau_p^{-1} - \tau_t^{-1}}{D} \, .
\label{eq:gamma'_t}
\end{equation}
For the sake of completeness we quote the mean $e^+$ lifetime for reaction-controlled trapping from both in- and outside:
\begin{equation}
\label{eq:tauq_extended_rate}
\overline{\tau}= \displaystyle{\tau_t \Bigl(1 - \Bigl[\frac{r_0}{R} \Bigr]^3 \Bigr)
\frac{\tau_t^{-1} + \frac{3 \alpha r_0^2}{R^3-r_0^3}}{\tau_f^{-1} + \frac{3 \alpha r_0^2}{R^3-r_0^3}}
+ \tau_t \Bigl(\frac{r_0}{R} \Bigr)^3 \, \, \frac{\tau_t^{-1} + \frac{3 \beta}{r_0}}{\tau_p^{-1} + \frac{3 \beta}{r_0}}}
\, .
\end{equation}
A further extension for taken into account additional $e^+$ trapping at point defects
inside the matrix (Sect.~\ref{sec:vacancies})
and inside the precipitates (in analogy to the GB model \cite{Oberdorfer09}) is straightforward, so that
the corresponding equations have not to be stated explicitly.
\section{\label{discussion} Discussion}
\subsection{\label{sec:voids}
Voids, clusters, small precipitates}
The presented model with the exact solution of diffusion-reaction
controlled trapping at voids (or other extended spherical defects like clusters and small precipitates)
and competitive transition-limited
trapping at vacancy-type defects yields closed-form expressions
for the mean positron lifetime $\overline{\tau}$ [Eq. (\ref{eq:tau_q_general})]
and for the relative intensities $I_t$ [Eq. (\ref{eq:I_t_general})]
and $I_v$ [Eq. (\ref{eq:I_v})] of the $e^+$ lifetime components
$\tau_t$ and $\tau_v$ of the void and the vacancy
trapped states, respectively.
We start the discussion considering exclusively diffusion-reaction
controlled trapping at voids (Sect. \ref{sec:general}).
The model contains as limiting cases both the solution of the simple trapping model
(Sect. \ref{sec:rate_limit}) and the one of the entirely diffusion-limited trapping (Sect. \ref{sec:diffusion_limit}).
The mean $e^+$ lifetime $\overline{\tau}$ [Eq. (\ref{eq:tauq})] and the intensity
$I_t$ [Eq. (\ref{eq:I_t})] in dependence of the radius $R$ of the diffusion sphere
are compared in Fig. \ref{fig:2} with the two limiting cases. Note, that $R$ is related to the the void concentration
[Eq. (\ref{eq:C_t})].
For illustration the following characteristic e$^+$ annihilation parameters are used:
a free $e^+$ lifetime $\tau_f=160$~ps as typical for aluminium,
a $e^+$ lifetime $\tau_t=400$~ps as typical for voids \cite{Eldrup03},
a $e^+$ diffusion coefficient $D=2 \times 10^{-5}$~m$^2$s$^{-1}$, a void radius $r = 3$~nm, and
a specific e$^+$ trapping rate $\alpha = 3 \times 10^3$~ms$^{-1}$
reported by Dupasquier et al. \cite{Dupasquier93} for interfaces in Al. For surfaces of Al
a value $\alpha = 7.6 \times 10^3$~ms$^{-1}$ was calculated by Nieminen and Lakkonnen \cite{Nieminen79a}.
Using an atomic volume $\Omega$ for Al of $\Omega^{-1} = 6 \times 10^{28}$~m$^{-3}$,
$\alpha = 3 \times 10^3$~ms$^{-1}$ corresponds to a trapping rate $\sigma_t = 2\times 10^{16}$ s$^{-1}$
[Eq. \ref{eq:sigma_t}] which is similar to that deduced by Bentzon and Evans \cite{Bentzon90} for voids in Mo.\footnote{
A value $\sigma_t = 4\times 10^{16}$ s$^{-1}$ is deduced from
the trapping rate of $3.2 \times 10^9$~s$^{-1}$ at 300~K and
a void number density of $5.3 \times 10^{21}$~m$^{-3}$
quoted in \cite{Bentzon90}.}
Both $\overline{\tau}$ (Fig. \ref{fig:2}a) and $I_t$ (Fig. \ref{fig:2}b) exhibit the characteristic sigmoidal increase
from the free state to the saturation-trapped state with decreasing $R$, i.e., increasing void concentration $C_t$.
Compared to the exact solution of the present model, the standard trapping model
and the limiting case of entirely diffusion-limited trapping
show qualitatively the same trend for $\overline{\tau}$ and $I_t$.
However, both special cases systematically overestimate $\overline{\tau}$ and $I_t$, i.e., predict
stronger trapping since either the rate-limiting effect or the diffusion-limiting effect are neglected in these
approximations. For instance, if one would determine the void concentration from a typical, experimentally measured intensity $I_t$ of
45\,\% \cite{Nambissan1989}, a concentration 36\,\% too low would be deduced from the standard trapping model
compared to the exact theory for the parameter set according to Fig. \ref{fig:2}b.
The deviations of the two limiting cases from the exact solution become even more clear when
the ratios of the trap component intensities of the limiting and exact solution is considered
as shown in the upper part of Fig. \ref{fig:3}.
The deviation from the exact solution substantially increases with decreasing intensity, i.e., with
decreasing void concentration. In this low concentration regime, the deviations
attain a factor of ca. 1.5 (reaction limit) or larger than 3 (diffusion limit)
for the present set of parameters, i.e., the entirely diffusion-limiting case
deviates in this example more strongly than the reaction-limited case.
Diffusion limitation gets even more pronounced when $e^+$ diffusivity is reduced, e.g., due to
scattering at lattice imperfections.
Regarding the opposite side of high defect concentrations,
Fig. \ref{fig:3} (upper part) nicely demonstrates that deviations from the exact theory
vanishes upon approaching $e^+$ saturation trapping since in this regime
kinetic effects tends to become irrelevant.
\subsubsection{\label{sec:effective_rate} Comparison with effective rate approach}
Next we compare the present model with approximations according to which
diffusion limitation is taking account in the standard trapping model by means of a diffusion-limited trapping rate
\cite{Seeger74,Bentzon90}:
\begin{equation}
K_{diff} = \frac{4\pi r_0 D}{\Omega} \times C_t \,.
\label{eq:K_diffusion}
\end{equation}
The case of both transition- and diffusion-limited trapping, is treated in this approximation by means of
the effective trapping rate \cite{Seeger74,Bentzon90}
\begin{equation}
K_{eff} = \frac{K_{diff} \sigma_t C_t}{K_{diff} + \sigma_t C_t}
\label{eq:K_eff}
\end{equation}
with $\sigma_t$ and $C_t$ according to equations (\ref{eq:sigma_t}) and (\ref{eq:C_t}), respectively.
We note that the diffusion-limited trapping rate according to eq. (\ref{eq:K_diffusion}) is also included in the present model; in fact $K_{diff}$ is identical to the pre-factor of $I_t$ for entirely diffusion limited trapping [Eq. (\ref{eq:I_t_diffusion})] when the
subtrahend $r_0^3$ in the nominator, which is associated with the defect volume, is omitted.
In figure \ref{fig:4} the concentration dependence of the relative intensity
$I_t$ of the $e^+$ lifetime component $\tau_t$
in voids is shown for the exact models of diffusion-reaction [Eq. (\ref{eq:I_t})] or
entire diffusion limitation [Eq. (\ref{eq:I_t_diffusion})] in comparison with the corresponding approximations
using the above mentioned effective or diffusion-trapping rates [Equations (\ref{eq:K_diffusion}), (\ref{eq:K_eff})]
with the simple trapping model [Eq. (\ref{eq:I_t_rate})].
Although the effective-rate approximations of the diffusion limitation describe the sigmoidal curve fairly well,
deviations from the exact diffusion models are also apparent, e.g., for the example, $I_t = 45$\,\%, mentioned above
the deviation in concentration is ca. 7\,\% compared to the exact diffusion-reaction theory.
The deviations become clearer once more when we consider the intensity ratio of the effective-rate model and the exact
theory, as plotted in the lower part of Fig. \ref{fig:3}.
Remarkably, since the effectice trapping rate $K_{eff}$ is lower than both the reaction-trapping rate
$\sigma_t C_t$ and the diffusion-trapping rate $K_{diff}$,
the intensity $I_t$ deduced from the effective trapping model is smaller than the exact value.
Deviations from the full model occur throughout the entire intensity regime, although these
deviations are less pronounced compared to the two limiting cases
(fully reaction- or diffusion limited, upper part of Fig. \ref{fig:3}).
For applications in the analysis of experimental data,
the accuracy of the effective rate approach [equation~(\ref{eq:K_eff})] can be assessed
by plotting the intensity ratio (lower part of Fig. \ref{fig:3}) for the respective parameter set.
Irrespectively whether deviations of the effective-rate approach are strong or minor only,
the present model founded on diffusion-reaction theory is that which covers the underlying physics
most accurately.
\subsubsection{\label{sec:competitive} Competitive trapping at point defects}
Now, we discuss the general case that in addition to diffusion-reaction
controlled trapping at voids also competitive transition-limited trapping
at vacancy-type defects in the lattice occurs (Sect. \ref{sec:vacancies}).
The relative intensities of the void component $I_t$ [Eq. (\ref{eq:I_t_general})] and
of the vacancy component $I_v$ [Eq. (\ref{eq:I_v})] is plotted in figure \ref{fig:5}
in dependence of void concentration $C_t$ (a) and vacancy concentration $C_v$ (b),
for a given fixed $C_v$ or $C_t$, respectively. For the
vacancy-type defect a $e^+$ lifetime component $\tau_v=250$~ps and
a specific trapping rate $\sigma_v=4\times10^{14}$~s$^{-1}$ \cite{Schaefer87} is assumed; the other parameters
are the same as used above.
The competitive $e^+$ trapping at voids and vacancy-type defects becomes evident.
For a given vacancy concentration the
intensity $I_t$ of the void increases and the
intensity $I_v$ of the vacancy component decreases with increasing
void concentration due to the increasing fraction of e$^+$ that reaches the
voids (Fig.~\ref{fig:5}.a). Likewise, for a given void concentration,
$I_v$ increases and $I_t$ decreases with increasing vacancy concentration
(Fig. \ref{fig:5}.b).
\subsubsection{\label{sec:gb} Comparison with $e^+$ trapping at grain boundaries}
In the end of this subsection (\ref{sec:voids}), the results of the present model on diffusion-reaction limited $e^+$ trapping
at extended spherical defects will briefly be compared with the corresponding model
of $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ \cite{Wuerschum96, Oberdorfer09}.
Whereas in the latter case the surface of the diffusion sphere
with area $4 \pi R^2$ acts as $e^+$ trap, in the present case with voids of radius $r_0$, the trapping active area
$4 \pi r_0^2$ is much smaller. Moreover, the trapping rate $3 \alpha /R$ for grain boundary trapping \cite{Oberdorfer09}
decreases much more slowly with increasing $R$ compared to the trapping rate $3 \alpha r_0^2 / (R^3-r_0^3)$ of
spherical extended defects with radius $r_0$ [Eq. \ref{eq:K}].
This is the reason why diffusion limitation
affects the kinetics of $e^+$ trapping at grain boundaries more strongly than in the case of voids
which is nicely demonstrated in Fig. \ref{fig:6} where the exact solutions are compared with those of infinite diffusivities.
In Fig. \ref{fig:6} the mean $e^+$ lifetime according to the exact solutions and those of the standard rate theory
for the two types of extended traps are plotted.
The exact solution for $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ reads
\cite{Wuerschum96, Oberdorfer09}
\begin{equation}
\label{eq:tauq_GB}
\overline{\tau} = \tau_f + (\tau_t-\tau_f) \times
\frac{3 \alpha L(\gamma_0 R)}{R \gamma_0
\Bigl\{ \alpha + \gamma_0 D L(\gamma_0 R) \Bigl\}} \text{ with } L (z) = \coth z - \frac{1}{z}\, .
\end{equation}
The more stronger deviation between the exact solution and the rate theory in the case of grain boundary trapping is obvious (Fig. \ref{fig:6}).
\subsection{\label{sec:composite}
Larger precipitates: $e^+$-trapping from both sides of precipitate$-$matrix interface }
In Sect.~(\ref{sec:extended}) we extended the model for applying it to larger precipitates taking into account free $e^+$ annihilation
within the precipitate. The $e^+$ trapping from the precipitate into the precipitate$-$matrix interface is handled
either by rate theory, for special cases where the precipitate radius is well below the $e^+$ diffusion length, or else
by diffusion-reaction theory, for the more general case that the precipitate radius is in the range of or larger than
the $e^+$ diffusion length.
With this extension the present model is applicable to a wide variety of structurally complex scenarios, namely to all type
of composite structures where spherical precipitates are embedded in a matrix irrespective of the size and the number density of
the precipitates.
Whereas for extended defects with smaller size, which were discussed in Sect.~(\ref{sec:voids}),
the deviations between the exact model and the rate theory may be of less relevance since the
trapping active area $4 \pi r_0^2$ is small,
for larger precipitates the diffusion-limitation in any case gets relevant owing
to the much larger trapping active area, similar as for $e^+$ trapping at GBs
(see Fig.~\ref{fig:6}).
This is demonstrated in Fig.~\ref{fig:7}, where the variation of the mean $e^+$ lifetime with radius $R$
is compared for four different solutions, namely diffusion-limitation of trapping into the
precipitate$-$matrix interface from both the matrix and the precipitate, from the matrix only, and for
entirely reaction-limited trapping from both sides with standard-trapping rate or with effective diffusion-limited trapping rate.
The latter is obtained by replacing in equation~(\ref{eq:tauq_extended_rate}) the standard-trapping rates by
the effective diffusion-limited trapping rate according to equation~(\ref{eq:K_eff}), i.e.,
$3 \alpha r_0^2 (R^3-r_0^3)^{-1}$ by $3 \alpha D r_0^2 R^{-3} (\alpha r_0 +D)^{-1}$ and
$3 \beta r_0^{-1}$ by $3 \beta D r_0^{-1} (\beta r_0 +D)^{-1}$.
In contrast to the case of small extended defects (Fig.~\ref{fig:6}), for larger precipitates
(example $r_0=100$~nm) substantial deviations between the solutions occur for the entire concentration regime
if the diffusion-limitation is neglected (Fig.~\ref{fig:7}).
Even the rate approach with effective diffusion-limited trapping rate,
which at least for small extended defects is a reasonable approximation (Sect.~\ref{sec:effective_rate}, Fig.~\ref{fig:4}),
turns out to be completely inadequate for the larger precipitate size.
The deviations are much less if the diffusion-limitation is only neglected
for the trapping from the precipitate into the interface, since the precipitate size (in contrast to the precipitate distance) remains in the range of the $e^+$ diffusion length independent of the
precipitate concentration. Anyhow, for a precise description even for such small precipitate sizes,
the exact theory of diffusion- and reaction controlled trapping has to be applied for
the trapping from the interior of the precipitates.
Finally, we compare this model with that presented by Dryzek \cite{dryzek1999, dryzek2016}
for studying recrystallization in highly deformed metals. In that case recrystallized grains are embedded in a highly deformed matrix. Diffusion-limited $e^+$ trapping occurs from the grains into the matrix,
whereas within the matrix saturation trapping of $e^+$ prevails due to the high defect density.
In this sense, the model of Dryzek represents an extension of the diffusion-reaction theory for trapping at grain boundaries, where instead of GBs a surrounding deformed matrix is considered. The model presented here, represents a further extension where diffusion- and reaction-controlled trapping also from the matrix into the interfaces is considered.
\section{\label{conclusion} Conclusion}
The present model with the exact solution
of the diffusion-reaction theory for the $e^+$ trapping at
extended spherical defects
and competitive transition-limited trapping at atomic defects
yields a basis for the quantitative description of the
$e^+$ behaviour in materials with complex defect structure.
It could be shown that the model includes as special cases
the simple trapping model and the entirely diffusion-limited trapping,
but both of these limiting cases represent approximations, only.
For the full model, closed-form expressions were obtained for
the mean positron lifetime $\overline{\tau}$ and for the intensities
of the e$^+$ lifetime components associated with
trapping. This exact model allowed a quantitative assessment of the usual approach,
which takes diffusion limitation for the trapping at voids into account
by effective diffusion-trapping rates. The present closed-form solutions also
renders this effective rate approach unnecessary.
The presented theory goes even much far beyond existing models,
since it is not only applicable to small extended defects (such as voids or clusters), but also
to larger precipitates where positron trapping from the precipitates into the
precipitate$-$matrix interface is taken into consideration.
Therefore, the model presents the basis for studying all type
of composite structures where spherical precipitates are embedded in a matrix irrespective of their size and their number density.
\begin{acknowledgments}
The senior author (R.W.) dedicates this work to Alfred Seeger
whose numerous pioneering works also included modeling of positron annihilation.
This work was performed in the framework of the inter-university cooperation of TU Graz and Uni Graz on natural science (NAWI
Graz).
\end{acknowledgments}
\newpage
|
2,877,628,090,161 | arxiv | \section{Introduction}
The explosion of a supernova triggered by the collapse of a massive star produces several solar masses of stellar ejecta expanding at $\sim$10$^4$~km~s$^{-1}$ into the surrounding circumstellar (CSM) and interstellar (ISM) material. The resulting forward shock compresses and heats the gas to high temperatures, thus producing X-ray radiation. As the shock sweeps up material, the deceleration drives a reverse shock back into cold metal-enhanced ejecta, which are also heated to X-ray emitting temperatures. While in young historical supernova remnants (SNRs) the reverse shock is very close to the main blast wave and a significant fraction of the ejecta are still cold and unshocked, the reverse shock in evolved SNRs has had time to reach the SNR center, and therefore all the ejecta has been shocked and is expected to emit X-rays.
Several SNRs are characterized by a knotty ejecta structure, and very many clumps have been observed at different wavelengths in remnants of core-collapse supernovae (SNe), such as G292.0$+$1.8 \citep{park2004}, Puppis~A \citep{katsudaAPJ2008}, and Cas~A, where knots have also been detected beyond the main shock front \citep{hammelfesen2008,delaney2010}.
The Vela SNR represents a privileged target for studying the distribution of the ejecta and fragments detected beyond the forward shock front. It is considered to be the remnant of a Type~II-P SN explosion of a progenitor star with a mass lower than 25~M$_{\odot}$ \citep{gvaramadze1999}. Its age is estimated to be 11.4 kyr \citep{taylor1993}, and the distance to the SNR is about 250~pc \citep{bocchino1999,cha1999}. \cite{aschenbach1995} identified six ``shrapnel'' (labeled shrapnel A-F), which are X-ray emitting ejecta fragments with a characteristic boomerang shape protruding beyond the primary blast wave. Shrapnel A, B, and D have been studied in detail by \cite{tsunemi1999}, \cite{miyata2001}, \cite{katsudatsunemi2005,katsudatsunemi2006} and {yamaguchikatsuda2009}. These works have shown that the shrapnel may be divided into two categories. Shrapnel B and D have high O, Ne, and Mg abundances, while shrapnel A has a high Si abundance and weak emission from other elements. Consequently, \cite{tsunemikatsuda2006} have pointed out that the Si-rich shrapnel A must have been generated in a deeper layer of the progenitor than all the other shrapnel. Several bright ejecta knots have been discovered in the northern part of the remnant \citep{miceli2008}. The authors suggested that these knots would be shrapnel hidden inside the main shell by a projection effect, showing relative abundances similar to those found in shrapnel B and D.
In other core-collapse SNRs, the Si-rich ejecta may show a very peculiar jet-counterjet structure. Moreover, \cite{grichenersoker2017} suggested that jet-like features are common in many core-collapse SNRs. The well-known case of Cas~A has been studied in detail thanks to a very long {\it Chandra} observation \citep{hwang2004}
that shows a jet (with a weaker counterjet structure) composed mainly of Si-rich plasma. \cite{laming2006} have performed an X-ray spectral analysis of several knots in the jet and concluded that the origin of this interesting morphology lies in an explosive jet and it does not arise through interaction with a cavity or other peculiar structure of the ISM or CSM. Detailed 3D hydrodynamic simulations have shown that this jet can be explained as the result of velocity and density inhomogeneities in the ejecta profile of the exploding star with $\sim$2\% of the energy of the total energy budget of this remnant \citep{orlando2016}.
In this {\it Letter} we present the analysis of an {\it XMM-Newton} dedicated observation of shrapnel G, located in the southwestern edge of the Vela SNR. We investigate the Si-rich ejecta in this shrapnel, which lies on the same line as the line that connects the center of the shell and the northeastern shrapnel A in the plane of the sky (see Fig.~\ref{rosat}). Shrapnel A and G reveal the signature of a jet-counterjet Si-rich structure in the ejecta of the very old Vela SNR, reminiscent of the very young core-collapse SNR Cas~A.
\begin{figure}
\centering
\includegraphics[trim={0 0 30 0},clip,width=0.47\textwidth]{rosat_new_cb_cross.ps}
\caption{{\em ROSAT} All-Sky Survey image of the Vela SNR in the 0.44$-$2.04~keV energy range. Shrapnel A and G are indicated with white circles connected by a dashed line that crosses close to the Vela PSR, and the explosion point inferred from its age and proper motion (yellow cross).}
\label{rosat}
\end{figure}
\section{Observations and data analysis}
The Vela shrapnel G has been observed once with the European Photon Imaging Camera (EPIC) of the {\it XMM-Newton} satellite. This camera consists of three detectors, two MOS cameras, namely MOS1 and MOS2 \citep{turner2001}, and a PN camera \citep{struder2001}, which operate in the 0.3$-$10 keV energy range. The observation was performed on April 22, 2012 (Obs. ID 0675080101), with a medium filter in Prime Full Window observation mode. The exposures were 42~ks, 43~ks, and 41~ks for the MOS1, MOS2, and PN cameras, respectively.
We analyzed the data using {\it XMM-Newton} Science Analysis System (SAS) version 15.0.0 and calibration files available in December 2016. In order to avoid soft-proton contamination, the observations were filtered using the SAS task {\sc espfilt} , resulting in reduced Good Time Intervals (GTI) of approximately 28ks for MOS1, 29ks for MOS2, and 27ks for PN exposures. For the subsequent analysis, the event lists were filtered to retain only events that likely stem from X-ray photons: we selected {\sc flag==0} events with single and double {\sc pattern} by means of the {\sc evselect} task.
\section{Results}
\subsection{X-ray images}
To produce images in different energy bands, we performed a double background subtraction to take into account particle and X-ray background contamination. For this purpose, we used Filter Wheel Closed and Blank Sky files available at {\it XMM ESAC} webpages\footnote{https://www.cosmos.esa.int/web/xmm-newton/filter-closed\\http://xmm-tools.cosmos.esa.int/external/xmm\_calibration/\\background/bs\_repository/blanksky\_all.html} and adopted the procedure described in \citet{miceli2017}. We then performed a point-source detection by running the {\sc edetect\_chain} script. Events in circular regions of 15~arcsec around each detected source were removed from the filtered event files, as we are only interested in the diffuse emission of the shrapnel. We created background-subtracted images for each camera correcting for exposure and vignetting effects in different energy bands. Finally, we combined them by applying an adaptive smoothing by means of the {\sc emosaic} and {\sc asmooth} tasks.
In Fig.~\ref{rgb} we show the resulting composite X-ray image of the Vela shrapnel G obtained by combining the three EPIC exposures using a spatial binning of 4 arcsec. The soft band (0.3$-$0.6~keV) is shown in red, the medium band (0.6$-$1.3~keV) in green, and the hard (1.3$-$3.0~keV) band in blue. Black circles correspond to the subtracted point-sources. In the image, north is up and east is to the left. The shrapnel is indistinguishable from the background above 3.0~keV in the available data. As can be seen, the morphology of the shrapnel is fairly regular, showing two bright extended eastern (E) and western (W) regions. The X-ray emission shows a strong edge to the southwest (SW) coincident with a possible shock front of the shrapnel and weak elongated X-ray emission pointing to the northeast (NE), corresponding to the geometrical center of the Vela SNR.
In Fig.~\ref{silicon} we show a mosaiced X-ray map of the Si band (1.3$-$2.0~keV) with a spatial binning of 20 arcsec. The
overlaid yellow contours correspond to the X-ray emission in the whole 0.3$-$3.0~keV energy range. From this map it is evident that the photons originated in the Si band are spatially correlated with the total X-ray emission.
\begin{figure}
\centering
\includegraphics[trim={0 0 20 0},clip,width=0.47\textwidth]{rgb_new_cb_num.ps}
\caption{{\em XMM-Newton} false-color count-rate image of Vela shrapnel G. Red represents 0.3$-$0.6~keV, green 0.6$-$1.3~keV, and blue the 1.3$-$3.0~keV energy range. The overlaid yellow contours indicate the east (E) and west (W) spectral extraction regions. Dashed rectangles indicate the background regions (bkg). The image is background- and vignetting-corrected and point sources were removed.}
\label{rgb}
\end{figure}
\begin{figure}
\centering
\includegraphics[trim={0 0 20 0},clip,width=0.48\textwidth]{silicon_new_cb_num.ps}
\caption{Background- and vignetting-corrected count-rate map in the 1.3$-$2.0~keV energy range, corresponding to the Si band. The overlaid yellow contours represent the emission in the 0.3$-$3.0~keV range. The Si maxima match the two bright knots of X-ray emission well.}
\label{silicon}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=-90,trim={0 30 0 0},clip,width=0.45\textwidth]{spectraNEI.ps}
\caption{PN and MOS1/2 spectra of shrapnel G. Solid lines indicate the best-fit VNEI model (see Table~\ref{allspectable}). Lower panel shows the fit residuals.}
\label{spectra}
\end{figure}
\begin{table}
\caption{Spectral parameters of Vela shrapnel G.}
\renewcommand{\arraystretch}{0.9}
\begin{centering}
\begin{small}
\begin{tabular}{l | c | c}
\hline\hline
Model \& Parameters & {\bf VAPEC+VAPEC} & {\bf VNEI} \\
\hline
$N_\mathrm{H}$ [10$^{22}$~cm$^{-2}$] & $0.11\pm0.01$ & $0.022\pm0.007$\\
\hline
$kT_{1}$ [keV] & $0.194\pm0.002$ & $0.49\pm0.02$\\
Norm$_{1}$ [$\times 10^{-3}$] & $25\pm3$ & $2.0\pm0.3$\\
$\tau$ [$10^{10}$~s~cm$^{-3}$] & $-$ & $3.1\pm0.3$\\
\hline
$kT_{2}$ [keV] & $0.64\pm0.07$ & $-$\\
Norm$_{2}$ [$\times 10^{-3}$] & $0.66\pm0.08$ & $-$\\
\hline
O,(=N),(=C) [O$_\odot$] & $0.38\pm0.02$ & $0.47\pm0.05$\\
Ne [Ne$_\odot$] & $1.14\pm0.08$ & $1.33\pm0.10$\\
Mg [Mg$_\odot$] & $0.99\pm0.11$ & $0.92\pm0.12$\\
Si [Si$_\odot$] & $2.06\pm0.45$ & $2.24\pm0.43$\\
Fe [Fe$_\odot$] & $0.34\pm0.03$ & $0.29\pm0.04$\\
\hline
$\chi^{2}_{\nu}$ / d.o.f. & 1.50 / 450 & 1.41 / 451 \\
\hline
Flux (0.3$-$0.6~keV) & $4.49\pm0.06$ & $4.57\pm0.06$\\
Flux (0.6$-$1.3~keV) & $4.75\pm0.03$ & $4.73\pm0.03$\\
Flux (1.3$-$3.0~keV) & $0.33\pm0.02$ & $0.29\pm0.01$\\
\hline
Total Flux (0.3$-$3.0~keV) & $9.57\pm0.06$ & $9.59\pm0.06$\\
\hline
\end{tabular}
\end{small}
\label{allspectable}
\tablefoot{Normalizations are defined as 10$^{-14}/4\pi$D$^2\times {\rm EM}$, where ${\rm EM} = \int n_H\,n_e dV$ is the emission measure, $D$ is distance in [cm], $n_\mathrm{H}$ and $n_{\rm e}$ are the hydrogen and electron densities [cm$^{-3}$], and $V$ is the volume [cm$^{3}$]. Considering $D=250$~pc and a solid angle $\Omega=A/D^2=1.17\times10^{-5}$~sr for the spectral regions, we obtain an EM$/A=2.1\times10^{17}$~cm$^{-5}$ for the preferred VNEI model. Error values are 1$\sigma$ (68\%) confidence intervals for each free parameter. Fluxes are given in units of 10$^{-12}$~erg~cm$^{-2}$~s$^{-1}$ and solar abundances are taken from \cite{anders1989}.}
\end{centering}
\end{table}
\subsection{X-ray spectra}
Since the diffuse emission from the shrapnel G, which is an inhomogeneous extended source, occupies almost the full field of view, we applied the SAS {\sc evigweight} task to correct the event lists for vignetting effects. We obtained response and ancillary matrices using {\sc rmfgen} and {\sc arfgen} for a flat detector map, and we binned the spectra to obtain at least 25 counts per bin. We selected two polygonal spectral extraction regions named E and W, and background was extracted from two different regions where no significant emission from the shrapnel was detected (see Fig.~\ref{rgb}). The spectral analysis was performed using the XSPEC package \citep[Version 12.9.0,][]{arnaud1996} in the 0.3$-$2~keV band for the two MOS cameras and in the 0.3$-$1.5~keV band for the pn. The pn spectrum above 1.5~keV is dominated by the background, so we did not include it in our analysis to maximize the signal-to-noise ratio. We verified that our best-fit model does not change significantly by adding the pn data in the 1.5$-$2~keV band, although the indetermination of the best-fit parameters increases. Since we did not find a significant difference between the spectral fits of regions E and W, we combined them to improve the signal-to-noise ratio. Furthermore, we also checked that independently of the background region used, the best-fit values were consistent with each other. We inspected the background spectra and verified that they do not show any feature at the 1.85~keV Si band, which is visible only in the source spectra.
In Fig.~\ref{spectra} we show the background-subtracted X-ray spectra of the Vela shrapnel G for the three EPIC cameras (black for PN, red and green for MOS~1 and MOS~2, respectively). Errors are at 1$\sigma$ (68\%) confidence levels, and $\chi^{2}$ statistics are used. The X-ray spectrum of shrapnel G has a thermal origin, showing emission lines from \ion{O}{VII} (0.56~keV), \ion{O}{VIII} (0.65~keV), \ion{Ne}{IX} (0.92~keV), \ion{Mg}{XI} (1.35~keV), and Si (1.85~keV). We fit the spectra with different models of thermal emission from an optically thin plasma in collisional ionization equilibrium (APEC model) and in non-equilibrium of ionization (NEI model), taking into account the interstellar absorption \citep[PHABS,][]{balucinska1992}. The best fit is obtained for an NEI plasma with $kT=0.49\pm0.02$~keV. A two-temperature APEC plasma with $kT_1=0.194\pm0.02$~keV and $kT_2=0.64\pm0.07$~keV also gives a good fit. Non-solar abundances are required for both models to fit the emission lines, and their values are compatible between each other. The best-fit parameters are presented in Table~\ref{allspectable}. From now on, we focus on the single-temperature NEI model, which provides a significantly better fit to the spectra.
Before this study, a bright Si He$\alpha$ emission line could be detected only in the northeastern shrapnel A \citep{katsudatsunemi2006}. Remarkably, shrapnel G is located in the opposite southwestern region of shrapnel A. In contrast, shrapnel D \citep{katsudatsunemi2005} and the ejecta knots found by \cite{miceli2008} in the northern rim of Vela SNR showed no Si line and notably higher abundances of O, Ne, Mg and Fe than those found in shrapnel A \citep{katsudatsunemi2006} and G (this study). In particular, we found that in shrapnel G the Ne:Mg:Si:Fe:O abundances relative to O are 2.8:2.0:4.8:0.6:1 to be compared with 2.6:2.2:7.6:2.6:1 in shrapnel A, while shrapnel D and northern ejecta present Ne:Mg:Fe:O=2.1:2.2:0.2:1 and 2.5:3.2:0.5:1 and no Si line \citep{miceli2008}. Similar abundance patterns have also been observed by \cite{lamassa2008}, who found ejecta-rich plasma in the (projected) direction of the Vela PSR. In conclusion, shrapnel A and G are the only Si-rich shrapnel observed up to now in the Vela SNR. Furthermore, their plasma temperatures are consistent within 2$\sigma,$ and the ionization parameters are quite similar \citep[see Table~1 in][]{katsudatsunemi2006}, as are their projected sizes in the plane of the sky.
\section{Discussion}
In this {\it Letter} we presented a detailed study of the shrapnel G located in the SW region of the Vela SNR and showed the presence of Si-rich plasma in shrapnel G for the first time. We showed that shrapnel G has a very similar chemical composition to the geometrically opposite shrapnel A located in the NE edge of the remnant. While shrapnel B and D and all the other ejecta knots detected so far in Vela SNR have high O, Ne, and Mg abundances, shrapnel A and G have a high Si abundance and weak emission from other elements.
As a consequence of the nucleosynthesis expected in stellar evolution, \cite{tsunemikatsuda2006} pointed out that Si-rich shrapnel must be generated in deeper layers of the progenitor star. However, hydrodynamic simulations of the Vela shrapnel show that an unrealistically high initial density contrast is required for an inner shrapnel to overcome outer ejecta knots, if we assume that the ejecta velocity increases linearly with their distance from the center \citep{miceli2008}. A possible solution for this issue is that (part of) the Si-burning layer has been ejected with a higher initial velocity, for example, as a collimated jet. This idea was later confirmed by dedicated 3D simulations of Cas~A, showing that both density and velocity inhomogeneities are necessary to reproduce the observed Si-rich jet \citep{orlando2016}.
The line connecting shrapnel A and G passes almost exactly through the expansion center of Vela SNR (see Fig. \ref{rosat}) as determined from the geometry of the other shrapnel and the proper motion of the Vela PSR. This alignment strongly supports the possibility that these shrapnel pieces are part of a Si-rich jet-counterjet structure. Assuming that the size along the line of sight, $L$, is equal to the projected size of shrapnel G in the plane of the sky (1320~arcsec) at a distance of $D=250$~pc, we estimate a number density of $n=0.21$~cm$^{-3}$ for the X-ray emitting plasma, and a total mass of $M=0.008$~M$\odot$ (for an average atomic mass of $2.1\times10^{-24}$~g for solar abundances). Considering the projected distance from shrapnel G to the geometrical center of the SNR and an age of $\sim$11~kyr, we obtain a velocity of $\sim$1400~km~s$^{-1}$ , leading to a total average kinetic energy of $E=1.6\times10^{47}$~erg, which is a lower limit when we take into account that the velocity of the shrapnel is not constant in time, that our estimate accounts only for the projected velocity, and that the X-ray emitting mass is only a fraction of the initial mass \citep{miceli2013}. Interestingly, the estimated mass of shrapnel G is similar (within a factor of 5) to the mass of the post-explosion anisotropy responsible for the Si-rich jet observed in Cas~A \citep{orlando2016}. On the other hand, our lower limit on the kinetic energy is two orders of magnitude lower than the
energy estimated soon after the SN explosion for the jet of Cas~A \citep[$\sim 4\times10^{49}$~erg,][]{orlando2016}. This is mainly due to our estimated velocity of shrapnel G: a velocity higher
by a factor of 10 (which is expected soon after the SN explosion) would produce energies similar to the energy found in the jet of Cas~A. We conclude that the mass and energy inferred for shrapnel G is very similar to the values estimated for the jet of Cas~A \citep{orlando2016}.
The structure of the ejecta in a SNR contains the imprint of the metal-rich layers inside the progenitor star. This type of detailed analysis of spatially resolved ejecta may help to understand the processes occurring in the latest stage of stellar evolution on the onset of core-collapse SNe explosions. In this sense, Vela SNR is an ideal candidate for performing this type of studies because of its age and angular size in the sky. Dedicated observations in the X-ray band of the remaining shrapnel that has not been studied so far are required to probe the still poorly understood physics of core-collapse supernovae and the formation of collimated ejecta jets.
With existing X-ray telescopes like {\it XMM-Newton}, it is possible to study small parts of a large SNR like Vela in pointed observations. Since we are not able to cover the entire SNR, it is difficult to achieve an understanding of the object as a whole. The German telescope {\it eROSITA} \citep[extended ROentgen Survey with an Imaging Telescope Array,][]{merloni2012} on board the Russian Spektrum-Roentgen-Gamma (SRG) mission, which is planned to be launched in 2018, will perform an all-sky survey (eRASS) in the 0.3$-$10~keV band for the first time. Equipped with CCDs similar to those of {\it XMM-Newton}, we will be able to study the entire SNR with a similar spatial and spectral resolution as we have presented here for shrapnel G. With a total exposure of $\sim$3~ks, eRASS will yield $\sim$23000 net counts for shrapnel G, allowing us to constrain abundances with an accuracy of $\sim$20\%.
\begin{acknowledgements}
We are grateful to the referee for very constructive comments. The research leading to these results has received funding from the European Union Horizon 2020 Programme under the AHEAD project (grant agreement n. 654215). FG and AES are grateful for the hospitality of INAF Osservatorio Astronomico di Palermo members. FG, AES and JAC were supported by PIP 0102 (CONICET). JAC was also supported by Consejer\'{\i}a de Econom\'{\i}a, Innovaci\'on, Ciencia y Empleo of Junta de Andaluc\'{\i}a under grant FQM-1343, and research group FQM-322, as well as FEDER funds. MM and SO aknowledge support by the PRIN INAF 2014 grant ``Filling the gap between supernova explosions and their remnants through magnetohydrodynamic modeling and high performance computing''. MS acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the Heisenberg professor grant SA 2131/5-1.
\end{acknowledgements}
|
2,877,628,090,162 | arxiv | \section*{Abstract}
In this paper we present a numerical study of a mathematical model of spiking neurons introduced by Ferrari et al. in \cite{ferrari2018phase}.
In this model we have a countable number of neurons linked together in a network, each of them having a membrane potential taking value in the integers, and each of them spiking over time at a rate which depends on the membrane potential through some rate function $\phi$. Beside being affected by a spike each neuron can also be affected by leaking. At each of these leak times, which occurs for a given neuron at a fixed rate $\gamma$, the membrane potential of the neuron concerned is spontaneously reset to $0$. A wide variety of versions of this model can be considered by choosing different graph structures for the network and different activation functions. It was rigorously shown that when the graph structure of the network is the one-dimensional lattice with a hard threshold for the activation function, this model presents a phase transition with respect to $\gamma$, and that it also presents a metastable behavior. By the latter we mean that in the sub-critical regime the re-normalized time of extinction converges to an exponential random variable of mean 1. It has also been proven that in the super-critical regime the renormalized time of extinction converges in probability to 1. Here, we investigate numerically a richer class of graph structures and activation functions. Namely we investigate the case of the two dimensional and the three dimensional lattices, as well as the case of a linear function and a sigmoid function for the activation function. We present numerical evidence that the result of metastability in the sub-critical regime holds for these graphs and activation functions as well as the convergence in probability to $1$ in the super-critical regime.
\section{Introduction}
Informally the model we consider is as follows. $I$ is a countable set representing the neurons, and to each $i \in I$ is associated a set $\mathbb{V}_i$ of \textit{presynaptic neurons}. If you consider the elements of $I$ as vertices, and draw and edge from $j$ to $i$ whenever $j \in \mathbb{V}_i$, then you obtain the graph structure of the network. The \textit{membrane potential} of neuron $i$ is a stochastic process denoted $(X_i(t))_{t \geq 0}$ taking value in the set $\mathbb{N}$ of non-negative integers. Moreover, we associate to each neuron a Poisson process $(N^{\dagger}_i(t))_{t \geq 0}$ of some parameter $\gamma$, representing the \textit{leak times}. At any of these leak times the membrane potential of the neuron concerned is reset to $0$. Another point process $(N_i(t))_{t \geq 0}$ representing the \textit{spiking times} is also associated to each neuron, which rate at time $t$ is given by $\phi (X_i(t))$, where $\phi$ is the rate function. Whenever a neuron spikes its membrane potential is also reset to $0$ and the membrane potential of all of its post-synaptic neurons is increased by one (i.e. the neurons of the set $\{ j : i \in \mathbb{V}_j\}$). All the point processes involved are assumed to be are mutually independent.
\vspace{0.4 cm}
More formally, beside asking that $(N^{\dagger}_i(t))_{t \geq 0}$ be a Poisson process of some parameter $\gamma$, this is the same as saying that $(N_i(t))_{t \geq 0}$ is the point process characterized by the following equation $$\mathbb{E} (N_i (t)- N_i (s)|\mathcal{F}_s) = \int_s^t \mathbb{E} (\phi (X_i(u))|\mathcal{F}_s)du $$ where $$X_i(t) = \sum_{j \in \mathbb{V}_i}\int_{]L_i(t),t[}dN_j(s),$$ $L_i(t)$ being the time of the last event affecting neuron $i$ before time $t$, that is, $$L_i(t) = \sup \Big\{s \leq t : N_i(\{s\}) = 1 \text{ or } N^\dagger_i(\{s\}) = 1\Big\}.$$ $(\mathcal{F}_t)_{t\geq 0}$ is the standard filtration for the point processes involved here. See \cite{ferrari2018phase} for more details.
\vspace{0.4 cm}
In \cite{ferrari2018phase}, \cite{andre2019meta}, and \cite{andre2019determ} a specific version of the model above was studied. The graph structure chosen there was the one-dimensional lattice, i.e. $I = \mathbb{Z}$ with $\mathbb{V}_i = \{i-1,i+1\}$, moreover the activation function was chosen to be a hard threshold of the form $\phi(x) = \mathbbm{1}_{x > 0}$. In such a paradigm the rate of the point processes representing spiking times can only take two values: $0$ or $1$, depending on whether the membrane potential is positive or null. In this context whenever the membrane potential of a neuron is different from $0$ we say that the neuron is \textit{active}, otherwise we say that it is \textit{quiescent}. More generally we will only consider here function $\phi$ satisfying $\phi(0) = 0$ and $\phi(x) > 0$ for $x > 0$, so that we will keep this distinction between active and quiescent neurons.
\vspace{0.4 cm}
In \cite{ferrari2018phase} it was proven that in the case of the one-dimensional lattice with hard threshold the following theorem holds
\begin{theorem} \label{thm:phasetransition}
Suppose that for any $i \in \mathbb{Z}$ we have $X_i(0) \geq 1$. There exists a critical value $\gamma_c$ for the parameter $\gamma$, with $0 < \gamma_c < \infty$, such that for any $i \in \mathbb{Z}$
$$\mathbb{P} \Big( N_i([0,\infty[) \text{ } < \infty \Big) = 1 \text{ if } \gamma > \gamma_c$$
and
$$ \mathbb{P} \Big( N_i([0,\infty[) \text{ } = \infty \Big) > 0 \text{ if } \gamma < \gamma_c.$$
\end{theorem}
In words there is a critical point for the parameter $\gamma$, such that below this critical point each neuron stays active forever (with positive probability), and above it each neuron becomes quiescent if you wait long enough.
\vspace{0.4 cm}
The process as a whole of course never dies completely because of the fact that there is an infinite number of neurons, so that it doesn't makes sense to consider the extinction time. Nonetheless nothing prevents us to consider a finite version of this model. Suppose we're still in the case $\mathbb{V}_i = \{i-1,i+1\}$ and $\phi(x) = \mathbbm{1}_{x > 0}$ and for any $N \geq 0$ consider the system defined on the finite set $I_N = \{-N,\ldots N\}$ instead of the whole lattice. Then you can define the process $(\xi_N(t))_{t \geq 0}$ of the spiking rates of the system, that is, the process taking value in $\{0,1\}^{I_N}$ defined by ${\xi_N(t)}_i = \mathbbm{1}_{X_i(t) > 0}$. This is a Markov process, belonging to the category of interacting particle systems, and by classical results on Markov processes we know that it needs to reach the state $0^{I_N}$ - where all neurons are quiescent - in finite time. We can therefore consider the extinction time of this finite model, which we denote $\sigma_N$, and it is natural to ask about its distribution in each of the two phases distinguished by the theorem above. It was proven in \cite{andre2019meta} that the following holds.
\begin{theorem} \label{thm:metamodel}
There exists $\gamma'_c$ such that if $\gamma < \gamma'_c$, then we have the following convergence
$$\frac{\sigma_N}{\mathbb{E} (\sigma_N)} \overset{\mathcal{L}}{\underset{N \rightarrow \infty}{\longrightarrow}} \mathcal{E} (1).$$
In words, the re-normalized time of extinction converges in distribution to an exponential random variable of mean 1.
\end{theorem}
We know that $\gamma'_c < \gamma_c$, and the fact that the theorem is stated for some $\gamma'_c$ and not for the critical value $\gamma_c$ comes from essentially technical reasons related to the way the proof is built. We believe that this metastable result holds for the whole sub-critical region but it is not yet proven. Moreover, it was proven in \cite{andre2019determ} that we also have the following.
\begin{theorem} \label{thm:deterministic}
Suppose that $\gamma > 1$. Then the following convergence holds
$$\frac{\sigma_N}{\mathbb{E} (\sigma_N)} \overset{\mathbb{P}}{\underset{N \rightarrow \infty}{\longrightarrow}} 1.$$
\end{theorem}
We know that $\gamma_c < 1$, so that the result concerns a sub-region of the super-critical region, and as for the previous result, while to the best of our knowledge the result has been proven only for $\gamma > 1$, it is reasonable to expect that it holds in the whole super-critical region.
\vspace{0.4 cm}
The choice of a one-dimensional lattice for the graph of interaction and of a hard threshold for the activation function were initially essentially motivated by mathematical conveniency, and we're interested in checking that the results hold for a richer class of instantiations of the model. In this paper we investigate numerically cases for which we don't have yet any rigorous result. Namely, we investigate higher dimension lattices $\mathbb{Z}^d$ ($d = 2$ and $d = 3$) to show that the results related to the asymptotical distribution of the extinction time stated for the one-dimensional case in Theorem \ref{thm:metamodel} and Theorem \ref{thm:deterministic} remain valid for these graphs. The choice of such structure for the graph of interaction is common in the literature that is concerned with mathematical modeling of neural networks (see for example \cite{miranda1991self} and \cite{makarenkov1991self}), and it is justified by the fact that highly connected cortical regions, such as specific regions of the primate visual cortex \cite{rockland1983intrinsic}\cite{yoshioka1992intrinsic}, present some similarity with multidimensional lattices \cite{sporns2004motifs}. We also investigate the effect of changing the activation function to a linear function and to a sigmoid function. Both activation functions have been used in some form in stochastic models of biological neural networks (see for example \cite{brochini2016phase} and \cite{kinouchi2019stochastic}) as well as in artificial neural networks (see for example \cite{maass1997networks}).
\vspace{0.4 cm}
This paper is organized as follows. In Section \ref{algo} we give a description of the algorithm used for the simulations. In Section \ref{models} we describe the specific instantiations of the model that we investigate. In Section \ref{results} we present the results of our simulations. Finally in Section \ref{discussion} we discuss these results.
\section{Simulation algorithm}
\label{algo}
All simulations were done in Python. The algorithm used to simulate our system of spiking neurons can be informally described as follows:
\begin{itemize}
\item{\textbf{Initial configuration:}}
The network start with all neurons actives (by default membrane potential equal $1$), and for each neuron an initial spiking value is sampled from an exponential random variable of parameter $1$ (corresponding to the first atom of $(N_i (t))_{t \geq 0}$) and an initial leaking time value is sampled for each neuron from an another independent exponential random variable of parameter $\gamma$ (corresponding to the first atom of $(N_i^\dagger (t))_{t \geq 0}$).
\item{\textbf{Interaction:}}
The current time is set to be the smallest of the previously sampled values, the neuron $i$ associated with this value is found and the value of its membrane potential is set to $0$. In case the event considered is a spike the membrane potential of the neurons in the set $\{j: i \in \mathbb{V}_j\}$ (the set of post-synaptic neurons) is increased by one. The membrane potential of neuron $i$ being equal to $0$ we set the next spiking time for neuron $i$ to be infinite until further notice. If the event was a leaking then we sample an exponential random variable of parameter $\gamma$ and add it to the current time to get the next leaking time for neuron $i$. If the event was a spike then we sample an exponential random variable of parameter $\phi(X_j)$ for all post-synaptic neuron $j$ and add this value to the current time to get the next spiking time for these neurons.
\item{\textbf{Stopping condition:}}
The previous operation is iterated until all neurons are quiescent.
\end{itemize}
\vspace{0.4 cm}
More formally, our simulation algorithm can be described by the following pseudo-algorithm.
\vspace{0.4 cm}
\begin{algorithm}[H]
\caption{Simulate the system of spiking neurons and return the extinction time}\label{alg:sim}
\begin{algorithmic}[1]
\State $I$ the (finite) set of neurons.
\State $\phi$ the activation function.
\State $\gamma$ the rate of the leaking point processes.
\State $t$ the current time.
\State $\mathbb{V}_i$ the set of presynaptic neurons for neuron $i$.
\State $X_i$ the membrane potential of neuron $i$ at the current time.
\State $\sigma^\dagger_i$ the time of the next leaking for neuron $i$.
\State $\sigma_i$ the time of the next spike for neuron $i$.
\vspace{0.6 cm}
INITIALIZATION
\vspace{0.4 cm}
\State $t \gets 0$
\For{each $i$ in $I$}
\State $X_i \gets 1$
\EndFor
\For{each $i$ in $I$}
\State $\sigma^\dagger_i \gets \mathcal{E} (\gamma)$ \Comment{$\mathcal{E}$ denotes the realization of an exponential random variable}
\State $\sigma_i \gets \mathcal{E} (\phi(X_i))$
\EndFor
\vspace{0.6 cm}
SIMULATION
\vspace{0.4 cm}
\While{$\sum_{i \in I} X_i \neq 0$}
\vspace{0.2 cm}
\State $MinLeaking \gets \min_{i \in I} \sigma^\dagger_i$
\State $MinSpiking \gets \min_{i \in I} \sigma_i$
\vspace{0.2 cm}
\If{$MinLeaking < MinSpiking$}
\State $t=MinLeaking$
\State $i \gets \argmin_{j \in I} \sigma^\dagger_j$
\State $X_i \gets 0$
\State $\sigma_i \gets \infty$
\State $\sigma^\dagger_i \gets t + \mathcal{E} (\gamma)$
\Else
\State $t=MinSpiking$
\State $i \gets \argmin_{j \in I} \sigma_j$
\State $X_i \gets 0$
\State $\sigma_i \gets \infty$
\For{each $j$ such that $i \in \mathbb{V}_j$}
\State $X_j \gets X_j + 1$
\State $\sigma_j \gets t + \mathcal{E} (\phi(X_j))$
\EndFor
\EndIf
\EndWhile
\vspace{0.4 cm}
\State $\sigma_N \gets t$
\State \Return $\sigma_N$
\end{algorithmic}
\end{algorithm}
\newpage
\section{Models investigated}
\label{models}
In this section we specify the structure of the graph of interaction and the activation function we're interested in.
\subsection{Multi-dimensional lattices}
For the graph of the network we consider the lattices $\mathbb{Z}^1$, $\mathbb{Z}^2$ and $\mathbb{Z}^3$. For any $d \in \{1,2,3\}$, let $\| \cdot \|$ be the norm on $\mathbb{Z}^d$ given for any $j\in \mathbb{Z}^d$ by $$\|j\| = \sum_{k=1}^d |j_k|,$$ where $j_k$ is the k-th coordinate of $j$. The structure of the network is then given by $I = \mathbb{Z}^d$ and $\mathbb{V}_i = \{j \in I^d : \|i-j\| = 1\}$ for $i \in I$.
\vspace{0.4 cm}
Notice that by defining the set of presynaptic neurons as the set of the nearest neighbours we actually have $j \in \mathbb{V}_i$ if and only if $i \in \mathbb{V}_j$. In other words, for a given neuron the set of the presynaptic neurons and the set of the postsynaptic neurons are equal. For this specific choice the graph of interaction is therefore actually undirected.
\begin{figure}[H]
\center{\includegraphics[width=9cm]
{lattices.png}}
\caption{\label{lattices} One-dimensional and two-dimensional lattices. A directed arrow is drawn toward the black neuron from each of its presynaptic neurons.}
\end{figure}
\vspace{0.4 cm}
\subsection{Linear and sigmoid activation functions}
The activation function considered in \cite{andre2019meta}, \cite{ferrari2018phase} and \cite{andre2019determ} was the hard threshold $\phi(x) = \mathbbm{1}_{x > 0}$. This choice is convenient mathematically as the system then becomes an additive interacting particle system where any neuron can only have two possible values for the spiking rate at any time: $0$ or $1$. Nonetheless, from a biological point of view, a hard threshold is a rough choice, and we would like to consider smoother options.
\vspace{0.4 cm}
The first option we consider is a linear function of the simplest form: $\phi(x) = x$.
\vspace{0.4 cm}
The second option we consider is a somewhat more sophisticated sigmoid function of the following form
$$\phi(x) =
\begin{cases}
(1 + e^{-3x + 6})^{-1}& \text{ if } x > 0 ,\\
0 & \text{ if } x = 0.
\end{cases}$$
\vspace{0.4 cm}
Notice that we need to have $\phi (0) = 0$, in order to avoid spontaneous spiking (neuron with null membrane potential that spikes nonetheless). This is the reason why we force this value for the sigmoid function.
\section{Results}
\label{results}
We run simulations for instantiations of the system of spiking neurons consisting of all the possible combinations between the graphs and activation functions described above.
\subsection{Simulations with a fixed number of neurons }
\subsubsection{Multidimensional lattices and hard threshold}
For each of the three lattices the Algorithm \ref{alg:sim} described in Section \ref{algo} was run, with an activation function of the form $\phi(x) = \mathbbm{1}_{x > 0}$. Each of the simulations were run for two different values of $\gamma$, 10,000 times for each of these values, using a number of neurons of the order of 100. The mean of the time of extinction $\sigma_N$ was then computed using these data, and used to build the re-normalized histogram in each of these cases. The exact values for the size of the network and for the parameter $\gamma$ can be found in Table \ref{parametershard}.
\vspace{0.4 cm}
\begin{table}[h]
\center{\begin{tabular}{|c|c|c|c|}
\hline
Lattice & Number of neurons & Value of $\gamma$ & \multicolumn{1}{l|}{Figure} \\ \hline
$\mathbb{Z}$ & 101 & 0.34 & \ref{hist_hardthre_sub} \\
& & 0.85 & \ref{hist_hardthre_sup} \\ \hline
$\mathbb{Z}^2$ & 121 & 1.25 & \ref{hist_hardthre_sub} \\
& & 5.00 & \ref{hist_hardthre_sup} \\ \hline
$\mathbb{Z}^3$ & 125 & 1.80 & \ref{hist_hardthre_sub} \\
& & 6.00 & \ref{hist_hardthre_sup} \\ \hline
\end{tabular}
\caption{Values of the total number of neurons and of the parameter $\gamma$ used in the simulation for each of the three lattices.} \label{parametershard}}
\end{table}
\vspace{0.4 cm}
The resulting histograms are presented in Figure \ref{hist_hardthre_sub} and \ref{hist_hardthre_sup}.
\begin{figure}[H]
\center{\includegraphics[width=12 cm]
{Histogram_Hard_Sub.png}}
\caption{\label{hist_hardthre_sub} Histogram of the re-normalized time of extinction $\sigma_N$ for small values of gamma, and an activation function of the form $\phi(x) = \mathbbm{1}_{x > 0}$. In \textbf{a}, \textbf{b} and \textbf{c} the blue, green and gray bars are the histograms for the time of extinction in the one-dimensional lattice, two-dimensional lattice and three-dimensional lattice respectively. The red line is the exponential function $t \mapsto e^{-t}$, which corresponds to the density of an exponential law of parameter 1. The parameter $n$ corresponds to the number of neurons.}
\end{figure}
\begin{figure}[H]
\center{\includegraphics[width=12cm]
{Histogram_Hard_Sup.png}}
\caption{\label{hist_hardthre_sup} Histogram of the re-normalized time of extinction $\sigma_N$ for high values of gamma, and an activation function of the form $\phi(x) = \mathbbm{1}_{x > 0}$. In \textbf{a}, \textbf{b} and \textbf{c} the blue, green and gray bars are the histograms for the time of extinction in the one-dimensional lattice, two-dimensional lattice and three-dimensional lattice respectively. The parameter $n$ corresponds to the number of neurons.}
\end{figure}
\vspace{0.4 cm}
\subsubsection{Multidimensional lattices, linear function and sigmoid function}
The routine described above was repeated with the two other activation functions. Only the values of $\gamma$ change, which was necessary as changing the activation function must change the critical value of the system. These values are given in Table \ref{parameterslinsig}.
\begin{table}[!h]
\center{\begin{tabular}{|c|c|c|c|c|}
\hline
Lattice & Number of neurons & Activation function & Value of $\gamma$ & \multicolumn{1}{l|}{Figure} \\ \hline
$\mathbb{Z}$ & 101 & Linear & 0.42 & \ref{histlin} \\
& & Sigmoid & 0.028 & \ref{histsig} \\
& & Linear & 1 & \ref{histlin} \\
& & Sigmoid & 0.85 & \ref{histsig} \\ \hline
$\mathbb{Z}^2$ & 121 & Linear & 1.70 & \ref{histlin} \\
& & Sigmoid & 0.2 & \ref{histsig} \\
& & Linear & 5.00 & \ref{histlin} \\
& & Sigmoid & 1.7 & \ref{histsig} \\ \hline
$\mathbb{Z}^3$ & 125 & Linear & 1.90 & \ref{histlin} \\
& & Sigmoid & 0.09 & \ref{histsig} \\
& & Linear & 6.00 & \ref{histlin} \\
& & Sigmoid & 1.8 & \ref{histsig} \\ \hline
\end{tabular}
\caption{Values of the total number of neurons and of the parameter $\gamma$ used in the simulation for each of the three lattices.} \label{parameterslinsig}}
\end{table}
The resulting histograms are presented in figures \ref{histlin} and \ref{histsig}.
\begin{figure}[H]
\center{\includegraphics[width=15 cm]
{Histogram_Linear.png}}
\caption{\label{histlin} Histogram of the re-normalized time of extinction $\sigma_N$ for a linear activation function for each of the three lattices. On the left side are the histograms for small values of $\gamma$ and on the right side the histograms for high values of $\gamma$. The red line on the left side is the exponential function $t \mapsto e^{-t}$, which corresponds to the density of an exponential law of parameter 1. The parameter $n$ corresponds to the number of neurons.}
\end{figure}
\begin{figure}[H]
\center{\includegraphics[width=15 cm]
{Histogram_Sigmoidal.png}}
\caption{\label{histsig} Histogram of the re-normalized time of extinction $\sigma_N$ for a sigmoid activation function for each of the three lattices. On the left side are the histograms for small values of $\gamma$ and on the right side the histograms for high values of $\gamma$. The red line on the left side is the exponential function $t \mapsto e^{-t}$, which corresponds to the density of an exponential law of parameter 1. The parameter $n$ corresponds to the number of neurons.}
\end{figure}
\subsection{Simulations for a varying number of neurons}
To further investigate the behavior of the time of extinction in the super-critical regime, we've run a set of simulation for a (fixed) high value of $\gamma$ and for a varying number of neurons. Each element of the set consists in 1000 repetitions with $\gamma = 4$. The value of the size of the network varies from $11$ to $2000$. For each of these values we've estimated the mean and variance of the extinction time $\sigma_N$, and the variance of the re-normalized extinction time $\sigma_N / \mathbb{E}(\sigma_N)$. These simulations have been done in the one-dimensional lattices for all of the three activation functions
\vspace{0.4 cm}
The results of these simulations are presented in Figure \ref{varying_hard}, Figure \ref{varying_lin} and Figure \ref{varying_sig}.
\vspace{0.4 cm}
\begin{figure}[H]
\center{\includegraphics[width=12cm]
{Log_Hard.png}}
\caption{\label{varying_hard} Mean and variance of $\sigma_N$ and variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a linear activation function. In \textbf{a} the red dots represents the values of the estimated mean of $\sigma_N$ for varying numbers of neurons and the red line a logarithmic function fitted over the values of the mean (C = 0.32). In \textbf{b} the blue crosses represent the variance of $\sigma_N$ as the number of neurons increases. The blue dots in \textbf{c} represent the variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a varying number of neurons. All simulations were run with $\gamma = 4$.}
\end{figure}
\begin{figure}[H]
\center{\includegraphics[width=10cm]
{Var_Normalized_Linear.png}}
\caption{\label{varying_lin} Mean of $\sigma_N$ and variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a linear activation function. In \textbf{a} the red dots represents the values of the estimated mean of $\sigma_N$ for varying numbers of neurons and the red line a logarithmic function fitted over the values of the mean. The blue dots in \textbf{b} represent the variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a varying number of neurons. All simulations were run with $\gamma = 4$.}
\end{figure}
\begin{figure}[H]
\center{\includegraphics[width=10cm]
{Log_Sig.png}}
\caption{\label{varying_sig} Mean of $\sigma_N$ and variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a sigmoid activation function. In \textbf{a} the red dots represents the values of the estimated mean of $\sigma_N$ for varying numbers of neurons and the red line a logarithmic function fitted over the values of the mean. The blue dots in \textbf{b} represent the variance of the renormalized extinction time $\sigma_N / \mathbb{E} (\sigma_N)$ for a varying number of neurons. All simulations were run with $\gamma = 4$.}
\end{figure}
\section{Discussion}
\label{discussion}
\subsection{Sub-critical regime}
The histogram built from the simulations for which $\gamma$ is small (Figure \ref{hist_hardthre_sub} and left side of Figure \ref{histlin} and Figure \ref{histsig}) closely approximates the density of a mean $1$ exponential random variable. The fact that the result of the simulations remains identical in all the cases investigated (dimension one, two and three, with hard-threshold, linear function and sigmoid function) suggests that Theorem \ref{thm:metamodel} doesn't merely hold for the specific instantiation of the model for which is was proven, but for a wide class of systems.
\vspace{0.4 cm}
In the one dimensional case with hard threshold the fact that the histogram approximates the density of an exponentially distributed random variable is of course not a surprise as this what Theorem \ref{thm:metamodel} predicts asymptotically. nonetheless it gives us evidences that the approximation by an exponential law holds for relatively small networks (in the simulation concerned the number of neuron in the system is $101$). The number of neurons in animals varies from hundreds \cite{white1986structure} to billions \cite{lent2012many}, so that in our model the approximation by an exponential law is observed for all possible biologically realistic number of neurons. This indicates that, in networks where the randomness of the connections between neurons is not of interest, the model presented here might be an interesting choice for the investigation of metastable behaviors in dense cortical regions.
\vspace{0.4 cm}
\subsection{Super-critical regime}
The histograms built from the simulations with high values of $\gamma$ are visibly not approximating any exponential. Instead they show a distribution that is reminiscent of a gamma distribution with mass concentrated around 1 (See Figure \ref{hist_hardthre_sup} and right side of Figure \ref{histlin} and Figure \ref{histsig}).
\vspace{0.4 cm}
Moreover the evolution of the variance of the renormalized time of extinction (last graph in Figure \ref{varying_hard}, Figure \ref{varying_lin} and Figure \ref{varying_sig}) is seemingly converging toward $0$ as the number of neurons grows for all of the three instantiations investigated.
\vspace{0.4 cm}
For the simulation of the system with hard-threshold, these facts are not a surprise neither, as this is what Theorem \ref{thm:deterministic} predicts. Again the fact that the simulations of the systems with a linear and a sigmoid function for $\phi$ show similar results suggests that Theorem \ref{thm:deterministic} isn't only satisfied for $\phi(x) = \mathbbm{1}_{x > 0}$, but for a wide class of activation functions.
\vspace{0.4 cm}
Moreover the first graph in Figure \ref{varying_hard}, Figure \ref{varying_lin} and Figure \ref{varying_sig} gives us strong evidence that in each of the three instantiations the expectation of the time of extinction grows approximately like a logarithm (up to a multiplicative constant) with respect to the number of neurons. This fact is interesting in itself as the proof of Theorem \ref{thm:deterministic} (which can be found in \cite{andre2019determ}) relies on the fact that for the hard-threshold instantiation of the model we have the following convergence in the super-critical region (at least for $\gamma > 1$)
\begin{equation}
\label{expectlog}
\frac{\mathbb{E} (\tau_N)}{\log (2N + 1)} \underset{N \rightarrow \infty}{\longrightarrow} C,
\end{equation}
where $C$ is a strictly positive (and finite) constant. This is an additional hint that the behavior of the time of extinction should be qualitatively identical in the super-critical region as well for any of the choices we proposed here for the activation function.
\newpage
\section{Acknowledgments}
This work was produced as part of the activities of FAPESP Research, Disseminations and Innovation Center for Neuromathematics (Grant 2013/07699-0, S. Paulo Research Foundation). Morgan André is supported by a FAPESP scholarship (grant number 2017/02035-7), Cecilia Romaro (grant number 88882.378774/2019-01) and Fernando Araujo Najman (grant number 88882.377124/2019-01) are the recipient of PhD scholarships from the Brazilian Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
\bibliographystyle{acm}
|
2,877,628,090,163 | arxiv | \section{Introduction}
Exclusive reactions induced at high momentum transfer in few body
systems provide us with an original way to study the production and propagation of hadrons in cold nuclear matter. In very well defined parts of the phase space, the reaction amplitude develops a logarithmic singularity which enhances the cross section. Here, the reaction amplitude is on solid ground since it depends only on on-shell elementary amplitudes and on low momentum components of the nuclear wave function~\cite{La81}. On the one hand this allows the determination of the scattering cross section of short lived particles. On the other hand this allows the study of the propagation of exotic configurations such as the onset of Color Transparency.
The concept of Color Transparency follows from the underlying
structure of QCD: interactions between ``white'' objects depend on
their transverse size~\cite{MuXX,Fa90}. A hard scattering of the probe produces recoiling particles with small transverse size, whose the subsequent interactions in Nuclear Matter are reduced. There is no doubt that Color Transparency should occur. The question is where and when.
The difficulty resides in the fact that such an exotic configuration evolves quickly toward the asymptotic state of the detected hadron: in order to observe Color Transparency, the characteristic scale of this evolution should be larger, or comparable, to the size of the largest nuclei.
To date there is no convincing evidence for Color Transparency in photon and electron induced reactions. The
reason is that most of the attempts were performed in semi-exclusive
kinematics. In the A(e,e'p) reactions~\cite{EnXX,On95} for instance, it is very likely that the values of the only available hard scale, $Q^2$, are too low to observe color transparency in
the quasi-free kinematics channels, where the energy of the ejected
nucleon $T_p$ and the photon four-momentum are not independent
($T_p=Q^2/2M$). In the range of values of $Q^2$ accessible today, the life time of the small object is
of the order of the distance between nucleons rather than the nuclear
radius. For instance, at the highest $Q^2= 6$ GeV$^2$ where data exist,
the energy of the outgoing proton is only 3 GeV and its characteristic
evolution distance~\footnote{An educated guess of the formation length is $l_f=2\,\hbar c\,p/\Delta m^2$, with $\Delta m^2\sim 0.8$~GeV$^2$. When $p\sim T_p=3$~GeV, one gets $l_f=1.5$~fm} is no more than 1.5 fm, closer to the internucleonic distance rather than the size of the nucleus.
A signal has been reported in A(e,e'$\rho$) reactions, at Fermi Lab.~\cite{FNAL} and DESY~\cite{HERMES}. However, it comes from a subtle interplay between the attenuation of the hadronic component of the virtual photon in the entrance channel and the onset of color transparency in the exit channel (see e.g.~\cite{Ko02}). An experiment~\cite{HaXX} has been completed recently at Jefferson Laboratory (JLab) to disentangle these two effects: one has to wait the final analysis for a more definite answer.
The way to overcome these difficulties is to study reactions induced by
photons in few body systems: exclusive reactions allow the formation length of the hadron to be adjusted to the distance between
nucleons~\cite{La98,Fr97,Fr96}. The kinematics should be chosen so that the interactions of the emerging hadron with a second nucleon are maximal. This occurs when the produced hadron propagates on-shell and rescatters on a second nucleon at rest (triangular logarithmic singularity). A clear signal for color transparency would be the suppression of the final state interaction peak when the momentum transfer increases. This situation is more comfortable than in the more classical study of quasi elastic scattering of electrons from heavy nuclei, where one looks for a change of a flat level of attenuation of the outgoing nucleon, instead of the evolution of a well defined peak.
This conjecture~\cite{La98,Fr97,Fr96} has been tested in two studies of the $^2$H$(e,e^{\prime}p)n$ reaction that have been completed recently at JLab: the first~\cite{BoXX} with two magnetic spectrometers in Hall A; the second~\cite{EgXX} with the CEBAF Large Acceptance Spectrometer~\cite{CLAS} (CLAS) in Hall B. The preliminary results do not exhibit a signal of color transparency (within the experimental and theoretical uncertainties) in the range $2<Q^2<6$~GeV$^2$, when compared to the latest prediction~\cite{La05} of the various interaction mechanisms.
It may occur earlier in Exclusive Photo-production of Mesons. The
reason is that mesons are made up of two quarks. They may recombine more easily, through the exchange of one hard gluon only, and prepare a configuration with a small transverse extension which further evolves toward its asymptotic size. The second reason is that the hard scale is provided by the four momentum transfer $t$ which happens to define the interaction volume, while the virtuality $Q^2$ of the photon that is exchanged in $(e,e'p)$ reactions defines the scale of observation~\cite{La04}.
Indeed, a hint has been reported in the $^4$He($\gamma$,p$\pi^-$) and $^{16}$O($\gamma$,p$\pi^-$) channels recently studied at JLab~\cite{Du03,Ga96}. However, the signal is weak and again these are semi exclusive reactions. The signal must be confirmed by completely exclusive measurements.
\begin{figure}[h]
\begin{center}
\epsfig{file=ratio_tetar_500.eps,width=3.5in}
\caption{Color on line. Ratio of the total
to the quasi-free cross section of the $^2$H$(\gamma,\pi^-p)p$ reaction
against the angle of the recoiling proton whose the momentum is kept
constant at 500 MeV/c (top) and 200 MeV/c (bottom). The peaks labeled $\pi p$ and $pp$ correspond respectively to $\pi p$ and $pp$ on shell rescattering. The dotted curve corresponds to the quasi-free process. The kinematics is coplanar, and positive angles correspond to the emission of the pion and the recoiling proton on the same side of the photon.}
\protect\label{pi1}
\end{center}
\end{figure}
The simplest example is the reaction $^2$H$(\gamma,p\pi^-)p$ in the energy range
$4 < E_{\gamma} < 10$ GeV. For real photons the momentum transfer $t$,
between the incoming photon and the outgoing pion, sets the size of the
interaction volume. As can be seen in Fig.~\ref{pi1}, the on shell
rescattering peaks corresponding to $\pi p$ or $pp$ interactions are
clearly separated. Such a Logarithmic singularity has already been observed at lower
energies~\cite{Ar78}. At the top of each peak, the rescattering amplitude
is dominated by low momentum components of the deuteron wave function
and on mass shell elementary reaction amplitudes (see Ref.~\cite{La81}).
The elementary reaction
$n(\gamma,\pi^-)p$ is well reproduced by a model based on the
exchange of saturating Regge trajectories~\cite{La97}. The $\pi$-nucleon, as well as the nucleon-nucleon, elementary scattering amplitudes are almost entirely absorptive, and well under control at high energy~\cite{La72,PDG}. The rescattering amplitudes are therefore on solid grounds~\cite{La05}: the method can be calibrated at low values of the four momentum transfer $t$. A signal of color transparency would be the reduction of the $\pi p$ rescattering peak when $t$ increases. It could happen sooner in the strange sector, where the strange quark may play a special role.
Alternatively, the method will allow a determination of the cross section of hyperons ($\Lambda$, \ldots) or vector mesons ($\phi$, J/$\Psi$, \ldots) scattering with nucleons, or the interactions between other unstable hadrons. More generally it offers us a way to access the mechanisms of the formation of hadrons in cold nuclear matter.
I have already presented these conjectures in several conference talks~\cite{La98,La98a,La00} and prospect reports.
In the mean time the CLAS collaboration at JLab has completed, with an unprecedented statistics, a study~\cite{g10} of the interactions of a real photon beam (maximum energy 3.7 GeV) on a deuterium target: it provides us with a unique testing ground of these ideas and the method.
This article is not one more attempt to predict the size of the signal of Color Transparency: too many parameters are unknown and although we already know what are the qualitative expectations, only experiments will allow us to quantify the effect. This article is rather an attempt to provide a comprehensive recollection and update of the various matrix elements in the meson production sectors and to provide a base line calculation in a dedicated kinematical range, already accessible at JLab at 6 GeV and its further upgrade to 12 GeV, from which any deviation will be meaningful. The nucleon sector has already been revisited in ref.~\cite{La05}. Section~\ref{pi} deals with the pion production sector, section~\ref{strange} deals with the single kaon production sector, section~\ref{vector} deals with the vector meson production sector (more specifically $\phi$ and J/$\Psi$), section~\ref{ct} addresses issues in Color Transparency, section~\ref{pent} investigates possible implication in the search of exotics, while section~\ref{conc} concludes and summarizes the prospects.
\section{The $^2$H$(\gamma,\pi^-p)p$ reaction \label{pi}}
\subsection{The model \label{model}}
The model is a straightforward update of the diagrammatic approach~\cite{La81} which has been successful in the analysis of meson production reactions at lower energies (let's say in the resonance region). It is particularly well suited to evaluate the reaction amplitude near the singularities of the $S$-matrix. The kinematics, the elementary operators as well as the propagators are relativistic. The deuteron wave function corresponds to the Paris potential~\cite{PaXX}, but any modern wave function leads to very similar results in the momentum range covered by this study.
Let $k=(\nu,\vec{k})$, $p_D=(M_D,\vec{0})$, $p_{\pi}=(E_{\pi},\vec{p_{\pi}})$, $p_1=(E_1,\vec{p_1})$ and $p_2=(E_2,\vec{p_2})$ be the four momenta, in the Lab. system, of respectively the incoming photon, the target deuteron, the outgoing pion, the slow outgoing proton and the fast outgoing proton. The 5-fold fully differential cross section is related to the square of the coherent sum of the matrix elements as follows:
\begin{eqnarray}
\frac{d\sigma}{d\vec{p_1}[d\Omega_{\pi}]_{cm2}}&=& \frac{1}{(2\pi)^5}
\frac{|\vec{\mu}_{c.m.}|m^2}{24|\vec{k}|E_1Q_f}
\sum_{\epsilon,M,m_1,m_2}\left| \sum_{i=I}^{III}
\right.\nonumber \\
&&{\cal M}_i(\vec{k},\epsilon,M,\vec{p_{\pi}},\vec{p_1},m_1,\vec{p_2},m_2)
\nonumber \\
&&\left.
-{\cal M}_i(\vec{k},\epsilon,M,\vec{p_{\pi}},\vec{p_2},m_2,\vec{p_1},m_1)
\right|^2
\label{cross}
\end{eqnarray}
where $\epsilon$ is the polarization vector of the photon and where $M$, $m_1$ and $m_2$ are the magnetic quantum numbers of the target deuteron and the two outgoing protons respectively. The norm of the spinors is $\overline{u}u=1$. The amplitudes are computed in the Lab. frame. The antisymmetry between the two outgoing protons is insured by exchanging the role of $(\vec{p_1}, m_1)$ and $(\vec{p_2}, m_2)$ in the second amplitude. The cross section is differential in the Lab. three-momentum of proton 1, but in the solid angle of the pion expressed in the c.m. frame of the pair made by the pion and proton 2. In this frame, the momentum of the pion is $\vec{\mu}_{c.m.}$ and the total energy is $Q_f= \sqrt{(E_{\pi}+E_2)^2 -(\vec{p_{\pi}} +\vec{p_2})^2}$.
\begin{figure}[h]
\begin{center}
\hspace{-2cm}
\epsfig{file=graphs_ct.eps,width=4.in}
\caption{
The relevant mechanisms. I: Quasi-free. II: Meson-Nucleon rescattering. III: Nucleon-Nucleon rescattering}
\protect\label{graph}
\end{center}
\end{figure}
The cross section and the amplitudes are given for the case of a real photon induced reaction, which I consider in this study. They depend only on the transverse components $J_X$ and $J_Y$ of the hadronic current $\vec{J}$. In the case of a virtual photon beam, additional terms in the cross section are related, as outlined in ref.~\cite{La94}, to the longitudinal component of the hadronic current, $J_z$. The matrix elements are expressed as the scalar product $\vec{J}\cdot\vec{\epsilon}$ from which each component of the hadronic current can be deduced. In the following, I give the expressions of the dominant reaction amplitudes in Fig.~\ref{graph} and discuss their update relevant to the high energy domain accessible at JLab. I refer the reader to~\cite{La78} for more technical details.
\subsubsection{Quasi-free meson production}
The matrix element of the quasi-free amplitude (graph I in Fig.~\ref{graph}) takes the simple form:
\begin{eqnarray}
{\cal M}_I(\vec{k},\epsilon,M,\vec{p_{\pi}},\vec{p_1},m_1,\vec{p_2},m_2)=
\nonumber \\
i \sum_{m_n m_l m_s} \sum_{ls} (lm_lsm_s|1M)(\frac{1}{2}m_n\frac{1}{2}m_1|sm_s)
\nonumber \\
u_l(|\vec{p_1}|) Y_l^{m_l}(\vec{p_1})
T_{\gamma n}(\vec{p_2},m_2,-\vec{p_1},m_n)
\label{q_f}
\end{eqnarray}
where $u_0$ and $u_2$ are the $S$ and $D$ components of the deuteron Paris wave function~\cite{PaXX}, and where $T_{\gamma n}$ is the amplitude of the elementary $n(\gamma,\pi^-)p$ reaction. I use the on-shell expression (see Appendix) of the Regge amplitude of ref.~\cite{La97}, which is based on the exchange of the saturating Regge trajectories of the pion and the rho mesons. It leads to a good description of the differential cross section of the $p(\gamma,\pi^+)n$ reaction at large momentum transfer $-t$ in the photon energy range of JLab (around 4~GeV). As shown in Fig.~\ref{pi-}, it leads also to a fair accounting of the more recent JLab data~\cite{Zh04} in the $\pi^-$ channel. I refer the reader to~\cite{La97} for a throughout presentation of this Regge model and the choice of the coupling constants and parameters: I use the same in this study, except for the cut-off mass of the hadronic form factor, which I chose to be $\Lambda=0.7$~GeV$^2$, instead of 0.8~GeV$^2$ in ref.~\cite{La97}.
\begin{figure}[h]
\begin{center}
\epsfig{file=sig_pi-_4gev.eps,width=3.in}
\caption{Color on line.
The cross section of the elementary reaction $n(\gamma,\pi^-)p$ at $E_{\gamma}= 4$~GeV. The curve is the prediction of the Regge model. The dashed curve corresponds to linear trajectories. The full curves correspond to saturating trajectories, for two choices of the cut-off mass in the hadronic form factor. The data has been recently recorded at JLab~\cite{Zh04}.}
\protect\label{pi-}
\end{center}
\end{figure}
When the momentum, $\vec{p_1}$, of one of the proton is low only one amplitude dominates the cross section~(\ref{cross}), which takes the simple form~\cite{La81,La78}:
\begin{eqnarray}
\frac{d\sigma}{d\vec{p_1}d\Omega_{\pi}}=
(1+\beta_1 \cos\theta_1)\rho(|\vec{p_1}|)\frac{d\sigma}{d\Omega_{\pi}}
(\gamma n\rightarrow \pi^-p)
\label{cross_qf}
\end{eqnarray}
where $\beta_1=p_1/E_1$ and $\theta_1$ are the velocity and the angle of the spectator nucleon. This is nothing but the relation between the yield and the elementary cross section of the production of a pion on a nucleon which moves with the velocity $-\vec{\beta_1}$. The number of target nucleons is $\rho(|\vec{p_1}|)d\vec{p_1}$, where $\rho(|\vec{p_1}|)$ is the momentum distribution of the neutron in deuterium, while $(1+\beta_1 \cos\theta_1)$ is the flux of photons seen by the moving target nucleon.
\subsubsection{Meson-nucleon rescattering}
The matrix element of the pion-proton rescattering amplitude (graph II in Fig.~\ref{graph}) takes the form:
\begin{eqnarray}
{\cal M}_{II}(\vec{k},\epsilon,M,\vec{p_{\pi}},\vec{p_1},m_1,\vec{p_2},m_2)=
\nonumber \\
i \sum_{m_n m_p} (\frac{1}{2}m_n\frac{1}{2}m_p|1M)
\int \frac{d^3\vec{p}}{(2\pi)^3} \frac{u_0(p)}{\sqrt{4\pi}}
\frac{1}{q^2_{\pi}-m^2_{\pi}+i\epsilon}
\nonumber \\
\frac{m}{E_p}
T_{\gamma n}(\vec{p_2},m_2,-\vec{p},m_n)T_{\pi N}(\vec{p_1},m_1,\vec{p},m_p)
\nonumber \\
+ D \; \mathrm{wave}\ \mathrm{part} \;\;
\label{pi_rescat}
\end{eqnarray}
The integral runs on the three momentum of the spectator proton in the loop, which has been put on-shell, $p^{\circ}=E_p=\sqrt{\vec{p}^2+ m^2}$, by the integration over its energy $p^{\circ}$. It can be split in two parts:
\begin{eqnarray}
{\cal M}_{II} = {\cal M}_{II}^{on}+{\cal M}_{II}^{off}
\end{eqnarray}
The singular part of the rescattering integral runs between the minimum and maximum values of the momentum of the spectator proton in the loop for which the pion can propagate on-shell:
\begin{eqnarray}
p_{min}(p\pi)= \frac{P}{Q_s}E_{c.m.}- \frac{E}{Q_s}p_{c.m.}
\label{pmin_ppi}
\end{eqnarray}
\begin{eqnarray}
p_{max}(p\pi)= \frac{P}{Q_s}E_{c.m.}+ \frac{E}{Q_s}p_{c.m.}
\label{pmax_ppi}
\end{eqnarray}
where $E=E_{\pi}+E_1$, $\vec{P}=\vec{p_{\pi}}+\vec{p_1}$ and $Q_s=\sqrt{E^2-\vec{P}^2}$ are respectively the energy, the momentum and the mass of the scattering $\pi p$ pair. The momentum and energy of the spectator proton, in the c.m. frame of the $\pi p$ pair are
\begin{eqnarray}
p_{c.m.}= \frac{\sqrt{(Q^2_s-(m+m_{\pi})^2)(Q^2_s-(m-m_{\pi})^2)}}{2 Q_s}
\end{eqnarray}
\begin{eqnarray}
E_{c.m.}= \sqrt{p^2_{c.m.}+m^2}
= \frac{Q^2_s+m^2-m^2_{\pi}}{2 Q_s}
\end{eqnarray}
It takes the form:
\begin{eqnarray}
{\cal M}_{II}^{on}= \frac{\pi}{(2\pi)^3\sqrt{{4\pi}}} \sum_{m_nm_p}
\frac{1}{2P} (\frac{1}{2}m_n\frac{1}{2}m_p|1M)
\nonumber \\
\int_0^{2\pi} d\phi \int_{|p_{min}(p\pi)|}^{p_{max}(p\pi)} pu_0(p) dp
\frac{m}{E_p}
\left[T_{\gamma n}T_{\pi N}\right]_{q^2_{\pi}=m^2_{\pi}}
\nonumber \\
+ D \; \mathrm{wave}\ \mathrm{part} \;\;
\label{sing_pin}
\end{eqnarray}
The two dimensional integral is done numerically. It depends only on on-shell elementary amplitudes. The weight, $pu_0(p)$,
selects nucleons almost at rest in the deuterium when the lower bound, $p_{min}(p\pi)$, of the integral vanishes. This is the origin of the meson-nucleon scattering peak in Fig.\ref{pi1}, which is therefore on solid grounds.
The $\pi N$ scattering amplitude can be expressed as:
\begin{eqnarray}
T_{\pi N} = (m_1|f(Q_s,t_r)
+ g(Q_s,t_r) \vec{\sigma}\cdot \vec{k}_{\perp}|m_p)
\end{eqnarray}
where $t_r=(p_{\pi}-q_{\pi})^2$ is the four momentum transfer at the $\pi p$ recattering vertex and $\vec{k}_{\perp}= \vec{p_{\pi}}\times \vec{q_{\pi}}$ is the direction perpendicular to the scattering plane. At high energies ($Q_s> 2$~GeV) the central part dominates at forward angles and is almost entirely absorptive. It can be parameterized as follows:
\begin{eqnarray}
f(Q_s,t_r)=-\frac{Q_s p_{c.m.}}{m} (\epsilon + i)\sigma_{\pi^-p} \exp[\frac{\beta_{\pi}}{2}t_r]
\label{scat_pin}
\end{eqnarray}
Above $Q_s\sim 2$~GeV, the total cross section stays constant at the value $\sigma_{\pi^-p}= 30$~mb~\cite{PDG}, and the fit of the differential cross section at forward angles leads to a slope parameter $\beta_{\pi}=6$~GeV$^{-2}$~\cite{La72}. At high energy the ratio between the real and imaginary part of the amplitude is small~\cite{PDG} and I set it to zero in this study.
With such an absorptive amplitude it is easy to see, from eqs.~(\ref{q_f})
and~(\ref{sing_pin}), that the singular part of the rescattering amplitude interferes destructively with the quasi-free amplitude.
The principal part of the rescattering integral takes the form:
\begin{eqnarray}
{\cal M}_{II}^{off}= \frac{i}{(2\pi)^3\sqrt{{4\pi}}} \sum_{m_nm_p}
(\frac{1}{2}m_n\frac{1}{2}m_p|1M)
\nonumber \\
\oint \frac{d^3\vec{p} u_0(p)T_{\gamma n}T_{\pi N}}{q^2_{\pi}-m^2_{\pi}}
\frac{m}{E_p}
+ D \; \mathrm{wave}\ \mathrm{part} \;\;
\end{eqnarray}
It turns out that it vanishes~\cite{La78,La81} when $p_{min}(p\pi)=0$ at the top the $\pi N$ recattering peak and contributes little to its tails only, in Fig.~\ref{pi1} for instance.
Since the Regge amplitude $T_{\gamma n}$ varies rapidly, as $s^{\alpha(t_f)}$, with the total energy $s=(E_2+E_q)^2-(\vec{p_2}+\vec{q}_{\pi})^2$ and momentum transfer $t_f=(k-q_{\pi})^2$, it can not be factorized out of the integral which should be evaluated numerically. This is not a problem for its singular part: it is a two-fold integral which involves well defined on-shell quantities. Its principal part is a three fold integral which requires a good knowledge of the off-shell extrapolation of the elementary amplitude. Since its contribution is small near the singularity, I do not take it into account in this study, in order to save time in the Monte Carlo simulation in the full phase space (section~\ref{clas}). It will be taken be into account later, in the final analysis of experimental data.
\subsubsection{Nucleon-Nucleon rescattering}
The matrix element of the proton-proton rescattering amplitude (graph III in Fig.~\ref{graph}) takes the form:
\begin{eqnarray}
{\cal M}_{III}(\vec{k},\epsilon,M,\vec{p_{\pi}},\vec{p_1},m_1,\vec{p_2},m_2)=
\nonumber \\
i \sum_{m_n m_pm'_p} (\frac{1}{2}m_n\frac{1}{2}m_p|1M)
\int \frac{d^3\vec{p}}{(2\pi)^3} \frac{u_0(p)}{\sqrt{4\pi}}
\frac{1}{{p^{\circ}}'-E'_p+i\epsilon}
\nonumber \\
\frac{m}{E_p}
T_{\gamma n}(\vec{p'},m'_p,-\vec{p},m_n)
T_{pp}(\vec{p_2},m_2,\vec{p_1},m_1,\vec{p'},m'_p,\vec{p},m_p)
\nonumber \\
+ D \; \mathrm{wave}\ \mathrm{part} \;\;
\label{p_rescat}
\end{eqnarray}
The integral runs on the three momentum of the spectator proton in the loop, which has been put on-shell, $p^{\circ}=E_p=\sqrt{\vec{p}^2+ m^2}$, by the integration over its energy $p^{\circ}$. It can be split in two parts:
\begin{eqnarray}
{\cal M}_{III} = {\cal M}_{III}^{on}+{\cal M}_{III}^{off}
\end{eqnarray}
The singular part of the rescattering integral runs between the minimum and maximum values of the momentum of the spectator proton in the loop for which the struck proton can propagate on-shell:
\begin{eqnarray}
p_{min}(pp)= \frac{P}{W}E_{c.m.}- \frac{E}{W}p_{c.m.}
\label{pmin_pp}
\end{eqnarray}
\begin{eqnarray}
p_{max}(pp)= \frac{P}{W}E_{c.m.}+ \frac{E}{W}p_{c.m.}
\label{pmax_pp}
\end{eqnarray}
where $E=E_2+E_1$, $\vec{P}=\vec{p_2}+\vec{p_1}$ and $W=\sqrt{E^2-\vec{P}^2}$ are respectively the energy, the momentum and the mass of the scattering $p p$ pair. The momentum and energy of the spectator proton, in the c.m. frame of the $p p$ pair are
\begin{eqnarray}
p_{c.m.}= \frac{\sqrt{W^2-4m^2}}{2}
\end{eqnarray}
\begin{eqnarray}
E_{c.m.}= \sqrt{p^2_{c.m.}+m^2}
= \frac{W}{2}
\end{eqnarray}
As in the previous section, the singular part of the integral picks the low momentum components of the deuteron wave function, relies on on-shell elementary matrix elements and is maximum when $p_{min}(pp)=0$. The principal part vanishes under the rescattering peak, and contributes little to its tails: the situation is the same as in the $np$ rescattering sector of the $^2$H$(e,e^{\prime}p)n$ reaction (see {\it e.g.} Fig.~2 of~\cite{La05}).
The dependency upon $t=(k-p_{\pi})^2$ of the elementary photo-production amplitude $T_{\gamma n}$ is fixed by the external kinematics (see graph~III in Fig.~\ref{graph}) and not by the internal kinematics in the loop integral. Therefore it can be safely factorized out the integrals and evaluated assuming that the target nucleon is at rest in the deuteron, in which case $s= 2m\nu + m^2$. The integrals can be performed analytically, following the method outlined in~\cite{La78}. I have checked~\cite{La05} that this approximation is very close (within 10~\%) to the full evaluation of the integrals, in the rescattering peak region. This saves computing time and both the singular and principal parts have been retained: the proton rescattering peak is therefore slightly wider than the pion rescattering peak in Fig.~\ref{pi1}.
The proton-proton scattering amplitude is taken as:
\begin{eqnarray}
T_{pp} = (m_2m_1|\alpha + i\gamma (\vec{\sigma_1}+
\vec{\sigma_2})\cdot \vec{k}_{\perp}
\nonumber \\
+\;\mathrm{spin-spin} \;\mathrm{terms}\;|m^{\prime}_pm_p)
\label{NN}
\end{eqnarray}
where $\vec{k}_{\perp}$ is the unit vector perpendicular to the scattering plane.
Above 500 MeV, the central part $\alpha$ dominates. It is almost entirely
absorptive, and takes the simple form
\begin{equation}
\alpha = -\frac{Wp_{cm}}{2m^2} \;(\epsilon + i)\;\sigma_{NN} \;\exp[\frac{\beta_N}{2}t_r]
\label{abs_amp}
\end{equation}
Where $t_r=(p^{\prime}-p_1)^2$ is the four momentum transfer at the $pp$ scattering vertex. In the forward direction its imaginary part is related to the total cross section $\sigma_{NN}$, while the slope parameter $\beta_N$ is related to the angular distribution of NN scattering at forward angles. I use the same values as in~\cite{La05}. Note that the difference in the norm of eqs.~(\ref{abs_amp}) and~(\ref{scat_pin}) comes from the choice of the norm of the spinors, $\overline{u}u=1$.
\subsection{Coplanar kinematics}
Fig.~\ref{pi1} exhibits the salient predictions of the model. It corresponds to a coplanar kinematics, that can be achieved by detecting the pion and one of the proton with two well shielded magnetic spectrometers in Hall A or Hall C at JLab for instance. It shows the ratio of the full cross section to the quasi-free cross section, as function of the polar angle of the slow nucleon, $\theta_R=\theta_1$, when its momentum, $P_R=|\vec{p_1}|$, is kept constant at 200 MeV/c (lower curve) or 500 MeV/c (upper curve). The mass of the pair made of the pion and the fast (second) nucleon is kept constant at the value $W=\sqrt{(p_2+p_{\pi})^2}= 2.896$~GeV that corresponds to the absorption of a 4~GeV photon by a nucleon at rest. The four momentum transfer is also kept constant at the value $t=(k-p_{\pi})^2= -3$~GeV$^2$ which corresponds to the emission of the pion around $90^{\circ}$ in the $\pi p_2$ c.m. frame.
At high recoil momentum, rescattering mechanisms dominate over the quasi-free contribution. The top of the peaks corresponds to kinematics where an on-shell pion or nucleon can be produced on a nucleon at rest ($p_{\rm min}=0$), in the rescattering amplitude. The width of the peaks reflects the Fermi motion of the target nucleon. The physical picture is the following. The pion (resp. proton) is photo-produced on a neutron at rest in deuterium, propagates on-shell and rescatters on the spectator proton, also at rest in deuterium. Two body kinematics requires that the angle between the scattered pion (proton) and the recoiling proton is constant (strictly $90^{\circ}$ for $pp$ elastic scattering). Since the recoiling nucleon momentum is fixed, the angles which it makes with the total momenta $\vec{p_1}+\vec{p_{\pi}}$ or $\vec{p_1}+\vec{p_2}$ are also fixed: typically $70^{\circ}$. So, the $\pi p$ or the $pp$ rescattering peaks form a cone centered along the direction of the total momentum of the corresponding scattering pair. In coplanar kinematics, two peaks appear for each rescattering, depending whether or not the pion and the recoiling proton are emitted on the same side of the photon.
The difference in the height of each of these two peaks reflects the rapid variation with the photon energy of the elementary pion photo-production Regge cross sections: $s^{2\alpha(t)-2}$. Although the mass of the pair made of the pion and the fast proton has been kept constant in Fig.~\ref{pi1}, the incoming photon must also provide the energy of the slow recoiling nucleon: this depends on its direction of motion. For instance, at the top of the $\pi p$ peak that is located at the left in Fig.~\ref{pi1} the photon energy is $E_{\gamma}=3.432$~GeV, while it is $E_{\gamma}=5.436$~GeV at the top of the peak at the right. Since, in the rescattering amplitude, the photo-production occurs on a nucleon at rest the corresponding masses are respectively 2.705 and 3.328~GeV. The situation is the same at the top of the two $pp$ rescattering peaks.
\begin{figure}[h]
\begin{center}
\epsfig{file=pi_pr.eps,width=3.5in}
\caption{
Ratio of the total
to the quasi-free cross section of the $^2$H$(\gamma,\pi^-p)p$ reaction
against the momentum of the recoiling proton, at the top of the
$\pi p$ rescattering peak. The full line includes $\pi p$ scattering and $pp$ scattering (small effect). The dotted curve corresponds to the quasi-free process.}
\protect\label{pi2}
\end{center}
\end{figure}
At low recoil momentum, this effect is less dramatic since the energy difference is less important (it vanishes at $P_r= 0$!). Here the rescattering amplitudes interfere destructively with the quasi-free amplitude, consistently with unitary. Since the elementary $\pi p$ and $pp$ scattering amplitudes are dominantly absorptive in the energy range covered by this study, a part of the strength is shifted from the quasi-free channel to inelastic channels. Above $p_r=300$~MeV/c, rescattering contributions take over and dominate the cross section. Figs.~\ref{pi2} shows this evolution of the cross section
at the top of the $\pi p$ rescattering peak ($\theta_R=-50^{\circ}$ in Fig.~\ref{pi1}) with the recoil momentum
$P_R$.
\subsection{CLAS kinematics \label{clas}}
The CLAS~\cite{CLAS} set up at JLab allows to record events in the full available phase space and is well suited to make a survey of the cross section of the $^2$H$(\gamma,p\pi^-)p$ reaction and to exploit its features which we just discussed.
Three superconducting coils generate a toroidal field perpendicular to the photon beam axis and define six sectors where particles are detected by wire chambers and scintillators. The geometrical fiducial acceptance represents more than $2\pi$~sr. It covers a range of polar angles between $11^{\circ}$ and $140^{\circ}$, but the coils define six azimuthal regions where the detector is blind.
I have implemented the code which computes the cross section of the $^2$H$(\gamma,p\pi^-)p$ reaction in a Monte-Carlo code which generates events in the full fiducial acceptance of CLAS. I sample, with a flat distribution, the three-momentum $\vec{p_1}$ of the slow proton and the two angles $\cos\theta_2$ and $\phi_2$ of the fast proton. If each proton falls in the fiducial acceptance, which I take from ref.~\cite{Nic04}, I record the kinematics of the event in a database (namely an Ntuple in the CERN package PAW~\cite{PAW}) and I weight it with the corresponding differential cross section
\begin{eqnarray}
\frac{d\sigma}{dp_1d\Omega_{p_1}d\Omega_{p_2}}=J\times
\frac{d\sigma}{d\vec{p_1}[d\Omega_{\pi}]_{cm2}}
\end{eqnarray}
where $J$ is the relevant Jacobian
\begin{eqnarray}
J= \frac{Q_f |\vec{p_2}|^3|\vec{p_1}|^2}
{\mu_{c.m.}|E_{\pi}|\vec{p_2}|^2 -E_2\vec{p_{\pi}}\cdot\vec{p_2}|}
\end{eqnarray}
(see section~\ref{model} for the definition of momenta and energies.)
The events in the data base are then binned as the experimental data, with the same cuts. This is the most straightforward way to compare a theory with experiments, or to simulate experiments, that are carried out over a wide and complicated phase space.
\begin{figure}[h]
\begin{center}
\epsfig{file=panel_ppi_w29_6gev.eps,width=3.5in}
\caption{CLAS kinematics for $\pi p$ rescattering in the $^2$H$(\gamma,pp_R)\pi^-$ reaction. The beam end point is 6 GeV. The full histograms correspond to the full calculation, while the dashed histograms correspond to the quasi-free process only. See text for the description of the cuts which have been used in each window.}
\protect\label{panel_ppi}
\end{center}
\end{figure}
Figs.~\ref{panel_ppi} shows various observables which emphasize the pion nucleon rescattering sector. The real photon beam end point has been set to $E_{\gamma}=6$~GeV, but the mass of the fast proton pion pair has been restricted to the range $2.85<W_{p_2\pi}<2.95$~GeV to keep contact with Fig.~\ref{pi1}. Only events corresponding to large momentum transfer $-t>1$~GeV$^2$ has been retained. The panel shows cross sections integrated over the various bins, within the CLAS fiducial acceptance. The top parts show the distribution of the minimum momentum $p_{min}(p_1\pi)$, eq.~\ref{pmin_ppi}, of the spectator proton, in the pion nucleon scattering loop, for which the pion can propagate on-shell. On the left, the cut $400<P_R<600$~MeV/c has been applied on the momentum $p_1$ of the slow nucleon: the pion nucleon rescattering peak clearly appears at $p_{min}=0$. On the right, the cut $250<p_R<350$~MeV/c has been applied: rescattering effects are small here, consistently with Fig.~\ref{pi2}, and the shape of the distribution reflects the kinematics and the detector acceptance. This is a good reference point which emphasizes the quasi-free process.
A further cut $-50<p_{min}(p_1\pi)<50$~MeV/c has been applied in the bottom parts of Fig.~\ref{panel_ppi}: it emphasizes pion nucleon rescattering. The $t$ distribution is plotted on the left for either low recoil momentum ($\sim 300$~MeV/c) or the high recoil momentum ($\sim 500$~MeV/c) bands. The ratio of second ($p_R=500$~MeV/c) to the first ($p_R=300$~MeV/c) $t$ distribution is plotted on the right. In plane wave, it is nothing but the ratio of these high momentum to low momentum components of the nucleon momentum distribution in deuterium. The full ratio is really a measure of the evolution of the top of the pion nucleon rescattering peak with the four momentum transfer $t$, which fixes the hard scale. It is almost flat (the oscillations are due to the statistical accuracy of the Monte Carlo sampling) and provides us with a good starting point to look for deviations, especially at high $-t$, which could reveal the onset of color transparency for instance. The last bins ($-t>6$~GeV$^2$) in the $t$ distribution should be disregarded since they correspond to the kinematical limits where the detector acceptance differs strongly at low and high recoil momentum.
\begin{figure}[h]
\begin{center}
\epsfig{file=panel_pp_w29_6gev.eps,width=3.5in}
\caption{CLAS kinematics for $pp$ rescattering in the $^2$H$(\gamma,pp_R)\pi^-$ reaction. The beam end point is 6 GeV. The full histograms correspond to the full calculation, while the dashed histograms correspond to the quasi-free process only. See text for the description of the cuts which have been used in each window.}
\protect\label{panel_pp}
\end{center}
\end{figure}
Fig.~\ref{panel_pp} shows the same observables in the proton proton rescattering sector. Now, the minimum momentum $p_{min}(p_2p_1)$ is the lowest value, eq.~\ref{pmin_pp}, of the momentum of the spectator proton for which the other proton can propagate on-shell in the nucleon nucleon scattering loop. Again, the bins at the highest values of $-t$ should be disregarded since they lie at the kinematics limits. Also the statistical accuracy can be improved by running the Monte Carlo code with more events (but also longer!).
\begin{figure}[h]
\begin{center}
\epsfig{file=lego_sing_w29_6gev.eps,width=3.5in}
\caption{The joint distribution of singularities in $pp$ and $p\pi$ rescattering in the $^2$H$(\gamma,pp_R)\pi^-$ reaction. The beam end point is 6 GeV. The range of mass of the fast $p \pi$ pair is $2.85<W_{p_2\pi}<2.95$~GeV. The range of the four momentum transfer is $-t>1$~GeV$^2$. The range of the momentum of the slow proton is $400<P_R<600$~MeV/c.}
\protect\label{sing_bidim}
\end{center}
\end{figure}
In Fig.~\ref{panel_ppi}, $pp$ rescattering gives also a contribution below the $p\pi$ rescattering peak. Also, $p\pi$ rescattering gives a contribution under the $pp$ rescattering peak in Fig.~\ref{panel_pp}. These contaminations can be removed by cutting the overlapping region in the joint distribution of the rescattering singularities which is shown in Fig.~\ref{sing_bidim}.
In the Monte Carlo simulation, the choice have been made to detect the two protons. The advantage is that the efficiency to detect a proton in each sector of CLAS is very good (better than $90$~\%, and only a small correction has to be applied to the histograms before comparing them to experiment): this is particularly interesting when one selects events corresponding to large recoil momentum, of which the probability is small. But this prevents to record events with small recoil momentum, since CLAS cannot detect protons with momentum lower than $\sim 250$~MeV/c. For recording such events one must detect the $\pi^-$, which is bent inward, in the beam direction, by the magnetic field and also decay in flight. Its detection efficiency is much smaller that the one of a proton, but on the other hand the cross section is higher at low recoil momentum.
\subsection{Determination of the $\pi^-$ elementary production amplitude}
\begin{figure}[h]
\begin{center}
\epsfig{file=elem_lowpr.eps,width=3.5in}
\caption{Color on line. The c.m. cross section of the $n(\gamma,\pi^-)p$ reaction as extracted from the analysis of the $^2$H$(\gamma,\pi^-p)p$ reaction when the angle of the recoiling proton varies but its momentum is kept
constant at 10, 50, 100 and 200 MeV/c. The dotted curves correspond to the quasi-free process. The kinematics is coplanar, and only the part of the angular distribution, which corresponds to the emission of the pion and the recoiling proton on different side of the photon, is shown. Each curve is labeled by the value of the corresponding recoil momentum; on the left for the quasi-free result, on the right for the full calculation}
\protect\label{low_p}
\end{center}
\end{figure}
All the preceding discussion relies on the good knowledge of the cross section of the elementary reaction $n(\gamma,\pi^-)p$. It can be determined in the same data set~\cite{g10}, since no free neutron target exists. To that end, one has to detect the $\pi^-$ and the fast proton $p_2$, and restrict the analysis to small values of the spectator slow proton, let say $p_1< 100$~MeV/c.
Fig.~\ref{low_p} shows that rescattering corrections are less than $10$~\% up to recoil momentum around $50$~Mev/c. Above, the effects are larger. The neutron cross section is deduced from the deuterium cross section as follows
\begin{eqnarray}
(1+\beta_1 \cos\theta_1)\frac{d\sigma}{d\Omega_{\pi}}=
\frac{d\sigma}{d\vec{p_1}d\Omega_{\pi}}
\times \frac{1}{\rho(|\vec{p_1}|)}
\label{elem_Xsec}
\end{eqnarray}
where $\rho(|\vec{p_1}|)$ is the neutron momentum distribution in the deuteron. I kept the flux factor in order to quantify the corresponding slope of the quasi-free cross section.
The kinematics, $W=2.896$~MeV and $t=-3$~GeV$^2$, corresponds to one of the few data point~\cite{Zh04} recently measured in Hall A at JLab. It corresponds to the kinematics in Fig.~\ref{pi1}, as well as the average kinematics covered by this study, and indicates that the elementary amplitude is not far off the mark.
This experimental datum (and few others) has been obtained from the analysis of the $^2$H$(\gamma,\pi^-)$ reaction induced by a Bremsstrahlung photon beam, which averages the elementary cross section over the Fermi motion of the neutron in the deuterium target. Since the major contribution comes from neutron momentum in the range of $\sim 80$~Mev/c, interaction effects cannot be neglected. The analysis of the high statistics Hall B experiment~\cite{g10}, along the line of Fig.~\ref{low_p}, would be very welcome in order to enlarge the data set and improve its accuracy.
In order to keep contact with Fig.~\ref{pi1}, the cross sections at $p_R=p_1= 200$~MeV/c are also plotted in Fig.~\ref{low_p}. The ratio of the full calculation to the quasi-free one is plotted in Fig.~\ref{pi1}. When multiplied by the momentum distribution $\rho(|\vec{p_1}|)$ these curves become the fully differential cross sections $d\sigma/d\vec{p_1}d\Omega_{\pi}$. One notes that the quasi-free cross sections follow almost exactly the variation of the flux factor $1+\beta_1 \cos\theta_1$, but decreases a little when the recoil momentum $p_1$ increases, by 2\% at 100~MeV/c, 5\% at 200~MeV/c and no more than 10\% above. This is a consequence of the choice of the off-shell extrapolation of the elementary pion photo production amplitude (see Appendix) and a measure of the uncertainty on the determination of the quasi-free amplitude at high recoil momenta. However, rescattering amplitudes dominate here (by about a factor 5 and more, at the top of the rescattering peak) and they are driven by on-shell elementary matrix elements and the low momentum components of the nuclear wave function. Therefore the uncertainties in the off-shell extrapolation of the elementary amplitudes do not affect the cross section in the domains of the rescattering peaks.
\section{The strange sector \label{strange}}
The extension to the $^2$H$(\gamma,K^+\Lambda)n$ reaction is straightforward. The amplitude of the elementary reaction $p(\gamma,K^+)\Lambda$ is driven by the exchange of the Regge trajectories of the $K$ and the $K^*$~\cite{La97}. Besides trivial changes in the mass of the particles, the Regge trajectories and the coupling constants, the reaction amplitudes exhibit the same form as in the pion production sector (section~\ref{pi}).
The kaon is produced on the proton. Since the neutron and the $\Lambda$ in the final state are distinct particles, there is no need to antisymmetrize the reaction amplitude.
All the coupling constants and Regge propagators are given in ref.~\cite{La97}. I use the slope parameters $\beta_K=3$~GeV$^{-2}$~\cite{La72} and $\beta_\Lambda=\beta_{pn}$~\cite{La05}, and the cross sections~\cite{PDG}: $\sigma_{K^+n}= 18$~mb and $\sigma_{\Lambda n}=35$~mb in the $K^+n$ and $\Lambda n$ scattering amplitudes respectively. However these parameters are less known than in the $\pi N$ and $NN$ scattering sector, and their choice should be refined by the analysis of the $^2$H$(\gamma,K^+\Lambda)n$ reaction at low four momentum transfer $t$.
\begin{figure}[h]
\begin{center}
\epsfig{file=ratio_tetar_k.eps,width=3.5in}
\caption{Color on line. Ratio of the total
to the quasi-free cross section of the $^2$H$(\gamma,K^+\Lambda)n$ reaction
against the angle of the recoiling neutron whose the momentum is kept
constant at 500 MeV/c (top curves) and 200 MeV/c (bottom curves). The peaks labeled $K^+n$ and $\Lambda n$ correspond respectively to $K^+n$ rescattering and $\Lambda n$ on shell rescattering. The dotted curves correspond to the quasi-free process. The kinematics is coplanar, and positive angles correspond to the emission of the kaon and the recoiling neutron on the same side of the photon. The top panel corresponds to a rotating phase, while the bottom panel corresponds to no phase in Regge trajectories. }
\protect\label{kaon}
\end{center}
\end{figure}
Fig.~\ref{kaon} shows the ratio of the full cross section to the quasi-free cross section, as function of the polar angle of the slow neutron, $\theta_R=\theta_n$, when its momentum, $P_R=|\vec{n}|$, is kept constant at 200 MeV/c (lower curve) or 500 MeV/c (upper curve). The mass of the pair made of the $K^+$ and the $\Lambda$ is kept constant at the value $W=\sqrt{(p_{\Lambda}+p_{K^+})^2}= 2.896$~GeV that corresponds to the absorption of a 4~GeV photon by a nucleon at rest. The four momentum transfer is also kept constant at the value $t=(k-p_{K^+})^2= -3$~GeV$^2$ which corresponds to the emission of the kaon around $90^{\circ}$ in the $K^+ \Lambda$ c.m. frame.
At high recoil momentum, the pattern is the same as in the pion production sector (Fig.~\ref{pi1}). The height of the rescattering peaks are different simply because the hadronic cross sections and their slopes are different. At low recoil momentum however, the pattern is different in the $Kn$ scattering sector. The reason is that contrary to the $n(\gamma,\pi^-)p$ reaction Regge amplitudes, the $p(\gamma,K^+)\Lambda$ reaction Regge amplitudes exhibit a phase, $\exp [-\imath \pi \alpha(t)]$ (see the discussion in section~2.3.2 of ref.~\cite{La97}). In the $Kn$ rescattering integral, which selects nucleons almost at rest in deuterium, the average four momentum transfer $t$ is different than in the quasi-free production amplitude, where the target nucleon moves with a momentum fixed by the kinematics. This difference in $t$ changes the phase and compensates the destructive interference, between the quasi-free amplitude and the absorptive rescattering amplitude, around $200$~MeV/c. When the Regge phases are turned off, in the bottom part of Fig.~\ref{kaon}, the pattern becomes the same as in the $\pi^-$ production channel, in Fig.~\ref{pi1}.
This effect does not occur in the $\Lambda n$ rescattering sector, since $t$ is defined by the external kinematics and is the same in the rescattering amplitude and the rescattering amplitude (See Fig.~\ref{graph}).
\begin{figure}[h]
\begin{center}
\epsfig{file=elem_lowpr_k.eps,width=3.5in}
\caption{Color on line. The c.m. cross section of the $p(\gamma,K^+)\Lambda$ reaction as extracted from the analysis of the $^2$H$(\gamma,K^+\Lambda)n$ reaction when the angle of the recoiling neutron varies but its momentum is kept
constant at 10, 50, 100 and 200 MeV/c. The dotted curves correspond to the quasi-free process. The kinematics is coplanar, and only the part of the angular distribution, which corresponds to the emission of the Kaon and the recoiling neutron on different side of the photon, is shown. Each curve is labeled by the value of the corresponding recoil momentum; on the left for the quasi-free result, on the right for the full calculation}
\protect\label{low_p_kaon}
\end{center}
\end{figure}
Fig.~\ref{low_p_kaon} shows the value of the elementary kaon photo production cross section as extracted from a deuteron target with eq.~\ref{elem_Xsec} for different values of the recoiling neutron. As in the pion production sector the quasi-free cross section follows the variation of the flux factor $1+\beta_1 \cos\theta_1$ and departs from the free cross section ($p_R=10$~MeV/c) when the recoil momentum increases. The datum in Fig.~\ref{low_p_kaon} is one of the few experimental cross sections available at high momentum transfer~\cite{An76}
The reaction amplitudes rely on the elementary photo-production of kaon on a proton target. The Regge model leads to a fair agreement with the existing set of data around $E_{\gamma}= 4$~GeV (see~\cite{La97}). This data set can be greatly enlarged by the analysis of the high statistical accuracy experiment~\cite{g11} recently completed in CLAS, with a beam of real photon of $4$~GeV on a proton target.
To achieve the full exclusivity of kaon production off a deuterium target one needs to detect both the $K^+$ and the $\Lambda$, which can be identified by its decay into a $\pi^-p$ pair. CLAS is ideally suited to record such a three charged particle configuration. When the $\Lambda$ decay distribution is implemented in the Monte Carlo code, the simulation leads to histograms similar to those which have been obtained in the pion sector. Since the physical content is the same, they are not shown but will be compared with experiment with the same cuts when the analysis is completed.
Other channels can also be studied. Of particular interest is the $^2$H$(\gamma,K^{\circ}\Lambda)p$ reaction~\cite{Pav05} where all the (decay) particles in the final state are charged. As in the $\pi^-$ sector the cross section of the elementary reaction $n(\gamma,K^{\circ})\Lambda$ must be determined from the same data set, demanding a low momentum ($p<50$~MeV/c) spectator proton. Also $^2$H$(\gamma,K^+\Lambda^*)n$ channel should be considered, since a Regge model based on the exchange of the $K$ and $K^*$ mesons~\cite{Gui96} leads also to a good agreement of the differential cross-section of the elementary reaction $p(\gamma,k^+)\Lambda^*$ at $E_{\gamma}=3.5$~GeV. Any signal in the $K^+n$ scattering sector should be the same as in the $^2$H$(\gamma,K^+\Lambda)n$ reaction.
\section{Vector meson production \label{vector}}
Exclusive Vector Mesons production on few body systems is certainly very
promising. It allows to prepare a pair of quarks, with an adjustable
transverse size, and to study its interaction with a nucleon in well
defined kinematics. Furthermore, the coherence time (during which the
incoming photon oscillates into a $q\overline{q}$ pair) and the
formation time (after which this pair recombines into the final meson)
can be adjusted independently to the internucleonic distance.
A special emphasis should be put on $\phi$ and $J/\Psi$ mesons
production: not only these narrow states are more easy to identify, but
their flavor content, different from that of the ground state of cold
hadronic matter, makes them a promising probe.
\begin{figure}[h]
\begin{center}
\epsfig{file=phi_tetar.eps,width=3.5in}
\caption{Color on line. Ratio of the total
to the quasi-free cross section of the $^2$H$(\gamma,\phi p)n$ reaction
against the angle of the recoiling neutron whose the momentum is kept
constant at 500 MeV/c. The kinematic is coplanar and only the part, that corresponds to the emission of the meson and the recoiling neutron on different sides of the photon, is shown.}
\protect\label{phi}
\end{center}
\end{figure}
Fig.~\ref{phi} shows the expected signal in the $\phi$ photo-production channel, when the photon beam end point is $E_{\gamma}=6$~GeV. The model is a straightforward extension of the previous amplitudes. Again, the mass of the $p\phi$ fast pair and the four momentum transfer are set at $W_{p\phi}= 2.896$~MeV and $t=(k-p_{\phi})^2=-3$~GeV$^2$ respectively. The elementary amplitude~\cite{La00b,Ca02} is based on the exchange of two non perturbative gluons and uses a correlated nucleon wave function. It leads to a very good account of the $p(\gamma,\phi)p$ reaction~\cite{An00} recently measured at JLab at $E_{\gamma}=4$~GeV. The $\phi$ can be photo-produced on the proton as well as on the neutron: this has been taken into account in the model. The $pn$ scattering amplitude is defined according to ref.~\cite{La05}, while the $\phi n$ total cross section and slope parameter are respectively $\sigma_{\phi n}=20$~mb and $\beta=6$~Gev$^{-2}$. Again those quantities are almost unknown, and must be determined by the analysis of the $^2$H$(\gamma,\phi)pn$ reaction at low $-t$. The $\phi$ nucleon scattering cross section has been extracted from one experiment (see ref.~\cite{VMD}), while I have taken the universal slope for high energy diffractive processes.
Such a study may end up with a better understanding of the formation of vector
mesons in cold hadronic matter, and will be a reference for the study of vector meson production in heavy ion collisions.
\section{Color Transparency and hadrons propagation \label{ct}}
The expected effect of Color Transparency would be a reduction of the rescattering peak in Figs.~\ref{panel_ppi}, \ref{panel_pp}, \ref{kaon} and~\ref{phi} when the four momentum transfer $-t$ increases. The idea is that hard scattering mechanisms produce colorless dipoles with a small transverse size. Their scattering cross section is therefore expected to be reduced according to the square of the ratio of their transverse size to the transverse size of their asymptotic states.
In addition, the rescattering peaks are expected to be wider since the small configuration is not an eigenstate of the mass operator. It is rather a combination of particles which can be diffractly excited from the ejectile~\cite{Ni94}. In the rescattering integrals (eqs.~\ref{pi_rescat} and \ref{p_rescat}) the propagator of these excited states should be added to propagator of the meson or the baryon which rescatters. The corresponding singularities are closely related to the mass of each particle of the spectrum and lead to peaks at slightly different locations. The rescattering peaks will be spread according to the actual distribution of these states in the mass spectrum of the small configuration.
In previous reviews or prospect talks~\cite{La98,La98a,La00} I have used a toy model based on a geometrical expansion of the mini configuration of the ejected hadron. Now, it is superseded by the quantum diffusion model~\cite{Fr94,Ja96,Ni94,Ni92} which can be implemented in eqs.~\ref{pi_rescat} and \ref{p_rescat}. I defer this study until dedicated experiments are performed. Only experiments will tell us what is the relevant nature of the process which governs the formation and the evolution of such a small configuration.
This paper provides a base line calculation, under the assumption that normal particles rescatter. This is a unique situation, which relies on the evolution of a peak rather than on the level of attenuation of a flat cross section.
The key parameter is the time scale of the interaction. If it is long, the degrees of freedom are hadrons, and well defined unitary peaks appear according to the study presented in this paper. If it is short enough, the degrees of freedom are quarks or exotic objects and unitary peaks should be different or disappear.
Exclusive reactions at hight $t$ are certainly best suited to these studies: they define the small interaction volume where quarks may be the relevant degrees of freedom. The virtuality $Q^2$ of the photon exchanged in electron scattering defines the volume of observation. Playing with this two independent hard scales is the key to the understanding of these rare processes~\cite{La04}. It can be started at 6 GeV, but clearly higher energies (12 GeV and even above) are needed
\section{Exotics \label{pent}}
The rescattering peaks also offer us with a tool to determine the cross section of the interaction with nucleon of unstable or exotic particles, of which a beam is not available. Obvious examples are the photo-production of strange baryons ($\Lambda, \Sigma, \ldots$) of vector mesons ($\phi, J/\Psi, \ldots$).
They can also offer us with a way to disentangle elusive exotics, like pentaquarks for instance, and the physical background. One of the way to chase pentaquarks has been to determine the variation of the mass of the $K^+ n$ pair which is emitted in photo-reactions induced on nuclear~\cite{LEPS} or deuterium~\cite{Ste03} targets. Selecting the $K^+ n$ unitary rescattering peak, $|p_{min}(n K^+)|<100$~MeV/c for instance, in the reactions $^2$H$(\gamma,K^+\Lambda)n$ or $^2$H$(\gamma,K^+\Lambda^*)n$ would be the best way to master the physical background in the mass distribution of the $K^+ n$ pair.
The extension of the model presented in this paper is straightforward: simply the high energy description of the $K^+ n$ scattering should be replaced by a low energy description consistent with the existing data set. The contribution of resonant state, of a given width, could be added in order to set a limit of the production cross section of a possible exotic. Also, the decay distribution of the $\Lambda$ of the $\Lambda^*$ should be taken into account in the Monte Carlo simulation of the experiment. I defer this study to a forthcoming paper~\cite{La06}.
\section{Conclusion \label{conc}}
At the top of each unitary rescattering peak, the reaction mechanism is well under control. It depends on on-shell elementary matrix elements and involves the low momentum components of the nuclear wave function. This is a good starting point to access the interaction with nucleons of exotic objects and short lived particles.
This offers us a last chance to see and study Color Transparency in the present JLab energy range, more particularly in the strange quark sector. It also gives access to the determination of the cross sections of the scattering of vector and pseudo-scalar mesons on nucleons in a cold nuclear environment. Their knowledge is a key to the analysis of collisions between heavy ions at high energy.
For simplicity, all the numerical predictions in this study rely on a high energy description of the elementary matrix element. The photo-production amplitude is described by the exchange of the Regge trajectories of pseudo-scalar and vector mesons. The hadron scattering amplitudes are almost entirely absorptive. This treatment is already valid at kinematics that are achievable at JLab at $6$~GeV: mass of the meson nucleon system above $2$~GeV, relative kinetic energy between baryons above $0.5$ GeV. It will be even more valid in the kinematical range that will be accessible when the CEBAF energy is upgraded to $12$~GeV.
The method can be easily adapted at lower energies, by implementing the relevant description (phase shift expansion, for instance) of the elementary amplitudes~\cite{La06}. It may prove to be useful to predict the physical background and the production cross section of elusive particles, such as pentaquarks for instance.
\section*{Acknowledgement}
I acknowledge the warm hospitality at JLab where this work was completed. The Southern Universities Research Association (SURA) operates Thomas Jefferson National Accelerator Facility (JLab) for the US Department of Energy under Contract No DE-AC05-84ER40150. Parts of this work have been done before I left Saclay.
\section*{Appendix}
In ref.~\cite{La97}, the elementary photo-production amplitude has been expressed in terms of Dirac matrices and spinors. In this work, I have rewritten it in terms of Pauli matrices and spinors.
For on-shell nucleons, both expressions are equivalent. Both are Lorentz and gauge invariant. They are valid in any frame.
For off-shell nucleons, I made the choice to conserve the three momenta at each vertex, to conserve the energy in the invariant operator but to use the on-shell energy $E=\sqrt{p^2+m^2}$ in the normalization, $\sqrt{E+m}$, and the denominator, $E+m$, of Pauli spinors. This choice follows the time ordered expression of Feynman diagrams~\cite{Gro74}.
For off shell nucleons, gauge invariance is lost. However, in the rescattering peaks the target nucleon is almost at rest and the kicked nucleon is on shell. Therefore, the electromagnetic current is conserved in the dominant amplitudes that are considered in this paper.
For the sake of completeness I reproduce the demonstration~\cite{La77} that I gave about thirty years ago. The quasi free matrix element takes the form:
\begin{eqnarray}
{\cal M}_I= - \overline{u}(p_2,m_2) \Theta
\frac{\gamma\cdot n +m}{n^2- m^2} \overline{u}(p_1,m_1)
\Gamma_{\mu}(p_D,n) \phi_D^{\mu}(M)
\nonumber \\
{}
\end{eqnarray}
where $\Theta$ is the elementary photo-production operator, $\Gamma_{\mu}$ the $^2$H$np$ vertex function and $\phi_D^{\mu}$ the deuteron field.
Retaining only the positive energy part of the neutron propagator~\cite{Gro74} and neglecting its negative energy part, one gets:
\begin{eqnarray}
{\cal M}_I= -\frac{m}{E_n}\sum_{m_n}
\frac{\overline{u}(p_2,m_2) \Theta u(\overline{n},m_n)
\overline{u}(\overline{n},m_n)\overline{u}(p_1,m_1)}{n^{\circ}-E_n}
\nonumber \\
\Gamma_{\mu}(p_D,n) \phi_D^{\mu}(M)
\nonumber \\ {}
\end{eqnarray}
with $E_n=\sqrt{\vec{n}^2+m^2}\neq n^{\circ}= E_D-E_1$ and $\overline{n}=(\overline{n}^{\circ}=E_n,\overline{\vec{n}}=\vec{n})$.
Identifying $\Gamma_{\mu}(p_D,n) \phi_D^{\mu}(M)m/E_n(n^{\circ}-E_n)$ with the deuteron wave function and defining
\begin{eqnarray}
T_{\gamma n}(\vec{p_2},m_2,-\vec{p_1},m_n)=\frac{1}{-i}
\overline{u}(p_2,m_2) \Theta u(\overline{n},m_n)
\end{eqnarray}
one gets eq.~(\ref{q_f}). In terms of Pauli spinors, it takes the form:
\begin{eqnarray}
T_{\gamma n}(\vec{p_2},m_2,-\vec{p_1},m_n)=
(m_2|\imath\; \vec{\sigma}\cdot\vec{K}+ L|m_n)
\end{eqnarray}
which I use in this work. In the quasi-free amplitude, $\vec{n}=-\vec{p_1}$. In the rescattering amplitudes, a similar expression takes into account the actual nucleon momenta.
The vector and scalar part of the $\pi$ exchange amplitude of the elementary reaction $n(\gamma,\pi^-)p$ are:
\begin{eqnarray}
\vec{K_{\pi}}= eg_{\pi NN}\sqrt{2}\frac{\sqrt{(E_n+m)(E_2+m)}}{2m}
(t-m^2_{\pi}) {\cal P}^{{\pi}^-}_{\mathrm{Regge}}F_1(t)
\nonumber \\
\left[\left( \frac{\vec{n}}{E_n+m}-\frac{\vec{p_2}}{E_2+m}\right)
\left(\frac{(2\vec{p_2}-\vec{k})\cdot\vec{\epsilon}}{u-m^2}
-\frac{(2\vec{p_{\pi}}-\vec{k})\cdot\vec{\epsilon}}{t-m^2_{\pi}} \right)
\right.\nonumber \\
+\frac{\vec{\epsilon}}{u-m^2}\left(\nu-\frac{\vec{p_2}\cdot\vec{k}}{E_2+m}
-\frac{\vec{n}\cdot\vec{k}}{E_n+m}
\right. \nonumber \\ \left.
+\frac{\nu\vec{n}\cdot\vec{p_2}}{(E_n+m)(E_2+m)}\right)
\nonumber \\
+\frac{\vec{k}}{u-m^2}\left( \frac{\vec{n}\cdot\vec{\epsilon}}{E_n+m}
+\frac{\vec{p_2}\cdot\vec{\epsilon}}{E_2+m} \right)
\nonumber \\
-\frac{\vec{n}}{u-m^2} \frac{\nu\vec{p_2}\cdot\vec{\epsilon}}{(E_n+m)(E_2+m)}
\nonumber \\ \left.
-\frac{\vec{p_2}}{u-m^2} \frac{\nu\vec{n}\cdot\vec{\epsilon}}{(E_n+m)(E_2+m)}
\right]\nonumber \\
{}
\end{eqnarray}
\begin{eqnarray}
L_{\pi}= eg_{\pi NN}\sqrt{2}\frac{\sqrt{(E_n+m)(E_2+m)}}{2m}
(t-m^2_{\pi}) {\cal P}^{{\pi}^-}_{\mathrm{Regge}}F_1(t)
\nonumber \\
\frac{1}{u-m^2}
\left[ \frac{\vec{p_2}\times\vec{k}}{E_2+m}
+\frac{\vec{k}\times\vec{n}}{E_n+m}
-\frac{\nu\vec{p_2}\times\vec{n}}{(E_n+m)(E_2+m)}
\right]\cdot\vec{\epsilon}
\nonumber \\
{}
\end{eqnarray}
where $g^2_{\pi NN}/4\pi=14.5$, $t=(p_{\pi}-k)^2$ and $u=(p_2-k)^2$. I use the Regge propagator ${\cal P}^{{\pi^-}}_{\mathrm{Regge}}$ that corresponds to the $\pi$ saturating trajectory (as defined in ref.~\cite{La97}) and the dipole parameterization of the nucleon isovector form factor:
\begin{eqnarray}
F_1(t)=\frac{4m^2-2.79t}{(4m^2-t)(1-t/0.7)^2}
\end{eqnarray}
One recognizes the pure $\pi$ exchange amplitude (term in $1/(t-m^2_{\pi})$) and the part of the $u$-channel nucleon exchange amplitude (terms in $1/(u-m^2)$) which has been added to insure gauge invariance (see~\cite{La97}).
The vector and scalar part of the $\rho$ exchange amplitude of the elementary reaction $n(\gamma,\pi^-)p$ are:
\begin{eqnarray}
\vec{K_{\rho}}= e\frac{g_{\rho \pi\gamma}}{m_{\pi}}g_{\rho NN}\sqrt{2}
\frac{\sqrt{(E_n+m)(E_2+m)}}{2m} {\cal P}^{\rho}_{\mathrm{Regge}}F_1(t)
\nonumber \\
\left\{\frac{\vec{p_2}\times\vec{n}}{(E_n+m)(E_2+m)}
\left[(1+\kappa_v)V^{\circ}
\right.\right.\nonumber \\
\left.
+\frac{\kappa_v}{2m}(V^{\circ}(n^{\circ}+E_2)-\vec{V}\cdot(\vec{n}+\vec{p_2}))
\right]
\nonumber \\
\left.
(1+\kappa_v)\vec{V}\times\left[ \frac{\vec{p_2}}{E_2+m}
- \frac{\vec{n}}{E_n+m}\right]
\right\}
\nonumber \\
{}
\label{rho_vec}
\end{eqnarray}
\begin{eqnarray}
L_{\rho}= e\frac{g_{\rho \pi\gamma}}{m_{\pi}}g_{\rho NN}\sqrt{2}
\frac{\sqrt{(E_n+m)(E_2+m)}}{2m} {\cal P}^{\rho}_{\mathrm{Regge}}F_1(t)
\nonumber \\
\left\{
(1+\kappa_v)\left[V^{\circ}\left(1+\frac{\vec{n}\cdot\vec{p_2}}{(E_n+m)(E_2+m)}
\right)
\right. \right.\nonumber \\
\left.
-\vec{V}\cdot\left( \frac{\vec{n}}{E_n+m}+\frac{\vec{p_2}}{E_2+m}\right)
\right]
\nonumber \\
-\frac{\kappa_v}{2m} \left[1-\frac{\vec{n}\cdot\vec{p_2}}{(E_n+m)(E_2+m)} \right]
\nonumber \\
\left.
\left[ V^{\circ}(n^{\circ}+E_2)-\vec{V}\cdot(\vec{n}+\vec{p_2})\right]
\right\}
\nonumber \\
{}
\label{rho_scal}
\end{eqnarray}
I use the Regge propagator ${\cal P}^{\rho}_{\mathrm{Regge}}$, with the saturating trajectory of the $\rho$ meson, and the coupling constants $g_{\rho \pi\gamma}=$~0.103, $g^2_{\rho NN}/4\pi=$~0.92 and $\kappa_v=$~6.1, as in ref.~\cite{La97}.
The four vector $V^{\mu}=(V^{\circ},\vec{V})$ contains the dependency upon the polarization vector $\vec{\epsilon}$ of the photon, in the following way:
\begin{equation}
\begin{array}{rrrrcr}
(-kR_y,& 0,& \nu R_z-kR^{\circ},& -\nu R_y)
&\mbox{if}& \vec{\epsilon}=(1,0,0)
\\
(kR_x,& kR^{\circ}-\nu R_z,&0,& \nu R_x)
&\mbox{if}& \vec{\epsilon}=(0,1,0)
\\
(0,& \nu R_y,& -\nu R_x,&0)
&\mbox{if}& \vec{\epsilon}=(0,0,1)
\end{array}
\end{equation}
where $R^{\circ}= \nu-E_{\pi}$ and $\vec{R}\equiv (R_x,R_y,R_z)= \vec{k}-\vec{p_{\pi}}$ are respectively the energy and the three momentum of the exchanged $\rho$ meson.
For the $p(\gamma,K^+)\Lambda$ reaction, the $K^*$ meson amplitudes take the same form as the $\rho$ meson exchange amplitudes~(\ref{rho_vec}) and~(\ref{rho_scal}), besides trivial changes in the masses, coupling constants and propagators.
The $K$ exchange amplitude contains the part of the $s$-channel nucleon exchange amplitude which is strictly necessary to insure gauge invariance. It takes the form:
\begin{eqnarray}
\vec{K_{K}}= eg_{K N\Lambda}\sqrt{2}\frac{\sqrt{(E_n+m)(E_2+m_{\Lambda})}}
{\sqrt{4m m_{\Lambda}}}
(t-m^2_{K}) {\cal P}^{K^+}_{\mathrm{Regge}}
\nonumber \\
\left[\left( \frac{\vec{n}}{E_n+m}-\frac{\vec{p_2}}{E_2+m_{\Lambda}}\right)
\left(\frac{(2\vec{n}+\vec{k})\cdot\vec{\epsilon}}{s-m^2}
+\frac{(2\vec{p_{K}}-\vec{k})\cdot\vec{\epsilon}}{t-m^2_{K}} \right)
\right.\nonumber \\
+\frac{\vec{\epsilon}}{s-m^2}\left(\nu-\frac{\vec{p_2}\cdot\vec{k}}{E_2+m_{\Lambda}}
-\frac{\vec{n}\cdot\vec{k}}{E_n+m}
\right. \nonumber \\ \left.
+\frac{\nu\vec{n}\cdot\vec{p_2}}{(E_n+m)(E_2+m_{\Lambda})}\right)
\nonumber \\
+\frac{\vec{k}}{s-m^2}\left( \frac{\vec{n}\cdot\vec{\epsilon}}{E_n+m}
+\frac{\vec{p_2}\cdot\vec{\epsilon}}{E_2+m_{\Lambda}} \right)
\nonumber \\
-\frac{\vec{n}}{s-m^2} \frac{\nu\vec{p_2}\cdot\vec{\epsilon}}{(E_n+m)(E_2+m_{\Lambda})}
\nonumber \\ \left.
-\frac{\vec{p_2}}{s-m^2} \frac{\nu\vec{n}\cdot\vec{\epsilon}}{(E_n+m)(E_2+m_{\Lambda})}
\right]\nonumber \\
{}
\end{eqnarray}
\begin{eqnarray}
L_{K}= eg_{K N\Lambda}\sqrt{2}\frac{\sqrt{(E_n+m)(E_2+m_{\Lambda})}}
{\sqrt{4mm_{\Lambda}}}
(t-m^2_{K}) {\cal P}^{K^+}_{\mathrm{Regge}}
\nonumber \\
\frac{1}{s-m^2}
\left[ \frac{\vec{p_2}\times\vec{k}}{E_2+m_{\Lambda}}
+\frac{\vec{k}\times\vec{n}}{E_n+m}
-\frac{\nu\vec{p_2}\times\vec{n}}{(E_n+m)(E_2+m_{\Lambda})}
\right]\cdot\vec{\epsilon}
\nonumber \\
{}
\end{eqnarray}
where $n=(E_n,\vec{n})$ and $p_2=(E_2,\vec{p_2})$ now stand for the four momentum of the target proton and the outgoing $\Lambda$ respectively, and where $g_{K N\Lambda}^2/4\pi=$~10.6.
As in ref.~\cite{La97}, I use the $K$ and $K^*$ linear trajectories in the Regge propagators, and no hadronic form factor ($F_1(t)=1$).
|
2,877,628,090,164 | arxiv | \section{Introduction}
Unification of forces has been the most challenging task the science community has ever faced. So far that quest has successfully brought the electromagnetic, strong and weak forces under one roof. However the unification scheme hits a wall when one tries to incorporate in it the only other fundamental force, namely, gravity. There have been numerous attempts, so far, to incorporate gravity in the above picture as well, leading to a unified quantum theory of nature. This has resulted in a large number of candidate theories for quantum gravity, \emph{but} without much success. This issue, broadly speaking, originates from the peculiar fact that the energy scale associated with grand unified theories is $\sim \mathcal{O}(10^{3})\textrm{GeV}$, while the natural energy scale for gravity is the Planck scale $\sim \mathcal{O}(10^{18}) \textrm{GeV}$. This huge difference between the respective energy scales manifests itself into unnatural fine tunings in various physical parameters of the model, e.g., in the mass
of the Higgs Boson. Thus it seems legitimate to understand the origin of this fine tuning problem (known as the \textit{gauge hierarchy problem}) before delving into quantization of gravity
\cite{PerezLorenzana:2005iv,Csaki:2004ay,Sundrum:2005jf,Polchinski:1998rq,Polchinski:1998rr,
Rovelli:2004tv,Chakraborty:2017s}.
One such natural candidate for resolving the gauge hierarchy problem in this regard corresponds to extra spatial dimensions, which can bring down the Planck scale to the realm of grand unified theories. Such a possibility was considered in
\cite{ArkaniHamed:1998rs,Antoniadis:1998ig,Antoniadis:1990ew,Rubakov:1983bz,Rubakov:1983bb} where the extra dimensions were large enough, such that the volume spanned by them could suppress the Planck scale of the higher dimensional spacetime (known as \textit{bulk}) to the TeV scale. However this proposal harbours two conceptual drawbacks: Firstly, it seems that the problem of energy scale hierarchy has merely been transferred to another form, the volume hierarchy, e.g., if one wants to reduce the energy scale to $1~\textrm{TeV}$ the size of the extra dimensions would be $\sim 10^{11}~\textrm{m}$; and more importantly it treats the higher dimensional spacetime to be flat \cite{PerezLorenzana:2005iv}. The second one is indeed a serious issue, as gravity cannot be shielded and hence if it is present in four dimensions gravity is bound to propagate in higher spacetime dimensions as well. In order to cure this problem, Randall and Sundrum proposed a very natural solution to the hierarchy problem with warped
extra dimensions, where presence of gravity in higher dimensions forces the effective Planck scale to reduce to TeV scale in the four dimensional hypersurface we live in (known as \textit{brane})
\cite{Randall:1999ee,Garriga:1999yh}. This scenario has been extensively studied in the literature in the past years in various contexts, starting from black holes
\cite{Chamblin:1999by,Berti:2015itd,Konoplya:2008ix,Konoplya:2007jv,Berti:2009kk,Gregory:2008rf,Emparan:1999wa,Cook:2017fec,Bhattacharya:2016naa} and cosmology
\cite{Csaki:1999mp,Csaki:1999jh,Binetruy:1999ut,Ida:1999ui,Nojiri:2001ae,Nojiri:2002hz,Charmousis:2002rc,Germani:2002pt,Gravanis:2002wy,Brax:2004xh}
to particle phenomenology as well as possible signatures in Large Hadron Collider
(LHC) \cite{Goldberger:1999un,Davoudiasl:1999tf,Davoudiasl:2000wi}. A lot of attention has also been devoted to the higher curvature generalization of this scenario, obtained by introducing terms like $R^{2}$, $R_{abcd}R^{abcd}$ in the gravitational action, as well as to the stabilization of these extra dimensions
\cite{Goldberger:1999uk,Csaki:1999mp,Chacko:1999eb,Chakraborty:2014zya,Chakraborty:2013ipa,
Chakraborty:2016gpg}.
Even though LHC provides us an observational window for the existence of extra spatial dimensions, it is important to know if there exists any other observational tests that can either prove or disprove their existence independently. It is obvious that in order to probe these effects, one has to investigate high energy/high curvature regime, which can originate from either high energy collisions like in LHC or from physics near black holes. The second possibility opens up a few interesting observational avenues --- (a) the black hole continuum spectrum, originating from accretion disc around a supermassive black hole, (b) strong gravitational lensing around supermassive black holes and finally, (c) gravitational waves from collision of two massive black holes. We have already elaborated on the continuum spectrum from supermassive black holes and their implications regarding the presence of extra dimensions in \cite{Banerjee:2017hzw}, while strong gravitational lensing has been discussed in detail in \cite{
Chakraborty:2016lxo}. In this work we aim to address the third possibility, i.e., the effect of higher dimensions on gravitational waves, in light of the recent detections
\cite{Abbott:2017vtc,TheLIGOScientific:2016pea,Abbott:2016nmj,TheLIGOScientific:2016src,Abbott:2016blz} of the same in Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO). The whole process of collision between two black holes can broadly be divided into three categories --- inspiral phase, merger phase and ringdown phase. The first two phases are best described by a combination of post-Newtonian and numerical approaches
\cite{Asada:1996ya,Damour:2008qf,Damour:2001tu,Nagar:2011fx,Damour:2008gu,Baiotti:2010xh,Blanchet:2006zz,
Flanagan:2005yc,Centrella:2010mx,Ajith:2007kx,Buonanno:2000ef,Delsate:2014hba,Pai:2000zt,Bose:2005fm,Pretorius:2005gq,
Arun:2006yw}, which we leave for the future, concentrating here on the ringdown phase only. In this situation the quasi-normal mode frequencies play a very fundamental role in determining the ring down phase and in this work we will concentrate on deriving the quasi-normal mode frequencies for this higher dimensional scenario
\cite{Kanti:2005xa,Berti:2009kk,Konoplya:2011qq,Berti:2015itd,Toshmatov:2016bsb,Andriot:2017oaz}.
To understand the behaviour of quasi-normal mode frequencies in the context of higher spacetime dimensions, one can follow two possible approaches --- (a) One starts from the gravitational field equations in the bulk and then consider its perturbation around a bulk solution, which manifests itself as a black hole on the brane. This one we refer to as the \emph{bulk based approach}. (b) Otherwise, one projects the bulk gravitational field equations on the brane hypersurface resulting in an effective description of the brane dynamics inherited from the bulk, referred to as the \emph{brane based approach}. In this case as well one perturbs the effective gravitational field equations on the brane, around a given bulk solution representing again a brane black hole. Some aspects of this problem along the first line of attack has already been elaborated and explored in \cite{Seahra:2004fg,Clarkson:2005eb,Seahra:2009gw,Clarkson:2006pq,Clarkson:2005mg,Seahra:2009zz,Witek:2013ora}, while to our knowledge the second avenue is
hitherto unexplored. In this work, we wish to fill this gap by providing a thorough analysis of the second approach in relation to the black hole perturbation theory and possible discords with the bulk based approach. In particular, we will try to understand whether the results derived in \cite{Seahra:2004fg} using Cauchy evolution of initial data matches with our quasi-normal mode frequency analysis. Further for completeness we will present the Cauchy evolution for the brane based approach as well. This will not only help to contrast these two approaches but will also depict whether the quasi-normal mode analysis and the Cauchy evolution are compatible with each other. Besides providing yet another independent route towards understandings of higher dimensions, this will also be of significant interest to the gravitational wave community.
The paper is organized as follows: We start in \ref{Perturb_Effective} with a brief introduction of the effective equation formalism in the context of higher spatial dimensions and then we build up our gravitational perturbation equation based on the above. This has been applied in \ref{Sph_Symm} to derive the evolution equations for the master variables associated with spherically symmetric brane and possible effects from higher dimensions. In \ref{qnm_analysis} we have studied these perturbation equations in Fourier space and have derived the quasi-normal mode frequencies using the continued fraction method as well as the direct integration scheme. Using these quasi-normal mode frequencies the time evolution of the master variable has been determined for both the bulk and the brane based approach in \ref{Num_qnm}. \ref{consistency_Cauchy} deals with Cauchy evolution of the initial data and its possible harmony with the quasi-normal mode analysis. We conclude with a discussion and implications of the results obtained, in \ref{conclusion}. Some
detailed calculations pertaining to derivation of gravitational perturbation equation on the brane have been presented in \ref{App_A}, while those associated with continued fraction method have been elaborated in \ref{App_B}.
\emph{Notations and Conventions:} We will set the fundamental constant $c$ as well as the combination $GM$ to unity, where $M$ is the mass of the black hole. Indices running over all the bulk coordinates are denoted by uppercase Latin letters, while all the brane indices are denoted by Greek letters. Any geometrical quantity associated with the brane hypersurface alone is being denoted with a superscript $(4)$. Further, all the matrix valued quantities will be denoted by boldfaced letters. Finally the signature convention adopted in this work is the mostly positive one.
\section{Perturbing effective gravitational field equations on the brane}\label{Perturb_Effective}
We start this section by providing a very brief introduction to the effective gravitational field equations on the brane, which will be necessary for our later purposes. Since we are interested in signatures of higher dimensions only, it will be sufficient to work within the context of Einstein gravity in five spacetime dimensions, in which case the gravitational Lagrangian density is the five dimensional Ricci scalar $R$. Thus the five dimensional gravitational field equations will read $G_{AB}=8\pi G_{(5)}T_{AB}$, where $T_{AB}$ stands for the matter energy momentum tensor, which may be present in the bulk and $G_{(5)}$ is the five dimensional gravitational constant. In the specific context when the bulk energy momentum tensor is originating from a negative cosmological constant $\Lambda$, one arrives at the following static and spherically symmetric solution on the brane,
\begin{align}\label{GW_Eq01}
d&s_{\rm unperturbed}^{2}
\nonumber
\\
&=e^{-2ky}\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega ^{2}\right)+dy^{2}~,
\end{align}
with $f(r)=1-(2/r)$ and $k\propto \sqrt{-\Lambda}$. Note that from the perspective of a brane observer located on a $y=\textrm{constant}$ hypersurface, the spacetime structure on the brane is given by the Schwarzschild solution.
This raises the following interesting question: What happens to the gravitational field equations on the brane, given the gravitational field equations on the bulk? It has been answered for Einstein gravity in \cite{Shiromizu:1999wj} and has been extended recently to various other scenarios involving alternative gravity theories \cite{Chakraborty:2014xla,Chakraborty:2015bja,Chakraborty:2015taq}. The derivation goes as follows, one first chooses the brane hypersurface, say $y=0$, and determines the normal $n_{A}=\nabla _{A}y$, yielding the induced metric on the brane hypersurface to be $h_{AB}=g_{AB}-n_{A}n_{B}$, such that $n_{A}h^{A}_{B}=0$. Given the induced metric, one can introduce the notion of covariant derivative on a brane hypersurface and hence a notion of brane curvature using commutator between the brane covariant derivatives. This enables one to express the bulk curvature in terms of the brane curvature and extrinsic curvatures associated with the brane hypersurface. Further contractions will
enable one to relate the bulk Einstein's equations with curvatures on the brane, referred to as the effective gravitational field equations on the brane. The effective equations in vacuum brane differ from four dimensional Einstein's equations by an additional term inherited from the bulk Weyl tensor and takes the following form,
\begin{equation}\label{GW_Eq02}
~^{(4)}G_{\mu \nu}+E_{\mu \nu}=0~.
\end{equation}
Here $E_{\mu \nu}$ stands for a particular projection of the bulk Weyl tensor $C_{ABCD}$ on the brane hypersurface (commonly known as the electric part) given by,
\begin{equation}\label{GW_Eq03}
E_{\mu \nu}=C_{ABCD}e^{A}_{\mu}n^{B}e^{C}_{\nu}n^{D}~,
\end{equation}
where $n_{A}$ is the normalized normal introduced earlier and $e^{A}_{\mu}=\partial x^{A}/\partial y^{\mu}$ is the bulk to brane projector, with $x^{A}$ being the bulk coordinates and $y^{\mu}$ are the brane coordinates \cite{gravitation,Poisson}.
At this stage it is worth mentioning that in order to arrive at the above relation we have assumed that the bulk cosmological constant and the brane tension cancels each other, leading to a vanishing effective cosmological constant on the brane hypersurface \cite{Shiromizu:1999wj,Randall:1999ee}. The above cancellation has its origin in the fact that in the effective field equation the effective cosmological constant is the difference between bulk cosmological constant and brane tension, and this difference has to be zero for the stability of the background spacetime. Further, note that even though \ref{GW_Eq02} acts as the effective field equations on the brane, to solve it explicitly one does require information of the bulk, hidden in $E_{\mu \nu}$ through the bulk Weyl tensor.
There are two ways to solve this equation --- (a) Assume certain bulk geometry as ansatz (which for our case corresponds to \ref{GW_Eq01}) and then try to see what sort of brane configuration solves \ref{GW_Eq02}. (b) Take $E_{\mu \nu}$ as an arbitrary tensor and try to solve \ref{GW_Eq02} with $E_{\mu \nu}$ treated as a source: e.g., in the context of spherical symmetry one often divides $E_{\mu \nu}$ into an energy density (known as \emph{dark radiation}) and pressure (known as \emph{dark pressure}). Even though one can have very interesting results emerging from the second scenario \cite{Maartens:2001jx}, it has the drawback that the bulk metric remains unknown and in general it is not even clear whether there exists a bulk metric that would satisfy Einstein's equations in the bulk. Thus we will adopt the first scenario and shall take \ref{GW_Eq01} as the background metric which indeed satisfies \ref{GW_Eq02} as well
\cite{Dadhich:2000am,Maartens:2001jx,Harko:2004ui,Maartens:2010ar,Chakraborty:2014xla}.
This procedure must be contrasted with the perturbation of bulk Einstein's equations around the solution presented in \ref{GW_Eq01}, since in the case of effective field equations, the perturbation of bulk Weyl tensor will play a crucial role. Thus it is not at all clear a priori how the perturbed equations in the brane based approach will behave in contrast to the bulk based approach, even though they are being perturbed around the same solution. With this motivation in the backdrop, let us concentrate on perturbation of \ref{GW_Eq03} around the bulk metric $g_{AB}$ given in \ref{GW_Eq01}, such that,
\begin{align}\label{GW_Eq04}
g_{AB}\rightarrow g_{AB}+h_{AB}~.
\end{align}
Here $h_{AB}$ is the perturbed metric around $g_{AB}$ and all the expressions to follow will be evaluated to the first order in the perturbed metric $h_{AB}$ \footnote{In principle one should write down $g_{AB}\rightarrow g_{AB}+\epsilon~ h_{AB}$, with small $\epsilon$ and then keeping only terms linear in $\epsilon$.}.
It is also well known that not all the components of $h_{AB}$ are dynamical, there are redundant gauge degrees of freedom. These gauge choices must be made according to convenience of calculations. In this particular situation the following gauge conditions will turn out to be useful later on,
\begin{align}\label{GW_Eq05}
\nabla _{A}h^{A}_{B}=0;\qquad h^{A}_{A}=0;\qquad h_{AB}=h_{\alpha \beta}e^{\alpha}_{A}e^{\beta}_{B}~,
\end{align}
known as the Randall-Sundrum gauge. The usefulness of this gauge condition can also be anticipated from the fact that these imply $h_{AB}n^{A}=0$ and hence the perturbed bulk metric takes the following form,
\begin{equation}\label{GW_Eq06}
ds^{2}_{\rm perturbed}=\Big[q_{\alpha \beta}(y,x^{\mu})+h_{\alpha \beta}(y,x^{\mu})\Big]dx^{\alpha}dx^{\beta}+dy^{2}~,
\end{equation}
where $q_{\alpha \beta}$ solves \ref{GW_Eq02} and is given by
\begin{align}\label{GW_N01}
&q_{\alpha \beta}dx^{\alpha}dx^{\beta}
\nonumber
\\
&=e^{-2ky}\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\theta ^{2}+r^{2}\sin ^{2}\theta d\phi ^{2} \right)~.
\end{align}
Even though \ref{GW_Eq01} opts for $f(r)=1-(2/r)$, in the rest of the analysis we will keep $f(r)$ as general as possible. Then to linear order in the perturbed metric $h_{\alpha \beta}$ one can expand the four dimensional Einstein tensor as,
\begin{align}\label{GW_Eq07}
~^{(4)}G_{\mu \nu}\simeq ~^{(4)}G_{\mu \nu}^{(q)}&+~^{(4)}R^{(h)}_{\mu \nu}
\nonumber
\\
&-\frac{1}{2}q_{\mu \nu}~^{(4)}R^{(h)}-\frac{1}{2}h_{\mu \nu}~^{(4)}R^{(q)}~,
\end{align}
where terms with superscript $(q)$ denote that they have to be evaluated for the brane background metric $q_{\alpha \beta}$ given in \ref{GW_N01} and superscript $(h)$ implies that it has been evaluated for the perturbed metric $h_{\mu \nu}$. The index $(4)$ implies that these are all four dimensional geometrical quantities.
Another ingredient in the perturbation of effective brane based approach is the perturbation of the bulk Weyl tensor. For that one has to write down the bulk Weyl tensor in terms of the bulk Riemann, Ricci tensor and Ricci scalar and expand all of them to leading order in the gravitational perturbation $h_{\alpha \beta}$. The above procedure leads to,
\begin{widetext}
\begin{align}\label{GW_Eq08}
C_{ABCD}&=R_{ABCD}-\frac{1}{3}R_{AC}g_{BD}+\frac{1}{3}R_{AD}g_{BC}+\frac{1}{3}R_{BC}g_{AD}-\frac{1}{3}R_{BD}g_{AC}
+\frac{1}{12}R\left(g_{AC}g_{BD}-g_{AD}g_{BC}\right)
\nonumber
\\
&\simeq C^{(g)}_{ABCD}+\Bigg\lbrace R^{(h)}_{ABCD}-\frac{1}{3}R^{(g)}_{AC}h_{BD}-\frac{1}{3}R^{(h)}_{AC}g_{BD}
+\frac{1}{3}R^{(h)}_{AD}g_{BC}+\frac{1}{3}R^{(g)}_{AD}h_{BC}-\frac{1}{3}R^{(h)}_{BD}g_{AC}
\nonumber
\\
&-\frac{1}{3}R^{(g)}_{BD}h_{AC}+\frac{1}{3}R^{(h)}_{BC}g_{AD}+\frac{1}{3}R^{(g)}_{BC}h_{AD}
+\frac{1}{12}R^{(h)}\Big(g_{AC}g_{BD}-g_{AD}g_{BC}\Big)
\nonumber
\\
&+\frac{1}{12}R^{(g)}\Big(g_{AC}h_{BD}+h_{AC}g_{BD}-g_{AD}h_{BC}-h_{AD}g_{BC}\Big)\Bigg\rbrace~.
\end{align}
\end{widetext}
Here superscript $(g)$ denotes that the respective quantity is evaluated for the bulk background metric $g_{AB}$. Note that due to dependence of $q_{\alpha \beta}$ on extra dimensional coordinate $y$, quantities evaluated for the bulk metric will inherit $y$-derivatives of $q_{\alpha \beta}$ and hence will differ from their four-dimensional counterparts. Given the perturbation of bulk Weyl tensor the corresponding projection of the perturbed bulk Weyl tensor onto the brane hypersurface results in,
\begin{widetext}
\begin{align}\label{GW_Eq09}
E_{\mu \nu}&\simeq E_{\mu \nu}^{(g)}+\Bigg\lbrace R^{(h)}_{ABCD}e^{A}_{\mu}n^{B}e^{C}_{\nu}n^{D}-\frac{1}{3}R^{(h)}_{AC} e^{A}_{\mu}e^{C}_{\nu}
\nonumber
\\
&-\frac{1}{3}R^{(h)}_{BD}n^{B}n^{D}q_{\mu \nu}-\frac{1}{3}R^{(g)}_{BD}n^{B}n^{D}h_{\mu \nu}+\frac{1}{12}R^{(h)} q_{\mu \nu}+\frac{1}{12}R^{(g)}h_{\mu \nu}\Bigg\rbrace ~.
\end{align}
\end{widetext}
Note that in the above perturbation equation for the projected bulk Weyl tensor, the first order corrections to bulk Riemann, Ricci tensor and Ricci scalar appears. One can decompose all these perturbed quantities evaluated for the bulk metric in terms of the respective brane metric and extra dimensional contributions. This has been explicitly carried out in \ref{App_A} and ultimately leads to the following expression for the projected bulk Weyl tensor,
\begin{align}\label{GW_Eq18}
E_{\mu \nu}^{(h)}&=\frac{1}{6}~^{(4)}\square h_{\mu \nu}-\frac{1}{3}\partial _{y}^{2}h_{\mu \nu}-k\partial _{y}h_{\mu \nu}+\frac{1}{3}k^{2}h_{\mu \nu}
\nonumber
\\
&+\frac{1}{3}h^{\alpha}_{\beta}~^{(4)}R^{(q)\beta}_{~~~~\mu \alpha \nu}-\frac{1}{6}h^{\alpha}_{\mu}~^{(4)}R^{(q)}_{\alpha \nu}
\nonumber
\\
&-\frac{1}{6}h^{\alpha}_{\nu}~^{(4)}R^{(q)}_{\alpha \mu}+\frac{1}{12}~^{(4)}R^{(q)}h_{\mu \nu}~.
\end{align}
At this stage it is worth emphasizing that the gauge conditions elaborated in \ref{GW_Eq05}, take a simpler form in this context. In particular, the spatial part of the differential condition $\nabla _{A}h^{A}_{\mu}=0$, when expanded in terms of four dimensional quantities immediately yields $\nabla _{\nu}h^{\nu}_{\mu}=0$. Use of this relation and commutator of four dimensional covariant derivative results into $\nabla ^{\mu}E^{(h)}_{\mu \nu}=0$, as is evident from \ref{GW_Eq18} in the context of vacuum solutions.
One can also try to understand this result from a different perspective. Since we are perturbing around vacuum solutions, it follows from \ref{GW_Eq02} that $E^{(h)}_{\mu \nu}\propto ~^{(4)}G^{(h)}_{\mu \nu}$. Thus it immediately implies that $\nabla _{\mu}E^{(h)~\mu \nu}=0$ as it should be, by virtue of Bianchi identity. Finally collecting all the pieces from perturbation of bulk Weyl tensor elaborated in \ref{GW_Eq18} as well as perturbation of original Einstein tensor as in \ref{GW_Eq07} we obtain,
\begin{align}\label{GW_Eq19}
e^{2ky}&\Big\lbrace ~^{(4)}\square h_{\mu \nu}+2h_{\alpha \beta}~^{(4)}R^{\beta~\alpha}_{~\mu ~\nu}\Big\rbrace
\nonumber
\\
&+\left\lbrace -k^{2}h_{\mu \nu}+3k\partial _{y}h_{\mu \nu}+\partial _{y}^{2}h_{\mu \nu}\right\rbrace=0~.
\end{align}
In order to arrive at the above relation we have used the fact that $q_{\alpha \beta}=\exp(-2ky)g_{\alpha \beta}$, where in this particular situation $g_{\alpha \beta}$ is the Schwarzschild metric. Note that we have not used this fact explicitly anywhere in this section, except for assuming that $g_{\alpha \beta}$ must satisfy vacuum Einstein's equations on the brane. Further, all the geometrical quantities present in the above equation are evaluated for the brane metric $g_{\alpha \beta}$.
At this stage it is instructive to split the perturbation equations into parts depending on four dimensional spacetime and those depending on extra dimensions, such that, $h_{\alpha \beta}(y,x^{\mu})=h_{\alpha \beta}(x^{\mu})\chi (y)$. Following the separability of the perturbed metric, the above equations can also be decomposed into two parts, which for vacuum brane solution reduce to,
\begin{align}
e^{-2ky}\left\lbrace-k^{2}\chi+3k\partial _{y}\chi+\partial _{y}^{2}\chi\right\rbrace&=-\mathcal{M}^{2}\chi (y)~,
\label{GW_Eq20a}
\\
~^{(4)}\square h_{\mu \nu}+2h_{\alpha \beta}~^{(4)}R^{\alpha~\beta}_{~\mu ~\nu}-\mathcal{M}^{2}h_{\mu \nu}&=0~.
\label{GW_Eq20b}
\end{align}
Remarkably, the effect of the whole analysis is just the emergence of a massive gravitational perturbation modes. With $\mathcal{M}=0$, one immediately recovers the dynamical equation governing gravitational perturbation in a non-trivial background. As we will see later, \ref{GW_Eq20a} will lead to a series of masses denoted by $m_{n}$ and is called the $n$th Kaluza-Klein mode mass of gravitational perturbation. For each Kaluza-Klein mode, say of order $n$, there will be a solution $h_{\mu \nu}^{(n)}$ to \ref{GW_Eq20b}. When all these $n$ values are summed over one ends up with the full solution of the gravitational perturbation.
To summarize, we have started from the effective gravitational field equations on the 3-brane, which depends on the bulk Weyl tensor and hence on the bulk geometry. The main problem of this approach being, not all the components of the projected bulk Weyl tensor $E_{\mu \nu}$ are determined in terms of quantities defined on the brane. In particular, the transverse-traceless part of the projected bulk Weyl tensor, representing the graviton modes in the bulk spacetime, can not be determined. This is intimately related to the fact that the effective field equations on the brane are \emph{not} closed \cite{Shiromizu:1999wj}. In this work, we have circumvented this problem by using the gauge freedom for the gravitational perturbation. We have started with the Schwarzschild anti-de Sitter spacetime (as in \ref{GW_Eq01}) which identically satisfies the effective gravitational field equations on the brane. We then consider perturbation around this background, which certainly involves graviton modes propagating in
the bulk spacetime. However the use of Randall-Sundrum gauge (presented in \ref{GW_Eq05}) enables one to reduce the number of propagating degrees of freedom and hence the effective field equations (at least in the perturbative regime) becomes closed. Finally the method of separation of variables enables one to separate a four dimensional part from the extra dimensional one and arrives at \ref{GW_Eq20a} and \ref{GW_Eq20b} respectively. The presence of extra dimension essentially translates into the infinite tower of Kaluza-Klein modes as far as the propagation of gravitational wave in four dimension is considered.
Let us now emphasise the key differences between our approach and the bulk based one. Interestingly, \ref{GW_Eq20b} governing the evolution of gravitational perturbation of the four dimensional brane is identical to that of bulk based approach, while the eigenvalue equation, i.e., \ref{GW_Eq20a} determining the mass of graviton is different. Hence the Kaluza-Klein mass modes of graviton in the brane based approach will be different from that in the bulk based approach and hence will have interesting observational consequences in both high energy collision experiments as well as in propagation of gravitational waves. In this work we will mainly be interested in the effect of the mass term originating from \ref{GW_Eq20b}, in particular how it modifies the behaviour of perturbations in contrast to general relativity\ and also how the brane and bulk based approach differs. This is what we will concentrate on in the next sections.
\section{Specialising to spherically symmetric vacuum brane}\label{Sph_Symm}
We have described a general method for deriving the dynamical equations pertaining to gravitational perturbation, starting from the effective gravitational field equations on the brane in the previous section. We would now like to apply the above scenario in the context of black holes on the brane. In particular we are interested in perturbations around the background given by \ref{GW_Eq01}. Thus in this section, with the above scenario in the backdrop, we specialise to vacuum and spherically symmetric solution on the brane, such that, $g_{\alpha \beta}=\textrm{diag}(-f(r),f^{-1}(r),r^{2},r^{2}\sin ^{2}\theta)$. For the moment we concentrate on situations with arbitrary choices for $f(r)$, while later on we will choose a specific form for $f(r)$, namely, $f(r)=1-(2/r)$. Further being a vacuum solution, the Ricci tensor and Ricci scalar identically vanishes.
The main focus now will be understanding the evolution equation of the gravitational perturbation $h_{\mu \nu}$ before discussing the Kaluza-Klein modes. In general the perturbation $h_{\mu \nu}$ can depend on all the spacetime coordinates $(t,r,\theta,\phi)$. The spherical symmetry associated with this problem demands a separation between $(t,r)$ and $(\theta,\phi)$ part, which results into decomposition of the angular part into spherical harmonics. In particular, for the gravitational perturbation we obtain,
\begin{align}\label{GW_Eq21}
h_{\alpha \beta}^{(n)}=\sum _{l=0}^{\infty}\sum _{m=-l}^{l}\sum _{i=1}^{10}h_{i}^{(nlm)}(t,r)\left\{Y^{(i)}_{lm}\right\}_{\alpha \beta}(\theta,\phi)~,
\end{align}
where the perturbation $h_{\alpha \beta}$ have been broken up into ten independent parts, separated into $h_{i}^{(nlm)}$ depending on $(t,r)$ and the rest depending on the angular coordinates. Further, $n$ stands for the Kaluza-Klein mode index, while $l$ is the angular momentum and $m$ being its $z$-component. The quantities $\{Y_{lm}\}_{\alpha \beta}$ are the tensorial spherical harmonics in four spacetime dimensions. In order to define these tensor harmonics one should introduce the following normalised basis vectors,
\begin{align}\label{GW_Eq22}
t^{\alpha}&=\frac{1}{\sqrt{f(r)}}\left(\partial _{t}\right)^{\alpha};\qquad r^{\alpha}=\sqrt{f(r)}\left(\partial _{r}\right)^{\alpha};
\nonumber
\\
\theta ^{\alpha}&=\frac{1}{r}\left(\partial _{\theta}\right)^{\alpha};\qquad \phi ^{\alpha}=\frac{1}{r\sin \theta}\left(\partial _{\phi}\right)^{\alpha}~.
\end{align}
It is clear that they are orthogonal to each other, while the factors in the front ensures that they are normalised as well. Given this structure one can introduce an induced metric on the $(\theta,\phi)$ plane such that, $\mu_{\alpha \beta}=g_{\alpha \beta}+t_{\alpha}t_{\beta}-r_{\alpha}r_{\beta}$, leading to, $t^{\alpha}\mu _{\alpha \beta}=0=r^{\alpha}\mu_{\alpha \beta}$. One can also define an antisymmetric tensor $\epsilon _{\alpha \beta}=\theta _{\alpha}\phi _{\beta}-\phi_{\alpha}\theta _{\beta}$. Given this one can construct ten such irreducible representations, which include $t_{\alpha}t_{\beta}Y_{lm}$, $\mu_{\alpha \beta}Y_{lm}$ and so on involving no derivatives of $Y_{lm}$, as well as terms like $r_{(\alpha}\mu_{\beta) \rho}\nabla ^{\rho}Y_{lm}$, $t_{(\alpha}\mu_{\beta)\rho}\nabla ^{\rho}Y_{lm}$ etc. depending on derivatives of $Y_{lm}$. Among all these choices, three terms among the ten will depend on the antisymmetric combination $\epsilon _{\alpha \beta}$ and will pick up a term $(-1)^{l+1}$
under parity. These we will refer to as \emph{axial perturbations}. On the other hand, the remaining seven components will inherit an extra factor of $(-1)^{l}$ under parity transformation and are referred to as \emph{polar perturbations}. Thus the spherical harmonic decomposition of $h^{(n)}_{\alpha \beta}$ in \ref{GW_Eq21} can be further subdivided into axial and polar parts.
The above decomposition is useful in simplifying the algebra further. It is evident that the operators acting on $h_{\alpha \beta}$ in \ref{GW_Eq20b} are invariant under parity. Thus the solutions to \ref{GW_Eq20b} which are eigenfunctions of parity with different eigenvalues decouple from each other. Hence in the present scenario, the polar and axial perturbations differ from each other in parity eigenvalue and hence evolves independently of one another. Further two axial (or, polar) modes having different $l$ and $m$ values also have different eigenvalues under parity and hence they also decouple. Thus one can solve for the evolution of a given $l$ mode for axial (or, polar) perturbation separately.
Due to complicated nature of the polar perturbations, we content ourselves with the axial perturbations only. The angular part of the axial perturbations contain essentially three terms, two depending on single derivative of $Y_{lm}$, while the third one depends on double derivatives of $Y_{lm}$. Thus for $l=0$ all the axial modes identically vanishes and for $l=1$, the term involving double derivatives of $Y_{lm}$ does not contribute. Hence in what follows we will concentrate on the $l\geq 2$ scenario. In this case there are two master variables, which we will denote by $u_{n,l}$ and $v_{n,l}$ respectively and their evolution equations read,
\begin{align}
\mathcal{D}u_{n,l}+f(r)\Big\{m_{n}^{2}&+\frac{l(l+1)}{r^{2}}-\frac{6}{r^{3}} \Big\}u_{n,l}
\nonumber
\\
&+f(r)\frac{m_{n}^{2}}{r^{3}}v_{n,l}=0~,
\label{GW_Eq23a}
\\
\mathcal{D}v_{n,l}+f(r)\Big\{m_{n}^{2}&+\frac{l(l+1)}{r^{2}} \Big\}v_{n,l}+4f(r)u_{n,l}=0~.
\label{GW_Eq23b}
\end{align}
Here, $\mathcal{D}$ is the differential operator $\partial _{t}^{2}-\partial _{r_{*}}^{2}$, where $r_{*}$ is the tortoise coordinate defined using $f(r)$ as $dr_{*}=dr/f(r)$. Note that these two differential equations are coupled to each other and provides a complete set. The massless limit also turns out to be interesting. As far as $u_{n,l}$ is concerned \ref{GW_Eq23a} decouples and the corresponding potential reduces to the well known Regge-Wheeler form. The potential for $v_{n,l}$ resembles to that of an electromagnetic field. Note that an identical form for the equations were derived in \cite{Seahra:2004fg}, however from a different perspective. This is due to the fact explained in \ref{Perturb_Effective}, i.e, the evolution of gravitational perturbation equation is identical to \cite{Seahra:2004fg} modulo the Kaluza-Klein decomposition and hence the mass term.
\begin{table*}
\begin{center}
\caption{Numerical estimates of the first ten Kaluza-Klein mass modes correct to second decimal place for two possible choices of the inter-brane separation $d$ and bulk curvature scale $\ell$ have been presented for brane based approach. First \ref{GW_Eq29} has been solved for $z_{n}$ and the result has been presented in the second column. Incidentally, the solution for $z_{n}$ is insensitive to choices of $d/\ell$ as far as solutions accurate to second decimal places are considered. To avoid any instability present in the problems the inverse of bulk curvature scale has been chosen such that, the mass of lowest lying Kaluza-Klein mode is greater than or equal to $0.43$ in geometrised units.}\label{Table_01}
\begin{tabular}{p{3.5cm}p{1.5cm}p{4cm}p{4.5cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Kaluza-Klein Modes & ~~$z_{n}$ & ~~~~~~Associated Mass & ~~~~~~Associated Mass \\
& &($d/\ell=20; 1/\ell=6\times 10^{7}$) & ($d/\ell=30; 1/\ell=1.3\times 10^{12}$) \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
~~~~~~n=1 & 3.56 & ~~~~~~~~~ 0.44 & ~~~~~~~~~~ 0.43 \\
~~~~~~n=2 & 6.74 & ~~~~~~~~~ 0.83 & ~~~~~~~~~~ 0.82 \\
~~~~~~n=3 & 9.88 & ~~~~~~~~~ 1.22 & ~~~~~~~~~~ 1.20 \\
~~~~~~n=4 & 13.03 & ~~~~~~~~~ 1.61 & ~~~~~~~~~~ 1.58 \\
~~~~~~n=5 & 16.17 & ~~~~~~~~~ 2.00 & ~~~~~~~~~~ 1.98 \\
~~~~~~n=6 & 19.32 & ~~~~~~~~~ 2.39 & ~~~~~~~~~~ 2.35 \\
~~~~~~n=7 & 22.48 & ~~~~~~~~~ 2.78 & ~~~~~~~~~~ 2.73 \\
~~~~~~n=8 & 25.60 & ~~~~~~~~~ 3.17 & ~~~~~~~~~~ 3.11 \\
~~~~~~n=9 & 28.75 & ~~~~~~~~~ 3.56 & ~~~~~~~~~~ 3.50 \\
~~~~~~n=10 & 31.89 & ~~~~~~~~~ 3.94 & ~~~~~~~~~~ 3.88 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
Having discussed the scenario for gravitational perturbation, let us explore the higher dimensional effects, i.e., determination of the mass term by solving \ref{GW_Eq20a}. We will be concerned with the even parity eigenfunctions of \ref{GW_Eq20a}, as the derivation of effective field equations assume existence of $Z_{2}$ symmetry. Further, \ref{GW_Eq20a} being a second order differential equation will require two boundary conditions to uniquely arrive at the solution. Rather than imposing boundary conditions on $\chi(y)$ we will impose boundary conditions in $\partial _{y}\chi(y)$. Before engaging with the boundary conditions let us solve \ref{GW_Eq20a}, which on introduction of the new variable, $\zeta =e^{ky}$, becomes
\begin{align}\label{GW_Eq24}
\zeta ^{-2}\left\lbrace -k^{2}\chi+k^{2}\zeta ^{2}\frac{d^{2}\chi}{d\zeta ^{2}}+4k^{2}\zeta \frac{d\chi}{d\zeta}\right\rbrace +m^{2}\chi =0~,
\end{align}
where the following results have been used,
\begin{align}\label{GW_Eq25}
\frac{d\chi}{dy}=k\zeta \frac{d\chi}{d\zeta};
\qquad
\frac{d^{2}\chi}{dy^{2}}=k^{2}\zeta ^{2}\frac{d^{2}\chi}{d\zeta ^{2}}+k^{2}\zeta \frac{d\chi}{d\zeta}~.
\end{align}
One can further transform the above equation to a more manageable form by introducing yet another variable $\xi$, replacing $\zeta$, such that $m\zeta=\xi$ and the transformed version of \ref{GW_Eq24} takes the following form,
\begin{align}\label{GW_Eq26}
k^{2}\xi ^{2}\frac{d^{2}\chi}{d\xi ^{2}}+4k^{2}\xi \frac{d\chi}{d\xi}+\left(\xi ^{2}-k^{2}\right)\chi=0~.
\end{align}
The above equation is essentially Bessel's differential equation and hence it's two independent solutions in terms of modified Bessel functions of first and second kinds are
\begin{align}\label{GW_Eq27}
\chi (y)=e^{-\frac{3}{2}ky}\left[C_{1}J_{\nu}\left(\frac{me^{ky}}{k}\right)
+C_{2}Y_{\nu}\left(\frac{me^{ky}}{k}\right)\right]~,
\end{align}
with $\nu=\sqrt{13}/2$. The departure from bulk based approach should now be evident from the above analysis. The effect of higher dimensions is through the extra dimensional part of the gravitational perturbation, namely $\chi(y)$. This is certainly a discriminating feature between the bulk and the brane based approach, since the order of the Bessel functions appearing in these two approaches to determine the Kaluza-Klein mode masses are different \cite{Seahra:2004fg}. Thus it is clear that the mass spectrum of our model will be different when compared to the bulk based approach.
Let us briefly point out the reason behind the difference between Kaluza-Klein mode masses when one follows the brane-based approach, on the one hand, and the bulk-based approach, on the other hand. This is basically due to the difference in the gravitational field equations. For example, when perturbing the bulk gravitational field equations, the Weyl tensor plays no role. By contrast, the perturbation of the Weyl tensor plays a central role in the brane-based approach. Therefore, the basic field equations governing dynamics of gravity in the two approaches differ, but the Schwarzschild AdS spacetime is still a solution of both the field equations. Hence, even though the background solution is the same in both cases, the perturbations follow different dynamics pertaining to the fact that field equations themselves are different. This is why the Kaluza-Klein mode masses are also different. An analogy may be helpful here. For instance, the Schwarzschild solution is a solution of both Einstein gravity as well as
$f(R)
$ gravity. However, the field equations of both these theories are widely different. Thus, the perturbations about the Schwarzschild background will satisfy different evolution equations in these theories (see, for example \cite{Bhattacharyya:2017tyc,Pratten:2015rqa,Berti:2015itd}), like the scenario we are considering in this work. The fact that the field equations in the bulk and brane based approaches are different is known and is manifested in the fact that there exist solutions to the field equations in the brane-based approach, with no bulk correspondence whatsoever \cite{Dadhich:2000am,Chamblin:2000ra,Maartens:2001jx,Maartens:2010ar,Dadhich:2001ry}. This explains the difference in the masses of the Kaluza-Klein modes associated with the brane and the bulk based approaches respectively.
\begin{table*}
\begin{center}
\caption{Numerical estimates of the mass of first ten Kaluza-Klein modes have been presented for the bulk based approach, by solving \ref{GW_Eq29} for $\nu=2$. It is clear from \ref{Table_01} that the solution $z_{n}$ of \ref{GW_Eq29} is different in the bulk based approach in comparison to the brane based one. Among the two sets of choices for the inter-brane separation $d$ and bulk curvature scale $\ell$, one is identical to that of brane based approach, while the other slightly differs. Both these situations clearly depict the differences of the Kaluza-Klein mass modes in the brane and the bulk based approach.}
\label{Table_02}
\begin{tabular}{p{3.5cm}p{1.5cm}p{4cm}p{4.5cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Kaluza-Klein Modes & ~~$z_{n}$ & ~~~~~~Associated Mass & ~~~~~~Associated Mass \\
& & ($d/\ell=20$; $1/\ell=6\times 10^{7}$) & ($d/\ell=30$; $1/\ell=1.2\times 10^{12}$) \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
~~~~~~~n=1 & 3.83 & ~~~~~~~~~ 0.47 & ~~~~~~~~~~~ 0.43 \\
~~~~~~~n=2 & 7.01 & ~~~~~~~~~ 0.87 & ~~~~~~~~~~~ 0.79 \\
~~~~~~~n=3 & 10.18 & ~~~~~~~~~ 1.26 & ~~~~~~~~~~~ 1.14 \\
~~~~~~~n=4 & 13.33 & ~~~~~~~~~ 1.65 & ~~~~~~~~~~~ 1.50 \\
~~~~~~~n=5 & 16.46 & ~~~~~~~~~ 2.03 & ~~~~~~~~~~~ 1.85 \\
~~~~~~~n=6 & 19.61 & ~~~~~~~~~ 2.42 & ~~~~~~~~~~~ 2.20 \\
~~~~~~~n=7 & 22.76 & ~~~~~~~~~ 2.81 & ~~~~~~~~~~~ 2.56 \\
~~~~~~~n=8 & 25.91 & ~~~~~~~~~ 3.20 & ~~~~~~~~~~~ 2.91 \\
~~~~~~~n=9 & 29.05 & ~~~~~~~~~ 3.59 & ~~~~~~~~~~~ 3.26 \\
~~~~~~~n=10 & 32.19 & ~~~~~~~~~ 3.98 & ~~~~~~~~~~~ 3.61 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
To find the unknown coefficients $C_{1}$ and $C_{2}$ we need to impose boundary conditions and as emphasised earlier these conditions will be on derivatives of $\chi(y)$. To make the analysis at par with possible resolutions of the hierarchy problem, we assume the existence of another brane located at some $y=d$. Incidentally, the distance $d$ need not be constant but varying, known as radion field, whose stabilisation would lead to a non-zero inter-brane separation $d$ \cite{Goldberger:1999uk}. We have also neglected effects of brane bending, if any, by assuming that $d$ is a pure constant. Hence the boundary conditions imposed are given by, $[\partial _{y}+(\nu+(3/2)) k]\chi =0$ at $y=0$ and also on the other brane hypersurface at $y=d$. This leads to the following two algebraic equations satisfied by the two unknown coefficients $C_{1}$ and $C_{2}$ as
\begin{align}
C_{1}J_{\nu-1}(m/k)&+C_{2}Y_{\nu-1}(m/k)=0~,
\label{GW_Eq28a}
\\
C_{1}J_{\nu-1}(\{m/k\}e^{kd})&+C_{2}Y_{\nu -1}(\{m/k\}e^{kd})=0~.
\label{GW_Eq28b}
\end{align}
Using the first relation one can determine the ratio $C_{1}/C_{2}$ and hence the solution for $\chi(y)$ gets determined except for an overall normalisation. On the other hand, substitution of the same in \ref{GW_Eq28b} results into the algebraic equation
\begin{align}\label{GW_Eq29}
Y_{\nu-1}(m_{n}/k)J_{\nu-1}(z_{n})-J_{\nu-1}(m_{n}/k)Y_{\nu-1}(z_{n})=0~,
\end{align}
where $m_{n}=\{z_{n}k\}e^{-kd}$ yields an infinite series of solutions for the mass, where $n$ stands for a particular Kaluza-Klein mode. The masses for the first ten Kaluza-Klein modes have been presented in \ref{Table_01} for two different sets of choices of inter-brane separation $d$ and bulk curvature scale $\ell=1/k$. This has been achieved by first solving for $z_{n}$ using \ref{GW_Eq29} and then obtaining the Kaluza-Klein mass $m_{n}$.
To see clearly the difference between brane and bulk based approach, we have presented the masses of the first ten lowest lying Kaluza-Klein modes in the context of bulk based approach as well. This requires solving \ref{GW_Eq29} for $z_{n}$ with $\nu=2$. It is evident from \ref{Table_02} that the solutions for $z_{n}$ are completely different in the two scenarios. In particular the numerical values of $z_{n}$ in the brane based approach are lower than the corresponding numerical values in the bulk based approach. This results in lowering of the masses of Kaluza-Klein modes in the brane based approach, as evident from \ref{Table_01} and \ref{Table_02} for the choices $d/\ell=20$ and $\ell^{-1}=6\times 10^{7}$ in geometrised units. The numerical values are so chosen that they are in agreement with other constraints already present in this framework. For example, $d/\ell \geq 13$ is necessary to arrive at the desired warping required to get around the gauge hierarchy problem, while the table top experiment of Newton's law would
demand $1/\ell \geq 10^{7}(M/M_{\odot})$ (or, $\ell\leq 0.1~\textrm{mm}$), where $M_{\odot}$ is the solar mass \cite{Hoyle:2004cw,Long:2002wn,Smullin:2005iv}. This explains the choices of $d/\ell$ as well as that of $1/\ell$. Numerical estimates of the masses of the Kaluza-Klein modes for two such choices of $d/\ell$ and $1/\ell$ values have been presented in \ref{Table_01} and \ref{Table_02} respectively. Masses of these Kaluza-Klein modes will be used in the next section for determination of the quasi-normal modes for the brane black hole.
At this stage it is worth mentioning about the Gregory-Laflamme instability, which originates due to instability of the bulk metric under perturbation pertaining to long wavelength modes \cite{Gregory:2008rf,Gregory:2000gf,Gregory:2011kh,Lehner:2011wc,Frolov:2009jr}. The fact that there exist another brane at $y=d$ helps to evade the instability by providing a cutoff on the long wavelength modes. The separation $d$ between the two branes as well as the bulk curvature scale $\ell\sim 1/k$ (see \ref{GW_Eq01}) are also bounded by the fact that we have not seen any influence of the extra dimension on the gravitational interaction in our observable universe. The above instability essentially translates through $d$ and $\ell$ into the mass of the Kaluza-Klein modes and for $m_{n}\gtrsim 0.43$ the above instability can be avoided, which is also reflected in both the tables depicting masses of the Kaluza-Klein modes (see also \cite{Konoplya:2008yy}).
Finally, given a particular Kaluza-Klein mode $n$, one can determine the extra dimensional part of the gravitational perturbation as
\begin{align}\label{GW_Eq30}
\chi _{n}(y)=N_{n}\Bigg[&Y_{\nu-1}(m_{n}/k)J_{\nu}(\{m_{n}/k\} e^{ky})
\nonumber
\\
&-J_{\nu-1}(m_{n}/k)Y_{\nu}(\{m_{n}/k\} e^{ky})\Bigg]~.
\end{align}
Here $N_{n}$ is the overall normalisation factor and $\nu=\sqrt{13}/2$ is the order of the Bessel functions. Thus the complete solution to the gravitational perturbation can be written in the following form,
\begin{align}
h_{\alpha \beta}(t,r,\theta,\phi;y)&=\sum _{n=0}^{\infty}N_{n}\Big\{Y_{\nu-1}(m_{n}/k)J_{\nu}(\{m_{n}/k\} e^{ky})
\nonumber
\\
&-J_{\nu-1}(m_{n}/k)Y_{\nu}(\{m_{n}/k\} e^{ky})\Big\}
\nonumber
\\
&\times \sum _{l=0}^{\infty}\sum _{m=-l}^{l} \Bigg\{\sum _{i=1}^{7}P_{i}^{(nlm)}(t,r)\mathcal{P}^{(i)lm}_{\alpha \beta}(\theta,\phi)
\nonumber
\\
&+\sum _{i=1}^{3}A_{i}^{(nlm)}(t,r)\mathcal{A}^{(i)lm}_{\alpha \beta}(\theta,\phi)\Bigg\}~.
\end{align}
Here the first part is the contribution from extra dimensions, while the four dimensional effects have been divided into polar and axial perturbations respectively. The first seven are the polar perturbations, while the last three are the axial ones. As already emphasised earlier, these two contributions do not mix and hence one can treat them separately. We have already provided the evolution equations for the master variables associated with the axial perturbation in \ref{GW_Eq23a} and \ref{GW_Eq23b}, which we will solve next. The solution (or, evolution) can be obtained in two ways, by the calculation of quasi-normal modes, or, performing a fully numerical Cauchy evolution of the initial data. We have performed both these analysis in this work and shall present the calculation of quasi-normal modes in the next section before taking up the Cauchy evolution of initial data.
\section{The spectrum of associated quasi-normal modes}\label{qnm_analysis}
\begin{table*}
\begin{center}
\caption{Real and Imaginary parts of the quasi-normal mode frequencies have been presented. These are obtained from the \emph{brane based approach} with the following choice of parameters associated with the extra dimensions: $d/\ell=20; 1/\ell=6\times 10^{7}$. In particular, results for the first two Kaluza-Klein mass modes have been presented for four different choices of the angular momentum.}
\label{Table_03}
\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}p{3cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
& $m=0.44, l=2$ & & $m=0.83, l=2$ & \\
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Mode & Real & Imaginary & Real & Imaginary \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.467 & -0.051 & 0.396 & -0.038 \\
j=2 & 0.530 & -0.071 & 0.543 & -0.104 \\
j=3 & 0.378 & -0.197 & 0.183 & -0.168 \\
j=4 & 0.473 & -0.239 & 0.243 & -0.369 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.44, l=3$ & & $m=0.83, l=3$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.653 & -0.078 & 0.843 & -0.051 \\
j=2 & 0.708 & -0.084 & 0.708 & -0.134 \\
j=3 & 0.618 & -0.244 & 0.773 & -0.176 \\
j=4 & 0.562 & -0.437 & 0.576 & -0.327 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.44, l=4$ & & $m=0.83, l=4$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.847 & -0.086 & 0.951 & -0.064 \\
j=2 & 0.827 & -0.263 & 0.960 & -0.219 \\
j=3 & 0.791 & -0.451 & 0.896 & -0.393 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.44, l=5$ & & $m=0.83, l=5$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 1.043 & -0.090 & 1.123 & -0.076 \\
j=2 & 1.084 & -0.272 & -1.098 & -0.233 \\
j=3 & 1.002 & -0.460 & 1.051 & -0.404 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{Real and Imaginary parts of the first few quasi-normal mode frequencies have been depicted. These values are obtained starting from the \emph{brane based approach}, with the following choices of the extra dimensional parameters: $d/\ell=30; 1/\ell=1.3\times 10^{12}$. Results have been presented for two lowest lying Kaluza-Klein mass modes and for four choices of angular momentum associated with each modes.}
\label{Table_04}
\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}p{3cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
& $m=0.43, l=2$ & & $m=0.82, l=2$ & \\
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Mode & Real & Imaginary & Real & Imaginary \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.462 & -0.053 & 0.702 & -0.006 \\
j=2 & 0.527 & -0.072 & 0.385 & -0.041 \\
j=3 & 0.377 & -0.201 & 0.541 & -0.109 \\
j=4 & 0.471 & -0.241 & 0.253 & -0.326 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=3$ & & $m=0.82, l=3$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.650 & -0.079 & 0.796 & -0.036 \\
j=2 & 0.616 & -0.246 & 0.705 & -0.138 \\
j=3 & 0.679 & -0.261 & 0.770 & -0.179 \\
j=4 & 0.562 & -0.439 & 0.576 & -0.331 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=4$ & & $m=0.82, l=4$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.846 & -0.086 & 0.948 & -0.065 \\
j=2 & 0.826 & -0.264 & 0.906 & -0.204 \\
j=3 & 0.790 & -0.452 & 0.833 & -0.373 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=5$ & & $m=0.82, l=5$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 1.041 & -0.090 & 1.120 & -0.076 \\
j=2 & 1.027 & -0.272 & 1.095 & -0.234 \\
j=3 & 1.001 & -0.461 & 1.050 & -0.406 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{In this table we have presented numerical estimates for real and imaginary parts of the quasi-normal mode frequencies obtained from the \emph{bulk based approach}. The parameters characterising the bulk spacetime corresponds to: $d/\ell=20$; $1/\ell=6\times 10^{7}$. In this situation as well we have presented the quasi-normal mode frequencies for four possible choices of angular momentum given the two lowermost Kaluza-Klein mode masses.}
\label{Table_05}
\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}p{3cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
& $m=0.47, l=2$ & & $m=0.87, l=2$ & \\
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Mode & Real & Imaginary & Real & Imaginary \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.480 & -0.046 & 0.437 & -0.015 \\
j=2 & 0.540 & -0.067 & 0.542 & -0.087 \\
j=3 & 0.381 & -0.185 & 0.119 & -0.128 \\
j=4 & 0.477 & -0.231 & 0.242 & -0.318 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.47, l=3$ & & $m=0.87, l=3$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.660 & -0.076 & 0.862 & -0.045 \\
j=2 & 0.716 & -0.082 & 0.719 & -0.117 \\
j=3 & 0.623 & -0.239 & 0.785 & -0.163 \\
j=4 & 0.564 & -0.431 & 0.576 & -0.313 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.47, l=4$ & & $m=0.87, l=4$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.853 & -0.085 & 1.010 & -0.067 \\
j=2 & 0.831 & -0.259 & 0.920 & -0.193 \\
j=3 & 0.793 & -0.447 & 0.840 & -0.359 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.47, l=5$ & & $m=0.87, l=5$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 1.047 & -0.089 & 1.176 & -0.077 \\
j=2 & 1.032 & -0.269 & 1.108 & -0.227 \\
j=3 & 1.004 & -0.457 & 1.058 & -0.396 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{Real and Imaginary parts of the quasi-normal mode frequencies have been depicted in a bulk spacetime with the following set of parameters: $d/\ell=30$; $1/\ell=1.2\times 10^{12}$ in the \emph{bulk based approach}. The values have been presented for four choices of angular momentum, given the two lowest lying Kaluza-Klein modes.}
\label{Table_06}
\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}p{3cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
& $m=0.43, l=2$ & & $m=0.79, l=2$ & \\
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
Mode & Real & Imaginary & Real & Imaginary \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.462 & -0.053 & 0.672 & -0.006 \\
j=2 & 0.527 & -0.072 & 0.456 & -0.014 \\
j=3 & 0.377 & -0.201 & 0.542 & -0.087 \\
j=4 & 0.471 & -0.241 & 0.534 & -0.123 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=3$ & & $m=0.79, l=3$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.650 & -0.079 & 0.825 & -0.055 \\
j=2 & 0.616 & -0.246 & 0.696 & -0.149 \\
j=3 & 0.678 & -0.261 & 0.761 & -0.187 \\
j=4 & 0.562 & -0.439 & 0.666 & -0.377 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=4$ & & $m=0.79, l=4$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 0.846 & -0.086 & 0.937 & -0.067 \\
j=2 & 0.826 & -0.264 & 0.898 & -0.210 \\
j=3 & 0.790 & -0.452 & 0.829 & -0.382 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& $m=0.43, l=5$ & & $m=0.79, l=5$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
j=1 & 1.041 & -0.090 & 1.112 & -0.078 \\
j=2 & 1.027 & -0.272 & -1.089 & -0.238 \\
j=3 & 1.001 & -0.461 & 1.045 & -0.411 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
In this section we will investigate the characteristic frequencies, namely the quasi-normal modes associated with the propagation of massive Kaluza-Klein modes in the Schwarzschild geometry induced on the brane hypersurface. This is usually performed by going over to the frequency space, such that,
\begin{align}
u_{n,l}(t,r)&=\int d\omega~e^{-i\omega t} \psi_{n,l}(\omega,r)~,
\label{Eq_qnm_01a}
\\
v_{n,l}(t,r)&=\int d\omega~e^{-i\omega t} \phi_{n,l}(\omega,r)~.
\label{Eq_qnm_01b}
\end{align}
At this stage all possible frequencies are allowed, but as we will see later on, this is not the case. Only some specific set of frequencies are allowed, known as the quasi-normal mode frequencies and hence the above integral will be converted to a sum over all the quasi-normal mode frequencies. The single most important fact about this expansion is that the quasi-normal mode frequencies can be imaginary. Since we do not expect any runaway situations associated with this problem, thus $\textrm{Im}(\omega)<0$ are the allowed quasi-normal mode frequencies \cite{Leaver:1985ax,Chandrasekhar:1985kt,Nollert:1999ji,Kokkotas:1999bd,Pani:2013pma,Cardoso:2011xi,
Rosa:2011my,Konoplya:2008rq,Konoplya:2006br,Kanti:2005xa}. Before getting into the details of obtaining the quasi-normal mode frequencies in this context, let us briefly discuss about another prediction of \ref{GW_Eq23a} and \ref{Eq_qnm_01a}, namely late time wave tails. Since quasi-normal mode frequencies have a real as well as imaginary parts, it is exponentially suppressed and at late times ($t\rightarrow \infty$) it produces vanishing contribution. Therefore the wave tail, originating from existence of branch cut in the frequency integral of \ref{Eq_qnm_01a} dominates the late time behaviour of the gravitational perturbation $u_{n,l}(t)$. It turns out that the power law scaling of the perturbation modes have a universal behaviour. In particular, for massive gravitational modes, which includes the scenario presented in this work, the late time behaviour essentially corresponds to the following universal power law behaviour $u_{n,l}(t)\sim t^{-5/6}\sin (\omega t)$. Here the oscillation frequency $\omega$ depends on the mass of the
perturbation mode linearly. Thus the late time behaviour is essentially governed by the $t^{-5/6}$ universal factor. We will need this fact in the later parts of this work. For the moment being we will exclusively concentrate on the quasi-normal mode analysis.
In order to determine the quasi-normal mode frequencies one also needs to impose suitable boundary conditions on the solution space. These are --- (a) the quasi-normal mode must be ingoing at the black hole horizon and (b) these modes must be outgoing in the asymptotic regions. These conditions are best suited in terms of the tortoise coordinate $r_{*}$, defined as integral of $\{dr/f(r)\}$, in which the horizon corresponds to $r_{*}\rightarrow -\infty$, while the asymptotic region implies $r_{*}\rightarrow \infty$. Thus the condition that quasi-normal modes are ingoing at the horizon implies that $u_{n,l}(\omega,r_{*})$ as well as $v_{n,l}(\omega,r_{*})$ behave as $\exp(-i\omega r_{*})$ in the near horizon regime. A similar situation will exist for the asymptotic region as well. These boundary conditions will dictate the discrete values of the frequencies associated with the quasi-normal modes. These values will have three indices, the Kaluza-Klein mode index $n$, the angular momentum index $l$ and the quasi-normal mode index $p$. Having obtained the
corresponding quasi-normal modes one can substitute them back to \ref{Eq_qnm_01a} and \ref{Eq_qnm_01b} respectively and thus obtain the time evolution of the both $u_{n,l}(t,r_{*})$ and $v_{n,l}(t,r_{*})$. These estimates can then be compared with the Cauchy evolution problem and a match between the two will ensure correctness of our method presented here. Thus for completeness and consistency we will also present results for Cauchy evolution in the next section. We will mainly content ourselves with the continued fraction method but will briefly discuss the forward integration scheme as well.
\subsection{Continued fraction method}
The frequency spectrum associated with the quasi-normal modes can be obtained by starting with a suitable ansatz for $u_{n,l}(t,r)$ and $v_{n,l}(t,r)$ respectively. Given this ansatz one can try to obtain a power series solution associated with the differential equations presented in \ref{GW_Eq23a} and \ref{GW_Eq23b}, resulting in recursion relation between the coefficients of various terms in the power series. This recursion relation will be satisfied provided the quasi-normal mode frequencies are discrete. For this purpose we start with the following general form of the coupled differential equations,
\begin{align}
-\frac{\partial ^{2}u_{n,l}}{\partial t^{2}}+\frac{\partial ^{2}u_{n,l}}{\partial r_{*}^{2}}-f(r)&\left(m_{n}^{2}+\frac{l(l+1)r-6}{r^{3}}\right)u_{n,l}
\nonumber
\\
&-f(r)\frac{m_{n}^{2}}{r^{3}}v_{n,l}=0~,
\\
-\frac{\partial ^{2}v_{n,l}}{\partial t^{2}}+\frac{\partial ^{2}v_{n,l}}{\partial r_{*}^{2}}-f(r)&\left(m_{n}^{2}+\frac{l(l+1)}{r^{2}} \right)v_{n,l}
\nonumber
\\
&-4f(r)u_{n,l}=0~,
\end{align}
where $f(r)=1-(2/r)$. Subsequently eliminating derivatives with respect to $r_{*}$ in favour of $r$ and writing down the two master variables $u_{n,l}(t,r)$ and $v_{n,l}(t,r)$ as in \ref{Eq_qnm_01a} and \ref{Eq_qnm_01b} we obtain after simplifications,
\begin{align}
r(r&-2)\frac{d^{2}\psi_{n,l}}{dr^{2}}+2\frac{d\psi _{n,l}}{dr}+\frac{\omega ^{2}r^{3}}{r-2}\psi _{n,l}
\nonumber
\\
&-\left[m_{n}^{2}r^{2}+l(l+1)-\frac{6}{r}\right]\psi _{n,l}-\frac{m_{n}^{2}}{r}\phi _{n,l}=0~,
\label{Eq_qnm_02a}
\\
r(r&-2)\frac{d^{2}\phi_{n,l}}{dr^{2}}+2\frac{d\phi _{n,l}}{dr}+\frac{\omega ^{2}r^{3}}{r-2}\phi _{n,l}
\nonumber
\\
&-\left[m_{n}^{2}r^{2}+l(l+1)\right]\phi _{n,l}-4r^{2}\psi _{n,l}=0~.
\label{Eq_qnm_02b}
\end{align}
Having derived the basic equations governing $\psi_{n,l}$ and $\phi_{n,l}$ one normally writes down both these master variables in terms of various powers of $r$ and $(r-2)$, such that the boundary conditions at horizon and at asymptotic regions can be satisfied. Subsequently the remaining pieces of $\psi_{n,l}$ and $\phi_{n,l}$ are solved by using the power series method. The resulting recursion relation between the coefficients of these power series will also be coupled and it is only helpful to combine them into a single matrix equation with off-diagonal entries illustrating the coupling between the systems. Performing the same for the master variables involved here as well, one ends up with the following matrix equation for $j>0$, with integer $j$ as,
\begin{align}
\mathbf{P}_{j}\mathbf{V}_{j+1}+\mathbf{Q}_{j}\mathbf{V}_{j}+\mathbf{R}_{j}\mathbf{V}_{j-1}=0~.
\label{Eq_qnm_07}
\end{align}
Here, the coefficients $\mathbf{P}_{j}$, $\mathbf{Q}_{j}$ and $\mathbf{R}_{j}$ depend on the details of the system, i.e., on the parameters involved. The vector $\mathbf{V}_{j}$ on the other hand corresponds to a column matrix constructed out of the power series coefficients for $\psi_{n,l}$ and $\phi_{n,l}$, such that one obtains
\begin{align}
\mathbf{P}_{j}&=\left(\begin{array}{ll}
\alpha _{j} & 0\\
0 & \alpha _{j}
\end{array}\right)~,\qquad
\mathbf{Q}_{j}=\left(\begin{array}{ll}
\beta _{j}+3 & -\frac{m_{n}^{2}}{2}\\
-4 & \beta _{j}
\end{array}\right)~,
\nonumber
\\
\mathbf{R}_{j}&=\left(\begin{array}{ll}
\gamma _{j}-3 & \frac{m_{n}^{2}}{2}\\
0 & \gamma _{j}
\end{array}\right)~,
\end{align}
where the unknown coefficients $\alpha_{j}$, $\beta _{j}$ and $\gamma _{j}$ can be written in terms of the Kaluza-Klein mode mass and the quasi-normal mode frequency $\omega$ as,
\begin{align}
\alpha _{j}&=(j+1)(j+1-4i\omega)~,
\nonumber
\\
\gamma _{j}&=\left(j-1+\frac{(\omega -i\lambda)^{2}}{\lambda}\right)\left(j+1+\frac{(\omega -i\lambda)^{2}}{\lambda}\right)~,
\nonumber
\\
\beta _{j}&=-2j^{2}+\left(-2+\frac{8i\omega \lambda -2\omega ^{2}+6\lambda ^{2}}{\lambda} \right)j
\nonumber
\\
&-l(l+1)+\frac{1}{\lambda}\Big(3\lambda ^{2}-\omega ^{2}-12i\omega \lambda^{2}-4\lambda ^{3}
\nonumber
\\
&+4i\omega \lambda +12\lambda \omega ^{2}+4i\omega ^{3} \Big)~.
\end{align}
where, $\lambda=\sqrt{m_{n}^{2}-\omega ^{2}}$. The above recursion relation must be supplemented with the zeroth order recursion relation, which simply reads, $\mathbf{P}_{0}\mathbf{V}_{1}+\mathbf{Q}_{0}\mathbf{V}_{0}=0$. Given this one can use \ref{Eq_qnm_07} to replace $\mathbf{V}_{1}$ in terms of $\mathbf{V}_{0}$ and $\mathbf{V}_{2}$. Subsequently one can again replace $\mathbf{V}_{2}$ by higher order terms using \ref{Eq_qnm_07} repeatedly. This method of solving the matrix valued recursion relation presented in \ref{Eq_qnm_07} is known as the method of continued fraction. In this method, following the procedure outlined above one ends up with an equation of the form $\mathbf{M}\mathbf{V}_{0}=0$, where the matrix $\mathbf{M}$ reads,
\begin{align}
\mathbf{M}=\mathbf{Q}_{0}-\mathbf{P}_{0}\left[\mathbf{Q}_{1}-\mathbf{P}_{1}\{\mathbf{Q}_{2}+\mathbf{P}_{2}\mathbf{M}_{2}\}\mathbf{R}_{2} \right]^{-1}\mathbf{R}_{1}~.
\end{align}
Here $\mathbf{M}_{j}$ is a matrix which can be written in terms of $\mathbf{P}_{j+1}$, $\mathbf{Q}_{j+1}$, $\mathbf{R}_{j+1}$ and most importantly also depends on $\mathbf{M}_{j+1}$. Moreover the matrix $\mathbf{M}_{j}$ when acts on $\mathbf{V}_{j}$ yields $\mathbf{V}_{j+1}$. Thus in order for the matrix equation $\mathbf{M}\mathbf{V}_{0}=0$ to have non-trivial solutions for $\mathbf{V}_{0}$, one must have
\begin{align}\label{Eq_qnm_08}
\textrm{det}~\mathbf{M}=0~.
\end{align}
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.45]{Fig_QNM_01.pdf}~~
\includegraphics[scale=0.45]{Fig_QNM_02.pdf}
\caption{Real and imaginary parts of the quasi-normal mode frequencies have been plotted. The figure on the left corresponds to the quasi-normal mode frequencies associated with the lowest lying Kaluza-Klein mass modes in both the brane and bulk based approach. The curves at the bottom shows the $l=2$ case, while the curves at the top depicts the situation when $l=3$. The figure to the right illustrates an identical situation but for the next Kaluza-Klein mass modes. As evident from the curves, the imaginary part of quasi-normal mode frequencies are smaller in the case of bulk based approach resulting in less damping. We will confirm this behaviour in the later sections.}
\label{Fig_QNM}
\end{figure*}
In principle one needs to take into account an infinite number of terms to solve the above equation. However in practice one truncates $\mathbf{M}_{j}$ at some order $J$, and hence obtain all the lower order matrices starting from $\mathbf{M}_{J}$. Of course, at a later stage one needs to check the independence of the solution of \ref{Eq_qnm_08} explicitly on the truncation order $J$. We have solved the above matrix valued recursion relation using the continued fraction method discussed earlier in the symbolic manipulation package MATHEMATICA and have obtained the corresponding lowest lying quasi-normal mode frequencies associated with various Kaluza-Klein mode masses for different values of angular momentum. These values are listed in four tables. In \ref{Table_03} we present both the real and imaginary parts of the quasi-normal mode frequencies for the two lowest lying Kaluza-Klein mass modes associated with the following values: $d/\ell=20; 1/\ell=6\times 10^{7}$. It is clear that as the mass increases the imaginary part of the lowest quasi-normal mode
frequency decreases, while it increases with angular momentum. For example, when $l=2$, $\textrm{Im}~\omega=-0.05$ for $m_{1}=0.44$, while it becomes $-0.04$ as the mass increases to $m_{2}=0.83$. Hence more massive the Kaluza-Klein modes are, the quasi-normal mode functions are less and less damped --- a feature in complete agreement with the result of \cite{Seahra:2004fg}. While for $m=0.44$, the imaginary part of the lowest quasi-normal mode frequency will read $\textrm{Im}~\omega=-0.051$ for $l=2$, while it becomes $-0.078$ as the angular momentum increases to $l=3$. Thus with an increase of angular momentum the imaginary part of the quasi-normal mode frequency also increases. Hence among the modes with $l=2$ and $l=3$, the time evolution of the $l=3$ mode will be more damped in comparison to the $l=2$ one. This feature is also present in \ref{Table_04}, where the quasi-normal mode frequencies have been presented for a different choice of the ratio between brane separation and bulk curvature scale, namely for $d/\ell=30$ and $1/\ell=1.3\times 10^{12}$.
These numerical values are again chosen to be consistent with previous experimental bounds on $d$ and $\ell$ as explained earlier. In this case also as the mass of the Kaluza-Klein mode increases the imaginary part of the quasi-normal mode frequency decreases, while the increase of angular momentum has a reverse effect. For the same choices of the bulk parameters, the Kaluza-Klein mode masses for the brane based and the bulk based approach differs as evident from \ref{Table_01} and \ref{Table_02}. For example, in the situation where $d/\ell=20; 1/\ell=6\times 10^{7}$, the lowest lying Kaluza-Klein mode mass in brane based approach is $m_{1}=0.44$, while that in the bulk based approach being $m_{1}=0.47$. Hence the imaginary part of the quasi-normal mode frequency will be lower for the bulk based approach. This has interesting implications --- the axial perturbation generated from bulk Einstein's equations will decay in a slower pace in time when compared to the corresponding perturbation mode originated from effective field equations on the brane. This
situation has been clearly depicted in \ref{Table_05} and \ref{Table_06} respectively (see also \ref{Fig_QNM}). One can also check that the quasi-normal mode frequencies derived here indeed matches with those derived in the direct integration scheme which we will discuss next.
\subsection{Direct integration method}
In the previous section we have discussed one particular method of determining the quasi-normal mode frequencies associated with the perturbation of brane world black hole. However for completeness we present another supplementary method of computing the quasi-normal mode frequencies, which can be used along with the continued fraction method to correctly predict the quasi-normal mode frequencies. In this method, as the name suggests, one integrates directly from the horizon to the asymptotic region given the boundary conditions mentioned earlier. In this problem we have two master variables characterising the axial gravitational perturbation and satisfying two second order coupled ordinary differential equations (see \ref{Eq_qnm_03a} and \ref{Eq_qnm_03b} respectively in \ref{App_B}). The solution in the near horizon regime will have $e^{-i\omega r_{*}}$ times a power series around the horizon, while at infinity it will behave as $e^{-k_{\infty}r_{*}}$, where $k_{\infty}$ is the wave number in the asymptotic region. The asymptotic solution will
be characterised by a two-dimensional column vector $\{b_{\infty}^{(1)},b_{\infty}^{(2)}\}$, for which one can choose a suitable orthonormal system of basis vectors. Numerical integration of these differential equations from the horizon out to infinity will lead to a $(2\times 2)$ matrix $\mathbf{S}(m_{n},\omega)$, which can be expanded in the basis introduced above. Finally, setting the determinant of this matrix $\mathbf{S}$ to zero one can solve for the quasi-normal mode frequencies
\cite{Pani:2013pma,Rosa:2011my}.
Further note that, this method is particularly suited for determination of quasi-bound states, for which the leading order behaviour of the fields at infinity is well understood. However for the determination of quasi-normal mode frequencies one needs to extract additional, sub-dominant behaviour of the mode functions at infinity, which makes this approach prone to numerical errors. However if the imaginary part is small compared to the real part, one can determine the quasi-normal mode frequencies to sufficient accuracy. In practice one integrates these differential equation to some high value of radial distance and the result must be impermeable to any shift in this distance. Also one can supplement one of these methods by checking whether for a given Kaluza-Klein mode mass and angular momentum one obtains the same quasi-normal mode frequency from the other. We have explicitly checked that this is indeed the case, the values obtained from the continued fraction method is in good agreement with those obtained from the direct integration scheme as well.
This depicts the internal consistency of our model in a straightforward manner.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.65]{GR_Prediction_Actual.pdf}~~
\includegraphics[scale=0.65]{GR_Prediction_Log.pdf}
\caption{Time evolution of the master mode function $u_{n,l}(t)$ associated with axial gravitational perturbation for two different values of angular momentum $l$ in the context of general relativity\ have been depicted. The time scale has been normalised to the mass of the central hole, i.e., $t\rightarrow t/GM$. Moreover the figure in the left illustrates the actual evolution of the mode function with time, while the right one presents the same but in a Logarithmic scale. The amplitude of the mode function corresponding to $l=3$ is slightly smaller compared to the mode function having $l=2$, as evident from the right figure. In both of them the dotted one stands for mode function with $l=2$, while the continuous one is the mode function with $l=3$. We will contrast this scenario with the respective ones in the presence of extra dimensions.}
\label{Fig_01_GR}
\end{figure*}
\section{Numerical analysis of the quasi-normal modes}\label{Num_qnm}
The principal aim of this work was to determine the time evolution of the perturbations obtained from the effective gravitational field equations on the brane. Also, to contrast the same with the time evolution of perturbation derived from bulk Einstein's equations. One can achieve this by following two possible avenues --- (a) Obtaining the quasi-normal mode frequencies and hence obtaining the time evolution and (b) Solving the Cauchy evolution problem numerically and hence arrive at the evolution of the gravitational perturbation.
In this section we will follow the first method where the time evolution of the mode function $u_{n,l}(t)$ depicting axial gravitational perturbation will be presented, using the quasi-normal mode analysis performed in \ref{qnm_analysis}. For this purpose we will use \ref{Eq_qnm_01a}, where the integral over all frequencies will now be replaced by summation over all the quasi-normal mode frequencies. Thus our strategy will be as follows, we will use the numerically computed quasi-normal mode frequencies and then sum over them in order arrive at the time evolution for the mode function $u_{n,l}(t)$. Here we would like to reiterate the fact that $n$ stands for the Kaluza-Klein modes and $l$ is the angular momentum associated with the gravitational perturbation. For example, $u_{0,2}$ corresponds to the axial gravitational perturbation associated with angular momentum $l=2$ around a general relativity\ solution, while $u_{1,3}$ is the axial gravitational perturbation associated with the lowest lying Kaluza-Klein mode and with angular momentum $l=3$. In what follows using the
numerical values of quasi-normal mode frequencies we will present time evolution of $u_{n,l}(t)$ for a few low lying Kaluza-Klein modes with different choices of angular momentum $l$. These will be contrasted with the mode functions $u_{0,l}$ associated with general relativity.
Note that this process is inherently approximate, since in principle one should add all the quasi-normal mode frequencies in order to obtain the time evolution of the perturbation, while here we will consider a few lowest lying quasi-normal modes to perform the same. Even though this is certainly an approximate description, it will nevertheless provide the overall behaviour of the gravitational perturbation with time and the key features that will distinguish the scenario presented here from that in general relativity. More refined results can be obtained using the Cauchy evolution, which we will present in the next section. This will provide another self-consistency check of our formalism and hence of the associated results.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.65]{GR_vs_KK_A_12.pdf}~~
\includegraphics[scale=0.65]{GR_vs_KK_A_13.pdf}\\
\includegraphics[scale=0.65]{GR_vs_KK_12.pdf}~~
\includegraphics[scale=0.65]{GR_vs_KK_13.pdf}\\
\caption{Time evolution of the master mode function $u_{n,l}(t)$ for two different values of angular momentum both in the context of general relativity\ ($n=0$) as well as in the \emph{brane based approach} have been presented. All the figures are associated with the lowest lying Kaluza-Klein mode mass $m_{1}=0.44$ (see \ref{Table_01}) but for two different choices of the angular momentum. For brevity we have presented both --- (a) the figures have been drawn in a Logarithmic scale (in the below panel) and (b) the figures in actual scale (in the top panel). All these figures clearly bring out the key differences between these two scenarios. See text for more discussions.}
\label{Fig_02_GR_vs_KK_AL_123}
\end{figure*}
As a first step towards the same we will present the time evolution of the axial perturbation in the context of general relativity\ alone. This will set the stage for what to come next. This has been presented in \ref{Fig_01_GR}, where we have depicted how the mode functions evolve with time in the actual scale as well as in the Logarithmic scale. The advantage of the Logarithmic scale is that, it can enhance very tiny differences, while the disadvantage being, it will make large differences to appear as a small one. The left figure in \ref{Fig_01_GR} presents the actual time evolution of the $l=2$ and $l=3$ mode functions in general relativity, i.e., $u_{0,2}(t)$ and $u_{0,3}$ respectively, while the right one presents the same in Logarithmic scale. It is clear that there is appreciable difference between the two at earlier times, which gets washed out as the modes gradually decay down. On the other hand the Logarithmic plot shows exactly the opposite nature as explained earlier.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.65]{GR_vs_KKB_A_12.pdf}~~
\includegraphics[scale=0.65]{GR_vs_KKB_A_13.pdf}\\
\includegraphics[scale=0.65]{GR_vs_KKB_12.pdf}~~
\includegraphics[scale=0.65]{GR_vs_KKB_13.pdf}\\
\caption{The above figures depict the time evolution of the master mode function $u_{n,l}(t)$ in the \emph{bulk based approach} and has been contrasted with that in general relativity\ (n=0). In this case also behaviour of the mode functions in a Logarithmic scale as well as in the actual scale have been presented. The Kaluza-Klein mode mass associated with the master variable presented here corresponds to the lowest one with $m_{1}=0.47$ with the following parameters: $d/\ell=20$; $1/\ell=6\times 10^{7}$.}
\label{Fig_04_GR_vs_KKB_AL_123}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.65]{KK_vs_KKB_A_12.pdf}~~
\includegraphics[scale=0.65]{KK_vs_KKB_A_13.pdf}\\
\includegraphics[scale=0.65]{KK_vs_KKB_12.pdf}~~
\includegraphics[scale=0.65]{KK_vs_KKB_13.pdf}
\caption{The two scenarios presented in this work, namely the perturbation of effective four dimensional Einstein's equations or the perturbation of bulk Einstein's equations, have been illustrated in this figure. Both for the identical choice of the extra dimensional parameters, i.e., $d/\ell=20$; $1/\ell=6\times 10^{7}$. It is clear that the time evolution of the mode function $u_{n,l}(t)$ differ from each other in these two distinct approaches. This is primarily due to the difference between the Kaluza-Klein mode masses in these two approaches. See text for more discussions.}
\label{Fig_05_KK_vs_KKB_AL_123}
\end{figure*}
Returning back to our main goal, we have illustrated time variation of the perturbation associated with the lowest lying Kaluza-Klein mode having mass $m_{1}=0.44$ and have contrasted the same with general relativity\ in \ref{Fig_02_GR_vs_KK_AL_123}. The figures on the left depict time variation of the perturbation for $l=2$ in both actual and Logarithmic scale, while those on the right are for $l=3$. The main difference emerging from \ref{Fig_02_GR_vs_KK_AL_123} is that the damping time scale of the massive modes are much greater compared to those in general relativity. The same is true for the Logarithmic plots as well, where the fact that modes in general relativity\ are heavily damped in comparison to the massive modes is much pronounced. The features of the massive modes remain identical as one considers the second lowest Kaluza-Klein mode having mass $m_{2}=0.83$ as well. Here also the slower decay of the massive modes with time is the key distinguishing feature between general relativity\ and the higher dimensional model discussed here.
So far we have been discussing the time evolution of the gravitational perturbation starting from the effective gravitational field equations on the brane. At this stage let us try to understand the corresponding situation when the gravitational perturbation originating from the bulk Einstein's equations is being considered. As emphasised earlier this will be similar to the brane based approach but will have an associated Kaluza-Klein mass mode, which will be different. For example, as evident from \ref{Table_01} and \ref{Table_02} for the same choice of bulk parameters, i.e., $d/\ell=20$; $1/\ell=6\times 10^{7}$, the Kaluza-Klein mass spectrum will be different in the two scenarios. Thus in \ref{Fig_04_GR_vs_KKB_AL_123} we have presented the time evolution of the gravitational perturbation derived from the bulk based approach. Here also we observe the same key features, e.g., very slow decay of the perturbation in contrast to that in general relativity. Thus if the ring down phase of any black hole merger is being probed to intermediate
times, where the evolution of gravitational perturbation is still dominated by the quasi-normal modes, any departure from the general relativity\ prediction can possibly signal the existence of extra spacetime dimensions. Following the general trend, in \ref{Fig_04_GR_vs_KKB_AL_123} as well we have presented the time evolution in actual as well as in Logarithmic scale for two possible choices of angular momentum $l=2$ and $l=3$ respectively.
This enables one to compare the bulk and the brane based approach given the same bulk parameters. The resulting discord should be attributed to the difference between the masses of the respective Kaluza-Klein modes. As evident from \ref{Fig_05_KK_vs_KKB_AL_123} this difference is really very small unlike the situation with general relativity. Moreover since the masses of the Kaluza-Klein modes are higher in the bulk based approach they would decay slower. This can be clearly seen from both the Logarithmic plots in \ref{Fig_05_KK_vs_KKB_AL_123}, where the perturbation in the bulk based approach becomes larger than the brane based one at late times. Same features appear for both the angular momentums as well, however the difference is much smaller in higher angular momentum compared to the lower one.
All these features can also be seen for the second lowest Kaluza-Klein mode mass, $m_{2}=0.87$, in the bulk based approach.The time evolution of the corresponding gravitational perturbation in both actual and Logarithmic scale for two choices of the angular momentum shows very similar features when compared with the lowest lying Kaluza-Klein mass mode. As expected, the massive modes decay much slowly in comparison with general relativity. Further from the comparison of brane and bulk based approach for the second lowest lying Kaluza-Klein mode, one may infer that the difference only becomes sensible after a large time has elapsed and hence if the ring down phase can be probed minutely at very late times one may infer the preference of the bulk based approach over the brane based one or vice versa. However, the situation is not so simple and another subtle effect comes into play at late times, which corresponds to the wave tail. All the quasi-normal modes are inherently exponentially suppressed and hence at very late times their effects are
negligibly small. In this situation the wave tails enter the picture and in most of the cases the late time behaviour is essentially governed by the wave tail decaying only as a power law (this is identical to the ``bulk based approach" as well, see \cite{Seahra:2009gw}). Since the scaling of the power law is mostly universal, independent of the nature of fields and mass of the fields under consideration, both the bulk and brane based approach will decay by an identical power law behaviour. This will make the detection of these two different approaches by late time measurements extremely difficult.
\section{Consistency of the approach: Comparison with Cauchy evolution}\label{consistency_Cauchy}
The previous section illustrates the methods to determine the quasi-normal mode frequencies, using which we have obtained the time evolution of the perturbation mode $u_{n,l}$. This is so because, in the limit of $m_{n}\rightarrow 0$ (i.e., general relativity\ limit) this mode represents the axial gravitational perturbation, while the other essentially becomes a gauge degree of freedom. Hence we can compare the time evolution of $u_{n,l}$ with the respective one in general relativity\ and see the harmony as well as possible discord among the two. We have already performed the same in the previous section. However in principle one expects the above approach to match with the Cauchy evolution of the perturbation equations presented in \ref{Eq_qnm_03a} and \ref{Eq_qnm_03b} respectively in \ref{App_B}. This is what we will explore in this section.
For this purpose, we closely follow the analysis put forward in \cite{Konoplya:2011qq} but modifying it wherever necessary. Referring back to \ref{GW_Eq23a} and \ref{GW_Eq23b} as the key differential equations for the master variables, one can write them in a compact manner as
\begin{equation}\label{Eq_Cauchy_01}
\mathcal{D}\mathbf{\Psi}+\mathbf{V}(r)\mathbf{\Psi}=0~.
\end{equation}
Here, $\mathbf{\Psi}$ is a two dimensional column matrix constructed out of $u_{n,l}$ and $v_{n,l}$ respectively. Rather than working with the normal $(t,r)$ coordinates it is instructive to transform to the light cone coordinates. The transformation into the light-cone coordinates can be achieved by introduction of the null coordinates as: $u=t-r_{*}$, $v=t+r_{*}$. Use of these null coordinates modifies \ref{Eq_Cauchy_01} to
\begin{equation}
4\partial_u\partial_v\mathbf{\Psi}+\mathbf{V}(u,v)\mathbf{\Psi}=0~,
\end{equation}
where
\begin{equation}
\mathbf{\Psi}=
\begin{pmatrix} u_{n,l} \\ v_{n,l} \end{pmatrix}~, \qquad
\mathbf{V}=
\begin{pmatrix} V_{11} & V_{12}\\ V_{21} & V_{22} \end{pmatrix}~.
\end{equation}
Here all the matrix coefficients of $\mathbf{V}$ are dependent on the black hole solution on the brane and the Kaluza-Klein mode mass $m_{n}$.
For clarity, we have suppressed the functional dependence of the potential matrix $\mathbf{V}$ for the time being. To proceed further we need to introduce a notion of time evolution operator. For this purpose, we note that for an arbitrary function of time $f(t)$, the function given by $e^{h\partial_t}$ is the time evolution operator, in the sense that $e^{h\partial_t}f(t)=f(t+h)$. Thus in order to obtain the time evolution of the mode functions $\mathbf{\Psi}$, we apply the time evolution operator on $\Psi$. This yields,
\begin{equation}\label{Eq_Cauchy_02}
\mathbf{\Psi}(t+h)=e^{h\partial_t}\mathbf{\Psi}=e^{h\partial_u+h\partial_v}\mathbf{\Psi}~.
\end{equation}
\ref{Eq_Cauchy_02} can be written in a nice manner by expanding the right hand side, resulting into,
\begin{widetext}
\begin{align}\label{Eq_Cauchy_03}
&\mathbf{\Psi}(u+h,v+h)=\sum_{j=0}\frac{1}{j!}(h\partial_u)^{j}\sum_{k=0}\frac{1}{k!}(h\partial_v)^{k}\Psi(u,v)
\nonumber
\\
&=\left[e^{h\partial_u} + e^{h\partial_v} -1 +\frac{1}{2} h^2\partial_u\partial_v\left(1+\frac{h\partial_u}{2!} +\frac{{h\partial_u}^2}{3!}+\cdots\right) \left(1+\frac{h\partial_v}{2!} +\frac{{h\partial_v}^2}{3!}+\cdots \right)\right]\mathbf{\Psi}(u,v)
\nonumber
\\
&=\Bigg[e^{h\partial_u} + e^{h\partial_v}-1+\frac{1}{2}h^2\partial_u\partial_v
\Bigg\{\left(e^{h\partial_u}-{(h\partial_u)}^2\left(\frac{1}{2!}-\frac{2}{3!}\right)-\cdots\right)
+\partial _{u}\rightarrow \partial _{v}\Bigg\}\Bigg]\mathbf{\Psi}(u,v)
\nonumber
\\
&=\left[e^{h\partial_u} + e^{h\partial_v}-1+\frac{1}{2}h^2\partial_u\partial_v \left\{\left(e^{h\partial_u}+\mathcal{O}(h^2)\right) +\left(e^{h\partial_v}+\mathcal{O}(h^2)\right)\right\}\right]\mathbf{\Psi}(u,v)~.
\end{align}
\end{widetext}
The last expression of \ref{Eq_Cauchy_03} can be expanded immediately and hence finally we have,
\begin{align}
\mathbf{\Psi}(u+h,v+h)=&\mathbf{\Psi}(u+h,v)+\mathbf{\Psi}(u,v+h)-\mathbf{\Psi}(u,v)
\nonumber
\\
-\frac{h^2}{8}&\Big\{\mathbf{V}(u+h,v)\mathbf{\Psi}(u+h,v)
\nonumber
\\
&+\mathbf{V}(u,v+h)\mathbf{\Psi}(u,v+h)\Big\}~.
\end{align}
This can be thought of as an evolution equation in the light-cone coordinates $u$ and $v$. The interesting aspect of this formalism is that once initial data is specified in the $u,v$ coordinates, we need no additional boundary conditions, which is unlike the physical coordinates $t,r$ (or, for that matter $t,r_{*}$). We evolve the system with Gaussian initial data in $u$ and constant data in $v$. \ref{Fig_QNM_Cauchy} illustrates the numerical evolution of $\Psi$ as a function of time for different choices of the angular momentums and Kaluza-Klein mode masses obtained by numerically integrating the above evolution equation in light cone coordinates. Interestingly and as expected, it illustrates all the basic properties that we have already observed from a quasi-normal mode analysis. For example, in all the cases illustrated in \ref{Fig_QNM_Cauchy} it is clear that at intermediate times (i.e., when the spectrum of quasi-normal modes dominate the evolution of gravitational perturbation) the mode functions due to massive Kaluza-Klein modes
will dominate over those in general relativity. This is again due to the fact that the massive modes suffer much less damping compared to the respective ones in general relativity. The translation of the same in the quasi-normal mode language corresponds to the imaginary part of the quasi-normal mode frequency to be smaller for massive Kaluza-Klein modes compared to the modes in general relativity. Further the fact that as the mass of the Kaluza-Klein mode increases it experiences less and less damping is also borne out by both Cauchy evolution (see \ref{Fig_QNM_Cauchy}) and the quasi-normal mode analysis. Of course, there are minute differences present between both these methods, which have their origin in the initial conditions and the fact that the Cauchy evolution is more accurate compared to the quasi-normal mode analysis. All in all, the quasi-normal mode analysis and the Cauchy evolution of initial data provides a complete and consistent description of the time evolution of gravitational perturbation in presence of extra spatial dimensions.
Besides being consistent with the quasi-normal mode analysis, Cauchy evolution has more additional features to offer. The most important such feature is the presence of late time power law tail. Since at the intermediate stages, the contributions from quasi-normal mode dominates, the behaviour of the mode function $u_{n,l}(t)$ as presented in \ref{Fig_QNM_Cauchy} resembles those in \ref{Num_qnm}. However, if one can perform the Cauchy analysis for a sufficiently long time, gradually the contributions from quasi-normal modes become smaller compared to the late time tail. Thus the Cauchy evolution of the initial data for a longer time must result in the desired power law tail and will serve as another consistency check of our approach. Following this we have presented a long time Cauchy evolution of the perturbation equation in \ref{Cauchy_Tail}, which distinctly depicts the late time wave tail. As evident from \ref{Cauchy_Tail}, the mode function is initially dominated by the quasi-normal modes and hence decays
linearly in the Logarithmic scale. However in the late stages of Cauchy evolution the power law takes over and dominates the quasi-normal modes, thus presenting an almost constant-in-time behaviour of the same. Hence, the numerical analysis of the Cauchy evolution of the perturbation equation is completely consistent with theoretical expectation, providing one more consistency check of our formalism.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.45]{QNM_Cauchy2p44.pdf}~~
\includegraphics[scale=0.45]{QNM_Cauchyl2p44.pdf}\\
\includegraphics[scale=0.45]{QNM_Cauchy2p83.pdf}~~
\includegraphics[scale=0.45]{QNM_Cauchyl2p83.pdf}\\
\caption{The Cauchy evolution of the master variable $u_{n,l}(t)$ has been plotted for two different choices of the Kaluza-Klein mode masses for a given angular momentum. The figures in the top panel depicts the evolution of the master variable for $l=2$ and $m_{1}=0.44$, the lowest lying Kaluza-Klein mode with $d/\ell=20$ and $1/\ell=6\times 10^{7}$. The figure on the left is the actual variation of the master variable with time, while that on the right presents the same variation but in a Logarithmic scale. While the figures in the bottom panel illustrates the same, however for $l=2$ and Kaluza-Klein mode mass $m_{2}=0.83$. It is clear that as the mass increases the master variable becomes less and less damped in comparison with general relativity. Further we clearly observe that the overall features present in the Cauchy evolution of the master variable are identical to those obtained by the quasi-normal mode analysis, illustrating the internal consistency of both the methods adapted here.}
\label{Fig_QNM_Cauchy}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.45]{Lt_Cauchyl3p44.pdf}~~
\includegraphics[scale=0.45]{Lt_Cauchyl4p44.pdf}\\
\caption{The Cauchy evolution of the master variable $u_{n,l}(t)$ associated with the brane based approach has been plotted for two different choices of the angular momentum given a Kaluza-Klein mode mass to illustrate the late time behaviour. The figure in the left depicts the evolution of the master variable for $l=3$ and $m_{1}=0.44$, the lowest lying Kaluza-Klein mode with $d/\ell=20$ and $1/\ell=6\times 10^{7}$. The figure on the right is for $l=4$ and identical Kaluza-Klein mode mass. It is clear that as time progresses the power law tail dominates over the exponential damping due to quasi-normal modes. This once again illustrates the consistency of Cauchy evolution with the theoretical methods adapted here.}
\label{Cauchy_Tail}
\end{figure*}
\section{Discussion and concluding remarks}\label{conclusion}
\begin{table*}
\begin{center}
\caption{Frequencies (in Hz) of oscillation for the quasi-normal modes emanating from black holes having different masses have been depicted. Numerical estimates for the frequencies have been presented for general relativity\ as well as for the two lowest lying Kaluza-Klein modes with masses $m_{1}=0.43$ and $m_{2}=0.83$ respectively. It is also clear that the frequency of the modes increases with an increase in the $l$ value. It is clear that as the mass of the black hole increases the frequency decreases. Thus more massive the black hole is it is more problematic to detect in aLIGO. While the Kaluza-Klein modes have better chance of originating if the mass of the black hole increases, this leads to lowering of the frequency and hence have less chance of getting detected in aLIGO.}
\label{Table_07}
\begin{tabular}{p{2cm}p{3cm}p{3.5cm}p{3.5cm}}
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
& & Frequencies for $l=2$ & \\
\hline\noalign{\smallskip}
\hline\noalign{\smallskip}
$(M/M_{\odot})$ & General Relativity & KK Mode ($m_{1}=0.43$) & KK Mode ($m_{2}=0.82$) \\
\hline \noalign{\smallskip}
\hline \noalign{\smallskip}
$1$ & 24140 & 14930 & 12441 \\
$10$ & 2414 & 1493 & 1244.1 \\
$10^{2}$ & 241.4 & 149.3 & 124.4 \\
$10^{3}$ & 24.1 & 14.9 & 12.4 \\
$10^{4}$ & 2.4 & 1.5 & 1.2 \\
$10^{5}$ & 0.2 & 0.1 & 0.1 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
& & Frequencies for $l=3$ & \\
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
$1$ & 38779 & 21328 & 27856 \\
$10$ & 3877.9 & 2132.8 & 2785.6 \\
$10^2$ & 387.7 & 213.3 & 278.6 \\
$10^3$ & 38.7 & 21.3 & 27.8 \\
$10^4$ & 3.9 & 2.1 & 2.8 \\
$10^5$ & 0.3 & 0.2 & 0.3 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
\hline \noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}
In this work we set out to achieve three goals in a single framework --- (a) Effect of extra spatial dimensions on the gravitational perturbation and whether one can provide some possible observational signatures of the same in the ring down phase of black hole merger; (b) How the two possible methods to determine the gravitational perturbation on the brane, namely by either perturbing the bulk gravitational field equations or perturbing the \emph{effective} gravitational field equations on the brane, differ as far as the behaviour of the gravitational wave solution is concerned; (c) Whether the analysis using quasi-normal modes is consistent with the fully numerical Cauchy evolution of the initial data. We believe to have addressed all of them in a satisfactory manner in this work which we summarise below.
We have explicitly demonstrated that the existence of extra spatial dimensions indeed modifies the gravitational perturbation equation by essentially introducing a tower of massive perturbation modes in addition to the standard massless one. Thus the presence of massive gravitational perturbation modes is a definitive signature of the existence of higher dimensions. To see the consequences of the above we have discussed the behaviour of quasi-normal modes in this context. In particular, we have shown that for the massive modes the imaginary part of the quasi-normal mode frequencies are much small compared to those in general relativity. This has resulted in the time evolution of the massive gravitational perturbations to exhibit a weak decay rate in comparison to the massless modes as in general relativity. The above phenomenon opens up the observational window to probe the possible existence of higher dimensions using gravitational wave observation. If the ring down phase during the merger of two black holes is loud enough to be accurately measured for a sufficient amount of time (unlike the aLIGO-VIRGO observations to-date) it may be possible to detect any departure from the general relativity\ prediction and thus may lead to concrete observational signature for the existence of higher dimensions or may provide stringent constraints on the associated parameters. This will become feasible as the sensitivity of the aLIGO detectors are further improved or the space based gravitational wave detector LISA becomes operational. We will address the detailed observational aspects of this particular signature of extra dimension in light of the recent detection of gravitational waves at aLIGO in a future work.
The evolution equation for the gravitational perturbation obtained by perturbing the bulk field equations has already been derived in \cite{Seahra:2004fg}, while in this work we have derived the evolution equation by perturbing the \emph{effective} gravitational field equations on the brane hypersurface. From the structure of the equation itself, difference between these two approaches should be evident. In both the bulk based and the brane based approach the four dimensional perturbation equation looks identical with one crucial difference, namely, the masses associated with both the approaches are different. This is because the differential equation satisfied by the extra dimensional part are different in these two scenarios. This in turn leads to difference in the quasi-normal mode frequencies as evident from \ref{Fig_QNM}, the imaginary parts of the quasi-normal mode frequencies are smaller than the bulk based approach in comparison to the brane based one. Since the difference is small there is possibly no way in foreseeable future to observationally distinguish these two effects (see e.g., \ref{Fig_05_KK_vs_KKB_AL_123}) however theoretically there does exist a difference between these approaches. Naively speaking, this is due to the fact that a solution of the \emph{effective} gravitational field equation on the brane may not have any higher dimensional embedding.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.80]{Frequency.pdf}
\caption{This figure depicts how the oscillation frequency of the gravitational wave in the ring down phase changes as the mass of the black hole increases. For conveniences frequencies are plotted in Hz, while black hole mass is presented in solar units, but both on a Logarithmic scale. The oscillation frequencies have been plotted in the brane based scenarios for two lowest lying Kaluza-Klein mode masses with $l=2$. The same has been contrasted with the corresponding curve in general relativity. It is clear that for $M\sim 10^{3}M_{\odot}$ the frequencies associated with general relativity\ are well within the aLIGO frequency band, however for the massive Kaluza-Klein modes they are outside. Since these massive modes have better chance of getting detected in the high mass regime it is most likely that they may become observable once LISA is operational. See text for more discussions.}
\label{Fig_Frequency}
\end{figure*}
Finally the time evolution of the gravitational perturbation can be obtained by either performing a quasi-normal mode analysis or by performing a fully numerical Cauchy evolution. We have performed both in this work and they are found to match very well with each other. This is expected as well as necessary for internal consistency of any gravitational perturbation computation. In particular from the quasi-normal mode analysis we have learned that the massive modes decay much slowly in contrast with the massless general relativity\ modes (see e.g., \ref{Fig_02_GR_vs_KK_AL_123}), which is also confirmed by the Cauchy evolution (see e.g., \ref{Fig_QNM_Cauchy}). Thus keeping aside minute details overall behaviour of the time evolution of gravitational perturbation is identical whether one performs a quasi-normal mode analysis or complete Cauchy evolution.
Having described the consistency of the time evolution obtained by using quasi-normal modes as well as a fully numerical Cauchy evolution of the perturbation equations, let us comment on possible detectability of the scenario presented above. For that purpose it is important to know the frequencies associated with the gravitational perturbation modes. The corresponding frequencies can broadly be divided into two classes, those originating from the real part of the quasi-normal modes of the gravitational perturbation and the universal one present in the very late time region \cite{Seahra:2004fg,Rosa:2011my} originating from the power law tail. As far as the possible detectability of the scenario presented here in aLIGO-like detectors using the real parts of the quasi-normal modes is concerned, one can safely say that most likely it is not a feasible option. This is mainly due to two reasons. First, the frequencies associated with these Kaluza-Klein modes are smaller compared to general relativity\ (see \ref{Table_07} as well as
\ref{Fig_Frequency}). Furthermore these Kaluza-Klein modes are supposed to be excited in the strong gravity regime, i.e., when mass of the black hole is large. On the other hand, as the mass of the black hole increases the frequency also decreases. This adds to the issue of detectability of these Kaluza-Klein modes. As evident from \ref{Table_07}, for black hole mass $\sim$ $10^{3}M_{\odot}$ the frequency of a mode in general relativity\ is within the frequency band of aLIGO detectors. However the same is not true for the Kaluza-Klein modes where the frequencies are smaller and hence possibly outside the operational band of the aLIGO detectors. Nonetheless, all these frequencies pertaining to higher mass black holes are very well within the projected band of LISA and hence possibly detectable in near future (see \ref{Fig_Frequency}). The second point corresponds to the Signal-to-Noise ratio, since in order to detect the signal it is necessary to generate oscillations with a high Signal-to-Noise ratio. For this purpose as well we need collisions among heavier black holes (i.e., stronger gravity regime), so that higher order massive KK modes are excited. The frequency of these modes would correspondingly be lower and might get pushed out of the aLIGO frequency band but possibly be well within the LISA band.
\begin{figure*}[t!]
\centering
\includegraphics[scale=0.80]{Overall_Behaviour.pdf}
\caption{This figure depicts the allowed region in the $(d/\ell,M/\ell)$ plane along with the frequency associated with the very late time behaviour of the perturbation modes for black holes. The accessible region in the plot is obtained by imposing a few restrictions on the $d/\ell$ as well as $M/\ell$ values. These correspond to --- (a) the Gregory-Laflamme instability, (b) the scalar-tensor limit and finally (c) the restriction on $d/\ell$ necessary to solve the hierarchy problem. The frequencies of the late time behaviour of the perturbation modes have been depicted in the accessible region by means of a colour coding. The labels on the right hand colour bar corresponds to a logarithm of frequencies (in Hz) to the base 10. The green dotted and dashed-dotted lines correspond to frequencies of 10 Hz and 10 KHz respectively, the extremities of aLIGO band. Similarly, the blue broken lines correspond to the extremities of the LISA band. The solid green and blue lines correspond to a configuration with $\ell=1\mathrm{\mu m}$ and $M$ of $50 M_{\odot}$ and $10^5 M_{\odot}$ respectively. For a given $d/\ell$ ratio, as $1/\ell$ increases the frequency also increases, while for a given $1/\ell$, as $d/\ell$ increases the frequency decreases. Hence the aLIGO band is completely within the accessible region while the LISA band has substantial overlap with the unstable regions. See text for more discussions.}
\label{Fig_Overall}
\end{figure*}
While, the late time behaviour of these massive gravitational perturbation modes corresponds to a universal frequency, associated with the power-law tail of the wave mode and proportional to the Kaluza-Klein mode mass \cite{Seahra:2004fg,Rosa:2011my}. Thus the frequency gets determined in terms of the $d/\ell$ ratio and the curvature length scale $\ell$ . Using the expression for $n$th Kaluza-Klein mode mass, $m_{n}=z_{n}(1/\ell)\exp(d/\ell)$, where $z_{n}$ are the zeros of the Bessel function $J_{\sqrt{13}/2}(x)$ one immediately arrives at the desired expression for the late time frequency as a function of $d/\ell$ and $1/\ell$ respectively. Given this universal late time frequency, one immediately observes that as $d/\ell$ and $1/\ell$ increase, the frequency also increases and hence for a given $d/\ell$ ratio, the frequency will be in aLIGO band for a larger value of $1/\ell$, but will fall within the LISA band for smaller values of $1/\ell$ (as evident from \ref{Fig_Overall} as well). This introduces additional
complications in the detectability of these late time modes modulo the Gregory-Laflamme instability, which sets in for small $1/\ell$ values given a $d/\ell$ (see \ref{Fig_Overall}). At this point, it is interesting to note that given a black hole mass and a particular value of $\ell$, the frequency bands of aLIGO and LISA set natural observational bounds on $d/\ell$. In \ref{Fig_Overall}, we have considered two such scenarios, where the black hole masses are $50 M_{\odot}$ and $10^5 M_{\odot}$, while $\ell =1\mathrm{\mu m}$. The scenarios are depicted by the thick green and blue lines respectively. As is clearly evident from the plot the modes from the $50 M_{\odot}$ black hole can probe the range $17.0<d/\ell<23.9$. The $10^5 M_{\odot}$ black hole has only a limited probe for $d/\ell$, because of the unstable configurations. The upper limit on $d/\ell$ in this case is therefore set by the boundary of the unstable region, so that $33.8<d/\ell<34.8$. Thus as long as the universal frequency spectrum is
concerned, the late time behaviour of the Kaluza-Klein modes has a better chance of detection in aLIGO rather than in LISA as clearly depicted in \ref{Fig_Overall}. The feasibility of the above detection, however is being determined by the Signal-to-Noise ratio, which will be much less for aLIGO while will be favourable for LISA. Hence even in this case there will be a tussle between the accessible regions in the $(d/\ell,1/\ell)$ space and the Signal-to-Noise ratio making the detectability difficult for aLIGO detectors for the late time behaviour of the massive quasi-normal modes as well.
The above exercise also opens up a few future avenues to explore. We have discussed the effect of higher dimensions on the quasi-normal modes in this work, however it is possible to address the nature of quasi-bound states and in particular how the presence of extra dimensions affect them. This may provide another observational test bed for detection of higher spatial dimensions. Besides whether one can obtain similar results for the quasi-bound states from Cauchy evolution as well remains to be verified. Also a through analysis of this allowed region in light of the recent detection of gravitational waves in aLIGO can lead to possible constraints on the extra-dimensional parameter space. Moreover the effect of higher dimensions on neutron star equation of state parameter, tidal love numbers associated with a brane black hole can lead to exciting results which we are currently pursuing and will report elsewhere.
\section*{Acknowledgement}
Research of S.C. is supported by the SERB-NPDF grant (PDF/2016/001589) from SERB, Government of India and the research of S.S.G is partially supported by the SERB-Extra Mural Research grant (EMR/2017/001372), Government of India. This work was supported in part by NSF Grant PHY-1506497 and the Navajbai Ratan Tata Trust. The authors gratefully acknowledge discussions with David Hilditch, Emanuele Berti, Paolo Pani and Sanjeev Seahra. We also thank Shasvath Kapadia for carefully reading the manuscript and making useful suggestions. This work has been assigned the LIGO document number: LIGO-P1700295.
|
2,877,628,090,165 | arxiv | \section*{Acknowledgment}
All authors would like to thank Ms. Jing Lin, Mr. Wu Shi, and Zuwen Zhu for the design and implementation of the Pegasus-Mini.
\bibliographystyle{IEEEtran}
\section{Introduction}
\IEEEPARstart{Q}{uadruped} locomotion has been a vibrant research field within recent years. Although blindly legged locomtion is extensively studied \cite{rudin2021cat, kim2020dynamic, 8324642, luo2017advanced}, autonomous navigation beyond blind robust locomotion in the complicated outdoor environment has drawn more interest. To accomplish the outdoor navigation, it is required to actively perceive the environment and efficiently plan the locomotion path accordingly. Currently, autonomous navigation for quadruped locomotion is mainly focused on medium-scale or large-scale real robot platforms, which are able to offer relatively larger load capacity and space for high-performance computing equipment which is critical for the processing and computation of big data from the perception sensors such as Lidar, camera, etc.
One of the paradigm is the ANYmal series, a type of large-scale quadruped robot, which weighs around 30 $kg$. Its length, height and width are around 1 $m$, 0.8 $m$ and 0.5 $m$ respectively. The earliest ANYmal is equipped with a rotating Hokuyo UTM-30lx laser sensor for navigation and three intel NUC PC for computing \cite{7758092}. A traversability map is built for the path planning. RRT algorithm is adopted to optimize the path length and safety \cite{7759199}. A method is presented on ANYmal, which rebuilds the surrounding environment in form of an elevation map with a depth camera \cite{8392399}. This approach updates the probabilistic map step by step. In another implementation, ANYmal precisely quantifies the environment traversability with the consideration of various geometry factors like surface roughness, inclination, and height difference \cite{7759199}. These methods show impressive performance and high precision which brings convenience for the foothold selection and path planning. However, efficiency is relatively low in the implementation due to the time-consuming 3D reconstruction process. The navigation research on ANYmal mainly focuses on the traditional approaches, such as geometry feature, point cloud, elevation map, or occupancy grid map.
\begin{figure}[t]
\centering
\includegraphics[width = 3.4 in]{figures/concept.png}
\caption{Locomotion of a small\textcolor{blue}{-}scale quadruped robot, Pegasus-Mini, in a garden with the assistance of vision-based navigation.}
\label{fig:concept}
\end{figure}
Another well-known quadruped robot is HyQ, which weighs 85 $kg$ and measures approximate 1.0$\times$0.5$\times$0.98 $m$ (length$\times$width$\times$height) \cite{9133154}. A depth camera (Asus Xtion), a MultiSense SL sensor (2.6 $kg$), and two intel i5 processors are mounted on HyQ to realize a coupled framework consisting of motion planning, whole-body control, and terrain model. A real-time and dynamic foothold adaptation strategy based on visual feedback is also presented on HyQ \cite{8642374}. Similar work also includes \cite{8961842}, which improve the energy efficiency based on the perception of the environment. These quadruped robots are large-scale with higher load capacity and therefore are able to integrate a variety of navigation sensors and high-performance computing platforms. Real-time path planning for navigation based on images is not fully explored yet.
Among small-scale quadruped robots, MIT Mini-cheetah, as one of the canonical platforms, presents the application in obstacle avoidance during the navigation \cite{9196777}. MIT Mini-cheetah is 0.3 $m$ tall and 9 $kg$. The small body size limits the types and numbers of sensors and hence computing performance is discounted. Currently, only two cameras and a depth camera are mounted on MIT Mini-cheetah.
Compared with the existing research in quadruped navigation, vision-based navigation, especially using semantic segmentation, for real-time path planning on a small-scale quadruped robot platform is yet explored.
The contributions in this letter lie in the following twofold:
1) Implementation of a vision-based navigation using semantic segmentation on a lightweight computing architecture deployed on a small-scale quadruped robot.
2) Trajectory compensation method is proposed to enhance the success rate of the vision-based navigation for quadruped locomotion.
The rest of this letter is organized as follows. Related work is reviewed in Section II. The vision-based navigation method for quadruped locomotion is summarized in Section III. Semantic segmentation of garden scene is presented in Section IV. Section V proposes a trajectory compensation method. Section VI demonstrates the experiment results. This line of research is concluded in Section VII.
\section{Related Work}
Navigation based on semantic segmentation for quadruped locomotion is yet explored. Semantic segmentation methods are mostly based on deep learning techniques and are extensively studied in autonomous driving. The dominant deep learning models include ERFNet\cite{8063438}, FCN \cite{7298965}, SegNet \cite{7803544}, etc. To improve the performance of the semantic segmentation models, The DeepLab series applies atrous convolution to magnify the receptive field without increment of weight amount, to extract large-scale context between objects \cite{7913730, 2018Encoder}. The efficiency of These models is relatively low due to the high computational cost. However, ERFNet achieves promising performance on mobile hardware. A self-supervised learning method is applied to train the semantic segmentation model with the generation of the traversable and untraversable labels with the aid of LIDAR \cite{7989025}. With the sensor fusion method, the 2D perception results is able to be projected into 3D space for obtaining the semantic map. The perception in the autonomous driving field highly relies on the big data of the urban environment. However, there lacks of open-source data for the unstructured environment, which incurs challenges for the legged robotic navigation. Moreover, the image-based semantic perception methods only provide two-dimensional results, which implies that the 3D information obtained from LIDARs or RADARs is needed to build the 3D semantic mapping. These approaches suffer from the limited payload capacity of the small target platform, such as the small-scale quadruped robot.
\begin{figure}[t]
\centering
\setlength{\abovecaptionskip}{0.1 cm}
\includegraphics[width = 2.6 in]{figures/robot.png}
\caption{The small-scale quadruped robot Pegasus-Mini. The body clearance is 26 $cm$. The body length is 40 $cm$. Intel RealSense D435 camera is mounted in the front part of Pegasus-Mini to collect images for vision-based navigation.}
\label{fig:robot}
\end{figure}
A metric of terrain negotiation difficulty is defined and a self-supervised learning-based method is developed to predict terrain properties for ANYmal locomotion \cite{8627373}. The success is impressive, however, the force-torque sensor must be equipped on the feet of the robot, which is not satisfied by many low-cost and small-scale quadruped robots. Besides, new data must be established and more workload are necessitated for the data pre-processing. Another study explores to solve the traversability estimation issue from a new perspective \cite{9247267}. In this inspiring research, a self-supervised learning method is adopted to record the acoustic signals when the robot passes various kinds of terrains and the semantic segmentation network is trained. However, this network is yet deployed on quadruped robot platforms.
In this study, a small outdoor data set is collected and labelled. The semantic feature of the terrains via only images is utilized. A convolutional neural network (CNN) of semantic segmentation is trained with the open-source dataset. To deal with the domain shift problem, a new small outdoor dataset is collected and labelled with few labors. To enhance the robustness of the path planning method via semantic segmentation method, a simple and robust algorithm is devised to compensate the robot’s pose commanded by vision-based navigation. A quadruped robot, Pegasus-Mini is built, which is able to trot at a high frequency and exhibits adaptability and flexibility to the outdoor environment. A vision-based navigation method based on semantic segmentation, together with trajectory compensation, is demonstrated on Pegasus-Mini.
\section{System Overview}
\subsection{Pegasus-Mini Quadruped Robot}
The platform for the vision-based navigation test in this study is Pegasus-Mini, a small-scale quadruped robot, as shown in Fig. \ref{fig:robot}. Pegasus-Mini is electrically actuated with 12 degrees of freedom. It weighs 12 $kg$ and is 0.32 $m$ tall. A D435 camera is equipped in the front of the body. The lengths of the upper and lower leg are 0.206 $m$ and 0.228 $m$ respectively. Pegasus-Mini is able to run in a trotting gait at the height of 0.26 $m$, based on leg workspace and debugging experience.
Navigation algorithm runs on Nvidia Xavier NX, which is a low-power computer with a 6-core Carmal ARM V8.2 architecture CPU and 384 CUDA core, 48 Tensor core, 8 GB RAM. Ubuntu 18.04 with the ROS-Melodic works as the operating system. Nvidia Xavier NX is deployed for running the CNN model to segment the trail in the garden.
Locomotion is executed on an Intel UP board low-power single-board computer with a quad-core Intel Atom CPU, 4 GB RAM. Linux with RT patch works as the operating system. UP board is used to run the low-level controller, including MPC, WBC, and state estimator.
\subsection{Framework Overview}
In this section, a vision-based navigation method for a small-scale quadruped robot Pegasus-Mini is proposed. Terrain classification through training a neural network based on the CNN framework is adopted. The training algorithm enables the deployment of a light-weight visual perception system on the small scale quadruped robot Pegasus-Mini.
The developed framework for learning-based garden navigation is illustrated in Fig. \ref{fig:framework}. An off-the-shelf low cost camera, Intel RealSense D435 is adopted on Pegasus-Mini to collect the RGB images. Two datasets are utilized for training the CNN neural network. One dataset is collected from an open-sourced dataset and the other dataset is generated from the garden.
The CNN model takes the RGB images sensed by the camera D435 mounted in the front part of Pegasus-Mini's body and outputs the segmentation of the traverse path. Yaw motion and $y$ deviation estimation are calculated based on the traverse path's geometrical information. Under the condition that image segmentation fails, the trajectory planning compensation method is supplemented to enhance the success rate of traversability in the garden environment.
The path planner outputs the desired yaw angle velocity and linear velocity in $y$ direction. The desired velocity is fed into MPC and WBC to calculated the joint controller. The CNN model runs at 4 $Hz$. The path planner runs at 4 $Hz$. The MPC and WBC run at 0.5 $kHz$. The joint controller runs at 40 $kHz$.
Fig. \ref{fig:computing_architecture} presents the computing architecture deployed on Pegasus-Mini. Nvidia Xavier NX is used to run the CNN model to segment the trail in the garden. UP board is used to run the low-level controller, including MPC, WBC, and state estimator as shown in Fig. \ref{fig:framework}. Xavier NX communicates with the UP board through ethernet. IMU communicates with the UP board through USB 2.0 to feedback the posture information. Desired joint position, velocity, and torques are calculated in UP board and sent to the robot joint controller through the SPI interface. The operator is also able to directly send command using a remote control receiver through the UART interface.
\begin{figure}[t]
\centering
\setlength{\abovecaptionskip}{-0.1 cm}
\includegraphics[width = 3.45 in]{figures/framework.png}
\caption{Vision-based navigation framework for small scale quadruped robot Pegasus-Mini.}
\label{fig:framework}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 3.4 in]{figures/computing_architecture.png}
\caption{Computing architecture of the Pegasus-Mini quadruped robot. Nvidia Xavier NX is used to run the CNN model to segment the trail in the garden. UP board is used to run the low-level controller, including MPC, WBC, and state estimator as shown in Fig. \ref{fig:framework}.}
\label{fig:computing_architecture}
\end{figure}
\section{Semantic Segmentation of Garden Scene}
In this section, a CNN model based on ERFNet is adopted for the trail segmentation and classification in a garden. Domain adaption method, dataset, and training method will be introduced in the below subsections. The yaw motion and deviation in $y$ direction estimation based on the segmentation of trail will be also described.
\subsection{Domain Adaption and Network Training}
There exists abundant open-sourced datasets for semantic segmentation, about indoors or urban environment. However, no dataset contains only the unstructured environment. In order to solve the domain shift problem and make the Pegasus-Mini correctly perceive the garden environment, a small dataset is recorded about the garden environment. After collection, we only label the traversable area of the garden dataset with the rectangle box for two reasons. Firstly, the model needs to work in different kinds of gardens that have various backgrounds but similar path, so precise labeling with all the classes is unnecessary. Secondly, we hope to complete the labeling work with less human labors. Based on this idea we only spend an hour labeling the traversable path with the rectangle box, and the Fig. \ref{fig:dataset} shows examples of the label.
The new garden dataset is combined with the Cityscape dataset \cite{7780719} for the domain adaption. From another perspective, the Cityscape dataset provides the negative labels. To this end, the Cityscape dataset is relabelled from 30 classes into 3 classes: traversable path, untraversable path, and void. This corresponds to the garden dataset, therefore, these are able to be put together and only one-stage training is needed, to avoid the catastrophic forgetting problem during the training.
\begin{figure}[ht]
\centering
\includegraphics[width = 3.4 in]{figures/example.png}
\caption{Example of the manually collected dataset for domain adaption. Only some parts of the road class area (traversable path) are labelled with the consideration of saving human labors. The labelled area covers about 5$\%$~50$\%$ of the whole image.}
\label{fig:dataset}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 3.3 in]{figures/cnn_framework.png}
\caption{The network architecture of CNN model adopted in this study.}
\label{fig:cnn_framework}
\end{figure}
The adopted network architecture in this study is based on ERFNet, which is capable of achieving good performance in semantic segmentation while running in real-time on mobile hardware \cite{7995966}. Compared with the original network, the size of the input layer and the number of the channels are changed in the last output layer to fit the training dataset. In order to speed up the model and reduce the over-fitting problem, the network is simplified by shrinking the repeated bottleneck from 5 to 3 and from 3 to 2 in the encoder and decoder respectively. Besides, two skip connections are introduced between the intermediate decoder and encoder layers to improve the performance of the model. The overall structure of the network model is illustrated in Fig. \ref{fig:cnn_framework}.
Since the semantic segmentation is a pixel-wise classification task, the cross-entropy loss is employed in form of multi-class classification as (\ref{eq:1}). The dataset is reweighted so that the weight of the garden dataset and Cityscape dataset is set as 2:1, considering that the amount of garden images is much fewer than the urban images.
\begin{equation}
J(W) = - \frac{1}{N} \sum_{i=1}^{N} \sum_{j=1}^{C} y_{i,j}log (\hat{y}_{i, j}),
\label{eq:1}
\end{equation}
where $y_{i,j}$ and $\hat{y}_{i, j}$ are the ground truth class and predicted class of $n$ $th$ images respectively. The loss of $N$ images and $C$ classes are summarized.
\subsection{Dataset}
To train the network, two datasets are mainly used. The Cityscape semantic segmentation benchmark is a public dataset with a multi-sensor collection in Germany. The dataset includes more than 25000 annotated images of which about 5000 images are with fine annotations. The images are annotated with 30 classes of relevant objects and summarized with 8 groups. Classes like road and sidewalk can be considered as the traversable path in our case and objects like person, vegetation and terrain are classified as untraversable areas, which is nutritious for the training.
Additionally, in this study, a dataset is generated for domain adaption by controlling the robot to trot in the garden. The robot trotting speed is set at 0.7 $m/s$ with the gait frequency at 4 $Hz$, to minimize the influence of the locomotion controller. This dataset consists of only one scene, the garden, with the robot-view image recorded at 4 $Hz$. There are totally about 700 images with a resolution of 640 $\times$ 480. The first 100 images are selected as the validation sets and the rest as the training set.
\subsection{Training}
To improve the generalization performance of the model, we perform image augmentation on the input images. The images are randomly cropped from the Cityscape dataset to the resolution of 640 $\times$ 480 to fit the input. The following image pre-processing augmentations are applied to both two datasets:
1) Image normalization to [0, 1];
2) Randomly horizontal flip of the input images with probability 0.5;
3) Random rotation of images from [-5$^\circ$, 5$^\circ$].
Our model is trained using the Adam optimizer with a learning rate of $10^{-5}$. We choose the batch size of 8 and the epoch of 150. To deal with the overfitting problem, the L2 regularization and early stopping strategy are applied. An NVIDIA GeForce RTX 2080 GPU is adopted for all the training and evaluation of the models. Computation architecture is as shown in Fig. \ref{fig:computing_architecture}.
\subsection{Pose Adjustment}
The 2D perception result upstream is used for the garden navigation, to enable the robot to locomote in the middle of the path throughout. To this end, a pose adjustment algorithm is proposed for the garden navigation mission.
The output images are downsampled to reduce the computational burden. Based on the post-processed perception results, the midpoints are calculated along the path from the boundary on both sides. The poses of all the points are summed to get the averaged point. From the start point to the averaged point, the radial angle can be calculated to provide the yaw angle for the next movement, as shown in (\ref{eq:yaw_angle}).
\begin{equation}
\alpha = arctan \frac{ \sum_{i=1}^{N} (p_{ix} - p_{0x})}{ \sum_{i=1}^{N} (p_{iy} - p_{0y})},
\label{eq:yaw_angle}
\end{equation}
where $p_i$ and $p_0$ are the calculated midpoints and start point respectively. $\alpha$ is the yaw angle of the robot's pose.
The overall perception and pose adjustment process can run at more than 6 $Hz$ on the robot platform after optimization. Since the trotting frequency of the robot is 4 $Hz$, the update frequency of the perception is fixed at also 4 $Hz$. In this case, the robot does not miss the perception results, and the time latency between the visual input and trotting decision is always less than 0.25 $s$, which guarantees the timely and robust movement control.
With the assistance of the classification of terrains in the garden environment, the trail is able to be segmented and extracted to provide a traversability reference for Pegasus-Mini. The edge of the trail in the image of Intel RealSense D435 is calculated for the estimation of the desired trajectory of the quadruped robot. Vision-based navigation is cost-effective and computing efficient. 4 $Hz$ frame frequency of the classification satisfies the requirement of normal locomotion speed of quadruped robot. However, there exists instability of trail classification based on the learning method.
\section{Path Compensation Planning}
Section IV introduces the learning method using a neural network to segment trail in a garden environment. The method is capable of extracting the edge of the trail for estimation of the direction and the middle line of the trail. However, the learning-based method is not able to guarantee the success rate, especially when the quadruped robot happens to move to a new location where the scenario is not recognized due to the limitation of the training dataset. Pose estimation is extensively studied in legged locomotion and floating-base system \cite{luo2020estimation, luo2021modeling}. Considering the fact that the state of the quadruped robot does not jump suddenly but changes consistently, the consistency characteristic of the dynamics of the quadruped robot is utilized and a compensation method is proposed in this section to correct the issue existing in vision learning.
\begin{figure}[b]
\centering
\setlength{\abovecaptionskip}{0.0 cm}
\includegraphics[width = 3.4 in]{figures/segmentation_comparison.png}
\caption{In the left image, the grass is mistaken to be the trail and the correspondence desired yaw rotation and $y$ deviation of Pegasus-Mini are wrongly determined. In comparison, the right image provided a right desired yaw rotation and $y$ deviation.}
\label{fig:trail_comparison}
\end{figure}
As shown in Fig. \ref{fig:trail_comparison}, the trail segmentation is wrong. To compensate the mistaken trail classification, this study proposes a trajectory planning method to compensate the wrong desired yaw and $y$ deviation velocity. $y^t = [y^t_0, y^t_1, ..., y^t_{n-1}]^T$ is the vector of middle point sequence in the image collected at timestamp $t$. $y_i^t$ is the $i$ th middle point of two edges of the trail in the image. A polynomial fitting is adopted to calculate a smooth curve as the estimated path in $y$ direction.
\begin{equation}
y^t_p = [\beta_0^t, \beta_1^t p, \beta_2^t p^2, ... , \beta_{n-1}^t p^{n-1}]^T,
\end{equation}
\label{eq:y_t}
where $y^t_p$ is the estimation of the middle point sequence in $y$ direction at time stamp $t$. $p = 0, 1, ..., n-1$. $n$ is the order of the polynomial fitting.
\begin{equation}
y^t = P (\beta^t)^T,
\end{equation}
where $y^t = [y^t_0, y^t_1, ..., y^t_{n-1}]^T$, $\beta^t = [\beta_0^t, \beta_1^t, ... , \beta_{n-1}^t]^T$ and $P$ is:
\begin{equation}
P = \begin{bmatrix}
p_0 & p_1 & ... & p_{n-1}\\
p_0^2 & p_1^2 & ... & p_{n-1}^2 \\
... & ... & ... & ... \\
p_0^n & p_1^n & ... & p_{n-1}^n
\end{bmatrix}_{n \times n}.
\end{equation}
Therefore, $\beta^t$ is able to be calculated by:
\begin{equation}
\beta^t = (P^{-1} y^t)^T.
\end{equation}
\begin{figure}
\centering
\includegraphics[width = 3.4 in]{figures/fitting_2.png}
\caption{The polynomial fitting of the middle points estimated by CNN model. The green area corresponds to the grass terrain where Pegasus-Mini is not supposed to traverse.}
\label{fig:fitting}
\end{figure}
Ideally, in the condition in which there is no mistaken segmentation, the coefficients $\beta^{t_j}$ at $t_{j}$ time stamp is updated with $\beta^{t_{j+1}}$:
\begin{equation}
\beta_{p}^{t_{j+1}} = w_1 \beta_{p+1}^{t_{j+1}} + w_2 \beta_{p+1}^{t_j}
\label{eq:iteration}
\end{equation}
where $t_j$ and $t_{j+1}$ are the $j$ th and $j+1$ th time stamp respectively. $w_1$ and $w_2$ are the weights for each term and $w_1 + w_2 = 1$. $\beta_{p}^{t_{j+1}}$ is the updated $p$ th term in the coefficient vector $\beta$ for time stamp $t_{j+1}$.
If the segmentation fails, as shown in Fig. \ref{fig:fitting}, the disturbed point calculated from the wrong trail edge (as shown in Fig. \ref{fig:trail_comparison}) will drag the fitted curve away from the desired path trajectory. In this condition, the quadruped robot will walk into the grass, which is not expected to occur.
Similarly, the estimated yaw angle of quadruped locomotion is calculated through (\ref{eq:yaw_angle}).
\begin{equation}
\alpha_{t_j} = w_{\hat{\alpha}} \hat{\alpha}_{t_j} + w_{\alpha} \alpha,
\label{eq:yaw_est}
\end{equation}
where $\alpha_{t_j}$ is the updated yaw angle at $t_j$ time stamp. $w_{\hat{\alpha}}$ and $w_{\alpha}$ are the weights for each term. $w_{\hat{\alpha}} + w_{\alpha} = 1$.
The principle of trajectory compensation is shown in Fig. \ref{fig:fitting}. The history information during the last $n$ th time stamp is considered together with the updated new estimation at $n+1$ th time stamp. If the estimation $\alpha$ or $y$ deviates a lot from the history records, the weight for this updated term will be attenuated.
\section{Experiment}
In this section, the vision-based navigation method proposed in this study is tested. The training performances of different datasets are compared and evaluated. To validate the effectiveness of the vision-based navigation on the quadruped robot Pegasus-Mini, trail detection algorithm is run under different trotting speeds from 0.2 $m/s$ to 1.0 $m/s$. Three CNN models trained using Cityscape, garden, and Cityscape-garden are compared respectively. Learn method only and the learning method with path planning compensation are compared. In the next subsections, training performance and comparison results will be demonstrated.
\subsection{Training Results}
Training and validation loss results are as shown in Fig. \ref{fig:training_results}. In Fig. \ref{fig:training_results}, the blue line represents the training loss of the Cityscape dataset, the orange line represents the training loss of the garden dataset and the green line represents the training loss of the Cityscape and garden dataset. From Fig. \ref{fig:training_results}, it is demonstrated that the training performance based on three datasets is satisfactory.
\begin{figure}[t]
\centering
\includegraphics[width = 3.4 in]{figures/training_results.png}
\caption{Training loss and validation loss. The blue, orange and grey line represent the training and validation loss results based on models of Cityscape ($C$ dataset), garden ($G$ dataset) and Cityscape-garden ($CG$ dataset) respectively.}
\label{fig:training_results}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 3.0 in]{figures/prediction_results.png}
\caption{Qualitative prediction results of semantic segmentation trained with difference dataset. The blue, green and red regions represent the traversable area(road), untraversable area and sky respectively.}
\label{fig:prediction}
\end{figure}
In this test, the CNN models trained with three datasets are tested on the quadruped robot Pegasus-Mini. As shown in Fig. \ref{fig:prediction}, the left column represents the raw scenario images. The left second, third and fourth columns represent the segmentation results using datasets from Cityscape, garden, and Cityscape-garden respectively. For convenience, the models trained with Cityscape, garden, and Cityscape-garden datasets are termed as $C$, $G$, and $CG$ respectively.
\subsection{Path Planning With Compensation}
This subsection demonstrates the comparison results. The trail segmentation results using three datasets are compared under different trotting speeds. The vision-based navigation with learning method only and with the trajectory compensation are also compared. To test the effectiveness of the proposed vision-based navigation on quadruped locomotion, experiments are conducted in trotting gait at different speeds ranging from 0.4 to 1.0 $m/s$ which are common speeds for quadruped locomotion.
\begin{figure}[ht]
\centering
\setlength{\abovecaptionskip}{-0.1 cm}
\setlength{\belowcaptionskip}{-1 cm}
\includegraphics[width = 3.0 in]{figures/c_model_with_4_speeds.png}
\caption{The performance of trail segmentation using Cityscape dataset under trotting speed at 0.2, 0.4, 0.6 0.8 $m/s$ respectively.}
\label{fig:com_cityscape}
\end{figure}
\vspace{-4 mm}
\begin{figure}[ht]
\centering
\setlength{\abovecaptionskip}{-0.1 cm}
\setlength{\belowcaptionskip}{-1 cm}
\includegraphics[width = 3.0 in]{figures/g_model_with_4_speeds.png}
\caption{The performance of trail segmentation using garden dataset under trotting speed at 0.2, 0.4, 0.6, and 0.8 $m/s$ respectively.}
\label{fig:com_garden}
\end{figure}
\vspace{-4 mm}
\begin{figure}[ht]
\centering
\setlength{\abovecaptionskip}{-0.1 cm}
\setlength{\belowcaptionskip}{-1 cm}
\includegraphics[width = 3.0 in]{figures/cg_model_with_4_speeds.png}
\caption{The performance of trail segmentation using Cityscape-garden dataset under trotting speed at 0.2, 0.4, 0.6, and 0.8 $m/s$ respectively.}
\label{fig:com_cityscape_garden}
\end{figure}
\subsubsection{Results Under Different Trotting Speeds and CNN models}
In this study, a common gait, trotting, is selected to test the effectiveness of the deployment of vision-based navigation for quadruped locomotion. Four trotting speeds are set in the experiment, i.e. 0.2, 0.4, 0.6, and 0.8 $m/s$. CNN models trained with Cityscape and Cityscape-garden datasets are tested respectively.
Fig. \ref{fig:com_cityscape} shows two example images taken with $C$ model. Each row shows the trail segmentation results under four different trotting speeds. The segmentation results at four different trotting speeds do not provide a clear extraction of the trail. Overall performance is not very satisfactory.
Fig. \ref{fig:com_garden} shows the results taken with $G$ model under four trotting speeds. Each row shows the trail segmentation results in a certain scene under four different trotting speeds. Due to the small size of training the dataset of the garden, the visualized segmented images in Fig. \ref{fig:com_garden} is sparse.
\begin{figure}[ht]
\centering
\includegraphics[width = 3.0 in]{figures/model_with_compensation_comparison.png}
\caption{Vision-based navigation with and without trajectory compensation under four different speeds ranging from 0.4 to 1.0 $m/s$. The left column represents the failure of the segmentation of trail from surrounding grass. The right column represents the successful corrected segmentation when trajectory compensation is supplemented.}
\label{fig:_with_compensation_comparison}
\end{figure}
Fig. \ref{fig:com_cityscape_garden} shows results taken with $CG$ model under four trotting speeds. Each row shows the trail segmentation results under four different trotting speeds. Compared with the segmentation performances in Fig. \ref{fig:com_cityscape} and \ref{fig:com_garden} which correspond to the training results using $C$ and $G$ , the $CG$ is able to output better segmentation of trail, which is taken as the traversable path for the quadruped locomotion.
\begin{figure*}[ht]
\centering
\includegraphics[width = 6.9 in]{figures/screenshot.png}
\caption{The screenshots of the vision-based navigation of Pegasus-Mini in the garden. The time duration between each screenshot is 2 $s$.}
\label{fig:screenshot}
\end{figure*}
\subsubsection{Results of Trajectory Compensation}
In this subsection, vision-based navigation performances with and without trajectory compensation are compared. The comparison is demonstrated under four different speeds. Fig. \ref{fig:_with_compensation_comparison} shows the result under trotting speed of 0.4, 0.6, 0.8, and 1.0 $m/s$ respectively. The left column represents the wrong segmentation of the trail no matter what speed the quadruped robot trots at. In comparison, the right column shows the corrected trail segmentation when trajectory compensation is supplemented in the whole navigation framework.
The above experiment results demonstrate that the proposed vision-based navigation method is effective for the normal quadruped trotting gait in a garden environment. Despite the instability of the image segmentation for the path planning, a compensation method is supplemented to enhance the success rate of traversability in the garden environment. The screenshot of the vision-based navigation of our small-scale quadruped robot Pegasus-Mini is as shown in Fig. \ref{fig:screenshot}. It is noteworthy that the open-sourced dataset Cityscape is proven to be able to be generalized to the garden scene in this study. With a small dataset collected from a specific scene merged with the Cityscape dataset, the CNN model is able to be deployed in the field.
\section{Conclusion}
This study proposed a vision-based navigation method combining learning-based method and trajectory planning to enhance the traversability. The learning method is based on ERFNet which is extensively used for semantic segmentation. The open-sourced dataset Cityscape is combined with the dataset collected from our garden scene to train ERFNet. ERFNet is deployed on a small-scale quadruped robot Pegasus-Mini to accomplish the real-time terrain segmentation. The training performance is compared with Cityscape only and with Cityscape-garden. Cityscape model ($C$ model) and Cityscape-garden model ($CG$ model) are tested in a common quadruped gait, trotting, under different speeds ranging from 0.4 to 1.0 $m/s$. Test results demonstrate that the $CG$ model performs better in the trail segmentation in the garden scene. Different trotting speeds ranging from 0.4 to 1.0 $m/s$ have little disturbance to the images sensing. However, the CNN model is not able to guarantee the stable trail extraction. To tackle this issue, this study proposes a trajectory compensation method, in which the consistent history trajectory sequence is taken into account together with the updated estimation of the middle line of the trail and the yaw angle. The learning-based method for image processing and semantic segmentation combined with the trajectory compensation method is capable of increasing the success rate of traversability in a garden scene. The future work includes further increasing the success rate traversability of quadruped locomotion. |
2,877,628,090,166 | arxiv | \section{ALICE experiment at CERN}
\label{intro}
A Large Ion Collider Experiment (ALICE)~\cite{Carminati:2004fp,Alessandro:2006yt,ALICEjinst} is one of the four major experiments
currently operating on the beams of the Large Hadron Collider (LHC) at the
European Organization for Nuclear Research (CERN).
The aim of this experiment is to study the physics of strongly
interacting matter at extreme energy densities, where the formation of
a new phase of matter, the quark-gluon plasma, is expected.
For this purpose, the collaboration is carrying out series of comprehensive
measurements of hadrons, electrons, muons and photons produced in
the collisions of heavy nuclei. In addition, ALICE is studying
proton-proton interactions
both as a reference for the heavy-ion collisions and also in physics areas
where ALICE is competitive with other LHC experiments.
\section{Selected (Run I) results and their limitations}
An important part of the ALICE physics program is dedicated to high precision
measurements of charm and beauty production in heavy-ion collisions.
The ultimate goals of the heavy-flavour studies include:
\begin{itemize}
\item Thermalization of heavy quarks in the produced medium (by measuring
baryon-to-meson ratios for charm and beauty particles, ratios of yields
of heavy-flavour particles with strange content, and azimuthal flow
anisotropy for as many as possible heavy-flavour species).
\item Parton mass and colour-charge dependence of in-medium energy losses (by
measuring the momentum-dependent nuclear modification factors for B and D
mesons, and comparing them with those for light-flavour particles).
\end{itemize}
Using the data recorded during the LHC Run I (2009--2013),
the ALICE Collaboration has already published several important results
on the D-meson production in Pb--Pb collisions.
For example, the left panel of Figure~\ref{raa_v2} shows the
$\pt$-dependent nuclear modification factor for D-mesons
(average of D$^0$, D$^+$ and D$^{*+}$) compared to that
of pions and charged particles in the 0--10~\% centrality class.
There is an indication that production of D-mesons in Pb--Pb collisions at
lower $\pt$ is less suppressed than that of light-flavour particles.
The ALICE measurement of the second-order azimuthal flow harmonic $v_2$
is presented on the right panel of the Figure. The data suggest that the
$v_2$ for D-mesons is compatible with that of charged particles and is
larger than 0 over a wide momentum range. But, in both cases,
the current statistical and systematic uncertainties do not allow for firm
conclusions. In addition, the read-out rate capabilities and space-point
precision of the present Inner Tracking System (ITS) are not sufficient
to perform similar measurements with beauty particles (B-mesons), which are
important to prove the existence of quark-mass ordering:
$\RAA^{\rm B} > \RAA^{\rm D} > \RAA^{\rm charged}$ and
$v_2^{\rm B} < v_2^{\rm D} < v_2^{\rm charged}$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.43\textwidth]{raa}
\includegraphics[width=0.55\textwidth]{v2}
\caption{Left panel: Nuclear modification factor
$\RAA$ for D-mesons, pions and charged particles~\cite{Adam:2015sza}. Right panel: Amplitude of the
second-order flow harmonic $v_2$ for D-mesons and charged particles~\cite{Abelev:2013lca}.
Statistical and systematic uncertainties are represented by bars
and boxes respectively.
}
\label{raa_v2}
\end{figure}
At the same time, even though ALICE has already measured several
baryon-to-meson ratios with light-flavour
particles~\cite{Abelev:2013xaa,Adam:2015kca}, the extension
of these results towards the heavy-flavous sector is currently not possible.
The read-out rates and precision of track reconstruction provided
by the present ITS do not allow for detecting
the heavy-flavous baryons ($\Lambda_{\rm c}$ and $\Lambda_{\rm b}$)
in high-track-multiplicity environment of Pb--Pb collisions at the LHC.
\section{Upgrade of the Inner Tracking System}
The limitations of the present ITS will be radically reduced with
the planned upgrade~\cite{Abelevetal:2014dna}.
The general layout of the upgraded ITS
is shown in Figure~\ref{layout}.
The detector will be 1.5 m long, with the outer radius of 40 cm,
and the overall surface of about 10~m$^2$ will be covered with
silicon pixel sensors. This will be a ``12.6 Giga-pixel camera'',
with an estimated cost of about 13.6 million Swiss Francs.
\begin{figure}[htb]
\centering
\includegraphics[width=0.77\textwidth]{layout}
\caption{The general layout of the upgraded ITS detector.}
\label{layout}
\end{figure}
The detector will consist of seven layers,
with the first layer as close to the interaction point as 22.4~mm.
The material budget of the three innermost layers will be as low as
0.3~\% of a radiation length. Combined with the space-point precision
of about 5~$\mu$m, the expected track impact-parameter
resolution at $\pt\sim$1~GeV/$c$ will be about 20~$\mu$m, which is a factor 3
(in the transverse plane) and 5 (in the beam direction) better
as compared with the current situation.
The new detector will also allow for an efficient track reconstruction
down to very low $\pt$.
The simulations show that the tracking efficiency at $\pt\sim$0.1~GeV/$c$
will be a factor about 6 higher than it is now. This will be of a great
importance for planned di-electron measurements (see Table~\ref{tableReach})
and the detection of D$^*$ and D$_{\rm S}$ mesons.
All seven layers of the upgraded ITS will be equipped with the ALPIDE
chip~\cite{Abelevetal:2014dna}. Taking advantages of the
0.18~$\mu$m CMOS technology by TowerJazz~\cite{towerjazz}, this chip
combines the sensitive part and the read-out electronics within the same
piece of silicon.
The main characteristics of the chip are:
\begin{itemize}
\item overall chip size: $15\times 30$~mm$^2$;
\item pixel size: $29\times 27~\mu$m$^2$;
\item detection efficiency: $>99$~\%, with a
noise probability of $<10^{-6}$;
\item time resolution: about $2~\mu$s;
\item in-pixel discriminators and in-matrix address encoder with asynchronous sparsified readout;
\item power consumption: 40~mW/cm$^2$;
\item high radiation tolerance: tested up to 2.7 Mrad (TID), and
$1.7\times 10^{13}$~1~MeV~n$_{\rm eq}$/cm$^2$ (NIEL)
\end{itemize}
The upgraded ITS will be almost two orders of magnitude faster than now.
The read-out electronics of the detector will be able to register
events with a typical rate of 50~kHz and a few 100~kHz for minimum
bias Pb--Pb and pp collisions respectively.
The parts of the detector will have to be aligned with the precision
of better than 5~$\mu$m by means of dedicated software. The data coming
out of the detector will be reconstructed quasi-online, using
a dedicated Online-Offline (O$^2$) computer farm~\cite{O2}.
Overall, the upgrade of the Inner Tracking System will be accomplished
by the end of the LHC Long Shutdown 2 (2021).
\section{Physics with the upgraded ITS}
The expected physics reach for various
observables is summarized in Table~\ref{tableReach},
in terms of minimum accessible $\pt$ and of statistical uncertainties.
We consider a scenario with an integrated luminosity of
\SI{10}{\per\nano\barn}, fully used for minimum-bias Pb--Pb data collection,
and a low-magnetic-field run with \SI{3}{\per\nano\barn}
of integrated luminosity for the low-mass di-electron studies.
The case of the programme up to Long Shutdown 2 is shown for comparison.
In this case, a delivered luminosity of
\SI{1}{\per\nano\barn} is assumed, out of which 10\% is recorded with a
minimum-bias trigger.
In the upgrade case, the systematic
uncertainties will be reduced as well. This is because for many
measurement, due to the mentioned improvements in the track impact-parameter
resolution, the level of the combinatorial background will significantly be
be reduced, and the feed-down contribution from beauty to charm measurements
will be estimated directly, from the data themselves.
There are also several observables, like those involving the heavy-flavour
baryons and the low-momentum di-electrons, that were never accessible before,
but will become well within the experimental reach thanks to the unique
capabilities of the upgraded ALICE Inner Tracking System.
\begin{table}[htb]
\centering
\caption{Summary of the physics reach: minimum accessible $\pt$ and relative
statistical uncertainty in Pb--Pb collisions for an integrated luminosity of
\SI{10}{\per\nano\barn}. For heavy flavour, the statistical uncertainties
are given at the maximum between $\pt=\SI{2}{GeV/\it{c}}$ and $p_{\rm T}^{\rm min}$.
For elliptic flow measurements, the value of $v_2$ used to calculate
the relative statistical uncertainty $\sigma_{v_2}/v_2$ is given
in parenthesis. The programme up to Long Shutdown 2, with an integrated
luminosity of \SI{0.1}{\per\nano\barn} collected with minimum-bias trigger,
is shown for comparison.}
\label{tableReach}
\begin{tabular}{lcccc}
\toprule
& \multicolumn{2}{c}{Current, \SI{0.1}{\per\nano\barn} } & \multicolumn{2}{c}{Upgrade, \SI{10}{\per\nano\barn}
} \\
\cmidrule{2-3} \cmidrule{4-5}
Observable & $p_{\rm T}^{\rm min}$ & \multicolumn{1}{c}{statistical} & $p_{\rm T}^{\rm min}$ & \multicolumn{
1}{c}{statistical} \\
& (GeV/$c$) & \multicolumn{1}{c}{uncertainty} & (GeV/$c$) & \multicolumn{1}{c}{uncertainty} \\
\midrule
\multicolumn{5}{c}{Heavy Flavour} \\
\midrule
D meson $R_{\rm AA}$ & 1 & \SI{10}{\percent} & 0 & \SI{0.3}{\percent}\\
D$_{\rm s}$ meson $R_{\rm AA}$ & 4 & \SI{15}{\percent} & $<2$ & \SI{3}{\percent}\\
D meson from B $R_{\rm AA}$ & 3 & \SI{30}{\percent} & 2 & \SI{1}{\percent}\\
J/$\psi$ from B $R_{\rm AA}$ & 1.5 & ~~~\SI{15}{\percent} \tiny{($\pt$-int.)}
& 1 & \SI{5}{\percent}\\
B$^+$ yield & \multicolumn{2}{c}{not accessible} & 2 & \SI{10}{\percent} \\
$\Lambda_{\rm c}$ $\RAA$ & \multicolumn{2}{c}{not accessible} & 2 & \SI{15}{\percent}\\
$\Lambda_{\rm c}/{\rm D^0}$ ratio & \multicolumn{2}{c}{not accessible} & 2 & \SI{15}{\percent} \\
$\Lambda_{\rm b}$ yield & \multicolumn{2}{c}{not accessible} & 7 & \SI{20}{\percent} \\
D meson $v_2$ ($v_2 = 0.2$) & 1 & \SI{10}{\percent} & 0 & \SI{0.2}{\percent} \\
D$_{\rm s}$ meson $v_2$ ($v_{2}=0.2$) & \multicolumn{2}{c}{not accessible} & $<2$ & \SI{8}{\percent}\\
D from B $v_2$ ($v_2 = 0.05$) & \multicolumn{2}{c}{not accessible} & 2 & \SI{8}{\percent}\\
J/$\psi$ from B $v_2$ ($v_2 = 0.05$) & \multicolumn{2}{c}{not accessible} & 1 & \SI{60}{\percent}\\
$\Lambda_{\rm c}$ $v_2$ ($v_2 = 0.15$) & \multicolumn{2}{c}{not accessible} & 3 & \SI{20}{\percent} \\
\midrule
\multicolumn{5}{c}{Di-electrons} \\
\midrule
Temperature (intermediate mass) & \multicolumn{2}{c}{not accessible} & & \SI{10}{\percent} \\
Elliptic flow ($v_2 = 0.1$)~\cite{LOI} & \multicolumn{2}{c}{not accessible} & & \SI{10}{\percent} \\
Low-mass spectral function~\cite{LOI} & \multicolumn{2}{c}{not accessible} & 0.3 & \SI{20}{\percent} \\
\midrule
\multicolumn{5}{c}{Hypernuclei} \\
\midrule
$^{\,3}_{\Lambda}$H yield & 2 & 18\,\% & 2 & \SI{1.7}{\percent} \\
\bottomrule
\end{tabular}
\end{table}
|
2,877,628,090,167 | arxiv | \section{Introduction}
For a long time dissipation has mainly been considered as a nuisance which destroys the coherence of a quantum state. However, in the last decade dissipation has been turned into a tool and the dissipative attractor dynamics has been used to stabilize complex quantum states \cite{MuellerZoller2012}. Examples of such realizations are the stabilization of a Tonks-Girardeau gas of molecules \cite{SyassenDuerr2008} or the creation of entanglement in an ion chain \cite{BarreiroBlatt2011}.
With a proper engineering of the coupling to the reservoir, it is possible to drive the system towards a desired quantum state. Here we concentrate on setups which exhibit local particle losses. Such a setup is realized using cold atomic gases subjected to an electron beam \cite{GerickeOtt2007} or more recently using near-resonant optical tweezers \cite{CormanEsslinger2019}. In the setup with the electron beam the quantum Zeno effect has been realized \cite{BarmettlerKollath2011,BarontiniOtt2013} and more recently, the phenomenon of coherent perfect absorption has been observed \cite{MuellersOtt2018}.
Theoretically, this setup has been studied extensively for weakly interacting bosons \cite{BrazhnyiOtt2009,ShchesnovichKonotop2010,ZezyulinOtt2012} and for the Bose-Hubbard model \cite{BarmettlerKollath2011,ShchesnovichMogilevtsev2010,WitthautWimberger2011,KieferEmmanouilidisSirker2017} with respect to different properties including transport phenomena.
More recently, interacting fermions with a single-site dissipative defect causing particle losses were investigated theoretically \cite{FroemlDiehl2019, FroemlDiehl2019b,DamanetDaley2019} and experimentally \cite{CormanEsslinger2019, LebratEsslinger2019}. It was found that the quantized transport of fermions survives in the presence of dissipative quantum dots \cite{CormanEsslinger2019, LebratEsslinger2019}. Additionally, the interplay between dissipation strength, coherence and interaction leads to the emergence of the fluctuation induced quantum Zeno effect \cite{FroemlDiehl2019, FroemlDiehl2019b}, signaled by a reduced rate of the total particle loss. In particular, the particle loss at the Fermi momentum is totally suppressed for repulsive interactions and a metastable state arises, which cannot be described by a thermal state. As different approximations have been employed in the aforementioned work, the question remains whether in a full treatment of both, the interaction and the dissipation, this metastable non-thermal state survives.
In this article we address this question and establish the existence of a quasi-stationary regime in time by using numerically exact matrix product state (MPS) methods to simulate the dynamics of a system of spinless fermions with nearest-neighbour interaction exposed to local loss processes. We simulate the full non-equilibrium dynamics of the system with a variety of different interaction and dissipation strengths. In the following we first introduce the model in Sec.~\ref{sec:model} and explain the method we use to investigate this model in Sec.~\ref{sec:method}. In Sec.~\ref{sec:te} we present the time evolution of the system; we discuss the initial dynamics of the local density in Sec.~\ref{sec:ie}, the quantum Zeno effect in Sec.~\ref{sec:zeno} and the properties of the metastable solution in Sec.~\ref{sec:meta}.
\section{Model}\label{sec:model}
\begin{figure}
\includegraphics[width=\linewidth]{fig1.pdf}
\caption{Sketch of the set-up of interacting spinless fermions in a one-dimensional lattice that includes hopping between adjacent sites with amplitude $J$, nearest-neighbour interaction with strength $V$ and Markovian particle loss at the central site labeled as '0' with dissipation strength $\gamma$.}
\label{fig:setup}
\end{figure}
We consider spinless fermions on a one dimensional lattice subjected to a local loss as sketched in Fig.~\ref{fig:setup}. Fermions on neighbouring sites are interacting with each other. The Hamiltonian describing the system is given by,
\begin{eqnarray}\label{eq:Ham}
H=&-&J \sum_{l=-\frac{L-1}{2}}^{\frac{L-1}{2}-1} \left (c^\dagger_{l} c^{\phantom{\dagger}}_{l+1}+\mathrm{H.c.} \right ) + V \sum_{l=-\frac{L-1}{2}}^{\frac{L-1}{2}-1}n_{l} n_{l+1}\nonumber\\
&+&H_{\text{b}}.
\end{eqnarray}
For simplicity, we consider a chain of $L$ sites, where $L$ is an odd number. The operators $c_l^{\phantom{\dagger}}$ and $c_l^\dagger$ are the annihilation and creation operators for fermions at site $l$ and $n_l=c^\dagger_{l} c^{\phantom{\dagger}}_{l}$ is the local density operator for fermions at site $l$. The hopping amplitude and the interaction strength are denoted by $J$ and $V$, respectively. In order to decrease the boundary effects induced by the interaction the term
\begin{eqnarray}\label{eq:Ham_b}
H_{\text{b}}=&&V \left(n_{-\frac{L-1}{2}} + n_{\frac{L-1}{2}}\right) \frac{N}{L}
\end{eqnarray}
is added to the Hamiltonian. This term couples the density at the boundaries to the average value of the density $N/L$, where $N$ is the number of fermions, and therefore smoothens the boundary effects. For intermediate interaction $-2\le V/J \le 2$, the ground state of this model can be well described by a gapless Tomonaga-Luttinger liquid. At larger values of the interaction $ |V/J|> 2$, the systems enters a gapped phase and at repulsive interactions, charge density wave correlations are dominant \cite{Giamarchibook}.
The chain of fermions is subjected at its center to a local particle loss. In ultracold atomic gases such losses can be created by the application of an electron beam \cite{BarontiniOtt2013} or a local radio frequency flip \cite{PalzerKoehl2009}.
The system dynamics is described by a Lindblad master equation given by
\begin{align}\label{eq:Lindblad}
&\dot \rho(t) = -\frac{i}{\hbar} \left[ H, \rho(t) \right] + \gamma \left( c_0 \rho(t) c_0^\dagger - \frac{1}{2}\left\{\rho(t), c_0^\dagger c_0 \right\} \right),
\end{align}
where $\rho$ is the density matrix of the fermionic system. Lost particles do not reenter the system. The first term on the right hand side describes the Hamiltonian evolution of the system with the Hamiltonian given above. The second term introduces the dissipative loss process with the so-called jump operator given by $c_0$ at the central site ($l=0$) with an amplitude $\gamma$.
\begin{figure*}
\includegraphics[width=0.99\linewidth]{fig2.pdf}
\caption{\label{fig:ntime}Time evolution of the density profile starting from the ground state of $H$ with initial filling $N/L=(L-1)/(2L)$ after switching on the losses at $t=0$ for attractive, vanishing and repulsive interactions. The dotted line represents the extrapolated long-time metastable profiles, which emerge around the central site before the evolution is affected by boundary effects. Data is shown for a system of size $L=63$, different interaction strengths $V/J$, dissipation strength $\hbar \gamma/J=20$ and $10^3$ sampled trajectories.}
\label{fig:density_evolution}
\end{figure*}
\section{Method}\label{sec:method}
We numerically simulate the dynamics of this open quantum system starting from the ground state of the Hamiltonian $H$, which is obtained by using the density matrix renormalization group (DMRG) method in the formulation of matrix product states (MPS) \cite{Schollwoeck2011}. The dissipative nature of the system is taken into account using the Monte-Carlo wave function method \cite{DalibardMolmer1992, GardinerZoller1992, CarmichaelBook, BreuerPetruccione2002, Daley2014}. In this method the evolution of wave functions is calculated, rather than the evolution of the density matrix, at the cost of a stochastic sampling of many different trajectories. A single trajectory sample is created by a piecewise deterministic process in which a deterministic evolution, generated by a non-Hermitian Hamiltonian, here given by
\begin{equation}
H_{\mathrm{eff}} = H - i \frac{\gamma}{2} c_0^\dagger c_0^{\phantom{\dagger}},
\end{equation}
is stochastically interrupted by applications of the jump operator. The duration of the deterministic evolution until the next jump occurs is the so-called waiting time $\tau$. This time is sampled according to the cumulative distribution
\begin{equation}
P\left(\ket{\psi(t)}, \tau\right) = 1 - \left\lVert \exp\left( -i H_{\mathrm{eff}} \tau \right)\ket{\psi(t)}\right\rVert,
\end{equation}
specified by the decay of the norm of the evolving state, which originates from the non-unitary evolution \cite{BreuerPetruccione2002}.
The time-dependent expectation value of an observable is averaged over a sufficiently large number of stochastically sampled wave functions to get a desired accuracy. The evolution of the observable calculated by Monte-Carlo wave function method coincides with the results calculated by the evolution of the density matrix with Lindblad master equation (Eq.~\ref{eq:Lindblad}).
In our work, the deterministic part of the evolution is computed using time-dependent matrix product state (tMPS) methods, which efficiently approximate evolving quantum states while keeping a very high level of accuracy\cite{DaleyVidal2004, WhiteFeiguin2004}. This method relies on a truncation scheme, which is controlled by the convergence parameters of the truncation weight $\varepsilon$ and the bond dimension $D$ \cite{Schollwoeck2011, PaeckleHubig2019}. In the presented results we define a truncation goal $\varepsilon=10^{-12}$ that determines a maximum accepted truncation error, but never exceeds a defined maximum value for the bond dimension $D=300$. Additionally, within this method we approximate the time evolution operator of the effective Hamiltonian by a second order Trotter-Suzuki decomposition, which is controlled by the time step $\Delta t$ \cite{DaleyVidal2004, WhiteFeiguin2004}, chosen here as $\Delta t=0.05\hbar/J$ if not stated otherwise. We assure convergence of our results with respect to these parameters.
In this work, the stochastic sampling is preferred over the full density matrix evolution which can be performed within MPS methods by the so-called purification \cite{Schollwoeck2011} for three reasons. Firstly, the initial state being the ground state of an interacting problem is a potentially highly entangled state. Thus, it requires a very large bond dimension when formulated as a purified matrix product state making an accurate description very challenging. Secondly, the fact that the Markovian dissipation is only represented by a single jump operator makes the additional stochastic selection among many different operators at the jump times unnecessary. This will lead to smaller deviations between the trajectory samples. Thus the number of trajectories needed for the convergence of the observables is less compared to situations with many jump operators. Furthermore, due to the presence of the atom loss, the total number of particles is not conserved by the full Lindblad evolution of the density matrix, and all particle number sectors need to be taken into account. Contrary to that, in the stochastic simulation performed here, each part of the deterministic time-evolution with non-unitary Hamiltonian (Eq.~\ref{eq:Ham_b}) can be computed using the atom number as a conservation law to reduce the computational complexity. Only the applications of the jump operator change the number of atoms.
\section{Time evolution of the interacting system subjected to losses}\label{sec:te}
The local loss will lead to an emptying of the system around the central site. In a finite system, the long time steady state is the trivial empty system. However, we will discuss the initial and intermediate time dynamics, at which an interesting metastable state arises in the system.
\subsection{Initial evolution of the density distribution}\label{sec:ie}
\begin{figure}
\includegraphics[width=0.99\linewidth]{fig3.pdf}
\caption{\label{fig:Ntottime} Dependence of the evolution of the total particle number $N$ on the dissipation strength $\gamma$ for a system of size $L=15$, $N(t=0)=7$ and repulsive interaction strength $V/J=-1$ exhibiting two different time regimes. Dotted and dashed lines show exponential fits in the two time regions, where the exponents $1/\tau_{1,2}$ are associated with inverse time scales. We present an average over $10^4$ trajectories.}
\label{fig:Ntot_dissipationdependence}
\end{figure}
In Fig.~\ref{fig:ntime} we show the evolution of the spatial density distribution of the fermions starting from the ground state of the Hamiltonian with $(L-1)/2$ particles. At time $t=0$ the local losses are switched on. Fig.~\ref{fig:ntime} shows that already in the initial density profile, oscillations are present which are induced by the open boundaries. These oscillations are stronger at repulsive interactions and are smeared out for attractive ones. After the switch-on of the losses, the density on the central site decreases first very quickly as $n_0(t)\approx n_0(0)e^{-\alpha \gamma t}$, where $\alpha$ depends on the filling and the interaction strength, and then saturates to a metastable value at large dissipation. The saturation value $n_0(t\to\infty)$ depends on the dissipation and interaction strength. Here for the considered case of very strong dissipation $\gamma=20J/\hbar$ the saturation value of the density at central site is very low. This initial process of the depletion of the central site by the loss can also be seen in the total number of atoms lost from the system as shown in Fig.~\ref{fig:Ntottime}. For very short times, the decay is exponential, i.e.~ $N(t)=N(0)e^{- t/\tau_1}$ with the time-scale depending strongly on $\gamma$. As shown in Fig.~\ref{fig:tau1}, the initial time-scale $\tau_1$ has linear dependence on the inverse of the dissipation strength, i.e.~ $\tau_1\propto 1/\gamma$. The slope depends on the interaction strength and the filling. For weak dissipation strength this regime lasts for very long times and defines the main dynamics of the system on the time scale accessible to the simulation. In contrast, for strong dissipation strength this regime lasts for very short times and crosses over to a second regime.
\begin{figure}
\includegraphics[width=0.99\linewidth]{fig4.pdf}
\caption{\label{fig:tau1} Inverse dependence of the time scale $\tau_1$ determining the decay of $N(t)$ in the first time regime on the dissipation strength $\gamma$ for different interaction strengths. The time scales were extracted by exponential fits as shown in Fig.~\ref{fig:Ntot_dissipationdependence}. Dotted lines are linear fits in $1/\gamma$ and the dashed line is a guide to the eye. We considered $L=15$ sites, an initial particle number $N(t=0)=7$ and $10^4$ trajectories. The time step was chosen as $\Delta t J/\hbar=0.01$ for $\hbar\gamma/J>20$ and $\Delta t J/\hbar=0.05$ else.}
\end{figure}
\begin{figure}
\includegraphics[width=0.99\linewidth]{fig5.pdf}
\caption{Left panel: Time evolution of the density profile for comparably low interaction $V/J =0.1$ and low dissipation strength $\hbar\gamma/J=0.1$. The dashed line marks a $t\rightarrow\infty$ extrapolation extracted from an exponential fit with offset in the second time region. Right panel: Dependence of the metastable quasi-stationary state occupation of the central site on both dissipation and interaction strength represented by value at the longest time reachable by the simulation. A system of size $L=63$ has been considered that initially contains $N(t=0)=31$ particles using $2\cdot10^3$ trajectories.}
\label{fig:n0}
\end{figure}
\subsection{Spreading of the depletion and the Zeno effect}\label{sec:zeno}
\begin{figure}
\includegraphics[width=0.99\linewidth]{fig6.pdf}
\caption{Dissipative evolution of the density distribution for the parameters of Fig.~\ref{fig:density_evolution} showing a light cone structure bounded by an interaction dependent velocity $v$, indicated by the dashed line as a guide to the eye.}
\label{fig:lightcone}
\end{figure}
After the initial depletion of the central site, the tunneling processes between the different sites set in and lead to a reduction of the density in a region around the central site (Fig.~\ref{fig:ntime}). For strong dissipation, this is signaled by a much slower decay of the particle number (Fig.~\ref{fig:Ntottime}). The region which is depleted spreads linearly with time through the system and a particle current towards the central site is induced (Fig.~\ref{fig:lightcone}). The velocity $v$ of the spreading depends strongly on the interaction strength and the filling. A slower spreading is found for the attractively interacting system compared to the non-interacting system and a faster spreading for the repulsive case. A metastable state is formed within the cone of the spreading.
In the non-interacting case the spreading of the metastable state takes place by a reduction of the density. In contrast, the front of the spreading has very different nature for attractive or repulsive interactions. For the attractive case, a small density dip is followed by a high density peak which is propagating through the system. Behind this peak, the density is reduced and small density oscillations arise. For the repulsive interaction, the initially stronger density oscillations are perturbed, the density is reduced and in the metastable regime, strong density oscillations arise close to the lossy site.
\begin{figure}\label{fig:zeno}
\includegraphics[width=0.99\linewidth]{fig7.pdf}
\caption{Dependence of the time scale of the particle loss for intermediate times on the strength of the dissipative coupling $\gamma$ for different interaction strengths extracted from exponential fits of $\mathrm{e}^{-t/\tau}$ as shown in Fig.~\ref{fig:Ntot_dissipationdependence}.
Dotted (dashed) lines represent fits for an inverse (a linear) dependence of the time scale on the dissipation strength $\gamma$ for the corresponding interaction. The linear decrease of the time scale for large values of $\gamma$ is the key signature for the quantum Zeno effect. System and simulation parameters are the same as in Fig.~\ref{fig:Ntot_dissipationdependence}.}
\label{fig:zeno}
\end{figure}
The loss rate in the time-regime of the spreading is decreased for strong interaction and depends both on the dissipation and the interaction strength. In order to analyze this in more detail, we show in Fig.~\ref{fig:zeno} the time scale $\tau$ in this intermediate time regime before the light cone reaches the boundaries and finite size effects set in.
In Fig.~\ref{fig:zeno} the dependence of the dimensionless time scale $\tau J/\hbar$ on the dissipation $\hbar\gamma/J$ is plotted for different interaction strengths. In the small dissipation regime $\gamma \lesssim J/\hbar$ this time scale decreases with increasing $\gamma$ as $1/\gamma$. This is naively expected, as the total loss rate is mainly set by the amplitude of the dissipative coupling. In this regime, the tunneling dynamics is faster than the time-scale of the loss. Only a relatively weak dependence on the interaction strength exists and can be addressed to the different velocities of the spreading of the depletion of the density through the system. At larger repulsive interaction a slower time-scale is found.
In the strongly dissipative regime, the quantum Zeno effect occurs for all considered interaction strengths. The time scale of the loss increases again for higher dissipation proportional to $\gamma$. This increase of the time scale can be interpreted in the way that many measurements -- which correspond to the action of the dissipation -- freeze the quantum state of the system such that its evolution in time is prevented.
The occurrence of the quantum Zeno effect was previously predicted in an analogous setup with bosonic atoms \cite{BarmettlerKollath2011} and has been observed in Ref.~\cite{BarontiniOtt2013}. Our findings confirm also the findings for the non-interacting fermionic case and a different analytical approximation \cite{FroemlDiehl2019, FroemlDiehl2019b}.
We further see a strong dependence of the decay time scale on the interaction strength. Although all interactions exhibit the quantum Zeno effect, the decay time scale depends on the interaction strength. The attractive fermions decay slower compared to non-interacting fermions and the repulsive fermions decay faster.
\subsection{Properties of the metastable state}\label{sec:meta}
As we have seen at intermediate times, a long lived metastable state is reached in the surrounding of the lossy site. Some properties of this metastable state have been analyzed previously using different methods in Ref.~\cite{FroemlDiehl2019,FroemlDiehl2019b}.
This metastable state is very special in the sense that is does not resemble any ground or finite temperature state in equilibrium even considering the presence of a conservative potential at the site of the loss. This can already be seen from the real space density distribution. In the non-interacting case Fig.~\ref{fig:ntime}(b), a depletion of the density can be seen. The quasi-stationary value of the central site (Fig.~\ref{fig:n0}) shows a strong dependence on both the interaction and the dissipation strengths. Our results are roughly approximated by a linear behavior with the interaction strength for the considered parameter regime. A strong dissipative coupling leads to an almost complete emptying of the central site in the metastable state. Also a depletion around the central site becomes stable with the surprising occurrence of oscillations in space whose period does not vary with time. This is in contrast to the equilibrium properties of such a system. In equilibrium Friedel oscillations of the density are caused by an impurity which are of the form $n(x)\propto n_0\frac{\sin(2k_F x)}{x}$ where $n_{0}$ is the average density of the state and $k_F=\pi n_0$ is the Fermi wave vector. The oscillation period depends via the Fermi momentum linearly on the inverse of the filling $n_0$. Thus, a depletion in equilibrium would lead to an increased oscillation frequency. However, in the metastable state such a change of the oscillation frequency is not observed.
\begin{figure}
\includegraphics[width=0.99\linewidth]{fig8b.pdf}
\includegraphics[width=0.99\linewidth]{fig8.pdf}
\caption{Evolution of the momentum distribution represented by the Fourier transform of the density considering only $\tilde{L}=16$ sites on the right side of the defect in the center for $V/J=0.1$ (upper panel) and $V/J=1$ (lower panel). Results are presented for system size $L=63$, initial particle number $N(t=0)=31$, dissipation strength $\hbar \gamma/J=20$ and $2\cdot 10^3$ trajectory samples.}
\label{fig:nk}
\end{figure}
Let us turn to the evolution of the momentum distribution, which further underlines the non-equilibrium nature of this state. The prediction from the non-interacting system and from an approximate treatment of the interacting system is that the discontinuity at the Fermi momentum $k_F$ remains intact and a depletion of the lower momenta takes place \cite{FroemlDiehl2019,FroemlDiehl2019b}. This is another clear non-equilibrium signature of the metastable state, since at equilibrium $k_F$ behaves linearly with the filling. This is rationalized by the fact that the loss is local in real space, and therefore empties out all occupied momentum modes roughly homogeneously. For repulsive interactions the loss of particles at the Fermi momentum is suppressed due to the scattering off the Friedel oscillations. This effect was called the fluctuation induced quantum Zeno effect, since it is induced by the interaction. However, an important open question is whether the scattering induced by the interaction, once fully taken into account, will lead to a redistribution of the fermions towards a thermal state.
In Fig.~\ref{fig:nk} we show the momentum distribution $n_k$ corresponding only to sites from $[0,\tilde L - 1]$, where $\tilde L$ marks the extend of the metastable region right from the lossy site at the longest considered times. The momentum distribution is defined by $n_{k_m}=c^\dagger_{k_m} c^{\phantom{\dagger}}_{k_m}$, where $c_{k_m}$ denotes the Fourier transform of the fermionic operators
\begin{align}
&c_{k_m} = \frac{1}{\sqrt{\tilde L}} \sum_{j=1}^{\tilde L} c_j \mathrm{e}^{-i k_m j}, \\\notag
&\text{with } k_m = \frac{m\pi}{\tilde L+1}\text{ and } m \in \{1, 2, \ldots, \tilde L \}.
\end{align}
The distribution at time $t=0$ shows the Fermi step at the initial Fermi wave vector within the resolution of the finite system size. For small interaction strength of $V=0.1J$ and dissipation $\hbar \gamma= 20J$ fermions are lost at all momenta below the Fermi level. However, after a certain time a stable distribution arises for momenta around the initial Fermi momentum. In particular in this distribution, the Fermi step remains pinned at its initial value, whereas the occupation below the Fermi momentum is reduced. Additionally, a substructure just below the Fermi momentum sets in, which causes a small rise of the occupation around these values of the momentum.
For stronger interactions $V=J$, a similar evolution is seen. The stronger interaction can be recognized in a smearing out of the step in the initial momentum distribution. With time, the occupation of all momenta below the Fermi level is reduced and a rise is seen for the occupation just above the Fermi momentum. At intermediate times, the evolution reaches a long lived metastable state for momenta close to the initial Fermi surface. In this quasi-stationary occupation, the Fermi step is still located around its initial value. A slight increase of the occupation just below the Fermi level arises, which is more pronounced than in the case of weak interaction. Therefore, the metastable state keeps its non-equilibrium nature for intermediate times.
The slight rise close to the Fermi momentum could have its origin in the fluctuation induced quantum Zeno effect \cite{FroemlDiehl2019, FroemlDiehl2019b}. A more direct verification of the fluctuation induced quantum Zeno effect could be gained by determining transport properties of the metastable state, which goes beyond the scope of the current work.
\section{conclusion}\label{sec:conc}
We have investigated interacting spinless fermions subjected to a local loss by numerically exact calculations using MPS methods. Our numerically exact treatment for moderate system sizes complements the previous treatments relying on approximate methods for interacting fermions in the thermodynamic limit \cite{FroemlDiehl2019,FroemlDiehl2019b}. Initially an exponentially fast reduction of the density on the lossy site is found. In a second time regime the Zeno effect sets in for large dissipation strengths and causes a slow down of the losses. As typical for the Zeno regime, the time-scale for this second regime is inversely proportional to the dissipation strength and depends on the interaction strength. A metastable state is formed in a broad region around the impurity, which has no thermal counterpart. In particular, Friedel oscillations occur still at the period related to the initial Fermi momentum and the Fermi step in the momentum distribution remains at its initial position even though the density is strongly depleted. A slight increase of the momentum distribution close to the initial Fermi momentum is found, which hints at the existence of the fluctuation induced quantum Zeno effect expected for larger systems \cite{FroemlDiehl2019, FroemlDiehl2019b}. However, further investigations of transport properties are required to strengthen this support, and to identify the fluctuation induced quantum Zeno effect.
\section{Acknowledgments}
We thank H. Fr\"oml, A. Chiocchetta, and H.~Ott for fruitful discussions. We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 277625399 - TRR 185 project B3 and project number 277146847 - project C04,C05, under the DFG Collaborative Research Center (CRC) 1238 and under Germany's Excellence Strategy – Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 – 390534769 and the European Research Council (ERC) under the Horizon 2020 research and innovation programme, grant agreement No. 648166 (Phonton) and grant agreement No. 647434 (DOQS).
|
2,877,628,090,168 | arxiv |
\section{Introduction and informal summary of results}
Recently, there has been much interest in the performance of asymptotically infinite-width neural nets which afford efficient training, as optimization entails perturbing the weights minimally away from a Gaussian initialization --- the aptly named ``lazy training'' regime \citep{chizat2019lazy}. The focus of analysis in these works is the so-called \emph{neural tangent kernel} \citep{jacot2018NTK}, which arises from the linearization of the neural net around the initial values of the weights, and the key observation is that the trajectory of SGD on this linearized model can be shown to closely track that of SGD on the (appropriately rescaled) original model \citep{jacot2018NTK,allenzhu2019over,chizat2019lazy}.
It is useful in the setting of infinitely wide nets to reason about the evolution of weights during training as a gradient flow in the space of probability measures over the weights, as this so-called \textit{distributional dynamics} can be captured by a certain nonlinear PDE of McKean--Vlasov type \citep{kolokoltsov2010nonlinear}. This \textit{mean-field description} preserves the essential features of optimization landscape that are insensitive to the number of neurons \citep{chizat_bach18,mei2018meanfield}. A finite-size network in this case can then be conceived of as providing an empirical distribution over the weights, and, under mild regularity assumptions on the data and on the activation function, we can transfer results from the continuous-time, infinite-width setting to the discrete-time, finite-width setting \citep{mei2018meanfield,rotskoff2018meanfield,mei2019meanfield,sirignano2020meanfield_CLT,sirignano2020meanfield_LLN}. In particular, \cite{mei2019meanfield} have provided a mean-field perspective on the neural tangent kernel by phrasing it as a linear approximation to the nonlinear distributional dynamics in a specific short-time limit.
In this paper, we provide an alternative view of lazy training through the lens of mean-field theory, connecting it to \textit{entropic regularization} in the space of probability measures over the weights. Our analysis of this mean-field limit consists of two complementary parts: a static formulation, where we seek to minimize a certain free-energy objective; and a dynamic formulation, which elucidates the evolution towards the optimal distribution by a controlled perturbation of a suitably rescaled Brownian motion.
For the first (static) part, the free energy functional is a linear combination of the mean-field risk of a distribution and its Kullback--Leibler divergence from an isotropic Gaussian. The free energy, which has appeared in a different guise in the work of \citet{mei2018meanfield}, is parametrized by the variance $\tau$ of this Gaussian prior and by the regularization (inverse temperature) parameter $\beta$ that controls the trade-off between these two terms. We show that, under appropriate regularity conditions, the free energy has a unique minimizer and provide explicit upper bounds on both the KL divergence and the $L^2$ Wasserstein distance between this minimizer and the Gaussian prior.
The second (dynamic) part entails setting up a finite-time optimal stochastic control problem for a stochastic differential equation (SDE) on the space of weights, where the terminal cost function accounts for both $\beta$ (scaling) and $\tau$ (the diffusion parameter of the controlled SDE). The control law that achieves minimum quadratic running cost subject to a desired terminal density corresponds to the F\"ollmer drift \citep{follmer1985reversal,daipra1991reciprocal,lehec2013entropy,eldan2018diffusion} that also solves the entropic optimal transport problem for the optimal measure. Working with the F\"ollmer drift here acquires the additional complication that it depends on the law of the process that minimizes the free energy; due to this nonlinear dependence of the terminal cost on the target probability law, we have to employ the machinery of \textit{controlled McKean--Vlasov dynamics} \citep{carmona2015MKV}. The resulting distributional dynamics consists of two coupled PDEs, the forward (Fokker--Planck) equation that governs the evolution of the probability density of the weights and the backward (Hamilton--Jacobi--Bellman) equation that governs the evolution of the value function of the McKean--Vlasov control problem. By contrast, the distributional dynamics that is analyzed in previous works on the mean-field theory of neural nets involves only the forward (Fokker--Planck) PDE over an infinite time horizon.
Finally, we examine the extent to which SGD can approximate the optimal mean-field dynamics. We begin by viewing it as a discrete-time greedy approximation to the F\"ollmer drift. In the mean-field limit, this greedy approximation is furnished by the type of nonlinear dynamics analyzed by \citet{mei2019meanfield}, which makes for a ready comparison to SGD with $N$ neurons. In particular, we show that, under the usual assumptions, with probability at least $1-\delta$, SGD with step size $\eta$ tracks the optimal mean-field dynamics over $t\in [0,1]$ to within
\begin{align*}
O\left(\frac{\kappa}{N} + \kappa \sqrt{\frac{1}{N}\log \frac{N}{\delta}} + \kappa\beta + \kappa(1 + \sqrt{\eta}\beta)(1+\beta)e^{\kappa\beta} \left( \beta + \sqrt{\tau d \log \frac{Nd}{\delta \eta}}\right) \right),
\end{align*}
where $\kappa$ is polynomial in the Lipschitz parameters of the target function and the activation function.
Vis-\`a-vis the tunable parameters $\beta$ and $\tau$, whose ratio and magnitude both play a role in the optimal distribution, we can identify different regimes of interest. As long as $\beta \gg \tau$, the entropic regularization has little contribution to optimization, and the process is comparable to a (noise-free) gradient flow; indeed, it corresponds to the goal of optimizing with a slight bias towards low-complexity solutions. On the other hand, choosing $\beta \sim \sqrt{\tau d} = \varepsilon$, where $\varepsilon$ is the desired accuracy of approximation, corresponds to a strong Gaussian prior and thus militates toward having the initial Brownian motion primarily drive the dynamics, as is witnessed with the phenomenon of ``lazy training." The scaling properties of Brownian motion also confer another interpretation to the choice of $\tau$: the optimization process can be scaled down (w.r.t.\ drift and diffusion coefficients) by $\tau$, and the time interval rescaled from $[0,1]$ to $[0,\tau]$. In this case, $\tau$ corresponds to the duration of the optimization process, and choosing $\tau$ to be small equivalently forces solutions to be minimally perturbed away from the Gaussian prior.
\subsection{Related work}
The amenability of the neural tangent kernel (NTK) regime to the analysis of neural network optimization was brought to attention by the work of \cite{allenzhu2019over}, which showed that massively overparametrized shallow networks, i.e., those whose width approaches infinity, can efficiently learn functions represented by smaller networks; and that SGD finds these solutions in overparametrized networks in polynomial time and sample complexity. These solutions were observed to be close to the Gaussian initialization. \cite{ji2020transport} considered the problem of universal approximation by shallow ReLU NTK models; in doing so, they introduced the idea of constructing both finite- and infinite-width approximations to a given target function by applying a \textit{transport map} to the weights sampled from a Gaussian prior.
The mean-field framework closest to the one we explore is due to \cite{mei2019meanfield}, who phrased the learning process as a discrete-time approximation to the continuous-time distributional dynamics and dervied a dimension-free bound on the approximation error. The work of \cite{chizat_bach18} has cast neural network optimization as particle dynamics and employed the limiting mean-field view to circumvent difficulties in the analysis of highly non-convex functions; subsequently, \cite{chizat2019lazy} established theoretically that lazy training actually occurs. Our work brings these perspectives together by means of a control-theoretic dynamic formulation and provides an alternative perspective on lazy training through the lens of approximate entropic optimal transport.
\subsection{Problem setup}
We consider the problem of approximating a target function $f : \cX \to \mathbb{R}$ by a two-layer neural net with $N$ hidden-layer neurons. Here, $\cX$ is a Borel subset of $\mathbb{R}^p$, and the neural nets will take the form
\begin{align*}
\hat{f}_N(x; \bd{w}) = \frac{1}{N}\sum^N_{i=1} \act(x; w^i),
\end{align*}
where $x \in \mathbb{R}^p$ is the input (or feature) vector, $\bd{w} = (w^1,\ldots,w^N)$ is the $N$-tuple of weights $w^i \in \mathbb{R}^d$, and $\act : \mathbb{R}^p \times \mathbb{R}^d \to \mathbb{R}$ is an activation function. We will measure the accuracy of approximation using $L^2(\pi)$ risk, where $\pi$ is a fixed Borel probability measure supported on $\cX$:
\begin{align}\label{eq:L2risk}
R_N(\bd{w}) := \| f - \hat{f}_N(\cdot; \bd{w}) \|^2_{L^2(\pi)} = \int_\cX \pi(\dif x) |f(x)-\hat{f}_N(x;\bd{w})|^2.
\end{align}
As usual, it is expedient to express the risk \eqref{eq:L2risk} as
\begin{align}\label{eq:L2risk_alt}
R_N(\bd{w}) = R_0 + \frac{2}{N}\sum^N_{i=1} \tilde{f}(w^i) + \frac{1}{N^2}\sum^N_{i=1}\sum^N_{j=1} K(w^i,w^j),
\end{align}
where $R_0 := \mathbf{E}_\pi[|f(X)|^2]$, $\tilde{f}(w) := - \mathbf{E}_\pi[f(X)\act(X;w)]$, and $K(w,w') := \mathbf{E}_\pi[\act(X;w)\act(X;w')]$. The alternative form \eqref{eq:L2risk_alt} of the $L^2$ risk makes it apparent that it depends only on the empirical distribution of the weights (and, in particular, is invariant under permutations of the neurons). Moreover, if we define the mapping $\hat{f} : \cX \times \fP(\mathbb{R}^d) \to \mathbb{R}$ by
\begin{align*}
\hat{f}(x; \mu) := \int_{\mathbb{R}^d} \act(x; w)\mu(\dif w),
\end{align*}
and the associated $L^2(\pi)$ risk
\begin{align*}
R(\mu) := \| f - \hat{f}(\cdot;\mu)\|^2_{L^2(\pi)} = R_0 + 2\int_{\mathbb{R}^d} \tilde{f}\dif \mu + \int_{\mathbb{R}^d \times \mathbb{R}^d} K \dif\,(\mu \otimes \mu),
\end{align*}
then $\hat{f}_N(x;\bd{w}) = \hat{f}(x;\hat{\mu}_{\bd{w}})$ and $R_N(\bd{w}) = R(\hat{\mu}_{\bd{w}})$, where $\hat{\mu}_{\bd{w}} := \frac{1}{N}\sum^N_{i=1} \delta_{w^i}$ is the empirical distribution of $\bd{w}$. This lifting from finite populations of neurons to continual ensembles is the essence of the mean-field theory of neural nets. Our focus in this work will be on managing the trade-off between the risk $R(\mu)$ and the relative entropy (or Kullback--Leibler divergence) between $\mu$ and an isotropic Gaussian prior.
\subsection{Notation}
For $\tau > 0$, we will denote by $\gamma_\tau$ the centered Gaussian measure on $\mathbb{R}^d$ with covariance matrix $\tau I_d$. The space of Borel probability measures on $\mathbb{R}^d$ will be denoted by $\fP(\mathbb{R}^d)$, and $\fP_p(\mathbb{R}^d)$, for $p \ge 1$, will stand for the set of $\mu \in \fP(\mathbb{R}^d)$ with finite $p$th moment. The relative entropy (or Kullback--Leibler divergence) between $\mu,\nu \in \fP(\mathbb{R}^d)$ will be denoted by $D(\mu \| \nu)$. The $L^p$ Wasserstein distance between $\mu,\nu \in \fP_p(\mathbb{R}^d)$ is
\begin{align*}
\cW_p(\mu,\nu) := \left(\inf_{W \sim \mu, V \sim \nu} \mathbf{E} [\| W - V \|^p] \right)^{1/p},
\end{align*}
where the infimum is over all random elements $(W,V)$ of $\mathbb{R}^d \times \mathbb{R}^d$ with marginals $\mu$ and $\nu$, and $\| \cdot \|$ denotes the Euclidean $(\ell_2)$ norm on $\mathbb{R}^d$. Other notation will be introduced in the sequel as needed.
\section{Entropy-regularized risk in the mean-field limit}
As stated in the Introduction, we are interested in trading off the risk $R(\mu)$ and the relative entropy between $\mu$ and a Gaussian prior $\gamma_\tau$. In this section, we will formalize this trade-off via two complementary formulations: a static one, under which the optimal measure arises as the global minimizer of a suitable free energy functional, and a dynamic one, under which the optimal measure emerges as the solution of a certain optimal stochastic control problem.
\subsection{The static formulation}
Let us consider minimizing, over $\fP_2(\mathbb{R}^d)$, the following \textit{free energy} functional:
\begin{align}\label{eq:FE}
\FE_{\beta,\tau}(\mu) := \frac{1}{2}R(\mu) + \frac{\tau}{\beta} D(\mu \| \gamma_\tau),
\end{align}
where $\tau > 0$ is the variance of the isotropic Gaussian prior and the inverse temperature parameter $\beta > 0$ controls the strength of the entropic regularization term in \eqref{eq:FE}. We impose the following assumptions, which were also made by \citet{mei2019meanfield}:
\begin{assumption}\label{as:bounded} The target function $f : \mathbb{R}^p \to \mathbb{R}$ and the activation function $\act : \mathbb{R}^p \times \mathbb{R}^d \to \mathbb{R}$ are bounded: $\| f \|_\infty, \| \act \|_\infty \le \kappa_1$. Moreover, for each $w \in \mathbb{R}^d$ the gradient $\nabla_w \act(X;w)$ is $\kappa_1$-subgaussian when $X \sim \pi$.
\end{assumption}
\begin{assumption}\label{as:Lipschitz} The functions $\tilde{f}$ and $K$ are differentiable and Lipschitz-continuous, with Lipschitz-continuous gradients: $\| \nabla \tilde{f}(w)\| \le \kappa_2$, $\| \nabla K(w,\tilde{w})\| \le \kappa_2$, $\| \nabla \tilde{f}(w) - \nabla \tilde{f}(w') \| \le \kappa_2 \| w-w' \|$, $\| \nabla K(w,\tilde{w}) - \nabla K(w',\tilde{w}') \| \le \kappa_2 \|(w,\tilde{w})-(w',\tilde{w}') \|$.
\end{assumption}
Throughout the paper, we will use $\kappa$ to denote a generic quantity that grows like $O({\rm poly}(\max\{\kappa_1,\kappa_2\}))$ and $c$ to denote a generic absolute constant. The values of $\kappa$ and $c$ may change from line to line.
\begin{theorem}\label{thm:static} Under Assumptions~\ref{as:bounded} and \ref{as:Lipschitz}, the free energy \eqref{eq:FE} admits a unique minimizer $\mu^\star = \mu^\star_{\beta,\tau}$, such that the following hold:
\begin{enumerate}
\item $\mu^\star$ is absolutely continuous w.r.t.\ $\gamma_\tau$ and satisfies the {\em Boltzmann fixed-point condition}
\begin{align}\label{eq:Boltzmann_FP}
\mu^\star(\dif w) = \frac{1}{Z^\star}\exp\left(-\frac{\beta}{\tau}\Psi(w;\mu^\star)\right)\gamma_\tau(\dif w),
\end{align}
where the potential $\Psi : \mathbb{R}^d \times \fP_2(\mathbb{R}^d) \to \mathbb{R}$ is given by
\begin{align*}
\Psi(w; \mu) := \tilde{f}(w) + \int_{\mathbb{R}^d} K(w,\tilde{w})\mu(\dif \tilde{w})
\end{align*}
and $Z^\star = Z(\beta,\tau; \mu^\star)$ is the normalization constant.
\item The risk of $\mu^\star$ is bounded by
\begin{align}\label{eq:FE_upper_bound}
R(\mu^\star) \le 2\FE_{\beta,\tau}(\mu^\star) \le \inf_{\mu \in \fP_2(\mathbb{R}^d)} \left[R(\mu) + \frac{1}{\beta}M^2_2(\mu)\right] + \frac{\tau d}{\beta} \log(2\kappa\beta+1),
\end{align}
where $M^2_2(\mu) := \int_{\mathbb{R}^d}\|w\|^2 \mu(\dif w)$ is the second moment of $\mu$.
\item The relative entropy and the squared $L^2$ Wasserstein distance between $\mu^\star$ and the Gaussian prior $\gamma_\tau$ are bounded by
\begin{align}\label{eq:entropy_W2}
D(\mu^\star \| \gamma_\tau) \le \frac{\kappa \beta^2}{\tau} \qquad \text{and} \qquad \cW^2_2(\mu^\star,\gamma_\tau) \le \kappa\beta^2.
\end{align}
\item If $f = \hat{f}(\cdot;\mu^\circ)$ for some $\mu^\circ \in \fP_2(\mathbb{R}^d)$, then
\begin{align}\label{eq:realizable}
R(\mu^\star) \le \frac{2\tau}{\beta}D(\mu^\circ\|\gamma_\tau).
\end{align}
\end{enumerate}
\end{theorem}
\begin{remark}
One way to motivate the scaling of the entropy term in \eqref{eq:FE} by $\tau$ is to view \eqref{eq:FE} as an entropic relaxation of the Wasserstein-regularized risk
\begin{align}\label{eq:WFE}
\bar{\FE}_{\beta,\tau}(\mu) := \frac{1}{2}R(\mu) + \frac{1}{2\beta}\cW^2_2(\mu,\gamma_\tau).
\end{align}
Indeed, the inequality $\bar{\FE}_{\beta,\tau}(\mu) \le \FE_{\beta,\tau}(\mu)$ follows from Talagrand's Gaussian entropy-transportation inequality $2\cW^2_2(\mu,\gamma_\tau) \le \tau D(\mu \| \gamma_\tau)$ \citep{Bakry_Gentil_Ledoux_book}. While the quantity \eqref{eq:WFE} is perhaps more meaningful from the practical standpoint, the entropic regularization term in \eqref{eq:FE} is easier to work with.
\end{remark}
\begin{remark}\label{rem:MMN}
The free energy \eqref{eq:FE} can be expressed in terms of another free-energy functional introduced by \citet{mei2018meanfield}:
\begin{align*}
\FE_{\beta,\tau}(\mu) = \FE^{\sM\sM\sN}_{\beta/\tau,1/\beta}(\mu) + \frac{\tau d}{2\beta}\log(2\pi\tau),
\end{align*}
where
\begin{align*}
\FE^{\sM\sM\sN}_{\beta,\lambda}(\mu) := \frac{1}{2} \left[ R(\mu) + \lambda M^2_2(\mu)\right] - \frac{1}{\beta} h(\mu),
\end{align*}
and $h(\mu) := -\int_{\mathbb{R}^d} \dif \mu \log \frac{\dif\mu}{\dif\lambda}$ is the differential entropy of $\mu$ \citep{cover2006infotheory}, i.e., the (negative) relative entropy of $\mu$ w.r.t.\ the $d$-dimensional Lebesgue measure $\lambda$. (Without loss of generality, we may assume that the density of $\mu$ w.r.t.\ $\lambda$ exists; otherwise, we can convolve $\mu$ with the Gaussian measure $\gamma_\varepsilon$ for a suitably small $\varepsilon > 0$.) Note that $\FE_{\beta,\tau}(\cdot)$ is always nonnegative, while $F^{\sM\sM\sN}_{\beta,\lambda}(\cdot)$ may take negative values.
\end{remark}
\begin{remark} The Boltzmann fixed-point equation \eqref{eq:Boltzmann_FP} and the risk bound \eqref{eq:FE_upper_bound} have also been derived by \citet{mei2018meanfield} in the context of minimizing $\FE^{\sM\sM\sN}_{\beta,\lambda}$ over $\fP_2(\mathbb{R}^d)$, cf.\ Remark~\ref{rem:MMN}.
\end{remark}
From the risk bound \eqref{eq:FE_upper_bound}, it readily follows that
\begin{align*}
\lim_{\beta \to \infty} \inf_{\mu \in \fP_2(\mathbb{R}^d)} \FE_{\beta,\tau}(\mu) = \frac{1}{2}\inf_{\mu \in \fP_2(\mathbb{R}^d)} R(\mu)
\end{align*}
for any $\tau$ that is of order $o(\beta)$. However, a more intriguing message of Theorem~\ref{thm:static} is that it is also meaningful to consider the regime where both $\beta$ and $\tau$ are small, as long as the ratio $\beta/\tau$ is suitably large. For instance, let us pick a suitably small $\varepsilon > 0$ and take $\tau = \varepsilon^2/d$, $\beta = \sqrt{\tau d} = \varepsilon$. Then $\beta/\tau = d/\varepsilon$, and we see from \eqref{eq:entropy_W2} that the corresponding optimal distribution $\mu^\star$ satisfies
\begin{align*}
D(\mu^\star \| \gamma_\tau) \le \frac{\kappa \beta^2}{\tau} = \kappa d \qquad \text{and} \qquad \cW^2_2(\mu^\star,\gamma_\tau) \le \kappa \varepsilon^2.
\end{align*}
Moreover, in the realizable case, i.e., when the target function $f$ is equal to $\hat{f}(\cdot; \mu^\circ)$ for some $\mu^\circ \in \fP_2(\mathbb{R}^d)$, Eq.~\eqref{eq:realizable} gives
\begin{align*}
R(\mu^\star) \le \frac{2\varepsilon}{d} D(\mu^\circ \| \gamma_\tau),
\end{align*}
so $R(\mu^\star)$ will be on the order of $\varepsilon$ as soon as $D(\mu^\circ \| \gamma_\tau) = O(d)$. This regime can be viewed as a mean-field counterpart of the notion that, for certain target functions $f$, with high probability there exist good neural-net approximations near a random Gaussian initialization \citep{allenzhu2019over}. Moreover, such a good approximation can be obtained from the random Gaussian initialization by applying a transport mapping to the weights \citep{ji2020transport}. The following corollary of Theorem~\ref{thm:static} gives a precise statement:
\begin{corollary}\label{cor:transport} Let $\beta, \tau > 0$ be given, where $0 < \beta < \frac{1}{c\kappa_2}$. Then there exists a Lipschitz-continuous transportation mapping $T : \mathbb{R}^d \to \mathbb{R}^d$ such that all of the following holds with probability at least $1-\delta$ for a tuple $\bd{W} = (W^1,\ldots,W^N)$ of i.i.d.\ draws from $\gamma_\tau$:
\begin{enumerate}
\item The neural net with transported weights $\hat{f}_N(\cdot; T(\bd{W})) := \frac{1}{N}\sum^N_{i=1}\act(\cdot; T(W^i))$ satisfies
\begin{align}\label{eq:transport_risk}
\|f - \hat{f}_N(\cdot; T(\bd{W}))\|_{L^2(\pi)} \le \| f - \hat{f}(\cdot; \mu^\star_{\beta,\tau})\|_{L^2(\pi)} + \kappa \sqrt{\frac{\log(1/\delta)}{N}};
\end{align}
\item The transported weights $T(W^i)$ are uniformly close to the i.i.d.\ Gaussian weights $W^i$:
\begin{align}\label{eq:transport_distance}
\max_{i \in [N]} \| T(W^i) - W^i \| \le \kappa \beta + \kappa \sqrt{\tau(\log N + \log (1/\delta))};
\end{align}
\item The transported neural net $\hat{f}_N(\cdot; T(\bd{W}))$ and the random Gaussian neural net $\hat{f}_N(\cdot; \bd{W})$ are close in $L^2(\pi)$ norm:
\begin{align}\label{eq:transport_vs_init}
\left\| \hat{f}_N(\cdot; T(\bd{W})) - \hat{f}_N(\cdot; \bd{W}) \right\|^2_{L^2(\pi)} \le \kappa\beta+ \kappa \sqrt{\frac{\tau\log(1/\delta)}{N}}.
\end{align}
\end{enumerate}
\end{corollary}
\begin{remark} Note that the bounds in Corollary~\ref{cor:transport} scale with $\beta$ and $\sqrt{\tau}$. The choice of $\beta = \varepsilon$ and $\tau d = \varepsilon^2$ for a sufficiently small $\varepsilon > 0$ suffices to guarantee that the transported weights are, with high probability, $\varepsilon$-close to the randomly sampled Gaussian weights, while at the same time the ratio $\frac{\tau}{\beta} = \frac{\varepsilon}{d}$ moderates the effect of entropic regularization in \eqref{eq:FE}.
\end{remark}
\begin{remark} The mapping $T$ is, in fact, the optimal (Brenier--McCann) transportation mapping that pushes $\gamma_\tau$ forward to $\mu^\star$ and satisfies $\cW^2_2(\mu^\star,\gamma_\tau) = \mathbf{E}[\|T(W)-W\|^2]$ for $W \sim \gamma_\tau$. In particular, it has the form $T(w) = \nabla \varphi(w)$ for some convex function $\varphi : \mathbb{R}^d \to \mathbb{R}$ \citep{Villani_topics}.
\end{remark}
\subsection{The dynamic formulation}
We now show that we can introduce a stochastic dynamics in the space of weights that leads to $\mu^\star$ and is also optimal in a well-defined sense. Specifically, we will construct a flow of measures $\bd{\mu}^\star = \{\mu^\star_t\}_{t \in [0,1]}$ with densities $\rho^\star_t$, such that: (i) $\mu^\star_0 = \delta_0$ (the Dirac measure concentrated at the origin), (ii) $\mu^\star_1 = \mu^\star$ (the unique minimizer of the free energy $\FE_{\beta,\tau}$), and (iii) the evolution of $\bd{\mu}^\star$ is governed by a system of two coupled nonlinear PDEs,
\begin{subequations}
\begin{align}
\partial_t \rho^\star_t(w) &= \nabla_w \cdot \left(\rho^\star_t(w) \nabla_w V^\star(w,t)\right) + \frac{\tau}{2} \Delta_w \rho^\star_t(w) \label{eq:FPE_optimal}\\
\partial_t V^\star(w,t) &= - \frac{\tau}{2}\Delta_w V^\star(w,t) + \frac{1}{2} \| \nabla_w V^\star(w,t)\|^2 \label{eq:HJB_optimal}
\end{align}
\end{subequations}
for $w \in \mathbb{R}^d$ and $t \in [0,1]$, where \eqref{eq:FPE_optimal} has initial condition $\rho^\star_0(w) = \delta(w)$ (the Dirac delta function) and \eqref{eq:HJB_optimal} has terminal condition $V^\star(w,1) = \beta\big(R_0 + \int_{\mathbb{R}^d}\tilde{f}\dif\mu^\star_1 + \Psi(w; \mu^\star_1)\big)$. Here, the forward PDE \eqref{eq:FPE_optimal} is the Fokker--Planck equation, while the backward PDE \eqref{eq:HJB_optimal} is the Hamilton--Jacobi--Bellman equation.
\sloppypar Let $(\Omega,\cF,\{\cF_t\},\mathbf{P})$ be a probability space with a complete and right-continuous filtration $\{\cF_t\}_{t \ge 0}$, and let $\{B_t\}$ be a standard $d$-dimensional Brownian motion adapted to $\{\cF_t\}$. Given an admissible \textit{control}, i.e., any progressively measurable $\mathbb{R}^d$-valued process $\bd{u} = \{u_t\}_{t \ge 0}$ satisfying
\begin{align*}
\mathbf{E}\left[\int^1_0 \|u_t\|^2\dif t\right] < \infty,
\end{align*}
we consider the It\^o SDE
\begin{align}\label{eq:MKV_dynamics}
\dif W_t = u_t \dif t + \sqrt{\tau} \dif B_t, \qquad W_0 \equiv 0;\, t \in [0,1]
\end{align}
We wish to choose $\bd{u}$ to minimize the expected cost
\begin{align}\label{eq:MKV_cost}
J_{\beta,\tau}(\bd{u}) := \mathbf{E}\left[\frac{1}{2}\int^1_0 \|u_t\|^2 \dif t\right] + \frac{\beta}{2}R(\mu^{\bd{u}}_1),
\end{align}
where $\mu^{\bd{u}}_t$ is the probability law of $W_t$ for $t \in [0,1]$. To motivate this dynamic optimization problem, consider first the case of zero drift: $u_t \equiv 0$ for all $t \in [0,1]$. Then $W$ is simply the rescaled Brownian motion $\sqrt{\tau}B$, so that in particular $W_1 = \sqrt{\tau}B_1 \sim \cN(0,\tau I_d)$, and the resulting expected cost is proportional to the risk of the Gaussian prior $\gamma_\tau$. By adding a nonzero drift $\bd{u}$, we perturb the Brownian path $\sqrt{\tau}B_{[0,1]}$, and the expected cost \eqref{eq:MKV_cost} captures the trade-off between the strength of this perturbation and the $L^2(\pi)$ risk of $\hat{f}(\cdot; \mu^{\bd{u}}_1)$. In fact, we can think of the drift as inducing a transport mapping that acts on the entire Brownian path $\{\sqrt{\tau}B_t\}_{t \in [0,1]}$ --- integrating \eqref{eq:MKV_dynamics} gives
\begin{align}\label{eq:Tu_transport}
W^{\bd{u}}_1 &= \int^1_0 u_t \dif t + \sqrt{\tau}B_1 =: T^{\bd{u}}(\sqrt{\tau}B_{[0,1]}),
\end{align}
so the control cost in \eqref{eq:MKV_cost} penalizes those transport maps that take $\sqrt{\tau}B_{[0,1]}$ to a random vector far from $\sqrt{\tau}B_1$.
\begin{remark} A word of caution is in order here: While the transport map in Corollary~\ref{cor:transport} is the Brenier--McCann optimal transportation map from $\gamma_\tau$ to $\mu^\star$, the transport maps $T^{\bd{u}}$ defined in \eqref{eq:Tu_transport} map Brownian paths to random vectors in $\mathbb{R}^d$ and are not related to optimal transportation in the $L^2$ Wasserstein sense. As we will see later, however, the problem of minimizing \eqref{eq:MKV_cost} subject to \eqref{eq:MKV_dynamics} is closely related to the so-called {\em Schr\"odinger bridge} problem \citep{daipra1991reciprocal,lehec2013entropy,chen2016bridges,eldan2018diffusion}, a form of entropic optimal transportation \citep{reich2019assimilation}.
\end{remark}
The problem of minimizing \eqref{eq:MKV_cost} subject to \eqref{eq:MKV_dynamics} is an optimal control problem, where the first term on the right-hand side of \eqref{eq:MKV_cost} is the \textit{control cost}, while the second term is the \textit{terminal cost}. There is, however, a twist: in contrast to the standard stochastic control framework \citep{fleming1975control}, where the terminal cost is of the form $\mathbf{E}[g(W_1)]$ for some measurable function $g : \mathbb{R}^d \to \mathbb{R}$ and is therefore \textit{linear} in $\mu^{\bd{u}}_1$, here the terminal cost is a \textit{nonlinear functional} of the probability law $\mu^{\bd{u}}_1$ of $W_1$. Indeed, as elaborated in the proof of Theorem~\ref{thm:dynamic} below, we can express the risk $R(\mu)$ as $\mathbf{E}_\mu[\bar{c}(W;\mu)]$ for a certain deterministic function $\bar{c} : \mathbb{R}^d \times \fP_2(\mathbb{R}^d) \to \mathbb{R}$. In other words, the terminal cost is an expectation of a function of both the terminal state $W_1$ and the probability law $\mu^{\bd{u}}_1$ of the terminal state. Thus, the problem of minimizing the cost \eqref{eq:MKV_cost} subject to \eqref{eq:MKV_dynamics} is an instance of \textit{controlled McKean--Vlasov dynamics} in the sense of \citet{carmona2015MKV}.\footnote{As defined by \citet{carmona2015MKV}, a stochastic control problem is of McKean--Vlasov type if the system dynamics, the control cost, or the terminal cost depend not only on the state and on the control, but also on the probability law of the state.} In general, solving optimal control problems of McKean--Vlasov type is a fairly intricate affair that rests on solving a coupled system of forward (Fokker--Planck) and backward (Hamilton--Jacobi--Bellman) PDEs; a detailed treatment can be found in \citet{carmona2018vol1}. Somewhat surprisingly, though, in this case we can obtain an exact characterization for both the optimal control law and the optimal cost in terms of the minimizer $\mu^\star$ of the free energy $F_{\beta,\tau}$, and the Boltzmann fixed-point condition \eqref{eq:Boltzmann_FP} plays the key role in guaranteeing that the corresponding forward-backward system admits a solution.
\begin{theorem}\label{thm:dynamic} Let $\mu^\star$ be the (unique) minimizer of the free energy $\FE_{\beta,\tau}(\mu)$. Then the optimal controlled process solves the It\^o SDE
\begin{align}\label{eq:optimal_SDE}
\dif W_t = - \nabla_w V^\star(W_t,t)\dif t + \sqrt{\tau}\dif B_t, \qquad t \in [0,1];\, \, W_0 = 0,
\end{align}
where
\begin{align}\label{eq:value_function}
V^\star(w,t) := -\tau\log \mathbf{E}\Bigg[\exp\left(-\frac{\beta}{\tau} \Psi(B_\tau;\mu^\star)\right) \Bigg| B_{\tau t} = w\Bigg].
\end{align}
Moreover, under \eqref{eq:optimal_SDE}, $W_1$ is distributed according to $\mu^\star$, and the above optimal control achieves
\begin{align*}
\inf_{\bd{u}} J_{\beta,\tau}(\bd{u}) = \beta \FE_{\beta,\tau}(\mu^\star).
\end{align*}
\end{theorem}
\begin{remark} The control law in Eqs.~\eqref{eq:optimal_SDE}--\eqref{eq:value_function} is also optimal in the following sense \citep{daipra1991reciprocal,lehec2013entropy,eldan2018diffusion,eldan2018gausswidth}: Let $\fU(\mu^\star)$ denote the subset of all admissible drifts that obey the terminal condition $W^{\bd{u}}_1 \sim \mu^\star$. Then
\begin{align}\label{eq:Schrodinger}
\inf_{\bd{u} \in \fU(\mu^\star)}\mathbf{E}\left[\frac{1}{2}\int^1_0 \|u_t\|^2 \dif t \right] = \tau D(\mu^\star \| \gamma_\tau),
\end{align}
and the infimum is achieved by the drift in \eqref{eq:optimal_SDE}. The optimization problem in \eqref{eq:Schrodinger} is known as the {\em Schr\"odinger bridge problem} (see \citet{chen2016bridges} and references therein), and the optimal drift is also referred to as the {\em F\"ollmer drift} \citep{follmer1985reversal}.
\end{remark}
\noindent It is instructive to take a closer look at the structure of the F\"ollmer drift in \eqref{eq:optimal_SDE}. To start, a simple computation gives
\begin{align}
-\nabla_w V^\star(w,t) = -\frac{\beta \int_{\mathbb{R}^d} \nabla \Psi(v;\mu^\star)\exp\left(-\frac{\|v-w\|^2}{2\tau(1-t)}-\frac{\beta}{\tau}\Psi(v;\mu^\star)\right)\dif v}
{\int_{\mathbb{R}^d}\exp\left(-\frac{\|v-w\|^2}{2\tau(1-t)}-\frac{\beta}{\tau}\Psi(v;\mu^\star)\right)\dif v}.\label{eq:Follmer_explicit}
\end{align}
If we define a family of measures $\{ Q_{w,t} : w \in \mathbb{R}^d, t \in [0,1]\} \subseteq \fP(\mathbb{R}^d)$ by
\begin{align}\label{eq:Q_measures}
Q_{w,t}(A) &:= \frac{\int_{A} \exp\left(-\frac{\|v-w\|^2}{2\tau(1-t)}-\frac{\beta}{\tau}\Psi(v; \mu^\star)\right)\dif v}{\int_{\mathbb{R}^d} \exp\left(-\frac{\|v-w\|^2}{2\tau(1-t)}-\frac{\beta}{\tau}\Psi(v; \mu^\star)\right)\dif v},
\end{align}
where $A$ ranges over all Borel subsets of $\mathbb{R}^d$, then we can write \eqref{eq:Follmer_explicit} more succinctly as
\begin{align}\label{eq:Follmer_Gibbs}
-\nabla_w V^\star(w,t) = -\beta \int_{\mathbb{R}^d}\nabla \Psi(v;\mu^\star) Q_{w,t}(\dif v),
\end{align}
An inspection of \eqref{eq:Q_measures} reveals that the flow $t \mapsto Q_{0,t}$ interpolates between $\mu^\star$ at $t=0$ and $\delta_0$ at $t=1$, so the measures $Q_{0,t}$ get increasingly concentrated as $t$ approaches $1$. Moreover, it can be shown that the flow of random measures $\{Q_{W_t,t}\}_{t \in [0,1]}$ along the trajectory of \eqref{eq:optimal_SDE} satisfies
\begin{align}
Q_{W_t,t}(\cdot) = \mathbf{P}[W_1 \in \cdot|\cF_t] = \mathbf{P}[W_1 \in \cdot|W_t]
\end{align}
almost surely \citep[Lemma~11]{eldan2018gausswidth}. Using these facts, one can readily verify that the drift in \eqref{eq:optimal_SDE} can be written as
\begin{align}\label{eq:optimal_drift}
-\beta \int_{\mathbb{R}^d} \nabla \Psi(v; \mu^\star) Q_{W_t,t}(\dif v) \equiv - \beta \cdot \mathbf{E}[\nabla \Psi(W_1;\mu^\star)|W_t],
\end{align}
i.e., it is equal to the conditional mean of the scaled negative gradient $-\beta \nabla \Psi(W_1;\mu^\star)$ given $W_t$. Note, however, that the potential $\Psi(W_1; \mu^\star)$ is a function of both $W_1$ and of the marginal distribution of $W_1$. This provides a nice illustration of the ``nonlocal'' nature of optimal control laws in control problems of McKean--Vlasov type \citep{carmona2018vol1}.
Another noteworthy feature of the dynamic formulation of mean-field entropic regularization is that the variance parameter $\tau$ can be interpreted as the total \textit{time} that the dynamics is run. Indeed, by the scale invariance of the Brownian motion, $\{\sqrt{\tau}B_{t/\tau}\}_{t \ge 0}$ and $\{B_t\}_{t \ge 0}$ have the same process law. From this and from \eqref{eq:optimal_SDE}, it follows that the process $W^\tau_t := W_{t/\tau}$ is a solution of the SDE
\begin{align}\label{eq:optimal_SDE_rescaled}
\dif W^\tau_t &= - \frac{1}{\tau} \nabla_w V^\star(W^\tau_t,t/\tau) \dif t + \dif B_t, \qquad W_0 = 0;\, t \in [0,\tau]
\end{align}
where
\begin{align*}
V^\star(w,t/\tau) &= -\tau \log \mathbf{E}\left[\exp\left(-\frac{\beta}{\tau} \Psi(B_\tau; \mu^\star)\right)\Bigg|B_t = w\right]
\end{align*}
and where $W^\tau_\tau \sim \mu^\star$. Once again, this suggests that taking $\tau$ to be small and choosing $\beta \gg \tau$ is a sensible course of action if the overall goal is to optimize the $L^2(\pi)$ risk, while keeping the relative entropy $D(\mu \| \gamma_\tau)$ small. On the other hand, by interpreting $\tau$ as time, we uncover an alternative interpretation of the entropic regularization term in \eqref{eq:FE}: If we zero out the drift in \eqref{eq:optimal_SDE_rescaled}, then at $t=\tau$ we will end up with $B_\tau$, which has the Gaussian distribution $\gamma_\tau$; the role of the drift in \eqref{eq:optimal_SDE_rescaled} is to transport the Brownian path $B_{[0,\tau]}$ to $W^\tau_{[0,\tau]}$ with $W^\tau_\tau \sim \mu^\star$, and the minimum total ``energy'' $\mathbf{E}[\frac{1}{2}\int^\tau_0 \|u_t\|^2 \dif t]$ over all such transport maps is equal precisely to $D(\mu^\star \| \gamma_\tau)$. Thus, by choosing the variance parameter $\tau$, we are effectively controlling the duration of the optimization process that leads to $\mu^\star$.
\section{Stochastic gradient descent as a greedy heuristic}
In the preceding section, we have shown that, under suitable regularity conditions, there exists a unique probability measure $\mu^\star = \mu^\star_{\beta,\tau}$ that minimizes the free energy $\FE_{\beta,\tau}$ and thus achieves optimal trade-off between the $L^2(\pi)$ risk and the relative entropy w.r.t.\ the Gaussian prior $\gamma_\tau$. We have also shown that $\mu^\star$ naturally arises in the context of a stochastic control problem of McKean--Vlasov type; in particular, if we denote by $\mu^\star_t$ the probability law of $W_t$ in \eqref{eq:optimal_SDE}, then $\mu^\star_1 \equiv \mu^\star$. Now, in the spirit of mean-field theory, we can run $N$ independent copies $\{W^i_t\}_{t \in [0,1]}$ of \eqref{eq:optimal_SDE} in parallel and form the finite neural-net approximation
\begin{align*}
\hat{f}_N(x;\bd{W}_1) \equiv \frac{1}{N}\sum^N_{i=1} \act(x; W^i_1).
\end{align*}
Since the empirical distribution $\hat{\mu}^{(N)} = N^{-1}\sum^N_{i=1}\delta_{W^i_1}$ converges to $\mu^\star$ as $N \to \infty$, a standard concentration-of-measure argument can be used to show that $R_N(\bd{W}_1) \le R(\mu^\star) + O(1/\sqrt{N})$ with high probability.
This, however, runs up against the obvious difficulty --- the drift in the optimal McKean--Vlasov dynamics \eqref{eq:optimal_SDE} depends functionally on the target measure $\mu^\star$, which is not available in closed form, but only implicitly through the Boltzmann fixed-point condition \eqref{eq:Boltzmann_FP}. Moreover, even if $\mu^\star$ were somehow known, Eq.~\eqref{eq:optimal_drift} shows that the computation of the optimal drift involves averaging w.r.t.\ a family of Gibbs measures specified by \eqref{eq:Q_measures}, yet another highly nontrivial task. This stands in stark contrast to the usual mean-field framework, where a data-driven iterative algorithm, such as SGD or noisy SGD, is shown to track a suitable continuous-time dynamics in the space of measures (i.e., the mean-field limit). Thus, a question that naturally arises is whether it is at all possible to approximately track the optimal McKean--Vlasov dynamics \eqref{eq:optimal_SDE} by means of a practically implementable iterative scheme, at least in some restricted regime.
To get an idea of how one might go about this, let us once again examine the expression for the optimal drift in \eqref{eq:optimal_SDE} in terms of the Gibbs measures \eqref{eq:Q_measures}:
\begin{align}\label{eq:optimal_SDE_alt}
\dif W_t = - \beta \left(\int_{\mathbb{R}^d} \nabla \Psi(v; \mu^\star) Q_{W_t,t}(\dif v)\right) \dif t + \sqrt{\tau} \dif B_t,
\end{align}
where, as we had noted already, the random Gibbs measures $Q_{W_t,t}$ become increasingly concentrated around $W_t$ as $t \to 1$ (we provide a quantitative illustration of this in Lemma~\ref{lm:Gstar}). Thus, it is tempting to compare \eqref{eq:optimal_SDE_alt} against
\begin{align}\label{eq:greedy_SDE_1}
\dif \tilde{W}_t = - \beta \nabla \Psi(\tilde{W}_t ; \mu^\star) \dif t + \sqrt{\tau} \dif B_t.
\end{align}
This admittedly crude step replaces, at each time $t$, the average of $\nabla \Psi(\cdot; \mu^\star)$ w.r.t.\ the Gibbs measure $Q_{W_t,t}$ by the `highly likely' value $\nabla \Psi(W_t; \mu^\star)$. However, there is still the worrisome dependence on the target measure $\mu^\star$, so we take yet another bold step and replace \eqref{eq:greedy_SDE_1} with
\begin{align}\label{eq:greedy_SDE_2}
\dif \hat{W}_t = - \beta \nabla \Psi(\hat{W}_t; \mu_t) \dif t + \sqrt{\tau} \dif B_t, \qquad t \in [0,1]
\end{align}
where $\mu_t$ is the probability law of $\hat{W}_t$. Note that the drift in \eqref{eq:greedy_SDE_2} is a function of not only the current state $\hat{W}_t$, but also of its marginal probability law $\mu_t$, so \eqref{eq:greedy_SDE_2} is a McKean--Vlasov SDE (see, e.g., Section~4.2 of \cite{carmona2018vol1}). We can think of the dynamics in \eqref{eq:greedy_SDE_2} as a sort of a ``greedy approximation'' to the optimal SDE \eqref{eq:optimal_SDE_alt}, provided we can show that the paths of $W_t$ and $\hat{W}_t$ stay close to one another with high probability. The main result of this section is that this is indeed the case when both $\beta$ and $\tau$ are suitably small, while the ratio $\beta/\tau$ is large. (As we will see from the bound of Theorem~\ref{thm:SGD} below, the choice of $\tau = \varepsilon^2/d$ and $\beta = \sqrt{\tau d} = \varepsilon$ is a reasonable one.) In fact, we will show that, somewhat surprisingly, vanilla SGD with Gaussian initialization can closely track the optimal mean-field McKean--Vlasov dynamics with high probability.
We consider the usual set-up, where we receive a stream of i.i.d.\ data $(X_1,Y_1)$, $(X_2,Y_2), \ldots$, where $X_1,X_2,\ldots \stackrel{{\rm i.i.d.}}{\sim} \pi$ and $Y_k = f(X_k)$, and update the weights $\bd{W}_k = (W^1_k,\ldots,W^N_k)$ by running SGD with constant step size $\eta > 0$:
\begin{align}\label{eq:OG_SGD}
W^i_{k+1} = W^i_k + \eta \beta (Y_{k+1} - \hat{f}_N(X_{k+1},\bd{W}_k)) \nabla_w \act(X_{k+1}; W_k^i), \qquad k = 0,1,2,\ldots
\end{align}
for $i \in [N]$. The process is initialized with $W^i_0 \stackrel{{\rm i.i.d.}}{\sim} \gamma_\tau$. Following \citet{mei2018meanfield, mei2019meanfield}, we are assuming that each sample is visited exactly once. While we focus on the noiseless case $Y = f(X)$, we can always arrange things so that $f(X) = \mathbf{E}[Y|X]$. For the sake of simplicity, we assume that $\eta = 1/n$ for some $n \in \mathbb{N}$.
\begin{theorem}\label{thm:SGD} Let $\bd{\mu}^\star = \{\mu^\star_t\}_{t \in [0,1]}$ be the flow of measures along the optimal McKean--Vlasov dynamics, with $\mu^\star_0 = \delta_0$ and $\mu^\star_1 = \mu^\star$. Then, with probability at least $1-\delta$,
\begin{align}\label{eq:MKV_vs_SGD}
\max_{0 \le k \le n} |R_N(\bd{W}_k) - R(\mu^\star_{k\eta})|
& \le \frac{\kappa}{N} + \kappa \sqrt{\frac{1}{N}\log \frac{N}{\delta}} + \kappa\beta + \kappa(1 + \sqrt{\eta}\beta)(1+\beta)e^{\kappa\beta} \left( \beta + \sqrt{\tau d \log \frac{Nd}{\delta \eta}}\right).
\end{align}
\end{theorem}
\begin{remark} As before, we see that choosing $\beta \sim \sqrt{\tau d} = \varepsilon$ will ensure that SGD will track the optimal McKean--Vlasov dynamics to accuracy $O(\varepsilon)$ with high probability, provided the network has at least $N \sim \frac{1}{\varepsilon^2}$ neurons. Curiously, the bound is insensitive to the choice of the step size $\eta \le 1$, since the latter is scaled down by $\beta$ in \eqref{eq:OG_SGD}. This helps further elucidate the connection between lazy training and entropic regularization: The above choice of $\beta$ and $\tau$ grants us the uniform approximation guarantee for SGD vs.\ optimal McKean--Vlasov dynamics regardless of how many steps of SGD we take. If we express everything in terms of $N$, we obtain $\beta = \frac{1}{\sqrt{N}}$ and $\tau = \frac{1}{Nd}$, which corresponds to parameter choices used in practice.
\end{remark}
|
2,877,628,090,169 | arxiv | \section{Introduction}
The solar corona can be described as a low $\beta$ plasma at low densities and high
temperatures. With the presence of coronal magnetic fields, this leads
to plasma, where the magnetic pressure is higher than the
gas pressure.
Therefore, the plasma
motions are dominated by the magnetic field, and the plasma can organise
itself in accordance to the geometry of the magnetic field, e.g. closed
loop structures.
The hot plasma in the corona emits radiation in extreme UV and X-ray emission, making
it observable from space-based telescopes.
One of the major open questions concerning the solar corona is its
heating mechanism, i.e. why is the solar corona typically more than 100
times hotter than the photosphere.
One of the ideas explaining coronal heating is the field-line
braiding model by \cite{Par72,Par88}, in which magnetic energy is
released in form of nanoflares.
In this model, the magnetic footpoints of the loops are
irreversibly moved by the small-scale photospheric motions, get
braided in chromosphere and corona, where the reconnecting
field lines release magnetic energy through ohmic heating and
contribute to the thermal energy budget.
Three-dimensional magnetohydrodynamic simulations modelling the
solar corona were able to show that with this nanoflare heating
mechanism the basic temperature structure and its dynamics can be
reproduced \citep[e.g.][]{GN02,GN05a,BP11}.
These models are able to describe energy transport to the corona
consistent with the nanoflare model \citep{BP13}.
These types of simulations are further used to synthesise coronal
emission comparable with actual observations of the corona.
From these synthesised emission, one finds that these models are able
to reproduce the average Doppler shifts to some extent \citep{PGN04,PGN06,HHDC10} and
the formation of coronal loops, when using a data driven model with an
observed photospheric magnetic field \citep{BBP13,BBP14,WP19}.
Furthermore, these models were used to show that the coronal magnetic field structure
is close to a potential field \citep{GN05a,BP11,BSB18}, and therefore
nearly force free.
However, the force-free approximation, broadly used to
obtain coronal magnetic field with field extrapolations \citep[for a
review, we refer to][]{Wi08}, turns out to
be not always valid \citep{PWCC15} and fails to describe complex
current structures in coronal loops above emerging active regions \citep{WCBP17}.
Recently, \cite{Rempel17} showed that the solar corona can be heated
by a small-scale dynamo operating in the near-surface region of the convection zone
braiding the magnetic footpoints in the photosphere.
Therefore, these types of models are able to reproduce the main
properties of the solar corona on the resolved scale
\citep[e.g.][]{P15}.
One of the most important ingredients is the vertical Poynting flux at the
bottom of the corona \citep[e.g.][]{GN96,BP11,BBP15}.
Currently there are only a limited number of codes available which are used
for this kind of simulations.
One of the most used codes to simulate the solar corona is the {\sc
Bifrost} code \citep{GCHHLM2011}, which is based on
earlier work of \cite{GN02,GN05a,GN05b} and the {\sc Stagger} code
\citep{GN96}. In these simulations, the near-surface convection is
self-consistently included and produces realistic photospheric velocities.
Furthermore, the {\sc Bifrost} code includes a realistic treatment of
the chromosphere using a non-local thermal equilibrium description.
Another code is the {\sc MuRAM} code \citep{VSSCEL05,Rempel14}
that has been recently extended to the upper atmosphere
\citep{Rempel17}. Also, there, the photospheric motions are driven by
near-surface convection.
Apart of these codes there are other codes used for realistic modelling of
the solar corona \citep[e.g.][]{Abbett07,MMLL05,MMLL08,HSMJMTG}.
In this paper, we present an extension to the coronal model of the {\sc
Pencil Code}\footnote{{\tt http://github.com/pencil-code}} that
has been used successfully to describe the solar corona using either
observed magnetograms and a velocity driver mimicking the photospheric
motions \citep{BP11,BP13,BBP13} or flux emergence simulations
\citep{CPBC14,CPBC15} as input at the lower boundary instead of
simulating the near-surface convection.
However, \cite{CP18} developed a 2D model, where the near-surface
convection is included with a realistic treatment of the solar corona.
Simplified two-layer simulations of the convection zone and the corona
of the Sun and stars using the {\sc Pencil Code}~ have been successfully used to investigate the
dynamo-corona interplay \citep{WB14,WKKB16}, to self-consistently drive current
helicity ejection into the corona \citep{WB10,WBM11,WBM12,WKMB13} and the formation of
sunspot-like flux concentrations \citep{WLBKR13,WLBKR16,LWBKR17}.
To be able to compare the simulations of the solar corona with
observations of emissivities, one needs to use a realistic value of the
Spitzer heat conductivity. However, this puts a major constraint on the
time step in these simulations. For simulations with a grid spacing of around
$200\,$km the time step due to the Spitzer heat conductivity is around
$1\,$ms.
However, this can be significant lower, if one does not limit the
diffusion speed by the speed of light.
If one wants to study the dynamics on smaller scales and being
able to reduce the fluid and magnetic diffusivities, one needs to use a
higher resolution. The smaller grid spacing leads to even lower values
of the time step. As the time step decreases quadratically with the grid
spacing, the simulations become unfeasible for very high resolutions.
To circumvent this,
\cite{CPBC14}, for example, used a sub-stepping scheme and
\cite{Rempel17} used a non-Fourier scheme, where the hyperbolic
equation for the heat transport is solved.
Similar approaches have also been used in the dynamo community to describe
the non-local evolution of the turbulent electromagnetic force
\citep{BKM04,HB09,RB12,BC18}.
We present here a non-Fourier description of the Spitzer heat flux, that
has been recently implemented to the {\sc Pencil Code}, see
\Sec{sec:heatflux}. We compare the outcome of the simulations obtained with and without the
non-Fourier scheme, see \Sec{sec:results}.
Furthermore, we also compare these simulations to those using
the semi-relativistic Boris correction \citep{Boris1970} to the Lorentz
force that has been also recently implemented to the {\sc Pencil Code}
\citep{CP18} to limit the time step constraint due to the Alfv\'en speed,
see \Sec{sec:boris}.
\section{Setup}
\label{model}
The setup of the simulations is based on the model of \cite{BP11,BP13},
therefore a detailed description will not be repeated here.
We model a part of the solar corona in a Cartesian box ($x$,$y$,$z$) of
$100\times100\times60\,$ Mm$^3$ using a uniform grid.
The $z=0$ layer represents the solar
photosphere. We use $128\times128\times256$
grid points, corresponding to a resolution of $781\,{\rm km}$ in the
horizontal $234\,{\rm km}$ in the vertical direction.
We solve the compressible magnetohydrodynamic equations for the
density $\rho$, the velocity $\mbox{\boldmath $u$} {}$, the magnetic vector potential
$\mbox{\boldmath $A$} {}$ and the temperature $T$.
\begin{equation}
{{\rm D} {}\ln\rho\over{\rm D} {} t} = -\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $u$} {},
\end{equation}
\begin{equation}
{{\rm D} {}\mbox{\boldmath $u$} {}\over{\rm D} {} t} = -{\mbox{\boldmath $\nabla$} {} p\over\rho} +\mbox{\boldmath $g$} {} + {\mbox{\boldmath $J$} {}\times\mbox{\boldmath $B$} {}\over\rho}
+ {1\over\rho}\mbox{\boldmath $\nabla$} {}{\bm \cdot}2\nu\rho\mbox{\boldmath ${\sf S}$} {},
\end{equation}
\begin{equation}
{{\rm D} {}\ln T\over{\rm D} {} t} + \left(\gamma-1\right)\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $u$} {} =
{1\overc_{\rm V}\rho T} \left[\mu_0\eta\mbox{\boldmath $J$} {}^2 + 2\rho\nu\mbox{\boldmath ${\sf S}$} {}^2 -
\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $q$} {} + L \right]
\label{eq:temp}
\end{equation}
where we use a constant gravity $\mbox{\boldmath $g$} {}=(0,0,-g)$ with $g=274\,$m/s$^2$,
a rate of strain tensor
$\mbox{\boldmath ${\sf S}$} {}=1/2(u_{i,j}+u_{j,i})-1/3\delta_{ij}\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $u$} {}$ and a constant
viscosity $\nu$ throughout the domain. Additionally we use a shock
viscosity to resolve shocks formed by high Mach number flows
(see \citealt{HBM04} and \citealt{GSFSM13} for details regarding its
implementation).
The pressure $p=(k_{\rm B}/\mu\,{\rm m}_{\rm p})\rho T$ is given by the
equation of state of an ideal gas, where $k_{\rm B}$, $\mu$ and
$m_{\rm p}$ are the Boltzmann constant, the molecular weight and the
proton mass, respectively. The corresponding adiabatic index
$\gamma=c_{\rm P}/c_{\rm V}$ is 5/3 for a fully ionised gas, with the specific heats at constant pressure $c_{\rm P}$
and constant volume $c_{\rm V}$.
The heat flux $\mbox{\boldmath $q$} {}$ is given by anisotropic Spitzer heat conduction
\begin{equation}
\mbox{\boldmath $q$} {}=-K_0 \left({T\over [K]}\right)^{\!5/2} {\mbox{\boldmath $B$} {}\!\mbox{\boldmath $B$} {}\over \mbox{\boldmath $B$} {}^2} \mbox{\boldmath $\nabla$} {} T
\equiv -\mbox{\boldmath ${\cal K}$} {}\mbox{\boldmath $\nabla$} {} T,
\label{eq:Fouier_heatf}
\end{equation}
which only gives a contribution aligned with the magnetic field and
$K_0=2\times10^{-11}\,$W(mK)$^{-1}$ is the value derived by
\cite{Spitzer:1962} assuming a constant Coulomb logarithm. In general,
the Coulomb logarithm and therefore $K_0$ depends weakly on the coronal plasma density.
We limit the heat conductivity tensor such that the corresponding heat
diffusion speed ${\rm d} {} x/(|\mbox{\boldmath ${\cal K}$} {}|/\rhoc_{\rm P})$ is 10\% of the speed of light
with ${\rm d} {} x$ being the grid spacing.
For some of the runs we replaced this equation by the hyperbolic
equation of the non-Fourier heat flux, see \Sec{sec:heatflux}.
Additionally to the anisotropic Spitzer heat conduction, we apply an
isotropic numerical heat conduction, which
is proportional to $|\mbox{\boldmath $\nabla$} {}\ln T|$ and a heat conduction with a
constant heat diffusivity $\chi = K/c_{\rm P}\rho$.
These additions are used to describe the heat flux in the lower part of
the simulation, where the temperature is significantly lower and
therefore the Spitzer heat conductivity is significantly smaller than
in the corona. It also makes the simulation numerically more stable.
The radiative losses due to the optically thin part of the atmosphere
are described by $L=-n_{\rm e} n_{\rm H} Q(T)$, where $n_{\rm e}$ and
$n_{\rm H}$ are the electron and hydrogen particle densities. $Q(T)$
describes the radiative losses as a function of temperature following the
model of \cite{CCJA89}, for details see \cite{Bingert2009}.
To fulfil the exact solenoidality of the magnetic field
$\mbox{\boldmath $B$} {}=\mbox{\boldmath $\nabla$} {}\times\mbox{\boldmath $A$} {}$ at all times, we solve for the induction equation
in terms of the vector potential $\mbox{\boldmath $A$} {}$.
\begin{equation}
{\upartial \mbox{\boldmath $A$} {}\over\upartial t} = \mbox{\boldmath $u$} {}\times\mbox{\boldmath $B$} {} +\eta\nabla^2\mbox{\boldmath $A$} {},
\label{eq:aa}
\end{equation}
where we use the resistive gauge, i.e. arbitrary scalar field
$\phi$, which divergence can be added to the induction
equation is chosen to be $\phi=\eta\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $A$} {}$.
The currents are given by
$\mbox{\boldmath $J$} {}=\mbox{\boldmath $\nabla$} {}\times\mbox{\boldmath $B$} {}$ and $\eta$ is the magnetic diffusivity.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{bbz_VAR0}
\caption{Initial vertical magnetic field $B_z$ at the photospheric
layer $z=0$ (colour online).
}\label{fig:bbz0}
\end{center}
\end{figure}
\subsection{Initial and boundary conditions}
At the lower boundary, we use for the vertical magnetic field the
line-of-sight magnetic field from the active region
AR 11102, observed on the 30th of August with the Helioseismic and
Magnetic Imager \citep[HMI;][]{HMI} onboard of the Solar Dynamics Observatory
(SDO), see \Fig{fig:bbz0} for an illustration.
As an initial condition, we use a potential field extrapolation to
fill the whole box with magnetic fields.
For the temperature, we use an initial profile of a simplified
representation of the solar atmosphere, similar as in \cite{BP11}.
The density is calculated accordingly using hydrostatic
equilibrium. The velocities are initially set to zero.
The simulations are driven by a prescribed horizontal velocity field at
the lower boundary mimicking the pattern of surface convection.
As discussed in \cite{GN02,GN05a,Bingert2009} and \cite{BP11}, such a surface velocity
driver is able to reproduce the observed
photospheric velocity
spectrum in space and time.
To avoid the destruction of the magnetic field pattern caused by the
photospheric velocities, we apply the following to stabilise the
field: i) we lower the magnetic diffusivity in the two lowest grid
layers by a factor of 800 using cubic step function, ii) we
apply a quenching of velocities by a factor of 2, when magnetic pressure
is larger than the gas pressure and iii) we
interpolate between the current vertical magnetic field and the
initial one $B_z^{\rm int}$ at $z=0$ layer following
\begin{equation}
{\upartial B_z\over\upartial t}={1\over\tau_b} \left(B_z^{\rm int} -B_z\right),
\end{equation}
where $\tau_b=10\,$min is the relaxation time.
The quenching of photospheric velocities mimics the suppression
of convection in magnetised regions as observed on the solar
surface (see detailed discussion in \citealt{GN02,GN05a,Bingert2009} and
\citealt{BP11}).
We apply a potential field boundary condition at the bottom and top
boundary of box for the magnetic field.
The temperature and density are kept fix at the bottom boundary.
The temperature is kept constant and the heat flux is set to zero at the top boundary
allowing the temperature to vary in time.
At the top boundary, we set all velocity components to zero to prevent mass
leaving or entering the simulation and to suppress all flows near the top
boundary. The density in the lower part is high enough to serve as a mass reservoir.
All quantities are periodic in horizontal directions.
For the viscosity we choose $\nu=10^{10}\,{\rm m}^2/\,{\rm s}$ similar to
the Spitzer value for typical coronal temperatures and densities.
We set $\eta=2\times10^{10}\,{\rm m}^2/\,{\rm s}$ motivated by the numerical
stability of the simulations.
In the solar corona the magnetic Prandtl number $Pr_{\rm M}=\nu/\eta$
is around 10$^{10}$-10$^{12}$ and not 0.5 as in our simulations.
\subsection{Non-Fourier heat flux scheme}
\label{sec:heatflux}
To reduce the time step constraints due to the Spitzer heat conductivity,
we use a non-Fourier description and solve for the heat flux $\mbox{\boldmath $q$} {}$
\begin{equation}
{\upartial \mbox{\boldmath $q$} {}\over \upartial t} = -{1\over\tau_{\rm Spitzer}} \left(\mbox{\boldmath $q$} {} +\mbox{\boldmath ${\cal K}$} {} \mbox{\boldmath $\nabla$} {}
T\right),
\label{eq:qq}
\end{equation}
where $\tau_{\rm Spitzer}$ is the heat flux relaxation time, i.e. e-folding time for $\mbox{\boldmath $q$} {}$ to
approach $-\mbox{\boldmath ${\cal K}$} {} \mbox{\boldmath $\nabla$} {} T$. $\mbox{\boldmath ${\cal K}$} {}$ is the Spitzer
heat conductivity tensor, which has contributions only along the magnetic field.
This approach enables us to use a different time stepping constrain to
solve our equations.
Instead of using the time step of Spitzer heat conduction ${\rm d} {} t_{\rm Spitzer}={\rm d} {}
x^2/\gamma\chi_{\rm Spitzer}$ with $\chi_{\rm Spitzer}=|\mbox{\boldmath ${\cal K}$} {}|/\rhoc_{\rm P}$, we
find two new time step constraints
\begin{equation}
{\rm d} {} t_1={\rm d} {} x\sqrt{\left({\tau_{\rm
Spitzer}\over\gamma \chi_{\rm Spitzer}}\right)} \equiv {{\rm d} {} x\over c_{\rm
Spitzer}}, \hskip 10mm \text{and} \hskip 10mm {\rm d} {} t_2= \tau_{\rm Spitzer},
\label{eq:dt}
\end{equation}
where ${\rm d} {} t_1$ comes from the wave propagation speed $c_{\rm Spitzer}$.
To see this more clearly, we can rewrite (\ref{eq:qq}) in one
dimension $x$, with $q$ and $K$ being the one-dimensional
counterparts of $\mbox{\boldmath $q$} {}$ and $\mbox{\boldmath ${\cal K}$} {}$ as
\begin{equation}
{\upartial q\over \upartial t} = -{1\over\tau_{\rm Spitzer}} \left(q +K {\upartial T\over \upartial x} \right)\,.
\end{equation}
Then together with a simplified one-dimensional
version of (\ref{eq:temp}), where we only consider the heat flux term
\begin{equation}
{\upartial T\over \upartial t} = -{1\overc_{\rm V}\rho} {\upartial q\over \upartial x}\,,
\end{equation}
we can construct a wave equation for the temperature, namely
\begin{equation}
{\upartial^2 T\over \upartial t^2} = -{1\over\tau_{\rm Spitzer}}
{\upartial T\over \upartial t} + {\gamma\chi_{\rm
Spitzer}\over\tau_{\rm Spitzer}} {\upartial^2 T\over \upartial x^2},
\end{equation}
in which $c_{\rm Spitzer}$=$\sqrt{\gamma\chi_{\rm Spitzer}/\tau_{\rm Spitzer}}$
emerges as the propagation speed.
The two new time step constraints emerge from the pre-factors of the
terms on the right-hand side.
By certain choices of $\tau_{\rm Spitzer}$, we can significantly increase the
time step. Furthermore, because ${\rm d} {} t_1$ depends linear on the grid
spacing ${\rm d} {} x$, instead of quadric as ${\rm d} {} t_{\rm Spitzer}$, the
speed-up ratio ${\rm d} {} t_1/{\rm d} {} t_{\rm Spitzer}$ grows with higher
resolutions, which leads to a computational gain.
Both time step constraints are included in the CFL condition to
calculate the time step of the simulation.
${\rm d} {} t_1$ enters the time step calculation through the advective time ${\rm d} {} t_{\rm advec}$ step using:
\begin{equation}
{\rm d} {} t_{\rm advec}={{\rm d} {} x\over u_{\rm advec}} \hskip 8mm \text{with}\hskip 8mm u_{\rm
advec}=\max{\left(|\mbox{\boldmath $u$} {}| + \sqrt{c_{\rm s}^2+v_{\rm A}^2} + c_{\rm
Spitzer}\right)},
\label{eq:dtadvec}
\end{equation}
where $u_{\rm advec}$ is the advection speed and $c_{\rm s}$ the sound
speed.
The major part of the heat flux is concentrated in the transition
region, where the temperature gradient is high. This can lead to
strong gradients in the heat flux $\mbox{\boldmath $q$} {}$ itself. We, therefore,
normalise $\mbox{\boldmath $q$} {}$ by the density $\rho$ to decrease the heat flux in the
lower part of the transition region compared to the upper part.
The main motivation is to gain a better numerical
stability and be able to resolve stronger gradients in $\mbox{\boldmath $q$} {}$ better.
This results in a new set of equations, where
\begin{equation}
\tilde{\mbox{\boldmath $q$}}{}}{={\mbox{\boldmath $q$} {}/\rho}.
\end{equation}
We basically solve now for the energy flux per unit particle instead of the
energy flux density.
\begin{equation}
{\upartial \tilde{\mbox{\boldmath $q$}}{}}{\over \upartial t}= {1\over\rho}{\upartial
\mbox{\boldmath $q$} {}\over \upartial t} - \tilde{\mbox{\boldmath $q$}}{}}{{\upartial\ln\rho\over \upartial t} = -{1\over\tau_{\rm Spitzer}} \left(\tilde{\mbox{\boldmath $q$}}{}}{ +{\mbox{\boldmath ${\cal K}$} {}\over\rho} \mbox{\boldmath $\nabla$} {}
T\right) + \tilde{\mbox{\boldmath $q$}}{}}{\left(\mbox{\boldmath $u$} {}{\bm \cdot}\mbox{\boldmath $\nabla$} {}\ln\rho+\mbox{\boldmath $\nabla$} {}{\bm \cdot}\mbox{\boldmath $u$} {} \right),
\label{eq:qqt}
\end{equation}
where we use the continuity equation to derive the last term.
The term in the energy equation changes correspondingly
\begin{equation}
{\upartial \ln T\over \upartial t} = -{1\over Tc_{\rm V}}\left(\mbox{\boldmath $\nabla$} {}{\bm \cdot}\tilde{\mbox{\boldmath $q$}}{}}{ +
\tilde{\mbox{\boldmath $q$}}{}}{{\bm \cdot}\mbox{\boldmath $\nabla$} {}\ln\rho\right) + \cdots\ .
\end{equation}
This formulation does not change the time step constraints shown in (\ref{eq:dt}).
Instead of choosing $\tau_{\rm Spitzer}$ as a constant value in time and space, we also implemented an
auto-adjustment, where $\tau_{\rm Spitzer}$ can vary in space and
time. This allows the simulation to be more flexible and to be able
to optimise the time step.
The main idea to choose a reasonable value for $\tau_{\rm Spitzer}$ is that we
set the time scale of the heat diffusion to be the smallest of all
relevant time scales in this problem, i.e. the
heat diffusion is the fastest process. The next bigger time scale is
typically the Alfv\'en crossing time ${\rm d} {} t_{\rm vA} = {\rm d} {} x /v_{\rm A}$
with the Alfv\'en speed $v_{\rm A}=B/\sqrt{\mu_0\rho}$.
We want to keep the hierarchy of the time steps of each process in
place while lowering the time step as much as possible.
So we choose the time step of the heat diffusion to be always a bit lower than the
Alfv\'en time step, therefore the heat diffusion is still the fastest
process, but slower as before. For a fixed ratio between the ${\rm d} {}
t_1$ and ${\rm d} {} t_{\rm vA}$, we ``tie'' $\tau_{\rm Spitzer}$ to $v_{\rm A}$ and we set
\begin{equation}
{\rm d} {} t_1={{\rm d} {} t_{\rm vA}\over\sqrt{2}}\hskip 5mm \rightarrow\hskip 5mm
c_{\rm Spitzer}=\sqrt{2}v_{\rm A} \hskip 5mm\rightarrow \hskip 5mm \tau_{\rm
Spitzer}={\gamma\chi_{\rm Spitzer}\over 2v_{\rm A}^2}.
\label{eq:tau_auto1}
\end{equation}
On one hand, $\tau_{\rm Spitzer}$ would become very small in regions below the
corona, because there $\chi_{\rm Spitzer}$ has very low values due to
the low temperature and high density values. However, in these regions the
heat transport is mainly due to the isotropic heat transport. Low
values of $\tau_{\rm Spitzer}$ in these regions would cause a very small
time step, even though the Spitzer heat flux is not important for the
heat transport in these regions. Therefore, we choose the lower limit
to be the advective time step, which assures that $\tau_{\rm
Spitzer}$ will not affect the time step in these regions.
On the other hand we want to avoid $\tau_{\rm Spitzer}$ becoming too
large and therefore the heat transport getting less efficient,
i.e. $\mbox{\boldmath $q$} {}$ is still sufficiently close to $-\mbox{\boldmath ${\cal K}$} {} \mbox{\boldmath $\nabla$} {} T$. So, we
choose $\tau_{\rm Spitzer}^{\rm max}=100\,$s as a limit for $\tau_{\rm
Spitzer}$:
\begin{equation}
\min{\left({\rm d} {} t_{\rm vA}, {{\rm d} {} x\over\sqrt{c_{\rm s}^2+\mbox{\boldmath $u$} {}^2}}\right)} \le
\tau_{\rm Spitzer} \le \tau_{\rm Spitzer}^{\rm max}.
\label{eq:tau_auto2}
\end{equation}
To use the non-Fourier heat flux description in the {\sc Pencil Code}, one has to
add \texttt{HEATFLUX=heatflux} to \texttt{src/Makefile.local} and set
the parameters in name list \texttt{heatflux\_run\_pars} in \texttt{run.in}.
The relaxation time $\tau_{\rm Spitzer}$ can be either chosen freely and
the inverse is set by using \texttt{tau\_inv\_spitzer} or one can
switch on the automatically adjustment by using
\texttt{ltau\_spitzer\_va=T}, then \texttt{tau\_inv\_spitzer} sets the
value of 1/$\tau_{\rm Spitzer}^{\rm max}$.
\subsection{Semi-relativistic Boris correction}
\label{sec:boris}
Above an active region the magnetic field strength can be high while
the density is low leading to Alfv\'en speeds comparable to the speed of
light \citep[e.g.][]{CF13,Rempel17}.
This causes two major issues.
On one hand, the MHD approximation assuming non-relativistic phase speeds
is not valid anymore, i.e. we cannot neglect the displacement current.
On the other hand, the high values of the Alfv\'en speed reduce the
time step significantly.
To address these two issues we use a semi-relativistic correction of
the Lorentz force following the work of \cite{Boris1970} and
\cite{Gombosi02}, where we apply a semi-relativistic correction term to the
Lorentz force.
This has been used and successfully tested for the {\sc MuRAM} code in
\cite{Rempel17}. Here, we use the implementation discussed by
\cite{CP18}, who added this correction term to the {\sc Pencil
Code}.
There, the Lorentz force transforms to
\begin{equation}
{\mbox{\boldmath $J$} {}\times\mbox{\boldmath $B$} {}\over\rho} \hskip 4mm\rightarrow \hskip 4mm\gamma_{\rm A}^2 {\mbox{\boldmath $J$} {}\times\mbox{\boldmath $B$} {}\over\rho} +
\left(1-\gamma_{\rm A}^2\right)\left(\mbox{\boldmath $I$} {} -
\gamma_{\rm A}^2{\mbox{\boldmath $B$} {}\BB\over\mbox{\boldmath $B$} {}^2}\right)\left(\mbox{\boldmath $u$} {}{\bm \cdot}\mbox{\boldmath $\nabla$} {}\mbox{\boldmath $u$} {} + {\mbox{\boldmath $\nabla$} {} p\over\rho}
-\mbox{\boldmath $g$} {} \right),
\end{equation}
where $\gamma_{\rm A}^2=1/(1+v_{\rm A}^2/c^2)$ is the relativistic correction factor.
We note here that the correction term used here and in \cite{CP18} is
slightly different from the one used by \cite{Rempel17}, because
\cite{CP18} finds a more accurate way to approximate the inversion
of the enhanced inertia matrix. This leads to an additional $\gamma_{\rm A}^2$ in
front of $\mbox{\boldmath $B$} {}\BB/\mbox{\boldmath $B$} {}^2$.
If $v_{\rm A}\ll c$ and $\gamma_{\rm A}^2\approx 1$, we retain the normal Lorentz force
expression. For $v_{\rm A} \le c$, the Lorentz force is reduced and the
inertia is reduced in the direction perpendicular to the magnetic field.
As the enhance inertia matrix \citep{Rempel17} is originally on the
right-hand side of the momentum equation, i.e. under the time derivative and it is just approximated by a correction
term on the left-hand side, the semi-relativistic Boris correction does not
change the stationary solution of the system and therefore does not lead
to further correction terms in the energy equation.
To switch on the Boris correction in {\sc Pencil Code} one sets the
flag \texttt{lboris\_correction=T} in the name list
\texttt{magnetic\_run\_pars}.
The Boris correction describes the modification of the Lorentz force
in the situation, where the Alfv\'en speed becomes comparable to the speed
of light.
In other words the speed of light is a natural Alfv\'en speed limiter
and the Boris correction describes the modification close to this
limiter.
We can artificially decrease the value of the limiter to a value of
our choice and the Boris correction takes care of the corresponding
modifications.
This can significantly reduce the value of the Alfv\'en speed in our
simulations and allow us to enhance the Alfv\'en time step.
Unlike in \cite{CP18}, we use the Boris correction indeed to increase
the Alfv\'en time step, similar to what has been done by
\cite{Rempel17}.
As shown by \cite{Gombosi02}, the propagation speed can be quite
complicated, we choose a similar time step modification as in \cite{Rempel17}
\begin{equation}
{\rm d} {} t_{\rm vA} \hskip 4mm\rightarrow\hskip 4mm {\rm d} {} t_{\rm vA}
\sqrt{1+\left({v_{\rm A}^2\big/ c_{\rm A}^2}\right)^2},
\end{equation}
where $c_{\rm A}$ is the limiter. We choose for Set~B $c_{\rm A}=10\,000\,{\rm km/s}$, which
corresponds to a time step of ${\rm d} {} t_{\rm vA}\approx 20\,$ms for our simulations.
The limiter $c_{\rm A}$ can be set by using \texttt{va2max\_boris} in the name list
\texttt{magnetic\_run\_pars}.
The Boris correction can be used together with the automatic adjusted
relaxing time $\tau_{\rm Spitzer}$ in the non-Fourier heat flux
calculation: if one sets \texttt{va2max\_tau\_boris} in \texttt{heatflux\_run\_pars} to the same value as
\texttt{va2max\_boris} in \texttt{magnetic\_run\_pars}, then the code
modifies the Alfv\'en speed and the Alfv\'en time step used in
(\ref{eq:tau_auto1} and \ref{eq:tau_auto2}) accordingly.
\section{Results}
\label{sec:results}
We present here the results of three sets of runs, where we use
different values of the heat flux relaxation time $\tau_{\rm Spitzer}$
in combination with and without the Boris correction. In the first set,
containing only Run~R, we use the normal treatment of the Spitzer
heat flux without using the non-Fourier heat flux evolution and without the Boris correction.
In the second set, containing 4 runs (Set~H), we use the non-Fourier
heat flux
evolution with $\tau_{\rm Spitzer}$ between $10$ and $1000$ ms and the
automatically adjustment, see \Sec{sec:heatflux}.
In the third set, containing 7 runs (Set~B), we use the
semi-relativistic Boris correction with $c_{\rm A}=10\,000\,{\rm km/s}$ and
the non-Fourier heat flux evolution with $\tau_{\rm Spitzer}=10-1000\,$ms and the automatically adjustment. We also use one run (Ba2)
with even lower Alfv\'en speed limit of $c_{\rm A}=3\,000\,{\rm km/s}$.
An overview of the runs can be found in \Tab{runs}.
\begin{table}
\caption{Summary of the runs. $\tau_{\rm Spitzer}$ is the relaxation
time for non-Fourier heat flux description, see \Sec{sec:heatflux}, $\tau_{\rm Spitzer}=\infty$
stands for the use of standard Fourier heat flux, see (\ref{eq:Fouier_heatf}). $c_{\rm A}$ is the
Alfv\'en speed limit, used for the Boris correction, see
\Sec{sec:boris}; $c_{\rm A}=\infty$ stands for no Boris correction. ${\rm d} {} t$
indicates the averaged time step, ${\rm d} {} t_{v_{\rm A}}$ the averaged Alfv\'en
time step and ${\rm d} {} t_1$ and ${\rm d} {} t_2$ the average time step due to
the heat flux evolution, see (\ref{eq:dt}). For Run~R, ${\rm d} {} t_1={\rm d} {}
t_2={\rm d} {} t_{\rm Spitzer}$. All these quantities are determine
as an average in the quasi-stationary state. $t_{\rm cpu}$ is
wall clock time per time step per mesh point. For the timing
we use the SISU Cray XC40 supercomputing cluster at CSC.
$\Delta T_{\rm cor}=(\brac{T}_{\rm runs}-\brac{T}_{R})/\brac{T}_{R}$ is the mean temperature
deviation from the reference runs, taking as a horizontal and height ($z$=20-40 Mm) average.}
\vspace{1mm}
\small
\centering
\label{runs}
\begin{tabular}{lrrrrrrrr}
\hline\hline
Runs & $\tau_{\rm Spitzer}$ [ms] & $c_{\rm A}$ [km/s]& ${\rm d} {} t$ [ms]&${\rm d} {} t_{v_{\rm A}} $ [ms]&${\rm d} {} t_1$[ms]&${\rm d} {} t_2$ [ms]&$t_{\rm cpu}$ [$\mu$s]&$\Delta T_{\rm cor}$\\
\hline\hline
R & $\infty$ & $\infty$& 1.5& 2.7 & 1.7& 1.7& 4.2$\times 10^{-2}$& 0 \\
\hline
H001 & 10 & $\infty$& 1.1& 2.8 & 1.2 & 9.0& 4.6$\times 10^{-2}$&-5\% \\
H005 & 50 & $\infty$& 2.4& 3.0 & 4.4 & 45.0& 4.5$\times 10^{-2}$&-14\%\\
H1 & 1000 & $\infty$& 4.5& 5.2 & 13.0 &900.0& 4.6$\times 10^{-2}$&-18\% \\
Ha & auto & $\infty$ & 2.0& 2.6 & 2.5 & 2.0& 4.6$\times 10^{-2}$&-3\% \\
\hline
B001 & 10 & 10 000 & 0.5& 21.2 & 0.5 & 9.0& 4.6$\times 10^{-2}$&14\%\\
B002 & 20 & 10 000 & 2.8& 21.2 & 2.8 &18.0& 4.6$\times 10^{-2}$&-13\%\\
B005 & 50 & 10 000 & 3.4& 21.2 & 3.7& 45.0& 4.5$\times 10^{-2}$&-7\%\\
B01 & 100 & 10 000 & 4.4&21.2 & 4.7 & 90.0& 4.5$\times 10^{-2}$&4\%\\
B03 & 300 & 10 000 & 5.4&21.2 & 6.1&270.0& 4.5$\times 10^{-2}$&-0.3\%\\
B1 & 1000 & 10 000 & 8.0&21.2 & 10.0&900.0& 4.5$\times 10^{-2}$&-4\%\\
Ba & auto & 10 000 & 15.4&21.2 &15.0&19.5& 4.7$\times 10^{-2}$&14\%\\
Ba2 & auto & 3 000 & 47.6&70.6 &49.9 &64.9& 4.7$\times 10^{-2}$&18\%\\
\hline\hline
\label{tab1}
\end{tabular}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{ptimesteps}
\caption{Vertical distribution of time step
constraints for Run~Ba at time $t=200$ mins. We plot the time step due to Spitzer
heat conductivity ${\rm d} {} t_{\rm Spitzer}$ (green), due to the heat flux
${\rm d} {} t_1$ (black) and ${\rm d} {} t_2$ (blue), due to the Alfv\'en speed
${\rm d} {} t_{\rm vA}$ (red) and reduced Alfv\'en speed with the Boris
correction ${\rm d} {} t^{\rm Boris}_{\rm vA}$. The horizontal averaged values are shown with a
solid line and the minimum values at each height with a dashed line (colour online).
}\label{fig:ts}
\end{center}
\end{figure}
\subsection{Time steps}
As a first step we look at the time steps of all the runs in \Tab{runs}.
In Run~R the averaged time step in the saturated stage is around
1.5 ms. This time step is constrained by the Spitzer time step ${\rm d} {}
t_{\rm Spitzer}$, which is shown as ${\rm d} {} t_1={\rm d} {} t_2$ in \Tab{runs}.
The Alfv\'en time step ${\rm d} {} t_{\rm vA}$ is around twice as large.
In the Set~H, the code additionally solves the non-Fourier heat flux
equation that leads to an increased time step. However, the time
step is actually limited by the low Alfv\'en time step and therefore
the time step cannot be increased by a large factor.
In Set~H the largest speed-up factor is around 3.
For Run~H001, the value of $\tau_{\rm Spitzer}$ is low enough to have a
time step constraint of ${\rm d} {} t_1$ instead of ${\rm d} {} t_{\rm vA}$. However,
the runs reach a lower time step than in Run~R.
For values of the relaxing time $\tau_{\rm Spitzer}= 50-1000\ $ms
(Runs~H005 and H1), the time
step due to the heat flux is larger than the Alfv\'en time
step. This means that the physical process of heat redistribution is
even slower than the Alfv\'en speed.
This leads in Run~H1 to higher densities resulting in a lower Alfv\'en
speed and a higher ${\rm d} {} t_{\rm vA}$ see discussion in \Sec{sec:emis}.
Furthermore, Run~H1 only runs stable, if we increase the shock viscosity to 10 times higher values than in the other runs. This
will certainly lead to some additional differences independent of the
direct influence of the non-Fourier heat flux description.
When applying the auto-adjustment of $\tau_{\rm Spitzer}$ (Run~Ha), the time steps
${\rm d} {} t_1$ and ${\rm d} {} t_2$ are slightly smaller than ${\rm d} {} t_{\rm vA}$ and
limits the time step.
There the speed up is less than a factor of 2, but the heat
distribution is the fastest process in the system.
Using the non-Fourier heat flux description leads
usually to higher peak temperatures, because the temperature diffusion
is less efficient.
For the calculation of ${\rm d} {} t_1$ and ${\rm d} {} t_2$, the code uses the CFL
pre-factors of $0.9$ for both time steps, this results in ${\rm d} {}
t_2=0.9\, \tau_{\rm Spitzer}$.
As ${\rm d} {} t_1$ enters via (\ref{eq:dtadvec}), ${\rm d} {} t$ is often lower than
${\rm d} {} t_1$ and ${\rm d} {} t_{\rm vA}$ in our simulations.
To increase the time step further, we use the semi-relativistic
Boris correction in all runs of Set~B.
As shown in \Tab{runs}, ${\rm d} {} t_{\rm vA}$ significantly increases to 21.2
ms for Runs~B001-Ba and to 70.6 ms for Ba2.
This leads to a much larger speed-up factor of 10 for Run~Ba and more than 30 for Run~Ba2.
For Run~B001 to Run~B1 with $\tau_{\rm Spitzer}$=
10-1000 ms, ${\rm d} {} t_1$ is lower than ${\rm d} {} t_{\rm vA}$ and the time step can
be significantly reduced, while the heat distribution is the fastest
process in the system.
For Run~B1, we achieve a speed up of more than five, however we need to
use a comparable large value of $\tau_{\rm Spitzer}$, which as discussed
in \Sec{sec:emis} can lead to artefacts.
For Runs~Ba and Ba2, the auto-adjustment of $\tau_{\rm Spitzer}$ takes
care that ${\rm d} {} t_1<{\rm d} {} t_{\rm vA}$.
As discussed below, Run~Ba shows a good agreement with Run~R, whereas
Run~Ba2 tends to produce higher temperatures in the corona.
To get a better understanding of the calculation of the time step, we plot in \Fig{fig:ts} various contributions to the time
steps for Run~Ba.
Without the non-Fourier heat flux description and the Boris correction,
the time step is dominated by Alfv\'en time step ${\rm d} {} t_{\rm vA}$ and
the Spitzer time step ${\rm d} {} t_{\rm Spitzer}$.
The Boris correction reduces ${\rm d} {} t_{\rm vA}$ to ${\rm d} {} t^{\rm
Boris}_{\rm vA}$ mostly in the regions between 5 and 30 Mm.
The auto-adjustment of $\tau_{\rm Spitzer}$ sets ${\rm d} {} t_1$ to be always
slightly lower than ${\rm d} {} t^{\rm Boris}_{\rm vA}$.
Only below $z=5\,$Mm, ${\rm d} {} t^{\rm Boris}_{\rm vA}$ is small, because
there the temperature diffusion is dominated by the other heat diffusion
mechanism described in \Sec{model}.
It is clearly visible that the ${\rm d} {} t_1$ is significantly
higher than ${\rm d} {} t_{\rm Spitzer}$ (green line) and ${\rm d} {} t_{\rm vA}$
(red) without the Boris correction.
However, we note here that because of the non-Fourier heat flux
description we find higher peak temperatures in the simulation. This
results in a decrease of ${\rm d} {} t_{\rm Spitzer}$ in comparison with runs
without the non-Fourier heat flux description. In Run~R, ${\rm d} {} t_{\rm
Spitzer}$ is around 1.6 ms, where in Run~Ba, it is around a factor
of eight lower. Such a factor can be explained by change in
temperature by a factor of 2.3.
Using the non-Fourier heat flux evolution requires to solve (\ref{eq:qq})
or (\ref{eq:qqt}) meaning three additional equations. However, the
computational extra calculation time is around 10\%, which is very
small compared to the gain in time step reduction. Using the
semi-relativistic Boris-Correction does not seem to increase the
computation time significantly. Only if we use the auto-adjustment of
$\tau_{\rm Spitzer}$ together with the Boris correction we find an additional
2-3\% increase in the computation time, as shown in the last row of \Tab{runs}.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{phist_va}
\caption{2D histograms of the Alfv\'en speed over height $z$ for
Runs~R, Ba, Ba2. We plot the mean value with red solid line and the
median with a yellow solid. The dashed white-blue lines show the 25 and 75
percentiles, i.e. half of the data points are in between these
lines. The black dashed line
indicate the Alfv\'en speed limit $c_{\rm A}$ for the Boris correction (colour online).
}\label{fig:alfven}
\end{center}
\end{figure}
\subsection{Alfv\'en velocity with Boris correction}
Next, we look at the influence of the semi-relativistic Boris
correction on the Alfv\'en velocity $v_{\rm A}$.
In \Fig{fig:alfven}, we plot 2D histograms of $v_{\rm A}$ for Runs~R, Ba,
Ba2. For Run~R, the maximum speed reaches $v_{\rm A}=80\,000
\,{\rm km/s}$ at the lower part of the corona, where the density has decreased
significantly with height, but the magnetic field is still strong.
The median (yellow line) has its maximum at the same location with a
value around $v_{\rm A}=18\,000 \,{\rm km/s}$.
In Run~Ba, we have applied the Boris correction with $c_{\rm
A}=10\,000\,{\rm km/s}$.
Even though, this value is lower than the averaged and mean value in the
region of $z=5$--$20$ Mm, the velocity distribution does not change
significantly in comparison to Run~R. As a main effect of the Boris
correction, the peak velocity at the top of the distribution is
reduced, therefore the distribution becomes more compact.
This can be also seen from the changes in the mean and
median velocity. While the maximum of the mean is reduced from above
$v_{\rm A}=20\,000 \,{\rm km/s}$ of Run~R to nearly $v_{\rm A}=15\,000 \,{\rm km/s}$, the
median changes just slightly.
Also, the area between the 25 and 75 percentiles of the
Alfv\'en velocity population moves only slightly towards lower
values.
This make us confident that the Boris correction with $c_{\rm A}=10\,000\,{\rm km/s}$ does only reduce the
peak velocities and not the overall velocity structure; most of the
points are unaffected by the correction.
For Run~Ba2, we reduce the Alfv\'en speed limit to $c_{\rm
A}=3\,000\,{\rm km/s}$. This makes the velocity distribution even more
compact.
The maximum values are significantly reduced to $v_{\rm A}=35\,000 \,{\rm km/s}$,
and the mean and median values are also lower than in Runs~R, Ba.
However, setting $c_{\rm A}=3\,000\,{\rm km/s}$ does not mean that all the
velocities are lower than this value, it can be understood as a
significant reduction of the peak velocities and a transfer of the
velocity distribution to a much more compact form.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{temp_prof}
\caption{(a) Averaged temperature $\bra{T}$ as a function of height $z$
for Run~R.
(b) Ratio of the averaged temperature profile of all runs and Run~R
$\brac{T}_{\rm runs}/\brac{T}_{R}$ as a function of height $z$. The temperatures are
averaged horizontal as well as in time for the last quarter (1
hours) of the simulation. The color of the lines indicates the run
names in terms of $\tau_{\rm Spitzer}$, the solid lines are for
runs of Sets~R and H, and dashed lines for Set~B (colour online).
}\label{fig:temp_prof}
\end{center}
\end{figure}
\subsection{Structure of temperature and ohmic heating}
\label{sec:temp}
Next, we look at the horizontal averaged temperature profile over
height.
Even though the non-Fourier description of the heat flux can lead to higher
peak temperatures, the overall temperature structure should remain
roughly the same.
In \Fig{fig:temp_prof}, we plot the horizontal averaged temperature
profile over height for the reference Run~R in panel (a) and a
comparison with the other runs in panel (b).
The horizontal averaged temperature structure in Run~R shows a typical
behaviour of corona above an active region with medium magnetic field
strengths.
The plasma above $z=10\,$ Mm is heated self-consistently to averaged
temperatures of around 1 million kelvin.
This temperature profile is very similar to results of earlier work with
the {\sc Pencil Code}~ \citep[e.g.][]{Bingert2009,BP11,BP13, BBP13} and other groups
\citep[e.g.][]{GN02,GN05a, GN05b, GCHHLM2011}.
When comparing with the temperature profiles of the other runs, we
find no large differences. For most of the runs the deviation is not
more than 10\%. For some runs the largest difference occurs
in the transition region, where the temperature has a large gradient.
Higher temperature values in this region simply mean a slightly lower transition
region and lower values mean a slightly higher transition region.
Nearly all runs develop a lower or similar transition region location
as in Run~R. Only Runs~H005, H1, Ba2 develop a higher transition region.
This can be explained either by sub-dominance of the heat flux time
step (Runs~H005, H1) or the too low limit for the Alfv\'en speed, see
discussion below.
Only in Runs~B001, Ba and Ba2 the plasma is heated to 20\% higher
temperature in the upper corona in comparison with Run~R.
For Run~B001, this high temperature only occur at the end of the
simulation, see \Fig{fig:h_t_evo}.
In these runs, the heat diffusion might be not efficient enough to
transport heat to lower layers.
When we look at the temperature evolution over time, as plotted in
\Fig{fig:h_t_evo}(a), we find that each run shows a large variation in
time even though we have averaged horizontally and over 18-20 Mm.
This can be explained by the non-linear behaviour of the
system. Because of this reason temporal variations occurring in the other runs
appear not at the same time for all runs.
The difference between the runs is comparable with the time variation
of each run.
Therefore, to be able to compare the runs, we should look at
the time averaged quantities as done throughout this work.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{heat_temp_time}
\caption{Time evolution of the horizontal averaged temperature
$\bra{T}$ (a) and of the horizontal averaged heating rate
$\bra{\mu_0\eta\mbox{\boldmath $j$} {}^2}$ (b) at $z=18-22\,$Mm. Color coding is the
same as in \Fig{fig:temp_prof} (colour online).
}\label{fig:h_t_evo}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{heat_prof}
\caption{(a) Averaged ohmic heating rate $\bra{\mu_0\eta\mbox{\boldmath $j$} {}^2}$ as a function of height $z$
for Run~R.
(b) Ratio of the averaged ohmic heating rate of all runs and Run~R
$\brac{T}_{\rm runs}/\brac{T}_{R}$ as a function of height $z$. The ohmic heating rate are
averaged horizontal as well as in time for the last quarter (1
hours) of the simulation. The color coding the same as in \Figs{fig:temp_prof}{fig:h_t_evo} (colour online).
}\label{fig:heat_prof}
\end{center}
\end{figure}
Next, we look at the ohmic heating rate in all the runs.
The ohmic heating is the main process in this type of
simulations to heat the coronal plasma up to million K.
Also, here, we plot the horizontal averaged profile of Run~R in panel
(a) of \Fig{fig:heat_prof} and compare it with the other runs in \Fig{fig:heat_prof}(b).
The profile of the ohmic heating rate shows the typical behaviour of an
exponential decrease corresponding to two scale heights. Below the
corona the scale height is roughly 0.5 Mm, while in the corona the
scale height is around 5 Mm.
Also, this is consistent with earlier finding with this kind of
simulations by many groups \citep[e.g.][]{GN02,GN05a,
GN05b,Bingert2009, GCHHLM2011, BP11,BP13, BBP13}.
By comparing with the other sets of runs, we find that these agree well with
Run~R.
Only Run~B001 shows a large heating rate in the lower corona, which
comes here also from the last part of the simulation.
Runs~B01, Ba and Ba2 develop a higher heating rate in the
upper corona resulting in higher temperatures at this location ( see \Fig{fig:temp_prof}).
Small changes either in the scale height of the coronal heating or
in the location in the transition region can explain most of the
differences we find in the comparison with Run~R.
This explains also the temporal changes of the heating rate at
constant height, as shown in \Fig{fig:h_t_evo}(b).
The large variations in time of the heating rate can be attributed to
non-linear behaviour of the system. Even in Run~R, these variations are
large compared to the average. Small local changes in temperature and
density can also affect the heating rate. As the field is very close
to a potential field the currents are due to small perturbations from
the potential field. These perturbations can easily be affected by changes
in the plasma flow due to temperature and density fluctuations.
Furthermore, in such dynamical non-linear systems, changes
for example in the time step can affect also the realisation of the
velocity solution. Even when solutions are the same on a
statistical level, this can cause variations in the ohmic heating.
For these types of models, large variations in time of the ohmic heating
rate are a common feature \citep[e.g.][]{BP11,BP13} as small changes in local scale
height will lead to a large change in the heating rate.
Overall, the vertical horizontally averaged temperature and heating
structure of all runs agree well with Run~R.
\subsection{Emission signatures}
\label{sec:emis}
\begin{figure}
\begin{center}
\includegraphics[width=1.05\textwidth]{Emis_good}
\caption{Temperature and emission structure for Runs~R, Ha, Ba,
Ba2. We show the temperature averaged over the $y$ direction
and in time $t=180-240$ mins (left panel) together with the synthesised
emission comparable to the AIA 171 channel, representing emission at
around 1 MK, integrated in the $y$
direction (side view, middle panel) and in $z$ direction (top view,
right panel). The emission values represent the count rate of the
AIA instrument and has been averaged in time $t=180-240$ mins.
The red square indicate the region which is used to calculate the
temporal evolution in \Fig{fig:emis_ts} (colour online) (colour online).
}\label{fig:emis1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{pEmis_region}
\caption{Time evolution of emission averaged over a small region for
Run~R, Ha, Ba and Ba2. We plot the emission of the AIA 171
channel in the $y$ direction averaged over a small region ($x=23-27$ Mm,
$z=18-22$ Mm) as indicated by red boxes in \Fig{fig:emis1} middle
panels. The inlay shows the time evolution of the averaged emission
from 150 to 300 mins on a linear scale instead of logarithmic. The
color and style of the lines are the same as in \Fig{fig:temp_prof} (colour online).
}\label{fig:emis_ts}
\end{center}
\end{figure}
To further test how well the non-Fourier description of the heat flux
reproduce the Fourier description, we synthesise coronal emissivities
corresponding to the 171 \r{A} channel of Atmospheric
Imaging Assembly \citep[AIA;][]{AIA:2012} on board of SDO.
We choose this AIA channel because it can be potentially compared with
observations and represents well the plasma structure of around 1
million K by convolving the temperature and density structures.
This can work as a good test, weather or not coronal emission structures are
affected by the choice of heat flux description.
For this we calculate the emission following optical thin radiation approximation,
\begin{equation}
\epsilon=n_e^2 G(T),
\label{eq:emis}
\end{equation}
where $G(T)$ is the response function of the particular filter, we want
to synthesise. Because we compare our simulations among each
other and not to observation, we simplify $G(T)$ using a gaussian
distribution around a mean temperature $\log_{10} T_0$,
\begin{equation}
G(T) \propto \exp{\left[- \left({\log_{10} T - \log_{10}
T_0\over\Delta\log_{10} T_0}\right)^2\right]},
\label{eq:emis2}
\end{equation}
where $\Delta\log_{10} T_0$ is the temperature width used to mimic the temperature
response function.
We use $\log_{10} T_0=6\, \log_{10}\, $K and $\Delta\log_{10} T_0=0.2\,
\log_{10}\, $K for
synthesising the emission of the AIA 171 \r{A} channel.
To calculate the emission emitted from a certain direction, we perform
an integration along this direction. For the discussion below, we apply
an integration along the $y$ and $z$ directions, respectively.
In \Fig{fig:emis1}, we plot the temperature as a side view ($xz$) averaged
over $y$ and in time (180-240 min) together with the
synthesised emission integrated over the $y$ and $z$
directions also averaged in time (180-240 min) representing the AIA
171 channel for Runs~R, Ha, Ba, Ba2.
For these runs, we expect a good agreement with the reference run R,
because the value of $\tau_{\rm Spitzer}$ is regulated
automatically and therefore the time step is controlled by the heat flux,
i.e. ${\rm d} {} t_1$.
We find agreement between the Runs~R, Ha and Ba, but we find
slightly stronger emission structures in Run~Ba and slightly hotter
temperatures in Run~Ha. To illustrate the variation in time we show in \Fig{fig:emis_ts} the
time evolution of the emission in a small region of the simulation box.
We find a good agreement between Runs~R and Ha with variation in time
which are comparable with their difference.
Run~Ba takes a bit longer to saturate, but at around 220 min it also settles to values similar to
Runs~R and Ha. Run~Ba2 seems to saturate to a much higher emission
level than the other runs, which is already seen in \Fig{fig:emis1}.
For Run~Ba2, as pointed out in \Sec{sec:temp} and shown in first
column of \Fig{fig:emis1}, the corona is heated to
higher temperatures, i.e. the heat transport is less efficient.
We find larger temperatures mostly at the top of the corona inside the
loop structures. This leads also to higher emission in the AIA 171
channel than in the Run~R.
This might be an artefact from the low limit of the Alfv\'en speed through
the Boris correction in this run.
Even though the Alfv\'en speed limiter does not effect the
heat flux directly, it increases the heat flux time step and makes
the heat transport less efficient.
\begin{figure}
\begin{center}
\includegraphics[width=1.05\textwidth]{Emis_bad}
\caption{Temperature and emission structure for Runs~H005, H1, B03,
B1. The emission values represent the count rate of the
AIA instrument. We show the temperature averaged over the $y$ direction
and in time ($180-240$ mins, left panel) together with the synthesised
emission comparable to the AIA 171 channel, representing emission at
around 1 MK, integrated in the $y$
direction (side view, middle panel) and in $z$ direction (top view,
right panel). The emission values represent the count rate of the
AIA instrument and has been averaged in time ($180-240$ mins) (colour online).
}\label{fig:emis2}
\end{center}
\end{figure}
In \Fig{fig:emis2}, we show a few other runs, which are either dominated
by the Alfv\'en time step (Runs~H005 and H1) or use a constant value
of $\tau_{\rm Spitzer}$ (Runs~B03 and B1).
Run~H005 shows a similar emission structure than the Runs~R, Ha, Ba,
however the emission is slightly larger in the legs of the loop. Because
also here the temperatures are not significantly higher, the
difference is due to the slightly higher density in these regions.
For Run~H1, $\tau_{\rm Spitzer}$ is large and the time
step is controlled by the Alfv\'en speed instead of the of heat flux. This leads to larger temperatures and
therefore higher emission. However, also here the density in the corona
loops is larger than in Run~R, leading not only to higher emission,
but also to a larger Alfv\'en time step, see \Tab{tab1}.
Furthermore, the high shock viscosity needed to keep the run stable
will also have an influence on the solution.
In contrast, the time steps in Runs~B03 and B1 are controlled by the time step of
the heat flux (${\rm d} {} t_1$). There, as expected, we find similar emission
loop structures as in Run~R, Ha and Ba. They are slightly larger in
Run~B1 than in Run~B03.
This means that simulations using either the automatic adjustment or a constant
value of $\tau_{\rm Spitzer}$ reproduce the emission structure of
Run~R well, as long as the time step is still controlled by the heat flux
time step ${\rm d} {} t_1$, however the emission tends to be slightly larger.
However, for too low values of the Alfv\'en limiter
($c_{\rm A}=3000\,{\rm km/s}$) the
emission and temperature become much higher than in Run~R.
We note here that the AIA 171 channel is relatively broad
filter around the mean temperature and therefore hide some of the
differences between the runs. A more narrow filters for example used
on Hinode/EIS might reveal larger differences.
\section{Discussion and conclusion}
In this work we present the new implementation of a non-Fourier
description of the heat flux to the {\sc Pencil Code}. We discuss the advantages and the
limitations using the example of 3D MHD simulations of the solar corona.
The implementation of the auto-adjustment of $\tau_{\rm Spitzer}$ is
slightly different from the implementation used in \cite{Rempel17} in
the sense that we ensure the heat flux time step to be always by a square root
of two smaller than the Alfv\'en time step,
whereas in \cite{Rempel17} there is not such a factor.
Even though a detailed comparison was not conducted here, we see
indications that our choice leads to a better stability of the
simulations.
We find that using the non-Fourier description of the heat flux alone
allows for a small speed up, because in our case the time constraint of the
Alfv\'en speed is large. For simulations with a lower magnetic
field strength, we would expect a larger speed up.
If we choose a constant $\tau_{\rm Spitzer}$, so that
the heat flux time step is four times higher than the Alfv\'en time step,
the temperatures and the emission are significantly larger than in the
other runs. This seems to be an artefact of this choice of $\tau_{\rm Spitzer}$.
We further test the implementation of the semi-relativistic Boris
correction \citep{Boris1970} as a limiter for the Alfv\'en speed.
The implementation to the {\sc Pencil Code}~ is slightly different from the one used
by \cite{Rempel17} and \cite{Gombosi02}, see \cite{CP18} for details.
The Boris correction does not quench the Alfv\'en speed at all locations
to the limit chosen, it actually reduces the peak velocities, which
are not very abundant. Therefore, this correction makes the velocity
distribution much more compact. The lower the limit, the more compact is the
velocity distribution.
Using the Boris correction allows for a significant speed up of around 10.
For higher speed up, i.e., lower limit for Alfv\'en speed, the simulation
develops higher temperatures and emission signatures than the reference
run. The auto-adjustment together with Boris correction works very
well to reproduce the temperatures and emission structures of the
reference run with a speed up of around 10 (Run~Ba).
These results convince us that we can use the non-Fourier heat flux
description together with the Boris correction to acquire a significant speed up
of the simulation without losing a correct representation of the physical
processes within the solar corona in a statistical sense.
We find some differences between the solution with and without non-Fourier heat flux
description and the Boris correction. However, we are not interested
if the non-Fourier heat flux description is identical to Fourier heat
flux description in every time step at every specific
location. Instead, we are interested if the non-Fourier heat flux
description reproduced the Fourier heat flux description on a
statistical level. On the statistical level we find a very good agreement.
In the future, we are planning to use these implementations to perform
large-scale active region simulations similar as done by \cite{BBP13,BBP14},
which can be then run for a much longer time and allowing the study of
hot core loop formations.
A first attempt is already published \citep{WP19}.
Furthermore, this implementation allows us to perform parameter
studies to investigate the coronal response to different types of
active regions on the Sun and also on other stars.
Finally, through these improvements, we get closer to the possibility
to simulate a more realistic convection-zone-corona model as started
in \cite{WKMB12,WKMB13,WKKB16}.
\section*{Acknowledgements}
We thanks the anonymous referees for the useful suggestions, we also
thank Piyali Chatterjee for the implementation of the Boris correction
to the {\sc Pencil Code}~ and detailed discussions about it. We furthermore thank Hardi Peter for discussion about the
non-Fourier heat flux description and comments to the manuscript.
The simulations have been carried out on supercomputers at
GWDG, on the Max Planck supercomputer at RZG in Garching, in the facilities hosted by the CSC---IT
Center for Science in Espoo, Finland, which are financed by the
Finnish ministry of education.
J. W.\ acknowledges funding by the Max-Planck/Princeton Center for
Plasma Physics.
\markboth{\rm {J.~WARNECKE AND S.~BINGERT}}{\rm {GEOPHYSICAL $\&$ ASTROPHYSICAL FLUID DYNAMICS}}
\bibliographystyle{gGAF}
\markboth{\rm {J.~WARNECKE AND S.~BINGERT}}{\rm {GEOPHYSICAL $\&$ ASTROPHYSICAL FLUID DYNAMICS}}
|
2,877,628,090,170 | arxiv | \section{Introduction}
Tunneling is a fundamental effect in wave mechanics, which allows for entering
classically inaccessible regions.
While textbooks focus on tunneling through potential barriers, tunneling
processes in nature often take place in the absence of such energetic barriers.
Instead one observes dynamical tunneling \cite{DavHel1981, KesSch2011}
between classically disjoint regions in phase space.
In generic Hamiltonian systems dynamical tunneling usually occurs between
regions of regular and chaotic motion.
For a typical phase space of a mixed regular--chaotic system see
Fig.~\ref{fig:RAT_Intro}(b).
\begin{figure}[b!]
\begin{center}
\includegraphics[]{fig1.eps}
\caption{(color online) (a) Regular-to-chaotic decay rate $\gamma_{0}$
versus $1/h$ for the standard map at $\K=3.4$. The numerically determined
rates (gray dots) are compared to (the sum of incoherent terms of) the
predictions of Eq.~\eqref{eq:GammaPrediction} ([red] triangles) and
Eq.~\eqref{eq:GammaPredictionAlt} ([magenta] squares). (b) Phase space with
regular orbits (lines) and a chaotic orbit (dots) including a 6:2 nonlinear
resonance chain. (c) Like (b) with an integrable approximation ([red] lines) on
top.}
\label{fig:RAT_Intro}
\end{center}
\end{figure}
In particular, while a classical particle cannot traverse from the regular
to the chaotic region, a wave can tunnel from the regular to the chaotic region.
This regular-to-chaotic tunneling process manifests itself impressively in
chaos-assisted tunneling \cite{LinBal1990, BohTomUll1993}.
Until today the importance of regular-to-chaotic tunneling has been
demonstrated in numerous experiments, including optical microcavities
\cite{ShiHarFukHenSasNar2010, ShiHarFukHenSunNar2011,
YanLeeMooLeeKimDaoLeeAn2010, KwaShiMooLeeYanAn2015, YiYuLeeKim2015,
YiYuKim2016}, microwave billiards
\cite{DemGraHeiHofRehRic2000, BaeKetLoeRobVidHoeKuhSto2008, DieGuhGutMisRic2014,
GehLoeShiBaeKetKuhSto2015}, and cold atom systems \cite{Hen2001,
SteOskRai2001}.
A recent success being the experimental verification
\cite{KwaShiMooLeeYanAn2015, GehLoeShiBaeKetKuhSto2015} that tiny nonlinear
resonance chains within the regular region, as shown in
Fig.~\ref{fig:RAT_Intro}(b), indeed drastically enhance tunneling as predicted
in Refs.~\cite{UzeNoiMar1983, Ozo1984, BroSchUll2001, BroSchUll2002}.
Furthermore, regular-to-chaotic tunneling is expected to play an important role
for atoms and molecules in strong fields, as discussed in
Refs.~\cite{ZakDelBuc1998, BucDelZak2002, WimSchEltBuc2006}.
Motivated by these applications regular-to-chaotic tunneling is also a field of
intense theoretical research \cite{ShuIke1995, ShuIke1998, PodNar2003,
PodNar2005, EltSch2005, WimSchEltBuc2006, SheFisGuaReb2006, BaeKetLoeSch2008,
BaeKetLoeRobVidHoeKuhSto2008, ShuIshIke2008, ShuIshIke2009a, ShuIshIke2009b,
BaeKetLoeWieHen2009, BaeKetLoe2010, LoeBaeKetSch2010, MerLoeBaeKetShu2013,
HanShuIke2015, KulWie2016}, which is mainly focused on periodically driven
model systems with one degree of freedom.
Here, a major achievement is the combination of (i) direct
\cite{BaeKetLoeSch2008, BaeKetLoe2010} and (ii) resonance-assisted
\cite{UzeNoiMar1983, Ozo1984, BroSchUll2001, BroSchUll2002} regular-to-chaotic
tunneling in a single prediction \cite{LoeBaeKetSch2010, SchMouUll2011}.
This prediction shows that as function of decreasing
effective Planck's constant $h$ one has two corresponding regimes:
(i) Regular states localize on a single quantizing torus.
In this regime, tunneling is determined by direct transitions from this regular
torus into the chaotic region \cite{BaeKetLoeSch2008, BaeKetLoe2010}
which can be evaluated semiclassically using complex paths
\cite{MerLoeBaeKetShu2013}.
(ii) For even smaller $h$ a regular state, while still mostly concentrated
on the main quantizing torus, acquires resonance-assisted contributions
on further quantizing tori \cite{UzeNoiMar1983, Ozo1984, BroSchUll2001,
BroSchUll2002} located more closely to the border of the regular region,
see Fig.~\ref{fig:Istates}(c) for an illustration.
This resonance-assisted contribution dominates
tunneling to the chaotic region \cite{EltSch2005, LoeBaeKetSch2010,
SchMouUll2011}.
Thus, one observes a resonance-assisted enhancement of regular-to-chaotic
tunneling.
For an example of this enhancement see Fig.~\ref{fig:RAT_Intro}(a).
Note, that for much smaller $h$ there is even a third regime for which
regular states may localize within the resonance chain. This regime is not
considered here.
Despite the above achievements, a semiclassical evaluation of resonance-assisted
tunneling in mixed regular--chaotic systems remains an open problem.
In particular, the state of the art predictions \cite{LoeBaeKetSch2010,
SchMouUll2011} defy a semiclassical evaluation using the techniques developed
for integrable systems~\cite{DeuMou2010, DeuMouSch2013}.
More specifically, so far (i) an integrable approximation of the regular region
which ignores resonance chains is used to predict the magnitude of direct
tunneling transitions from quantizing tori towards the chaotic region
\cite{BaeKetLoeSch2008, BaeKetLoe2010}.
Subsequently, (ii) resonance-assisted contributions are taken into account
by perturbatively solving \cite{BroSchUll2002} an additional pendulum
Hamiltonian which models the relevant resonance chain \cite{LoeBaeKetSch2010,
SchMouUll2011}.
However, only a perturbation-free prediction, based on a single integrable
approximation which includes the relevant resonance chain will allow for a
semiclassical evaluation of resonance-assisted regular-to-chaotic tunneling in
the spirit of Refs.~\cite{DeuMou2010, DeuMouSch2013}.
In this paper we derive such perturbation-free predictions of resonance-assisted
regular-to-chaotic tunneling.
They are based on a new class of integrable approximations $H_{r:s}$
\cite{KulLoeMerBaeKet2014} which include the dominant $r$:$s$ resonance, see
Fig.~\ref{fig:RAT_Intro}(c).
In particular, the eigenvalue equation
\begin{align}
\label{eq:HrsEigenvalue}
\widehat{H}_{r:s} \ket{m_{\text{\intindex}}} = E_m \ket{m_{\text{\intindex}}},
\end{align}
of such integrable approximations $H_{r:s}$ provides eigenstates $\ket{m_{\text{\intindex}}}$
which model the localization of regular states on the regular phase-space
region, explicitly including the resonance-assisted contributions on multiple
quantizing tori, in a non-perturbative way.
Using such states allows for extending the results of
Refs.~\cite{BaeKetLoeSch2008, BaeKetLoe2010} to the case of resonance-assisted
tunneling.
In particular, the decay rates $\gamma_m$ of metastable states which localize on
the regular phase-space region and decay via regular-to-chaotic tunneling can be
predicted according to
\begin{align}
\label{eq:GammaPrediction}
\gamma_{m} \approx \Gamma_{m}(t=1) := \big\|\widehat{P}_{\mathcal{L}} \widehat{\map} \ket{m_{\text{\intindex}}}
\big\|^2.
\end{align}
Here $\widehat{\map}$ is the time evolution operator and $\widehat{P}_{\mathcal{L}}$ is a projector onto a
leaky region $\mathcal{L}$ located in the chaotic part of phase space.
We further show that regular-to-chaotic decay rates can be predicted with
similar accuracy, when using a simplified formula which no longer contains the
time-evolution operator.
Instead it evaluates only the probability of the state $\ket{m_{\text{\intindex}}}$ on the
leaky region $\mathcal{L}$
\begin{align}
\label{eq:GammaPredictionAlt}
\gamma_{m} \approx \Gamma_{m}(t=0) := \big\|\widehat{P}_{\mathcal{L}} \ket{m_{\text{\intindex}}} \big\|^2.
\end{align}
Both perturbation-free predictions, Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt}, give good results for the standard map,
see Fig.~\ref{fig:RAT_Intro}.
In that both predictions provide the foundation for future semiclassical
predictions of resonance-assisted regular-to-chaotic tunneling
\cite{FriMerLoeBaeKet}.
We remark that the prediction of Eq.~\eqref{eq:GammaPrediction} has previously
been evaluated semiclassically for integrable approximations without resonances
\cite{MerLoeBaeKetShu2013}, using the time-domain techniques of
Refs.~\cite{ShuIke1995, ShuIke1998, ShuIshIke2008, ShuIshIke2009a,
ShuIshIke2009b} giving predictions for direct regular-to-chaotic tunneling.
However, we believe that a future semiclassical prediction of resonance-assisted
regular-to-chaotic tunneling would be easier obtained from
Eq.~\eqref{eq:GammaPredictionAlt}, since it does not involve any time
evolution and thus allows for a semiclassical evaluation using the simpler
WKB-like techniques of Refs.~\cite{DeuMou2010, DeuMouSch2013}.
The paper is organized as follows:\
In Sec.~\ref{Sec:ExampleSystem} we introduce the standard map as a paradigmatic
Hamiltonian example system with a mixed phase space.
We further present regular-to-chaotic decay rates as a measure of
regular-to-chaotic tunneling and discuss their numerical evaluation.
In Sec.~\ref{Sec:RAT} we derive the predictions,
Eqs.~\eqref{eq:GammaPrediction} and \eqref{eq:GammaPredictionAlt}.
In Sec.~\ref{Sec:ResultsStandardMap} we illustrate how these predictions are
evaluated using the example of the standard map.
In Sec.~\ref{Sec:Results} we present our results and compare them to the
perturbative predictions of Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}.
In Sec.~\ref{Sec:Discussion} we discuss the main approximations and
limitations of our approach.
A summary and outlook is given in Sec.~\ref{Sec:Summary}.
\section{Example System}
\label{Sec:ExampleSystem}
In this paper we focus on periodically driven Hamiltonian systems with one
degree of freedom, which exhibit all generic features of a mixed phase space.
Classically, the stroboscopic map
\begin{align}
\label{eq:StroboscopicMap}
U: (q_n, p_n) \mapsto (q_{n+1}, p_{n+1}),
\end{align}
describes the evolution of positions and momenta, $(q,p)$, in phase space
from time $t=n$ to $t=n+1$ over one period of the external driving.
Quantum-mechanically, the time-evolution is given by the corresponding
unitary time-evolution operator $\widehat{\map}$.
In Sec.~\ref{Sec:StandardMap} we introduce the standard map as a paradigmatic
example of a periodically driven one-degree-of-freedom system with a mixed phase
space.
In Sec.~\ref{Sec:TunnelingRatesInTheStandardMap} we introduce regular-to-chaotic
decay rates $\gamma$, as the central object of our investigation.
Furthermore, we discuss their numerical computation.
Particular attention is paid to nonlinear resonance chains and their quantum
manifestations.
\subsection{Standard Map}
\label{Sec:StandardMap}
Classically, the standard map originates from a periodically kicked Hamiltonian
with one degree of freedom $H(q,p,t) = T(p) + V(q) \sum_{n\in\mathbb{N}}
\delta(n-t)$.
Here, $\delta(\cdot)$ is the Dirac delta function.
For the standard map $T(p)=p^2/2$ and $V(q) = \K/(2\pi)^2\cos{(2\pi q)}$, where
$\K$ is the kicking strength.
Its stroboscopic map $U$ \cite{Chi1979}, Eq.~\eqref{eq:StroboscopicMap}, in
its symmetrized version is given by
\begin{subequations}
\label{eq:SMap}
\begin{align}
q_{n+1} =&\, q_n + p_n + \frac{\K}{4\pi} \sin(2\pi q_n), \\
p_{n+1} =&\, p_n + \frac{\K}{4\pi} \sin(2\pi q_n) + \frac{\K}{4\pi}
\sin(2\pi q_{n+1}),
\end{align}
\end{subequations}
where $(q_n, p_n)$ represents a phase-space point in the middle of the $n$th
kick.
For convenience the standard map is considered on a torus $(q, p) \in [0,1[
\times [-0.5, 0.5[$ with periodic boundary conditions.
In this paper we mainly focus on kicking strength $\K=3.4$.
Here, the phase space exhibits a large regular region which is centered around
an elliptic fixed point, see Fig.~\ref{fig:RAT_Intro}(b).
As expected from the KAM theorem \cite{Kol1954, Arn1963, Arn1963b, Mos1962} the
regular region consists of one-dimensional invariant tori.
Along these tori orbits of regular motion rotate around the fixed point.
These tori are interspersed by nonlinear resonance chains, wherever $s$
rotations of a regular orbit match $r$ periods of the external driving
\cite{Bir1913, LicLie1983a, Chi1979}.
For example, the standard map at $\kappa=3.4$ has a dominant $r$:$s=6$:$2$
resonance, leading to the six regular sub-regions in
Fig.~\ref{fig:RAT_Intro}(b).
Note that we choose the numbers $r$ and $s$ in the ratio $r$:$s$ such that
$r$ is the number of sub-regions of the resonance.
The region of regular motion is embedded in a region of chaotic motion.
The quantum-mechanical analogue of the stroboscopic map $U$ is the unitary
time-evolution operator \cite{BerBalTabVor1979, HanBer1980, ChaShi1986,
KeaMezRob1999, DegGra2003b, Bae2003}
\begin{align}
\label{eq:Qmap}
\hspace*{-0.2cm} \widehat{\map} = \exp\left(-i\frac{V(\hat{q})}{2\hbar}\right)
\exp\left(-i\frac{T(\hat{p})}{\hbar} \right)
\exp\left(-i\frac{V(\hat{q})}{2\hbar}\right).
\end{align}
Here, $h = 2\pi\hbar$ is the effective Planck constant and $\hat{q}$ and $\hat{p}$
are the operators of position and momentum, respectively.
Similar to the classical case we consider $\widehat{\map}$ on a toric phase space, which
leads to grids in position and momentum space \cite{BerBalTabVor1979,
HanBer1980, ChaShi1986, KeaMezRob1999, DegGra2003b, Bae2003}
\begin{subequations}
\begin{align}
\label{eq:qnDef}
\overline{q}_n &= h (n+\theta_p),\quad \text{with}\quad
\overline{q}_{n}\in[0,1[\\
\label{eq:pnDef}
\overline{p}_n &= h (n+\theta_q),\quad \,\text{with}\quad
\overline{p}_{n}\in[-0.5,0.5[,
\end{align}
\end{subequations}
with $n\in \mathbb{N}$.
This implies that the inverse of the effective Planck constant is a natural
number $1/h=N \in \mathbb{N}$, giving the dimension of the Hilbert space.
For the standard map, we choose the Bloch phase $\theta_p=0$, while $\theta_q=0$
if $N$ is even and $\theta_q=0.5$ if $N$ is odd.
This gives the finite-dimensional time-evolution operator in position
representation
\begin{align}
\label{eq:StandardQmapOnTorus}
\braOpket{\overline{q}_n}{\widehat{\map}}{\overline{q}_k} \! &= \! \\ \nonumber
& \hspace*{-1.0cm}\frac{e^{-i\pi/4}}{\sqrt{N}}
\exp\!\left(i \,2\pi N \!
\left[- \frac{V(\overline{q}_n)}{2} +
\frac{(\overline{q}_n-\overline{q}_k)^2}{2}
- \frac{V(\overline{q}_k)}{2}\right]\!\right),
\end{align}
with $n,k = 0, \dots ,N-1$.
In the following it is fundamental that eigenstates of a
mixed regular--chaotic system can be
classified according to their semiclassical localization on the regular or
chaotic region, respectively.
More specifically, chaotic states spread across the chaotic region
\cite{Per1973, Ber1977b, Vor1979}, while regular states localize on a torus
$\tau_{m}$ of the regular region which has quantizing action \cite{Boh1913,
Boh1913b, Som1916}
\begin{align}
\label{eq:BohrSommerfeld}
J_{m} := \frac{1}{2\pi}\oint_{\tau_{m}} p(q)\: \text{d}q
= (m + 1/2) \hbar,
\end{align}
labeled by an index $m\in\mathbb{N}$.
In order to account for resonance-assisted tunneling it is further indispensable
to consider the finer structure of regular states.
In particular, it will be crucial that a regular state $m$ localizes not only on a
dominant quantizing torus $J_m$.
Instead, an $r$:$s$ resonance induces additional contributions on the tori
$J_{m+kr}$ with $k\in\mathbb{Z}$, see Refs.~\cite{UzeNoiMar1983, Ozo1984,
BroSchUll2001, BroSchUll2002, WisSarArrBenBor2011, Wis2014, WisSch2015} and
references therein.
\subsection{Regular-to-Chaotic Decay Rates in the Standard Map}
\label{Sec:TunnelingRatesInTheStandardMap}
In this section we introduce regular-to-chaotic decay rates $\gamma$ of an open
system for quantifying regular-to-chaotic tunneling.
Note that in closed systems chaos-assisted tunnel splittings
\cite{BohTomUll1993} are an often-used alternative \cite{BaeKetLoe2010}.
Our general approach for defining regular-to-chaotic decay rates proceeds in
three steps:\
(a) We introduce a leaky region $\mathcal{L}$ within the chaotic part of phase
space,
(b) we determine the \emph{decay rates} of its regular states, and
(c) we classify the corresponding decay rates as \emph{regular-to-chaotic
decay rates}.
Step (c) is justified because each regular state of the open system decays by
regular-to-chaotic tunneling towards the chaotic region and subsequently
entering the leaky region within the chaotic part of phase space.
More specifically, we proceed by (a) introducing a projector $\widehat{P}_{\mathcal{L}}$ which
absorbs probability on a phase-space region $\mathcal{L}$ within the chaotic
part of phase space.
Based on this projector and the unitary time-evolution operator $\widehat{\map}$ of the
closed system we define the time-evolution operator of the open system as
\begin{align}
\label{eq:UopenNum}
\widehat{\map}_{\text{o}} = (\hat{\mathbf{1}} - \widehat{P}_{\mathcal{L}})\hat{U}(\hat{\mathbf{1}} - \widehat{P}_{\mathcal{L}}).
\end{align}
(b) We solve its eigenvalue equation
\begin{equation}
\label{eq:UopenEigenvalueNum}
\widehat{\map}_{\text{o}}\ket{m} = \exp\left(i\phi_m-\frac{\gamma_m}{2}\right)\ket{m}.
\end{equation}
Here, $\ket{m}$ represents a metastable, right eigenvector of the sub-unitary
operator $\widehat{\map}_{\text{o}}$.
The corresponding eigenvalue is determined by an eigenphase $\phi_m$ and a
decay rate $\gamma_{m}$.
The latter describes the exponential decay of $\ket{m}$ in time.
(c) We assign to each regular state $\ket{m}$ a label $m$ according to its
dominant localization on the quantizing torus $J_m$ and refer to its decay
rate $\gamma_m$ as the regular-to-chaotic decay rate.
Specifically, for the standard map (a) we use
\begin{align}
\label{eq:LeakyRegion}
\!\!\mathcal{L} := \left\{(q,p) \; \left|\right. \;\; q <
q_l\;\;\;\text{or}\;\;\;q > q_r := 1- q_l \right\},
\end{align}
and define the projector
\begin{align}
\label{eq:Pabs}
\widehat{P}_{\mathcal{L}} \ket{q} = \chi(q)\ket{q} \quad \text{with }
\chi(q) = \left\{ \begin{array}{l l}
1 & \text{for } (q, \cdot) \in \mathcal{L}\\
0 & \text{for } (q, \cdot) \notin \mathcal{L}
\end{array}
\right. .
\end{align}
Here, we choose $q_l$ close to the regular--chaotic border.
This ensures that $\gamma_m$, which depends on the choice of the leaky region
$\mathcal{L}$, is dominated by tunneling from the regular towards the chaotic
region.
For a more detailed discussion see
Sec.~\ref{Sec:BeyondRTCTunneling}.
(b) We compute the finite-dimensional matrix representation of $\widehat{\map}_{\text{o}}$ for each
value of $1/h\in\mathbb{N}$.
To this end we set all those entries in Eq.~\eqref{eq:StandardQmapOnTorus}
equal to zero, for which either $\overline{q}_{n}$ or $\overline{q}_{k}$ are in
the leaky region $\mathcal{L}$.
We diagonalize the resulting $\widehat{\map}_{\text{o}}$ numerically.
(c) The regular-to-chaotic decay rates $\gamma_{m}$ are labeled according
to the dominant localization of $\ket{m}$ on the quantizing tori $J_m = \hbar
(m + 1/2)$.
We present the numerically obtained regular-to-chaotic decay rates
$\gamma_{0}$ of the standard map at $\kappa=3.4$ as a function of the inverse
effective Planck constant ([gray] dots) in Fig.~\ref{fig:RAT_Intro}(a).
The numerical results are consistent with the expectations due to
Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}:\
(i) For $1/ h \lesssim 35$ the state $\ket{0}$ localizes on the torus
$J_0$ such that the direct tunneling from $J_0$ to $\mathcal{L}$
dominates.
In this regime, $\gamma_0$ decreases exponentially for decreasing $h$ which
is a characteristic feature of direct transitions, see Ref.~\cite{HanOttAnt1984,
BaeKetLoeSch2008, BaeKetLoe2010, MerLoeBaeKetShu2013}.
(ii) In the regime $1/ h \gtrsim 35$ tunneling is enhanced by the $6$:$2$
resonance.
For $35 \lesssim 1/h \lesssim 80$ the resonance contribution of the state
$\ket{0}$ on $J_{6}$ is significant such that direct tunneling transition
from $J_{6}$ to $\mathcal{L}$ dominates $\gamma_m$.
This leads to a peak at $1/h=53$, where the state $\ket{0}$ has half its
weight on $J_{6}$.
Finally, for $1/h\gtrsim 80$ the resonance contribution of $\ket{0}$ on
$J_{12}$ is significant such that direct tunneling from $J_{12}$ to
$\mathcal{L}$ dominates the decay rate $\gamma_m$, with a peak at
$1/h=98$.
In Fig.~\ref{fig:Results}(a,c) we show similar numerical rates ([gray] dots)
for the standard map at $\K=2.9$ and $\K=3.5$ with a dominating $10$:$3$ and
$6$:$2$ resonance, respectively.
\section{Perturbation-Free Predictions of Resonance-Assisted Regular-to-chaotic
Tunneling}
\label{Sec:RAT}
In this section we derive the perturbation-free predictions for
resonance-assisted regular-to-chaotic decay rates.
In Sec.~\ref{Sec:DerivationWithTimeEvolution} we derive
Eq.~\eqref{eq:GammaPrediction} which uses the time-evolution operator.
In Sec.~\ref{Sec:DerivationWithoutTimeEvolution} we derive
Eq.~\eqref{eq:GammaPredictionAlt} which does not use the time-evolution
operator.
\subsection{Derivation of Eq.~\eqref{eq:GammaPrediction} with Time Evolution}
\label{Sec:DerivationWithTimeEvolution}
The starting point for deriving Eq.~\eqref{eq:GammaPrediction} is the definition
of the regular-to-chaotic decay rate $\gamma_m$ from the appropriate eigenvalue
problem.
We use the same definitions as for the numerical determination of
regular-to-chaotic decay rates, see Eqs.~\eqref{eq:UopenNum} and
\eqref{eq:UopenEigenvalueNum} of Sec.~\ref{Sec:TunnelingRatesInTheStandardMap}.
They are repeated for convenience, namely a general sub-unitary operator
\begin{align}
\label{eq:Uopen}
\widehat{\map}_{\text{o}} &:= (\hat{\mathbf{1}} - \widehat{P}_{\mathcal{L}})\hat{U}(\hat{\mathbf{1}} - \widehat{P}_{\mathcal{L}}),
\end{align}
and its eigenvalue equation
\begin{align}
\label{eq:UopenEigenvalue}
\widehat{\map}_{\text{o}}\ket{m} &= \exp\left(i\phi_m-\frac{\gamma_m}{2}\right)\ket{m}.
\end{align}
Here, the unitary operator $\widehat{\map}$ describes the time evolution of a mixed
regular--chaotic system over one unit of time.
Furthermore, $\widehat{P}_{\mathcal{L}}$ is a projection operator which absorbs probability on the
leaky region $\mathcal{L}$ within the chaotic part of phase space.
For decay rates of such systems, it can be shown, that the following formula
applies, see App.~\ref{App:Derivation} for details,
\begin{equation}
\label{eq:GammaPredOpen}
\gamma_{m} = - \log\left(1 - \big\|\widehat{P}_{\mathcal{L}} \widehat{\map} \!\ket{m}
\big\|^2\right)\!
\stackrel{\gamma_m \ll 1}{\approx} \!\big\|\widehat{P}_{\mathcal{L}} \widehat{\map} \!\ket{m} \big\|^2,
\end{equation}
i.\,e., a regular-to-chaotic decay rate $\gamma_m$ (for which $\gamma_m\ll1$) is
given by the probability transfer from the regular state $\ket{m}$ into the
leaky region $\mathcal{L}$ via the unitary time-evolution operator $\widehat{\map}$.
Equation~\eqref{eq:GammaPredOpen} is as such not useful, since it still contains
the unknown eigenvector $\ket{m}$.
In particular, it would require to solve Eq.~\eqref{eq:UopenEigenvalue} which defines
$\gamma_m$ in the first place.
Hence, we proceed in the spirit of Refs.~\cite{BaeKetLoeSch2008, BaeKetLoe2010},
i.\,e., we approximate $\ket{m}$ using the eigenstates $\ket{m_{\text{\intindex}}}$ of an
integrable approximation $H_{r:s}$, leading to our prediction
Eq.~\eqref{eq:GammaPrediction}.
The novel point of this paper is the use of an integrable approximation
$H_{r:s}$, which includes the dominant $r$:$s$ resonance.
This ensures that $\ket{m_{\text{\intindex}}}$ models not only the localization of $\ket{m}$
on the main quantizing torus $J_m$ but also accounts for the
resonance-assisted contributions on the tori $J_{m+kr}$.
Precisely this extends Eq.~\eqref{eq:GammaPrediction}, as previously used in
\cite{BaeKetLoeSch2008, BaeKetLoe2010} for direct tunneling, to the regime
of resonance-assisted regular-to-chaotic tunneling in a non-perturbative way.
\subsection{Derivation of Eq.~\eqref{eq:GammaPredictionAlt} without Time Evolution}
\label{Sec:DerivationWithoutTimeEvolution}
In this section we derive Eq.~\eqref{eq:GammaPredictionAlt}.
It predicts regular-to-chaotic decay rates from the localization of the mode
$\ket{m_{\text{\intindex}}}$ on the leaky region $\mathcal{L}$.
In contrast to Eq.~\eqref{eq:GammaPrediction} it does not use the
time-evolution operator.
In that, Eq.~\eqref{eq:GammaPredictionAlt} is an ideal starting point for future
semiclassical predictions of regular-to-chaotic decay rates
\cite{FriMerLoeBaeKet} in the spirit of Refs.~\cite{DeuMou2010, DeuMouSch2013}.
In particular, it avoids the complications which arise in a semiclassical
evaluation of Eq.~\eqref{eq:GammaPrediction} due to the time-evolution operator.
We further remark that predictions like Eq.~\eqref{eq:GammaPredictionAlt} are
common for open systems.
For regular-to-chaotic decay rates they have heuristically been used,
e.\,g. in Refs.~\cite{BroSchUll2002, PodNar2003, PodNar2005, SchMouUll2011}.
Here, the main purpose of deriving Eq.~\eqref{eq:GammaPredictionAlt} is to
explicitly point out the involved approximations.
The derivation starts from an alternative definition of the sub-unitary
time-evolution operator
\begin{align}
\label{eq:UopenAlt}
\widehat{\map}'_{\text{o}} := \widehat{\map} (\hat{\mathbf{1}} - \widehat{P}_{\mathcal{L}}),
\end{align}
which satisfies the eigenvalue equation
\begin{align}
\label{eq:UopenAltEigenvalue}
\widehat{\map}'_{\text{o}}\ket{m'} = \exp\left(i\phi_{m}-\frac{\gamma_{m}}{2}\right)\ket{m'}.
\end{align}
Compare with Eqs.~\eqref{eq:Uopen} and \eqref{eq:UopenEigenvalue}.
As shown in App.~\ref{App:Isospectrality} the operators $\widehat{\map}_{\text{o}}$ and
$\widehat{\map}'_{\text{o}}$ are isospectral.
Therefore, they exhibit the same eigenvalues, which give rise to the same
regular-to-chaotic decay rates $\gamma_m$.
Furthermore, the corresponding normalized right eigenvectors can be transformed
into each other, see App.~\ref{App:Isospectrality}.
We find,
\begin{align}
\label{eq:SameLocalization}
\ket{m} =
\frac{1}{\exp\left(i\phi_{m}-\frac{\gamma_{m}}{2}\right)}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m'},
\end{align}
which implies that $\ket{m}$ and $\ket{m'}$ localize on the quantizing tori
$J_{m+kr}$ of the regular region with equal probability (for $\gamma_m\ll1$).
On the other hand, $\ket{m'}$ is the time-evolved mode $\ket{m}$ according to
\begin{align}
\label{eq:TimeEvolvedMode}
\ket{m'} = \widehat{\map} \ket{m}.
\end{align}
Inserting Eq.~\eqref{eq:TimeEvolvedMode} into Eq.~\eqref{eq:GammaPredOpen} gives
\begin{align}
\label{eq:GammaPredOpenAlt}
\gamma_{m} = - \log\left(1 - \big\|\widehat{P}_{\mathcal{L}} \ket{m'} \big\|^2\right)
\stackrel{\gamma_{m}\ll1}{\approx} \big\|\widehat{P}_{\mathcal{L}} \ket{m'} \big\|^2,
\end{align}
which shows that a regular-to-chaotic decay rate $\gamma_m$ (for which
$\gamma_m\ll1$) is equivalent to the probability to find $\ket{m'}$ on the leaky
region $\mathcal{L}$.
Similar to Eq.~\eqref{eq:GammaPredOpen}, Eq.~\eqref{eq:GammaPredOpenAlt} is as such not
helpful, because it still contains the eigenvector $\ket{m'}$.
In particular, it would require to solve Eq.~\eqref{eq:UopenAltEigenvalue} which
defines $\gamma_m$ in the first place.
Hence, we approximate the mode $\ket{m'}$ using the more accessible eigenstates
$\ket{m_{\text{\intindex}}}$ of an integrable approximation $H_{r:s}$, leading to our
prediction Eq.~\eqref{eq:GammaPredictionAlt}.
Here, the key point is again the use of integrable approximations $H_{r:s}$
which includes the relevant $r$:$s$ resonance.
Therefore, $\ket{m_{\text{\intindex}}}$ models not only the localization of $\ket{m'}$ on
the main quantizing torus $J_m$ but also its resonance-assisted contributions
on the tori $J_{m+kr}$.
Precisely this allows for predicting resonance enhanced regular-to-chaotic
decay rates from Eq.~\eqref{eq:GammaPredictionAlt} in a non-perturbative
way.
An application of the predictions, Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt}, for the standard map is demonstrated in
Sec.~\ref{Sec:ResultsStandardMap}.
The key approximation, i.\,e., modeling metastable regular states $\ket{m}$ (or
$\ket{m'}$) in terms of eigenstates $\ket{m_{\text{\intindex}}}$ of an integrable approximation
$H_{r:s}$, is discussed in Sec.~\ref{Sec:DiscussionStates}.
Moreover, a comparison of the non-perturbative predictions,
Eqs.~\eqref{eq:GammaPrediction} and \eqref{eq:GammaPredictionAlt}, to the
perturbative predictions of Refs.~\cite{LoeBaeKetSch2010,
SchMouUll2011} is given in Sec.~\ref{Sec:ResultsPerturbation}.
\section{Perturbation-free Prediction of Tunneling in the Standard Map}
\label{Sec:ResultsStandardMap}
In this section we illustrate our approach by applying it to the standard map.
In Sec.~\ref{Sec:DominantResonance}, we determine the $r$:$s$ resonance which
dominates tunneling.
In Sec.~\ref{Sec:IntegrableApproximation}, we set up an integrable approximation
including the nonlinear resonance chain using the iterative canonical
transformation method \cite{LoeLoeBaeKet2013, KulLoeMerBaeKet2014} as presented
in Ref.~\cite{KulLoeMerBaeKet2014}.
In Sec.~\ref{Sec:Quantization}, we quantize the integrable approximation and
determine its eigenstates $\ket{m_{\text{\intindex}}}$ from Eq.~\eqref{eq:HrsEigenvalue}.
Finally, the results will be discussed in the next section,
Sec.~\ref{Sec:Results}.
\subsection{Choosing the Relevant Resonance}
\label{Sec:DominantResonance}
In order to apply our prediction it is crucial to first identify the $r$:$s$
resonance which dominates the tunneling process.
A detailed discussion as to which resonance dominates tunneling in which regime,
can be found in Ref.~\cite{LoeBaeKetSch2010}.
Here, we focus on the $r$:$s$ resonance of lowest order~$r$, which
dominates the numerically and experimentally relevant regime where
$\gamma>10^{-15}$.
The area covered by the sub-regions of such a resonance can be very small, see
the inset of Fig.~\ref{fig:Results}(a).
Therefore, it is necessary to search for resonances systematically.
To this end we determine the frequencies of orbits within the regular region, as
described in Ref.~\cite{KulLoeMerBaeKet2014}.
We then identify the $r$:$s$ resonance of lowest order $r$, by searching for
the rational frequencies $2\pi s/r$ with smallest possible denominator.
Specifically, for the standard map parity implies that $r$ has to be an even
number in order to reflect the correct number of subregions forming the
resonance chain.
For the examples we consider in this paper we find a dominant
$10$:$3$ resonance for $\K=2.9$ and a dominant $6$:$2$ resonance
for both $\K=3.4$ and $\K=3.5$.
\subsection{Integrable Approximation of a Regular Region including a Resonance
Chain}
\label{Sec:IntegrableApproximation}
In order to determine an integrable approximation of the regular region which
includes the dominant $r$:$s$ resonance, we use the method introduced in
Ref.~\cite{KulLoeMerBaeKet2014}.
Here, we briefly summarize the key points.\
The integrable approximations $H_{r:s}(q, p)$ of
Ref.~\cite{KulLoeMerBaeKet2014} is generated in two steps.
First the normal-form Hamiltonian is defined as
\begin{subequations} \label{eq:HrsActAng-all}
\begin{align}
\label{eq:HrsActAng}
\hspace*{-0.4cm} \mathcal{H}_{r:s}(\theta, I) &= \mathcal{H}_{0}(I) +
2V_{r:s} \left(\frac{I}{I_{r:s}}\right)^{r/2}\!\!\!\cos(r\theta + \phi_0),\\
\label{eq:H0}
\mathcal{H}_{0}(I) &= \frac{(I-I_{r:s})^{2}}{2 M_{r:s}} +
\sum_{n=3}^{N_{\text{disp}}} h_{n}(I-I_{r:s})^{n}.
\end{align}
\end{subequations}
It contains the essential information on the regular region in the co-rotating
frame of the resonance.
This Hamiltonian is precisely the effective pendulum Hamiltonian used in
Ref.~\cite{LoeBaeKetSch2010, SchMouUll2011}.
Here, $\mathcal{H}_{0}(I)$ is a low order polynomial, chosen such that its
derivative fits the actions and frequencies of the regular region in the
co-rotating frame of the resonance.
The action of the resonant torus is $I_{r:s}$.
The parameters $M_{r:s}$ and $V_{r:s}$ are determined from the size of the
resonance regions in the mixed system as well as the stability of its central
orbit \cite{EltSch2005}.
Finally, $\phi_0$ is used to control the fix-point locations of the resonance
chain.
In a second step, a canonical transformation
\begin{align}
\label{eq:CanTrans}
\mathcal{T}: (\theta, I) \mapsto (q,p)
\end{align}
is used to adapt the tori of the effective pendulum Hamiltonian to the shape of
the regular region in $(q,p)$-space, giving the Hamilton function
\begin{align}
\label{eq:Hrs}
H_{r:s}(q,p) = \mathcal{H}_{r:s}(\mathcal{T}^{-1}(q,p)).
\end{align}
The transformation $\mathcal{T}$ is composed of:\
(i) a harmonic oscillator transformation to the fixed point of the regular
region $\mathcal{T}^{0}$, Eq.~\eqref{eq:CanTransInit}, which provides a rough
integrable approximation and
(ii) a series of canonical near-identity transformations $\mathcal{T}^{1}, ...,
\mathcal{T}^{N_{\mathcal{T}}}$, Eq.~\eqref{eq:CanTransIter}, which improve the
agreement between the shape of tori of the mixed system and the integrable
approximation.
Note that a successful prediction of decay rates requires an integrable
approximation which provides a smooth extrapolation of tori into the chaotic
region \cite{BaeKetLoeSch2008, BaeKetLoe2010}, see insets of
Fig.~\ref{fig:Results}.
This is ensured by using simple near-identity transformations
$\mathcal{T}^{1}, ..., \mathcal{T}^{N_{\mathcal{T}}}$, i.\,e., low orders $N_{q},
N_{p}$ in Eq.~\eqref{eq:CanTransIter}.
For further details the reader is referred to Ref.~\cite{KulLoeMerBaeKet2014}
and Appendix~\ref{App:IntegrableApproximation}, where it is described how the
integrable Hamiltonians for the standard map at $\K=2.9$, $\K=3.4$ and $\K=3.5$,
see insets of Fig.~\ref{fig:Results}, are generated.
\subsection{Quantization of the Integrable Approximation}
\label{Sec:Quantization}
In the following, we summarize the quantization procedure for the integrable
approximation.
The details are discussed in App.~\ref{App:IntegrableApproximationQuantum}.
In its final form, this quantization procedure is almost identical to the
approach presented in Ref.~\cite{LoeBaeKetSch2010}.
It consists of two steps:\
(Q1) The integrable approximation without resonance is used to construct states
which localize along a single quantizing torus of the regular region.
(Q2) The mixing of states, localizing along a single quantizing torus, is
described by solving the quantization of the effective pendulum Hamiltonian,
Eq.~\eqref{eq:HrsActAng-all}, introduced in Ref.~\cite{SchMouUll2011}.
Combining (Q1) and (Q2) gives the sought-after eigenstate $\ket{m_{\text{\intindex}}}$ of the
integrable approximation which includes the resonance.
More specifically:\
(Q1) We use the canonical transformation, Eq.~\eqref{eq:CanTrans}, in order to
define the function $I(q,p)$.
Its contours approximate the tori of the regular phase-space region, ignoring
the resonance chain.
It thus resembles the role of the integrable approximation, previously used in
Refs.~\cite{BaeKetLoeSch2008, BaeKetLoe2010, LoeBaeKetSch2010}.
The Weyl-quantization of this function on a phase-space torus gives a Hermitian
matrix
\begin{align}
\label{eq:WeylI}
&\braOpket{\overline{q}_n}{\hat{I}}{\overline{q}_m} = \frac{1}{2N}
\sum_{l=0}^{2N-1}
\exp\left(\frac{i}{\hbar}(\overline{q}_n\!-\overline{q}_m)\,\overline{p}_{\frac{
l } {2 }}
\right)\times \\
&\quad \quad\left[I\!\left(\frac{\overline{q}_n\!+\overline{q}_m}{2},
\overline{p}_{\frac{l}{2}}\right) + (-1)^l
I\!\left(\frac{\overline{q}_n \! +\overline{q}_m \!+ M_q}{2},
\overline{p}_{\frac{l}{2}}\right)\right]\!. \nonumber
\end{align}
Solving its eigenvalue equation gives states $\braket{\overline{q}_l}{I_n}$
which localize along a single contour of $I(q,p)$ with quantizing action $I_n
= \hbar(n+1/2)$.
These states model the localization of states along the tori of quantizing
action $J_n$ in the mixed system.
For an illustration see Fig.~\ref{fig:Istates}(a,b).
\begin{figure}[tb!]
\begin{center}
\includegraphics[]{fig2.eps}
\caption{(color online)
(a, b) Husimi representation of $\ket{I_n}$ for the standard map at $\K=3.4$
for (a) $n=0$ and (b) $n=6$ at $1/h=53$. Regular tori (gray lines) and
chaotic orbits (dots) illustrate the phase space. The quantizing tori of
$H_{r:s}$ for $V_{r:s}=0$ are shown by a thick (white) line.
(c) Approximate mode $\ket{m_{\text{\intindex}}}$ for $m=0$.}
\label{fig:Istates}
\end{center}
\end{figure}
(Q2) In the second step, we model the mixing of states
$\braket{\overline{q}_l}{I_n}$ due to the nonlinear resonance chain.
To this end, we follow Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011} and
consider the quantization of the effective pendulum Hamiltonian,
Eq.~\eqref{eq:HrsActAng-all}, given by
\begin{align}
\label{eq:HrsActAngIBasis}
\braOpket{I_m}{\widehat{\mathcal{H}}_{r:s}}{I_n} &=
\mathcal{H}_0(I_n)\,\delta_{m,n} +
V_{r:s} \left(\frac{\hbar}{I_{r:s}}\right)^{r/2} \times \\
& \hspace*{-1.5cm} \left(e^{-i \phi_{0}}
\sqrt{\frac{n!}{(n-r)!}}\,\delta_{m,n-r} +
e^{i \phi_{0}}\sqrt{\frac{(n+r)!}{n!}}\,\delta_{m,n+r}\right). \nonumber
\end{align}
Solving this eigenvalue problem gives the sought-after state in the basis of
quantizing actions $\braket{I_n}{m_{\text{\intindex}}}$.
Note, that the matrix in Eq.~\eqref{eq:HrsActAngIBasis} couples basis states
$\ket{I_n}$ and $\ket{I_{n'}}$ only if $|n'-n|=kr$.
Thus, the coefficients $\braket{I_{n}}{m_{\text{\intindex}}}$ are non-zero, only if $n = m +
kr$.
This is called the selection rule of resonance-assisted tunneling.
Combining (Q1) and (Q2) results in the mode expansion
\begin{align}
\label{eq:ModeExpansion}
\braket{\overline{q}_l}{m_{\text{\intindex}}} = \sum_{k}\braket{\overline{q}_l}{I_{m +
kr}} \braket{I_{m+kr}}{m_{\text{\intindex}}}.
\end{align}
For an illustration of a state $\ket{m_{\text{\intindex}}}$ see Fig.~\ref{fig:Istates}(c).
Note that its Husimi-function exhibits exactly the morphology discussed in
Ref.~\cite{Wis2014}.
We now make a couple of remarks:\
(a) We use the above quantization procedure, rather than directly applying the
Weyl-rule to $H_{r:s}(q,p)$, Eq.~\eqref{eq:Hrs}, in order to explicitly enforce
the selection rule of resonance-assisted tunneling.
(b) The ad-hoc two step quantization scheme avoids the problem of defining the
quantum counterpart for the canonical transformations $\mathcal{T}^{1}, ...,
\mathcal{T}^{N_{\mathcal{T}}}$, Eq.~\eqref{eq:CanTransIter}, used in the classical
construction of the integrable approximation, see
App.~\ref{App:IntegrableApproximationQuantum} for details.
(c) The above quantization is almost identical to the procedure used in
Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}.
This allows for a direct comparison to the results of
Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}, see
Sec.~\ref{Sec:ResultsPerturbation}.
(d) The quantization procedure cannot determine the relative phase between
the terms in the mode expansion of Eq.~\eqref{eq:ModeExpansion}.
In order to understand the relative phase recall:\
(i) The coefficient vector $\braket{I_{m+kr}}{m_{\text{\intindex}}}$ is determined by solving
the eigenvalue problem of Eq.~\eqref{eq:HrsActAngIBasis}.
Hence, it is determined up to a global phase $\xi_{m}$.
(ii) The coefficient vectors $\braket{\overline{q}_l}{I_{m+kr}}$ are determined
by solving the eigenvalue problem of Eq.~\eqref{eq:WeylI}.
Hence, each coefficient vector is determined up to a global phase
$\varphi_{m+kr}$.
Therefore:
(i) Changing the phase of the coefficient vector $\braket{I_{m+kr}}{m_{\text{\intindex}}}$
in Eq.~\eqref{eq:ModeExpansion} changes the global phase of
$\braket{\overline{q}_l}{m_{\text{\intindex}}}$.
This has no consequences for predicting decay rates.
However, (ii) changing the phases $\varphi_{m+kr}$ of each coefficient vector
$\braket{\overline{q}_l}{I_{m+kr}}$, changes the relative phase of
contributions in Eq.~\eqref{eq:ModeExpansion}.
This changes the interference between the contributions to the sum in
Eq.~\eqref{eq:ModeExpansion} and affects the predicted decay rates.
So far the phase issue was avoided by neglecting interference terms in the
tunneling predictions \cite{LoeBaeKetSch2010, SchMouUll2011}.
For the symmetrized standard map, we propose to define the phases as follows:\
(i) Eq.~\eqref{eq:HrsActAngIBasis} gives a real symmetric matrix.
This allows for choosing real coefficients $\braket{I_{n}}{m_{\text{\intindex}}}$ such that
$\braket{I_{m}}{m_{\text{\intindex}}}>0$.
(ii) Eq.~\eqref{eq:WeylI} also gives a real symmetric matrix.
This allows for choosing real coefficients $\braket{\overline{q}_l}{I_{n}}$.
Choosing the sign of these coefficients is discussed in
App.~\ref{App:IntegrableApproximationQuantum}.
The main idea is to exploit the eigenstates of the harmonic oscillator which
approximates the central fixed point of the regular region.
For these harmonic oscillator states the relative phase is well-defined.
Then we choose the sign of $\braket{\overline{q}_l}{I_{n}}$ such that its
overlap with the corresponding eigenstate of the harmonic oscillator is
positive, Eq.~\eqref{eq:ModeAligning}.
\section{Results}
\label{Sec:Results}
We now apply the above procedure to the standard map at $\K=2.9$, $3.4$, and
$3.5$.
This gives eigenstates $\ket{m_{\text{\intindex}}}$ which we insert into our predictions,
Eqs.~\eqref{eq:GammaPrediction} and \eqref{eq:GammaPredictionAlt}.
The necessary time-evolution operator, used in Eq.~\eqref{eq:GammaPrediction},
is given by Eq.~\eqref{eq:StandardQmapOnTorus}.
The projector is defined by Eq.~\eqref{eq:Pabs} using $q_l=0.27, 0.26, 0.25$,
respectively.
The results are shown in Fig.~\ref{fig:Results}.
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig3.eps}
\caption{(color online) Decay rates for the standard map at (a) $\K=2.9$,
(b) $\K = 3.4$, and (c) $\K=3.5$ versus the inverse effective Planck constant
$1/h$. Numerically determined rates (dots) are compared to predicted rates,
using Eq.~\eqref{eq:GammaPrediction} ([red] triangles) and
Eq.~\eqref{eq:GammaPredictionAlt} ([magenta] squares). The insets show the
corresponding phase space with regular tori ([gray] lines) and chaotic orbits
(dots) with tori of the integrable approximation ([red] lines).}
\label{fig:Results}
\end{center}
\end{figure}
The numerically determined rates and the
predicted rates are overall in good qualitative agreement.
In both cases they deviate from the exact numerical rates by at most two orders
of magnitude.
In that the accuracy of the perturbation-free predictions,
Eq.~\eqref{eq:GammaPrediction} and Eq.~\eqref{eq:GammaPredictionAlt}, is
equivalent to perturbative predictions from Refs.~\cite{LoeBaeKetSch2010,
SchMouUll2011}.
This establishes Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt} as state of the art perturbation-free predictions
of resonance-assisted regular-to-chaotic tunneling.
See Sec.~\ref{Sec:ResultsPerturbation} for a detailed comparison.
\subsection{Incoherent Predictions and Quantum Phase}
\label{Sec:ResultsIncoherent}
As discussed in Sec.~\ref{Sec:Quantization} our quantization scheme cannot
determine the relative phases between the contributions of
Eq.~\eqref{eq:ModeExpansion} for a system without time-reversal symmetry.
In the following, we discuss the consequences of such an undetermined phase for
the prediction of decay rates.
To this end we summarize our predictions, Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt}, in the following compact form
\begin{align}
\label{eq:GammaPredictionCompact}
\Gamma_{m}(t) := \big\|\widehat{P}_{\mathcal{L}} \widehat{\map}^{t} \ket{m_{\text{\intindex}}} \big\|^2,
\end{align}
where $t=1$ denotes the prediction based on time-evolution and $t=0$ denotes
the prediction without time evolution.
Now we insert the mode expansion, Eq.~\eqref{eq:ModeExpansion}, and average
over the undetermined phases $\varphi_{m+kr}$ of the coefficient vectors
$\braket{\overline{q}_l}{I_{m+kr}}$.
This gives the incoherent prediction
\begin{align}
\label{eq:GammaPredictionCompactIncoherent}
\Gamma_{m}^{\text{inc}}(t) := \sum_{k} \Gamma_{m,m+kr}^{\text{diag}}(t)
\end{align}
where the diagonal term $\Gamma_{m,n}^{\text{diag}}(t)$ is the contribution of the
state $\ket{I_n}$ to the incoherent prediction as
\begin{align}
\label{eq:GammaPredictionDiagonal}
\Gamma_{m,n}^{\text{diag}}(t) := \left|\braket{I_{n}}{m_{\text{\intindex}}}\right|^{2}
\Gamma_{n}^{\text{d}}(t)
\end{align}
and
\begin{align}
\label{eq:GammaPredictionCompactDirekt}
\Gamma_{n}^{\text{d}}(t) := \big\|\widehat{P}_{\mathcal{L}} \widehat{\map}^{t} \ket{I_n} \big\|^2.
\end{align}
is the rate of direct regular-to-chaotic tunneling as previously introduced in
Refs.~\cite{BaeKetLoeSch2008, BaeKetLoe2010}.
The results based on Eq.~\eqref{eq:GammaPredictionCompactIncoherent} are
shown in Fig.~\ref{fig:ResultsIncoherent}.
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig4.eps}
\caption{(color online) Decay rates for the standard map at (a) $\K=2.9$,
(b) $\K = 3.4$, and $\K=3.5$ versus the inverse effective Planck constant
$1/h$.
Numerically determined rates (dots) are compared to rates, predicted from
incoherent terms according to Eq.~\eqref{eq:GammaPredictionCompactIncoherent},
with $\Gamma_{m}^{\text{inc}}(1)$ ([red] pluses) and $\Gamma_{m}^{\text{inc}}(0)$ ([magenta]
crosses).
We further show predictions according to Eq.~\eqref{eq:GammaPrediction}
([gray] triangles) and Eq.~\eqref{eq:GammaPredictionAlt} ([gray] squares).
\label{fig:ResultsIncoherent}}
\end{center}
\end{figure}
As expected the incoherent predictions,
Eq.~\eqref{eq:GammaPredictionCompactIncoherent}, and the full predictions,
Eq.~\eqref{eq:GammaPredictionCompact}, agree very well in the regime where a
single diagonal contribution dominates, i.\,e., in the regime of direct
tunneling as well as the peak region.
However, in between these regions there are always two diagonal contributions of
similar magnitude, which can interfere.
It is in these regions that we observe clear deviations between the predictions
of Eq.~\eqref{eq:GammaPredictionCompact} and the incoherent predictions of
Eq.~\eqref{eq:GammaPredictionCompactIncoherent}.
In particular, for $\K=3.4$ and $\K=3.5$ the prediction of
Eq.~\eqref{eq:GammaPredictionCompact} predicts destructive interference, while
the incoherent results describe the numerical rates much better.
These results highlight the relevance of the phase factor $\varphi_{m+kr}$ for
obtaining an accurate description of decay rates even between the
resonance-assisted tunneling peaks.
In previous studies of resonance-assisted tunneling in systems with a mixed
phase space \cite{LoeBaeKetSch2010} this phase factor has been ignored by
directly employing the incoherent predictions.
Hence, a satisfactory theoretical treatment of the phase factor $\varphi_{m+kr}$
does so far not exist.
Clearly, our current approach is also insufficient.
The precise reason is not clear to us.
We expect that exploiting the symmetry of the integrable approximation in order
to find a real representation of the approximate mode
$\braket{\overline{q}_l}{m_{\text{\intindex}}}$ is too naive.
In particular, because it is used for approximating the metastable state
$\braket{\overline{q}_l}{m}$ of the open standard map, which can never admit an
entirely real representation.
For a detailed discussion of this point see Sec.~\ref{Sec:ErrorAnalysis}.
Another possibility is that the phase factor in a non-integrable system is
beyond an integrable approximation.
\subsection{Perturbative Predictions}
\label{Sec:ResultsPerturbation}
In this section, we compare our results to the perturbative
predictions of Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}.
This perturbative prediction is obtained by approximating the coefficient
$\braket{I_{m+kr}}{m_{\text{\intindex}}}$ in the incoherent prediction
Eq.~\eqref{eq:GammaPredictionCompactIncoherent} by solving
Eq.~\eqref{eq:HrsActAngIBasis} perturbatively, \cite{SchMouUll2011},
\begin{align}
\label{eq:PerturbationCoefficients}
\braket{I_{m+kr}}{m_{\text{\intindex}}} \approx \mathcal{A}_{m,m+kr}^{(r:s)} := \prod_{l=1}^{k}
\frac{\braOpket{I_{m+lr}}{\widehat{\mathcal{H}}_{r:s}}{I_{m+(l-1)r}}}{\mathcal{H
} _ { 0 } (I_m) - \mathcal{H}_{0}(I_{m+kr})},
\end{align}
Note that $\mathcal{H}_{0}(I)$ is considered in the co-rotating frame.
This leads to
\begin{align}
\label{eq:GammaPredictionCompactPerturbation}
\Gamma_{m}^{\text{per}}(t) := \sum_{k} \left|\mathcal{A}_{m,m+kr}^{(r:s)}\right|^{2}
\Gamma_{m}^{\text{d}}(t).
\end{align}
A slight difference of the above expression as compared to
Ref.~\cite{LoeBaeKetSch2010, SchMouUll2011} is the use of the projector $\widehat{P}_{\mathcal{L}}$
rather than a projector on the whole chaotic region.
Thus our prediction eliminates a free parameter from the perturbative
predictions of Refs.~\cite{LoeBaeKetSch2010, SchMouUll2011}.
The results of the perturbative predictions are presented in
Fig.~\ref{fig:ResultsPerturbation}.
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig5.eps}
\caption{(color online) Decay rates for the standard map at (a) $\K=2.9$,
(b) $\K = 3.4$, and (c) $\K=3.5$ versus the inverse effective Planck constant
$1/h$.
Numerically determined rates (dots) are compared to rates, predicted
perturbatively according to Eq.~\eqref{eq:GammaPredictionCompactPerturbation}
with $\Gamma_{m}^{\text{per}}(1)$ ([red] pluses) and $\Gamma_{m}^{\text{per}}(0)$ ([magenta]
crosses).
We further show the prediction based on incoherent terms, according to
Eq.~\eqref{eq:GammaPredictionCompactIncoherent} with $\Gamma_{m}^{\text{inc}}(1)$
([gray] pluses) and $\Gamma_{m}^{\text{inc}}(0)$ ([gray] crosses).}
\label{fig:ResultsPerturbation}
\end{center}
\end{figure}
They agree with the prediction obtained from
Eq.~\eqref{eq:GammaPredictionCompactIncoherent}, with the slight difference
that the perturbative results deviate around the peak region.
We conclude this section with a short list of advantages and disadvantages of
the perturbation-free and perturbative predictions:\
(i) The perturbation-free framework, Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt}, as well as their incoherent version,
Eq.~\eqref{eq:GammaPredictionCompactIncoherent}, predict numerical rates with
similar accuracy as the perturbative framework of Refs.~\cite{LoeBaeKetSch2010,
SchMouUll2011}.
(ii) One advantage of the perturbative prediction is the possibility to evaluate
the terms $\braket{I_{m}}{m_{\text{\intindex}}}$ analytically,
Eq.~\eqref{eq:PerturbationCoefficients}.
Yet, for practical use even the perturbative approach requires an integrable
approximation for predicting the direct rates $\Gamma_{m}^{\text{d}}$.
Hence, both predictions are equally challenging in their implementation.
(iii) Another advantage of the perturbative prediction is the possibility to
include multiple resonances into
Eq.~\eqref{eq:GammaPredictionCompactPerturbation}, which is not yet
possible for the perturbation-free predictions presented in this paper.
Note that this restriction is not too severe, because decay rates in the
experimentally and numerically accessible regimes ($\gamma>10^{-15}$) are
typically affected by a single resonance only.
Nevertheless, an extension of the perturbation-free results to the
multi-resonance regime is of theoretical interest and requires normal-form
Hamiltonians $\mathcal{H}_{r:s}$ which include multiple resonances.
(iv) The main advantage of the perturbation-free framework is that it provides
the foundation for deriving a future semiclassical prediction of
resonance-assisted regular-to-chaotic tunneling \cite{FriMerLoeBaeKet}.
\section{Discussion}
\label{Sec:Discussion}
In this section, we discuss several aspects of our results in detail.
In Sec.~\ref{Sec:BeyondRTCTunneling} we discuss the dependence of decay rates on
the choice of the leaky region.
In Sec.~\ref{Sec:DiscussionStates} we compare the metastable states $\ket{m}$
and $\ket{m'}$ to the eigenstate $\ket{m_{\text{\intindex}}}$ of an integrable approximation.
In Sec.~\ref{Sec:ErrorAnalysis} we analyze the approximation of $\ket{m}$ and
$\ket{m'}$ via $\ket{m_{\text{\intindex}}}$ more systematically.
In Sec.~\ref{Sec:PeakPosition} we comment on the predictability of peaks.
\subsection{Dependence of Decay Rates on the Leaky Region}
\label{Sec:BeyondRTCTunneling}
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig6.eps}
\caption{(color online) Numerically determined decay rates $\gamma_0$ of the
standard map at $\K=3.4$ versus the inverse effective Planck constant $1/h$
for $q_l=0.26$ ([gray] dots) and $q_l=0.1$ ([magenta] squares). (b, c) Phase
space with shaded areas showing the leaky regions corresponding to
$q_l$.
}
\label{fig:DiscussionLeakyRegionK34}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig7.eps}
\caption{(color online) Same as Fig.~\ref{fig:DiscussionLeakyRegionK34} for
$\K=2.9$ with $q_l=0.27$ ([gray] dots) and $q_l=0.1$ ([magenta] squares)}
\label{fig:DiscussionLeakyRegionK29}
\end{center}
\end{figure}
This paper focuses entirely on situations where the leaky region $\mathcal{L}$
is chosen close to the regular--chaotic border region.
However, in generic Hamiltonian systems like the standard map, the chaotic
region is interspersed with partial barriers \cite{KayMeiPer1984a,
KayMeiPer1984b}.
This leads to sticky motion in a hierarchical region surrounding the regular
region.
Furthermore, the chaotic component might be inhomogeneous and exhibit slow
classical transport.
In view of these classical phenomena, it is not surprising that the numerical
decay rates of the standard map, defined via Eqs.~\eqref{eq:UopenEigenvalue},
depend on the choice of the leaky region via the parameter $q_l$.
In order to illustrate this phenomenon, we show the numerically determined decay
rate $\gamma_{0}$ of the standard map for two choices of the leaky region
and two different $\K$ parameters in Figs.~\ref{fig:DiscussionLeakyRegionK34}
and \ref{fig:DiscussionLeakyRegionK29}, respectively.
In Fig.~\ref{fig:DiscussionLeakyRegionK34} we show results for the standard map
at $\K=3.4$.
Here, we compare (i) the regular-to-chaotic decay rates obtained for
$q_l=0.26$ (parameter used in this paper, [gray] dots) to (ii) decay rates
obtained for $q_l=0.1$ ([magenta] squares).
While the decay rates for $q_l=0.26$ exhibit a rather smooth behavior the decay
rates for $q_l=0.1$ clearly exhibit additional oscillations and some overall
suppression.
An even stronger deviation between regular-to-chaotic decay rates with
varying leaky regions is observed in Fig.~\ref{fig:DiscussionLeakyRegionK29}
for the standard map at $\K=2.9$.
Here, (i) the decay rates as obtained for $q_l=0.27$ (parameter used in this
paper, [gray] dots) are compared to (ii) the decay rates obtained for $q_l=0.1$
([magenta] squares).
In addition to oscillations, the decay rates for $q_l=0.1$ exhibit a clear
suppression of their average value.
The origin of these deviations is unclear.
The suppression of decay rates for leaky regions far from the regular--chaotic
border could be due to slow transport through an inhomogeneous chaotic region
from the regular--chaotic border towards the leaky region.
So far a quantitative prediction of decay rates with leaky regions far from the
regular--chaotic border remains an open problem.
While varying the leaky region $\mathcal{L}$ close to the regular--chaotic
border can be accounted for by our approach, predicting decay rates with leaky
region far from the regular--chaotic border is beyond our framework.
In particular, while we observe that the numerical decay rates stabilize when
pushing the leaky region away from the regular--chaotic border, the predicted
rates continue to decrease exponentially.
So far the best approach for dealing with this problem is to use an effective
prediction \cite{LoeBaeKetSch2010, SchMouUll2011}.
To this end one argues that the numerical decay rate would not change much upon
pushing the boundary of the leaky region $\mathcal{L}$ beyond some effectively
enlarged regular region $\mathcal{R}_{\text{eff}}$.
See Ref.~\cite{SchMouUll2011} for a discussion of $\mathcal{R}_{\text{eff}}$.
Successively one would approximate the projector onto the leaky region
$\mathcal{L}$ in our predictions by the projector onto the complement of the
effectively enlarged regular region $\mathcal{L}_{\text{eff}}$.
This would result in an effective prediction $\Gamma_{m}^{\text{eff}}$.
Yet, there are several problems with such effective predictions:\
(a) Even though there are semiclassical arguments to define the effectively
enlarged regular region in terms of partial barriers \cite{SchMouUll2011},
replacing the leaky region $\mathcal{L}$ with some effective region
$\mathcal{L}_{\text{eff}}$ introduces an effective parameter to the prediction.
(b) Throughout this paper, we used leaky regions $\mathcal{L}$ which were
almost tangential to the effectively enlarged regular regions discussed in
Ref.~\cite{SchMouUll2011}.
Hence, replacing the region $\mathcal{L}$ with the effective region
$\mathcal{L}_{\text{eff}}$ would not give results which are too far away from
the predictions discussed in this paper, i.\,e., even the effective predictions
$\Gamma_{m}^{\text{eff}}$ clearly deviates from numerically determined decay
rates with leaky regions far from the regular region.
(c) Even when using $\mathcal{L}_{\text{eff}}$ as a free fit parameter the
effective prediction $\Gamma_{m}^{\text{eff}}$ can at most capture the average
behavior of numerical decay rates with leaky region far from the
regular--chaotic border.
In particular, the oscillations observed for the numerical rates in
Figs.~\ref{fig:DiscussionLeakyRegionK34} and \ref{fig:DiscussionLeakyRegionK29}
which span up to four orders of magnitude cannot be accounted for even by an
effective theory.
Note that, accurately predicting decay rates based on
Eqs.~\eqref{eq:GammaPrediction} and \eqref{eq:GammaPredictionAlt}, even for
leaky regions far from the regular region, requires modes $\ket{m_{\text{\intindex}}}$ which
model the localization of $\ket{m}$ and $\ket{m'}$ even in the chaotic region.
We expect that this is beyond the framework of an integrable approximation.
\subsection{Metastable States and Integrable Eigenstates}
\label{Sec:DiscussionStates}
We now discuss the key approximation of our predictions.
To this end we compare the metastable states $\ket{m}$ and $\ket{m'}$
to the corresponding approximate state $\ket{m_{\text{\intindex}}}$,
which originates from an integrable approximation $H_{r:s}$ including
the relevant resonance.
We focus on a typical example using the states $m = m' = m_{\text{\intindex}} = 0$ of the
standard map at $\K=3.4$ with $1/h=55$ close to the first resonance peak in
Fig.~\ref{fig:RAT_Intro}(c).
The absolute squared values of the states in position representation are
shown in Fig.~\ref{fig:DiscussionStates}.
Here we compare (a) $\ket{m}$ to $\ket{m_{\text{\intindex}}}$, (b) $\ket{m'}$ to $\ket{m_{\text{\intindex}}}$,
and (c) $\ket{m'} = \widehat{\map}\ket{m}$ to $\widehat{\map}\ket{m_{\text{\intindex}}}$, depicting them by
(gray) dots and (magenta) squares, respectively.
\begin{figure}[tb]
\begin{center}
\includegraphics[]{fig8.eps}
\caption{(color online) For the standard map at $\K=3.4$ with $1/h = 55$
we compare the position representation of
(a) $|\langle q|m\rangle|^2$ to $|\langle q|m_{\text{\intindex}}\rangle|^2$,
(b) $|\langle q|m'\rangle|^2$ to $|\langle q|m_{\text{\intindex}}\rangle|^2$, and
(c) $|\langle q|\widehat{\map}|m\rangle|^2$ to $|\langle q|\widehat{\map}|m_{\text{\intindex}}\rangle|^2$,
depicting them by (gray) dots and (magenta) squares, respectively, for $m = m' =
m_{\text{\intindex}} = 0$. The dashed lines mark the positions $q_l$ and $1-q_l$ of the leaky
region, as given in the text.}
\label{fig:DiscussionStates}
\end{center}
\end{figure}
As a first conclusion we see that the metastable states are well approximated by
their integrable partners within the non-leaky region, i.\,e., the region
between the dashed lines in Figs.~\ref{fig:DiscussionStates}(a-c).
In particular, both the metastable states and their integrable approximations
exhibit the generic structure which is determined by the regular region and the
dominant $6$:$2$ resonance \cite{BroSchUll2001, BroSchUll2002}:\
(i) A main Gaussian-like hump at $q=0.5$ marks the main localization of the
modes on the torus $J_0$.
(ii) The decrease of the hump is interrupted at two side humps, which correspond
to the resonance-assisted contribution of each mode on the torus $J_6$.
From there, the Gaussian-like exponential decrease continues towards the leaky
region, which is outside the dashed lines in
Figs.~\ref{fig:DiscussionStates}(a-c).
As a second conclusion from Fig.~\ref{fig:DiscussionStates} we infer that beyond
the regular--chaotic border, i.\,e., within the leaky region the metastable
states deviate from their integrable counter parts.
Here, the integrable states continue to decrease exponentially.
In contrast, the state $\ket{m}$ vanishes, see Fig.~\ref{fig:DiscussionStates}(a),
while the state $\ket{m'}=\widehat{\map}\ket{m}$, Eq.~\eqref{eq:TimeEvolvedMode},
does not decrease much slower, see Fig.~\ref{fig:DiscussionStates}(b,c).
Finally, we emphasize that $\ket{m'}$ and $\ket{m_{\text{\intindex}}}$ agree for positions
close to the regular--chaotic border.
Furthermore, these contributions dominate the probability of $\ket{m'}$ and
$\ket{m_{\text{\intindex}}}$ on the leaky region.
Precisely this ensures that replacing $\ket{m'}$ in the exact prediction,
Eq.~\eqref{eq:GammaPredOpenAlt}, by $\ket{m_{\text{\intindex}}}$ results in a meaningful
prediction according to Eq.~\eqref{eq:GammaPredictionAlt}.
An analogous argument explains why replacing $\widehat{\map}\ket{m}$ in the exact result,
Eq.~\eqref{eq:GammaPredOpen}, by $\widehat{\map}\ket{m_{\text{\intindex}}}$ gives meaningful predictions
according to Eq.~\eqref{eq:GammaPrediction}.
\subsection{Error Analysis}
\label{Sec:ErrorAnalysis}
In this section, we investigate the approximation of the metastable states
$\ket{m}$ in the exact result Eq.~\eqref{eq:GammaPredOpen} via the mode
$\ket{m_{\text{\intindex}}}$ in Eq.~\eqref{eq:GammaPrediction} from the perspective of
Eq.~\eqref{eq:ModeExpansion}, i.\,e.,
(i) we investigate the basis states $\ket{I_n}$ and (ii) the expansion
coefficients $\braket{I_n}{m_{\text{\intindex}}}$.
We focus on the standard map at $\kappa=3.4$.
(i) In order to investigate our basis set $\ket{I_n}$, we expand the metastable
state $\ket{m}$ in this basis and insert this expansion into the exact
result~\eqref{eq:GammaPredOpen}.
This gives
\begin{align}
\label{eq:GammaPredOpenExpanded}
\gamma_{m} &= \sum_{n} \left|\braket{I_n}{m}\right|^{2} \big\|\widehat{P}_{\mathcal{L}} \widehat{\map}
\ket{I_n} \big\|^2 \\ \nonumber
&+ \sum_{n,n'} \braket{m}{I_{n'}}
\braOpket{I_{n'}}{\widehat{\map}^{\dagger}\widehat{P}_{\mathcal{L}}^2\widehat{\map}}{I_{n}}\braket{I_n}{m}.
\end{align}
Since the diagonal terms
\begin{align}
\label{eq:GammaOpenDiag}
\gamma_{m,n}^{\text{diag}} := \left|\braket{I_n}{m}\right|^{2}\big\|\widehat{P}_{\mathcal{L}} \widehat{\map}
\ket{I_n} \big\|^2 = \left|\braket{I_n}{m}\right|^{2}
\Gamma_{n}^{\text{d}}(1)\end{align}
provide a bound to the off-diagonal terms according to Cauchy's inequality
\begin{align}
\left|\braket{m}{I_{n'}}
\braOpket{I_{n'}}{\widehat{\map}^{\dagger}\widehat{P}_{\mathcal{L}}^2\widehat{\map}}{I_{n}}\braket{I_n}{m}\right| \le
\sqrt{\gamma_{m,n}^{\text{diag}} \gamma_{m,n'}^{\text{diag}}}
\end{align}
we can interpret them as a way to quantify the contribution of the $n$th basis
state $\ket{I_n}$ to the decay rate $\gamma_{m}$.
In that $\gamma_{m,n}^{\text{diag}}$ takes a similar role as the contribution
spectrum,
discussed in Ref.~\cite{HanShuIke2015}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[]{fig9.eps}
\caption{(color online) Error analysis for the standard map at $\K=3.4$.
(a,b,c) The numerically determined rates ([gray] dots) and (d) the numerically
determined phases $\text{Arg}(\braket{I_n}{0})$ for $n=0,1,2$ (lines) are shown
versus the inverse effective Planck constant $1/h$.
(a) The contributions $\gamma_{0,n}^{\text{diag}}$, Eq.~\eqref{eq:GammaOpenDiag}, are
shown by lines.
(b) The reduced prediction, Eq.~\eqref{eq:GammaPredOpenExpanded} with
$n,n'\in\{0,6,12\}$ is shown by (red) triangles.
(c) The contributions $\gamma_{0,n}^{\text{diag}}$ of Eq.~\eqref{eq:GammaOpenDiag}
(lines), are compared to $\Gamma_{0,n}^{\text{diag}}(1)$ of
Eq.~\eqref{eq:GammaPredictionDiagonal} (markers) for $n=0,6,12$.
(d) The phases $\text{Arg}(\braket{I_n}{0})$ (lines) and
$\text{Arg}(\braket{I_n}{0_{\text{int}}})$ (markers) are compared for $n=0,6,12$.
[Phases close to zero are slightly shifted for $n=6,12$ for visibility.]
}
\label{fig:ErrorAnalysis}
\end{center}
\end{figure}
In Fig.~\ref{fig:ErrorAnalysis}(a), we consider all contributions
$\gamma_{0,n}^{\text{diag}}$ (lines) in comparison with the
decay rate $\gamma_{0}$ (dots) for the standard map at $\K=3.4$.
While most contributions are two to three orders of magnitude smaller than
$\gamma_{0}$, we find that the contributions $\gamma_{0,0}^{\text{diag}}$,
$\gamma_{0,6}^{\text{diag}}$, and $\gamma_{0,12}^{\text{diag}}$ dominate.
In order to further test whether the modes $\ket{I_{n}}$ with $n=0,6,12$ are
sufficient for describing $\gamma_{0}$ we sum the contributions $n,n'\in
\{0,6,12\}$ of the dominant terms in Eq.~\eqref{eq:GammaPredOpenExpanded}.
This gives the red curve of Fig.~\ref{fig:ErrorAnalysis}(b).
From this numerical observations we conclude that a reasonable description of
$\gamma_{0}$ can be extracted using an approximate mode exclusively
composed of states $\ket{I_n}$ with $n=0,6,12,...$, as used in
Eq.~\eqref{eq:ModeExpansion}.
However, it should be noted that the difference between $\gamma_{0}$ and its
reduced version, based on contributions $n,n'\in \{0,6,12\}$ in
Eq.~\eqref{eq:GammaPredOpenExpanded}, is already of the order of $\gamma_{0}$
itself.
See the region $70<1/h<100$ of Fig.~\ref{fig:ErrorAnalysis}(b) in
particular.
Hence, reducing the metastable state $\ket{m}$ to an approximate mode
$\ket{m_{\text{\intindex}}}$ using only basis states $\ket{I_{m+kr}}$ as in
Eq.~\eqref{eq:ModeExpansion} can at best provide a reasonable backbone for
describing the structure of $\gamma_{0}$.
On the other hand, for our example a prediction of $\gamma_{0}$ where the
remainder is smaller than the decay rate based on a reduced set of basis states
$\ket{I_n}$ is only possible when summing over many additional contributions,
even including $n\neq m+kr$.
The precise origin of such contributions $\gamma_{m,n}^{\text{diag}}$ with
$n\neq m+kr$ is currently under debate \cite{HanShuIke2015}:\
From the framework of resonance-assisted tunneling~\cite{BroSchUll2001,
BroSchUll2002, LoeBaeKetSch2010, SchMouUll2011}, we expect that the overlap
$\braket{I_n}{m}$ vanishes for $n\neq m+kr$.
Hence, one might argue that the contributions $\gamma_{m,n}^{\text{diag}}$ with $n\neq
m+kr$ arise in our example only because our basis $\ket{I_n}$ is insufficiently
accurate to decompose $\ket{m}$ according to the theoretical expectation of
resonance-assisted tunneling.
On the other hand, the authors of of Ref.~\cite{HanShuIke2015} observe
non-vanishing contributions $\braket{I_n}{m}$ also for $n\neq m+kr$ even for a
near-integrable situation, where an excellent integrable approximations exist.
They argue that non-vanishing $\braket{I_n}{m}$ should always occur and claim
their treatment is beyond the current framework of resonance-assisted tunneling.
Independent of the origin of the non-zero contributions
$\gamma_{m,n}^{\text{diag}}$ for $n\neq m+kr$, their theoretical description
is beyond the scope of this paper.
In our examples the irrelevance of these contributions is ensured by choosing
leaky regions close to the regular--chaotic border.
However, for leaky regions far from the regular--chaotic border the
contributions $\gamma_{m,n}^{\text{diag}}$ with $n\neq m+kr$ become relevant.
(ii) In the next step we evaluate the errors introduced by replacing the
expansion coefficients $\braket{I_{m+kr}}{m}$ by $\braket{I_{m+kr}}{m_{\text{\intindex}}}$ in
Eq.~\eqref{eq:GammaPredOpenExpanded}.
We focus on the corresponding diagonal contributions $\gamma_{m,m+kr}^{\text{diag}}$
and $\Gamma_{m,m+kr}^{\text{diag}}$, which represent the squared norm of the expansion
coefficients $\braket{I_{m+kr}}{m}$ and $\braket{I_{m+kr}}{m_{\text{\intindex}}}$ up to a
multiplication by the direct rate $\Gamma_{m+kr}^{\text{d}}(1)$.
See lines and symbols in Fig.~\ref{fig:ErrorAnalysis}(c), respectively.
From this data we conclude that the norm of $\braket{I_{m+kr}}{m_{\text{\intindex}}}$ provides
a reasonable approximations for the norm of the expansion coefficients
$\braket{I_{m+kr}}{m}$.
The deviations before each peak could be due to neglecting the higher order
action dependencies discussed in Ref.~\cite{SchMouUll2011} in the Hamilton
function of Eq.~\eqref{eq:HrsActAng-all}.
Furthermore, we expect that the slightly broader peaks in the numerical rates
$\gamma_{0,kr}^{\text{diag}}$ as compared to the sharper peaks of
$\Gamma_{0,kr}^{\text{diag}}(1)$ observed for the integrable approximation, are
related to the openness of the mixed system.
Finally, in Fig.~\ref{fig:ErrorAnalysis}(d) we compare the phases
$\text{Arg}(\braket{I_{m+kr}}{m})$ and $\text{Arg}(\braket{I_{m+kr}}{m_{\text{\intindex}}})$, for $m=0$ and
$k=0,1,2$, respectively.
Here, $\text{Arg}(\cdot)\in(-\pi,\pi]$ is the principal value of the complex argument
function.
Note that the global phase of $\ket{m}$ is fixed by setting
$\text{Arg}(\braket{I_{m}}{m})=0$.
The phases $\text{Arg}(\braket{I_{m+kr}}{m_{\text{\intindex}}})$ are fixed as described in
App.~\ref{App:IntegrableApproximationQuantum}.
While the phases of $\text{Arg}(\braket{I_{m+kr}}{m_{\text{\intindex}}})$ jump from $\pi$ to zero
upon traversing the peak for decreasing $1/h$ (change from destructive to
constructive interference) their counterparts for $\text{Arg}(\braket{I_{m+kr}}{m})$
seem to follow this jump in a smoothed out way.
Compare symbols and lines in Fig.~\ref{fig:ErrorAnalysis}(d).
We attribute this phase detuning to the openness of the system, i.\,e.:\
(a) The symmetries of the integrable approximation allow for choosing a real
representation of the coefficient $\braket{I_{m+kr}}{m_{\text{\intindex}}}$.
Its phase can thus only take values
$\text{Arg}(\braket{I_{m+kr}}{m_{\text{\intindex}}})\in\{0,\pi\}$.
In contrast (b) the mode $\ket{m}$ originates from an open system and thus the
coefficient $\braket{I_{m+kr}}{m}$ are usually complex such that
$\text{Arg}(\braket{I_{m+kr}}{m})$ might take any value.
While the deviation between the numerically determined phases $\text{Arg}( \braket{
I_{m+kr}}{m})$ and the theoretically predicted phases $\text{Arg}(\braket{ I_{m+kr}}{
m_{\text{\intindex}}})$ are seemingly small in Fig.~\ref{fig:ErrorAnalysis}(d), their deviation
has huge effects on the predicted decay rate, i.\,e.:\
(a) Eq.~\eqref{eq:GammaPrediction} predicts destructive interference of the
diagonal terms in the region before each peak.
This leads to strong deviations from the numerical decay rate, see
Fig.~\ref{fig:Results}(b).
On the other hand, (b) already the minimal detuning of
$\text{Arg}(\braket{I_{m+kr}}{m})$ from our prediction
$\text{Arg}(\braket{I_{m+kr}}{m_{\text{\intindex}}})$ is sufficient to lift the destructive
interference.
We assume that this explains why the incoherent prediction,
Eq.~\eqref{eq:GammaPredictionCompactIncoherent}, as illustrated in
Fig.~\ref{fig:ResultsIncoherent}, describe the numerical rates
much better than predictions according to, Eq.~\eqref{eq:GammaPrediction}, see
Fig.~\ref{fig:ResultsIncoherent}.
\subsection{Predictability of Peak Positions}
\label{Sec:PeakPosition}
Finally, we discuss the predictability of peak positions.
To this end we recall that $\mathcal{H}_{0}(I)$ in Eq.~\eqref{eq:H0}, is
determined by fitting its derivative to the numerically determined actions and
frequencies $(\bar{\omega}, \bar{J})$ of the regular phase-space region in
the co-rotating frame.
For an illustration see Fig.~\ref{fig:dispersion}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[]{fig10.eps}
\caption{(color online) For the standard map at $\K=3.4$ we show (a) the fit
of $\mathcal{H}'_{0}(I)$ (line) to the actions and frequencies of the regular
region $(\bar{J},\bar{\omega})$ (crosses).
(b) The function $\mathcal{H}_{0}(I)$ is shown as a (red) line. The two (black)
dots show $(I_{m}, \mathcal{H}_{0}(I_{m}))$ and $(I_{m+kr},
\mathcal{H}_{0}(I_{m+kr}))$ at $1/h=53$.
(a, b) The dotted line shows the position of $\bar{J}_{\text{max}}$.
}
\label{fig:dispersion}
\end{center}
\end{figure}
In particular, the data of the mixed system has a maximal action
$\bar{J}_{\text{max}}$, see (gray) dotted line in Fig.~\ref{fig:dispersion}.
Hence, $\mathcal{H}_{0}$ can be well controlled in the regular region
$I<\bar{J}_{\text{max}}$.
However, for $I>\bar{J}_{\text{max}}$ the function $\mathcal{H}_{0}$ is only an
extrapolation to the chaotic region.
Furthermore, the integrable approximation predicts a peak for $\gamma_{m}$
\cite{BroSchUll2001, BroSchUll2002, LoeBaeKetSch2010, SchMouUll2011}, if
\begin{align}
\mathcal{H}_{0}(I_m) = \mathcal{H}_{0}(I_{m+kr}),
\end{align}
where $I_m=\hbar(m+1/2)$ and $I_{m+kr}=\hbar(m+kr+1/2)$.
This resonance conditions follows from Eq.~\eqref{eq:PerturbationCoefficients}.
However, for all examples presented in this paper the resonant torus
$I_{m+kr}$ is always located outside of the regular region, where
$\mathcal{H}_{0}(I)$ is only given by an extrapolation.
See Fig.~\ref{fig:Istates}(c) for an example of this situation.
The (black) dots in Fig.~\ref{fig:dispersion}(b) show the corresponding
situation for $\mathcal{H}_{0}(I)$.
In such a situation our approach cannot guarantee an accurate prediction of the
peak position.
Usually, this problem is not too severe and the extrapolation is good enough.
An example where this problem appears can be seen in the second peak of
Fig.~\ref{fig:Results}(a) where the peak of the numerical decay rates and the
predicted rates is shifted by $1/h=1$.
\section{Summary and Outlook}
\label{Sec:Summary}
In this paper we present two perturbation-free predictions of resonance-assisted
regular-to-chaotic decay rates, Eqs.~\eqref{eq:GammaPrediction} and
\eqref{eq:GammaPredictionAlt}.
Both predictions are based on eigenstates $\ket{m_{\text{\intindex}}}$ of an integrable
approximation $H_{r:s}$, Eq.~\eqref{eq:HrsEigenvalue}.
The key point is the use of an integrable approximation $H_{r:s}$ of the mixed
regular--chaotic system which includes the relevant nonlinear resonance chain.
Therefore $\ket{m_{\text{\intindex}}}$ models the localization of regular modes on the regular
region, including resonance-assisted contributions in a non-perturbative way.
This allows for extending the validity of Eq.~\eqref{eq:GammaPrediction},
previously used for direct tunneling in Refs.~\cite{BaeKetLoeSch2008,
BaeKetLoe2010}, to the regime of resonance-assisted tunneling.
Furthermore, we introduce a second prediction,
Eq.~\eqref{eq:GammaPredictionAlt}, which no longer requires the time-evolution
operator.
Instead it allows for predicting decay rates using the localization of the
approximate mode on the leaky region.
In that Eq.~\eqref{eq:GammaPredictionAlt} provides an excellent foundation for a
future semiclassical prediction of resonance-assisted regular-to-chaotic
decay rates \cite{FriMerLoeBaeKet} in the spirit of Refs.~\cite{DeuMou2010,
DeuMouSch2013}.
The validity of the presented approach is verified for the standard map, where
predicted and numerically determined regular-to-chaotic decay rates show
good agreement.
Finally, we list future challenges:\
(a) The presented approach is so far limited to periodically driven systems
with one degree of freedom.
An extension to autonomous or periodically driven systems with two or more
degrees of freedom is an interesting open problem.
(b) The perturbation-free approach applies to the experimentally and
numerically relevant regime, where a single resonance dominates
regular-to-chaotic tunneling.
Its extension to the semiclassical regime where multiple-resonances affect
tunneling is of theoretical interest.
(c) The suppression of decay rates due to partial barriers is so far treated by
choosing leaky regions close to the regular--chaotic region.
Explicitly predicting the additional suppression of decay rates due to slow chaotic
transport through inhomogeneous chaotic regions remains an open question.
\begin{acknowledgments}
We gratefully acknowledge fruitful discussions with
J{\'e}r{\'e}my Le Deunff,
Felix Fritzsch,
Yasutaka Hanada,
Hiromitsu Harada,
Kensuke Ikeda,
Martin K{\"o}rber,
Steffen L\"ock,
Amaury Mouchet,
Peter Schlagheck,
and
Akira Shudo.
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) Grant No.\
BA 1973/4-1.
N.M.\ acknowledges successive support by JSPS (Japan) Grant No.\ PE 14701 and
Deutsche Forschungsgemeinschaft (DFG) Grant No.\ ME 4587/1-1.
\end{acknowledgments}
\begin{appendix}
\section{Derivation of Eq.~\eqref{eq:GammaPredOpen}}
\label{App:Derivation}
In this appendix we derive Eq.~\eqref{eq:GammaPredOpen} starting from
Eqs.~\eqref{eq:Uopen} and \eqref{eq:UopenEigenvalue}.
Taking the norm of the eigenvalue equation~\eqref{eq:UopenEigenvalue}
for a normalized state $\ket{m}$ one finds
\begin{align}
\exp{\left(-\gamma_{m}\right)}
&= \big\|\widehat{\map}_{\text{o}}\ket{m}\big\|^{2}
= \left\langle m \right| \widehat{\map}_{\text{o}}^{\dagger}\widehat{\map}_{\text{o}} \ket{m} \\
&\hspace*{-1.7cm}
= \left\langle m \right| (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})^{\dagger}
\widehat{\map}^{\dagger}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})^{\dagger}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map}
(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m}, \nonumber
\end{align}
where in the last step the definition of $\widehat{\map}_{\text{o}}$, Eq.~\eqref{eq:Uopen}, is used.
We simplify this expression using
\begin{align}
\label{eq:ProjectiveInvariance}
(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m} = \ket{m},
\end{align}
which follows from Eqs.~\eqref{eq:Uopen} and \eqref{eq:UopenEigenvalue}, giving
\begin{align}
\exp{\left(-\gamma_{m}\right)}
&= \left\langle m \right|
\widehat{\map}^{\dagger}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})^{\dagger}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map} \ket{m}.
\end{align}
Finally, exploiting the idempotence and hermiticity of the projector $\widehat{P}_{\mathcal{L}}$ gives
\begin{align}
\exp{\left(-\gamma_{m}\right)}
&= \left\langle m \right| \widehat{\map}^{\dagger}\widehat{\map}\ket{m}
- \left\langle m \right| \widehat{\map}^{\dagger} \widehat{P}_{\mathcal{L}} \widehat{\map} \ket{m} \nonumber\\
&= 1 - \|\widehat{P}_{\mathcal{L}}\widehat{\map}\ket{m}\|^{2},
\end{align}
where in the last step the unitarity of $\widehat{\map}$ is used. From
this follows the expression for regular-to-chaotic tunneling
rates, Eq.~\eqref{eq:GammaPredOpen}.
\section{Isospectrality}
\label{App:Isospectrality}
In this appendix, we demonstrate the isospectrality of the sub-unitary operators
$\widehat{\map}_{\text{o}}$ and $\widehat{\map}'_{\text{o}}$ as defined by Eqs.~\eqref{eq:Uopen} and
\eqref{eq:UopenAlt}, respectively.
Furthermore, we discuss the transformation relating their eigenmodes.
For convenience, we repeat the corresponding eigenvalue equations
\eqref{eq:UopenEigenvalue} and \eqref{eq:UopenAltEigenvalue}
\begin{align}
\label{eq:UopenEigenvalueAppendix}
\widehat{\map}_{\text{o}} \ket{m} &= \lambda_m \ket{m}, \\
\label{eq:UopenAltEigenvalueAppendix}
\widehat{\map}'_{\text{o}} \ket{m'} &= \lambda_{m}' \ket{m'},
\end{align}
where the eigenvalues have been denoted by $\lambda_m$ and
$\lambda_m'$.
We now demonstrate the isospectrality of $\widehat{\map}_{\text{o}}$ and $\widehat{\map}'_{\text{o}}$.
To this end we show:\
(a) For each eigenstate $\ket{m}$ of $\widehat{\map}_{\text{o}}$ with eigenvalue
$\lambda_m$, $\widehat{\map}\ket{m}$ is an eigenstate of $\widehat{\map}'_{\text{o}}$ with the same eigenvalue
$\lambda_m$
\begin{align}
\widehat{\map}'_{\text{o}}\widehat{\map}\ket{m}
&\stackrel{\eqref{eq:UopenAlt}}{=}\widehat{\map}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map}
\ket{m} \\
&\stackrel{\eqref{eq:ProjectiveInvariance}}{=} \widehat{\map}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map}
(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})
\ket{m} \\
&\stackrel{\eqref{eq:Uopen}}{=} \widehat{\map}\widehat{\map}_{\text{o}}\ket{m} \\
&\stackrel{\eqref{eq:UopenEigenvalueAppendix}}{=} \lambda_m \widehat{\map}
\ket{m}.
\end{align}
This further shows that the normalized eigenmode $\ket{m}$ of $\widehat{\map}_{\text{o}}$ with
eigenvalue $\lambda_m$ gives a normalized eigenmode $\ket{m'}$ of $\widehat{\map}'_{\text{o}}$
with eigenvalue $\lambda_m$ according to Eq.~\eqref{eq:TimeEvolvedMode}.
(b) For each eigenstate $\ket{m'}$ of $\widehat{\map}'_{\text{o}}$ with eigenvalue
$\lambda_m'$, the state $(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m'}$ is an eigenstate of $\widehat{\map}_{\text{o}}$
with
the same eigenvalue $\lambda_m'$
\begin{align}
\widehat{\map}_{\text{o}} (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m'}
&\stackrel{\eqref{eq:Uopen}}{=}(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map}
(\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})^2 \ket{m'} \\
&\;= (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map} (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\ket{m'} \\
&\stackrel{\eqref{eq:UopenAlt}}{=} (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})\widehat{\map}'_{\text{o}}\ket{m'} \\
&\stackrel{\eqref{eq:UopenAltEigenvalueAppendix}}{=} \lambda_m' (\hat{\mathbf{1}}-\widehat{P}_{\mathcal{L}})
\ket{m'}.
\end{align}
This further shows that for non-zero eigenvalue $\lambda_m'$ the normalized
eigenmode $\ket{m'}$ of $\widehat{\map}'_{\text{o}}$ gives a normalized eigenmode $\ket{m}$ of
$\widehat{\map}_{\text{o}}$ according to Eq.~\eqref{eq:SameLocalization}.
\section{Details of the Integrable Approximation}
\label{App:IntegrableApproximation}
In this appendix we summarize some technical aspects on the integrable
approximation.
Computational details of the classical integrable approximation as well as
slight changes as compared to Ref.~\cite{KulLoeMerBaeKet2014} are given in
Sec.~\ref{App:IntegrableApproximationClassical}.
Details of the quantization are discussed in
Sec.~\ref{App:IntegrableApproximationQuantum}.
\subsection{Details of the Classical Integrable Approximation}
\label{App:IntegrableApproximationClassical}
We now summarize the modifications of the algorithm described in
\cite{KulLoeMerBaeKet2014} in order to account for the symmetries of our system.
Then we give a list of relevant computational parameters.
\subsubsection{Symmetrization}
In agreement with Ref.~\cite{KulLoeMerBaeKet2014} the canonical transformation
$\mathcal{T}$, Eq.~\eqref{eq:CanTrans}, is composed of (i) an initial canonical
transformation
\begin{align}
\label{eq:CanTransInit}
\mathcal{T}^{0}: (\theta, I) \mapsto (Q,P)
\end{align}
which provides a rough integrable approximation of the regular phase-space
region and (ii) a series of canonical near-identity transformations
\begin{align}
\label{eq:CanTransIter}
\mathcal{T}' \equiv \mathcal{T}^{N_{\mathcal{T}}}\circ...\circ\mathcal{T}^{1}: (Q, P)
\mapsto (q,p),
\end{align}
which improve the agreement between the shape of the tori of the mixed system
and the integrable approximation.
In contrast to Ref.~\cite{KulLoeMerBaeKet2014} we use the symmetrized standard
map in this paper.
In order to account for this symmetry, we specify the canonical transformation,
Eq.~\eqref{eq:CanTransInit}, as
\begin{align}
\label{eq:CanTransInitHO}
\mathcal{T}^{0}:
\begin{pmatrix}
\theta \\ I
\end{pmatrix}
\mapsto
\begin{pmatrix}
Q \\ P
\end{pmatrix}
=
\begin{pmatrix}
q^{\star} + \sqrt{2I/\sigma}\cos(\theta) \\
p^{\star} - \sqrt{2I\sigma}\sin(\theta)
\end{pmatrix}
\end{align}
Here, $(q^{\star}, p^{\star}) = (0.5, 0.0)$ are the coordinates of the central
fixed point in the standard map.
The parameter $\sigma$ is determined from the stability matrix of the standard
map at $(q^{\star}, p^{\star})$
\begin{align}
\mathcal{M} =
\begin{pmatrix}
1-\frac{\K}{2} & 1 \\
- \K (1 - \frac{\K}{4}) & 1-\frac{\K}{2}
\end{pmatrix}
\end{align}
as \cite{LoeLoeBaeKet2013}
\begin{align}
\sigma^{2} =
\frac{\left|1+\frac{\K}{2}\right|-\left|1-\frac{\K}{2}\right|}{
\left|1+\frac{\K}{2}\right|+\left|1-\frac{\K}{2}\right|}.
\end{align}
Furthermore, the symmetry of the systems is imposed on the transformations
$\mathcal{T}^{1}, ..., \mathcal{T}^{N_{\mathcal{T}}}$, Eq.~\eqref{eq:CanTransIter},
by specifying their generating function as
\begin{align}
\label{eq:CanTransFamily}
F^{a}(q,p') &=\\ \nonumber
qp' + &\sum_{n=1}^{N_{q}}\sum_{m=1}^{N_{p}} a_{m,n}
\sin(2\pi n[q-q^{\star}]) \sin(2\pi m[p'-p^{\star}]),
\end{align}
rather than using the more general form of
Ref.~\cite[Eq.~(31)]{KulLoeMerBaeKet2014}.
\subsubsection{Algorithmic Overview}
(Ai) We determine the parameters $I_{r:s}$, $M_{r:s}$, $V_{r:s}$, $\phi_{0}$,
Eq.~\eqref{eq:HrsActAng-all} as described in Ref.~\cite{EltSch2005}.
(Aii) We determine $\mathcal{H}_{0}(I)$, Eq.~\eqref{eq:H0}, by fitting it to
$N_{\text{disp}}$ tuples of action and frequency $(\bar{J},\bar{\omega})$
describing the tori of the regular region in the co-rotating frame of the
resonance.
(Aiii) We determine the near-identity transformations of
Eq.~\eqref{eq:CanTransIter}.
Initially, this requires sampling of the regular region using $N_{\text{ang}}$
points along $N_{\text{tori}}$ tori.
The invertibility of the near-identity transformations in a certain
phase-space region is ensured by rescaling the coefficients $a_{m,n}\mapsto
\eta a_{m,n}$ in Eq.~\eqref{eq:CanTransFamily} using a damping factor $\eta$.
If $N_{q}$, $N_{p}$ in Eq.~\eqref{eq:CanTransFamily} are too large, the tori of
the integrable approximation form curls and tendrils in the chaotic region.
In that case the integrable approximation cannot predict decay rates.
We control this problem by choosing the largest possible parameters $N_{q}$,
$N_{p}$ for which the tori of the integrable approximation provide a smooth
extrapolation into the chaotic phase-space region.
After a finite amount of steps $N_{\mathcal{T}}$, the canonical transformations
do not improve the agreement between the regular region and the integrable
approximation.
At this point we terminate the algorithm.
\subsubsection{Computational Parameters}
\label{App:IntegrableApproximationListOfParameters}
In the following we list the important parameters of the integrable
approximation.
For $\K=2.9$ we use $I_{r:s}=0.009223$, $M_{r:s}=0.06243$, $V_{r:s}=-1.655 \cdot
10^{-7}$, and $\phi_{0}=\pi$.
For $\mathcal{H}_{0}$ in Eq.~\eqref{eq:H0} we used $N_{\text{disp}}=4$ and
fit its derivative to $N_{\text{disp}}=120$ tori of noble frequency.
We use $N_{\mathcal{T}}=40$ near-identity transformations,
Eq.~\eqref{eq:CanTransIter}, generated from Eq.~\eqref{eq:CanTransFamily} with
$N_{q}=N_{p}=2$ and coefficients rescaled by $\eta=0.05$.
The regular region was sampled using $N_{\text{ang}}=200$ points along
$N_{\text{tori}}=120$ tori, equidistantly distributed in action.
For $\K=3.4$ we use $I_{r:s}=0.01026$, $M_{r:s}=-0.047$, $V_{r:s}=-1.612 \cdot
10^{-5}$, and $\phi_{0}=0$.
For $\mathcal{H}_{0}$ in Eq.~\eqref{eq:H0} we use $N_{\text{disp}}=6$ and fit
its derivative to $N_{\text{disp}}=120$ tori, equidistantly distributed in
action.
We use $N_{\mathcal{T}}=15$ near-identity transformations,
Eq.~\eqref{eq:CanTransIter}, generated from Eq.~\eqref{eq:CanTransFamily} with
$N_{q}=N_{p}=2$ and coefficients rescaled by $\eta=0.25$.
The regular region is sampled using $N_{\text{ang}}=300$ points along
$N_{\text{tori}}=120$ tori, equidistantly distributed in action.
For $\K=3.5$ we use $I_{r:s}=0.01244$, $M_{r:s}=-0.048$, $V_{r:s}=-2.98 \cdot
10^{-5}$, and $\phi_{0}=0$.
For $\mathcal{H}_{0}$ in Eq.~\eqref{eq:H0} we use $N_{\text{disp}}=4$ and fit
its derivate to $N_{\text{disp}}=120$ tori, equidistantly distributed in
action.
We use $N_{\mathcal{T}}=15$ near-identity transformations,
Eq.~\eqref{eq:CanTransIter}, generated from Eq.~\eqref{eq:CanTransFamily} with
$N_{q}=N_{p}=2$ and coefficients rescaled by $\eta=0.25$.
The regular region is sampled using $N_{\text{ang}}=300$ points along
$N_{\text{tori}}=120$ tori, equidistantly distributed in action.
\subsubsection{Robustness}
After fixing all parameters as described above the final integrable
approximation might differ, depending on the sampling of the regular region.
In order to show that this does not affect the final prediction, we evaluate
Eq.~\eqref{eq:GammaPrediction}, for three integrable approximations which are
based on slightly different sets of sample points.
The result is illustrated in Fig.~\ref{fig:Robustness}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[]{fig11.eps}
\caption{(color online) Decay rates $\gamma_{0}$ for the standard map at
$\K=3.4$ versus the inverse effective Planck constant.
Numerically determined rates ([gray] circles) are compared to predicted rates
according to Eq.~\eqref{eq:GammaPrediction} ([colored] symbols) based on three
slightly different integrable approximations.
}
\label{fig:Robustness}
\end{center}
\end{figure}
It shows that the prediction is clearly robust.
\subsection{Derivation of Quantization}
\label{App:IntegrableApproximationQuantum}
In the following we sketch the basic ideas leading to the quantization
procedure presented in Sec.~\ref{Sec:Quantization}.
To this end we first present the quantization of the Hamilton-function obtained
after the transformation $\mathcal{T}^{0}$, Eqs.~\eqref{eq:CanTransInit} and
\eqref{eq:CanTransInitHO}, in Sec.~\ref{App:IntegrableApproximationQuantumT0}.
In Sec.~\ref{App:IntegrableApproximationQuantumT} we present how we extend
these results to the full transformation $\mathcal{T}$.
\subsubsection{Quantization after $\mathcal{T}^{0}$}
\label{App:IntegrableApproximationQuantumT0}
To quantize the Hamilton-function $H_{r:s}^{(0)}(Q,P)$
obtained after the canonical transformation $\mathcal{T}^{0}$,
Eqs.~\eqref{eq:CanTransInit} and \eqref{eq:CanTransInitHO},
we follow Ref.~\cite{SchMouUll2011} by starting with the
transformed Hamilton-function
\begin{align}
H_{r:s}^{(0)}(Q,P) &= \mathcal{H}_{0}\left(\frac{Q^2 + P^2}{2}\right) +
\frac{V_{r:s}}{\left(2I_{r:s}\right)^{r/2}} \times \\ \nonumber
& \left[\exp(i\phi_0) \left(\sigma^{1/2}[Q-q^{\star}] -
i\frac{P}{\sigma^{1/2}}\right)^r\right. \\ \nonumber
&\left.+ \exp(-i\phi_0) \left(\sigma^{1/2}[Q-q^{\star}]
+ i\frac{P}{\sigma^{1/2}}\right)^r \right].
\end{align}
In order to quantize this function we replace the coordinates $(Q,P)$ by
operators
\begin{subequations}
\begin{align}
Q \mapsto \widehat{Q}\\
P \mapsto \widehat{P}
\end{align}
\end{subequations}
and demand the usual commutation relation
\begin{align}
\label{eq:CommutatrQP}
[\widehat{Q}, \widehat{P}] = i\hbar.
\end{align}
This allows for introducing the corresponding ladder operators as
\begin{subequations}
\label{eq:LadderoperatorQP}
\begin{align}
\widehat{a} &:=
\frac{1}{(2\hbar)^{1/2}}\left(\sigma^{1/2}[\widehat{Q}-q^{\star}] +
i\frac{\widehat{P}}{\sigma^{1/2}}\right)\\
\widehat{a}^{\dagger} &:=
\frac{1}{(2\hbar)^{1/2}}\left(\sigma^{1/2}[\widehat{Q}-q^{\star}] -
i\frac{\widehat{P}}{\sigma^{1/2}}\right)
\end{align}
\end{subequations}
which admit the commutator
\begin{align}
\label{eq:CommutatorLadderQP}
[\widehat{a}, \widehat{a}^{\dagger}] = 1,
\end{align}
such that we get the number operator
\begin{align}
\label{eq:numberOpQP}
\widehat{n} := \widehat{a}^{\dagger}\widehat{a}.
\end{align}
Based on these operators the quantization of $H_{r:s}^{(0)}$ takes the form
\cite{SchMouUll2011}
\begin{align}
\label{eq:HrsQP}
\widehat{H}_{r:s}^{(0)} &=
\mathcal{H}_{0}(\widehat{I}) \\ \nonumber
&+ V_{r:s}\left(\frac{\hbar}{I_{r:s}}\right)^{\frac{r}{2}}
\left[\widehat{a}^{\dagger^{r}}\exp(i\phi_0) +
\widehat{a}^{r}\exp(-i\phi_0)\right],
\end{align}
where
\begin{align}
\widehat{I} := \hbar(\widehat{n} + 1/2)
\end{align}
is the operator replacing the unperturbed action $I$.
Finally, in order to define the basis states, we identify them with the
eigenstates of the number operator leading to
\begin{align}
\label{eq:eigenvalueNumberOpQP}
\widehat{I}\Ket{I_n^{(0)}} &= I_n \Ket{I_n^{(0)}},
\end{align}
where the eigenvalues become quantizing actions $I_n=\hbar(n+1/2)$ and the basis
states $\Ket{I_n^{(0)}}$ fulfill
\begin{subequations}
\label{eq:basisstatesQP}
\begin{align}
\widehat{a}\Ket{I_0^{(0)}} &= 0\\
\Ket{I_n^{(0)}} &= \frac{1}{\sqrt{n!}}\widehat{a}^{\dagger^{n}}\Ket{I_0^{(0)}}.
\end{align}
\end{subequations}
With respect to this position basis $\Ket{I_n^{(0)}}$ become the eigenstates of
the harmonic oscillator
\begin{align}
\label{eq:basisstatesQP_Qrepr}
\!\!\!\BraKet{Q}{I_n^{(0)}} =
\left(\frac{\sigma}{\pi\hbar}\right)^{\frac{1}{4}} \!\!\!\!\frac{1}{\sqrt{2^n
n!}} \;\text{H}_{n}\!\left(\!\!\frac{Q}{\sqrt{\hbar/\sigma}}\!\!\right)
\exp{\!\left(\!\!-\frac{\sigma Q^2}{2\hbar}\right)},
\end{align}
where $\text{H}_{n}(\cdot)$ are the Hermite polynomials.
\subsubsection{Quantization after $\mathcal{T}$}
\label{App:IntegrableApproximationQuantumT}
Our final goal is of course to obtain the quantization of the Hamilton-function
$H_{r:s}(q,p)$ which is related to $H_{r:s}^{(0)}(Q,P)$ via the canonical
transformation $\mathcal{T}'$, Eq.~\eqref{eq:CanTransIter}.
In order to obtain its quantization we assume that $\mathcal{T}'$
quantum-mechanically corresponds to a unitary operator
$\widehat{U}_{\mathcal{T}'}$ which has the following properties:
\begin{subequations}
\label{eq:UT}
\begin{align}
\widehat{U}_{\mathcal{T}'}^{-1} &= \widehat{U}_{\mathcal{T}'^{-1}}\\
\widehat{Q}' &=
\widehat{U}_{\mathcal{T}'}\widehat{Q}\widehat{U}_{\mathcal{T}'}^{-1}\\
\widehat{P}' &=
\widehat{U}_{\mathcal{T}'}\widehat{P}\widehat{U}_{\mathcal{T}'}^{-1}
\end{align}
\end{subequations}
Such an operator exists at least within a semiclassical approximation
\cite{Bog1992}.
Note that $\widehat{Q}', \widehat{P}'$ represent the operators $\widehat{Q},
\widehat{P}$ within the final coordinate frame $(q,p)$.
However, they must not be confused with the operators $\widehat{q}, \widehat{p}$
which give rise to the position and momentum basis in the final coordinate
frame $(q,p)$.
In particular, while $\widehat{q}\ket{q}=q\ket{q}$, $\widehat{Q}\ket{q}\neq
q\ket{q}$.
Under the above assumption the transformed operators preserve the commutation
relation
\begin{align}
\label{eq:Commutatrqp}
[\widehat{Q}', \widehat{P}'] = i\hbar.
\end{align}
Furthermore, we get the transformed ladder operators as
\begin{subequations}
\label{eq:Ladderoperatorqp}
\begin{align}
\widehat{a}' &:=
\widehat{U}_{\mathcal{T}'}\widehat{a}\widehat{U}_{\mathcal{T}'}^{-1}\\
\widehat{a}'^{\dagger} &:=
\widehat{U}_{\mathcal{T}'}\widehat{a}^{\dagger}\widehat{U}_{\mathcal{T}'}^{-1}
\end{align}
\end{subequations}
which admit the same commutator
\begin{align}
\label{eq:CommutatorLadderqp}
[\widehat{a}', \widehat{a}'^{\dagger}] = 1,
\end{align}
such that we get the transformed number operator
\begin{align}
\label{eq:numberOpqp}
\widehat{n}' =
\widehat{U}_{\mathcal{T}'}\widehat{n}\widehat{U}_{\mathcal{T}'}^{-1},
\end{align}
and the transformed action operator
\begin{align}
\widehat{I}' := \hbar(\widehat{n}' + 1/2).
\end{align}
Based on these operators we can define the transformation of the quantization of
$H_{r:s}^{(0)}(Q,P)$ which we identify with the quantization of
$H_{r:s}(q,p)$.
It takes the form \cite{SchMouUll2011}
\begin{align}
\label{eq:Hrsqp}
\widehat{H}_{r:s} &=
\mathcal{H}_{0}(\widehat{I}') \\ \nonumber
&+ V_{r:s}\left(\frac{\hbar}{I_{r:s}}\right)^{\frac{r}{2}}
\left[\widehat{a}'^{\dagger^{r}}\exp(i\phi_0) +
\widehat{a}'^{r}\exp(-i\phi_0)\right].
\end{align}
Finally, in order to define the basis states $\ket{I_n}$, we identify them
with the eigenstates of the number operator $\widehat{n}'$, such that
\begin{align}
\label{eq:eigenvalueNumberOpqp}
\widehat{I}'\ket{I_n} &= I_n \ket{I_n}
\end{align}
with the basis states $\ket{I_n}$ which admit the property
\begin{subequations}
\label{eq:basisstatesqp}
\begin{align}
\widehat{a}'\ket{I_0} &= 0\\
\ket{I_n} &= \frac{1}{\sqrt{n!}}\widehat{a}'^{\dagger^{n}}\ket{I_0}.
\end{align}
\end{subequations}
Evaluating $\widehat{H}_{r:s}$, Eq.~\eqref{eq:Hrsqp} in the basis of
$\ket{I_n}$, based on Eqs.~\eqref{eq:basisstatesqp} gives the matrix
representation of Eq.~\eqref{eq:HrsActAngIBasis}.
Finally, for connecting $\widehat{H}_{r:s}$ and $\widehat{\map}$ we require the basis
states with respect to the basis $\ket{q}$.
To this end, one can show from the above equations that
\begin{align}
\label{eq:basisstates_transformed_qp}
\Ket{I_n} &= \widehat{U}_{\mathcal{T}'}\Ket{I_n^{(0)}},
\end{align}
such that
\begin{align}
\label{eq:QuantumCanonicalTrafoOfStates}
\BraKet{q}{I_n} = \int \text{d}Q \BraOpKet{q}{\widehat{U}_{\mathcal{T}'}}{Q}
\BraKet{Q}{I_n^{(0)}}.
\end{align}
In principle, the operator $\BraOpKet{q}{\widehat{U}_{\mathcal{T}'}}{q'}$ can be
evaluated semiclassically, using the techniques described in
Ref.~\cite{Bog1992}.
However, this does not give an analytical closed form result and its evaluation
is numerically extremely tedious.
Furthermore, $\widehat{U}_{\mathcal{T}'}$ is usually so close to an identity
transformation such that a semiclassical evaluation of
$\BraOpKet{q}{\widehat{U}_{\mathcal{T}'}}{Q}$ contains too many turning points.
Hence, we take an alternative approach, which is numerically feasible:\
(i) We recognize that the states $\ket{I_n}$ are the eigenstates of the operator
$\widehat{I}$, originating from the phase-space coordinate $I$.
(ii) We define the function $I(q,p)$ which is obtained after the full canonical
transformation $\mathcal{T}$.
(iii) We define the Weyl-quantization of this function on a phase-space torus
giving the hermitian matrix of Eq.~\eqref{eq:WeylI}.
(iv) We diagonalize this matrix numerically, yielding the states
$\BraKet{\overline{q}_l}{I_n}$.
Finally, obtaining the modes $\BraKet{\overline{q}_l}{I_n}$ from an eigenvalue
equation comes at the cost that their relative phase (usually ensured via
Eq.~\eqref{eq:basisstatesqp} or alternatively via Eqs.~\eqref{eq:basisstatesQP}
and \eqref{eq:basisstates_transformed_qp}) is lost.
For the standard map we try to restore this phase by exploiting the symmetry of
Eq.~\eqref{eq:WeylI}, which for our system becomes a real symmetric matrix.
In that we can ensure that the coefficient vector $\BraKet{\overline{q}_l}{I_n}$
can be chosen real.
Finally, we fix the sign of this coefficient vector, by aligning
it with the mode $\BraKet{Q}{I_n^{(0)}}$ defined via
Eq.~\eqref{eq:basisstatesQP_Qrepr}.
This means, we choose the sign of the coefficient vector
$\BraKet{\overline{q}_l}{I_n}$ such that the following relation is
fulfilled
\begin{align}
\label{eq:ModeAligning}
\sum_{n} \BraKet{I_n}{\overline{q}_l}
\left.\left[\BraKet{Q}{I_n^{(0)}}\right]\right|_{Q=\overline{q}_l} > 0.
\end{align}
This assumes that the unitary operator representing the quantum
canonical transformation in Eq.~\eqref{eq:QuantumCanonicalTrafoOfStates} is
sufficiently close to an identity transformation
$\widehat{U}_{\mathcal{T}'}\approx1$.
\end{appendix}
|
2,877,628,090,171 | arxiv | \section{Supplemental Materials}
\pagenumbering{arabic}
\renewcommand{\thepage}{S-\arabic{page}}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{page}{1}
\makeatletter
\renewcommand{\theequation}{S\arabic{equation}}
\renewcommand{\thefigure}{S\arabic{figure}}
\section{I. Bogoliubov-de Gennes Quasiparticle Spectrum}
In the Nambu basis $\hat{\Psi}_\mathbf{k}=[\hat{c}_{\mathbf{k},\uparrow},\hat{c}_{\mathbf{k},\downarrow},\hat{c}^\dagger_{-\mathbf{k},\uparrow},\hat{c}^\dagger_{-\mathbf{k},\downarrow}]^T$, the Hamiltonian Eq. \eqref{eq:Ham} takes the form $\hat{H}=\frac{1}{2}\sum_{\mathbf{k}}\hat{\Psi}^\dagger_{\mathbf{k}}H_{\text{BdG}}(\mathbf{k})\hat{\Psi}_{\mathbf{k}}$, where the one-body Bogoliubov-de Gennes (BdG) Hamiltonian is
\begin{equation}
H_{\text{BdG}}(\mathbf{k})=\begin{bmatrix}
h(\mathbf{k})&\Delta\\
\Delta&-h^*(-\mathbf{k})
\end{bmatrix}.
\end{equation}
The quasi-particle energy spectrum is determined by the eigenvalues of $H_{\text{BdG}}(\mathbf{k})$,
\begin{align}
E_{\pm,\pm}(\mathbf{k})=\pm \bigg[&|h(\mathbf{k})|^2+\mu^2+\Delta_0^2\pm 2\sqrt{\mu^2|h(\mathbf{k})|^2+\Delta_0^2h_z^2(\mathbf{k})}\bigg]^{\frac{1}{2}},
\end{align}
where $|h(\mathbf{k})|^2=\sum_i h_{i}^2(\mathbf{k})$. The quasi-particle spectrum consists of four bands. In the lightly-doped and weak-coupling regime, $|t|\gg |\mu| \gg |\Delta_0|$, there are two linear band crossings at zero-energy in the reduced Brillouin zone ($-\pi\leq k_{x,y}<\pi$, $0\leq k_z< \pi$) located at $\vec{k}_{N/S}\approx \left(0,0,k_{N/S}\right)$, where $k_N=K_0+ q_{\text{node}}$ and $k_S=K_0-q_{\text{node}}$, $q_{\text{node}}=\sqrt{q_F^2+(\Delta_0/\hbar v_F)^2}$, and the Fermi wavevector $q_F\equiv \mu/(\hbar v_F)$.
\section{II. Wavefunction of the Zero Mode Without Projection}
We study the vortex problem in a monopole superconductor and investigate the zero-energy vortex core states.
Assuming the vortex line is along the $z$-axis, $k_z$ is conserved because of translation symmetry along the vortex line.
Define the operator $\hat{\psi}_{k_z\sigma}(\vr_\parallel)=(1/\sqrt{A}) \sum_{k_x,k_y}e^{i(k_x x+k_y y)}\hat{c}_{\mathbf{k}\sigma}$, where $A$ is the area of the system on the $x$-$y$ plane. For simplicity, we use a low-energy continuum model around $\pm K_0$ for the band Hamiltonian,
\begin{equation}
h_{\pm \mathbf{K}_{0,z}+q_z \hat{\mathbf{z}}}(\vr_\parallel)
=\hbar
v_F(-i\partial_x\sigma_x-i\partial_y\sigma_y\pm q_z\sigma_z)-\mu I,
\end{equation}
where $q_z$ is measured from the band Weyl points
momenta $\pm \mathbf{K_0}$.
The BdG Hamiltonian reads
\begin{equation}
\hat{H}_{\text{BdG}} = \sum_{k_z>0} \int d^2 r_\parallel
\hat{\Psi}^\dagger_{k_z} (\mathbf{r}_\parallel)
h_{\text{BdG}}(k_z,\mathbf{r}_\parallel)
\hat{\Psi}_{k_z}( \mathbf{r}_\parallel),
\end{equation}
where the four-component Nambu spinor operator is
$\hat{\Psi}^\dagger_{k_z}(\mathbf{r}_\parallel) =
\Big[\hat{\psi}^\dagger_{k_z,\uparrow}(\mathbf{r}_\parallel),
\hat{\psi}^\dagger_{k_z,\downarrow}(\mathbf{r}_\parallel),
\hat{\psi}_{-k_z,\uparrow}(\mathbf{r}_\parallel),
\hat{\psi}_{-k_z,\downarrow}(\mathbf{r}_\parallel) \Big]^T$,
and the summation over $k_z$ only covers $k_z$.
The matrix kernel $h_{\text{BdG}}(k_z,\mathbf{r}_\parallel)$
is defined as
\begin{equation}
h_{\text{BdG}}(k_z,\vr_\parallel)=
\begin{bmatrix}
h_{k_z}(\vr_\parallel)&\Delta(\vr_\parallel)i \sigma_y\\
-\Delta^*(\vr_\parallel)i\sigma_y&-h_{-k_z}^*(\vr_\parallel)
\end{bmatrix}.
\end{equation}
In order to obtain the quasi-particle excitation spectrum, we perform a Bogoliubov transformation
to solve for the eigenfunction $\psi_{n,k_z}(\mathbf{r}_\parallel)$
\begin{align}
h_{\text{BdG}}(k_z,\mathbf{r}_\parallel)\psi_{n,k_z}(\mathbf{r}_\parallel)
=E_{n,k_z}\psi_{n,k_z}(\mathbf{r}_\parallel),
\end{align}
where $n$ runs over all the eigenstates. Then $\hat{H}_{\text{BdG}}$ is diagonal,
$\hat{H}=\sum_{n,k_z}E_{n,k_z}\hat{\gamma}^\dagger_{n,k_z}
\hat{\gamma}_{n,k_z}$.
We seek the zero-energy vortex bound state solutions to the BdG equation. Note that the Hamiltonian possesses the particle-hole symmetry $\mathcal{C}h_{\text{BdG}}(k_z,\mathbf{r}_\parallel)\mathcal{C}^{-1} =-h_{\text{BdG}}^*(-k_z,\mathbf{r}_\parallel)=-h_{\text{BdG}}^*(k_z,\mathbf{r}_\parallel)$,
where $\mathcal{C}=\tau_xK$ and $K$ is the complex conjugation operator. Therefore, for every solution $\psi_{n,k_z}(\mathbf{r}_\parallel)$ to
$H_{\text{BdG}}(k_z,\mathbf{r}_\parallel)$ with energy $E_{n,k_z}$, there exists another solution $\mathcal{C}\psi_{n,k_z}$ with energy $-E_{n,k_z}$. The zero-energy solutions can be arranged to satisfy $\psi_{0,k_z}=\mathcal{C}\psi_{0,k_z}$.
In other words, we look for the zero-energy solutions of the form $\psi_{0,k_z}(\mathbf{r}_\parallel)=[u_{0,k_z\uparrow}(\mathbf{r}_\parallel),u_{0,k_z\downarrow}(\mathbf{r}_\parallel),
u^*_{0,k_z\uparrow}(\mathbf{r}_\parallel),u^*_{0,k_z\downarrow}(\mathbf{r}_\parallel)]^T$.
For every momentum slice at $k_z=+K_0+q_z$, we consider the exponentially decaying solution corresponding to a vortex bound state,
\begin{equation}\label{eq: exact exponential}
\psi_{0,k_z}(\vr_\parallel)=e^{-\frac{1}{\hbar v_F}\int_0^{r_\parallel} d\rho^\prime\Delta(\rho^\prime)}\chi_{k_z}(\vr_\parallel).
\end{equation}
When $|q_z|<q_F$, $\chi_{k_z}(\vr_\parallel)$ is solved analytically as
\begin{equation}\label{eq: exact J}
\chi_{K_0+q_z}(\vr_\parallel)=\mathcal{N}_{1,k_z}
\begin{bmatrix}
A e^{-i\frac{\pi}{4}}J_0(k_{F,\parallel}r_\parallel)\\
B e^{i\frac{\pi}{4}}e^{i\phi_\vr}J_1(k_{F,\parallel}r_\parallel)\\
A e^{i\frac{\pi}{4}}J_0(k_{F,\parallel}r_\parallel)\\
B e^{-i\frac{\pi}{4}}e^{-i\phi_\vr}J_1(k_{F,\parallel}r_\parallel)
\end{bmatrix},
\end{equation}
where $A=\sqrt{1+q_z/q_F}$, $B=\sqrt{1-q_z/q_F}$, $J_{0,1}(z)$ are the zeroth and first order Bessel functions, respectively, and $k_{F,\parallel}=\sqrt{\left|q_F^2-q_z^2\right|}$ is the in-plane component of the Fermi wavevector.
When $q_F<|q_z|<q_{\text{node}}$, the solutions for $\psi_{0,k_z}$ are non-oscillating, with
\begin{equation}
\label{eq: exact I}
\chi_{K_0+q_z}(\vr_\parallel)=\mathcal{N}_{2,k_z}
\begin{bmatrix}
C e^{-i\frac{\pi}{4}}I_0(k_{F,\parallel}r_\parallel)\\
D e^{i\frac{\pi}{4}}e^{i\phi_\vr}I_1(k_{F,\parallel}r_\parallel)\\
C e^{i\frac{\pi}{4}}I_0(k_{F,\parallel}r_\parallel)\\
D e^{-i\frac{\pi}{4}}e^{-i\phi_\vr}I_1(k_{F,\parallel}r_\parallel)
\end{bmatrix},
\end{equation}
where $I_{0,1}(z)$ are respectively the zeroth and first order modified Bessel functions of the first kind, and $C=\sqrt{1+q_z/q_F} \left(\sqrt{-1-q_z/q_F}\right)$ and $D=-\sqrt{-1+q_z/q_F}\left(\sqrt{1-q_z/q_F}\right)$ for $q_z>q_F$ $(q_z<-q_F)$.
When $|q_z|>q_{\text{node}}$, there is no zero-energy solution.
\subsection{Normalization Constant}
The states $\psi_{0,k_z}(\vr_\parallel)$ in Eq. \eqref{eq: exact exponential} are normalized according to $\int d^2 \vr_{\parallel} \psi^\dagger_{0,k_z}(\vr_{\parallel})\psi_{0,k_z}(\vr_{\parallel})=1$. When $|q_z|<q_F$, the normalisation constant is
given by
\begin{equation}
|\mathcal{N}_{1,k_z}|^2=\frac{1}{4\pi}\frac{1}{A^2S_0+B^2S_1},
\end{equation}
where $S_\nu$ is the integral defined by
\begin{align}
S_\nu=\int_0^\infty dr_{\parallel}e^{-\frac{2}{\hbar v_F}\int_0^{r_{\parallel}}dr^\prime \Delta(r^\prime)}J_\nu^2(k_{F,\parallel}r_{\parallel})\approx \int_0^\infty dr_{\parallel} e^{-2r_{\parallel}/\xi}J_\nu^2(k_{F,\parallel}r_{\parallel}),
\end{align}
with $\xi=\hbar v_F/\Delta_0$. The integral can be evaluated to give \cite{Cheng2010,Abramowitz1948}
\begin{align}
S_\nu&=\frac{k_{F,\parallel}^{2\nu}\xi^{2\nu+2}}{4^{2\nu+1}}\frac{\Gamma(2\nu+2)}{[\Gamma(\nu+1)]^2}F\left( \nu+\frac{1}{2},\nu+\frac{3}{2};2\nu+1;-k_{F,\parallel}^2\xi^2\right),
\end{align}
where $\Gamma(z)$ is the gamma function and $F(a,b,c;z)$ the hypergeometric function.
When $q_F<|q_z|<q_{\text{node}}$, the normalization constant is
\begin{equation}
|\mathcal{N}_{2,k_z}|^2=\frac{1}{4\pi}\frac{1}{C^2T_0+D^2T_1},
\end{equation}
where $T_\nu$ is the integral defined by
\begin{align}
T_\nu=\int_0^\infty dr_{\parallel}e^{-\frac{2}{\hbar v_F}\int_0^{r_{\parallel}}dr^\prime \Delta(r^\prime)}I_\nu^2(k_{F,\parallel}r_{\parallel})\approx \int_0^\infty dr_{\parallel} e^{-2\frac{\Delta_0}{\hbar v_F}r}I_\nu^2(k_{F,\parallel}r_{\parallel}).
\end{align}
Evaluation of the integral gives
\begin{align}
T_\nu&=\frac{k_{F,\parallel}^{2\nu}\xi^{2\nu+2}}{4^{2\nu+1}}\frac{\Gamma(2\nu+2)}{[\Gamma(\nu+1)]^2}F\left( \nu+\frac{1}{2},\nu+\frac{3}{2};2\nu+1;k_{F,\parallel}^2\xi^2\right).
\end{align}
\section{III. Mapping to the (1+1)d Dirac Hamiltonian}
In this section we derive the effective $(1+1)d$ Dirac Hamiltonian Eq. \eqref{eq:monopoleDirac} in the main text. We follow the method presented in \cite{Tewari2007,Tewari2010} and generalize it to our case of a monopole superconductor.
\subsection{Weyl Hamiltonian}
The kinetic Hamiltonian is given by the Weyl band structure Hamiltonian
Eq. \eqref{eq:WeylHam}.
Because only low-energy excitations are of interest, the sum over $\mathbf{q}$
can be restricted to momenta near the two FSs,
\begin{eqnarray}\label{eq: Weyl + and -}
\hat{H}_{\text{Weyl}}&=&\hat{H}_{\text{Weyl},+}+\hat{H}_{\text{Weyl},-}
\nonumber\\
\hat{H}_{\text{Weyl},\pm}&=&\sum_{\mathbf{q},\sigma,\sigma^\prime}\hat{c}^\dagger_{\pm \mathbf{K}_0+\mathbf{q},\sigma}h_{\pm,\sigma\sigma^\prime}(\mathbf{q})\hat{c}_{\pm \mathbf{K}_0+\mathbf{q},\sigma^\prime},
\end{eqnarray}
where $h_{\pm}(\mathbf{q})=\hbar v_F(q_x\sigma_x+q_y\sigma_y\pm q_z\sigma_z)-\mu I_\sigma$, as defined in the main text. $\hat{H}_{\text{Weyl},\pm}$ are the parts of the band structure Hamiltonian near the nodes $\pm \mathbf{K}_0$. It is convenient to exploit the cylindrical symmetry of the vortex by utilizing cylindrical coordinates. By keeping $q_z$ discrete and taking the continuum limit in the $q_x$ and $q_y$ directions, the Hamiltonians $\hat{H}_{\text{Weyl},\pm}$ can be recast into
\begin{equation}
\hat{H}_{\text{Weyl},\pm}=
\frac{1}{(2\pi)^2}\sum_{q_z,\sigma,\sigma^\prime}\int_{k_{F,\parallel}(q_z)-\Lambda}^{k_{F,\parallel}(q_z)+\Lambda}
q_\parallel d q_\parallel \int_0^{2\pi} d\phi_{\mathbf{q}} \hat{c}^\dagger_{\pm K_0+q_z,\sigma}(q_\parallel,\phi_{\mathbf{q}})h_{\pm,\sigma\sigma^\prime}(q_\parallel,\phi_{\mathbf{q}})\hat{c}_{\pm K_0+q_z,\sigma^\prime}(q_\parallel,\phi_{\mathbf{q}}),
\end{equation}
where $q_\parallel=\sqrt{q_x^2+q_y^2}$ and $\phi_{\mathbf{q}}=\arctan\frac{q_y}{q_x}$ is the azimuthal angle of $\mathbf{q}$. Here $k_{F,\parallel}(q_z)\equiv\sqrt{q_F^2 -q_z^2}$ is the in-plane Fermi wavevector for the momentum space cut at $q_z$ and $\Lambda$ is a momentum cutoff. The operators $\hat{c}_{k_z,\sigma}(q_\parallel,\phi_{\mathbf{q}})$ satisfy the fermionic anti-commutation relation
\begin{eqnarray}
\Big\{\hat{c}_{k_z,\sigma}(q_\parallel,\phi_{\mathbf{q}}), \hat{c}^\dagger_{k_z^\prime,\sigma^\prime}(q_\parallel,\phi_{\mathbf{q}}) \Big\}
= (2\pi)^2\delta_{k_z,k_z^\prime}\delta_{\sigma,\sigma^\prime}
\frac{1}{q_\parallel}\delta(q_\parallel-q'_\parallel)\delta(\phi_{\mathbf{q}}-\phi_{{\mathbf{q}}'}).
\end{eqnarray}
As a result of spin-orbit coupling, the $z$-component of the orbital and spin angular momenta, $m_l$ and $m_s$, are not good quantum numbers, while the total angular momentum in the $z$ direction, $m_j=m_l+m_s$, is. It is therefore convenient to decompose the fermion operator $\hat{c}^\dagger_{k_z,\sigma}(q_\parallel,\phi_{\mathbf{q}})$ in angular momentum channels with definite $m_j$:
\begin{align}
\begin{bmatrix}
\hat{c}_{k_z,\uparrow}(q_\parallel,\phi_{\mathbf{q}})\\
\hat{c}_{k_z,\downarrow} (q_\parallel,\phi_{\mathbf{q}})
\end{bmatrix}
=\frac{1}{\sqrt{q_\parallel}}\sum_{m_j}\begin{bmatrix}
e^{i\left(m_j-\frac{1}{2} \right)\phi_{\mathbf{q}}}\hat{c}_{k_z,\uparrow,m_j}(q_\parallel)\\
e^{i\left(m_j+\frac{1}{2} \right)\phi_{\mathbf{q}}}\hat{c}_{k_z,\downarrow,m_j}(q_\parallel)
\end{bmatrix},
\end{align}
where $m_j$ takes on half-integer values and $\hat{c}_{k_z,\sigma,m_j}(q_\parallel)$
satisfies the anticommutation relation
\begin{eqnarray}
\acomm{\hat{c}_{k_z,\sigma,m_j}(q_\parallel)}
{\hat{c}^\dagger_{k_z^\prime,\sigma^\prime,m_j^\prime}(q'_\parallel)}=
2\pi\delta_{k_z,k_z^\prime}\delta_{\sigma,\sigma^\prime}\delta_{m_j,m_j^\prime}
\delta(q_\parallel-q'_\parallel).
\end{eqnarray}
Performing the partial wave decomposition on $\hat{H}_{\text{Weyl},+}$ and integrating over the azimuthal angle, it becomes
\begin{equation}
\hat{H}_{\text{Weyl},+}=\sum_{q_z}\sum_{m_j}\int \frac{dq_\parallel}{2\pi}
\begin{bmatrix}
\hat{c}^\dagger_{K_0+q_z,\uparrow,m_j}(q_\parallel)&\hat{c}^\dagger_{K_0+q_z,\downarrow,m_j}(q_\parallel)
\end{bmatrix}\begin{bmatrix}
-\mu +\hbar v_F q_z&\hbar v_F q_\parallel\\
\hbar v_F q_\parallel &-\mu -\hbar v_Fq_z
\end{bmatrix}\begin{bmatrix}
\hat{c}_{K_0+q_z,\uparrow,m_j}(q_\parallel)\\
\hat{c}_{K_0+q_z,\downarrow,m_j}(q_\parallel)
\end{bmatrix}.
\end{equation}
For fixed $q_z$ and $m_j$, the Hamiltonian can be diagonalized by the unitary transformation
\begin{align}
\begin{bmatrix}
\hat{c}_{K_0+q_z,\uparrow,m_j}(q_\parallel)\\
\hat{c}_{K_0+q_z,\downarrow,m_j}(q_\parallel)
\end{bmatrix}=
\begin{bmatrix}
\cos\frac{\theta_{\mathbf{q}}}{2}& -\sin\frac{\theta_{\mathbf{q}}}{2} \\
\sin\frac{\theta_{\mathbf{q}}}{2} & \cos\frac{\theta_{\mathbf{q}}}{2}
\end{bmatrix}
\begin{bmatrix}\label{eq:unitary1}
\hat{f}_{K_0+q_z,-,m_j}(q_\parallel)\\
\hat{f}_{K_0+q_z,+,m_j}(q_\parallel)
\end{bmatrix},
\end{align}
where $\theta_{\mathbf{q}}$ is defined via $q_z=q\cos\theta_{\mathbf{q}}$ and $q_\parallel=q\sin\theta_{\mathbf{q}}$ with $q=\sqrt{q_z^2+q^2_\parallel}$. Because the transformation is unitary, the operators $\hat{f}_{K_0+q_z,m_j,\mp}(q_\parallel)$ inherit the fermionic anticommutation relations from $\hat{c}_{K_0+q_z,\sigma,m_j}(q_\parallel)$. In terms of these operators, the Hamiltonian reads
\begin{equation}
\hat{H}_{\text{Weyl},+}=\sum_{q_z}\sum_{m_j}\int \frac{d q_\parallel}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{K_0+q_z,-,m_j}(q_\parallel)&\hat{f}^\dagger_{K_0+q_z,+,m_j}(q_\parallel)
\end{bmatrix}\begin{bmatrix}
-\mu +\hbar v_F q &0\\
0&-\mu -\hbar v_F q
\end{bmatrix}\begin{bmatrix}
\hat{f}_{K_0+q_z,-,m_j}(q_\parallel)\\
\hat{f}_{K_0+q_z,+,m_j}(q_\parallel)
\end{bmatrix}.
\end{equation}
The Hamiltonian is projected onto the FS by only keeping states near the
Fermi energy, leading to
\begin{equation}\label{eq: Weyl +}
\hat{H}_{\text{Weyl,+}}=\sum_{q_z}\sum_{m_j}
\int_{k_{F,\parallel}(q_z)-\Lambda}^{k_{F,\parallel}(q_z)+\Lambda} \frac{d q_\parallel}{2\pi} \hat{f}^\dagger_{K_0+q_z,-,m_j}(q_\parallel)\left(-\mu+\hbar v_F q \right)\hat{f}_{K_0+q_z,-,m_j}(q_\parallel).
\end{equation}
A similar analysis can be performed for $\hat{H}_{\text{Weyl},-}$. After decomposing into angular momentum channels and performing the unitary transformation
\begin{align}\label{eq:unitary2}
\begin{bmatrix}
\hat{c}_{-K_0-q_z,\uparrow,m_j}(q_\parallel)\\
\hat{c}_{-K_0-q_z,\downarrow,m_j}(q_\parallel)
\end{bmatrix}=
\begin{bmatrix}
-\sin\frac{\theta_{\mathbf{q}}}{2} &\cos\frac{\theta_{\mathbf{q}}}{2} \\
\cos\frac{\theta_{\mathbf{q}}}{2}& \sin\frac{\theta_{\mathbf{q}}}{2}
\end{bmatrix}
\begin{bmatrix}
\hat{f}_{-K_0-q_z,-,m_j}(q_\parallel)\\
\hat{f}_{-K_0-q_z,+,m_j}(q_\parallel)
\end{bmatrix},
\end{align}
the Hamiltonian becomes
\begin{equation}
\hat{H}_{\text{Weyl},-}=\sum_{q_z}\sum_{m_j}\int \frac{d q_\parallel}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{-K_0-q_z,-,m_j}(q_\parallel)&\hat{f}^\dagger_{-K_0-q_z,+,m_j}(q_\parallel)
\end{bmatrix}\begin{bmatrix}
-\mu -\hbar v_F q &0\\
0&-\mu +\hbar v_F q
\end{bmatrix}\begin{bmatrix}
\hat{f}_{-K_0-q_z,-,m_j}(q_\parallel)\\
\hat{f}_{-K_0-q_z,+,m_j}(q_\parallel)
\end{bmatrix}.
\end{equation}
Projecting onto the FS, the Hamiltonian reads
\begin{equation}\label{eq: Weyl -}
\hat{H}_{\text{Weyl},-}=\sum_{q_z}\sum_{m_j}
\int_{k_{F,\parallel}(q_z)-\Lambda}^{k_{F,\parallel}(q_z)+\Lambda} \frac{d q_\parallel}{2\pi}
\hat{f}^\dagger_{-K_0-q_z,+,m_j}(q_\parallel)\left(-\mu+\hbar v_F q \right)\hat{f}_{-K_0-q_z,+,m_j}(q_\parallel).
\end{equation}
\subsection{Pairing Hamiltonian}
The mean-field pairing Hamiltonian with a vortex of winding number $+1$ imposed is
\begin{equation}
\hat{H}_{\Delta}=\sum_{k_z>0,\sigma,\sigma^\prime} \int d^2r_\parallel
~\Delta(\vr_\parallel)e^{i\phi_\vr} \hat{\psi}^\dagger_{k_z,\sigma}(\vec{r}_\parallel)[i\sigma_y]_{\sigma\sigma^\prime}\hat{\psi}^\dagger_{-k_z,\sigma^\prime}(\vec{r}_\parallel)+\text{H.c.},
\end{equation}
where $\Delta(r_\parallel)=\Delta_0\tanh(r_\parallel/\xi)$. Again the sum over $k_z$ only covers $k_z>0$.
Switching to the momentum representation, $\hat{\psi}^\dagger_{k_z,\sigma}(\vec{r}_\parallel)= \int \frac{d^2\mathbf{q}_{\parallel}}{(2\pi)^2} e^{-i\vec{q}_\parallel\cdot\vec{r}_\parallel}\hat{c}^\dagger_{k_z,\sigma}(\vec{q}_\parallel)$, the pairing Hamiltonian becomes
\begin{align}
\hat{H}_{\Delta}&=\sum_{k_z>0,\sigma,\sigma^\prime}\int \frac{d^2\mathbf{q}_\parallel}{(2\pi)^2} \frac{d^2\mathbf{q}^\prime_\parallel}{(2\pi)^2}
\underbrace{\int d^2r\Delta(\rho_\vr)e^{i\phi_\vr}
e^{-i(\vec{q}_\parallel+\vec{q}_\parallel^\prime)\cdot\vec{r}_\parallel}}_{I(\vec{q}_\parallel+\vec{q}^\prime_\parallel)}
\hat{c}^\dagger_{k_z,\sigma}(q_\parallel,\phi_\mathbf{q})[i\sigma_y]_{\sigma\sigma^\prime}
\hat{c}^\dagger_{- k_z,\sigma^\prime}(q_\parallel^\prime,\phi_{\mathbf{q}'})+ \text{H.c.}\nonumber.
\end{align}
The integral $I(\mathbf{q}_\parallel+\mathbf{q}'_\parallel)$ has been evaluated in Refs. \cite{Tewari2007,Tewari2010}, which we repeat here for completeness. Writing the integral in cylindrical coordinates as
\begin{equation}
I(\mathbf{q}_\parallel+\mathbf{q}'_\parallel)=\int d^2r_\parallel \Delta(r_\parallel)e^{i\phi_{\vr}}
e^{-i |\mathbf{q}_\parallel+\mathbf{q}'_\parallel| r_\parallel\cos\left(\phi_{\vr}-\phi_{\mathbf{q}+\mathbf{q}^\prime}\right)}
\end{equation}
and making the change of variables $\phi_\vr\to \phi_\vr+\phi_{\mathbf{q}+{\mathbf{q}^\prime}}+\frac{\pi}{2}$, it becomes
\begin{align}
I(\mathbf{q}_\parallel+\mathbf{q}'_\parallel)&=e^{i\left(\phi_{\mathbf{q}+{\mathbf{q}^\prime}}+\frac{\pi}{2} \right)}
\int r_\parallel dr_\parallel \Delta(r_\parallel)
\int d\phi_\vr e^{i|\mathbf{q}_\parallel+\mathbf{q}_\parallel^\prime|r_\parallel\sin\left( \phi_\vr+\phi_\vr\right)}.
\end{align}
Using the integral representation for Bessel functions of integer order $J_n(z)=\frac{1}{2\pi}\int_{-\pi}^\pi d\tau e^{i(z\sin \tau-n\tau)}$, the $\phi_\vr$ integral can be written as $2\pi J_{-1}(|\mathbf{q}_\parallel+\mathbf{q}^\prime_\parallel|r_\parallel)$. Finally, performing the integral over $r_\parallel$, we obtain
\begin{equation}
I(\mathbf{q}_\parallel+\mathbf{q}'_\parallel)=-2\pi i\Delta_0 \frac{q_{\parallel}e^{i\phi_\mathbf{q}}+q_{\parallel}^\prime e^{i\phi_{{\mathbf{q}^\prime}}}} {|\mathbf{q}_\parallel+\mathbf{q}'_\parallel|^3}.
\end{equation}
\
Substituting in the integral, the pairing Hamiltonian becomes
\begin{align}
\hat{H}_{\Delta}=-2\pi i\Delta_0 \sum_{k_z>0}\int \frac{d^2 \mathbf{q}_\parallel}{(2\pi)^2}
\frac{d^2 \mathbf{q}^\prime_\parallel}{(2\pi)^2} \frac{q_{\parallel}e^{i\phi_\mathbf{q}}+q_{\parallel}'e^{i\phi_{{\mathbf{q}^\prime}}}}{|\mathbf{q}_\parallel+\mathbf{q}'_\parallel|^3}
[\hat{c}^\dagger_{k_z,\uparrow}(\mathbf{q}_\parallel)\hat{c}^\dagger_{- k_z,\downarrow}(\mathbf{q}'_\parallel)-(\uparrow\leftrightarrow\downarrow)]+\text{H.c.}.
\end{align}
Because $|\mathbf{q}_\parallel+\mathbf{q}_\parallel^\prime|=\sqrt{q_\parallel^2+q_\parallel^{'2}-2q_\parallel q'_\parallel \cos(\phi_{\mathbf{q}}-\phi_{{\mathbf{q}^\prime}})}$ is periodic in $\phi_{\mathbf{q}}-\phi_{{\mathbf{q}^\prime}}$, we can expand it in angular momentum channels:
\begin{align*}
\frac{1}{|\mathbf{q}_\parallel+\mathbf{q}_\parallel^\prime|^3}=
\sum_{n}u_n(q_\parallel,q_\parallel^\prime)e^{in(\phi_\mathbf{q}-\phi_{{\mathbf{q}^\prime}})},
\end{align*}
where $n$ takes on integer values. Since $|\mathbf{q}_\parallel+\mathbf{q}_\parallel^\prime|$ is invariant under swapping $q_\parallel$ and $q_\parallel^\prime$, the Fourier coefficients satisfy $u_n(q_\parallel,q_\parallel^\prime)=u_{n}(q_\parallel^\prime,q_\parallel)$. Furthermore, because it only depends on $\cos(\phi_\mathbf{q}-\phi_{{\mathbf{q}^\prime}})$, which is even in $\phi_\mathbf{q}-\phi_{{\mathbf{q}^\prime}}$, we have $u_{n}(q_\parallel,q_\parallel^\prime)=u_{-n}(q_\parallel,q_{\parallel}^\prime)$ and real Fourier coefficients. Expanding the fermion operators in angular momentum channels as well and performing the angular integrals, the pairing Hamiltonian
becomes
\begin{align}
\hat{H}_{\Delta}&=-2\pi i \Delta_0
\sum_{k_z}\sum_n\int \frac{d q_\parallel}{2\pi} \frac{d q'_\parallel}{2\pi}
\sqrt{q_\parallel q'_\parallel} u_{n}(q_\parallel,q'_\parallel) \bigg[ q_\parallel
\hat{c}^\dagger_{k_z,\uparrow,n+\frac{3}{2}}(q_\parallel)
\hat{c}^\dagger_{-k_z,\downarrow,-n-\frac{1}{2}}(q'_\parallel)
+q'_\parallel \hat{c}^\dagger_{k_z,\uparrow,n+\frac{1}{2}}(q_\parallel)
\hat{c}^\dagger_{-k_z,\downarrow,-n+\frac{1}{2}}(q'_\parallel)
\nonumber \\
&-q_\parallel \hat{c}^\dagger_{k_z,\downarrow,n+\frac{1}{2}}(q_\parallel)\hat{c}^\dagger_{-k_z,\uparrow,-n+\frac{1}{2}}(q_\parallel^\prime)-q_\parallel ^\prime \hat{c}^\dagger_{k_z,\downarrow,n-\frac{1}{2}}(q_{\bf p})\hat{c}^\dagger_{-k_z,\downarrow,-n+\frac{3}{2}}(q_\parallel^\prime)\bigg]+\text{H.c.}.
\label{eq:pairing}
\end{align}
The $m_j=\frac{1}{2}$ channel ($n=-1$ in first term, $n=0$ in second and third, and $n=1$ in fourth) in Eq. \eqref{eq:pairing} pairs with itself and is decoupled from the rest. The Hamiltonian for this angular momentum channel reads
\begin{equation}
\hat{H}_{\Delta,m_j=\frac{1}{2}}
=-i\sum_{k_z} \int \frac{d q_\parallel}{2\pi} \frac{dq^\prime_\parallel}{2\pi}
\left[h_\Delta(q_\parallel,q_\parallel^\prime)\hat{c}^\dagger_{k_z,\uparrow,\frac{1}{2}}(q_\parallel)\hat{c}^\dagger_{-k_z,\downarrow,\frac{1}{2}}(q_\parallel^\prime) -h_{\Delta}(q_\parallel^\prime,q_\parallel)\hat{c}^\dagger_{k_z,\downarrow,\frac{1}{2}}(q_\parallel)\hat{c}^\dagger_{-k_z,\uparrow,\frac{1}{2}}(q_\parallel^\prime)\right]+\text{H.c.},
\end{equation}
where
\begin{equation}
h_{\Delta}(q_\parallel,q_\parallel^\prime)=2\pi \Delta_0 \sqrt{q_\parallel q_\parallel^\prime} \left[q_\parallel u_{-1}(q_\parallel,q_\parallel^\prime)+q_\parallel^\prime u_0(q_\parallel,q_\parallel^\prime) \right].
\end{equation}
In what follows we deal exclusively with the $m_j=\frac{1}{2}$ angular momentum channel so the $m_j$ index will henceforth be neglected.
Since only the low-energy excitations are of interest, we project the pairing Hamiltonian onto the FSs. Using the transformations in Eqs. \eqref{eq:unitary1} and \eqref{eq:unitary2}, we write $\hat{c}^\dagger_{k_z,\uparrow}(q_\parallel)\approx\cos\frac{\theta_\mathbf{q}}{2}\hat{f}^\dagger_{k_z,-}(q_\parallel)$, $\hat{c}^\dagger_{k_z,\downarrow}(q_\parallel)\approx\sin\frac{\theta_\mathbf{q}}{2}\hat{f}^\dagger_{k_z,-}(q_\parallel)$, $\hat{c}^\dagger_{-k_z,\uparrow}(q_\parallel^\prime)\approx \cos\frac{\theta_{{\mathbf{q}^\prime}}}{2}\hat{f}^\dagger_{-k_z,+}(q_\parallel^\prime)$, and $\hat{c}^\dagger_{-k_z,\downarrow}(q_\parallel^\prime)\approx \sin\frac{\theta_{{\mathbf{q}^\prime}}}{2}\hat{f}^\dagger_{-k_z,+}(q_\parallel^\prime)$. Making these substitutions, the Hamiltonian becomes
\begin{equation}\label{eq: pair1}
\hat{H}_{\Delta}=-i\sum_{k_z}\int\frac{dq_\parallel}{2\pi}\frac{dq^\prime_\parallel}{2\pi}A(q_\parallel-q_\parallel^\prime)\hat{f}^\dagger_{k_z,-}(q_\parallel)\hat{f}^\dagger_{-k_z,+}(q_\parallel^\prime)+\text{H.c.},
\end{equation}
where
\begin{equation}
A(q_\parallel-q_\parallel^\prime)=h_{\Delta}(q_\parallel,q_{\parallel}^\prime)\cos\frac{\theta_\mathbf{q}}{2}\sin\frac{\theta_{\mathbf{q}^\prime}}{2}-h_{\Delta}(q_\parallel^\prime,q_\parallel) \sin\frac{\theta_\mathbf{q}}{2}\cos\frac{\theta_{\mathbf{q}^\prime}}{2}
\end{equation}
is a real function that is antisymmetric under the exchange of $q_\parallel$ and $q_\parallel^\prime$.
\subsection{Fourier Transform}
Finally, combining the above results, we can obtain an expression for the full Hamiltonian $\hat{H}=\hat{H}_{\text{Weyl}}+\hat{H}_{\Delta}$. Using Eqs. \eqref{eq: Weyl + and -}, \eqref{eq: Weyl +}, \eqref{eq: Weyl -}, and \eqref{eq: pair1}, $\hat{H}$ takes the form
\begin{align}
\hat{H}&=\sum_{q_z}\int_{-\Lambda}^{\Lambda} \frac{dq_\parallel}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{k_z,-}(q_\parallel )&\hat{f}_{-k_z,+}(q_\parallel)
\end{bmatrix}\begin{bmatrix}
\hbar v_F (q-q_F)
&0 \\
0&
-\hbar v_F (q-q_F)
\end{bmatrix}
\begin{bmatrix}
\hat{f}_{k_z,-}(q_\parallel)\\
\hat{f}^\dagger_{-k_z,+}(q_\parallel)
\end{bmatrix}\nonumber\\
&+\sum_{q_z}\int_{-\Lambda}^{\Lambda} \frac{dq_\parallel}{2\pi}
\int_{-\Lambda}^{\Lambda}
\frac{dq_{\parallel}^\prime}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{k_z,-}(q_\parallel )&\hat{f}_{-k_z,+}(q_\parallel)
\end{bmatrix}\begin{bmatrix}
0
&-iA(q_\parallel -q'_\parallel) \\
iA(q_\parallel -q'_\parallel)&
0
\end{bmatrix}
\begin{bmatrix}
\hat{f}_{k_z,-}(q'_\parallel)\\
\hat{f}^\dagger_{-k_z,+}(q'_\parallel)
\end{bmatrix}.
\end{align}
We expand $\hbar v_F (q-q_F)=\hbar \tilde{v}_F \delta q_\parallel$,
where $\tilde{v}_F = v_F\sin\theta_{\mathbf{q}}$ and $\delta q_\parallel=q_\parallel -k_{F,\parallel}(q_z)$, i.e.,
$\delta q_\parallel$ is set to zero at the Fermi radius at a fixed $q_z$.
Then we have
\begin{align}
\hat{H}&=\sum_{q_z}\int_{-\Lambda}^\Lambda \frac{d \delta q_\parallel}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{k_z,-}(\delta q_\parallel )&\hat{f}_{-k_z,+}( \delta q_\parallel)
\end{bmatrix}\begin{bmatrix}
\hbar \tilde{v}_F\delta q_\parallel
&0 \\
0&
- \hbar \tilde{v}_F\delta q_\parallel
\end{bmatrix}
\begin{bmatrix}
\hat{f}_{k_z,-}( \delta q_\parallel)\\
\hat{f}^\dagger_{-k_z,+}(\delta q_\parallel)
\end{bmatrix}\nonumber\\
&+\sum_{q_z}\int_{-\Lambda}^\Lambda \frac{d \delta q_\parallel}{2\pi}
\int_{-\Lambda}^\Lambda \frac{d\delta q'_\parallel}{2\pi}
\begin{bmatrix}
\hat{f}^\dagger_{k_z,-}(\delta q_\parallel )&\hat{f}_{-k_z,+}( \delta q_\parallel)
\end{bmatrix}\begin{bmatrix}
0
&-iA(\delta q_\parallel - \delta q'_\parallel) \\
iA(\delta q_\parallel - \delta q'_\parallel)&
0
\end{bmatrix}
\begin{bmatrix}
\hat{f}_{k_z,-}( \delta q'_\parallel)\\
\hat{f}^\dagger_{-k_z,+}(\delta q'_\parallel)
\end{bmatrix}.
\end{align}
Since we are interested in the low-energy physics, we send the momentum cutoff $\Lambda\to \infty$.
Define the Fourier transformations
\begin{align}
\hat{f}_{k_z,-}(\delta q_\parallel)&=\int_{-\infty}^\infty dx e^{-i \delta q_\parallel x} \hat{f}_{k_z,-}(x),&
\hat{f}_{-k_z,+}(\delta q_\parallel)&=\int_{-\infty}^\infty dx e^{-i \delta q_\parallel x} \hat{f}_{-k_z,+}(x).
\end{align}
In this new representation, the Hamiltonian is
\begin{equation}
\hat{H}=\sum_{q_z}\int dx\left[-i\hbar \tilde{v}_F
\hat{\phi}^\dagger_{k_z}\tau_z\partial_x\hat{\phi}_{q_z}+ \hat{\phi}^\dagger_{k_z}\tau_y m(x)\hat{\phi}_{k_z}\right],
\end{equation}
where $\hat{\phi}_{k_z}(x)=\left[\hat{f}_{k_z,-}(x),
\hat{f}^\dagger_{-k_z,+}(-x) \right]^T$, and the mass function
\begin{equation}
m(x)\equiv\int \frac{d \delta q_\parallel}{2\pi} A( \delta q_\parallel )e^{i\delta q_\parallel x}
\end{equation}
is an odd function in $x$ because $A(\delta q_\parallel)$ is an odd function
in $\delta q_\parallel$.
This is the derivation of Eq. \eqref{eq:monopoleDirac} in the main text.
\end{document}
|
2,877,628,090,172 | arxiv | \section*{Supplementary Material}
\renewcommand\thefigure{S\arabic{figure}}
\setcounter{figure}{0}
\section{Quantum computation details}
\subsection{Encoding}
To go from fermionic variables to spin variables to write the Hamiltonian, one must provide an encoding scheme. Here we use the straightforward Jordan-Wigner encoding, mapping the creation/annihilation operators as:
\begin{align}
& c^{\dagger}_j \rightarrow \hat{Z}_1 \otimes \hat{Z}_2 \otimes ... \otimes \hat{Z}_{j-1} \otimes \frac{1}{2} (\hat{X}_j + i \hat{Y}_j) \\
& c_j \rightarrow \hat{Z}_1 \otimes \hat{Z}_2 \otimes ... \otimes \hat{Z}_{j-1} \otimes \frac{1}{2} (\hat{X}_j - i \hat{Y}_j)
\end{align}
where the chains of $\hat{Z}$ operators ensure fermionic anticommutation.
\subsection{Ansatz circuits}
\subsubsection{Low-Depth Circuit Ansatz (LDCA) circuit}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{MG_decomposition.pdf}
\caption{Decomposition of a 2-qubit rotation gate. Here, $P, Q \in X, Y, Z$ are rotation axes and $U_X=RY(\frac{\pi}{2}), U_Y=RZ(\frac{\pi}{2}) RY(\frac{\pi}{2})$ and $U_Z=I$.
}
\label{fig:RPQ_gate}
\end{figure}
The Low-Depth Circuit Ansatz\cite{dallaire-demers_low-depth_2019} circuit (exemplified in its 8-qubit, 1-cycle version on Figure \ref{fig:circuits}) is inpired from a special class of circuits
meant to prepare uncorrelated (or "gaussian") states (see Refs \onlinecite{jozsa_matchgates_2008} and \onlinecite{dallaire-demers_application_2020}).
The building block of uncorrelated states preparation is the sequence of 2-qubit rotations $R_{YY}(\theta_1) R_{XX}(\theta_2) R_{YX}(\theta_3) R_{XY}(\theta_4)$, with
\begin{equation}
R_{PQ}^{(i,j)}(\theta) = e^{i \theta P_i \otimes Q_j}
\end{equation}
arranged in so-called matchgate cycles that alternatively connect qubits $i, i+1$ and $i+1, i+2$.
To endow the output state with a non-gaussian (i.e correlated) character, a $R_{ZZ}$ rotation is inserted in the $YY$-$XX$-$YX$-$XY$ sequence.
Decomposing the 2-qubit gates into CNOT and one-qubit rotation gates (see Figure \ref{fig:RPQ_gate}), the 1-cycle version of the LDCA circuit with 8 qubits we have used to prepare our embedded models' ground states has initially a total count of 1108 gates, including 280 CNOT gates. Each matchgate sequence of five two-qubit rotation gates comprises twenty-five one-qubit gates (up to the cancellation due to the successive application of $U_P$ and $U_P^{\dagger}$) and ten two-qubit gates. Yet, it was shown that a two-qubit unitary could be written with at most fifteen one-qubit gates and three CNOT gates \cite{vatan_optimal_2004}. We thus applied circuit recompilation techniques \cite{martiel_architecture_2020} to lower this count and achieved a count of 324 gates among which 112 are CNOT gates.
\subsubsection{Multi-Reference Excitations Preserving (MREP) circuit}
The Multi-Reference Excitations Preserving (MREP) ansatz is a custom-made ansatz that starts from a multireference state and then distributes the fermionic excitations among the orbitals.
We design the sequence of the first few gates (one-qubit gates and subsequent CNOT gates) by duplicating the gate patterns that we use for the single-site case (see section~\ref{sec:Nc1} below), which are themselves inspired by Ref.~\onlinecite{sugisaki_quantum_2019}.
This effectively corresponds to starting with a circuit that would capture exactly the ground state of two disconnected (without intra-dimer hopping) single-site impurity problems.
The second part of the MREP circuit is composed of layers of $U_F$ blocks, each of which contains cone-shaped patterns of so-called fSim gates\cite{kivlichan_quantum_2018, Foxen2020}, defined as\cite{arute_quantum_2019}:
\begin{equation}
\mathrm{fSim}(\theta,\phi)=\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & \cos\theta & -i\sin\theta & 0\\
0 & -i\sin\theta & \cos\theta & 0\\
0 & 0 & 0 & e^{-i\phi}
\end{array}\right].
\end{equation}
These gates are native to transmon-qubit architectures such as the Sycamore chip.
From the point of view of fermionic physics, fSim gates can create gaussian (i.e uncorrelated) states when $\phi = 0$ (they then belong to the set of so-called "matchgates" that are universal for simulating uncorrelated fermions).
Conversely, with $\phi \neq 2n\pi $ ($n\in \mathbb{Z}$), they create states with a non-gaussian character.
Importantly, they do not modify the number of fermions in the wavefunction, which is why they are described as "excitation-preserving".
Thus, the fSim "cones" allow for the spreading of the excitations among the orbitals. This is somewhat akin to turning on the intradimer hopping.
In the main text, we chose a number of 4 layers of fSim gates. This choice is dictated by the numerical observation that more layers induced only a marginal improvement of the VQE energies.
\subsection{Classical optimization} \label{subsec:classical_optim}
In this subsection, we focus on the classical optimizer used within the VQE procedure described in Fig.~\ref{fig:method}(c).
The circuits are optimized using Python's scipy optimization package with the BFGS method, allowing for up to 10000 iterations of the algorithm.
To account for the dependence of the accuracy of the VQE optimization to the initialization of the circuit's parameters, three runs were made for each configuration, and one given random set of parameters was used as a first guess twice: for a subsequent noise-free optimization as well as a noisy one.
\section{Noise models}\label{sec:noise_models}
In this section, we describe the noise models used to simulate noisy QPUs.
NISQ processors experience many kinds of errors, among which the most prominent ones are gate noise, idling noise and State Preparation and Measurement (SPAM) errors.
In this work, we deliberately choose a simple noise model with only gate noise and adjust the level of this gate noise to match the error levels measured in Randomized Benchmarking experiments on current NISQ processors.
More specifically, we work with a simple depolarizing noise model: the density matrix $\rho$ evolves after a 1-qubit process according to the channel
\begin{equation}
\mathcal{E}^{(1)}(\rho) = (1-p_1)\rho + \frac{p_1}{3}\left( X \rho X + Y \rho Y + Z \rho Z \right),
\end{equation}
which can be interpreted as a random Pauli operation being inserted with depolarization probability $p_1$, and the expected gate operation occurring alone with probability $(1-p_1)$.
Likewise, two-qubit gate errors are modelled as a two-qubit depolarizing channel that consists of the tensor product of the 1-qubit channels with depolarizing probability $p_2$.
The $p_1$ and $p_2$ probabilities are taken to reproduce randomized-benchmarking error levels.
Starting from the relationship between average process fidelity and the coefficient $p_0$ of the identity in the Kraus decomposition of the channel \cite{magesan_characterizing_2012},
\begin{equation}
\mathcal{F}_{\mathrm{ave}} \equiv 1-\epsilon_\mathrm{RB} = \frac{p_0d+1}{d+1},
\end{equation}
with $d=2^n$ the dimension of the subspace that is acted on by the channel, we must set $p_0$ so that
\begin{equation}
p_0 = 1-\left(1+\frac{1}{d} \right)\epsilon_\mathrm{RB}.
\end{equation}
Applying this formula to the one-qubit process, we get
\begin{equation}
p_1 = \frac{3}{2} \epsilon_\mathrm{RB}^{(1)},
\end{equation}
and for the two-qubit process:
\begin{equation}
p_2 = 1-\sqrt{1-\frac{5}{4}\epsilon_\mathrm{RB}^{(2)}}.
\end{equation}
The specific values we choose for $\epsilon_\mathrm{RB}^{(1)}$ and $\epsilon_\mathrm{RB}^{(2)}$ are those measured through randomized benchmarking performed on Google's Sycamore chip (see Supplementary Material of Ref.~\onlinecite{arute_quantum_2019}):
\begin{align}
& \epsilon_\mathrm{RB}^{(1)} = 0.16\%,\\
& \epsilon_\mathrm{RB}^{(2)} = 0.6 \%.
\end{align}
\section{The Rotationally Invariant Slave Boson method}\label{sec:RISB}
In this section, we elaborate on the embedding method used to solve the Hubbard model in the main text.
\subsection{Formalism}
There are a priori several embedding methods that could be used to map the original lattice problem, the Hubbard model, onto an effective problem of reduced dimensionality. Prominent choices include, ranging from the most accurate (and computationally expensive) to the simplest, Dynamical Mean Field Theory (DMFT), Rotationally-Invariant Slave Bosons (RISB, which is equivalent to the Gutzwiller method used in Ref.~\onlinecite{yao_gutzwiller_2021}), and Density-Matrix Embedding Theory (DMET). DMET can essentially be regarded as a simplified version of RISB, which itself can be regarded as a low-energy version of DMFT \cite{ayral_dynamical_2017}.
Here, as briefly argued in the main text, we choose to resort to RISB: on the one hand, as a low-energy simplification of DMFT, it does not require the computation of the full time-dependent Green's function, nor the use of a large number of uncorrelated bath orbitals, both of which would be costly in terms of the depth and width (respectively) of the ansatz quantum circuits required by the VQE step.
On the other hand, RISB, contrary to DMET, gives access to the quasiparticle renormalization factor $Z$, a key quantity to get insights into the properties of correlated materials.
We note that this approach shares commonalities with "Energy-Weighted DMET"\cite{Fertitta2018, Fertitta2019}, which interpolates between DMET and DMFT by requiring an increasing number of bath levels to progressively describe the full dynamics of the self-energy. This approach was implemented for the $N_c=1$ case in Ref.~\onlinecite{tilly_reduced_2021}.
We use the RISB method as introduced by Ref.~\onlinecite{lechermann_rotationally-invariant_2007} as a rotationally-invariant generalization of the work by Kotliar and Ruckenstein \cite{kotliar_new_1986}. It allows to properly handle multi-orbital problems, whether the orbitals denote true atomic orbitals or, as in the current work, the $2 N_c$ site orbitals of a unit cell (counting spin degeneracy).
We use a recent refinement of RISB \cite{lanata_phase_2015} where the free energy functional---a priori a highly nonlinear function of the slave-boson amplitudes $\Phi_{An}$---is made quadratic in the amplitudes, at the expense of adding Lagrange multipliers and additional variables.
The resulting six-variable free energy is extremized to find the six Lagrange equations of the problem:
\begin{subequations}
\begin{align}
& \Delta_{\alpha\beta}^{p}=\sum_{\boldsymbol{k}\in\mathrm{BZ},i\omega}\left[i\omega-R \varepsilon_{\boldsymbol{k}}R^{\dagger}-\lambda+\mu\right]_{\beta\alpha}^{
-1}e^{i\omega0^{+}},\label{eq:Delta_p-lagrange}\\
& \sum_{\mu}\left[\left(\Delta^{p}(1-\Delta^{p})\right)^{1/2}\right]_{\alpha\mu}\mathcal{D}_{\beta\mu}\label{eq:D_lagrange}\\
& \;=\sum_{\boldsymbol{k}\in\mathrm{BZ},i\omega}\left[\left\{ \varepsilon_{\boldsymbol{k}}R^{\dagger}\right\} \left[i\omega-R\varepsilon_{\boldsymbol{k
}}R^{\dagger}-\lambda+\mu\right]^{-1}\right]_{\beta\alpha}e^{i\omega0^{+}},\nonumber \\
& \lambda_{\alpha\beta}^{c}=-\lambda_{\alpha\beta}\nonumber\\
&\;-\sum_{\gamma\delta\eta}\left\{ \mathcal{D}_{\gamma\delta}R_{\eta\gamma}\frac{\partial\left[\left(\Delta^{p}(1-\Delta^{p})\right)^{1/2}\right]_{\eta\delta}}{\partial\Delta_{\alpha\beta}^{p}}+c.c\right\} ,\label{eq:lambda_c-lagrange}\\
& \hat{H}_{\mathrm{emb}}|\Phi\rangle=E_0|\Phi\rangle,\label{eq:H_imp-1}\\
& \langle\Phi|f_{\beta}f_{\alpha}^{\dagger}|\Phi\rangle=\Delta_{\alpha\beta}^{p},\label{eq:F1}\\
& \langle\Phi|c_{\alpha}^{\dagger}f_{\beta}|\Phi\rangle=R_{\gamma\alpha}\left[\left(\Delta^{p}(1-\Delta^{p})\right)^{1/2}\right]_{\gamma\beta}.\label{eq:F2}
\end{align}
\end{subequations}
Here, $\varepsilon_{\boldsymbol{k}}$ is the free dispersion on a square lattice tiled with $N_c=2$-site unit cells (as in e.g Ref.~\onlinecite{lee_rotationally_2019}). $i\omega$ denotes Matsubara frequencies and $\boldsymbol{k}\in \mathrm{BZ}$ denotes discretized points in the first Brillouin zone.
$H_\mathrm{emb}$ is the impurity model defined in Eq.~\eqref{eq:Hemb} of the main text, with $2 N_c$ correlated orbitals ("impurities") and $2 N_c$ bath orbitals.
Technically, the appearance of an impurity model in RISB comes from a reinterpretation of the slave-boson amplitudes $\Phi_{An}$ as the coefficients of the Schmidt decomposition of a ket $|\Phi\rangle$ defined on a Hilbert space that is a tensor product of the original $2 N_c$ fermionic degrees of freedom with an additional $2 N_c$ bath degrees of freedom\cite{lanata_phase_2015}.
The Greek indices are compound indices $\alpha = (i, \sigma)$ with $i=1\dots N_c$ and $\sigma=\uparrow,\downarrow$.
We refer the reader to Ref.~\onlinecite{ayral_dynamical_2017} for a derivation of these equations and an explanation of the meaning of the $\Delta^p$, $\mathcal{D}$, $\lambda^c$ variables.
As explained in the main text, $R$ and $\lambda$ turn out to be low-energy parameterizations of the lattice self-energy via Eq.~\eqref{eq:selfenergy}.
To solve these equations, we reformulate them as a root problem as in e.g Ref.~\onlinecite{ayral_dynamical_2017}: we seek to find the roots $R$ and $\lambda$ of the functions
\begin{subequations}
\begin{align}
\mathcal{F}^{(1)}[R,\lambda] & \equiv\langle\Phi|f_{\beta}f_{\alpha}^{\dagger}|\Phi\rangle-\Delta_{\alpha\beta}^{p}\label{eq:F1_def},\\
\mathcal{F}^{(2)}[R,\lambda] & \equiv\langle\Phi|c_{\alpha}^{\dagger}f_{\beta}|\Phi\rangle-R_{\gamma\alpha}\left[\left(\Delta^{p}(1-\Delta^{p})\right)^{1/2}\right]_{\gamma\beta},\label{eq:F2_def}
\end{align}
\end{subequations}
where $\mathcal{F}^{(1)}$ and $\mathcal{F}^{(2)}$ are implicit functions
of $R$ and $\lambda$ via the above equations.
The computational bottleneck of the root-solving procedure is the solution of the impurity model, Eq.~\eqref{eq:H_imp-1} that is needed to compute the 1-RDM elements $\langle\Phi|f_{\beta}f_{\alpha}^{\dagger}|\Phi\rangle$ and $\langle\Phi|c_{\alpha}^{\dagger}f_{\beta}|\Phi\rangle$ and then $\mathcal{F}^{(1)}$ and $\mathcal{F}^{(2)}$.
While it is usually solved with a classical impurity solver (by e.g exact diagonalization of $H_\mathrm{emb}$), we propose to solve it approximately using the hybrid quantum-classical VQE method combined with the NOization procedure described in the next section (section \ref{sec:NOization}).
\subsection{Implementation details}
For the solution of the RISB equations, we discretize the first Brillouin zone with a regular two-dimensional mesh with $32 \times 32$ points. We use an inverse temperature of $\beta = 300$ (instead of $\beta = \infty$, in order to smoothen the summation). We take $t = -0.25$, so that the half-bandwidth is unity.
While the $R$, $\lambda$, $\Delta_p$, $\mathcal{D}$, and $\lambda^c$ variables are in principle $2 N_c \times 2 N_c$ matrices, the absence of spin-flip terms in the Hamiltonian and the enforcement of paramagnetism allows to simplify all those matrices as $A_{i\sigma, j \sigma'} = \tilde{A}_{ij} \delta_{\sigma \sigma'}$.
Finally, the mirror symmetry of the $N_c=2$ unit cell obtained by periodizing the lattice implies, in the absence of symmetry breaking, that in the symmetry-adapted basis one has $\tilde{A}_{ij} =
\left[\begin{array}{cc}
a & b\\
b & a
\end{array}\right]
$ so that for each of these matrices has only two degrees of freedom.
We use these symmetry properties to simplify the computations. We refer the reader to Refs~\onlinecite{lanata_efficient_2012, lanata_phase_2015} for more details on these symmetry considerations.
\subsection{Convergence details}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RpRm.pdf}
\caption{Trajectory of the two components $R_+$ and $R_-$ of $R$ along the $N_c=2$ RISB minimization procedure based on the VQE optimization of the MREP ansatz in Natural Orbitals, with (right column) and without (left column) noise, for $U=0.05, 1, 2$. The black star materializes the classical RISB solution. The colormap enables to visualize the progression from the first guess (deepest red shade) to the last iteration (in purple).}
\label{fig:R}
\end{figure}
While the solution of RISB equations can be formulated as a root-solving problem, we found that the use of an approximate impurity solver caused the root-solving procedure to fail.
We therefore turned the root-solving procedure into a minimization problem, in which $||F_1||^2+||F_2||^2$ is minimized.
We used the Nelder-Mead algorithm to solve this minimization problem.
We allow for up to 100 iterations of the algorithm, taking the classical solution $(R, \lambda)$ at $U-0.05$ as a starting point.
To ensure we indeed get a minimization, we plotted the cost function the evolution of the cost function as evaluated along the minimization procedure (see the inset in Fig.~\ref{fig:RISB_result}, where it was considered that the convergence had been reached even though the whole computational budget was not necessarily consumed). In the absence of noise, the MREP ansatz-based does provide a clear RISB minimization of the cost function for $U=0.05$ and $U=1$. For $U=2$, after a regime in which the cost function diminishes in average, there is a plateau regime affected by spikes where the cost function gets very high. We checked that these spikes were caused by sensitively larger VQE errors at these points. The convergence is still manifest in the presence of noise, although less pronounced. On the other hand, the LDCA ansatz gives rise to a noisy RISB loop that does not converge. We conclude that the latter ansatz is ruled out for RISB at cluster size $N_c=2$ with current noise levels.
The trajectory of the components of $R$ along the minimization is also presented, on Fig.\ref{fig:R}. We see that one of the components accurately converges whereas the other one gets deviated.
Note that to accelerate the computation, we carried out the RISB procedure directly in the exact NO basis, obtained by diagonalizing the Hamiltonian.
We checked that the NOization procedure does not impede the RISB convergence by running the optimization for $U=1$ with the NOization layer (see Fig.~\ref{fig:conv_NO_VS_NOization}).
The VQE output state for which a 1-RDM is computed and diagonalized is the lowest-energy lying state obtained by running five VQE optimizations corresponding to five different, random initializations for the MREP ansatz. For the LDCA ansatz, only one run is considered to limit the computational overhead. This is not expected to impede the procedure since there is little dependence to the initial condition according to Fig.~\ref{fig:VQE}(b).
Fig.~\ref{fig:RISB_result} display results of the RISB procedure in terms of quasi-particle weights $Z_{\pm} = |R_{\pm}|^2$ and non-trivial static self-energy shift elements $\widetilde{\lambda}_{\pm} = \lambda_{\pm} - \varepsilon_{\mathrm{loc}}$ (where $\varepsilon_{\mathrm{loc}} \equiv \sum \limits_{\boldsymbol{k} \in BZ}\varepsilon_{\boldsymbol{k}} - \mu$).
\section{Natural Orbitalization}\label{sec:NOization}
In this section, we give more details about the Natural Orbitalization (NOization) method introduced in the main text.
\subsection{Formalism}
Our goal is to express our problem in the basis of the Natural Orbitals. However, this basis can only be computed once the exact ground state---which we are looking for---is known.
It is thus not directly accessible.
Nonetheless, it is possible to approximately rotate into this basis. We do this by applying the VQE procedure several times, using the variational approximation to the ground state provided by the VQE algorithm at step $k$ to update the orbital basis in which VQE at step $k+1$ is run.
More specifically, provided the optimal VQE state $\ket{\psi(\vec{\theta}^{*(k)})}$ returned by the VQE procedure, we use the quantum computer to compute the 1-RDM
\begin{equation}
D_\mathrm{emb} [\psi(\vec{\theta}^{*(k)}), c^{(k) \dagger}, c^{(k)}] \equiv \langle \psi(\vec{\theta}^{*(k)}) | c^{(k) \dagger}_i c^{(k)}_j |\psi(\vec{\theta}^{*(k)}) \rangle,
\end{equation}
and classically compute the transformation $V^{(k)}$ that diagonalizes it:
\begin{equation}
D_\mathrm{emb} [\psi(\vec{\theta}^{*(k)}), c^{(k) \dagger}, c^{(k)}]_{pq} = V^{(k)}_{p\alpha} n_{\alpha} V^{(k)\dagger}_{\alpha q}.
\end{equation}
where the Einstein convention is used on repeated indices.\\
This matrix is used to update the orbital basis as:
\begin{equation}
c^{(k+1)\dagger}_{\alpha} = \sum_p V^{(k)}_{p\alpha} c^{(k)\dagger}_p,
\end{equation}
which corresponds to the following transformation on the Hamiltonian's coefficients:
\begin{align}
&h^{(k+1)}_{pq}=V^{(k)}_{p'p}h^{(k)}_{p'q'}(V^{(k)\dagger})_{qq'} \\
&h^{(k+1)}_{pqrs}=V^{(k)}_{p'p}V^{(k)}_{q'q}h^{(k)}_{p'q'r's'}(V^{(k)\dagger})_{rr'}(V^{(k)\dagger})_{ss'}.
\end{align}
Note that since in general the accuracy of the VQE procedure is sensitive to the initial tuning of the parameters, we may want to consider the best VQE run out of several ones corresponding to different initializations. The plots of Fig.~\ref{fig:NOization} were however obtained by running one single VQE from a random initialization for each point, and may thus be slightly impacted by this effect.
\subsection{RISB convergence and NOization: data}
A display of RISB convergence with the effective NOization procedure against the RISB convergence exhibited running VQE in the exact NO basis is presented in Fig.~\ref{fig:conv_NO_VS_NOization} for $U=1$, showing that the procedure does indeed work in the context of RISB minimization (although the cost function converges to a slightly higher value).
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{convergence_NOized.pdf}
\caption{Convergence of the cost function for $N_c=2$ RISB at $U=1$ with the MREP ansatz without noise, in the exact NO basis (blue line) and in an effective NOization setup in which the NO basis is approximated iteratively (orange line).}
\label{fig:conv_NO_VS_NOization}
\end{figure}
\section{Single-site results}\label{sec:Nc1}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{multi_ref_circuit.pdf}
\caption{The Multi-Reference (MR) state preparation circuit ($N_c=1$ case).}
\label{fig:MR_circuit}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Nc1_RISB.pdf}
\caption{Evolution of the quasiparticle weight $Z$ as a function of $U$ for $N_c=1$ (top) and distance to the analytical solution (bottom) for the Multi-Reference ansatz (MR, this work) compared with the HEA ansatz.
}
\label{fig:Z_Nc1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Nc1_RISB_optim_landscape.pdf}
\caption{Optimization landscape $(R, \lambda$) around the analytical solution for $U=0.1$, $N_c=1$. The blue cross materializes the solution. From (a) to (d), each point is computed using the parameter shift rule to tune the MR circuit with depolarizing noise levels set at an increasing fraction of the Sycamore chip's levels. (e): landscape associated with a full noise MR circuit optimization made with the COBYLA algorithm. (f): landscape obtained using a $RY$ gates-product ansatz, optimized with COBYLA with full noise on.}
\label{fig:optim_ls_Nc1}
\end{figure}
While the main text focuses on impurity models with $N_c=2$ correlated sites, which in principle allows to obtain a coarse-grained space dependence of the correlation parameters $R$ and $\lambda$, it is also instructive to look at the single-site case ($N_c=1$), where the impurity model is said to be purely local and $R$ and $\lambda$ are merely scalar numbers. This case was addressed in a quantum-classical fashion in Refs~\onlinecite{keen_quantum-classical_2020,rungger_dynamical_2020,yao_gutzwiller_2021,tilly_reduced_2021}. We recall that we are still solving for the properties of a large or infinite-dimensional Hubbard model defined in Eq.~\eqref{eq:Hubbard}, a large $N_c$ giving information on more and more extended correlation effects. We refer the reader to Ref.~\onlinecite{lee_rotationally_2019} for a study of the $N_c$-dependence of RISB.
\subsection{Circuit}
For the paramagnetic RISB embedding model at $N_c=1$ at half-filling, one can write an exact ansatz circuit to prepare the natural-orbital ground state. Indeed, it is easy to show that in this case, the "natural ground state"'s 1-RDM is of the form
\begin{equation}
\label{eq:RDM_NO}
D_\mathrm{emb} = \begin{pmatrix}
n_0 & 0& 0& 0\\
0& n_0& 0& 0\\
0& 0& 1-n_0& 0\\
0& 0& 0& 1-n_0
\end{pmatrix}
\end{equation}
The 1-RDM in this case characterizes uniquely the ground state: since the embedded Hamiltonian only has real coefficients, the ground state is defined by ${4 \choose 2}=6$ real coefficients and the 1-RDM's non-diagonal vanishing entries impose 6 independent equations they must satisfy. Note that this does not happen for $N_c=2$ (70 coefficients but only 27 independent equations). Such 1-RDM states can be prepared by the minimal "Multi-Reference" (MR) circuit shown in Fig.~\ref{fig:MR_circuit}, with $\theta=2\,\mathrm{arcsin}(\sqrt{n_0})$.
\subsection{Optimization}
On top of its low gate count, the MR circuit has the very useful property that it only comprises one parameter, carried by a rotation gate. As a consequence, its optimization only requires three energy measurements in virtue of the parameter-shift rule \cite{nakanishi_sequential_2020}, instead of a proper VQE optimization.
\subsection{NOization}
One apparent drawback of such a minimal circuit is that since it only produces diagonal 1-RDMs, it cannot be used in a NOization procedure. As an alternative, one could resort to a dressed Hamiltonian strategy \cite{mizukami_orbital_2020}, in which the transformation to the Natural Orbitals is determined by minimizing the expectation value of a Hamiltonian dressed with a variational orbital transformation whose parameters must be optimized on top of the parameter of the circuit. This strategy would also allow for the use of the small ansatz circuit that works in NO without adding further noise effects as the added computational burden only lies on the CPU, but would require full-fledged VQE with 17 parameters (one circuit parameters and sixteen matrix components for the Hamiltonian dressing).
This would however not be advantageous over the strategy corresponding to previous work that consists in using the small HEA ansatz proposed in Ref.~\onlinecite{keen_quantum-classical_2020} as this latter ansatz has 3 CNOT gates too, and only 8 parameters.
Hopefully, the transformation to the NO basis in this context is almost always the same, so that it can be determined once and for all. We determine it by optimizing the HEA ansatz for $U=0, D=-0.4, \lambda_c=0.004$ (these embedding parameters being typical of those found along the RISB procedure for low $U$) with 10 sequential optimization cycles of the \textit{Rotosolve} algorithm \cite{ostaszewski_structure_2021} (that leverages the parameter-shift rule) and diagonalizing the 1-RDM associated with the optimal state (we test for 5 random initializations to limit the risk of dealing with an optimized state corresponding to a local minimum). We also test for the effect of a very simple error mitigation strategy, the zero-noise extrapolation \cite{he_zero-noise_2020}, here in the form of linear extrapolation with only 1 CNOT pair insertion.
\subsection{Results}
We display the results obtained with the technique described above in the noise-free case as well as the noisy and noisy mitigated cases on Figure \ref{fig:Z_Nc1}. Results obtained with the hardware-efficient ansatz (HEA\cite{keen_quantum-classical_2020}) in presence of error mitigation are also plotted as a reference to the state of the art against which our strategy is evaluated. The number of points in the Brillouin zone was set to 40 $\times$ 40 whereas the inverse temperature is $\beta=200$. The number of Nelder-Mead iterations is limited to 20.
Without noise, this strategy yields very accurate results, except for numerical instabilities at the Mott transition. When switching the noise on, the quasi-particle weight is at first underestimated but as $U$ grows, the error vanishes and the agreement with analytical data remains excellent for values of $U$ that are not too small. The poorer performances in the noisy regime at low $U$ are also observed at $N_c=2$. Here they correspond to a shift in the minimum of the optimization landscape that increases as the noise level grows, as can be seen on Figure \ref{fig:optim_ls_Nc1} (a)-(d). It may be due to the fact that the ansatz is overly complicated with regards to the state that must be prepared, so that noise has a greater impact. For instance here for $U=0$ the 3 CNOT gates are superfluous as the ground state corresponds to $\theta=0$, so that they add unnecessary noise. This intuition is supported by the fact that the noisy optimization of a 'product' ansatz, with a $RY$ gate applied on each qubit, better reproduces the noise-free, exact optimization landscape (see Fig.~\ref{fig:optim_ls_Nc1}(e)), and that an adverse effect of the parameter tuning method is ruled out (compare (d) and (e) in Fig.~\ref{fig:optim_ls_Nc1}). Overall, we observe a substantial enhancement in the accuracy of the results compared with the HEA ansatz, with a far more accurate rendering of the Mott transition. This comes with the additional benefit of necessitating a lot fewer shots, since the VQE optimization of the HEA circuit requires here to evaluate the cost function 240 times and is run 5 times to avoid local minima.
The strength of this statement is only midly mitigated by the fact that the NO Hamiltonian has a greater number of Pauli terms, since the increase is only of a factor $7-8$ (from 7 terms to 52 for $U>0$). All in all, the gain in terms of shots can be evaluated as:
\begin{equation}
\frac{n_{\mathrm{evals}}^{\mathrm{HEA}}}{n_{{\mathrm{evals}}}^{\mathrm{MR}}} = \frac{5 \times 8 \times 10 \times 3 \times 7}{1 \times 1 \times 1 \times 3 \times 52} \simeq 54,
\end{equation}
which is a substantial advantage in the context of an effective computation on quantum hardware.
\bibliographystyle{apsrev4-1}
|
2,877,628,090,173 | arxiv | |
2,877,628,090,174 | arxiv | \section{Introduction}
The first detailed calculations of the final, neutrino-cooled burning stages prior to core collapse
of massive stars were done by \cite{rsz67},
with simplified energy generation rates, and by \cite{wda68} using a small (24 species)
network.
Because of the extreme demands upon computer resources then available, simplified
energy generation rates were used in subsequent calculations; see \cite{wda96}
for references to the early work.
This work showed two issues which have not yet be resolved: (1) the definition of convective zone boundaries involves incomplete physics (by construction
mixing length theory ignores gradients, \cite{spie71}), and (2) the stability of the burning in a convective region has not been demonstrated, only assumed.
While thermal instability (thermal {\em stability} implies a global balance between nuclear heating and neutrino cooling in the convective zone) was considered \citep{wda72a,wda96}), dynamical instability (i.e., instability related to fluid flow) was not.
In this paper we will show that both these issues are significant,
and that they can be resolved by 3D numerical simulations and theory.
Almost all previous progenitor models for core collapse have focused on thermal
behavior and quasi-static mixing, which are described by the evolution of
the temperature and the composition variables. The dynamic behavior of the
stellar plasma includes and is dominated by the velocity fields, which not only imply mixing, but also possible change in the stellar structure. The star is not necessarily a quasi-static object, but may have significant fluctuations in its variables. The dynamical behavior, found here for simultaneous, multi-shell
burning, will drive entrainment \citep{ma07b} at convective shell boundaries,
changing the nucleosynthesis yields and the size of the Fe core at collapse.
The presupernova structure of a massive star consists of a core, mantle, and envelope. The envelope is extended, composed of H and He, and may have been removed prior to core collapse by wind-driven mass loss, or tidal stripping by a companion. The mantle is composed of burning shells of C, Ne, O and Si; these shells are convective, interact nonlinearly, contain most of the nucleosynthesis products ejected, and may smother a neutrino-driven explosion. The core is composed of Fe-peak nuclei, and its mass is determined by its entropy and electron fraction. Lower entropy and electron fraction give smaller cores, which are easier to explode by neutrino transport mechanisms. Core collapse mechanisms for explosion are sensitive to core mass, mantle density, and rotation; all are sensitive to the treatment of turbulence.
The simulations described here involve the simultaneous action of C, Ne,
O and Si burning shells.
The oxygen shell is special, because (1) formation of electron-positron pairs
softens the equation of state, aiding the formation of large amplitude motion,
(2) the large abundance of oxygen and its relatively large energy release per
unit mass provide a large thermonuclear energy reservoir, and (3) oxygen burning,
unlike silicon burning,
has little negative feedback from quasi-equilibrium to damp flashes (see below).
In Section~2 we discuss the historical context of progenitor models of core collapse
supernovae, focusing on issues of mixing, causes of time-dependence and multi-dimensionality, prospects for development of better computational tools,
and the differences between 2D and 3D simulations.
In Section~3 we describe our 2D simulations of progenitor evolution with multiple,
simultaneously active, burning shells (C, Ne, O, Si), and discuss some new phenomena which appear.
In Section~4 we summarize the implications for evolution prior to core collapse.
In Section~5 we focus on several problems in observational astrophysics which need to be reconsidered in light of our results.
In Section~6 we summarize the major conclusions, and outline the necessary features of new 3D simulations which will be required to quantitatively resolve the issues presented by the 2D simulations.
\section{Brief Historical review of 1D, 2D, and 3D Models}
\subsection{1D Models: Mixing}
By the term {\em advection} we mean the transport of a parcel of matter by
a large-scale flow; by {\em diffusion} we mean the transport of a parcel of matter
by a random walk of small-scale motions (often microscopic motion of ions
in the stellar plasma).
The mass (baryon) flux due to advection is
\begin{equation}
{\bf F_m} = {\bf u} \rho,
\end{equation}
where $\bf u$ is the fluid velocity vector and $\rho$ is the mass (baryon) density.
The flux of any scalar variable is related to this flux by a factor of the
density of the variable per unit mass (baryon). For example, if the mass (baryon)
fraction of nuclear species $i$ is $X_i$, the flux of this species is
\begin{equation}
{\bf F_i} = {\bf u} \rho X_i = {\bf F_m} X_i.
\end{equation}
The rate of change due to such fluxes involves a divergence, as in the continuity equation,
\begin{equation}
\partial \rho / \partial t = - {\bf \nabla \cdot F_m}.
\end{equation}
Thus, 1D advection involves a first order spatial differential operator.
If the considered volume contains heterogeneous (turbulent) matter,
the flux may vary over
the corresponding surface. If that volume is a zone in a stellar evolution computation,
then the fluxes must be averaged over the heterogeneity, and some knowledge
(or assumption) regarding the smaller scale structure is required. Consider the
simple example of only an inflow and an outflow, and variation only in the
vertical direction. The net flux for species $X_i$ is then
\begin{equation}
F_{net}(X_i) = - (F_m X_i )_{out} + (F_m X_i)_{in}.
\end{equation}
Suppose that $F_{out} = F_{in} \equiv \rho u$, so $F_{out}-F_{in}= 0$,
which automatically satisfies
mass (baryon) conservation, so that
\begin{equation}
F_{net}(X_i) = -\rho u \Big [ (X_i)_{out} - (X_i)_{in} \Big]. \label{advectx}
\end{equation}
In this case the flux (of composition $i$) is proportional to the negative of the
difference in abundance in the up and down flows\footnote{A more realistic
and relevant case for stellar turbulent convecction would allow different speeds in up and down flows; that complication is not necessary here.}.
For diffusion, the number flux follows from Fick's law,
\begin{equation}
{\bf \mathcal F_d}(X_i) = - {1\over3} \lambda v {\bf \nabla} N_i,
\end{equation}
and is proportional to the gradient of a number density $N_i$. Here $\lambda$ is
a mean-free-path and $v$ is a speed of diffusing particles.
The number density for species $i$ may be written as
$N_i = \rho {\cal A} X_i/A_i$, where ${\cal A} = 1/m_{amu}$ is Avogadro's number (the inverse of the atomic mass unit) and
$A_i$ is the number of nucleons in species $i$.
The mass (baryon) flux is ${\bf F_d}(X_i) = A_i m_{amu} {\bf \mathcal F_d}(X_i) $.
Thus,
\begin{equation}
{\bf F_d}(X_i) = -{1 \over 3} \lambda v {\bf \nabla} (\rho X_i). \label{diffusionx}
\end{equation}
If we consider the divergence of the flux, diffusion implies a second order spatial differential operator, in contrast to advection which, as we saw, implies first order. This is a fundamental mathematical difference.
\cite{wzw78,whw02} introduce an effective diffusion coefficient $D$ to model convection:
\begin{equation}
{d Y_i \over dt} = {\partial \over \partial m}\Big [ (4 \pi r^2)^2 \rho^2 D {\partial Y_i \over \partial m } \Big ].
\end{equation}
In the case of a region unstable to convection by the Ledoux criterion, they take
$D_c \equiv v_c \ell/3$, where $v_c$ is the velocity estimated by mixing-length
theory \citep{bv58,clay83} and $\ell$ is the mixing length. Thus if
\begin{equation}
\lambda v = \ell v_c, \label{wzwapprox}
\end{equation}
then Equation~ \ref{diffusionx} is recovered.
The use of Equation~\ref{diffusionx} requires that $ \lambda \ll \Delta r$,
where $\Delta r$ is a zone size.
For a turbulent cascade, $\ell \gg \Delta r$, which is inconsistent with the previous
requirement.
The treatment of convection as diffusion is essentially an algorithmic interpolation
procedure, but has
an intrinsic contradiction in physics. This arises because turbulent flow has two
facets: a flow at large scale $\ell$ which does most of the transport, and a flow at small scale $\lambda$ which does the dissipation, and $\ell \gg \lambda$. The diffusion
approximation might be used for the small scale flow, but that is irrelevant to the
transport problem, which is dominated by the large scale flow \citep{amy09}.
Notice the rough similarity between Equation~\ref{advectx}, which represents the underlying fluid dynamics \citep{ll59}, and Equation~\ref{diffusionx}, which represents the proposed approximation.
Taking the density outside the differential operator and using a finite difference
representation of Equation~\ref{diffusionx},
\begin{equation}
{\bf F_d(X_i) } \sim -\Big [ {1 \over 3} \lambda v \rho / \Delta r \Big ] \Big [ (X_i)_{out}- (X_i)_{in} \Big ].
\end{equation}
For this to be consistent with 1D advection (Equation \ref{advectx}) we must have
a diffusion rate that is dependent upon the zoning! While perhaps useful
algorithmically, this is not clear conceptually; see also the discussion of
\cite{mm00}, who also express doubts concerning the approximation of advection by diffusion.
Mathematically, diffusion is a procedure which maximally smooths
gradients, so that a more realistic procedure may be expected to exhibit
less smoothing.
While based on poor physics, the diffusion model of convection had the virtue that it allowed numerical prediction of yields.
Much effort has recently gone into extending stellar evolution to include rotational
effects \citep{zahn92,cdp95,mm00,tas,whw02}. Evolution of a rotating star will develop differential rotation in general, and shear flow. Similarly, deceleration of
convective plumes will develop shear flow. In stars both flows will be turbulent. Convection and rotation have an underlying similarity not reflected in stellar evolution theory.
The review of \cite{mm00} gives a clear presentation of the various approximations
involved in reducing the full fluid dynamic equations to a simpler, more easily solvable set. It is now possible to test these approximations by numerical
simulation in both shearing box \citep{am09iau, sg10} and whole star domains
\citep{bp09,bbbmt10}; see also the theoretical developments of \cite{balbus09,bw10}. A theoretically sound approach must treat the shear from
differential rotation and the shear from convective plumes on a consistent basis,
reflecting their underlying physical similarity \citep{turner73}.
\subsection{Time Variation\label{time-dep}}
\noindent{\bf The $\epsilon$-mechanism.}
The $\epsilon$-mechanism \citep{ledoux41,ledoux58} for driving stellar pulsations
by nuclear burning was examined by \cite{wda77} for Si burning in 1D geometry,
using the simple energy generation rate proposed by \cite{bcf68}. In this case,
explicit 1D hydrodynamic simulations gave nuclear-energized pulsations (see
Figure~6 in \cite{wda77}). There were two high frequency modes
(period ~$0.1$ seconds and $1$ second) and a lower frequency convective mode
(turnover time $\sim20$ seconds). The intermediate frequency mode was
related to the Si flash.
Computation with a realistic nuclear network subsequently showed that the highest frequency (``acoustic") pulsations would be strongly damped due to the
quasi-equilibrium nature of Si burning; an increase in temperature gave an increase
in free alpha particles (and neutrons and protons), which required energy, and
resisted the increase in temperature (with an almost 180 degree phase shift, meaning
that there is strong negative feedback to resist changes).
However, this damping process does not apply to O, Ne or C burning, which
have little or no quasi-equilibrium behavior, and in principle could drive pulsations
more vigorously.
\noindent{\bf The $\tau$-mechanism.}
\cite{am11b}, in study of the 3D simulations of \cite{ma07b}, found that the bursts
in turbulent kinetic energy were related to similar fluctuations in the \cite{lorenz}
model of a convective roll; this is the now-famous strange attractor.
This instability mechanism, called the $\tau$-mechanism for ``turbulence", is
expected to be a general property of stellar convection zones, and probably
is the cause of the fluctuations in luminosity observed in irregular variables
(e.g., Betelgeuse, see \cite{am11a,am11b}).
This process is independent of the temperature and density dependence of the
thermonuclear heating, and thus is distinctly different from the $\epsilon$-mechanism.
Unlike the $\epsilon$-mechanism, the $\tau$-mechanism is not a linear instability, but is inherently nonlinear.
In appropriate circumstances however, the $\tau$-mechanism may couple to nuclear burning, making pulsations more violent, giving a more complex, combined $\epsilon+ \tau$ mechanism.
The $\tau$ mechanism involves a {\em non-linear} instability, unlike the linear
instabilities discussed by \cite{gold97}.
In a detailed analysis, \cite{mbh06} solve the {\em linear} perturbation
equations for several massive stars prior to core collapse, and based upon
the $\epsilon$ mechanism alone, find inadequate driving to cause such violent behavior as is actually shown in the non-linear simulations presented below.
The analysis of \cite{mbh06} is perfectly correct within framework of the linear assumption, but the assumption fails.
The linear approximation is not valid in this case because turbulence is not a linear
perturbation of the system. \cite{lorenz} showed that chaotic behavior in a convective roll is due to the {\em nonlinear} interaction between temperature gradients (both vertical and horizontal) and convective speed. Solutions which are initially close to each other will diverge exponentially as time passes.
Thus the interplay between pulsation and
turbulent convection is not captured by traditional linear perturbation analysis
of stellar pulsations. This is an interesting theoretical result, suggesting
that linear perturbative methods for pulsations \citep{cox80,unno89} may require reexamination when turbulent convection is important (as those authors feared).
\subsection{History of Multi-dimensional Progenitor Models}
There have been relatively few multi-dimensional simulations of core collapse
progenitors, but there has been a rich context of efforts on turbulence and
stellar hydrodynamics \citep{pw94,pw00,fma89}, on the helium flash
\citep{deupree84,deupree96,deupree03,dearborn06,mocak08,mocak09,mocak10},
and on turbulent MHD with rotation (e.g., m-dwarf simulations by
\cite{browning08}), as well as extensive work on the Sun with the ASH code
(e.g., \cite{bbbmt10})
and on stellar atmospheres pioneered by Nordlund and Stein (e.g., \cite{nsa}). This context
has speeded the development of tools and helped determine their reliability.
The first effort at simulating oxygen burning \citep{wda94} helped define the shell burning problem with regard to required resolution, but suffered from the use of sectors so narrow in angle that boundary effects affected the flow.
\begin{deluxetable}{lrrrrr}
\tablecaption{Two-dimensional Progenitor Simulations\label{table1}}
\tabletypesize{\small}
\tablewidth{390pt}
\tablehead{ \colhead{Reference} &
\colhead{$a$} & \colhead{$b$} & \colhead{$c$} & \colhead{$d$} & \colhead{$e$}
}
\startdata
zoning & 256x64 & 256x64 & 172x60 & 800x320 & 800x320 \\
code & PROMETHEUS & PROMETHEUS & VULCAN & PROMPI & PROMPI \\
eos$\rm^f$ & wda & wda & wda & TS & TS \\
network & 12 & 123 & 12 & 25 & 37 \\
burning & O & Si &O & C,Ne,O & C,Ne,O,Si \\
core & Si & Si & Si & Si & Fe\\
duration(s) & 300 & ~200 & 1200 & 2500 & 600 \\
\enddata
\tablenotetext{a}{\cite{ba98,ba94}}
\tablenotetext{b}{\cite{ba97b}, a small inert spherical boundary surrounded by Si.}
\tablenotetext{c}{\cite{aa00}}
\tablenotetext{d}{\cite{ma06}, energy release verified against a 177 nuclei network}
\tablenotetext{e}{\cite{meakin06} and this paper, energy release verified against a 177 nuclei network}
\tablenotetext{f}{See \cite{ta99} and
\cite{ts00}.}
\end{deluxetable}
\placetable{1}
Table~\ref{table1} summarizes aspects of early 2D simulations of progenitor models
in comparison to the present work.
The first 2D hydrodynamic simulations of core-collapse progenitors (oxygen burning) showed striking new phenomena: mixing beyond formally stable boundaries, hot spots of burning due to $\rm C^{12}$ entrainment, and inhomogeneity in neutron excess \citep{ba94}. Further work \citep{ba98}
confirmed that convection was too dynamic to be well represented by diffusion-like
algorithms, that large density perturbations (8\%) formed at convective
boundaries, and that gravity waves were vigorously generated by the flow.
Extension to Si burning \citep{ba97b} with a 123
isotope network showed similar highly dynamic
behavior and significant inhomogeneity in neutron excess.
With an entirely different 2D code (VULCAN), \cite{aa00} extended the evolution of
the oxygen shell of \cite{ba98} to later times, and discovered that the extensive
wave generation at the convective boundaries induced a slow mixing in the
bounding non-convective regions.
\cite{kwg03} investigated shell oxygen burning in 3D with an anelastic hydrodynamics code \citep{gg84}, and found small density and pressure perturbations (less than 1\%). The boundary conditions were impermeable and stress-free, so that convective overshoot could not be studied. They concluded
that, contrary to previous work listed above (done with 2D compressible codes),
the behavior was not very dynamic and could be described by the MLT algorithms
used in 1D stellar evolution codes (e.g., \cite{wzw78}). \cite{ma07a}, using 3D compressible hydrodynamics, showed that the differences were due to the choice of (unrealistic) boundary conditions that \cite{kwg03} used. Inside the convection zone,
away from the boundaries, the Glatzmaier code gave results in good agreement with
the compressible code. However, as stressed in \cite{ba98},
fluid boundaries allow surface waves to build to large amplitudes ($\delta \rho / \rho \sim 0.1$), so that the hard boundaries used in \cite{kwg03} distorted the physics. The good agreement between the anelastic and the compressible solution {\em within} the convection zone, and the agreement between the stable layer dynamics given by the compressible fluid code and
analytic solutions to the non-radial wave equation, indicate that the compressible
hydrodynamic techniques are robust for this problem, even for Mach numbers below 0.01. An anelastic code with the correct boundary conditions should give the same result; the flow is subsonic.
With the PROMPI code \citep{meakin06}, several new series of computations were done. \cite{ma07a,ma07b} presented the first 3D calculations of the phase of shell
O burning with full physics (i.e., compressibility, nuclear network, real equation of state, appropriate boundaries, etc.).
\cite{ma06} calculated in 2D the oxygen burning shell, and both C and
Ne burning shells above this.
Here we present simulations \citep{meakin06} with similar microphysics, extended to include the silicon burning shell as well, so that C, Ne, O and Si burning occur on the grid, but only in 2D.
\subsection{Future Prospects: Beyond 2D}
Progress will not be a simple progression reflecting the growth of computational
resources, but also of theoretical understanding.
\cite{am11b} have shown a connection between the Lorenz model of a convective
roll and the bursty pulses in turbulent kinetic energy seen originally in the
\cite{ma07b} simulations, which seem to have a similar strange attractor.
The Lorenz model is a low order (3 variable) dynamical system, and its physical
identification with the 3D simulations suggests how we may project the essence
of the 3D solutions onto 1D for stellar evolution and dynamics (``321D").
This will give an immediate improvement over MLT as well as better initial models
for numerical simulation; see below. The physical basis is strengthened
mathematically by use of the Karhunen-Lo\`eve or proper orthogonal decomposition (POD; see \cite{hlb96})
of the 3D numerical data set of \cite{ma07b}. Preliminary results show that roughly half of the turbulent kinetic energy is in the single lowest order empirical eigenmode,
supporting the idea that low order dynamical systems may be used to describe the complexities of time dependent turbulent flow in stellar convection zones.
The dynamical phases prior to core collapse may not attain a statistically
averaged state, so that these methods (321D and KL decomposition), while
promising for earlier evolution, may not be the optimum tools for the pre-collapse
itself, where large eddy simulations (ILES, \cite{boris}) in 3D are needed.
However, use of the theoretical tools (321D and KL decomposition) provides
a means of estimating the range of fluctuations about an given ILES simulation,
which may be tested insofar as computational resources allow, for other initial
conditions.
\subsection{Comparison of 2D and 3D Simulations}
\cite{ma07b} show a precise comparison of two simulations, which use the same microphysics, initial model and code,
and differ only in dimensionality, which changes from 2D to 3D. The simulations are
discussed there as ``core convection" (see Figure~4 therein).
The topology of the convective flow is significantly different
between 2D and 3D models: the 3D convective flow is dominated by small plumes and eddies while the 2D flow is much more laminar, and dominated by large vortices
(``cyclones") which span the depth of the layer.
The 2D vortices trap material which is slow to mix with surrounding matter;
in 3D the flows become unstable
and matter mixes more completely. The wave motions in the stable layers do
not have an identical morphology in 2D and 3D, and the velocity amplitudes are much larger in 2D.
The differing behavior is due to the constraint of geometry, and the law of angular momentum conservation, which forces the formation of large cyclonic patterns in 2D that are unstable in 3D. The turbulent cascade moves from small scale to large (cyclones) in 2D, but from
large to small (Kolmogorov cascade) in 3D.
Cyclonic behavior at large scales is physically reasonable for the Earth's
atmosphere, for example, which because of its restriction in height, is approximately two-dimensional\footnote{The density scale height in the vertical direction is small compared to the width of a typical cyclonic system seen on the evening news. Oxygen would be required in an unpressurized aircraft at 35,000 feet, which is a measure of the ``height" of the atmosphere. This is less that one percent of the width of a large cyclonic system. This is a very flat domain, and with rotation, is dominated by geostrophic (i.e., 2D) flow patterns at the large scales.}. In stars there is no physical constraint to enforce 2D motion, so that 2D simulations of stars are not realistic, merely computationally less expensive.
In \cite{ma07b}, 3D simulations were done for a single cell in the O shell; the computational demands for simultaneous multi-shell burning are more extreme
(but almost feasible). As a first step we present 2D simulations, which though
flawed, are instructive.
We describe the first extended results for four simultaneous burning shells (Si burning included, \cite{meakin06}), and interpret them with the benefit of extensive analysis of 3D O burning results.
Taken with Figure~4 from \cite{ma07b}, they provide clues about the nature of the true behavior to be expected in 3D.
\section{2D Simulations of Multiple Simultaneously Burning Shells}
\subsection{Starting Point}
The initial conditions are a fundamental problem for numerical simulations which explore the development of instabilities to large amplitude. Subtle biases in
the initial state might make the subsequent development misleading. Use of
a 1D initial model, with no reliable description of the turbulent velocity field or the
extent and position of the boundaries of the convection, is a cause for worry, but is
the best we can do at present.
The 2D simulations started from a model obtained from a sequence by P. A. Young (private communication; \cite{fy08} have evolved this model to core collapse
in 1D, and to 3D explosions).
The initial state was a 1D model of a 23 $M_\odot$\ star, mapped to 2D; it is at the latest
stages of evolution, about one hour prior to core collapse.
Turbulent convection developed from very small perturbations ($\sim 10^{-3}$ or less, due to mapping onto a different grid) in the unstable regions. The convection rapidly evolved to a
dynamic state with far larger fluctuations, independent of the initial perturbations.
The model was
similar to that used in \cite{ma07b}, except the computational volume is deeper,
reaching down past the Si burning shell, the aspect ratio is larger (a full quadrant
was calculated), the abundance gradients were smoother (more like a diffusion
approximation for convection; see \S~2.1), and the onset of core collapse was much nearer ($\sim1$ hour).
\begin{figure}
\figurenum{1}
\includegraphics[angle=-90,scale=0.5]{zn.ps}
\caption{Thermonuclear reaction network (37 nuclei) used for burning in C, Ne, O, and Si shells. Each box represents a nucleus; see text for details.}
\label{fig0}
\end{figure}
The computational domain had an inner boundary representing
the Fe core and extended well beyond the active burning regions to the
edge of the He core. The boundary conditions in angle were periodic and
in radius were reflecting, as in \cite{ma06,ma07a,ma07b}.
The equation of state was that of \cite{ts00}, which
accurately describes the effects of partial relativistic degeneracy of electrons
and of the formation of electron-positron pairs, and is very similar \citep{ta99}
to the equation of state used in previous simulations listed above.
Figure~\ref{fig0} shows the nuclear reaction network used, which contained 38 species (37 isotopes plus electrons).
To deal with increasing neutron excess it was extended to $\rm Ni^{62}$, which
corresponds to ${\rm Z/A} \sim 0.45$. During Si burning, the nuclei having $\rm Z \ge 22$ approach a local nuclear statistical equilibrium (they become a quasi-equilibrium group). The results of this network were compared to
those of a 177 isotope network; it reproduced both the energy generation and
the increase in neutron excess predicted by the larger network (see discussion
of nucleosynthesis and of increase in neutron excess during silicon burning in \cite{wda96}). The nuclear burning was directly coupled to the fluid flow by the method of operator splitting.
\subsection{Results}
\begin{figure}
\figurenum{2}
\plotone{fig1.eps}
\caption{A snapshot of the structure of C, Ne, O, and Si shells surrounding the Fe-core of a pre-collapse progenitor of 23 $M_\odot$\ star, about 500 seconds after the constraint of spherical symmetry has been removed. The left side shows the abundance of $\rm Si^{28}$ while the right side shows the net energy generation rate.}
\label{fig1}
\end{figure}
Figure~\ref{fig1} shows the structure of a quadrant of the core of the $23 \msol$ star, with
an iron core (shown as a white semi-circle) in the center. The computational
domain includes burning shells of Si, O, Ne and C, in order of increasing radius.
The left quadrant displays abundance contours of $\rm Si^{28}$ (dark blue is
high abundance, white is low). The inner light region is a burning shell
which is depleting its Si. Above this is a dark ring which is unburned Si.
Further out is a medium blue layer which contains the O burning shell,
in which Si is being produced. At the top of this layer are seen streams of
very light blue, denoting entrainment of matter which has not been contaminated by oxygen burning (mostly C and Ne). Finally
there is a very light blue layer from which the streams came, and which
has no enhancement of Si above its original value.
The right quadrant shows contours of energy generation rate, in units of $10^{13}$
erg/s. Note that the second quadrant is presented as a rotation about the vertical axis; this helps identify corresponding matter in the two different variables
which are mirror images.
The inner ring is again the Si burning shell. It has both strong heating (yellow)
and strong cooling (blue) at the same radius, that is, the burning does not possess
spherical symmetry. This is enclosed by a green ring, the Si rich layer, which has
milder neutrino cooling, and is no longer spherically symmetric because of ``hot
spots" of burning in entrained (descending) plumes. Beyond this is a ring
of red and yellow which is highly dynamic: the O burning shell itself.
Finally, above this wisps and plumes of heating are beginning to be seen; these are due to C and Ne burning in entrained matter which is rich in these fuels.
These models have a low Damk\"ohler number \citep{damk},
$D_a = \tau_{turbulence}/\tau_{react} \ll 1$, where the time scale for turbulent flow
is $\tau_{turbulence}$ and the reaction time is $\tau_{react}$. In this limit
a complex mixing model is not needed to describe the burning, unlike
burning fronts in type Ia supernova models which require a more complicated
description.
Here fuel is transported into regions of higher pressure,
compresses and heats, burns, expands and cools, and is buoyantly transported back to lower pressure. Changes in composition due to turbulent mixing are much faster than those due to nuclear burning, so that a sub-grid flame model is not required.
\begin{figure}
\figurenum{3}
\plotone{fig2.eps}
\caption{Snapshots of the structure of C, Ne, O, and Si shells surrounding the Fe-core of
a pre-collapse progenitor of 23 $M_\odot$\ star. Three different times are shown,
$t_f$ = 0, 61, 83 seconds (from top, 0\ s, to bottom, 83\ s)
after our fiducial model (see text). The left panels (blue) show abundance of
$\rm Si^{28}$, while the right panels show energy generation rate and
convective speed, respectively.}
\label{fig2}
\end{figure}
\placefigure{3}
The initial model was strictly spherically symmetric, and had to develop a self-consistent turbulent velocity field to carry the heat from nuclear burning away from
the burning regions.
During this mild transient phase, asymmetries begin to develop slowly. Figure~\ref{fig1} shows the structure after this development, and at the beginning of significant
deviation from the usually assumed spherical symmetry. Because of the time
needed for the initial spherical model to develop a realistic and consistent convective flux, there is ambiguity regarding the ``initial time"; we simply define
a ``fiducial time" ($t_f = t-t_0$ where $t_0 \sim 345$ seconds into the 2D simulations) at which the model is still fairly spherical but has a realistic convective flux, and we count elapsed time after that.
Figure~\ref{fig2} shows three snapshots of the structure at different times after the
fiducial time: $t_f$ = 0, 61, and 83 seconds. The left column represents the same variables as shown in Figure~1. The right column shows the $\rm Si^{28}$ abundance as before in the left panel, but in the right panel the energy generation rate is replaced by the turbulent speed in units of $10^7$ cm/s.
Distortions in the O burning shell are obvious. The equation of state in this
region is affected strongly by the thermal production of an equilibrium abundance
of electron-positron pairs, so that the effective adiabatic exponent drops below\footnote{See Table~5 in \cite{amy09}, and recall that
$\nabla_{ad} = (\Gamma_2-1)/\Gamma_2 \approx 1/4$ across the whole O burning convective zone.}
$\Gamma_1 \sim 4/3$. Similarly $\Gamma_4 \equiv 1 + E/PV \sim 4/3$;
This means the local contribution to the gravitational binding energy, which is proportional to $\Gamma_4 - 4/3$, is small.
This is a common property of oxygen burning in stars \cite{wda68,wda96}. The restoring force for stable stratification is weak, allowing large amplitude distortions with little energy cost.
The decrease in $\Gamma_1$ and $\Gamma_4$ are due to the need to provide
the rest mass for newly formed electron-positron pairs. At low temperature,
$kT \ll 2 m_e c^2$, the number density of pairs relative to the charge density
of ions decreases exponentially with decreasing temperature, and is negligible.
As temperature approaches $ T \sim 2 \times 10^9 $ degrees Kelvin, the
effect on the equation of state becomes largest. At higher temperatures,
the increase in mass of pairs is less relative to the thermal energy $kT$,
so that these gammas approach that of an extreme relativistic gas,
$\Gamma_1 = \Gamma_4 = 4/3$. Oxygen burning ($\rm O^{16}$ fusion) in stars
occurs at $T \sim 2 \times 10^9$ degrees Kelvin, so that this burning stage is
most influenced by the effects of electron-positron pair production on the equation of state.
Consider first the left column in Figure~\ref{fig2}. The top panel (t=0) is relatively
symmetric, but as time passes, the middle and lower panel show increasing
distortion, especially visible at the interface between the outer, light blue layer
and the middle, medium blue layer.
The streams of light blue inside
the medium blue represents entrainment of matter with little Si, that is, C and
Ne fuel. This corresponds to the flame structures seen in the right side of the
left column,
which are due to C and Ne burning (note similarity in shapes on left and right sides in the left column).
Similar entrainment is occurring at the top of the Si burning convective region;
the outer edge of the light blue inner ring is rippled due to bursts of burning. The amplitude
of these variations is smaller due to the stiffer equation of state here.
The right side of the right column shows the turbulent convective speed. The large
structures are the oxygen burning convective zone. A smaller convection zone
may be seen surrounding the Fe core, due to the Si burning shell. The C-Ne layer,
lying outside the oxygen burning shell, illustrated the effect of a low-order mode.
In the top and bottom panels, there is little motion, while in the middle panel
the amplitude of the motion is near maximum. The lighter red areas at
about 30 degrees and 70 degrees from vertical correspond to nodes in the
modal velocity. Because of symmetry about the equator (horizontal) we have
four nodes in 180 degrees, or an $\ell = 4$ mode being dominant. Odd values
of $\ell$ are suppressed by the domain size and symmetry imposed by our boundary condition, so this is the lowest order possible in this simulation;
it has a period of about 60 seconds, but is mixed with other, weaker modes.
A movie of the simulation shows a dramatic change as this mode turns the speed on and off as we move from top to middle to bottom.
\begin{figure}
\figurenum{4}
\plotone{fig3.eps}
\caption{Structure of C, Ne, O, and Si shells surrounding the Fe-core of
a pre-collapse progenitor of 23 $M_\odot$\ star. Snapshots at times $ t_f$ = 115, 247, and 307 seconds (top to bottom) after our fiducial model (see text). The format is the same as in Figure~\ref{fig2}. The eruption has become strongly non-linear, as the bottom panels show.}
\label{fig3}
\end{figure}
\placefigure{4}
Figure~\ref{fig3} shows the same variables at t = 115, 247, and 307 seconds, after several, increasingly vigorous ``sloshes".
The distortion in the O shell has grown, with large amplitude motions
whose wave-form is
characteristic of $g$-modes. Entrainment of C and Ne increases.
Fluctuations in Si burning become more vigorous and distort the Si layer
although quasi-equilibrium damps explosive excursions.
The Si shell burning occurs in dynamically forming and disrupting cells,
qualitatively similar in nature to the 3D cell studied in \cite{am11b}.
A pink tinge indicates weak but widespread burning of
carbon in the outer layers of the computational domain.
At $t_f = 307$ seconds what is best described as an ``eruption" is occurring.
The distortion in the O shell (medium blue bulge) continues to grow;
it is due to strong wave motion, powered by the nuclear burning.
In the bottom panel, the thickness of the O shell convective layer varies by more than a factor of three as a function of angle; it is thin at the equator and thick at a 45 degree angle. This corresponds to an eruption at that angle.
Entrainment is correspondingly increased, with consequent burning.
Waves "slosh" back and forth. The
computational domain and boundary conditions restrict the lowest order modes.
The Si burning shell
is affected by the behavior of the O shell.
There appears to be no evidence of a slacking in growth of the dynamic
behavior, and extensive mixing is occurring. We ended the simulation because the
extent of mixing is so great that it reaches the grid boundaries, and the
simulation domain becomes inadequate.
The assumption of spherically symmetric structure has clearly failed,
as has the assumption of quasi-hydrostatic structure.
\section{Implications}
Figures~3-4 show the breakdown of the assumption of spherical symmetry,
which has been the basis of supernova progenitor models, and much of the interpretation of observational data of supernovae and young supernova remnants. The $\ell = 4$ spherical harmonic is dominant because the computational domain had only one quadrant and periodic boundaries; a full hemisphere would have allowed the $\ell = 1$ mode to appear.
These simulations, of a stage only one hour before core collapse, show more violent behavior than previous 2D simulations of multiple (CNeO) shells
\citep{ma06}, or 3D simulations with a single burning shell \citep{ma07b}. They become more violent in less time; compare durations
in Table~1.
There are several reasons for this behavior. (1) This stage is late, only an hour
prior to collapse. (2) The 3D O shell simulations had a
smaller computational domain, so that low order modes were restricted by periodic
boundary conditions. (3) The difference between the 2D three shell (CNeO) and four
shell (CNeOSi) simulations seems to be due to an additional interaction between the O and the
Si shells, which are both active.
Entrainment of new fuel causes mixing, which is countered by heating from burning, which changes entropy deficits in down-flows to entropy excesses, and halts the down-flow motion. This gives a
natural layering mechanism for the highly dynamic system, a {\em dynamic layering}.
Obviously the compositional structure, and
the predicted yields, depend upon this effect, which has been ignored (or treated
in a diffusion rather than advection algorithm) in all
1D models of progenitor evolution.
Inter-zonal mixing also would have an impact on yields.
Strong wave generation is observed. Such waves may become compressional (mixed mode, \cite{ma06}) as they propagate
into the strong density gradient. The waves will dissipate in non-convective regions, causing heating and slow mixing there, and to the extent that the wave heating is faster than radiative diffusion (which is very slow), expansion will occur. These effects are large enough to be seen in the simulations, but need further
quantification to determine their relative importance.
The Fe core (here a static boundary condition)
will contract, giving increasingly dynamic behavior. Such a dynamic approach to core-collapse has not been
investigated in multi-dimensional, multi-shell simulations, but may have interesting consequences (see below). The duration of the 2D CNeOSi simulation was $\sim 600$ seconds,
while the time for core collapse, neutrino trapping,
and rebound shock are together a few seconds \citep{wda96}. The observed diffusion time for neutrinos from SN1987A was comparable, also a few seconds. Because time for core collapse is fast in comparison to the period associated with O shell dynamics, the shell structure may be caught in a distorted state by the explosion shock, {\em giving a non-spherical remnant
even if the explosion shock were perfectly spherical} (which it is unlikely to be;
\cite{kif03,kif06}).
The 2D simulations show much more active dynamical behavior than suggested
by linear perturbation analysis \citep{mbh06}. In \S\ref{time-dep} this was
traced back to the treatment of turbulent convection in the linear
analysis. \cite{lorenz} showed that convection has an intrinsic nonlinear instability;
it arises from advection, which gives product terms (XY and XZ in his
notation); these are products of the velocity amplitude with the horizontal
and vertical temperature fluctuations respectively \citep{am11b}.
This provides an explicit example in which
linear perturbative methods for pulsations \citep{cox80,unno89} require modification when turbulent convection is present. Study of low order dynamical
models, such as \cite{lorenz}, will provide insight into the nature of this general problem.
It is unlikely that 1D progenitor models are realistic;
in addition to unlikely {\em geometry}, they ignore the vigorous {\em dynamic
behavior}, which becomes manifest in 2D and 3D simulations, in which it is not forbidden as it is in 1D. There may be eruptions prior to core
collapse, there will be large amplitude distortions away from spherical, ``onion''-skin
structure, and there may be modifications of the supernova shock by explosive oxygen burning. Not only can explosion shocks be non-spherical, but the progenitor mass
they propagate through can be asymmetric as well.
The distortions in the progenitor and those induced by the explosion shock
may leave an imprint on the abundances in the supernova and its ejecta.
Of course, rotation will have its own effects which we will have to disentangle.
\section{Some Issues to Reconsider}
\noindent{\bf Progenitor Structure and Fall-back.}
Mechanisms of explosion and fall-back are all predicated upon the validity of the detailed structure of the 1D progenitor models.
The old problem of non-explosion of supernova models \citep{wda96} has been alleviated by multi-dimensional simulations of collapsing cores (e.g., \cite{bldom06,bmh09,wjm10}). More
realistic initial models will bring further changes.
For example, the explosion
and remnant formation could be modified if an eruption occurred prior to and during core collapse. Similarly, expansion of the mantle surrounding the Fe core
could be caused by dynamic burning such as that illustrated in Figures~3-4.
It would reduce the ram pressure of infall and the mass to be
photo-dissociated, making it easier to eject matter for a given core explosion mechanism. See \cite{bbb82,bethe90,wda96}.
Changes in the mass and entropy of the collapsing core will affect its dynamics.
It has long been apparent that rotational effects too should play a role
\citep{fh46,fh55}. The historical focus on non-rotating collapse was merely because the non-rotating problem was feasible with computing resources then available. All of these characteristics of the progenitor model might be modified significantly by the use of 3D initial models.
The asymmetry and the rotational structure of the progenitor can be significantly
modified by burning shell interactions, as can the rate of fallback, which depends upon the mantle structure and rotation around the Fe core. See
\cite{fryer99,fbh96,fyh06} for a discussion of fallback and black hole formation.
Core collapse is a converging flow (density increases); explosion is a diverging one (density decreases). Asymmetries
in convergent flow grow, which is a problem for inertial-confinement fusion
efforts \citep{lindl}. In divergent slow, asymmetries decrease in importance, and
the flow tends toward the spherically symmetric similarity solutions \citep{sedov}.
Because of the growth of asymmetry during collapse, it is important to have
realistic estimates of the asymmetry in initial models of progenitors; 1D models
are spherical, so that the seeds of asymmetry are introduced numerically or
arbitrarily (e.g., \cite{wjm10,bmh09,nbblo}).
The 2D simulations above suggest that
progenitors prior to collapse develop large asymmetries in the O shell. What
about the Fe core, which is what collapses? Si burning is active is a layer
of convective cells, so that the asymmetries will tend to average out \citep{am11b}.
Simply scaling from Figure~\ref{fig3} suggests asymmetries in the Si shell of order
of a few percent or more. In the absence of simulations including the Fe core
in a dynamical way, we have no plausible quantitative estimates of asymmetries in the Fe core itself prior to collapse. Turbulence in O and Si shells
will drive fluctuations \citep{am11b}; the resulting motion will induce fluctuations
in the URCA shells in the core, and affect the change of entropy and electron fraction there (see \citep{wda96}, \S11 and \S12). This in turn will affect the mass of the core at collapse, and thus the parameters of the collapse mechanism.
Our understanding of the coupled physics of the dynamics of the core as
it approaches collapse is still uncomfortably vague.
\noindent{\bf Neutron Star Kicks.}
For core collapse progenitors, solitary or in binaries, the internal structure becomes
increasingly inhomogeneous in radial density as evolution procedes, and evolves toward a condensed core and extended mantle structure (Fig.~10.4 in \cite{wda96}).
As the core plus mantle mass of these objects approaches the Chandrasekhar mass from above, this tendency increases. Supernovae of type SNIb and SNIc
have light curves which require that they have such masses just prior to collapse. Consider a simple model in which there is a
point-like core inside an extended mantle. If the core is not located at the center
of mass, then the mantle must be displaced from the center of mass in the opposite
direction. More of the mantle mass lies to one side of the core, so there is a net
gravitational force which pulls the core back toward the center of mass (which does
not move). Similarly the core exerts an equal and opposite pull on the mantle to bring the mantle back toward the center of mass.
With no dissipation, an oscillation would ensue.
The motion of the core relative to the fluid in the mantle generates waves
in the mantle material, providing a means for dissipation, so that in the absence of
driving, the oscillation of the core and mantle about the center of mass would
be damped, and settle to a state in which both the core and mantle are centered
on the center of mass.
If there were a driving mechanism for core-mantle oscillation, there would be
an asymmetry due to the displacement of core and mantle relative to the
center of mass. The core collapse would give an off-center explosion within the mantle, even if the core collapse gave a perfectly spherical explosion shock
relative to the center of mass of the core.
Figures~3-4 above suggest that multiple shell burning may provide a driving mechanism for core-mantle oscillation; a computational domain containing an
entire hemisphere would have allowed an $\ell=1$ mode to develop, which is
even more suitable for driving such oscillations. Moreover, there will be a difference in the strength of
driving depending upon the mass of the O burning shell; low mass shells will be
less effective at driving the heavier core. Slow accretion onto an ONeMg white
dwarf, or evolution to collapse by electron-capture, would occur by O burning in
a low mass shell. However, evolution to core collapse by the instability of a more
massive Fe core generally occurs concurrently with more massive O burning shells, such as described above. \cite{vdh10} has suggested,
on the basis of data on double neutron stars in the galaxy, that formation of neutron stars from the collapse of ONeMg cores might occur with almost no kick velocity at
birth, while neutron stars formed by Fe core collapse would receive a large
space velocity at birth. See the review by \cite{kvw08} for background and references.
The discussion above provides a physical mechanism for the empirical
suggestion of van den Heuvel; the size of the kick velocity at collapse will depend
upon the mass of the oxygen shell surrounding the core, and is driven by the
dynamics of multiple shell burning.
\cite{wjm10} find more vigorous kicks from collapse of Fe cores because the
explosion develops sooner in ONeMg core, and the longer term instability
in the post bounce behavior has less time to be effective.
However, the collapse simulations to date have used small seed perturbations
which may be unrealistic (they are much smaller than those
found in Figures~3-4).
In addition, the simulations shown in Figures~3-4
assume a {\em static} Fe core, and thus underestimate the total effect:
asymmetry in the Fe core itself should have an
important effect on the collapse and bounce, as mentioned above.
A relatively
massive mantle may itself affect the post-bounce behavior of the explosion
by setting the initial condition which results in symmetry breaking
(see below), which in the long term gives the hydrodynamical kick \citep{wjm10}.
Either way, the higher kick velocity is associated with a more massive mantle,
which provides the mechanism for symmetry breaking, and a complete
picture needs to be developed.
\noindent{\bf Early $\gamma$-rays.} Gamma rays from the decay of $\rm Ni^{56}$
were observed in SN1987A before they were expected, based upon 1D progenitor models \citep{abkw89}.
A strong O shell eruption, if hot enough to produce some Ni, followed by convective buoyancy and penetrative convection {\em prior to collapse}, would explain the early detection of gamma-rays, with no new hypotheses.
Alternatively, explosive burning during the
passage of the ejection shock would give an uneven distribution of fresh $\rm Ni^{56}$.
Either way, some $\rm Ni^{56}$ would be moved out further than in a spherically symmetric model, allowing earlier escape of $\gamma$-rays.
\noindent{\bf Young SN Remnants}.
Young SN remnants have not yet mixed with the interstellar medium, and contain
abundance information about the progenitor.
The dynamic nature of pre-collapse evolution adds a new consideration to
attempts to connect progenitor models to
observations of young supernova remnants, such as Cas~A \citep{yeafr08}.
For example, the puzzling inversion of Fe relative to Si found by \cite{hughes00} in Cas~A could easily be explained by vigorous dynamics of the O shell prior to core collapse.
The spherical 1D models are likely to be an inadequate basis for interpreting
observational data (e.g., \cite{fesen88,jds09}), which may now be reanalyzed
with a broader insight. Aspherical shock waves
from the collapsed core become more spherical as they propagate outward
\citep{wjm10}, so that even a very non-spherical collapse may be ineffective at
producing asymmetries in the O shell. However, pre-existing asymmetries in the O shell, already significant, will be enhanced by explosive burning as the shock passes.
\section{Summary}
The evolution of core-collapse progenitors is likely to be strongly dynamic,
non-spherical, and may have extensive inter-shell mixing. These effects are
ignored in existing progenitor models.
The cyclonic patterns typical of 2D simulations are unstable in 3D, breaking apart and becoming the turbulent cascade of Richardson and Kolmogorov. This enhances damping, and results in the lower velocities seen in 3D relative to 2D simulations. However, simulations in 3D will have essentially the same driving
mechanisms as in 2D. Based upon existing simulations (e.g., \cite{ma07b}), it is unlikely that the increased damping will eliminate dynamic behavior entirely. The increased damping may be able to moderate the eruptions seen in 2D, so that a set of quasi-steady dynamic pulses develops and continues until
the core collapses. Alternatively, the increased damping may be inadequate to prevent continued growth of the instability, so that eruptions such as seen in Figure~\ref{fig3} will develop anyway, at a later time. The ultimate behavior would then be decided well into the non-linear regime. An extreme case would be an explosion
powered by O burning prior to collapse of the Fe core. The observed light curve would depend upon the mass, kinetic energy and amount of ejected $\rm Ni^{56}$
\citep{wda96}.
We need full 3D simulations to determine the quantitative impact of these new phenomena.
From the discussion above it is possible to determine the
features needed for such simulations:
\begin{itemize}
\item {\bf Full $4\pi$ steradians,} including the whole core, to get the lowest order fluid modes ($\ell=1$), rotation, and low order MHD modes,
\item {\bf Real EOS,} to capture effect of electron-positron pairs and relativistic partial degeneracy,
\item {\bf Network,} for realistic burning of C, Ne, O and especially Si with e-capture in a dynamic environment,
\item {\bf Multiple shells,} (C, Ne, O, Si) to get shell interactions,
\item {\bf Sufficient resolution,} to get turbulence
and to calculate coherent structures (ILES), and
\item {\bf Compressible fluid dynamics,} to get mixed mode waves, and possible eruptions.
\end{itemize}
Low Mach number solvers such as MAESTRO \citep{maestro} may be useful, if generalized to include a dynamic background (the core evolution accelerates),
or applied to earlier stages of neutrino cooled evolution (which may be strongly subsonic).
This is a challenging combination of constraints, but such computations are becoming feasible.
If we scale from the \cite{ma07b} 3D simulation, the increased solid angle gives a factor of 85, the increase in radius a factor of 2, for a total increase of 170. A further
increase in the radius of another factor of 2 would increase the computational load
by a factor of 340, and would allow investigation of eruptions further into the
strongly nonlinear regime than shown here.
However that simulation was done on
a small Beowulf cluster ($\sim100$ cpus, which were slower than available now).
This factor of 340 from the increased computational domain is more than balanced
by the increased computational power available with top level machines. Doing
an equivalent simulation, but in 3D, is feasible. More difficult is including the
Fe core, which requires a different grid near the origin, but this has already been
solved in different ways by several groups (see \cite{wpj03,dearborn06,wjm10}).
The computational demands, of a 3D simulation of the evolution of a progenitor into hydrodynamic core collapse, seems to be no worse than the computational demands of a single 3D core collapse calculation through bounce.
\begin{acknowledgements}
This work was supported in part by NSF Grant 0708871,
NASA Grant NNX08AH19G at the University of Arizona, and
by ARC DP1095368 (J. Lattanzio, P. I.) at Monash University, Melbourne, Australia.
Dr. T. Janka, Dr. E. M\"uller, Prof. F. Timmes and Prof. P. A. Young provided
detailed comments and questions which helped us tighten the presentation.
One of us (DA) wishes to thank
Prof. Remo Ruffini of ICRAnet, and Prof. John Lattanzio
of CSPA, Monash University,
Peter Wood of Australian National University / Mount Stromlo
Observatory, and the
Aspen Center for Physics for their hospitality and support.
\end{acknowledgements}
\eject
|
2,877,628,090,175 | arxiv | \section{Introduction}
\noindent Let \(X\) be a compact connected CW complex, let \(\widetilde{X}\) be the universal covering and let \(\overline{X}\) be a finite sheeted Galois covering. In this paper we will define the \emph{alpha numbers} \(\alpha_p(\overline{X}) \in \R\) in terms of the singular value decomposition of the \(p\)-th cellular differential of \(\overline{X}\). Intuitively, the definition of \(\alpha_p(\overline{X})\) in terms of singular values mimics the definition of \emph{Novikov--Shubin numbers} \(\alpha^{(2)}_p(\widetilde{X})\) in terms of spectral distribution functions. A natural question then asks whether the Novikov--Shubin numbers can be recovered asymptotically from the net of alpha numbers \((\alpha_p(\overline{X_i}))_{i \in F}\) of all finite Galois coverings of \(X\). We show that the answer is yes if the fundamental group contains a cyclic subgroup of finite index.
\begin{theorem} \label{thm:approxvcycspaceversion}
Suppose \(\pi_1(X)\) is virtually cyclic and \(\alpha_p(\widetilde{X}) < \infty^+\). Then
\[ \alpha^{(2)}_p(\widetilde{X}) = \limsup\limits_{i \in F} \alpha_p(\overline{X}_i). \]
\end{theorem}
\noindent Moreover, we construct a CW complex \(X\), obtained from \(S^1 \vee S^2\) by attaching one 3-cell, such that \(\alpha_3^{(2)}(\widetilde{X}) = 1\) but \(0 < \liminf_{i \in F} \alpha_3(\overline{X}_i) \le \frac{1}{2}\).
\subsection{The definition of alpha numbers}
To construct the numbers \(\alpha_p(\overline{X}_i)\), we will have to take a close look on the definition of Novikov--Shubin numbers. In doing so, let us go over from spaces to matrices which seem to form the appropriate setting for the approximation theory of \(L^2\)-invariants.
Let \(G\) be a countable, discrete group and let \(A \in M(r,s; \C G)\) be a matrix inducing the right multiplication operator \(r_{AA^*}^{(2)} \colon (\ell^2 G)^r \rightarrow (\ell^2 G)^r\) given by \(x \mapsto xAA^*\). Here the matrix \(A^*\) is obtained from \(A\) by transposing and applying the canonical involution \((\sum \lambda_g g)^* = \sum \overline{\lambda_g} g^{-1}\) to the entries. Let \(\{ E^{AA^*}_\lambda \}_{\lambda \ge 0}\) be the family of equivariant spectral projections obtained from \(r^{(2)}_{A A^*}\) by Borel functional calculus, \(E^{A A^*}_\lambda = \chi_{[0,\lambda]}(r^{(2)}_{AA^*})\), where \(\chi_{[0,\lambda]}\) is the characteristic function of the interval \([0,\lambda]\). Recall that the group von Neumann algebra \(\mathcal{N}(G)\) of \(G\) comes endowed with a canonical finite, faithful, normal trace \(\tr_{\mathcal{N}(G)}\) which extends diagonally to equivariant operators of \((\ell^2 G)^r\).
\begin{definition}
The function \(F_A \colon [0,\infty) \rightarrow [0,\infty)\) given by \(\lambda \mapsto \tr_{\mathcal{N}(G)}E^{AA^*}_{\lambda^2}\) is called the \emph{spectral distribution function} of the matrix \(A\). The \emph{upper Novikov--Shubin number} of \(A\) is given by
\[ \overline{\alpha}^{(2)}(A) = \limsup_{\lambda \rightarrow 0^+} \frac{\log(F_A(\lambda) - F_A(0))}{\log \lambda} \in [0,\infty] \]
unless \(F_A(\lambda) = F_A(0)\) for some \(\lambda > 0\) in which case we set \(\alpha^{(2)}(A) = \infty^+\). The \emph{lower Novikov--Shubin number} \(\underline{\alpha}^{(2)}(A)\) of \(A\) is defined similarly with ``\(\liminf\)'' in place of ``\(\limsup\)''. We say that \(A\) has the \emph{limit property} if \(\overline{\alpha}^{(2)}(A) = \underline{\alpha}^{(2)}(A)\). In this case we simply call this common value the \emph{Novikov--Shubin number} \(\alpha^{(2)}(A)\).
\end{definition}
\noindent The formal symbol ``\(\infty^+\)'' indicates a spectral gap at zero. We adopt the convention that \(c < \infty < \infty^+\) for all \(c \in [0,\infty)\). Novikov--Shubin numbers thus capture the polynomial growth rate near zero of the spectral distribution function \(F_A\). More precisely, if there are constants \(C, d, \varepsilon > 0\) such that \(C^{-1} \lambda^d \le F_A(\lambda) - F_A(0) \le C \lambda^d\) for \(\lambda \in [0, \varepsilon)\), then \(A\) has the limit property and \(\alpha^{(2)}(A) = d\). We should say that while the distinction between upper and lower Novikov--Shubin numbers is already contained in \cite{Gromov-Shubin:vonNeumannSpectra}, the (somewhat arbitrary) decision that \(\alpha^{(2)}(A)\) should mean \(\underline{\alpha}^{(2)}(A)\) has become accepted in the literature.
Now let \(G\) be \emph{residually finite} meaning there exists a \emph{residual system} \((G_i)_{i \in I}\), an inverse system of finite index normal subgroups directed by inclusion over a directed set \(I\) with trivial total intersection. We obtain matrices \(A_i \in M(r,s; \C (G/G_i))\) from \(A\) by applying the canonical projections \(\C G \rightarrow \C(G/G_i)\) to the entries. Set \(n_i = [G \colon G_i]\). Then the group algebra \(\C(G/G_i)\) embeds as a subalgebra of \(M(n_i, n_i; \C)\) by means of the left regular representation of the finite group \(G/G_i\). Accordingly, we can view \(A_i\) as lying in \(M(r n_i, s n_i; \C)\). So we can consider the positive \emph{singular values}
\[ \sigma_1(A_i) \ge \cdots \ge \sigma_{r_i}(A_i) > 0 \]
of \(A_i\) given by \(\sigma_j(A_i) = \sqrt{\lambda_{j,i}}\) where the \(\lambda_{j,i}\) are the positive eigenvalues of \(A_iA_i^*\) in non-ascending order and \(r_i = \rank_\C A_i\). We denote the \emph{multiplicity} of \(\sigma_j(A_i)\) as \(m_j(A_i) = \dim_\C \ker(A_i A_i^* - \lambda_{j,i})\) and set \(m_{r_i + 1}(A_i) = \dim_\C\ker(A_i A_i^*)\). With this data, the spectral distribution function \(F_{A_i}\) can be described as a monotone, right continuous step function with jumps at the singular values \(\sigma_j(A_i)\) and jump size \(\frac{m_j(A_i)}{n_i}\). It is known that these step functions approximate the spectral distribution function \(F_A\). More precisely,
\[ F_A(\lambda) = \lim_{\delta \rightarrow 0^+} {\textstyle \limsup\limits_{i \in I}}\, F_{A_i}(\lambda + \delta) = \lim_{\delta \rightarrow 0^+} {\textstyle \liminf\limits_{i \in I}}\, F_{A_i}(\lambda + \delta) \]
as is proven in \cite{Lueck:Approximating}*{Theorem~2.3.1} for residual chains (when \(I\) is totally ordered), the proof for residual systems being similar.
So we might want to think about the values \(F_{A_i}(\sigma_j(A_i)) = \sum_{k \ge j} \frac{m_k(A_i)}{n_i}\) as experimental samples of the function of interest \(F_A\). To extract the growth rate of \(F_A\) from these samples we do what every physicist would do: we measure the slope of the regression line through the doubly logarithmic scatter plot of the samples. The sample that is most valuable for our purposes is given by the first positive singular value \(\sigma^+(A_i) = \sigma_{r_i}(A_i)\) with multiplicity \(m^+(A_i) = m_{r_i}(A_i)\).
\begin{definition}
The \emph{alpha number} of a nonzero \(A_i \in M(r,s; \C(G/G_i))\) is
\[ \alpha(A_i) = \frac{\log\frac{m^+(A_i)}{[G \colon G_i]}}{\log \sigma^+(A_i)} \in \R. \]
\end{definition}
\noindent Choosing the first positive singular value in the above definition serves a double purpose. Firstly, this makes sure that the growth behavior close to zero is reflected because \(\lim_i \sigma_+(A_i) = 0\) whenever \(\alpha^{(2)}(A) < \infty^+\). Secondly, since therefore \(\log \sigma_+(A_i)\) tends to \(-\infty\), the alpha number ultimately measures the slope of the line through the origin which is parallel to the regression line and hence has the same slope. Finally note that the embedding \(\C (G/G_i) \subset M(n_i,n_i; \C)\) as a subalgebra is unique up to conjugating with a permutation matrix and a diagonal matrix with entries \(\pm 1\). Any two resulting embeddings \(M(r, s; \C (G/G_i)) \subset M(r n_i, s n_i; \C)\) are thus conjugate by a unitary transformation which leaves the singular value decomposition unaffected. This shows that the alpha number is well-defined.
\subsection{Approximating Novikov--Shubin numbers by alpha numbers}
The canonical example of a residual system is the \emph{full residual system} \((G_i)_{i \in F}\) of \emph{all} finite index normal subgroups of \(G\). We ask the following question.
\begin{question} \label{question:approxmatrixversion}
Let \(G\) be a residually finite group, let \(\Q \subset F \subset \C\) be a field and let \(A \in M(r,s; F G)\). Suppose that \(\overline{\alpha}^{(2)}(A) < \infty^+\). Is it true that
\begin{enumerate}[(a)]
\item \label{item:sup} \(\overline{\alpha}^{(2)}(A) = \limsup_{i \in F} \alpha(A_i)\)?
\item \label{item:inf} \(\underline{\alpha}^{(2)}(A) = \liminf_{i \in F} \alpha(A_i)\)?
\end{enumerate}
\end{question}
\noindent In this paper we answer Question~\ref{question:approxmatrixversion} for virtually cyclic groups.
\begin{theorem} \label{thm:approxvcycmatrixversion}
Let \(G\) be a virtually cyclic group and let \(\Q \subset F \subset \C\) be an arbitrary field. Then the answer to Question~\textup{\ref{question:approxmatrixversion}\,\eqref{item:sup}} is positive and the answer to Question~\textup{\ref{question:approxmatrixversion}\,\eqref{item:inf}} is negative.
\end{theorem}
\noindent We remark that the related \emph{approximation conjecture for Fuglede--Kadison determinants} \cite{Lueck:Survey}*{Conjecture~6.2} is likewise only known for virtually cyclic groups \cite{Schmidt:DynamicalSystems}. Though the class of groups is small, the proof of Theorem~\ref{thm:approxvcycmatrixversion} is nontrivial and requires number theoretic input. Here also lies the reason for the symmetry breaking answer which at first glance might come as a surprise. It is the existence of infinitely many good rational approximations to a given irrational number which tears the lower limit apart from the upper one. But for virtually cyclic \(G\) it is easy to see that every \(A \in M(r,s;\C G)\) has the limit property. So even for virtually cyclic groups the equality \(\alpha^{(2)}(A) = \limsup_{i \in F} \alpha(A_i)\) cannot be improved to \(\alpha^{(2)}(A) = \lim_{i \in F} \alpha(A_i)\). However, for \(F = \Q\) we can show that \(\liminf_{i \in F} \alpha(A_i)\) is always positive as a consequence of a result in transcendence theory. We will discuss this in a moment but first let us return from matrices to spaces and explain that the case \(F = \Q\) of Theorem~\ref{thm:approxvcycmatrixversion} gives Theorem~\ref{thm:approxvcycspaceversion} and the example below it.
Let \(X\) be a connected finite CW complex with \(G = \pi_1(X)\) residually finite. Choosing a cellular basis of \(X\) gives rise to an isomorphism that identifies the \(p\)-th cellular chain module \(C_p(\widetilde{X})\) of the universal covering with the standard left \(\Z G\)-module \((\Z G)^{N_p}\). Here \(N_p\) is the number of \(p\)-cells of \(X\) or, equivalently, the number of \(G\)-equivariant \(p\)-cells of the \(G\)-CW complex \(\widetilde{X}\). Under this isomorphism the \(G\)-equivariant differential \(d_p \colon C_p(\widetilde{X}) \rightarrow C_{p-1}(\widetilde{X})\) of the chain complex \(C_*(\widetilde{X})\) is represented by right multiplication with a matrix \(A(\widetilde{X}, p) \in M(N_p,N_{p-1}; \Z G)\). We define the \emph{\(p\)-th Novikov--Shubin number} of \(\widetilde{X}\) as \(\alpha^{(2)}_p(\widetilde{X}) = \alpha^{(2)}(A(\widetilde{X}, p))\). Note that in \cite{Lueck:L2Invariants}*{Definition~2.16, p.\,81} one restricts the induced operator \(d_p \colon \ell^2(G)^{N_p} \rightarrow \ell^2 (G)^{N_{p-1}}\) to the orthogonal complement of \(\im d_{p+1}\) to make sure the spectral distribution function takes the value \(b^{(2)}_p(\widetilde{X})\) at zero. For the Novikov--Shubin numbers this is of course irrelevant.
Given a finite index normal subgroup \(G_i \subset G\) we can construct the finite covering space \(\overline{X}_i\) with deck transformation group \(G/G_i\) as \(G_i \backslash \widetilde{X}\). The chosen cellular basis of \(X\) identifies \(C_p(\overline{X}_i) \cong (\Z (G/G_i))^{N_p}\) and the differential \(d^i_p \colon C_p(\overline{X}_i) \rightarrow C_{p-1}(\overline{X}_i)\) is thus represented by right multiplication with a matrix \(A(\overline{X}_i, p)\) which coincides with the matrix \(A(\widetilde{X},p)_i\) obtained from \(A(\widetilde{X}, p)\) by applying the canonical projection \(\Z G \rightarrow \Z (G/G_i)\) to the entries. We define the \emph{\(p\)-th alpha number} of \(\overline{X}_i\) as \(\alpha_p(\overline{X}_i) = \alpha(A(\overline{X}_i, p))\). Both Novikov--Shubin numbers and alpha numbers are well-defined because the isomorphisms \(C_p(\widetilde{X}) \cong (\Z G)^{N_p}\) and \(C_p(\overline{X}_i) \cong (\Z (G/G_i))^{N_p}\) are unique up to unitaries.
With these definitions it is immediate that Theorem~\ref{thm:approxvcycmatrixversion} implies Theorem~\ref{thm:approxvcycspaceversion}. It is moreover well-known that matrices in \(M(r, s; \Z G)\) can be realized as cellular differentials of \(G\)-CW complexes, compare \cite{Lueck:L2Invariants}*{Lemma~10.5, p.\,371}. In this way the counterexample we will construct for Question~\ref{question:approxmatrixversion}\,\eqref{item:inf} translates to the example mentioned below Theorem~\ref{thm:approxvcycspaceversion}.
\subsection{The role of the coefficient field}
This realization of matrices over \(\Z G\) as differentials of based \(G\)-CW complexes is why Theorem~\ref{thm:approxvcycspaceversion} is actually equivalent to (a positive answer to) Question~\ref{question:approxmatrixversion}\,\eqref{item:sup} for \(F = \Q\). Similarly, the aforementioned determinant approximation conjecture \cite{Lueck:Survey}*{Conjecture~6.2} is formulated for coefficients in \(\Q\). It is remarkable that for coefficients in \(\C\) the statement of the determinant approximation conjecture is wrong, even in the case of a \((1 \times 1)\)-matrix over \(\C[\Z]\), see \cite{Lueck:L2Invariants}*{Example~13.69, p.\,481}. This is just one instance showing that coefficients matter for approximation questions. In the ``topological case'' \(F = \Q\), there are results in the theory of linear forms in (two) logarithms which are of value to us. They allow at least the conclusion that \(\liminf_{i \in F} \alpha(A_i)\) is positive, as it should be, because so is every \(\alpha^{(2)}(A)\).
\begin{theorem} \label{thm:liminfpositive}
Let \(G\) be a virtually cyclic group and let \(A \in M(r, s; \Q G)\) with \(\alpha^{(2)}(A) < \infty^+\). Then \(\liminf_{i \in F} \alpha(A_i) > 0\).
\end{theorem}
\noindent For Theorem~\ref{thm:approxvcycspaceversion} this says that while it can happen that \(\liminf_{i \in F} \alpha_p(\overline{X}_i) < \limsup_{i \in F} \alpha_p(\overline{X}_i)\), at least we have \(\liminf_{i \in F} \alpha_p(\overline{X}_i) > 0\). In fact, the number theory involved gives something stronger than Theorem~\ref{thm:liminfpositive}, namely the existence of some \(D > 0\) such that \(\liminf_{i \in F} \alpha(A_i) \ge \frac{\alpha^{(2)}(A)}{D+1}\) together with some explicit bounds for the constant \(D\) in terms of degree and height of a certain polynomial associated with \(A\). For the precise statement see Corollary~\ref{cor:boundonliminf}.
\subsection{Outline and organization of the paper}
Our proofs of Theorem~\ref{thm:approxvcycmatrixversion} and Theorem~\ref{thm:liminfpositive} rely on methods from Diophantine approximation and transcendence theory. Since these are topics that tend to fall short in a typical topologist's curriculum, we give a brief recap in Section~\ref{section:preliminaries} and recall the theorems of Dirichlet, Kronecker, Gelfond--Schneider and a baby version of Baker's theorem. We also fix the terminology we use in the context of nets.
In Section~\ref{section:onebyone} we start with the proof of Theorem~\ref{thm:approxvcycmatrixversion}. As a warm-up we consider the case of the easiest polynomial \(p(z) = z-1\) and show that Dirichlet's theorem easily answers Question~\ref{question:approxmatrixversion}\,\eqref{item:inf} in the negative. To answer Question~\ref{question:approxmatrixversion}\,\eqref{item:sup} affirmatively, we then move on with the case of a \((1 \times 1)\)-matrix over the group ring \(\C[\Z]\). It turns out that again one runs into a problem of Diophantine approximation: Can one find a sequence of regular \(i\)-gons whose vertices are far away from given elements of the unit circle? Solving this problem amounts to understanding how the rational dependency of coordinates of a torus point determines the closure of its \(\Z\)-orbit. This is what Kronecker's theorem accomplishes.
In Section~\ref{section:rbys} we perform the passage to \((r \times s)\)-matrices over \(\C[\Z]\). The methods are singular value inequalities and another simple but effective tool that is widely employed in Diophantine approximation: the pigeon hole principle.
Section~\ref{section:virtuallycyclic} reduces the general case of a virtually cyclic group to the case of the group \(\Z\) and thereby finishes the proof of Theorem~\ref{thm:approxvcycmatrixversion}.
Finally, Section~\ref{section:liminfpositive} discusses the case of rational coefficients. The little Baker theorem and thus the theory of bounding linear forms in (two) logarithms is what allows in this case the conclusion of Theorem~\ref{thm:liminfpositive}.
\subsection{Acknowledgements}
I am indebted to Yann Bugeaud, Wolfgang L\"uck, Malte Pieper, Henrik R\"uping, Roman Sauer and Thomas Schick for helpful conversations.
\section{Preliminaries}
\label{section:preliminaries}
\subsection{Some facts from Diophantine approximation} \label{subsection:diophantine}
For a real number \(x\) let \(\|x\|\) denote the distance to the closest integer. It is easy to see that the usual triangle equality \(\| x + y \| \le \|x\| + \|y\|\) holds. From this it follows that \(\|nx\| \le \lvert n \rvert \|x\|\) for any \emph{integer} \(n\). Dirichlet famously concluded the following result from the pigeon hole principle.
\begin{theorem}[Dirichlet, {\raise.17ex\hbox{$\scriptstyle\sim$}}1840] \label{thm:Dirichlet}
Given real numbers \(l_1, \ldots, l_u\) and a natural number \(N\), there is \(1 \le q \le N\) such that \(\|q l_i\| \le N^{-\frac{1}{u}}\) for all \(i = 1, \ldots, u\).
\end{theorem}
\noindent Dirichlet's theorem will be key for constructing a counterexample to Question~\ref{question:approxmatrixversion}\,\eqref{item:inf} in Section~\ref{section:onebyone}. We are moreover interested in an inhomogeneous variant of this problem of simultaneous Diophantine approximation: If additionally real numbers \(x_1, \ldots, x_u\) and \(\varepsilon > 0\) are given, does there exist \(q \in \Z\) with \(\|ql_i - x_i\| < \varepsilon\) for all \(i = 1, \ldots, u\)? The answer cannot be an unconditional ``yes'' because there might be integers \(A_1, \ldots, A_u\) with the property that the linear combination \(\sum_{i=1}^u A_i l_i\) is an integer as well. If the desired conclusion held true, we would get
\[ \| A_1 x_1 + \cdots + A_u x_u \| = \| A_1 (q l_1 - x_1) + \cdots + A_u (q l_u - x_u) \| \le (\lvert A_1 \rvert + \cdots + \lvert A_u \rvert) \varepsilon \]
which says that \(\sum_{i=1}^u A_i x_i\) is an integer, too. The good news is that this necessary condition is also sufficient.
\begin{theorem}[Kronecker, 1884] \label{thm:Kronecker}
Let \(l_1, \ldots, l_u\) and \(x_1, \ldots, x_u\) be real numbers. The following are equivalent:
\begin{enumerate}[(i)]
\item For every \(\varepsilon > 0\) there is \(q \in \Z\) such that \(\|q l_i - x_i\| < \varepsilon\) for \(i = 1, \ldots, u\).
\item For every \(u\)-tuple \((A_1, \ldots, A_u) \in \Z^u\) with the property that \(\sum_{i=1}^u A_i l_i\) is an integer, the linear combination \(\sum_{i=1}^u A_i x_i\) is an integer as well.
\end{enumerate}
\end{theorem}
\noindent A proof can be found in \cite{Cassels:Diophantine}*{Theorem~IV, p.\,53}. We remark that Kronecker's theorem is usually given in a slightly more general version where the real numbers \(l_i\) are replaced by linear forms but as of now we do not need this. Kronecker's theorem will become handy for understanding torus orbits in Section~\ref{section:onebyone}.
\begin{theorem}[Gelfond--Schneider, 1934] \label{thm:Gelfond-Schneider}
Let \(\alpha_1, \alpha_2 \in \overline{\Q}\) be different from \(0\) and~\(1\) such that (some fixed values of) \(\log \alpha_1\) and \(\log \alpha_2\) are linearly independent over \(\Q\). Then \(\log \alpha_1\) and \(\log \alpha_2\) are linearly independent over \(\overline{\Q}\).
\end{theorem}
\noindent This theorem has the equivalent formulation that for \(\alpha_1, \alpha_2\) as above and additionally \(\alpha_2\) irrational, any value of \(\alpha_1^{\alpha_2}\) is transcendental. As such, it yields the positive answer to Hilbert's seventh problem. For applications to Diophantine equations not only the nonvanishing of the \emph{linear form in two logarithms}
\[ \Lambda = b_1 \log \alpha_1 + b_2 \log \alpha_2 \]
is important but also explicit lower bounds on \(\Lambda\) in terms of the heights and degrees of \(b_1, b_2 \in \overline{\Q}\) are relevant. For our purposes it is enough to consider the special case where \(b_1\) and \(b_2\) are rational integers.
\begin{theorem} \label{thm:littlebaker}
Let \(\alpha_1, \alpha_2 \in \overline{\Q}\) be different from \(0\) and~\(1\) and let \(b_1, b_2\) be rational integers such that \(\Lambda \neq 0\). Set \(B = \max \{\lvert b_1 \rvert, \lvert b_2 \rvert\}\). Then there is a constant \(D\) depending only on the heights and degrees of \(\alpha_1\) and \(\alpha_2\) such that
\[\lvert \Lambda \rvert > B^{-D}. \]
\end{theorem}
\noindent It is hard to track down where exactly in the involved history of bounding logarithms in linear forms the theorem in this formulation was included for the first time. Gelfond already gave the weaker estimate \(\lvert \Lambda \vert > C e^{-(\log B)^\kappa}\) with improvements on the constant \(\kappa \) over two decades \citelist{\cite{Gelfond:first} \cite{Gelfond:second} \cite{Gelfond:third}}. But the above theorem is definitely a special case of Baker's celebrated theorem from 1966-1967, see \cite{Baker:LinearForms}*{Theorem~2} for a strong version and information on the constant \(D\). Let us refer to any \(D = D(\alpha_1, \alpha_2) \ge 1\) satisfying the inequality of the theorem as a \emph{Baker constant} of the pair \((\alpha_1, \alpha_2)\). Theorem~\ref{thm:littlebaker} will be crucial for the proof of Theorem~\ref{thm:liminfpositive} in Section~\ref{section:liminfpositive}.
\subsection{Nets and cluster points} \label{subsection:nets}
The finite index normal subgroups of a group and thereby the finite Galois coverings of a space are natural examples of \emph{directed sets}. A set \(I\) is called \emph{directed} if it comes with a reflexive, transitive binary relation ``\(\le\)'' such that any two elements \(a, b \in I\) have a common \emph{upper bound} \(c \in I \) with \(a \le c\) and \(b \le c\). A function from a directed set \((I, \le)\) to a topological space \(X\) is called a \emph{net} in \(X\). If \((x_i)_{i \in I}\) is a net in \(X\), then a point \(c \in X\) is called a \emph{cluster point} if for every neighborhood \(U\) of \(c\) and for every \(i \in I\) there exists \(j \ge i\) with \(x_j \in U\). The set of cluster points is closed. In the special case \(X = \R\) we define \(\limsup_{i \in I} x_i\) and \(\liminf_{i \in I} x_i\) as the largest and the smallest cluster point, respectively. Here, we also allow the values \(\pm \infty\) as cluster points in the natural way, so that both \(\limsup_{i \in I} x_i\) and \(\liminf_{i \in I} x_i\) are guaranteed to exist. If the latter two are equal, we say the net is \emph{convergent} and write \(\lim_{i \in I} x_i\) for the common value. Alternatively, we clearly have the description
\[ \liminf_{i \in I} x_i = \sup_{i \in I} \inf_{i \le j} x_j \quad \text{and} \quad \limsup_{i \in I} x_i = \inf_{i \in I} \sup_{i \le j} x_j. \]
For the set of natural numbers \(\N\) we will have occasion to deal with two different directions. One is the usual total order ``a \(\le\) b'' in which all the above notions reduce to the familiar ones from sequences. The other is divisibility ``\( a \mid b\)'' and arises when we identify \(\N\) with the full residual system \(F\) of the group \(\Z\). We should clarify the relation between the resulting upper and lower limits in order to dispel any possible confusion from the very start.
\begin{lemma} \label{lemma:sequencenet}
Let \(a \colon \N \rightarrow \R\) be a function which we interpret either as the sequence \((a_i)_{i \ge 0}\) or as the net \((a_i)_{i \in F}\). Then
\[ \liminf_{i \rightarrow \infty} a_i \le \liminf_{i \in F} a_i \le \limsup_{i \in F} a_i \le \limsup_{i \rightarrow \infty} a_i \]
where each inequality can be strict.
\end{lemma}
\begin{proof}
Let \(c \in \R\) be a cluster point of the net \((a_i)_{i \in F}\). By definition this means that for all \(\varepsilon > 0\) and for all \(k \in F = \N\) there is \(l \in \N\) such that \(\lvert a_{kl} - c \rvert < \varepsilon\). In particular, we obtain a subsequence \((a_{i_k})_{k \ge 0}\) of \((a_i)_{i \ge 0}\) which converges to \(c\). Thus any cluster point of the net \((a_i)_{i \in F}\) is a cluster point of the sequence \((a_i)_{i \ge 0}\). This gives the two outer inequalities of the lemma.
Consider the example \(a_i = (-1)^i\). Then the leftmost inequality is strict for \((a_i)\) and the rightmost inequality is strict for \((-a_i)\). To see that the middle inequality can be strict, consider \(a_i = (-1)^{N_i}\) where \(N_i\) is the number of prime factors of \(i\).
\end{proof}
\section{The case of a single Laurent polynomial} \label{section:onebyone}
\noindent In this section we give a proof of Theorem~\ref{thm:approxvcycmatrixversion} for \(r = s = 1\). Consider an element \(A \in M(1,1; \C[\Z])\). The full residual system is given by \(G_i = i\Z\) for \(i \in F = \N\) directed by divisibility. We identify the group ring \(\C[\Z]\) with the ring of Laurent polynomials \(\C[z, z^{-1}]\). Moreover, Fourier transform identifies the Hilbert space \(\ell^2(\Z)\) with \(L^2(S^1, \mu)\), the space of square integrable complex valued functions on the unit circle with respect to the probability Haar measure~\(\mu\), factoring out those function which vanish almost everywhere.
\subsection{Two examples} \label{subsection:twoexamples} Let us sneak up on the proof by considering the first nontrivial case \(A = (p(z))\) with \(p(z) = z-1\). The operator \(r^{(2)}_{AA^*}\) is then given by multiplying functions with \(|z-1|^2\). We have \(\alpha^{(2)}(A) = 1\) as can be seen from the proof of \cite{Lueck:L2Invariants}*{Lemma~2.58, p.\,101}. By finite Fourier transform, the matrices \(A_iA_i^* \in M(1,1;\C[\Z / i\Z]) \subset M(i,i; \C)\) are diagonal with entries \(|\zeta_i^k - 1|^2\) where \(\zeta_i\) is one of the two primitive \(i\)-th roots of unity that enclose the smallest angle with \(1 \in \C\), where \(k = 0, \ldots, i-1\) and say \(i \ge 3\). Thus we have \(\sigma^+(A_i) = |\zeta_i - 1| = 2 \sin(\frac{\pi}{i})\) and \(m^+(A_i) = 2\). By L'H\^opital's rule and substituting \(x = \frac{\pi}{i}\) the ordinary limit of the sequence \((\alpha(A_i))_{i \ge 0}\) is
\[ \lim_{i\rightarrow\infty} \alpha(A_i) = \lim_{i\rightarrow\infty} \frac{\log\left(\frac{2}{i}\right)}{\log\left(2\sin\left(\frac{\pi}{i}\right)\right)} = \lim_{i\rightarrow\infty} \frac{i \tan\left(\frac{\pi}{i}\right)}{\pi} = \lim_{x \rightarrow 0^+} \frac{\tan(x)}{x} = 1. \]
By Lemma~\ref{lemma:sequencenet} the net \((\alpha(A_i))_{i \in F}\) has limit \(\lim_{i \in F} \alpha(A_i) = 1\) as well. So in this simplest possible case of Question~\ref{question:approxmatrixversion} the answer is ``yes'' for both part \eqref{item:sup} and part \eqref{item:inf}.
Now we can already give the counterexample for Question~\ref{question:approxmatrixversion}\,\eqref{item:inf}. Consider \(A = (p(z))\) with the polynomial \(p(z) = 5z^2 -6z + 5\). The roots of \(p(z)\) are given by \(a = \frac{3}{5} + \frac{4}{5} \ima\) and its complex conjugate. Let \(l \in (0,1)\) be determined by \(a = e^{2 \pi \ima l}\). Since \(a\) is not a root of unity, the number \(l\) is irrational. Let \(K\) be a positive integer. Then Theorem~\ref{thm:Dirichlet} provides us with a sequence of positive integers \((i_j)\) such that \(0 < \|i_j K l\| \le \frac{1}{i_j}\). This implies that we can find a \(K i_j\)-th root of unity \(\xi_{K i_j}\) with \(0 < |\xi_{K i_j} - a| \le 2 \sin(\frac{\pi}{K i_j^2}) \le \frac{2 \pi}{K i_j^2}\). For sufficiently large \(j\) we obtain
\[ \sigma^+(A_{K i_j}) \le |p(\xi_{K i_j})| = 5 |\xi_{K i_j} - \overline{a}| |\xi_{K i_j} - a| \le 5 \cdot 2 \cdot \frac{2 \pi}{K i_j^2} \]
which gives
\[ \alpha(A_{K i_j}) \le \frac{\log\left(\frac{2}{K i_j}\right)}{\log\left(\frac{20 \pi}{K i_j^2}\right)} \]
hence \(\inf_{K \mid i} \alpha(A_i) \le \frac{1}{2}\). Thus \(\liminf_{i \in F} \alpha(A_i) = \sup_{K \in F} \inf_{K | i} \alpha(A_i) \le \frac{1}{2}\) whereas \(\alpha^{(2)}(A) = 1\).
\subsection{General Laurent polynomials} \label{subsection:generallaurentpolynomials} Still let \(G = \Z\) but now let \(A = (p(z))\) for a general Laurent polynomial
\[ p(z) = c z^k \prod_{r=1}^s (z-a_r)^{\mu_r} \]
with \(c \in \C\), \(k \in \Z\) and the distinct roots \(a_r \in \C^*\) of \(p(z)\) of multiplicities \(\mu_r\). We rearrange the roots of \(p(z)\) so that \(a_1, \ldots, a_u \in S^1\) and \(a_{u+1}, \ldots, a_s \notin S^1\) for some \(0 \le u \le s\). By \cite{Lueck:L2Invariants}*{Lemma~2.58, p.\,100} and its proof we have that \(\alpha^{(2)}(p(z)) = \frac{1}{\max\{\mu_1, \ldots, \mu_u\}}\) if \(u \ge 1\) and \(\alpha^{(2)}(p(z)) = \infty^+\) otherwise.
To compute the alpha number of \(A_i\) in the case \(u \ge 1\), note that the singular values of \(A_i \in M(i,i; \C)\) are given by \(|p(\zeta_i^k)| = |c| \prod_{r=1}^s |\zeta_i^k - a_r|^{\mu_r}\) for \(k = 0, \ldots, i-1\). Let \(d > 0\) and \(D > 0\) be given by the minimum and the maximum, respectively, of \(\prod_{r=u+1}^s |z-a_r|^{\mu_r}\) for \(z \in S^1\). Let \(r_0 \le u\) be an index such that \(a_{r_0}\) is a root on the unit circle of maximal multiplicity \(\mu_0 = \mu_{r_0}\). If \(a_{r_0}\) is an \(i\)-th root of unity, we have \(|\zeta_i^k - a_{r_0}| = 2 \sin(\frac{\pi}{i})\) where \(\zeta_i^k\) is either of the two \(i\)-th roots of unity adjacent to \(a_{r_0}\). If \(a_{r_0}\) is not an \(i\)-th root of unity, then it lies in the open circle segment above one particular edge of the regular \(i\)-gon so that \(|\zeta_i^k - a_{r_0}| < 2 \sin(\frac{\pi}{i})\) for either of the two roots of unity \(\zeta_i^k\) spanning the segment. In any case, we obtain that there exists \(0 \le k \le i-1\) with
\[ |p(\zeta_i^k)| \le |c| D 2^\mu \left(\sin\left(\frac{\pi}{i}\right)\right)^{\mu_0} \]
where \(\mu = \mu_1 + \cdots + \mu_u\). Let us merge the constants to \(K = |c| D 2^\mu\). Since \(\sigma^+(A_i) \le |p(\zeta_i^k)|\) and \(m^+(A_i) \ge 1\), we have
\[ \alpha(A_i) \le \frac{\log\left(\frac{1}{i}\right)}{\log\left(K \left(\sin\left(\frac{\pi}{i}\right)\right)^{\mu_0}\right)} = \frac{\log\left(\frac{1}{i}\right)}{\mu_0 \log\left(K^{\frac{1}{\mu_0}} \sin\left(\frac{\pi}{i}\right)\right)}. \]
A computation similar to the one in Section~\ref{section:preliminaries} gives \(\limsup_{i \rightarrow \infty} \alpha(A_i) \le \frac{1}{\mu_0}\), thus also \(\limsup_{i \in F} \alpha(A_i) \le \frac{1}{\mu_0}\) by Lemma~\ref{lemma:sequencenet}. To show equality (in both cases) it remains to identify \(\frac{1}{\mu_0}\) as a cluster point of the net \((\alpha(A_i))_{i \in F}\). This is the tricky part.
Note that the notation \(\|x\|\) from Section~\ref{section:preliminaries} still makes sense and is well-defined for \(x \in \mathbb{R} / \mathbb{Z} = \mathbb{T}\). The same two inequalities from before hold true and even better, the term \(\|x - y\|\) for \(x, y \in \mathbb{T}\) defines a metric inducing the given topology on \(\mathbb{T}\).
\begin{proposition} \label{prop:middlengonedge}
For all points \(z_1, \ldots, z_u \in S^1 \subset \C\) on the circle there is \(0<R<\frac{1}{2}\) such that for each positive integer \(K\) there are infinitely many positive integers \(i_j\) such that for all \(t = 1, \ldots, u\) and for all \( k = 1, \ldots, K i_j\) either
\[ z_t = \zeta^k_{K i_j} \quad \text{or} \quad |z_t-\zeta^k_{K i_j}| \ge 2 \sin \left(\frac{R\pi}{K i_j}\right) \]
where \(\zeta_{K i_j}\) is a fixed primitive \(K i_j\)-th root of unity.
\end{proposition}
\begin{proof}
For what comes next it is preferable to think of the \(u\)-dimensional torus as the additive group \(\mathbb{T}^u = (\mathbb{R} / \mathbb{Z})^u\). Accordingly, let us change the notation for the point \((z_1, \ldots, z_u)\) in \((S^1)^u\) to \(L = (L_1, \ldots, L_u)\) in \(\mathbb{T}^u\) so that \((z_1^n, \ldots, z_u^n)\) corresponds to \(nL = (nL_1, \ldots, nL_u)\). The point \(L\) defines a homomorphism of \(\Z\)-modules (abelian groups) \(\varphi_L \colon \Z^u \rightarrow \mathbb{T} = \R/\Z\) sending \((a_1, \ldots, a_u) \in \Z^u\) to \(\sum_{j=1}^u a_j L_j \in \mathbb{T}\). Let \(\{A_1, \ldots, A_k \} \subset \Z^u\) be a basis of the free submodule \(\ker \varphi_L\) of \(\Z^u\). Considering these basis elements as the columns of a \((u \times k)\)-matrix \(A\), they define a homomorphism \(\R^u \rightarrow \R^k\) where we write elements of \(\R^u\) and \(\R^k\) as row vectos and multiply them from the right with \(A\). This homomorphism descends to a homomorphism \(\psi_A \colon \mathbb{T}^u \rightarrow \mathbb{T}^k\). Theorem~\ref{thm:Kronecker} says precisely that the \(\Z\)-orbit \( B_L = \{ nL \in \mathbb{T}^u \, | \, n \in \Z \} \) of \(L\) in the \(u\)-torus \(\mathbb{T}^u\) has closure \(\overline{B_L} = \ker(\psi_A)\). It follows from this description that \(\overline{B_L} \cong \mathbb{T}^v \oplus \Z / m \Z\) for some \(m \ge 1\), compare also \cite{Abbaspour-Moskowitz:Basic}*{Corollary~4.2.5, p.\,209}. Here the dimension \(v\) is one less than the dimension of the \(\Q\)-vector space generated by \(1, \widetilde{L_1}, \ldots, \widetilde{L_u}\) where each \(\widetilde{L_t}\) is some lift of \(L_t\) from \(\mathbb{T}\) to \(\R\). Therefore \(v\), depending on \(L\), can take any value between zero and \(u\). For the moment, let us assume \(v \ge 1\). Since the quotient \(\overline{B_L} / \overline{B_L}^0 \cong \Z / m \Z\) by the unit component is generated by \(L + \overline{B_L}^0\), it follows that \(\overline{B_{mL}} = \overline{B_L}^0 \cong \mathbb{T}^v\). Let
\[ \mathbb{T}_{mL} = \{ (x_1, \ldots, x_u) \in \mathbb{T}^u \,|\, x_t = [0] \text{ if } mL_t = [0] \} \]
be the unique minimal subtorus obtained from \(\mathbb{T}^u\) by setting fixed coordinates to zero under the side condition that it still contains \(\overline{B_L}^0\). It is then of course necessary that \(1 \le v \ \le \ l = \dim \mathbb{T}_{mL} \ \le \ u\). In what follows we will delete the zero coordinates from \(\mathbb{T}_{mL}\).
We can choose \(0 < R < \frac{1}{2}\) so small that the interior of the centered cube
\[ K_R = \{ (x_1, \ldots, x_l) \in \mathbb{T}_{mL} \,|\, {\textstyle \|x_t\| \ge R \text{ for all } t = 1, \ldots, l} \}. \]
contains \([mL]\) and therefore intersects \(\overline{B_L}^0\) in the nonempty set \(U_L\). Next we claim that for every nonzero \(K \in \Z\) we have \(\overline{B_{KmL}} = \overline{B_{mL}} = \overline{B_L}^0\).
Indeed, the inclusion \(\overline{B_{KmL}} \subset \overline{B_{mL}}\) is clear. For the other inclusion we note that \(\overline{B_{mL}} = \overline{B_L}^0 \cong \mathbb{T}^v\) is a torus, hence is divisible. Thus for given \(x \in \overline{B_{mL}}\) and \(\varepsilon > 0\) there is \(y =(y_1, \ldots, y_u) \in \overline{B_{mL}}\) such that \(Ky = x\) and there is \(N \in \Z\) such that \(\|N m L_t - y_t\| < \frac{\varepsilon}{\lvert K \rvert}\) for all \(t = 1, \ldots, u\). It follows that \(\|N (KmL_t) - x_t\| = \|K(NmL_t - y_t)\| < \varepsilon\), hence \(x \in \overline{B_{KmL}}\).
Since \(U_L\) is open in \(\overline{B_L}^0\), it contains infinitely many \(\Z\)-translates of \(K m L\). Note moreover that \(U_L = -U_L\), so we can pick a sequence \(i_j\) of positive integer multiples of \(m\) such that \(i_j K L \in U_L\) for all \(j\).
By construction we have that for each \(i_j\) either \(K i_j L_t = [0]\), meaning that \(z_t \in S^1\) is a \(K i_j\)-th root of unity, or \(\|K i_j L_t \| \ge R\), meaning that \(z_t\) encloses an angle of at least \(\frac{2 \pi R}{K i_j}\) with any \(K i_j\)-th root of unity. This gives the assertion for \(v \ge 1\). In case \(v = 0\) we have \(L_t \in \Q / \Z\) for all \(t = 1, \ldots, u\) or in other words each \(z_t \in S^1\) is some \(k_t\)-th root of unity. In that case setting \(i_j = j \lcm (k_1, \ldots, k_u) \) does the trick for arbitrary \(0 < R < \frac{1}{2}\).
\end{proof}
\noindent To see that \(\frac{1}{\mu_0}\) is a cluster point of the net \((\alpha(A_i))_{i \in F}\), for any given positive integer \(K\) we have to construct a sequence of positive integers \(i_j\) such that \(\lim_{j \rightarrow \infty} \alpha(A_{K{i_j}}) = \frac{1}{\mu_0}\). So let the number \(0 < R < \frac{1}{2}\) and the sequence \((i_j)\) be specified by \(a_1, \ldots, a_u \in S^1\) and by \(K\) according to Proposition~\ref{prop:middlengonedge}. We now ask for a lower bound on \(\sigma^+(A_{K i_j})\). Let \(\delta = \min \{ \frac{1}{2}, \eta \}\) where \(\eta\) is the minimum of the pairwise Euclidean distances of the points \(\{a_1, \ldots, a_u\} \subset S^1\). Let \(\xi_{K i_j}\) be (one of) the \(K i_j\)-th root(s) of unity for which \(\sigma^+(A_{K i_j}) = \lvert p(\xi_{K i_j}) \rvert\). For sufficiently large \(j\), there must be one and only one root \(a_r\) on \(S^1\) within the open \(\delta\)-ball around \(\xi_{K i_j}\), where \(r = r(j)\) depends on \(j\). So if \(a_{r(j)}\) is not a \(K i_j\)-th root of unity, we have \(\lvert \xi_{K i_j} - a_{r(j)} \rvert \ge 2 \sin \frac{R \pi}{K i_j}\) and if \(a_{r(j)}\) is a \(K i_j\)-th root of unity, we even have \(\lvert \xi_{K i_j} - a_{r(j)} \rvert \ge 2 \sin \frac{\pi}{K i_j}\). For sufficiently large \(j\), this gives
\begin{align*}
|p(\xi_{K i_j})| \ge |c| d \delta^{\mu -\mu_{r(j)}} \left(2\sin\left(\frac{R\pi}{K i_j}\right)\right)^{\mu_{r(j)}} \ge |c| d \delta^\mu 2^{\mu_0} \left(\sin\left(\frac{R\pi}{K i_j}\right)\right)^{\mu_0}.
\end{align*}
Since \(p\) is a polynomial, the function \(t \mapsto \lvert p(e^{2\pi \ima t})\rvert\) is strictly monotonic on small half-open intervals starting at the zeros and the function is bounded from below outside these intervals. Thus for large \(j\) we have \(m^+(A_{K i_j}) \le 2u\). (Note that we use the symbol ``\(\ima\)'' for the imaginary unit whereas the symbol ``\(i\)'' is reserved for indices.) The same computation as above shows \(\liminf_{j\rightarrow \infty} \alpha(A_{K i_j}) \ge \frac{1}{\mu_0}\), thus \(\lim_{j\rightarrow \infty} \alpha(A_{K i_j}) = \frac{1}{\mu_0}\). This answers Question~\ref{question:approxmatrixversion}\,\eqref{item:sup} affirmatively for the case \(G = \Z\) and \(r = s = 1\).
\section{The case of a matrix of Laurent polynomials}
\label{section:rbys}
\noindent For a general matrix \(A \in M(r,s; \C[\Z])\) with arbitrary \(r, s\) we notice that the ring of Laurent polynomials \(\C[\Z]\), being a localization of the polynomial ring \(\C[z]\), is a principal ideal domain. Therefore \(A\) can be transformed into \emph{Smith normal form}. This means there are invertible matrices \(S \in M(r,r; \C[\Z])\) and \(T \in M(s,s; \C[\Z])\) such that \(SAT\) is an \((r \times s)\)-matrix of block form \(\left( \begin{smallmatrix} P & 0 \\ 0 & 0 \end{smallmatrix} \right)\) where \(P\) is a diagonal matrix with entries \(p_1(z), \ldots, p_k(z)\).
The (Laurent) polynomials \(p_1(z), \ldots, p_k(z)\) are called the \emph{invariant factors} and satisfy the relation \(p_l \mid p_{l+1}\). Multiplying \(S\) or \(T\) by a diagonal matrix with nonzero constant polynomials as entries, if need be, we can and will additionally assume that \(|p_{l+1}(z)| \le |p_l(z)|\) for all \(z \in S^1\) and \(l=1, \ldots, k-1\). By \cite{Lueck:L2Invariants}*{Lemma~2.11\,(9), p.\,77, and Lemma~2.15\,(1), p.\,80} we get
\[ \alpha^{(2)}(A) = \alpha^{(2)}(SAT) = \min_{l=1,\ldots, k} \{ \alpha^{(2)}(p_l(z)) \} = \alpha^{(2)}(p_k(z)). \]
The last equality holds because the maximal multiplicity of a root on the unit circle can only increase from \(p_l\) to \(p_{l+1}\). The following proposition thus reduces Question~\ref{question:approxmatrixversion} for the \((r \times s)\)-matrix \(A\) to the same question for the \((1 \times 1)\)-matrix \((p_k(z))\). The latter was treated in the preceding section.
\begin{proposition} \label{prop:rbystoonebyone} Suppose \(\alpha^{(2)}(A) < \infty^+\). Then we have
\[ \liminf_{i \in F} \alpha(A_i) = \liminf_{i \in F} \alpha(p_k(z)_i) \ \ \text{and} \ \ \limsup_{i \in F} \alpha(A_i) = \limsup_{i \in F} \alpha(p_k(z)_i) \]
and the same statement holds replacing ``\(i \in F\)'' with ``\(i \rightarrow \infty\)''.
\end{proposition}
\noindent The proof requires some labor. We prepare it with a lemma that captures those properties of the functions \(t \mapsto \lvert p_l(e^{\ima 2 \pi t}) \rvert\) that are relevant for computing alpha numbers.
\begin{lemma} \label{lemma:growthnearroots}
Let \(p_1(z), \ldots, p_k(z) \in \C[z,z^{-1}]\) be complex Laurent polynomials. Then there exists \(0 < \varepsilon < 1\) and there exist constants \(d, D > 0\) such that for every polynomial \(p_l(z)\)
\begin{enumerate}[(i)]
\item \label{item:estimatearoundroots} we have the inequality
\[ d|t|^{\mu} \le |p_l(a e^{\ima 2 \pi t})| \le D|t|^{\mu} \]
for every root \(a\) of \(p_l(z)\) on \(S^1\) and each \(t \in (-\varepsilon, \varepsilon)\) where \(\mu\) is the multiplicity of \(a\),
\item \label{item:monotonearoundroots} the function \(|p_l(ae^{\ima 2\pi t})|\) is monotone decreasing for \(t \in (-\varepsilon, 0]\) and monotone increasing for \(t \in [0, \varepsilon)\) for every root \(a\) of \(p_l(z)\) on \(S^1\),
\item \label{item:estimateoutsideroots} the function \(|p_l(e^{\ima 2\pi t})|\) is bounded from below by \(d \varepsilon^{\mu_0}\) on the complement of all open \(\varepsilon\)-balls around the roots of \(p_l(z)\) on \(S^1\) where \(\mu_0\) is the maximal multiplicity among all the roots of all polynomials \(p_1(z), \ldots, p_k(z)\).
\end{enumerate}
\end{lemma}
\begin{proof}
Let \(a \in S^1\) be a root of \(p_l(z)\) of multiplicity \(\mu\). Let \(0 < \delta < 2\) be so small that \(p(z)\) has no second root in \(B_\delta(a)\), the closed \(\delta\)-ball around \(a\). Let \(d' > 0\) and \(D' > 0\) be given by the minimum and maximum, respectively, of \(\left| \frac{p(z)}{(z-a)^\mu}\right|\) for \(z \in B_\delta(a)\). Set \(\varepsilon = \frac{1}{\pi} \arcsin(\frac{\delta}{2})\), so that in particular \(\varepsilon\) is bounded from above by \(\frac{1}{2}\), and set \(d = d' 4^\mu\) and \(D = D' (2 \pi)^\mu\). Then for \(|t| < \varepsilon\) we have
\begin{align*}
|p_l(a e^{\ima 2 \pi t})| & \le D' |ae^{\ima 2 \pi t} - a|^\mu = D' |e^{\ima 2 \pi t} - 1|^\mu = D' 2^\mu |\sin(\pi t)|^\mu \\
& \le D' (2\pi)^\mu|t|^\mu = D |t|^\mu
\end{align*}
and similarly
\[ |p_l(a e^{\ima 2 \pi t})| \ge d' 2^\mu |\sin(\pi t)|^\mu \ge d' 4^\mu |t|^\mu = d |t|^\mu. \]
We repeat this construction for all the remaining roots of \(p_l(z)\) on \(S^1\) and for all the remaining polynomials. The minimal occurring \(\varepsilon\) and \(d\) together with the maximal occurring \(D\) will then work for all roots and polynomials and gives \eqref{item:estimatearoundroots}. It is clear that since \(p_l(z)\) is a polynomial, we can additionally achieve \eqref{item:monotonearoundroots} and \eqref{item:estimateoutsideroots} by making \(\varepsilon\) smaller, if necessary.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:rbystoonebyone}.]
For any two matrices \(M, N \in M(n,n; \C)\) we have the inequalities of singular values for each \(t = 1, \ldots, n\)
\begin{align}
\label{eq:ineqsingvalues1} \sigma_n(M) \sigma_t(N) & \le \sigma_t(MN) \le \sigma_1(M) \sigma_t(N) \\
\label{eq:ineqsingvalues2} \sigma_t(M) \sigma_n(N) & \le \sigma_t(MN) \le \sigma_t(M) \sigma_1(N)
\end{align}
as given for instance in \cite{Hogben:HandbookLinAl}*{24.4.7\,(c), p.\,24-8}. Here, as usual, the singular values are listed in nonincreasing order. Of course the second inequality follows from the first because \(\sigma_t(M) = \sigma_t(M^\top)\). We apply these inequalities to our setting as follows. Let \(m = \max \{r, s\}\) and view the matrices \(A_i\) as lying in \(M(mi,mi; \C)\) by embedding \(A_i\) in the upper left corner of an \((mi \times mi)\)-matrix, filling up the remaining entries with zeros. If \(r < s\) we consider \(S_i\) as an element of \(\textup{GL}(mi; \C)\) by overwriting the upper left block of an \((mi \times mi)\)-identity matrix with \(S_i\) and similarly for \(T_i\) in place of \(S_i\) if \(r > s\). Since both \(S\) and \(T\) are invertible over the group ring \(\C[\Z]\), it follows from \cite{Lueck:L2Invariants}*{Lemma~13.33, p.\,466} that the spectrum of \(r^{(2)}_{SS^*}\) and \(r^{(2)}_{TT^*}\) is contained in \([C^{-1}, C]\) for some \(C \ge 1\). Since the operator norm of the projection map \(L^1(G) \rightarrow L^1(G/G_i)\) is bounded by one, it follows that the eigenvalues of \((SS^*)_i\) and \((TT^*)_i\) are likewise constrained to lie within \([C^{-1}, C]\). Therefore \(C^{-\frac{1}{2}} \le \sigma_t(S_i), \sigma_t(T_i) \le C^{\frac{1}{2}}\) for each \(t = 1, \ldots, mi\) so that the inequalities~\eqref{eq:ineqsingvalues1} and~\eqref{eq:ineqsingvalues2} give
\begin{equation} \label{eq:boundedsingvalue} C^{-1} \sigma_t((SAT)_i) \le \sigma_t(A_i) \le C \sigma_t((SAT)_i). \end{equation}
The special case \(t = \rank_\C (A_i)\) gives
\begin{equation} \label{eq:boundedsmallestsingvalue} C^{-1} \sigma^+((SAT)_i) \le \sigma^+(A_i) \le C \sigma^+((SAT)_i). \end{equation}
Next we show that there is \(M > 0\) such that
\begin{equation} \label{eq:boundedmultiplicities} 1 \le m^+(A_{i}) \le M \end{equation}
for all sufficiently large \(i\). To this end, let \(r_i = \rank_\C A_i\) so that \(\sigma^+(A_i) = \sigma_{r_i}(A_i)\). Let \(\mu_0\) be the maximal occurring multiplicity among the roots of \(p_k(z)\) on \(S^1\). Let \(\varepsilon > 0\) and \(d, D > 0\) be the constants from Lemma~\ref{lemma:growthnearroots} applied to the polynomials \(p_1(z), \ldots, p_k(z)\) which form the diagonal of the matrix \(SAT\). Pick a positive integer
\begin{equation} \label{eq:constantk} K > \left(\frac{C^2 D}{d}\right)^{\frac{1}{\mu_0}} + 1 \end{equation}
and set \(\delta = \frac{d}{D} \varepsilon^{\mu_0}\). Now we consider \(i\) so large that at least \(2K\) of the \(i\)-th roots of unity lie in any open \(\delta\)-ball around any point on \(S^1\). By Lemma~\ref{lemma:growthnearroots}\,\eqref{item:estimatearoundroots} and \eqref{item:monotonearoundroots}, evaluating the function \(|p_l(z)|\) in the \(2K\) roots of unity closest to any root \(a \in S^1\) gives values smaller than \(D (\frac{d}{D} \varepsilon^{\mu_0})^\mu \le d \varepsilon^{\mu_0}\) where \(\mu\) was the multiplicity of \(a\). So if \(N\) denotes the sum of the number of distinct roots of each \(p_l(z)\), then by Lemma~\ref{lemma:growthnearroots}\,\eqref{item:estimateoutsideroots} the first \(2KN\) (positive) singular values of \((SAT)_i\) are given by evaluating some \(|p_l(z)|\) within the \(\varepsilon\)-ball of some root. By the pigeon hole principle there is one root \(a \in S^1\) of some \(p_l(z)\) such that \(K\) singular values among the smallest \(2KN\) singular values of \((SAT)_i\) are given by evaluating \(|p_l(z)|\) at the \(K\) closest \(i\)-th roots of unity on one side of the root \(a\). Again we denote the multiplicity of \(a\) by \(\mu\). Using the monotonicity asserted by Lemma~\ref{lemma:growthnearroots}\,\eqref{item:monotonearoundroots} this gives
\[ \sigma_{{r_i} - 2NK} ((SAT)_i) \ge \left|p_l\left(a e^{\pm \ima \frac{2 \pi}{i} (K - 1)}\right)\right|. \]
Applying Lemma~\ref{lemma:growthnearroots}\,\eqref{item:estimatearoundroots} and inequality~\eqref{eq:constantk} we get
\begin{align*}
\left|p_l\left(a e^{\pm \ima \frac{2 \pi}{i} (K - 1)}\right)\right| \ge d\left(\frac{K-1}{i}\right)^\mu \ge d\left(\frac{K-1}{i}\right)^{\mu_0} > C^2 D \left(\frac{1}{i}\right)^{\mu_0}
\end{align*}
Let \(a_0 \in S^1\) be any root of \(p_k(z)\) with multiplicity \(\mu_0\). There is an \(i\)-th root of unity \(\xi_i \ne a_0\) which encloses an angle of at most \(\frac{2\pi}{i}\) with \(a_0\). Applying Lemma~\ref{lemma:growthnearroots}\,\eqref{item:estimatearoundroots} again we obtain
\[ C^2 D \left(\frac{1}{i}\right)^{\mu_0} \ge C^2 \left|p_k\left(a_0 e^{\pm \ima \frac{2 \pi}{i}}\right)\right| \ge C^2 |p_k(\xi_i)| \ge C^2 \sigma^+((SAT)_i). \]
So setting \(M = 2NK\) we have \(\sigma_{r_i - M} ((SAT)_i) > C^2 \sigma^+((SAT)_i)\) for every large enough \(i\). From inequality~\eqref{eq:boundedsingvalue} we conclude
\[ \sigma_{r_i - M}(A_i) \ge C^{-1} \sigma_{r_i - M}((SAT)_i) > C \sigma^+((SAT)_i) \ge \sigma^+(A_i) \]
which proves inequality~\eqref{eq:boundedmultiplicities}.
Finally note that the inequality \(|p_{l+1}(z)| \le |p_l(z)|\) gives \(\sigma^+((SAT)_i) = \sigma^+((p_k(z))_i)\). Inequalities~\eqref{eq:boundedsmallestsingvalue} thus yields
\begin{equation} \label{eq:boundonalphanumbers}
\frac{\log\left(\frac{m^+(A_i)}{i}\right)}{\log (C^{-1}\sigma^+((p_k(z))_i))} \le \alpha(A_i) \le \frac{\log\left(\frac{m^+(A_i)}{i}\right)}{\log (C\sigma^+((p_k(z))_i))}.
\end{equation}
We can rewrite the outer terms as
\[ \frac{\log\left(\frac{m^+(A_i)}{i}\right)}{\log (C^{\pm 1}\sigma^+((p_k(z))_i))} = \frac{\log\left(\frac{m^+(p_k(z)_i)}{i}\right) + \log\left(\frac{m^+(A_i)}{m^+(p_k(z)_i)}\right)}{\log\left(\sigma^+(p_k(z)_i)\right)\left(1 \pm \frac{\log C}{\log(\sigma^+(p_k(z)_i))}\right)}. \]
Since the multiplicities are bounded according to inequality~\eqref{eq:boundedmultiplicities}, we see from this that for an increasing sequence of positive integers \((i_j)\) we have \(\lim_{j \rightarrow \infty} \alpha(A_{i_j}) = c\) if and only if \(\lim_{j \rightarrow \infty} \alpha(p_k(z)_{i_j}) = c\). As a consequence the sequences \((\alpha(A_i))_{i \ge 0}\) and \((\alpha(p_k(z)_i))_{i \ge 0}\) share the same set of cluster points. Considering integer sequences of the form \((K i_j)\) for any positive integer \(K\), the same goes for the nets \((\alpha(A_i))_{i \in F}\) and \((\alpha(p_k(z)))_{i \in F}\). This clearly implies the proposition.
\end{proof}
\noindent This answers Question~\ref{question:approxmatrixversion}\,\eqref{item:sup} affirmatively for the case \(G = \Z\).
\section{The case of a virtually cyclic group}
\label{section:virtuallycyclic}
\noindent Finally let \(G\) be infinite virtually cyclic so that \(G\) contains an infinite cyclic subgroup \(Z \le G\) with \([G \colon Z] = n < \infty\). By going over to the normal core, if need be, we can and will assume that \(Z\) is a normal subgroup.
We choose representatives \(g_i \in G\) such that \(Z \backslash G = \{ Z g_1, \ldots, Z g_n\}\). Let \(A \in M(r,s; \C G)\). Right multiplication with \(A\) defines a homomorphism \((\C G)^r \rightarrow (\C G)^s\) of left \(\C G\)-modules. If we consider \(\C G\), the free left \(\C G\)-module of rank one, as a left \(\C Z\)-module, then it is free of rank \(n\) and a basis is given by \(g_1, \ldots, g_n \in \C G\). Accordingly, viewing right multiplication with \(A\) as a homomorphism \((\C Z)^{rn} \rightarrow (\C Z)^{sn}\) of left \(\C Z\)-modules, it is given by right multiplication with the matrix \(\res^Z_G(A) \in M(rn, sn; \C Z)\) that results from \(A\) by replacing the \((p,q)\)-th entry \(\sum_{g \in G} \lambda^{p,q}_g g\) with the \((n \times n)\)-matrix over \(\C Z\) whose \((u,v)\)-th entry is \(\sum_{h \in Z} \lambda^{p,q}_{g_u^{-1} h g_v} h\) for \(1 \le u, v \le n\).
Let \(Z_i\) be the unique subgroup of \(Z\) with \([Z : Z_i] = i\). Then \([G : Z_i] = ni\) and \(Z_i\) is normal in \(G\) because \(Z_i\) is characteristic in \(Z\).
\begin{proposition} \label{prop:equalmatrices}
We have \(\res^Z_G(A)_i = A_i\) as elements in \(M(r n i, s n i; \C)\).
\end{proposition}
\begin{proof}
We pick representatives \(Z_i \backslash Z = \{ Z_i h_1, \ldots, Z_i h_i \}\) and verify that for \(1 \le p \le r\) and \(1 \le q \le s\) as well as \(1 \le u, v \le n\) we have
\[ (\res^Z_G(A)_i)_{(p-1)n + u,(q-1)n + v} = \sum_{l=1}^i \left( \sum_{h \in Z_i} \lambda_{g_u^{-1} h h_l g_v}^{p,q} \right) Z_i h_l. \]
Multiplication with a fixed coset \(Z_i h_k\) gives
\[ Z_i h_k \sum_{l=1}^i \left( \sum_{h \in Z_i} \lambda_{g_u^{-1} h h_l g_v}^{p,q} \right) Z_i h_l = \sum_{l=1}^i \left( \sum_{h \in Z_i} \lambda_{g_u^{-1} h_k^{-1} h h_l g_v}^{p,q} \right) Z_i h_l. \]
Hence \(\res_G^Z(A)_i\) is realized over \(\C\) by replacing the entry at \(((p-1)n + u,(q-1)n + v)\) with a (circulant) \((i \times i)\)-matrix whose \((k,l)\)-th entry is \(\sum_{h \in Z_i} \lambda_{g_u^{-1} h_k^{-1} h h_l g_v}^{p,q}\).
To realize \(A_i\) as a matrix over \(\C\) we now use our chosen representatives to list the cosets of \(Z_i \backslash G\) in this order as
\[\{ Z_i h_1 g_1, \ldots, Z_i h_i g_1, \quad \ldots \quad , Z_i h_1 g_n, \ldots, Z_i h_i g_n\}. \]
Again we compute for \(1 \le p \le r\) and \(1 \le q \le s\) as well as \(1 \le u, v \le n\) and \(1 \le k \le i\) that
\[ Z_i h_k g_u \sum_{g \in G} \lambda^{p,q}_g Z_i g = \sum_{g \in G} \lambda^{p,q}_{(h_k g_u)^{-1} g} Z_i g = \sum_{v = 1}^n \sum_{l = 1}^i \sum_{h \in Z_i} \lambda^{p,q}_{g_u^{-1} h_k^{-1} h h_l g_v} Z_i h_l g_v. \]
Thus \(A_i\) is realized over \(\C\) by replacing the \((p,q)\)-th entry with the \((n i \times n i)\)-matrix whose entry at \(((u-1)i + k, (v-1)i + l)\) is \(\sum_{h \in Z_i} \lambda^{p,q}_{g_u^{-1} h_k^{-1} h h_l g_v}\). Thus the \(\C\)-matrices \(\res_G^Z(A)_i\) and \(A_i\) coincide.
\end{proof}
\begin{proposition} \label{prop:equallimits}
Let \(F(Z)\) and \(F(G)\) denote the full residual systems of \(Z\) and \(G\), respectively. Suppose \(\alpha^{(2)}(A) < \infty^+\), then
\[ \liminf_{i \in F(Z)} \alpha(A_i) = \liminf_{i \in F(G)} \alpha(A_i) \quad \text{and} \quad \limsup_{i \in F(Z)} \alpha(A_i) = \limsup_{i \in F(G)} \alpha(A_i). \]
\end{proposition}
\begin{proof}
Let \(c\) be a cluster point of the net \((\alpha(A_i))_{i \in F(Z)}\) and let \(H \trianglelefteq G\) be a finite index normal subgroup representing some element in \(F(G)\). Then there are upper bounds \(j\) of \(H \cap Z\) in \(F(Z) \subset F(G)\) with \(\alpha(A_j)\) arbitrarily close to \(c\). Conversely, let \(c\) be a given cluster point of the net \((\alpha(A_i))_{i \in F(G)}\) and consider \(Z_i \trianglelefteq Z\). Then \(Z_i\) represents an element in \(F(G)\), thus there are upper bounds \(j\) of \(Z_i\) in \(F(G)\), which actually lie in \(F(Z)\), with \(\alpha(A_j)\) arbitrarily close to \(c\). Thus the set of cluster points agrees for the nets \((\alpha(A_i))_{i \in F(Z)}\) and \((\alpha(A_i))_{i \in F(G)}\) which in particular implies the proposition.
\end{proof}
\noindent Now we are in the position to complete the proof of our main result.
\begin{proof}[Proof of Theorem~\ref{thm:approxvcycmatrixversion}] It follows from \cite{Lueck:L2Invariants}*{Theorem~1.12\,(6), p.\,22} that for the spectral distribution functions we have \(F_{\res^Z_G(A)}(\lambda) = n F_A(\lambda)\), hence \(\alpha^{(2)}(A) = \alpha^{(2)}(\res^Z_G(A))\). Together with the preceding section, Proposition~\ref{prop:equalmatrices} and Proposition~\ref{prop:equallimits} we obtain
\begin{align*}
\alpha^{(2)}(A) &= \alpha^{(2)}(\res^Z_G(A)) = \limsup_{i \in F(Z)} \alpha(\res^Z_G(A)_i) = \\
&= \limsup_{i \in F(Z)} \alpha(A_i) = \limsup_{i \in F(G)} \alpha(A_i).
\end{align*}
This answers Question~\ref{question:approxmatrixversion}\,\eqref{item:sup} in the affirmative for \(F = \C\) and thus for any subfield. In Section~\ref{subsection:twoexamples} we gave an example answering Question~\ref{question:approxmatrixversion}\,\eqref{item:inf} in the negative for \(F = \Q\) and thus for every larger field.
\end{proof}
\section{The lower limit of alpha numbers} \label{section:liminfpositive}
\noindent In this final section we give the proof of Theorem~\ref{thm:liminfpositive}. Recall our definition of Baker constants from the end of Section~\ref{subsection:diophantine}.
\begin{theorem} \label{thm:circlerunner}
Let \(a \neq 1\) be an algebraic number on the unit circle and let \(D\) be a Baker constant of the pair \((a, -1)\). Then for all \(n \ge 2\) with \(a^n \neq 1\) we have \(\lvert a^n - 1 \rvert \ge \frac{n^{-D}}{2} \).
\end{theorem}
\begin{proof}
The principal value logarithm satisfies \(\lvert \log (1 + z) \rvert \le 2 \lvert z \rvert\) for \(\lvert z \rvert \le \frac{1}{2}\) and is additive up to some integer multiple of \(2 \pi \ima\). If \(\lvert a^n - 1 \rvert > \frac{1}{2}\), there is nothing to prove. Otherwise we have
\[ 1 \ge 2 \lvert a^n - 1 \rvert \ge \lvert \log(a^n) \rvert = \lvert n \log a + 2 \pi \ima k \rvert = \lvert n \log a + 2k \log(-1) \rvert, \]
so if \(a^n \ne 1\), Theorem~\ref{thm:littlebaker} gives \(2 \lvert a^n - 1 \rvert \ge \max \{ n, 2 \lvert k \rvert \}^{-D}\). Moreover, the inequalities \(1 \ge \lvert n \log a + 2 \pi \ima k \rvert\) and \(\lvert \log a \rvert \le \pi\) imply
\[ \lvert k \rvert \le \frac{1 + n \lvert \log a \rvert}{2 \pi} \le \frac{1}{2 \pi} + \frac{n}{2} \]
which is equivalent to \(\lvert k \rvert \le \frac{n}{2}\) because \(k\) and \(n\) are integers. Thus we obtain \(2 \lvert a^n - 1 \rvert \ge n^{-D} \) as desired.
\end{proof}
\noindent The mere existence of some \(D > 0\) giving the estimate of the theorem also serves as the main ingredient for \cite{Lueck:L2Invariants}*{Lemma~13.53, p.\,478}. The latter is just the \((1 \times 1)\)-case of the Fuglede--Kadison determinant approximation conjecture for the group \(\Z\). We recapped a proof here, however, in order to identify the constant \(D\) as the Baker constant in Theorem~\ref{thm:littlebaker}. This has the virtue that the many estimates on \(D\) in the literature lead to explicit lower bounds on our \(\liminf_{i \in F} \alpha(A_i)\) as we will see in the subsequent corollary. We admit that the practical value of these bounds is limited because the values for \(D\) given in the literature are typically astronomic. The constant in \cite{Baker:LinearForms}*{Theorem~2}, for example, is \(D = (32d)^{400}\) times a logarithmic function in the height of \(a\), where \(d\) is the degree of \(a\).
\begin{corollary} \label{cor:boundonliminf}
Let \(G\) be a virtually cyclic group and let \(A \in M(r,s;\Q G)\) with \(\alpha^{(2)}(A) < \infty^+\). Choose an infinite cyclic normal subgroup \(Z \trianglelefteq G\) of finite index and let \(p_k(z)\) be the maximal invariant factor of \(\res^G_Z(A)\). We denote the zeros of \(p_k(z)\) on \(S^1\) by \(a_1, \ldots, a_u\) and let \(D\) be the maximal occurring Baker constant \(D = D(a_t, -1)\) for \(a_t \neq 1\). Then
\[ \liminf_{i \in F} \alpha(A_i) \ge \frac{\alpha^{(2)}(A)}{1 + D}. \]
\end{corollary}
\begin{proof}
Again let \(\mu_0\) be the maximal multiplicity amongst the roots \(a_1, \ldots, a_u\) of the polynomial \(p_k(z)\) which lie on \(S^1\). As explained in the previous two sections we have
\[ \alpha^{(2)}(A) = \alpha^{(2)}(\res^Z_G(A)) = \alpha^{(2)}(p_k(z)) = \textstyle \frac{1}{\mu_0}. \]
Fix \(\varepsilon > 0\) and consider \(i \ge 2^{\frac{1}{\varepsilon}}\). Let \(\zeta_i\) be a primitive \(i\)-th root of unity. For every \(a_t\) which is not an \(i\)-th root of unity, Theorem~\ref{thm:circlerunner} gives us
\(\lvert a_t^i - 1 \rvert \ge \frac{1}{2} i^{-D} \ge i^{-(D + \varepsilon)}\) and therefore
\begin{equation} \label{eq:dplusone}
\lvert a_t - \zeta_i^l \vert = \frac{\lvert a_t^i - 1 \rvert}{\left\lvert \sum_{j = 0}^{i-1} a_t^{i-j-1} \zeta_i^{lj} \right\rvert} \ge \frac{1}{i^{D+1 + \varepsilon}}
\end{equation}
for every \(l = 0, \ldots, i-1\). Let \(\xi_i\) be the (or an) \(i\)-th root of unity for which \(\sigma^+(p_k(z)_i) = \lvert p_k(\xi_i) \rvert\). Let \(c\), \(d\), \(\mu\) and \(\delta\) be the constants from below the proof of Proposition~\ref{prop:middlengonedge}. As before, for large enough \(i\) there is one and only one root \(a_{r(i)}\) of \(p_k(z)\) with multiplicity \(\mu_{r(i)}\) that lies within the open \(\delta\)-ball around \(\xi_i\). If \(a_{r(i)}\) is an \(i\)-th root of unity and \(i\) is large enough, then \(\xi_i\) must be one of the two \(i\)-th roots of unity adjacent to \(a_{r(i)}\) so that we get
\begin{equation} \label{eq:ifrootofunity} \lvert a_{r(i)} - \xi_i \rvert = 2 \sin \left( \frac{\pi}{i} \right) \ge \frac{1}{i}. \end{equation}
So in any case, either from equation~\eqref{eq:dplusone} or from equation~\eqref{eq:ifrootofunity}, we get
\[ \sigma^+(p_k(z)_i) = \lvert p_k(\xi_i) \rvert \ge \frac{\lvert c \rvert d \delta^{\mu - \mu_{r(i)}}}{i^{(D+1+\varepsilon)\mu_{r(i)}}} \ge \frac{\lvert c \rvert c d \delta^\mu}{i^{(D+1 + \varepsilon) \mu_0}}. \]
Since again \(m^+(p_k(z)_i) \le 2u\) for large \(i\), it follows that
\[ \liminf_{i \rightarrow \infty} \alpha(p_k(z)_i) \ge \frac{1}{(D+1+\varepsilon) \mu_0} = \frac{\alpha^{(2)}(A)}{(D+1+\varepsilon)}. \]
with arbitrary \(\varepsilon > 0\). Lemma~\ref{lemma:sequencenet}, Proposition~\ref{prop:rbystoonebyone}, Proposition~\ref{prop:equalmatrices} and Proposition~\ref{prop:equallimits} finish the proof.
\end{proof}
\noindent Of course, this also completes the proof of Theorem~\ref{thm:liminfpositive}.
\begin{bibdiv}[References]
\begin{biblist}
\bib{Abbaspour-Moskowitz:Basic}{book}{
author={Abbaspour, H.},
author={Moskowitz, M.},
title={Basic Lie theory},
publisher={World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ},
date={2007},
pages={xvi+427},
isbn={978-981-270-699-7},
isbn={981-270-669-2},
review={\MRref{2364699}{2008i:22001}},
doi={10.1142/6462},
}
\bib{Baker:LinearForms}{article}{
author={Baker, A.},
title={The theory of linear forms in logarithms},
conference={
title={Transcendence theory: advances and applications},
address={Proc. Conf., Univ. Cambridge, Cambridge},
date={1976},
},
book={
publisher={Academic Press, London},
},
date={1977},
pages={1--27},
review={\MRref{0498417}{58 \#16543}},
}
\bib{Cassels:Diophantine}{book}{
author={Cassels, J. W. S.},
title={An introduction to Diophantine approximation},
series={Cambridge Tracts in Mathematics and Mathematical Physics, No. 45},
publisher={Cambridge University Press},
place={New York},
date={1957},
pages={x+166},
review={\MRref{0087708}{19,396h}},
}
\bib{Gelfond:first}{article}{
author={Gelfond, A.},
title={On the approximation of transcendental numbers by algebraic numbers},
journal={Dokl. Akad. Nauk SSSR},
volume={2},
date={1935},
pages={177--182},
}
\bib{Gelfond:second}{article}{
author={Gelfond, A.},
title={On the approximation of algebraic numbers by algebraic numbers and the theory of transcendental numbers},
journal={Izv. Akad. Nauk SSSR},
volume={5--6},
date={1939},
pages={509--518},
}
\bib{Gelfond:third}{article}{
author={Gelfond, A.},
title={On the algebraic independence of transcendental numbers of certain classes},
journal={Uspehi Mat. Nauk SSSR},
volume={5},
date={1949},
pages={14--48},
}
\bib{Gromov-Shubin:vonNeumannSpectra}{article}{
author={Gromov, M.},
author={Shubin, M. A.},
title={von Neumann spectra near zero},
journal={Geom. Funct. Anal.},
volume={1},
date={1991},
number={4},
pages={375--404},
issn={1016-443X},
review={\MRref{1132295}{92i:58184}},
doi={10.1007/BF01895640},
}
\bib{Hogben:HandbookLinAl}{collection}{
title={Handbook of linear algebra},
series={Discrete Mathematics and its Applications (Boca Raton)},
editor={Hogben, L.},
edition={2},
publisher={CRC Press, Boca Raton, FL},
date={2014},
pages={xxx+1874},
isbn={978-1-4665-0728-9},
review={\MRref{3013937}{}},
}
\bib{Lueck:Approximating}{article}{
author={L{\"u}ck, W.},
title={Approximating $L^2$-invariants by their finite-dimensional
analogues},
journal={Geom. Funct. Anal.},
volume={4},
date={1994},
number={4},
pages={455--481},
issn={1016-443X},
review={\MRref{1280122}{95g:58234}},
doi={10.1007/BF01896404},
}
\bib{Lueck:L2Invariants}{book}{
author={L{\"u}ck, W.},
title={$L^2$-invariants: theory and applications to geometry and
$K$-theory},
series={Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A
Series of Modern Surveys in Mathematics [Results in Mathematics and
Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]},
volume={44},
publisher={Springer-Verlag},
place={Berlin},
date={2002},
pages={xvi+595},
isbn={3-540-43566-2},
review={\MRref{1926649}{2003m:58033}},
}
\bib{Lueck:Survey}{article}{
author={L{\"u}ck, W.},
title={Survey on approximating \(L^2\)-invariants by their classical counterparts},
date={2015},
status={eprint},
note={\arXiv{1501.07446v1}},
}
\bib{Schmidt:DynamicalSystems}{book}{
author={Schmidt, K.},
title={Dynamical systems of algebraic origin},
series={Progress in Mathematics},
volume={128},
publisher={Birkh\"auser Verlag, Basel},
date={1995},
pages={xviii+310},
isbn={3-7643-5174-8},
review={\MRref{1345152}{97c:28041}},
}
\end{biblist}
\end{bibdiv}
\end{document}
|
2,877,628,090,176 | arxiv | \section{Introduction}
\label{sec:intro}
One of the primary goals of cosmology is to accurately measure
the expansion history and the growth of structure in the Universe.
Many of the cosmological probes used for this purpose can be
classified as standard candles or standard rulers.
The most obvious examples are
type Ia Supernovae and the baryon acoustic oscillation feature in galaxy
correlation functions, which in an unperturbed Friedmann-Robertson-Walker
(FRW) Universe directly measure the geometry
and expansion history of the Universe \cite{RiessEtal,PerlmutterEtal,EisensteinEtal,2dF}. Beyond the background cosmology,
cosmological perturbations affect the apparent scale of rulers, which can
be used as a probe of structure in the Universe. In fact, from this
point of view, standard rulers
comprise a much larger set of observations: for example, galaxy redshift
surveys measure the correlation function of galaxies, which
is then compared with predictions based on a cosmological model; in other
words, the correlation length of galaxies (or any characteristic scale in their
correlation function) serves as a standard ruler.
Weak lensing shear, measured using galaxy ellipticities, uses the fact
that galaxies sizes measured along fixed directions are on average equal.
On the other hand, lensing magnification measurements rely on the fact
that galaxies have a characteristic luminosity (standard candle) and/or
size (standard ruler). Of course, in the latter three cases the ``ruler'' has
a large amount of scatter, so that one might call it a ``statistical ruler''.
Another example of this kind is lensing reconstruction on
diffuse backgrounds such as the cosmic microwave background (CMB)
or 21cm emission from the dark ages \cite{LewisChallinor06,ZahnZaldarriaga,LuPen}.
In this approach one uses the
intrinsic correlation pattern of the background, which is known statistically,
to reconstruct the distortion from the observed pattern.
There is a simple, unified description of these various cosmological
probes: we observe photons from two different directions and redshifts,
which correspond to a known physical scale (e.g., the comoving sound horizon
at recombination, or the characteristic size of a galaxy). In this paper,
we study in a general covariant setting which underlying properties of the
spacetime can be measured with an ideal standard ruler, working to linear
order in perturbations. Since we
have six parameters to vary when scanning over photon arrival directions
and redshifts, we can measure six degrees of freedom. These can be
interpreted as the components of a metric (of Euclidean signature)
mapping apparent coordinate
distances into actual physical separations at the source.
It is useful to further decompose these components into parts
parallel to the line of sight (longitudinal), transverse, and
mixed longitudinal-transverse parts. This is equivalent to a decomposition
into scalars, vectors, and tensors in the \emph{two-dimensional} subspace
perpendicular to the photon 4-momentum and the observer's four-velocity; i.e. we
classify components
in terms of their transformation properties under a rotation around the
line of sight. Note that this is independent of the usual
decomposition of metric perturbations on \emph{three-dimensional}
spatial hypersurfaces (i.e., in terms of the transformation of
plane-wave metric perturbations under a rotation around the $k$-vector).
We will denote the latter (``3-scalars'' and so on) as $C,\,B_i,\,A_{ij},...$,
and the former (``2-scalars'' etc) as $\mathcal{C},\,\mathcal{B}_i,\,\mathcal{A}_{ij}, ...$.
The transverse components of the distortion, $\mathcal{A}_{ij}$, are
perhaps best known. They correspond to the magnification (2-scalar) and
shear (2-tensor).
We show that in general one can also measure a longitudinal
scalar and the two components of a vector on the sphere.
On scales much smaller than the horizon, an effective Newtonian
description is sufficient, and this is what essentially all previous
studies are based on. However,
upcoming surveys will probe scales approaching
the horizon, and an interpretation of these data sets can in principle
be hampered by gauge ambiguities. In the case of
the correlation of galaxy density contrast,
this issue has attracted significant interest and has
recently been resolved
\cite{yoo/etal:2009,challinor/lewis:2011,bonvin/durrer:2011,gaugePk}.
The unified treatment presented here resolves
these issues for the wide set of cosmological observables mentioned above.
More precisely, we obtain general
coordinate-independent and gauge-invariant results for all observables,
including the shear and magnification.
The differential equation (\emph{optical equation})
governing the magnification and shear was first derived in \cite{sachs:1961}.
The magnification has been derived to first order in \cite{sasaki:1987}.
The shear has been derived to second order in conformal-Newtonian gauge
in \cite{BernardeauBonvinVernizzi}, while \cite{PitrouEtal} derive the shear for
general backgrounds. To the best of our knowledge,
the expression for the observable shear
written in a general gauge is presented here for the first time.
Further, all expressions are valid on the full sky.
Our approach naturally includes the ``metric shear'' contribution
introduced in \cite{DodelsonEtal}, and we provide a straightforward physical
interpretation of our result.
In addition, we show how the (2-)vector observable uncovered here
can be decomposed into $E$- and $B$-modes in analogy with the shear,
corresponding to polar and axial vector parts. As in the case of shear
and CMB polarization, (3-)scalar perturbations do not contribute to the $B$-mode,
while (3-)tensor perturbations contribute. This in principle offers another
avenue to search for a stochastic gravitational wave background in
large-scale structure, since no scalar perturbations contribute at linear
order. However, a spectroscopic data set is likely
necessary to reconstruct the vector component with an interesting
signal-to-noise.
Apart from the linear treatment of metric perturbations, we
make two further simplifying assumptions: first, we assume ``small rulers''
in the sense that rulers subtend a small apparent angle and redshift interval.
Wide-angle effects are likely negligible for almost all applications (the
large-scale BAO feature being perhaps the most important exception).
A treatment of wide-angle effects necessarily involves a detailed model
of the survey geometry, which is clearly beyond the scope of this paper.
The second assumption is that any scatter or variation in the actual
(``intrinsic'') physical scale of the standard ruler is uncorrelated
with large-scale perturbations. This will not hold true in general,
since the physical systems used as rulers will be affected by their
large-scale environment. One well-known example is the intrinsic
alignment contribution to shear correlations \cite{CatelanKamionkowskiBlandford}.
Further examples include
the distortion of correlation
functions by large-scale tidal fields \cite{PenEtal},
or by a non-Gaussian coupling of the density field to primordial
degrees of freedom \cite{jeong/kamionkowski:2012}.
Since these ``intrinsic effects'' depend on the physics of the given ruler,
we refrain from discussing them here, as they would distract from the
generality of the rest of the results.
Finally, while we focus on general standard rulers here, the case of
standard candles is directly related to our results. This is because
the relation between angular diameter distance $D_A$ and luminosity
distance $D_L$,
\begin{equation}
D_L = (1+\tilde{z})^2 D_A,
\label{eq:DLDA}
\end{equation}
where $\tilde{z}$ is the observed redshift, holds in a general spacetime
and for any source (this is a consequence of photon phasespace conservation).
Thus, the magnification measured for standard candles is identical
to the magnification for standard rulers which we will derive here.
However, as discussed, standard rulers can
measure five additional degrees of freedom not accessible to standard candles.
Our results are of immediate relevance to recent studies which consider
the $B$-modes of the cosmic shear as possible probe of an inflationary
gravitational wave background \cite{DodelsonEtal,Dodelson10,paperII},
and to studies that propose to use the high-redshift 21cm emission
for the same purpose \cite{MasuiPen,BookMKFS}. In particular,
our expressions can be used directly to construct optimal estimators
on the full sky for
searching for the imprint of gravitational waves in a three-dimensional
field such as the 21cm background.
We also use many of the results derived here in two recent papers studying
the impact of gravitational waves on the observed large-scale structure
\cite{paperI,paperII}.
The outline of the paper is as follows: we begin in \refsec{not} by
introducing our metric convention and useful notation. The general expression
for the mapping from apparent size to true physical size of the ruler
is derived in \refsec{ruler}. We then decompose the contributions
into longitudinal and transverse parts in \refsec{SVT}. The following
three sections deal with these different parts consecutively.
We discuss and conclude in \refsec{disc}. The appendix contains
a large amount of additional reference material on multipole expansions of
higher spin functions, perturbed photon geodesic equation, and various
test cases applied to our results.
\section{Notation}
\label{sec:not}
In a general gauge, the perturbed FRW metric is given by
\ba
ds^2 = a^2(\eta)\Big[&
-(1+2A) d\eta^2 - 2B_i d\eta dx^i
\nonumber\\
& + \left(\d_{ij}+h_{ij}\right) dx^i dx^j \Big],
\label{eq:metric}
\ea
where we have assumed a spatially flat Universe (curvature can be
included straightforwardly, at the expense of some extra notation).
Here, $\eta$ denotes conformal time.
Often, the spatial part is further expanded as
\begin{equation}
h_{ij} = 2D \d_{ij} + 2 E_{ij},
\label{eq:DE}
\end{equation}
where $E_{ij}$ is traceless.
We shall also present the most interesting results in two popular
gauges: the synchronous-comoving (sc) gauge, where $A = 0 = B_i$, so that
\begin{equation}
ds^2 = a^2(\eta)\left[- d\eta^2
+ \left(\d_{ij}+h_{ij}\right) dx^i dx^j \right];
\label{eq:metric_sc}
\end{equation}
and the conformal-Newtonian (cN) gauge, where $B_i = 0 = E_{ij}$. In the
latter case, we denote $A = \Psi$, $D = \Phi$, conforming
with standard notation, so that
\begin{equation}
ds^2 = a^2(\eta)\left[- (1+2\Psi) d\eta^2
+ (1+2\Phi) \d_{ij} dx^i dx^j \right].
\label{eq:metric_cN}
\end{equation}
We also denote the background FRW metric (in the absence of perturbations) as
$\bar g_{\mu\nu} = a^2(\eta) \eta_{\mu\nu}$.
It is useful to define projection operators parallel and perpendicular
to the observed line-of-sight direction $\hat{n}^i$,
so that for any spatial vector $X^i$ and tensor $E_{ij}$,
\ba
X_\parallel \equiv\:& \hat{n}_i X^i, \nonumber\\
E_\parallel \equiv\:& \hat{n}_i \hat{n}_j E^{ij}, \nonumber\\
X_\perp^i \equiv\:& \mathcal{P}^{ij} X_j \nonumber\\
\mathcal{P}^{ij} \equiv\:& \d^{ij} - \hat{n}^i \hat{n}^j.
\label{eq:proj1}
\ea
Correspondingly, we define projected derivative operators,
\ba
\partial_\parallel \equiv\:& \hat{n}^i\partial_i,{\rm ~and} \nonumber\\
\partial_\perp^i \equiv\:& \mathcal{P}^{ij} \partial_j.
\label{eq:proj2}
\ea
Note that $\partial_\perp^i,\,\partial_\parallel$ and $\partial_\perp^i,\,\partial_\perp^j$
do not commute. Further, we find
\begin{equation}
\partial_j \hat{n}^i = \partial_{\perp j} \hat{n}^i = \frac1\chi \mathcal{P}_j^{\ i},
\end{equation}
where $\chi$ is the norm of the position vector so that $\hat{n}^i = x^i/\chi$.
Note that $\hat{n}^i$ and $\partial_\parallel$ commute.
More expressions can be found in \S~II of \cite{gaugePk}.
Finally, it proves useful to decompose the quantities defined on the
sphere, i.e. as function of the unit line-of-sight vector $\hat{\v{n}}$,
in terms of their properties under a rotation around $\hat{\v{n}}$. In particular,
consider an orthonormal coordinate system $(\v{e}_1,\v{e}_2,\hat{\v{n}})$.
If we rotate the coordinate system around $\hat{\v{n}}$ by an angle $\psi$,
so that $\v{e}_i \to \v{e}'_i$,
then the linear combinations $\v{m}_\pm \equiv (\v{e}_1\mp i\,\v{e}_2)/\sqrt{2}$
transform as
\begin{equation}
\v{m}_\pm \to \v{m}'_\pm = e^{\pm i\psi} \v{m}_\pm.
\label{eq:spin1}
\end{equation}
We say that a general function $f(\hat{\v{n}})$ is spin-$s$ if it transforms
under the same transformation as
\begin{equation}
f(\hat{\v{n}}) \to f(\hat{\v{n}})' = e^{i s\psi} f(\hat{\v{n}}).
\end{equation}
An ordinary scalar function on the sphere is clearly spin-0, while the
unit vectors $\v{m}_\pm$ defined above are spin$\pm1$ fields. More details
can be found in \refapp{spinSH}. This decomposition is particularly
useful for deriving multipole coefficients and angular power spectra.
We also define
\ba
X_\pm \equiv\:& m_\mp^i X_i \nonumber\\
E_\pm \equiv\:& m_\mp^i m_\mp^j E_{ij}
\label{eq:Xpm}
\ea
for any 3-vector $X_i$ and 3-tensor $E_{ij}$.
For the quantitative results shown in \reffig{Cl}, we assume
a flat $\Lambda$CDM cosmology with $h=0.72$, $\Omega_m=0.28$, a scalar
spectral index $n_s=0.958$ and power spectrum normalization at $z=0$ of
$\sigma_8 = 0.8$.
\section{Standard ruler}
\label{sec:ruler}
In the absence of perturbations, photon geodesics are given by straight lines
in conformal coordinates,
\begin{equation}
\bar{x}^\mu(\chi)
=
\left(
\eta_0-\chi, \hat{\v{n}}\chi
\right),
\label{eq:geod_conf}
\end{equation}
where we have chosen the comoving distance $\chi$ as affine parameter.
Correspondingly, for a photon arriving from a direction $\hat{\v{n}}$ with
redshift $\tilde{z}$, we assign an ``observed'' position of emission $x^\mu$ given by
\ba
\tilde x^0 =\:& \eta_0 - \tilde{\chi} \nonumber\\
\tilde x^i =\:& \hat{n}^i\,\tilde{\chi} \nonumber\\
\tilde{\chi} \equiv\:& \bar{\chi}(\tilde{z}),
\label{eq:xtilde}
\ea
where $\bar{\chi}(\tilde{z})$ denotes the comoving distance-redshift relation in the
background Universe. Here we have chosen the observer to reside at the
spatial origin without loss of generality. The coordinate time of the observer
who is assumed to be comoving is fixed by the condition of a fixed proper time $t_0$ at observation. On the other hand, the actual spacetime point of emission, denoted
with $x^\mu$, is displaced from the observed positions by
$\Delta x^\mu$ (see also \reffig{sketch}),
\begin{equation}
x^\mu = \tilde x^\mu + \Delta x^\mu(\hat{\v{n}},\tilde{z}).
\end{equation}
Further, we will need the scale factor at emission. It is related
to the inferred emission scale factor $\tilde a \equiv 1/(1+\tilde{z})$ by
\begin{equation}
\frac{a(x^0(\hat{\v{n}},\tilde z))}{\tilde a} = 1 + \Delta\ln a(\hat{\v{n}},\tilde z).
\label{eq:Dlnadef}
\end{equation}
At first order, $\Delta\ln a = \Delta z/(1+\tilde z)$, where $\Delta z$
is the difference between the observed redshift and the redshift that would
be observed in an unperturbed Universe. Note that the latter quantity
is gauge-dependent.
We will give explicit expressions for $\Delta\ln a$ and $\Delta x^\mu$ in \refsec{SVT}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{sketch2.eps}
\caption{Illustration of the apparent and actual standard ruler. Photons
arrive out of the observed directions $\hat{\v{n}},\,\hat{\v{n}}'$ and with
observed redshifts $\tilde{z},\,\tilde{z}'$. The apparent positions are indicated
by $\tilde x^\mu,\,\tilde x'^\mu$, while the true positions are at
$x^\mu,\,x'^\mu$, perturbed by the displacements $\Delta x^\mu,\,\Delta x'^\mu$
(whose magnitude is greatly exaggerated here).
$\tilde r$ is the apparent size of the ruler, while $r_0$ is the true
ruler.
\label{fig:sketch}}
\end{figure}
The displacements $\Delta\ln a$, $\Delta x^\mu$ are not observable (they depend on which
gauge, or frame, the spacetime perturbations are described in).
However, we will construct them in such a way that for a local source,
i.e. for photons emitted an infinitesimal distance away from the observer,
the perturbations vanish\footnote{That is, up to an additive constant to $\Delta\ln a$ and $\Delta x^0$ which enforces the condition that the observer is at a fixed proper time.}:
\begin{equation}
\Delta\ln a,\ \Delta x^\mu \stackrel{\tilde{z} \to 0}{=} 0.
\end{equation}
In order to determine actual observables, we consider the case of a
standard ruler. A standard ruler exists if we can identify two
spacetime points which are separated by a fixed spacelike distance $r_0$.
What we observe is the \emph{apparent} size
at which this ruler appears in a given direction $\hat{\v{n}}$ and redshift
$\tilde z$. Let $\hat{\v{n}},\tilde z$ and $\hat{\v{n}}',\tilde z'$ denote the observed
coordinates of the ``end points'' of the ruler, and $\tilde\v{x}$ and
$\tilde\v{x}'$ the apparent spatial positions inferred through \refeq{xtilde}.
The \emph{inferred} physical separation is then given by
\begin{equation}
\tilde r^2 = \tilde a^2 \d_{ij} (\tilde x^i - \tilde x'^i) (\tilde x^j - \tilde x'^j),
\label{eq:rt1}
\end{equation}
where $\tilde a = 1/(1+\tilde{z})$ is the observationally inferred scale factor
at emission (\reffig{sketch}).
We now have to carefully consider what the condition of a standard
ruler in cosmology means. A useful, physically motivated definition is
that it corresponds to a fixed spatial scale as measured by local
observers which are comoving with the cosmic fluid; precisely,
the spatial part of the four-velocity $u^\mu$ of these observers is given by
\begin{equation}
v^i = \frac{T^i_{\ 0}}{\rho + p}.
\label{eq:vcom}
\end{equation}
We are mostly interested in applications to the large-scale structure during
matter domination; in this case, the cosmic fluid is simply matter
(dark matter + baryons), and there is no ambiguity in this definition.
In synchronous-comoving gauge, \refeq{vcom} yields $v^i = 0$.
Further, in the following we assume the ruler scale is fixed.
An evolving ruler is considered in \cite{Tpaper}.
This definition can also be phrased as that the length of the ruler
is defined on a surface of
constant proper time of comoving observers. This proper time corresponds
to the ``local age'' of the Universe.
The separation of the two endpoints
of the ruler, $x^\mu,\,x'^\mu$, projected onto this hypersurface
should thus be equal to the fixed scale $r_0$:
\ba
\left[g_{\mu\nu}(x^\alpha) + u_\mu(x^\alpha) u_\nu(x^\alpha)\right]
(x^a-x'^a) (x^b - x'^b) = r_0^2,
\label{eq:r01}
\ea
where $g_{\mu\nu} + u_\mu u_\nu$ is the metric projected perpendicular
to $u_\mu$, the four-velocity of the comoving observers
(note that $u_\mu u^\mu = -1$). Here and throughout, we
will assume for simplicity that the ruler is ``small'', i.e.
it subtends a small angle, and
redshift interval ($|\tilde z-\tilde z'| \ll \tilde z$). This entails
$\d\tilde x^i \d\tilde x_i \ll \tilde{\chi}^2$, and that we can simply
evaluate the metric and
four-velocity at either end-point (corrections involve higher powers of
$x^\mu - x'^\mu$).
The four-velocity of comoving observers, whose spatial components
are fixed by \refeq{vcom}, is given by
\ba
u^\mu =\:& a^{-1} \left(1-A,\, v^i \right) \nonumber\\
u_\mu =\:& a \left(-1-A,\, v_i - B_i \right),
\label{eq:umu}
\ea
where we consider $v^i$ to be first order (as the metric perturbations).
In the following, we will assume sources to be comoving as well, i.e. to
follow \refeq{umu}. It is straightforward to generalize the
treatment to different source velocities.
Using \refeq{metric} and
\refeq{umu}, we have
\ba
g_{\mu\nu} + u_\mu u_\nu
=\:& a^2 \left(\begin{array}{cccc}
0 & & -v_i & \\
& & & \\
- v_i & & \d_{ij} + h_{ij} & \\
& & &
\end{array}\right).
\ea
With this, \refeq{r01} yields
\ba
& -2 \tilde a^2 v_i \left\{ \d\tilde x^0 \d\tilde x^i + \d\tilde x^0 [\Delta x^i -\Delta x'^i] + \d\tilde x^i [\Delta x^0 - \Delta x'^0 ] \right\} \nonumber\\
&+ g_{ij}(x^\alpha) \bigg\{ \d\tilde x^i \d\tilde x^j + \d\tilde x^i [ \Delta x^j - \Delta x'^j ]
+ [ \Delta x^i - \Delta x'^i ]\d\tilde x^j
\bigg\}\nonumber\\
& = r_0^2,
\label{eq:r02}
\ea
where $\Delta x^\mu = \Delta x^\mu(\hat{\v{n}},\tilde{z})$, $\Delta x'^\mu = \Delta x^\mu(\hat{\v{n}}',\tilde{z}')$,
and the components of the \emph{apparent} separation vector are
\begin{equation}
\d\tilde x^\mu = \tilde x^\mu - \tilde x'^\mu.
\label{eq:dxmu}
\end{equation}
In order to evaluate the spatial metric $g_{ij}(x^\alpha)$ at the
location of the ruler, we use \refeq{Dlnadef} to obtain at first order
\begin{equation}
g_{ij}(x^\alpha) = \tilde a^2\left[\left(1 + 2 \Delta\ln a \right) \d_{ij} + h_{ij} \right].
\end{equation}
We now again make use of the ``small ruler'' approximation, so that
\begin{equation}
\Delta x^i - \Delta x'^i \simeq \d\tilde x^\alpha \frac{\partial}{\partial\tilde x^\alpha}
\Delta x^i.
\end{equation}
Like any vector, we can decompose the spatial part of the apparent
separation $\d\tilde x^i$ into parts parallel and transverse to the line
of sight:
\ba
\d\tilde x_\parallel \equiv\:& \hat{n}_i \d\tilde x^i \nonumber\\
\d\tilde x_\perp^i \equiv\:& \mathcal{P}^i_{\ j} \d\tilde x^j = \d\tilde x^i - \hat{n}^i \d\tilde x _\parallel.
\ea
In the correlation function literature, $\d\tilde x_\parallel,\,|\d\tilde\v{x}_\perp|$
are sometimes referred to as $\pi$ and $\sigma$, respectively. Then,
\ba
\d\tilde x^\alpha \frac{\partial}{\partial\tilde x^\alpha} =\:&
(\d\tilde x^0 \partial_\eta + \d\tilde x_\parallel \partial_\parallel)
+ \d\tilde x_\perp^i \partial_{\perp\,i},
\ea
where we have similarly defined $\partial_\parallel = \hat{n}^i \partial_i$,
$\partial_{\perp\,i} = \mathcal{P}^{\ j}_i \partial_j$. Since the observed
coordinates $\tilde x^\mu$ by definition satisfy the light cone condition
with respect to the unperturbed FRW metric,
we have $\d\tilde x^0 = -\d\tilde x_\parallel$ in the small-angle approximation.
Thus,
\ba
\d\tilde x^0 \partial_\eta + \d\tilde x_\parallel \partial_\parallel
=\:& \d\tilde x_\parallel (\partial_\parallel - \partial_\eta) \nonumber\\
=\:& \d\tilde x_\parallel \frac{\partial}{\partial\tilde{\chi}} = \d\tilde x_\parallel
H(\tilde{z}) \frac{\partial}{\partial\tilde{z}},
\ea
where $\partial/\partial\tilde{\chi}$ is the derivative with respect to the affine
parameter at emission. We thus have
\begin{equation}
\d\tilde x^\alpha \frac{\partial}{\partial\tilde x^\alpha} =
\d\tilde x_\parallel \partial_{\tilde{\chi}} + \d\tilde x_\perp^i \partial_{\perp\,i}.
\end{equation}
Working to first order in perturbations, we then obtain
\ba
r_0^2 - \tilde r^2 =\:& 2 \Delta\ln a\: \tilde r^2
+ \tilde a^2 h_{ij} \d\tilde x^i \d\tilde x^j \nonumber\\
+& 2 \tilde a^2 \left( v_\parallel \d\tilde x_\parallel^2 + v_{\perp\,i} \d\tilde x_\perp^i \d\tilde x_\parallel\right)
\nonumber\\
+& 2\tilde a^2 \d_{ij} \d\tilde x^{i} \left(\d\tilde x_\parallel \partial_{\tilde{\chi}}
+ \d\tilde x_\perp^k \partial_{\perp\,k} \right) \Delta x^{j}.
\label{eq:rt2}
\ea
All terms are straightforward to interpret: there are the perturbations
to the metric (both from the metric perturbation $h_{ij}$ and the
perturbation to the scale factor at emission); the contribution $\propto v$
from the projection from fixed-$\eta$ to fixed-proper-time
hypersurfaces; and the difference in
the spatial displacements of the endpoints of the ruler.
\section{Scalar-vector-tensor decomposition on the sky}
\label{sec:SVT}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{ruler.eps}
\caption{Illustration of the distortion of standard rulers
due to the longitudinal (2-)scalar $\mathcal{C}$, (2-)vector $\mathcal{B}$, and
transverse components, magnification $\mathcal{M}$ and shear $\gamma$.
The first row shows the projection onto the sky plane, while the
second (third) row show the projection onto the line-of-sight and
$x_\perp^1$ ($x_\perp^2$) axes, respectively. In case of $\mathcal{B}$ and $\gamma$,
we only show one of the two components. See also Fig.~3 in \cite{sachs:1961}.
\label{fig:SVT}}
\end{figure}
It is useful to separate the contributions to \refeq{rt2} in terms of the
observed longitudinal and transverse displacements. For some
applications, only the transverse displacements are relevant. This
is the case for diffuse backgrounds without redshift resolution,
such as the CMB or the cosmic infrared background, and largely the case
for photometric galaxy surveys. On the other hand, spectroscopic
surveys and redshift-resolved backgrounds such as the 21cm emission
from high-redshifts are able to measure the longitudinal displacements
as well.
Noting that
$\tilde r^2 = \tilde a^2[\d\tilde x_\parallel^2 + (\d\tilde\v{x}_\perp)^2]$,
and taking the square root of \refeq{rt2}, we obtain the
relative perturbation to the physical scale of the ruler as
\ba
\frac{\tilde r - r_0 }{\tilde r} =\:&
\mathcal{C} \frac{(\d\tilde x_\parallel)^2}{\tilde r_c^2} + \mathcal{B}_i \frac{\d\tilde x_\parallel \d\tilde x_\perp^i}{\tilde r_c^2}
+ \mathcal{A}_{ij} \frac{\d\tilde x_\perp^i \d\tilde x_\perp^j}{\tilde r_c^2},
\label{eq:r03}
\ea
where we have defined
$\tilde r_c \equiv \tilde r/\tilde a$ as the apparent comoving size of the ruler.
The quantities multiplying $\mathcal{C},\,\mathcal{B}_i,\,\mathcal{A}_{ij}$ are thus simply geometric
factors. The coefficients are given by
\ba
\mathcal{C} =\:& - \Delta\ln a
- \frac12 h_\parallel - v_\parallel - \partial_{\tilde{\chi}} \Delta x_\parallel
\nonumber\\
\mathcal{B}_i =\:& -\mathcal{P}_i^{\ j} h_{jk} \hat{n}^k - v_{\perp i} - \hat{n}^k \partial_{\perp\,i} \Delta x_k - \partial_{\tilde{\chi}} \Delta x_{\perp i}
\nonumber\\
\mathcal{A}_{ij} =\:&
- \Delta\ln a\: \mathcal{P}_{ij}
- \frac12 \mathcal{P}_i^{\ k}\mathcal{P}_j^{\ l} h_{kl} \nonumber\\
& - \frac12 \left(\mathcal{P}_{jk} \partial_{\perp\,i} + \mathcal{P}_{ik} \partial_{\perp\,j}\right) \Delta x^k, \label{eq:coeff}
\ea
where $\Delta x_\parallel,\,\Delta x_\perp^i$ are the parallel and
perpendicular components of the displacements $\Delta x^i$.
Note that while we
have assumed that the ruler is small (i.e. $\d\tilde x^i \ll \tilde{\chi}$), the
expressions for $\mathcal{C},\,\mathcal{B}_i,\,\mathcal{A}_{ij}$ are valid on the
full sky. \reffig{SVT} illustrates the distortions induced by these
components.
Observationally, we have 6 free parameters (assuming accurate redshifts
are available): the
location of one point $\hat{\v{n}},\,\tilde{z}$, and the separation vector described by
$\d\tilde x^i$ (with $\d\tilde x^0$ being fixed by the light cone condition).
Using these, we can measure a (2-)scalar on the sphere, $\mathcal{C}$, a $2\times2$
symmetric matrix, $\mathcal{A}_{ij}$, and a 2-component vector on the sphere, $\mathcal{B}_i$.
As a symmetric matrix on the sphere, $\mathcal{A}_{ij}$ has a scalar component, given
by the trace $\mathcal{M} \equiv \mathcal{P}^{ij} \mathcal{A}_{ij}$ (\emph{magnification}), and two
components of the traceless
part which transform as spin-2 fields on the sphere (\emph{shear} ${}_{\pm 2}\gamma$
as defined in \refeq{shear1} below).
These quantities are observable and gauge-invariant, while any of the individual
contributions in \refeq{coeff} are not in general. Note that we cannot measure
any of the anti-symmetric components, such as the rotation. This is because
we have not assumed the existence of any preferred directions in the
Universe. If there is a primary spin-1 or higher spin field, such as
the polarization in case of the CMB, then a rotation can be measured as
it mixes the spin$\pm 2$ components (see, e.g. \cite{gluscevic/etal:2009}).
In the next sections we study these three terms in turn.
For reference, we now give the explicit expressions for the
displacements $\Delta x^i$ and $\Delta\ln a$. They are defined such that
$\Delta x^i = 0 = \Delta\ln a$ for a local source, i.e. for $\tilde{z} \approx 0$, up to a shift in the observer's time coordinate.
The details of the derivation are presented in \refapp{geod}.
Separating into line-of-sight and transverse parts, we have
\ba
\Delta x_\parallel =\:& \int_0^{\tilde{\chi}} d\chi\left[ A - B_\parallel - \frac12 h_\parallel
\right]
- \frac{1+\tilde{z}}{H(\tilde{z})} \Delta \ln a \nonumber\\
& -\int_0^{t_0} A(\v{0},t) dt \label{eq:Dxpar}\\
\Delta x_\perp^i =\:& \left[\frac12 \mathcal{P}^{ij} (h_{jk})_o\, \hat{n}^k + B^i_{\perp o} - v^i_{\perp o}\right] \tilde{\chi} \label{eq:Dxperp}\\
& + \int_0^{\tilde{\chi}} d\chi \bigg[
- B_\perp^i - \mathcal{P}^{ij} h_{jk}\hat{n}^k \nonumber\\
&\hspace*{1.6cm} + (\tilde{\chi}-\chi)\bigg\{
- \partial_\perp^i A + \hat{n}^k \partial_\perp^i B_k \nonumber\\
&\hspace*{3.4cm} + \frac12 (\partial_\perp^i h_{jk})\hat{n}^j\hat{n}^k
\bigg\}\bigg] \nonumber\\
=\:& \left[\frac12 \mathcal{P}^{ij} (h_{jk})_o\, \hat{n}^k + B^i_{\perp o} - v^i_{\perp o}\right] \tilde{\chi} \label{eq:Dxperp2}\\
& - \int_0^{\tilde{\chi}} d\chi \bigg[
\frac{\tilde{\chi}}{\chi} \left(
B_\perp^i + \mathcal{P}^{ij} h_{jk}\hat{n}^k\right)
\nonumber\\
&\hspace*{1.6cm}+(\tilde{\chi}-\chi)\partial_\perp^i
\left( A - B_\parallel - \frac12 h_\parallel\right)
\bigg].\nonumber
\ea
The perturbation to the scale factor at emission is given by
\ba
\Delta\ln a =\:& A_o - A + v_\parallel - v_{\parallel o}
+ \int_0^{\tilde{\chi}} d\chi\left[
- A' + \frac12 h_{\parallel}' + B_\parallel'\right] \nonumber\\
& - H_0 \int_0^{t_0} A(\v{0},\bar\eta(t)) dt\,.
\label{eq:Dlna}
\ea
Here, a subscript $o$ indicates quantities evaluated at the observer,
while primes denote derivatives with respect to $\eta$.
Note the appearance of the scalar quantity
$A - B_\parallel - \frac12 h_\parallel$ in \refeqs{Dxpar}{Dlna}.
This is the ``lensing potential'' $\Phi-\Psi$ in conformal-Newtonian gauge,
written in the general gauge \refeq{metric}. The term in the second lines of \refeq{Dxpar} and \refeq{Dlna} comes from requiring the observer to lie at a fixed proper time, rather than at fixed scale factor or coordinate time, which are gauge-dependent quantities. While these terms only contribute to the monopole of $\mathcal{C}$ and $\mathcal{M}$, which
are typically not observable, they are essential for cross-checking the result with test cases and against gauge-transformations.
In particular, in the two popular gauges
introduced in \refsec{ruler}, \refeq{Dlna} becomes
\ba
(\Delta\ln a)_{\rm sc} =\:& \frac12 \int_0^{\tilde{\chi}} d\chi\:h_\parallel'
\label{eq:Dlna_sc}\\
(\Delta\ln a)_{\rm cN} =\:& \Psi_o - \Psi + v_\parallel - v_{\parallel o}
+ \int_0^{\tilde{\chi}} d\chi \left[ \Phi' - \Psi'\right] \nonumber\\
& - H_0 \int_0^{t_0} \Psi(\v{0},\bar\eta(t)) dt\,. \label{eq:Dlna_cN}
\ea
The latter result clearly shows the ``Sachs-Wolfe'', ``Doppler'',
and ``integrated Sachs-Wolfe'' contributions, along with the coordinate
time perturbation at the observer fixing the proper time.
\section{Longitudinal scalar}
\label{sec:C}
The longitudinal component can be simplified to become
\ba
\mathcal{C} =\:& - \Delta\ln a
\left[1 - H(\tilde{z}) \frac{\partial}{\partial\tilde{z}}\left( \frac{1+\tilde{z}}{H(\tilde{z})}\right)
\right] \nonumber\\
& - A - v_\parallel + B_\parallel \nonumber\\
& + \frac{1+\tilde{z}}{H(\tilde{z})} \left(
- \partial_\parallel A + \partial_\parallel v_\parallel
+ B_\parallel' - v_\parallel' + \frac12 h_{\parallel}' \right).
\label{eq:C}
\ea
The first line contains the contributions due to the fact that the
scale factor at emission is perturbed from $1/(1+\tilde{z})$, and due to the evolution of the distance-redshift relation. The second
line contains the perturbations from the metric at the source location
($-A$) and the projection from coordinate-time to proper-time hypersurfaces
($B_\parallel-v_\parallel$). Finally, the contributions from the line-of-sight
derivative of the line-of-sight displacements ($\propto (1+\tilde{z})/H(\tilde{z})$)
are given in the third line.
Note the term $\partial_\parallel v_\parallel$, which is the dominant term
on small scales in the conformal-Newtonian gauge. This term is
also responsible for the leading-order redshift distortions \cite{Kaiser87}.
Apart from the pertubation to the scale factor at emission,
$\mathcal{C}$ does not involve any integral terms; this is expected since $\mathcal{C}$
is the only term remaining if the two lines of sight coincide ($\hat{\v{n}}=\hat{\v{n}}'$).
In this case, the two rays share the same path from the closer of the
two emission points, and no quantities integrated along the line of sight
can contribute to the perturbation of the ruler.
Restricting to the synchronous-comoving and conformal-Newtonian gauges,
respectively, we obtain
\ba
(\mathcal{C})_{\rm sc} =\:& - (\Delta\ln a)_{\rm sc}
\left[1 - H(\tilde{z}) \frac{\partial}{\partial\tilde{z}}\left( \frac{1+\tilde{z}}{H(\tilde{z})}\right)
\right] \nonumber\\
& + \frac{1+\tilde{z}}{2 H(\tilde{z})}h_{\parallel}'.
\label{eq:C_sc}\\
(\mathcal{C})_{\rm cN} =\:& - (\Delta\ln a)_{\rm cN}
\left[1 - H(\tilde{z}) \frac{\partial}{\partial\tilde{z}}\left( \frac{1+\tilde{z}}{H(\tilde{z})}\right)
\right] \nonumber\\
& - \Psi - v_\parallel + \frac{1+\tilde{z}}{H(\tilde{z})} \left(
- \partial_\parallel\Psi + \partial_\parallel v_\parallel
- v_\parallel' + \Phi' \right).
\label{eq:C_cN}
\ea
Note that in case of the sc-gauge expression,
the redshift-space distortion term is included in the last term, through
$h_\parallel'/2 = D' + \partial_\parallel^2 E'$. \reffig{Cl} shows
the angular power spectrum of $\mathcal{C}$ due to standard adiabatic
scalar perturbations in a $\Lambda$CDM cosmology (the details of the
calculation are given in \refapp{calc}). Clearly,
$\mathcal{C}$ is of the same order as the matter density contrast in
synchronous-comoving gauge on all scales. In particular,
the velocity gradient term dominates over all other contributions.
Due to the different dependence on the angle with the line of sight,
the projection kernel of $\mathcal{C}$ is proportional to $\partial_x^2 j_l(x)$,
while that of $\d_m^{\rm sc}$ is $\propto j_l(x)$. The former favors larger
$x$ at a given $l$, and thus leads to a relative suppression as the
slope of the matter power spectrum changes at $k \gtrsim 0.01 \:h/{\rm Mpc}$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{Cl_ruler.eps}
\caption{Angular power spectra of the different standard ruler perturbations
produced by a standard scale-invariant power spectrum of curvature perturbations:
$\mathcal{C}$, $E$-mode of $\mathcal{B}_i$, $E$-mode of the shear, and magnification $\mathcal{M}$.
All quantities are calculated for a non-evolving ruler and a sharp source
redshift of $\tilde{z} = 2$.
For comparison, the thin dotted line shows the angular power spectrum at $z=2$ of the matter
density field in synchronous-comoving gauge. Note that all quantities
shown here, except for $\d_m^{\rm sc}$, are gauge-invariant and (in principle)
observable.
\label{fig:Cl}}
\end{figure}
\section{Vector}
\label{sec:vector}
Next, we have the two-component vector
\ba
\mathcal{B}_i =\:& -\mathcal{P}_i^{\ j}h_{jk}\hat{n}^k - v_{\perp i} - \partial_{\perp\,i} \Delta x_\parallel - \partial_{\tilde{\chi}} \Delta x_{\perp\,i}
+ \frac{\Delta x_{\perp\,i}}{\tilde{\chi}}
\nonumber\\
=\:& - v_{\perp i} + B_{\perp i} + \frac{1+\tilde{z}}{H(\tilde{z})}\partial_{\perp i}\Delta\ln a,
\label{eq:Bi}
\ea
where we have inserted projection operators for clarity (these are trivial
since $\mathcal{B}_i$ is contracted with $\d\tilde x_\perp^i$).
As expected, this vector involves the transverse derivative of the
line-of-sight displacement and the line-of-sight derivative of the
transverse displacement. Note that these two quantities are
\emph{not} observable individually.
Using the spin$\pm1$ unit vectors $\v{m}_\pm$,
$\mathcal{B}_i$ can be decomposed into spin$\pm1$ components:
\ba
\mathcal{B}_i =\:& {}_{+1}\mathcal{B} m_+^i + {}_{-1}\mathcal{B} m_-^i \nonumber\\
{}_{\pm 1}\mathcal{B} \equiv\:& m_\mp^i \mathcal{B}_i
= - v_{\pm} + B_{\pm} + \frac{1+\tilde{z}}{H(\tilde{z})}\partial_{\pm}\Delta\ln a,
\label{eq:Bpm}
\ea
where we have used the notation of \refeq{Xpm}.
Similar to before, we can specialize this general result to the
synchronous-comoving and conformal-Newtonian gauges:
\ba
({}_{\pm 1}\mathcal{B})_{\rm sc} =\:& \frac{1+\tilde{z}}{2 H(\tilde{z})}\int_0^{\tilde{\chi}}d\chi\frac{\chi}{\tilde{\chi}}\partial_\pm h_\parallel'
\label{eq:Bpm_sc}\\
({}_{\pm 1}\mathcal{B})_{\rm cN} =\:& - v_\pm + \frac{1+\tilde{z}}{H(\tilde{z})}\partial_{\pm}\Delta\ln a \nonumber\\
=\:& -v_\pm + \frac{1+\tilde{z}}{H(\tilde{z})}\Bigg(-\partial_\pm \Psi + \partial_\pm
[v_\parallel - v_{\parallel o}] \nonumber\\
& \hspace*{2.5cm} + \int_0^{\tilde{\chi}} d\chi\frac{\chi}{\tilde{\chi}} \partial_\pm(\Phi'-\Psi')
\Bigg).
\label{eq:Bpm_cN}
\ea
On small scales, the dominant contribution to $\mathcal{B}_i$ comes from the
transverse derivative of the line-of-sight component of the velocity
$\partial_\pm v_\parallel$, which
is of the same order as the tidal field.
Applying the spin-lowering operator $\bar\eth$ to ${}_1\mathcal{B}$ (see \refapp{spinSH}) yields
a spin-zero quantity, which can be expanded in terms of the usual
spherical harmonics\footnote{This is of course equivalent to expanding
${}_1\mathcal{B}$ in terms of spin-1 spherical harmonics.}. We then obtain
the multipole coefficients of $\mathcal{B}$ as
\ba
a^{\mathcal{B}}_{lm}(\tilde{z}) =\:& - \sqrt{\frac{(l-1)!}{(l+1)!}} \int d\Omega\: \left[\bar\eth\,{}_1\mathcal{B}(\hat{\v{n}},\tilde{z})\right] Y^*_{lm}(\hat{\v{n}}).
\label{eq:aBlm}
\ea
An equivalent result is obtained for $\eth {}_{-1} \mathcal{B}$. In general, the
multipole coefficients $a^{\mathcal{B}}_{lm}$ are complex, so that we can decompose
them into real and imaginary parts,
\begin{equation}
a^{\mathcal{B}}_{lm} = a^{\mathcal{B} E}_{lm} + i\,a^{\mathcal{B} B}_{lm}.
\end{equation}
One can easily show (\refapp{spinSH}) that under a change of parity
$a^{\mathcal{B} E}_{lm}$ transform as the spherical harmonic coefficients of a
vector (parity-odd), whereas $a^{\mathcal{B} B}_{lm}$, picking up an additional
minus sign, transforms as those of a pseudo-vector (parity-even).
These thus correspond to the polar (``$E$'') and
axial (``$B$'') parts of the vector $\mathcal{B}_i$.
As required by parity, scalar perturbations do not contribute to
the axial part $a^{\mathcal{B} B}_{lm}$ (this is shown explicitly in \refapp{vector}).
Thus, a measurement of the vector component $\mathcal{B}_i$ of standard ruler distortions
offers an additional possibility to probe tensor modes with large-scale
structure, as tensor modes do contribute to $a^{\mathcal{B} B}_{lm}$ (\refapp{vector}).
Thus, in principle the axial component of $\mathcal{B}_i$ could
be of similar interest for constraining
tensor modes as weak lensing $B$-modes \cite{paperII}, though one
likely requires accurate redshifts to measure $\mathcal{B}_i$ to sufficient accuracy.
We leave a detailed investigation of this for future work.
The power spectrum of the $E$-mode of $\mathcal{B}$ due to standard
scalar perturbations is shown in \reffig{Cl} (see \refapp{calc}).
While the dominant
contribution to $\mathcal{C}$ is $\propto k_\parallel^2/k^2\, \d^{\rm sc}_m(\v{k},\tilde{z})$
for a given Fourier mode of the matter density contrast in
synchronous-comoving gauge (\refsec{C}), the corresponding contribution to $\mathcal{B}$ is
$\propto k_\perp k_\parallel/k^2\, \d^{\rm sc}_m(\v{k},\tilde{z})$. Even though
approximate scaling arguments suggest that $C_\mathcal{C}(l)$, $C^{EE}_{\mathcal{B}}(l)$
should scale roughly equally with $l$, we see that
$C_{\mathcal{B}}(l)$ scales faster with $l$ for $l \lesssim 500$. The reason is
that the projection kernel for the $E$-mode of $\mathcal{B}$
($\propto (\partial_x j_l)/x$)
is relatively suppressed with respect to that of $\mathcal{C}$ ($\propto \partial_x^2 j_l$) at
large $x/l$. Since $l \lesssim 500$ corresponds to a typical $k \lesssim 10^{-2} \:h/{\rm Mpc}$
at the source redshift, where $P_m(k) \propto k$, larger $x/l$ are favored
for progressively smaller $l$, leading to a more rapid decrease of $C_{\mathcal{B}}(l)$
towards smaller $l$. This suppression is thus fundamentally a consequence of the shape
of the matter power spectrum.
\section{Transverse tensor: shear and magnification}
Finally, we have the purely transverse component,
\ba
\mathcal{A}_{ij} =\:& - \Delta\ln a \: \mathcal{P}_{ij}
- \frac12 \mathcal{P}_i^{\ k} \mathcal{P}_j^{\ l} h_{kl} \nonumber\\
& - \partial_{\perp\,(i} \Delta x_{\perp\,j)}
- \frac{1}{\tilde{\chi}} \Delta x_\parallel \mathcal{P}_{ij},
\label{eq:Aij}
\ea
where we have again inserted projection operators for clarity (note that
$\mathcal{P}_{ij}$ serves as the identity matrix on the sphere). As
a symmetric matrix on the sphere, $\mathcal{A}_{ij}$ has a scalar component, given
by the trace $\mathcal{A}$, and two components of the traceless part which transform
as spin-2 fields on the sphere. The trace corresponds to the change in
area on the sky subtended by two perpendicular standard rulers. Thus,
it is equal to the magnification $\mathcal{M}$ (see also \reffig{SVT}).
The two components of the traceless
part correspond to the shear $\gamma$. If we choose a fixed coordinate
system $(\v{e}_\theta,\v{e}_\phi,\hat{\v{n}})$, we can thus write
\begin{equation}
\mathcal{A}_{ij} = \left(\begin{array}{cc}
\mathcal{M}/2 + \gamma_1 & \gamma_2 \\
\gamma_2 & \mathcal{M}/2 - \gamma_1
\end{array}\right).
\label{eq:Aijcoord}
\end{equation}
Below, we will derive magnification and shear without reference to a fixed
coordinate system.
\subsection{Magnification}
\label{sec:mag}
Taking the trace of \refeq{Aij} yields
\ba
\mathcal{M} \equiv\:& \mathcal{P}^{ij} \mathcal{A}_{ij} \nonumber\\
=\:& - 2\Delta\ln a
- \frac12 \left(h^i_{\ i} - h_\parallel\right)
+ 2\hat\k - \frac{2}{\tilde{\chi}} \Delta x_\parallel
\label{eq:mag}.
\ea
The magnification is directly related to the fractional perturbations in
distances (see \cite{HuiGreene,BonvinEtal06}) through
\begin{equation}
\frac{\Delta D_L}{D_L} = \frac{\Delta D_A}{D_A} = -\frac12 \mathcal{M},
\end{equation}
where the first equality for the luminosity distance follows from \refeq{DLDA}.
The contributions to the magnification are straightforwardly interpreted
as coming from the conversion of coordinate distance to physical scale at
the source (from the perturbation to the scale factor $\Delta\ln a$ and the
metric at the source projected perpendicular to the line of sight,
$h^i_{\ i}-h_\parallel$); from the fact that the entire ruler is moved
closer or further away by $\Delta x_\parallel$; and finally from the coordinate
convergence $\hat\k$ defined through
\begin{equation}
\hat\k = -\frac12 \partial_{\perp\,i} \Delta x_\perp^i.
\end{equation}
This term dominates the other contributions to $\mathcal{M}$ on small scales. However,
the coordinate convergence is a gauge-dependent quantity;
see for example App.~B2 in \cite{gaugePk}. For the general
metric \refeq{metric} it is given by
\ba
\hat\k =\:& -\frac12\left[\frac12 \left((h^i_{\ i})_o - 3 (h_\parallel)_o \right)
- 2(B_\parallel - v_\parallel)_o
\right] \label{eq:kappaG}\\
& +\frac12 \int_0^{\tilde{\chi}} d\chi\,\Bigg[
\partial^k_{\perp} B_k
- \frac{2}{\chi} B_\parallel
+ (\partial_\perp^l h_{lk}) \hat{n}^k \nonumber\\
& \hspace*{2cm} + \frac{1}{\chi} \left(h^i_{\ i} - 3 h_\parallel\right)
\nonumber\\
& \hspace*{2cm} + (\tilde{\chi}-\chi)\frac{\chi}{\tilde{\chi}} \nabla_\perp^2\left\{
A - B_\parallel - \frac12 h_\parallel \right\}
\Bigg]. \nonumber
\ea
In conformal-Newtonian gauge, it
assumes its familiar form,
\ba
(\hat\k)_{\rm cN} =\:& - v_{\parallel o}
+ \frac12 \int_0^{\tilde{\chi}} d\chi\,\frac{\chi}{\tilde{\chi}}
(\tilde{\chi}-\chi)
\nabla_\perp^2 \left(\Psi - \Phi\right),
\label{eq:kcN}
\ea
with an additional term $-v_{\parallel o}$ contributing to the dipole
of $\hat\k$ only, which corresponds to the relativistic beaming effect
at linear order. An explicit expression for the magnification
in general gauge is straightforward to obtain, however it becomes lengthy.
Here we just give the results for the synchronous-comoving and
conformal-Newtonian gauges. Using \refeq{DE} for synchronous-comoving gauge,
$(h^i_{\ i} - h_\parallel)/2 = 2 D - E_\parallel$, and we obtain
\begin{equation}
(\mathcal{M})_{\rm sc} =- 2 (\Delta\ln a)_{\rm sc} - 2 D
+ E_\parallel + 2 (\hat\k)_{\rm sc} - \frac2{\tilde{\chi}} \Delta x_\parallel.
\label{eq:magsc}
\end{equation}
Since $(\Delta\ln a)_{\rm sc} = \d z$ defined in \cite{gaugePk}, we see
that we thus recover the covariant magnification, $\mathcal{M} = \d\mathcal{M}$, as derived
using an independent approach in \cite{gaugePk}.
In conformal-Newtonian gauge [\refeq{metric_cN}],
we have $(h^i_{\ i} - h_\parallel)/2 = 2\Phi$,
so that the magnification in this gauge becomes
\ba
\left(\mathcal{M}\right)_{\rm cN} =\:& \left[- 2 + \frac2{a H \tilde{\chi}}\right] (\Delta\ln a)_{\rm cN} - 2 \Phi
+ 2(\hat\k)_{\rm cN} \nonumber\\
& - \frac{2}{\tilde{\chi}} \int_0^{\tilde{\chi}} d\chi\: (\Psi-\Phi)
+ \frac2{\tilde{\chi}} \int_0^{t_0} dt\: \Psi(\v{0},t)\,.
\label{eq:magcN}
\ea
The last term here is a pure monopole and thus usually absorbed in the ruler
calibration (since $r_0$ can rarely be predicted from first principles without
any dependence on the background cosmology). Nevertheless, including this term ensures that gauge modes (for example superhorizon metric perturbations) do not affect the observed magnification. In particular, in \refapp{magtest} we apply two
test cases to \refeq{magcN} where the monopole and dipole contributions (including $v_{\parallel o}$) become important.
\subsection{Shear}
\label{sec:shear}
We now consider the traceless part of $\mathcal{A}_{ij}$, given by
\ba
\gamma_{ij}(\hat{\v{n}}) \equiv\:& \mathcal{A}_{ij} - \frac12\mathcal{P}_{ij} \mathcal{M} \nonumber\\
=\:&
-\frac12 \left(\mathcal{P}_i^{\ k} \mathcal{P}_j^{\ l}
- \frac12 \mathcal{P}_{ij} \mathcal{P}^{kl} \right) h_{kl}
\nonumber\\
& - \partial_{\perp (i} \Delta x_{\perp\,j)}
- \mathcal{P}_{ij} \hat\k.
\label{eq:shear1}
\ea
Here, the terms $\propto \mathcal{P}_{ij}$ in \refeq{Aij} drop out.
The terms in the second line here is what commonly is regarded as the shear,
i.e. the trace-free part of the transverse derivatives of the transverse
displacements. The first term on the other hand is important to
ensure a gauge-invariant result. This is the term referred to as
``metric shear'' in \cite{DodelsonEtal}. Its physical significance
becomes clear when constructing the Fermi normal coordinates for
the region containing the standard ruler.
Consider a region of spatial extent $R$, say
centered on a given galaxy, with $R$
assumed to be much larger than the scale of individual galaxies.
We can construct orthornomal Fermi normal coordinates \cite{Fermi,ManasseMisner}
around the center of this region, which follows a timelike geodesic,
by choosing the origin to be located at the center of the region at all times,
and the time coordinate to be the proper time of this geodesic.
The spacetime in these Fermi coordinates $(t_F, x^i_F)$ then
becomes Minkowski, with corrections going as $x_F^2/R_c^2$ where $R_c$
is the curvature scale of the spacetime.
Thus, as long as these corrections to the metric are negligible,
there is no preferred direction in this frame, and the size of the standard ruler
has to be (statistically) independent of the orientation. The most
obvious example is galaxy shapes, which are used for cosmic shear
measurements. In the Fermi frame, galaxy orientations are random.
Note that the Fermi coordinates are uniquely
determined up to three Euler angles. The statement that galaxy orientations
are random in this frame is thus coordinate-invariant.
As an example, consider the case where we have a purely spatial metric perturbation
(cf. \refeq{metric}) at a fixed time. We can then expand around the
origin,
\begin{equation}
h_{ij}(\v{x}) = h_{ij}(0) + h_{ij,k}(0) x^k.
\label{eq:gg}
\end{equation}
Higher order terms are suppressed by $(x/R_c)^2$. Now, consider coordinates
given by
\begin{equation}
a^{-1} x_{F}^i
= x^i + \frac12 h_{ij}(0) x^j +
\frac14 \left[2h_{ij,k}(0) - h_{jk,i}(0)\right] x^j x^k.
\label{eq:transf}
\end{equation}
In these coordinates, the metric becomes
\begin{equation}
g^{F}_{\mu\nu} = \eta_{\mu\nu} + \mathcal{O}(x_{F}^2).
\label{eq:gF}
\end{equation}
Thus, it is in terms of the coordinates $x^i_{F}$ that galaxies
should be isotropically oriented on average, \emph{not} in terms
of the cosmological coordinates $x^i$. Correspondingly, in order
to obtain the shear relative to the Fermi frame, we need to add the
transformation \refeq{transf} to the displacements $\Delta x^i$:
\begin{equation}
\Delta x^i \rightarrow \Delta x^i + \frac12 h_{ij}(0) x^j
+
\frac14 \left[2h_{ij,k}(0) - h_{jk,i}(0)\right] x^j x^k.
\end{equation}
With these new displacements, the transverse derivative of the
transverse displacement becomes
\ba
\partial_{\perp (i} \Delta x_{\perp\,j)} \rightarrow \partial_{\perp (i} \Delta x_{\perp\,j)} + \frac12 \mathcal{P}_i^{\ k} \mathcal{P}_j^{\ k} h_{kl} + \mathcal{O}(h_{ij,k} x^k),
\label{eq:shearF}
\ea
where the last term is suppressed by the size of the ruler over the
wavelength of the metric perturbation, and is thus negligible in the
small-ruler approximation. We see that \refeq{shearF} agrees exactly
with the result derived above, \refeq{shear1} (after subtracting the
trace of \refeq{shearF}). In other words,
the shear derived in the standard ruler formalism (\refsec{ruler}) is
equivalent to the statement that the ruler is isotropic in its Fermi
frame, the additional term coming from the transformation from global
coordinates to the local Fermi coordinates. This additional term
was introduced in \cite{DodelsonEtal} as ``metric shear'', with a similar
motivation as given here. In our case, this term is naturally included in the
standard ruler formalism.
$\gamma_{ij}$ is a symmetric trace-free tensor on the sphere, and can thus
be decomposed into spin$\pm 2$ components (in analogy to the
polarization of the CMB). Following \refapp{spinSH} (see also \cite{Hu2000})
we can write $\gamma_{ij}$ as
\ba
\gamma_{ij} =\:& {}_2\gamma\, m_+^i m_+^j + {}_{-2}\gamma\, m_-^i m_-^j \nonumber\\
{}_{\pm 2}\gamma =\:& m_\mp^i m_\mp^j \gamma_{ij},
\label{eq:sheardecomp}
\ea
where ${}_{\pm2}\gamma$ are spin$\pm2$ functions on the sphere (in analogy
to the combination of Stokes parameters $Q \pm i U$).
We obtain for the shear components
\begin{widetext}
\ba
{}_{\pm 2}\gamma =\:& -\frac12 h_\pm - m_\mp^i m_\mp^j \partial_{\perp i} \Delta x_{\perp j} \nonumber\\
=\:& -\frac12 h_\pm - \frac12 (h_{\pm})_o - \int_0^{\tilde{\chi}} d\chi \Bigg[
\left(1 - 2\frac\chi{\tilde{\chi}}\right)
\left[m_\mp^k \partial_\pm B_k
+ (\partial_\pm h_{lk}) m_\mp^l \hat{n}^k\right]
- \frac{1}{\tilde{\chi}}h_{\pm}
\label{eq:shear2}\\
&\hspace*{4.2cm} +(\tilde{\chi}-\chi)\frac{\chi}{\tilde{\chi}} \Bigg\{
- m_\mp^i m_\mp^j \partial_i \partial_j A
+\hat{n}^k m_\mp^i m_\mp^j \partial_i \partial_j B_k
+ \frac12 m_\mp^i m_\mp^j(\partial_i \partial_j h_{kl})\hat{n}^k\hat{n}^l
\Bigg\}\Bigg].
\nonumber
\ea
\refeq{shear2} is valid in any gauge. We can now specialize to
the synchronous-comoving (sc) and conformal-Newtonian (cN) gauges:
\ba
\left({}_{\pm 2}\gamma\right)_{\rm sc} =\:&
-\frac12 h_\pm - \frac12 (h_{\pm})_o - \int_0^{\tilde{\chi}} d\chi\, \Bigg[
\left(1-2\frac{\chi}{\tilde{\chi}}\right) (\partial_\pm h_{kl}) m_\mp^k \hat{n}^l
- \frac{1}{\tilde{\chi}} h_\pm
\label{eq:shear_sc}\\
& \hspace*{4.4cm} + (\tilde{\chi}-\chi)\frac{\chi}{\tilde{\chi}}
\frac12 (m_\mp^i m_\mp^j \partial_i\partial_j h_{lk})\hat{n}^l\hat{n}^k
\Bigg] \nonumber\\
\left({}_{\pm 2} \gamma\right)_{\rm cN} =\:&
\int_0^{\tilde{\chi}} d\chi\, (\tilde{\chi}-\chi)\frac{\chi}{\tilde{\chi}}
m_\mp^i m_\mp^j \partial_i\partial_j
\left(\Psi - \Phi \right).
\label{eq:shear_cN}
\ea
\end{widetext}
In case of the cN gauge, we have used that $h_{ij} = 2\Phi \d_{ij}$,
and thus $h_\pm = 0$. We see that \refeq{shear_cN} recovers the
``standard'' result; in other words, there are no additional
relativistic corrections to the shear \emph{in this gauge}. This is not
surprising following our arguments above: in conformal-Newtonian gauge,
the transformation \refeq{transf} from global coordinates to the local Fermi
frame is isotropic since $h_{ij} = 2\Phi \d_{ij}$. Thus, it does not
contribute to the shear. Note however
that only scalar perturbations are included in this gauge; when
considering vector or tensor perturbations, one has to use a different
gauge, for example synchronous-comoving gauge
(see \cite{paperII} for a study of tensor perturbations). Thus,
\refeq{shear2} and \refeq{shear_sc} are important new results.
In \refapp{test}, we apply several test cases to the shear in
synchronous-comoving gauge, \refeq{shear_sc}, in order
to verify that it is gauge-invariant and correctly reproduces known results.
In particular, we consider a Bianchi~I cosmology which induces a shear
due to the anisotropic angular diameter distance. We also show that
\refeq{shear_sc}, when restricted to scalar perturbations, does not
produce $B$-mode shear.
\reffig{Cl} shows the angular power spectrum of shear and magnification
due to scalar perturbations
for a sharp source redshift $\tilde{z} = 2$ (see \refapp{calc}).
For $l\gtrsim 10$, the results
follow the familiar relation $C_{\mathcal{M}}(l) = 4 C^{EE}_\gamma(l)$, valid when
all relativistic corrections to the magnification become irrelevant
so that $\mathcal{M} \simeq 2\hat\k$.
These corrections slightly increase the magnification for small $l$.
We also see that $\gamma$ and $\mathcal{M}$ are suppressed with respect to $\mathcal{C}$
and $\mathcal{B}$ (on smaller scales), at least when the latter are evaluated
for a sharp source redshift. This is a well-known consequence of
the projection with the broad lensing kernel, leading to a cancelation
of modes that are not purely transverse (see e.g. \cite{JeongSchmidtSefusatti}).
\section{Discussion}
\label{sec:disc}
Over the past decade, cosmology has benefited from a vast increase
in the available data, which have been exploited through a broad variety of methods
to constrain the history of structure in the Universe. Clearly,
this calls for a rigorous investigation of what quantities precisely
are observable in the relativistic setting. Some observables have
been investigated previously, most notably the number density of tracers
and the magnification. Here, we have presented a unified relativistic analysis
of ``standard rulers'', where a standard ruler simply means there is
an underlying physical scale which we compare the observations to.
This treatment applies to lensing measurements through galaxy ellipticities,
sizes and fluxes, or through standard candles, to distortions of
cosmologial correlation functions,
and to lensing of diffuse backgrounds.
We show that in this framework, for ideal measurements, one can
measure six degrees of freedom: a scalar corresponding to purely
line-of-sight effects; a vector (on the sphere) which corresponds
to mixed tranverse/line-of-sight effects; and a symmetric transverse
tensor on the sphere which comprises the shear and magnification.
We obtain general, gauge-invariant expressions for the six observable
degrees of freedom, valid on the full sky. These constitute the main
result of the paper and are given in \refeq{C}, (\ref{eq:Bpm}),
(\ref{eq:mag}), and (\ref{eq:shear2}).
The vector component and the shear admit a decomposition into
$E/B$-modes. The $B$-modes are free of all scalar contributions
(including lensing as well as redshift-space distortions) at the
linear level, making them ideal probes to look for tensor perturbations.
As an application of our results, we study the shear induced by
tensor modes (gravitational waves) in \cite{paperII}.
The logical next step is to construct estimators for these degrees of
freedom, based on measurements of the density field of tracers
(such as galaxies, the Lyman-$\alpha$ forest, 21cm emission, and so on).
We will leave this for future work.
\acknowledgments
We would like to thank Yanbei Chen, Scott Dodelson, Olivier Dor\'e,
Sam Gralla,
Chris Hirata, Wayne Hu, Bhuvnesh Jain, Marc Kamionkowski, Eiichiro Komatsu,
David Nichols, Samaya Nissanke, Enrico Pajer, and Matias Zaldarriaga for helpful discussions.
FS thanks Masahiro Takada and the Kavli-IPMU, University of Tokyo,
for hospitality.
FS is supported by the Gordon and Betty Moore Foundation at Caltech.
\begin{widetext}
|
2,877,628,090,177 | arxiv | \section{Introduction}
\label{sec:introduction}
Most of the light mesons listed in the Particle Data Group (PDG) \cite{pdg} can be understood as quark-antiquark ($\bar{q}q$) states \cite{isgur}, see also e.g. the review in\ Ref. \cite{klemptrev}. Quark-antiquark mesons, also denoted as conventional mesons, are grouped into nonets. This is a consequence of flavor symmetry, $U_{V}(3)$: in the limit of equal masses of the light quark flavors $u$, $d$, and $s$, the QCD Lagrangian is invariant under their interchange. In reality, this symmetry is explicitly broken by non-equal bare quark masses, mostly due to the fact that the strange quark $s$ is sizable heavier than the up and down quarks $u$ and $d$ \cite{pdg}.\\
In the limit of vanishing light quark masses, the QCD Lagrangian exhibits also chiral symmetry, $U_{R}(3)\times U_{L}(3)$. Quark-antiquark nonets with equal total angular momentum $J$ but with opposite parity $P$ are connected by chiral transformations. For instance, scalar ($1^3 P_0$ , $J^{PC}=0^{++}$) and pseudoscalar mesons ($1^1 S_0$ , $J^{PC}=0^{-+}$) as well as vector
($1^3 S_1$ , $J^{PC}=1^{--}$) and axial-vector mesons ($1^3 P_1$ , $J^{PC}=1^{++}$) are chiral partners, e.g. Ref. \cite{dick}. In addition to explicit breaking, chiral symmetry is -- even more importantly -- also broken spontaneously, $U_{R}(3) \times U_{L}(3) \rightarrow U_{V}(3)$: pseudoscalar mesons (e.g., the pions) are the corresponding quasi-Goldstone bosons.\\
Tensor mesons ($1^3 P_{2}$ , $J^{PC}=2^{++}$) are another example of a very well-understood $\bar{q}q$ nonet: their decays fit nicely into this scheme \cite{isgur,burakovsky,tensor,ciricgliano}. The chiral partners of tensor mesons, the pseudotensor mesons ($1^1 D_{2}$, $J^{PC}=2^{-+}$), are not so well understood, e.g. Refs. \cite{wang,bing} and Refs. therein. The standard assignment \cite{pdg,isgur} contains the isotriplet state $\pi_{2}(1670),$ the isodoublet states $K_{2}(1770),$ and the isoscalar states $\eta_2(1645)$ and $\eta_2(1870)$. We plan to study the decays of these resonances in order to test the validity of this assignment and to investigate the mixing in the isoscalar sector. To this end, we build two effective interaction Lagrangians which describe the decays of pseudotensor states into vector-pseudoscalar and into tensor-pseudoscalar pairs. The isotriplet and isodoublet states $\pi_{2}(1670)$ and $K_{2}(1770)$ fit well into the $\bar{q}q$ picture. However, in the isoscalar sector the situation is not that simple: the only possible way to understand the experimental results of $\eta_2(1645)$ and $\eta_2(1870)$ listed in the PDG is a very large mixing in the isoscalar-pseudotensor sector. In fact, the mixing angle between the purely nonstrange and strange states turns out to be about $-42^{\circ}$ in our model, which is very similar to the one in the pseudoscalar sector (leading to the famous mixing among $\eta$ and $\eta^{\prime}(958)$, e.g. Ref. \cite{AmelinoCamelia:2010,feldmann,thooftanomaly}.) However, while the large pseudoscalar-isoscalar mixing is related to the chiral anomaly, such an analogous strong mixing in the pseudotensor sector, if confirmed, would require a careful analysis in order to be understood.\\
In addition to the mixing angle, in the isoscalar sector there are some conflicting experimental informations about the ratio $[\eta_{2}(1870) \rightarrow a_{2}(1320)\,\pi]/[\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta]$ that we will discuss. From an experimental point of view it is expected that the GlueX \cite{gluex1,gluex2,gluex3} and CLAS12 \cite{clas12} experiment at Jefferson Lab will soon start taking data (photoproduction of mesons by photon-nucleon scattering) and can possibly shed light on pseudotensor states in general, with special attention to the $\eta_2(1870)$ state. Indeed, one of the main motivations of the present work is to draw attention on the existing problems in understanding pseudotensor states. The hope is that new and better experimental data from Jlab will be available in the upcoming years.\\
Another interesting -- although yet only theoretical -- pseudotensor state is the pseudotensor glueball. According to lattice QCD its mass is supposed to be about $3$ GeV \cite{mainlattice,gregory}. A simple modification of our interaction Lagrangians allows to make some predictions concerning this putative state. This might help to identify possible glueball candidates.\\
The article is organized as follows: in Sec.\ \ref{sec:two} we discuss $\bar{q}q$ nonets, their microscopic currents and quantum numbers; then, we introduce the two effective Lagrangians mentioned above and the corresponding decay widths. In\ Sec.\ \ref{sec:results} we present the results for the decays of isotriplet and isodoublet members of the pseudoscalar nonet. In\ Sec.\ \ref{sec:isoscalarmixing} we turn to the isoscalar sector: after showing that a small mixing angle cannot be correct, we show that a large negative mixing angle allows to understand all the existing data. Within this context, we describe also some puzzling entries in the PDG.\ In Sec.\ \ref{sec:glueball} we concentrate on the decays of a (yet hypothetical) pseudoscalar glueball. Finally, in\ Sec.\ \ref{sec:conclusions} we turn to the conclusions and outlooks.
\section{The model}
\label{sec:two}
In this Section we describe the model for the decays of pseudotensor mesons. First, we introduce the relevant quark-antiquark nonets and provide a brief repetition of some aspects of the quark-antiquark assignments. In the second step, the effective Lagrangians for the present model are presented.
\subsection{Quark-antiquark nonets}
The underlying structure of conventional meson fields is described via $\bar{q}q$ currents. First, we briefly review their quantum numbers and illustrate their construction scheme.\\
The total spin of a $\bar{q}q$ bound state can be either $S=0$ or $S=1$, whereas the angular momentum is not limited: $L = 0\, \text{(ground state)},\, 1 ,\, 2 ,\, 3 ,\, \ldots\,$. The total angular momentum $\vec{J}=\vec{L}+\vec{S}$ is restricted by $\left\vert L-S\right\vert\leq J\leq L+S$. Parity and charge conjugation quantum numbers are given by
\begin{align}
P & =(-1)^{L+1}\,, && C = (-1)^{L+S} \,,
\end{align}
(where -- strictly speaking -- only diagonal members of a given multiplet are $C$-eigenstates). As usual, mesons are grouped into nonets (see below) corresponding to a well-defined combination $J^{PC}$ \cite{pdg}. In Tab.\ (\ref{tab:quarkcurrents}) we present various combinations $J^{PC}$ together with the values of $L,$ $S,$ the old spectroscopic notation $n$ $^{2S+1}L_{J},$ and the corresponding microscopic currents generating them (here, the quark fields reads $q_{i}$, where $i \in \{u,\,d,\,s\}$). We recall that $n$ is the radial quantum number ($n=1$ for all the cases under considerations), $L = S ,\, P ,\, D ,\,, \ldots\,$, and $\overleftrightarrow{\partial}_{\mu}=\overrightarrow{\partial}_{\mu}-\overleftarrow{\partial}_{\mu}$.
\begin{table}[H]
\centering\renewcommand{\arraystretch}{2.0}
\begin{tabular}[c]{|c|cc|c|c||c|}
\hline
Meson & $n^{2S+1}L_J$ & $J^{PC}$ & $S$ & $L$ & Hermitian quark current operators \\
\hline
pseudoscalar & $1^1 S_0$ & $0^{-+}$ & $0$ & \multirow{2}{*}{$0$} & $P_{ij} = \bar{q}_j \, i \gamma^{5} \, q_i$ \\
\cline{1-4} \cline{6-6}
vector & $1^3 S_1$ & $1^{--}$ & $1$ & & $V_{ij}^{\mu} = \bar{q}_j \, \gamma^{\mu} \, q_i$ \\
\hline \hline
pseudovector & $1^1 P_1$ & $1^{+-}$ & $0$ & \multirow{4}{*}{$1$} & $P_{ij}^{\mu} = \bar{q}_j \, \gamma^{5} \overleftrightarrow{\partial}^{\mu} \, q_i$ \\
\cline{1-4} \cline{6-6}
scalar & $1^3 P_0$ & $0^{++}$ & $1$ & & $S_{ij} = \bar{q}_j \, q_i$ \\
\cline{1-4} \cline{6-6}
axial vector & $1^3 P_1$ & $1^{++}$ & $1$ & & $A_{ij}^{\mu} = \bar{q}_j \, \gamma^{5} \gamma^{\mu} \, q_i$ \\
\cline{1-4} \cline{6-6}
tensor & $1^3 P_2$ & $2^{++}$ & $1$ & & $X_{ij}^{\mu\nu} = \bar{q}_j \, i \Big[ \gamma^{\mu} \overleftrightarrow{\partial}^{\nu} + \gamma^{\nu} \overleftrightarrow{\partial}^{\mu} - \frac{2}{3}\, \tilde{G}^{\mu\nu} \overleftrightarrow{\slashed{\partial}} \Big] \, q_i$ \\
\hline \hline
pseudotensor & $1^1 D_2$ & $2^{-+}$ & $0$ & \multirow{4}{*}{$2$} & $T_{ij}^{\mu\nu} = \bar{q}_j \, i \Big[ \gamma^{5} \overleftrightarrow{\partial}^{\mu} \overleftrightarrow{\partial}^{\nu} - \frac{2}{3}\, \tilde{G}^{\mu\nu} \overleftrightarrow{\partial}_{\alpha} \overleftrightarrow{\partial}^{\alpha} \Big] \, q_i$ \\
\cline{1-4} \cline{6-6}
excited vector & $1^3 D_1$ & $1^{--}$ & $1$ & & $S_{ij}^{\mu} = \bar{q}_j \, \overleftrightarrow{\partial}^{\mu} \, q_i$ \\
\cline{1-4} \cline{6-6}
axial tensor & $1^3 D_2$ & $2^{--}$ & $1$ & & $B_{ij}^{\mu\nu} = \bar{q}_j \, i \Big[ \gamma^{5} \gamma^{\mu} \overleftrightarrow{\partial}^{\nu} + \gamma^{5} \gamma^{\nu} \overleftrightarrow{\partial}^{\mu} - \frac{2}{3}\, \tilde{G}^{\mu\nu} \gamma^{5} \overleftrightarrow{\slashed{\partial}} \Big] \, q_i$ \\
\cline{1-4} \cline{6-6}
spin-3 tensor & $1^3 D_3$ & $3^{--}$ & $1$ & & \ldots \\
\hline
\end{tabular}
\caption{Types of different mesons and their corresponding quantum numbers. We use the projector $\tilde{G}^{\mu\nu} = \eta^{\mu\nu} - \frac{k^\mu k^\nu}{k^2}$.}
\label{tab:quarkcurrents}
\end{table}
We now introduce the matrices of the nonets relevant in the present work. The nonet of pseudoscalar mesons is given by
\begin{align}
P & =
\begin{pmatrix}
\frac{\eta_{N} + \pi^{0}}{\sqrt{2}} & \pi^{+} & K^{+} \\
\pi^{-} & \frac{\eta_{N} - \pi^{0}}{\sqrt{2}} & K^{0} \\
K^{-} & \bar{K}^{0} & \eta_{S}
\end{pmatrix} \,, \label{eq:pseudoscalar_nonet}
\end{align}
where $\eta_{N}=\sqrt{1/2}\,(\bar{u}u+\bar{d}d)$ is the pure non-strange state and $\eta_{S}=\bar{s}s$ the pure strange state. The identifications of the fields with physical resonances are listed in Tab.\ (\ref{tab:resonances_states}), while the transformation properties under parity, charge conjugation, and flavor transformations are reported in Tab.\ (\ref{tab:transformations}). The isoscalar mixing angle $\beta_{p} = -40.5^{\circ}$ in the non-strange-strange pseudoscalar sector is taken from \cite{AmelinoCamelia:2010}. The large maxing is linked to the chiral anomaly, see e.g. Refs. \cite{klemptrev,feldmann,dick}.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}[c]{|l|l|}
\hline
Physical resonances & Nonet $q\bar{q}$-states \\
\hline \hline
$\pi$ & $\hphantom{-}\pi$ \\
$K$ & $\hphantom{-}K$ \\
$\eta$ & $\hphantom{-}\eta_{N} \cos\beta_{p} + \eta_{S} \sin\beta_{p}$ \\
$\eta^{\prime}(958)$ & $- \eta_{N} \sin\beta_{p} + \eta_{S} \cos\beta_{p}$ \\
\hline
$\rho(770)$ & $\hphantom{-}\rho$ \\
$K^{*}(892)$ & $\hphantom{-}K^{*}$ \\
$\omega(782)$ & $\hphantom{-}\omega_{N} \cos\beta_{v} + \omega_{S} \sin\beta_{v}$ \\
$\phi(1020)$ & $-\omega_{N} \sin\beta_{v} + \omega_{S} \cos\beta_{v}$ \\
\hline
$a_{2}(1320)$ & $\hphantom{-}a_{2}$ \\
$K_{2}^{*}(1430)$ & $\hphantom{-}K_{2}^{*}$ \\
$f_{2}(1270)$ & $\hphantom{-} f_{2,N} \cos\beta_{t} + f_{2,S} \sin\beta_{t}$ \\
$f^{\prime}_{2}(1525)$ & $- f_{2,N} \sin\beta_{t} + f_{2,S} \cos\beta_{t}$ \\
\hline
$\pi_{2}(1670)$ & $\hphantom{-} \pi_{2}$ \\
$K_{2}(1770)$ & $\hphantom{-}K_{2}$ \\
$\eta_{2}(1645)$ & $\hphantom{-} \eta_{2,N} \cos\beta_{pt} + \eta_{2,S} \sin\beta_{pt}$ \\
$\eta_{2}(1870)$ & $- \eta_{2,N} \sin\beta_{pt} + \eta_{2,S} \cos\beta_{pt}$ \\
\hline
\end{tabular}
\caption{Assignment of physical resonances to $\bar{q}q$-states in the model.}
\label{tab:resonances_states}
\end{table}
Similarly, vector mesons are arranged in the vector nonet given by
\begin{align}
V^{\mu} & =
\begin{pmatrix}
\frac{\omega_{N}^{\mu} + \rho^{0\mu}}{\sqrt{2}} & \rho^{+\mu} & K^{\ast+\mu} \\
\rho^{-\mu} & \frac{\omega_{N}^{\mu} - \rho^{0\mu}}{\sqrt{2}} & K^{\ast0\mu} \\
K^{\ast-\mu} & \bar{K}^{\ast0\mu} & \omega_{S}^{\mu}
\end{pmatrix} \,, \label{eq:vector_nonet}
\end{align}
where $\omega_{N}$ is purely non-strange and $\omega_{S}$ purely strange. Identifications and transformation properties are provided by Tab.\ (\ref{tab:resonances_states}) and Tab.\ (\ref{tab:transformations}). The isoscalar-vector mixing angle is very small: $\beta_{v}=-3.8^{\circ}$ \cite{pdg}. The physical states $\omega$ and $\phi$ are dominated by non-strange and strange components, respectively.\\
We now turn to tensor and pseudotensor states (for a recent review on mathematical aspects of tensor fields, see Ref. \cite{Koenigstein:2015} and references therein). The tensor meson nonet reads
\begin{align}
X^{\mu\nu} & =
\begin{pmatrix}
\frac{f_{2,N}^{\mu\nu} + a_{2}^{0\mu\nu}}{\sqrt{2}} & a_{2}^{+\mu\nu} & K_{2}^{\ast+\mu\nu} \\
a_{2}^{-\mu\nu} & \frac{f_{2,N}^{\mu\nu} - a_{2}^{0\mu\nu}}{\sqrt{2}} & K_{2}^{\ast0\mu\nu} \\
K_{2}^{\ast-\mu\nu} & \bar{K}_{2}^{\ast0\mu\nu} & f_{2,S}^{\mu\nu}
\end{pmatrix} \,. \label{eq:tensor_nonet}
\end{align}
In analogy to the vector case, the mixing angle between the pure non-strange $f_{2,N}$ and the pure strange $f_{2,S}$ is small: $\beta_{t}=3.2^{\circ}$, see Ref. \cite{pdg}. Finally, the nonet of pseudotensor states -- which constitutes the main subject of the present work -- is given by
\begin{align}
T^{\mu\nu} & =
\begin{pmatrix}
\frac{\eta_{2,N}^{\mu\nu} + \pi_{2}^{0\mu\nu}}{\sqrt{2}} & \pi_{2}^{+\mu\nu} & K_{2}^{+\mu\nu} \\
\pi_{2}^{-\mu\nu} & \frac{\eta_{2,N}^{\mu\nu} - \pi_{2}^{0\mu\nu}}{\sqrt{2}} & K_{2}^{0\mu\nu} \\
K_{2}^{-\mu\nu} & \bar{K}_{2}^{0\mu\nu} & \eta_{2,S}^{\mu\nu}
\end{pmatrix} \,, \label{eq:pseudotensor_nonet}
\end{align}
see again Tab.\ (\ref{tab:resonances_states}) and Tab.\ (\ref{tab:transformations}) for the physical content and transformations of (pseudo)tensor mesons.\\
The mixing angle $\beta_{pt}$ is -- at first -- unknown. Naively, one would expect a small mixing angle (similarly to the vector and the tensor mesons; alternatively, one may use the Okubo formula, which yields $\beta_{pt} \approx 14.8^{\circ}$ derived from (15.9) in \cite{pdg}). However, a small mixing angle does not lead to acceptable results within our model. A large and negative mixing angle -- as in the pseudoscalar sector -- is needed, see Sec.\ \ref{sec:isoscalarmixing}.\\
\\
Other mesonic nonets can be constructed in the same way, see for instance Refs. \cite{dick,florianlisa,julia}. They are omitted here, since they do not enter the decays under consideration.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}[c]{|c|c|c|c|}
\hline
Nonet & Parity $(P)$ & Charge congugation $(C)$ & Flavour $(U_{V}(3))$ \\
\hline \hline
$P$ & $-P(t,-\vec{x})$ & $P^{t}$ & $U P U^{\dagger}$ \\
\hline
$V^{\mu}$ & $V_{\mu}(t,-\vec{x})$ & $-(V^{\mu})^{t}$ & $U V^{\mu}U^{\dagger}$ \\
\hline
$X^{\mu\nu}$ & $X_{\mu\nu}(t,-\vec{x})$ & $(X^{\mu\nu})^{t}$ & $U X^{\mu\nu} U^{\dagger}$ \\
\hline
$T^{\mu\nu}$ & $-T_{\mu\nu}(t,-\vec{x})$ & $(T^{\mu\nu})^{t}$ & $U T^{\mu\nu} U^{\dagger}$ \\
\hline
\end{tabular}
\caption{Transformation properties of the pseudoscalar (\ref{eq:pseudoscalar_nonet}), the vector (\ref{eq:vector_nonet}), the tensor (\ref{eq:tensor_nonet}), and pseudotensor (\ref{eq:pseudotensor_nonet}) nonets under charge conjugation $C$, parity $P,$ and flavour transformations $U_{V}(3)$. Notice the position of the Lorentz indices for parity transformations, since spatial and time-like indices do not transform identically.}
\label{tab:transformations}
\end{table}
\subsection{The interaction Lagrangians and tree-level decay width}
\label{sec:themodel}
Using the nonets (\ref{eq:pseudoscalar_nonet}) - (\ref{eq:pseudotensor_nonet}) introduced in the previous subsection, we construct two effective interaction Lagrangians which describe the decays of pseudotensor mesons.\\
The first Lagrangian $\mathcal{L}_{TVP}$ includes the coupling of pseudotensor mesons to vector-pseudoscalar pairs,
\begin{align}
\mathcal{L}_{TVP} & = c_{TVP}\, \mathrm{Tr} \big\{ T_{\mu\nu} \big[ V^{\mu} ,\, (\partial^{\nu}P) \big] _{-} \big\} \,, \label{eq:lagtvp}
\end{align}
where $c_{TVP}$ denotes an (at first) unknown dimensionless coupling constant and $[\,,]_{-}$ is the commutator.\\
The second interaction Lagrangian $\mathcal{L}_{TXP}$ contains the coupling of pseudotensor mesons to tensor-pseudoscalar pairs,
\begin{align}
\mathcal{L}_{TXP} & = c_{TXP}\, \mathrm{Tr} \big( T_{\mu\nu} \big\{ X^{\mu\nu} ,\, P \big\}_{+} \big) \,, \label{eq:lagtxp}
\end{align}
where $c_{TXP}$ is an (at first) unknown coupling constant with dimension energy and $\{\,,\}_{+}$ is the anti-commutator.\\
Both Lagrangians are invariant under $CPT$- , Poincar\'{e}- and flavour-transformations listed in Tab.\ (\ref{tab:transformations}). The explicit form of the Lagrangians are presented in App.\ \ref{app:lagrangians}. Hence, the total Lagrangian, which specifies our model, is given by
\begin{align}
\mathcal{L}_{T-\text{total}} & = \mathcal{L}_{\text{kin}} + \mathcal{L}_{TVP} + \mathcal{L}_{TXP} \,, \label{eq:ltot}
\end{align}
where $\mathcal{L}_{\text{kin}}=\frac{1}{2} \mathrm{Tr}\big[(\partial_{\mu}P)(\partial^{\mu}P)\big] + \ldots$ contains the usual kinetic terms of all relevant nonets (for the tensor fields, see again Ref. \cite{Koenigstein:2015}), while the interaction term is the sum of the two interaction Lagrangians presented above.\\
The interaction Lagrangians outlined above can be considered as the dominant terms in the large-$N_{c}$ and flavour breaking expansions. For instance, the full Lagrangian for the interaction of the pseudotensor field with tensor and pseudoscalar field takes the form
\begin{align}
\mathcal{L}_{TXP}^{full} & = c_{TXP}\, \mathrm{Tr} \big(T_{\mu\nu} \big\{ X^{\mu\nu} ,\,P \big \}_{+} \big) + c_{TXP}\, \mathrm{Tr} \big( \hat{\delta}\, T_{\mu\nu} \big\{ X^{\mu\nu},\,P \big\}_{+} \big) + \label{ltxpfull} \vphantom{\frac{1}{2}}
\\
& \quad\, + \tilde{c}_{TXP}\, \mathrm{Tr} \big( T_{\mu\nu} \big)\, \mathrm{Tr} \big( \big\{ X^{\mu\nu},\,P\big \}_{+} \big) + \tilde{c}_{TXP}\, \mathrm{Tr} \big( \hat{\delta}\, T_{\mu\nu} \big)\, \mathrm{Tr} \big( \big\{X^{\mu\nu},\,P\big\}_{+} \big) + \ldots \vphantom{\frac{1}{2}} \nonumber
\end{align}
where the matrix $\hat{\delta} = \mathrm{diag} ( 0,\, \delta_{d},\,\delta_{s} )$ describes isospin violation. Following \cite{Amsler:1995td} and denoting $V$ as the potential responsible of the creation of a quark-antiquark pair from the QCD vacuum, we find,
\begin{align*}
& \delta_{d} =\frac{ \langle 0 | V | \bar{d} d \rangle }{\langle 0 | V | \bar{u} u \rangle } - 1 \simeq 0 &&\text{(isospin violation can be neglected to a very good level of accuracy),}
\\
& \delta_{s} = \frac{ \langle 0 | V | \bar{s}s \rangle }{\langle 0 | V | \bar{u}u \rangle } - 1 \lesssim 0.2 && \text{(small but eventually important for a precise description).}
\end{align*}
A similar value for $\delta_{s}$ has been obtained in Refs.\ \cite{Giacosa:2004ug,isgur}. Moreover, according to Refs.\ \cite{largen1,largen2,largen3} we expect
\begin{align}
c_{TXP} & \propto N_{c}^{-1/2} \,,
\\
\tilde{c}_{TXP} & \propto N_{c}^{-3/2} \,.
\end{align}
The first term in Eq.\ (\ref{ltxpfull}) is the dominant one: due to the present experimental status this is the only term in the pseudeotensor-tensor-pseudoscalar sector which is used in the numerical calculations of the present work. Then, the second and the third term of Eq.\ (\ref{ltxpfull}) generate decay amplitudes proportional to $c_{TXP}\,\delta_{s}$ and $\tilde{c}_{TXP}$ respectively, hence they generate corrections of the order of 5-10\%, which should be taken into account when better experimental data will be available.\ Further terms such as the fourth term generate a contribution of the type $\tilde{c}_{TXP}\, \delta_{s},$ hence they are suppressed twice, because of large-$N_{c}$ and flavour symmetry violation. Their influence is expected to be about $1\%$, hence negligible.\\
For what concerns the interaction term of pseudotensor mesons with pseudoscalar and vector fields the expansion is similar:
\begin{align}
\mathcal{L}_{TVP}^{full} & = c_{TVP}\, \mathrm{Tr} \big\{ T_{\mu\nu} \big[V^{\mu},\,(\partial^{\nu}P)\big]_{-} \big\} + c_{TVP}\, \mathrm{Tr} \big\{\hat{\delta}\, T_{\mu\nu} \big[V^{\mu},\,(\partial^{\nu}P)\big]_{-} \big\} \vphantom{\frac{1}{2}} \label{ltvpfull}
\\
& \quad\, + \tilde{c}_{TVP}\, \mathrm{Tr} \big\{ \hat{\delta}\, T_{\mu\nu}\big\}\, \mathrm{Tr} \big\{ \hat{\delta}\, \big[V^{\mu},\,(\partial^{\nu}P)\big]_{-} \big\} + \ldots \vphantom{\frac{1}{2}} \nonumber
\end{align}
As before, the first term dominates and the only term of Eq.\ (\ref{ltvpfull}) considered in this paper. There is however a difference between Eq.\ (\ref{ltxpfull}) and Eq.\ (\ref{ltvpfull}): due to the anti-commutator in the latter, some terms do not appear in the expansion. The next-to-leading contribution is the second term $\propto c_{TVP}\, \delta_{s}$ (breaking of flavour symmetry). Furthermore, one has terms $\propto \tilde{c}_{TVP}\,\delta_{s}$ such as the third one which suppressed in $N_{c}$ and the flavour expansions.\\
\\
We now turn to the tree-level decay widths. The tree-level decay widths derived from the two Lagrangians $\mathcal{L}_{TVP}$ and $\mathcal{L}_{TXP}$ read
\begin{align}
\Gamma_{T\rightarrow VP}^{tl}(m_{T},\,m_{V},\,m_{P}) & = \frac{k_f}{8\pi\,m_{T}} \frac{g_{TVP}^{2}}{15} \Big( 2\, \frac{k_f^{4}}{m_{V}^{2}} + 5\, k_f^2 \Big)\, \Theta( m_{T}-m_{V}-m_{P} ) \,, \label{eq:tvp}
\end{align}
and
\begin{align}
\Gamma_{T\rightarrow XP}^{tl}(m_{T},\,m_{X},\,m_{P}) & = \frac{k_f}{8\pi\,m_{T}} \frac{g_{TXP}^{2}}{45} \Big( 4\, \frac{k_f^{4}}{m_{X}^{4}} + 30\,\frac{k_f^{2}}{m_{X}^{2}} + 45 \Big)\, \Theta( m_{T}-m_{X}-m_{P} ) \,. \label{eq:txp}
\end{align}
Above, $m_{T}$ is the mass of a (decaying) pseudotensor fields, while $m_{V}$, $m_{P},$ and $m_{X}$ are the masses of the vector, pseudoscalar, and tensor states (the decay products). $\Theta(x)$ denotes the Heaviside step-function. The quantities $g_{TVP}$ and $g_{TXP}$, which are proportional to the coupling constants $c_{TVP}$ and $c_{TXP}$, have to be calculated for each decay channel separately by using the expressions in App.\ \ref{app:lagrangians}. Finally, the function $k_f \equiv k_{f}(m_{T},\,m_{V},\,m_{P})$ is the modulus of the momentum $\vec{k}_f$ of one of the outgoing particles. Its analytic expression in Eq.\ (\ref{eq:tvp}) reads:
\begin{align}
k_{f} ( m_{T} ,\, m_{V} ,\, m_{P} ) & = \frac{1}{2\,m_{T}} \sqrt{ m_{T}^{4} + ( m_{V}^{2} - m_{P}^{2} )^{2} - 2\, m_{T}^{2}\, ( m_{V}^{2} + m_{P}^{2} ) } \,, \label{eq:kf}
\end{align}
while its expression in Eq.\ (\ref{eq:txp}) is obtained by substituting $m_{V}\rightarrow m_{X}.$\\
These decay widths (\ref{eq:tvp}) and (\ref{eq:txp}) are derived via Feynman rules under the use of the polarization vectors (tensors) and their corresponding completeness relations. For a derivation of the unpolarized invariant decay amplitudes see App.\ \ref{app:deacyamplitudes}. \newline Calculations are performed at tree-level, that is NLO effects due to quantum loops are not considered here. Loops are expected to be small when the ratio $\Gamma/(M-M_{th})$ is sufficiently small ($\leq0.2$, where $\Gamma$ and $M$ are the decay width and the mass of the decaying state and $M_{th}$ is the lowest threshold \cite{lupo}). This condition is fulfilled for the pseudotensor mesons under study.
\section{Results for $\pi_{2}(1670)$ and $K_{2}(1770)$}
\label{sec:results}
As a first step, we determine the coupling constants $c_{TVP}$ and $c_{TXP}$ using two -- well known -- experimental decay widths,
\begin{align}
\pi_{2}(1670) & \rightarrow \rho(770)\,\pi \,, && \text{and} && \pi_{2}(1670) \rightarrow f_{2}(1270)\,\pi \,.
\end{align}
Taking into account that for these two decays one has $g_{TVP}^{2} = 2\,(\sqrt{2}\,c_{TVP})^{2}$ and $g_{TXP}^{2}=2\,\cos\beta_{t}\,c_{TXP}^{2}$, the use of Eqs.\ (\ref{eq:tvp}), (\ref{eq:txp}), and (\ref{eq:tvplagexp}) implies that [see Tab.\ (\ref{tab:ergebnisohneisoscalar})],
\begin{align}
c_{TVP}^{2} & = 11.9 \pm 1.6 \,, && c_{TXP}^{2} = (15.1 \pm 1.0) \cdot 10^{6} \, \, \text{MeV}^{2} \,. \label{eq:cs}
\end{align}
The quoted errors are solely determined by the experimental errors on the decay widths (mass errors are small and negligible for our accuracy).\\
Once the coupling constants are known, all the isotriplet and isodoublet decays of pseudotensor states are uniquely fixed. The results are shown in Tab.\ (\ref{tab:ergebnisohneisoscalar}).
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}[c]{|l|c|c|}
\hline
Decay process & Theory (MeV) & Experiment (MeV) \\
\hline \hline
$\pi_{2}(1670) \rightarrow\rho(770)\,\pi$ & $80.6 \pm 10.8$ & $80.6 \pm 10.8$ \\
\hline
$\pi_{2}(1670) \rightarrow f_{2}(1270) \,\pi$ & $146.4 \pm 9.7$ & $146.4 \pm 9.7$ \\
\hline \hline
$\pi_{2}(1670) \rightarrow\bar{K}^{*}(892)\,K + c.c. $ & $11.7 \pm 1.6$ & $10.9 \pm 3.7$ \\
\hline
$\pi_{2}(1670) \rightarrow\bar{K}_{2}^{*}(1430)\,K + c.c. $ & $0$ & \\
\hline
$\pi_{2}(1670) \rightarrow f^{\prime}_{2}(1525) \,\pi$ & $0.1 \pm0.1$ & \\
\hline
$\pi_{2}(1670) \rightarrow a_{2}(1320) \,\pi$ & $0$ & not seen \\
\hline
$\pi_{2}(1670) \rightarrow a_{2}(1320) \,\eta$ & $0$ & \\
\hline
$\pi_{2}(1670) \rightarrow a_{2}(1320) \,\eta^{\prime}(958) $ & $0$ & \\
\hline \hline
$K_{2}(1770) \rightarrow\rho(770)\,K $ & $22.2 \pm3.0$ & \\
\hline
$K_{2}(1770) \rightarrow\bar{K}^{*}(892)\,\pi$ & $25.5 \pm3.4$ & seen \\
\hline
$K_{2}(1770) \rightarrow\bar{K}^{*}(892)\,\eta$ & $10.5 \pm1.4$ & \\
\hline
$K_{2}(1770) \rightarrow\bar{K}^{*}(892)\,\eta^{\prime}(958) $ & $0$ & \\
\hline
$K_{2}(1770) \rightarrow\omega(782)\,K$ & $8.3 \pm1.1$ & seen \\
\hline
$K_{2}(1770) \rightarrow\phi(1020)\,K$ & $4.2 \pm0.6$ & seen \\
\hline
$K_{2}(1770) \rightarrow a_{2}(1320)\,K$ & $0$ & \\
\hline
$K_{2}(1770) \rightarrow\bar{K}_{2}^{*}(1430)\,\pi$ & $84.5 \pm5.6$ & dominant \\
\hline
$K_{2}(1770) \rightarrow\bar{K}_{2}^{*}(1430)\,\eta$ & $0$ & \\
\hline
$K_{2}(1770) \rightarrow\bar{K}_{2}^{*}(1430)\,\eta^{\prime}(958)$ & $0$ & \\
\hline
$K_{2}(1770) \rightarrow f_{2}(1270)\,K$ & $5.8 \pm0.4$ & seen \\
\hline
$K_{2}(1770) \rightarrow f^{\prime}_{2}(1525)\,K$ & $0$ & \\
\hline
\end{tabular}
\caption{Decays of $I=1$ and $I=1/2$ pseudotensor states. The first two
entries were used to determine the coupling constants of the model, see Eq.\ (\ref{eq:cs}). The total decay widths are
$\Gamma^{\text{tot}}_{\pi_{2}(1670)} = (260 \pm9)\,\,\text{MeV}$ and $\Gamma^{\text{tot}}_{K_{2}(1770)} = (186 \pm 14) \,\, \text{MeV}$.}
\label{tab:ergebnisohneisoscalar}
\end{table}
Following comments are in order:
\begin{enumerate}
\item We recall that the first two entries of Tab.\ (\ref{tab:ergebnisohneisoscalar}) were used to calculate the coupling constants of the model, see Eq.\ (\ref{eq:cs}). These values are extracted from the PDG using the branching ratio quoted by the PDG: $56.3 \pm 3.2 \%$ for $\pi_{2}(1670) \rightarrow f_{2}(1270)\,\pi$ and $31 \pm 4 \%$ for $\pi_{2}(1670) \rightarrow \rho\,\pi.$ However, a closer inspection of the performed experiments on the resonance $\pi_{2}(1670)$ shows that new experiments are needed to improve experimental knowledge. For instance, the ratio $\Gamma(\rho\,\pi) / [0.565\, \Gamma(f_{2}(1270)\,\pi)]$, whose PDG quoted average is $0.97\pm0.09,$ was determined only by two distinct results: $0.76\pm0.07\pm0.10$ by \cite{Chung:2002pu} and $1.01\pm0.05$ by \cite{Barberis:1998in}. While consistent among each other, a new experimental determination would be very welcome for our theoretical approach, since the determination of the coupling constant follows from these experimental values.
\item The prediction for the decay channel $\pi_{2}(1670) \rightarrow K^{\ast}(892)\,K$ is in agreement with the present value quoted by the PDG. However, it should be stressed that this value is extracted from a single experiment \cite{Armstrong:1981kc}: a new experimental determination of this quantity is then compelling for a better comparison.
\item In our model $\pi_{2}(1670)$ does not couple to the $a_{2}(1320)\,\pi$ channel, see Eq.\ (\ref{eq:txplagexp}). Experimentally this decay could also be excluded to a good accuracy (for more experimental details see Ref.\ \cite{kuhn}).
\item The experimental total decay width of $\pi_{2}(1670)$ reads $\Gamma_{\pi_{2}(1670)}^{\text{exp,tot}} = (260 \pm 9) \,\,$MeV. The value in our model is $\approx 238.8\,\,$MeV. The reason why the sum of the theoretical and experimental decay modes in our model is slightly smaller than the experimental total width is due to the fact that the model does not include the decay $\pi_{2}(1670)\rightarrow f_{0}(500)\,\pi,$ which is $\approx 26\,\,$MeV. (Other channels, which are not present in our model, would also contribute. Nevertheless they are negligibly small \cite{pdg}). Taking this into account, the model is consistent with the total width.
\item Experimentally, the decay channel $K_{2}(1770)\rightarrow\bar{K}_{2}^{\ast}(1430)\,\pi$ is dominant. The total decay width of $K_{2}(1770)$ is $\Gamma_{K_{2}(1770)}^{\text{exp,tot}}=(186\pm14)\,\,\text{MeV}$. Theoretically, the $K_{2}(1770)\rightarrow\bar{K}_{2}^{\ast}(1430)\,\pi$ decay mode is the dominant one ($84.5\pm5.6$ MeV).
\item The full theoretical decay width of $K_{2}(1770)$ amounts to $(162.0 \pm 15.4)\,\,$MeV which is compatible with the experimental value.
\item Various branching ratios of $K_{2}(1770)$ can be calculated from Tab.\ (\ref{tab:ergebnisohneisoscalar}). At present, experimental results are missing (no average or fit is quoted by PDG \cite{pdg}). In this respect, our approach makes predictions for new future experimental measurements. In this context, it will also be possible to determine the mixing angle of the bare $1^{1}D_{2}$ and $1^{3}D_{2}$ configurations into the physical $K_{2}(1770)$ and $K_{2}(1820)$ states [so far, $K_{2}(1770)$ is dominated by $1^{1}D_{2}$].
\item Most of the decays which are predicted by our model to vanish were consistently not seen in experiments. Yet, the decays $\pi_{2}(1670) \rightarrow a_{2}(1230)\,\eta$ and $K_{2}(1270) \rightarrow K^{\ast}(892)\,\eta$ are expected to be small but not zero. They were not yet measured, hence they represent a test of our approach as soon as new experimental data will be available.
\end{enumerate}
\section{The Isoscalar sector: $\eta_{2}(1645)$ and $\eta_{2}(1870)$}
\label{sec:isoscalarmixing}
In this section we present the results of the isoscalar sector of the pseudotensor mesons. In Subsec.\ \ref{subsec:small_strange_nonstrange_mixing} we show that a small mixing of $\eta_{2,N}$ and $\eta_{2,S}$ is not capable to describe the data. Thus, in Subec.\ \ref{subsec:large_strange_nonstrange_mixing} we allow for a large mixing angle: a value of about $-40^{\circ}$ turns out to be favored. In Subsec.\ \ref{subsec:branching} we discuss the experimental status of branching ratios with particular attention to $[\eta_{2}(1870) \rightarrow a_{2}(1320)\,\pi]/[\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta]$, for which conflicting experimental results exist.
\subsection{Small strange-nonstrange mixing angle}
\label{subsec:small_strange_nonstrange_mixing}
First, we present the results of the decay widths of the isoscalar $I=0$ pseudotensor states in Tab.\ (\ref{tab:ergebnisisoscalar}) for a vanishing mixing angle ($\beta_{pt}=0^{\circ}$) and for a small but non-vanishing angle ($\beta_{pt}=14.8^{\circ},$ obtained via the Okubo formula \cite{pdg}). Both cases lead to inconsistent results. In fact, the decay $\eta_{2}(1645)\rightarrow a_{2}(1320)\,\pi$ turns out to be larger than $300\,\,$MeV, which is not acceptable, since it is sizable larger than the total experimental width $\Gamma_{\eta_{2}(1645)}^{\text{exp,tot}}=(181\pm11)\,\,$MeV. These results hold for any small mixing angle ($\left\vert \beta_{pt}\right\vert \lesssim30^{\circ}$).
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}[c]{|l|c|c|c|}
\hline
Decay process & Theory (MeV) & Theory (MeV) & Experiment (MeV) \\
& $(\beta_{pt} = 14.8^{\circ})$ & $(\beta_{pt} = 0.0^{\circ})$ & \\
\hline \hline
$\eta_{2}(1645) \rightarrow\bar{K}^{*}(892)\,K + c.c. $ & $3.2 \pm0.4$ & $8.6 \pm 1.1$ & seen \\
\hline
$\eta_{2}(1645) \rightarrow a_{2}(1320)\,\pi$ & \textcolor{red}{$315.6 \pm 21.2$} & \textcolor{red}{$337.8 \pm 22.6$} & \\
\hline
$\eta_{2}(1645) \rightarrow\bar{K}_{2}^{*}(1430)\,K + c.c. $ & $0$ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f_{2}(1270)\,\eta$ & $0$ & $0$ & not seen \\
\hline
$\eta_{2}(1645) \rightarrow f_{2}(1270)\,\eta^{\prime}(958) $ & $0$ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f^{\prime}_{2}(1525)\,\eta$ & $0$ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f^{\prime}_{2}(1525)\,\eta^{\prime}(958) $ & $0$ & $0$ & \\
\hline \hline
$\eta_{2}(1870) \rightarrow\bar{K}^{*}(892)\,K + c.c. $ & $60.1 \pm8.0$ & $45.6 \pm6.1$ & \\
\hline
$\eta_{2}(1870) \rightarrow a_{2}(1320)\,\pi$ & \textcolor{red}{$32.2 \pm 2.1$} & \textcolor{red}{$0$} & \\
\hline
$\eta_{2}(1870) \rightarrow\bar{K}_{2}^{*}(1430)\,K + c.c. $ & $0$ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta$ & $2.5 \pm0.2$ & $0.1 \pm0.1$ & \\
\hline
$\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta^{\prime}(958) $ & $0$ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f^{\prime}_{2}(1525)\,\eta$ & $0$ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f^{\prime}_{2}(1525)\,\eta^{\prime}(958) $ & $0$ & $0$ & \\
\hline
\end{tabular}
\caption{Decays of $I=0$ pseudo-tensor states. The total decay widths are $\Gamma_{\eta_{2}(1645)}^{\text{tot}}=(181\pm11)\,\,\text{MeV}$ and $\Gamma_{\eta_{2}(1870)}^{\text{tot}}=(225\pm14)\,\,\text{MeV}$.}
\label{tab:ergebnisisoscalar}
\end{table}
In addition, the following comments are in order:
\begin{enumerate}
\item The PDG reports the following ratios for $\eta_{2}(1645)$:
\begin{align}
\Gamma^{\text{exp}} (\bar{K}\,K\,\pi)/\Gamma^{\text{exp}}(a_{2}(1320)\,\pi) & = 0.07 \pm 0.03 \,, \vphantom{\frac{1}{2}}
\\
\Gamma^{\text{exp}} (a_{2}(1320)\,\pi)/\Gamma^{\text{exp}}(a_{0}(980)\,\pi) & = 13.1 \pm 2.3 \,. \vphantom{\frac{1}{2}}
\end{align}
Thus, the channel $\eta_{2}(1645)\rightarrow a_{2}(1320)\,\pi$ is indeed expected to be dominant. This fact is well captured by the theoretical results of Tab.\ (\ref{tab:ergebnisisoscalar}), which however overshoot the data. Moreover, the experiment shows that $\Gamma^{\text{exp}}(\bar{K}\,K\,\pi) << \Gamma^{\text{exp}}(a_{2}(1320)\,\pi)$: this feature is in agreement with the experiment upon identifying $\bar{K}\,K\,\pi\approx\bar{K}^{\ast}(892)\,K+c.c.$.
\item For what concerns $\eta_{2}(1870)$, the PDG reports $\Gamma^{\text{exp}} (a_{2}(1320)\,\pi) /\Gamma^{\text{exp}}(f_{2}(1270)\,\eta) = 1.7\pm0.4,$ i.e. nonzero. This result excludes $\beta_{pt}=0.0^{\circ}$ (for which the theoretical ratio would vanish). Yet, for $\beta_{pt}=14.8^{\circ}$ a large ratio is obtained ($\approx12$). However, while all experiments do agree that this ratio is sizable, there is a clear disagreement on its actual magnitude. In the next two subsections [(\ref{subsec:large_strange_nonstrange_mixing}) and (\ref{subsec:branching})] we describe this issue in detail.
\item For what concerns $K^{\ast}(892)K+c.c.$ of $\eta_{2}(1870)$, there is also a disagreement with the theoretical results of Tab.\ (\ref{tab:ergebnisisoscalar}) and the experiment. Namely, the theoretical predictions are large. This is understandable, because for a small $\beta_{pt}$ the resonance $\eta_{2}(1870)$ is dominated by its $\bar{s}s$ component. However, the kaonic channel has not been seen in the experiments. Also this aspect is analyzed in the upcoming subsection.
\end{enumerate}
In conclusion, various inconsistencies with experimental data exist: a small mixing angle must be rejected.
\subsection{Large strange-nonstrange mixing angle}
\label{subsec:large_strange_nonstrange_mixing}
Models based on mesonic $\bar{q}q$ nonets and flavour symmetry have proven to be successful, as the clear example of tensor mesons shows \cite{tensor}. Yet, the results of the previous section shows that the case of pseudotensor mesons is more complicated than expected. As an immediate next step, we leave the mixing angle $\beta_{pt}$ unconstrained and test if there is an -- even large -- value which allows for a correct description of all known experimental data.\\
\\
In Fig.\ (\ref{fig:plots}), upper panel, we plot the theoretical decay widths $\eta_2(1645)\rightarrow a_{2}(1320)\,\pi$ and $\eta_2(1870)\rightarrow a_{2}(1320)\,\pi$ as a function of $\beta_{pt}.$ Only for $\beta_{pt}\approx\pm40^{\circ}$ the theoretical decay of $\eta_2(1645)$ and $\eta_2(1870)$ are comparable with the corresponding total experimental widths. Namely, for values of $\beta_{pt} \in [-40^{\circ},\,+40^{\circ}]$ the decay $\eta_2(1645)\rightarrow a_{2}(1320)\,\pi$ exceeds clearly the total decay width of $\eta_2(1645)$ [which is constrained experimentally to be $(181\pm11)\,\,$MeV], while for $\beta_{pt} \in [-90^{\circ},\,-45^{\circ}] \cup [+45^{\circ},\,+90^{\circ}]$ the decay $\eta_2(1645)\rightarrow a_{2}(1320)\,\pi$ exceeds the total decay width of $\eta_2(1870)$ [which is given experimentally by $(225\pm14)\,\,$MeV]. Hence, a large mixing angle $\beta_{pt}\approx\pm40^{\circ}$ is a necessary condition for a consistent description of data.
\begin{figure}[H]
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[scale=0.65]{plot_winkel1.pdf}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\hspace{-5.5mm}
\includegraphics[scale=0.65]{plot_winkel2.pdf}
\end{subfigure}
\caption{Upper panel: $\eta_2(1645)\rightarrow a_{2}(1320)\,\pi$ and $\eta_2(1870)\rightarrow a_{2}(1320)\,\pi$ as function of $\beta_{pt}.$ The bands correspond to the total decay widths of $\eta_2(1645)$ and $\eta_2(1870)$. Lower panel: $\eta_2(1645)\rightarrow K^{\ast}(892)\,K,$ $\eta_2(1870)\rightarrow K^{\ast}(892)\,K,$ and $\eta_2(1870)\rightarrow f_{2}(1270)\,\eta$ as function of $\beta_{pt}.$ Only for $\beta_{pt} \approx -40^{\circ}$ the decay $\eta_2(1870)\rightarrow K^{\ast}(892)\,K$ is suppressed. For further details, see discussion in the main text.}
\label{fig:plots}
\end{figure}
In Fig.\ (\ref{fig:plots}), lower panel, we show three further decay channels as function of $\beta_{pt}$: $\eta_2(1645)\rightarrow K^{\ast}(892)\,K,$ $\eta_2(1870)\rightarrow K^{\ast}(892)\,K,$ and $\eta_2(1870)\rightarrow f_{2}(1270)\,\eta.$ In particular, the channel $\eta_2(1870)$ $\rightarrow K^{\ast}(892)\,K$ is crucial for our purposes: a negative and large mixing angle, $\beta_{pt}\approx(-40^{\circ},-50^{\circ}),$ is needed to obtain a small partial decay width (this is a consequence of destructive interference). This is necessary to be in agreement with the fact that this decay channel has not been seen in experiments.\\
Combining the results of both panels, we realize that a large and negative mixing angle is the only possibility. It is instructive to make a definite choice: this is done in Tab.\ (\ref{tab:ergebnisisoscalar-42}) for the illustrative value $\beta_{pt}=-42^{\circ}$. It is visible that all theoretical results are in good agreement with the experimental data.
\begin{table}[H
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}[c]{|l|c|c|}
\hline
Decay process & Theory (MeV) & Experiment (MeV) \\
& $(\beta_{pt} = -42^{\circ})$ & \\
\hline\hline
$\eta_{2}(1645) \rightarrow\bar{K}^{*}(892)\,K + c.c. $ & $24.7$ & seen \\
\hline
$\eta_{2}(1645) \rightarrow a_{2}(1320)\,\pi$ & $186.5$ & \\
\hline
$\eta_{2}(1645) \rightarrow\bar{K}_{2}^{*}(1430)\,K + c.c. $ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f_{2}(1270)\,\eta$ & $0$ & not seen \\
\hline
$\eta_{2}(1645) \rightarrow f_{2}(1270)\,\eta^{\prime}(958) $ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f^{\prime}_{2}(1525)\,\eta$ & $0$ & \\
\hline
$\eta_{2}(1645) \rightarrow f^{\prime}_{2}(1525)\,\eta^{\prime}(958) $ & $0$ & \\
\hline\hline
$\eta_{2}(1870) \rightarrow\bar{K}^{*}(892)\,K + c.c. $ & $3.3$ & \\
\hline
$\eta_{2}(1870) \rightarrow a_{2}(1320)\,\pi$ & $221.0$ & \\
\hline
$\eta_{2}(1870) \rightarrow\bar{K}_{2}^{*}(1430)\,K + c.c. $ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta$ & $9.4$ & \\
\hline
$\eta_{2}(1870) \rightarrow f_{2}(1270)\,\eta^{\prime}(958) $ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f^{\prime}_{2}(1525)\,\eta$ & $0$ & \\
\hline
$\eta_{2}(1870) \rightarrow f^{\prime}_{2}(1525)\,\eta^{\prime}(958) $ & $0$ & \\
\hline
\end{tabular}
\caption{Decays of $I=0$ pseudotensor states. The total decay widths are $\Gamma_{\eta_{2}(1645)}^{\text{tot}}=(181\pm11)\,\,\text{MeV}$ and $\Gamma_{\eta_{2}(1870)}^{\text{tot}}=(225\pm14)\,\,\text{MeV}$.}
\label{tab:ergebnisisoscalar-42}
\end{table}
Some additional remarks are in order:
\begin{enumerate}
\item Our model is based on exact flavour symmetry and only large-$N_{c}$ dominant terms are retained. An agreement at the $5 - 10 \%$ level is expected. In order to achieve a better precision, one should include further terms (see above).
\item The experimental total width of $\eta_2(1645)$ is $(181\pm11)\,\,$MeV, while the the theoretical result for $\beta_{pt}=-42^{\circ}$ reads $211$ MeV [from Tab.\ (\ref{tab:ergebnisisoscalar-42})]. Taking into account the uncertainty on the coupling constant (keeping $\beta_{pt}=-42^{\circ}$ fixed), one obtains $(211.2\pm13)\,\,$MeV. Obviously, this is only an underestimation of the full error, but it shows that theory and experiment are compatible. The use of remark (1) would improve the agreement.
\item The experimental total width of $\eta_2(1870)$ is $(225\pm14)\,\,$MeV; the corresponding theoretical width for $\beta_{pt}=-42^{\circ}$ reads $233.7\,\,$MeV. Including errors on the coupling constants implies $(233.7\pm15)\,\,$MeV. Also in this case, there is agreement of theory and experiment.
\item The suppressed decay of $\eta_2(1870)$ into $K^{\ast}(892)\,K$ is the result of a destructive interference due to a large and negative strange-nonstrange mixing angle (similar to the one in the pseudoscalar sector).
\end{enumerate}
In conclusion, it is possible to interpret the resonances $[\pi_{2}(1670),\,K_{2}(1770),\,\eta_{2}(1645),\,\eta_{2}(1870)]$ as the ground-state $\bar{q}q$ pseudotensor mesons nonet if a large and negative mixing angle of about $-42^{\circ}$ is considered. In this respect, there is -- at the stage of the present experimental knowledge -- no need to include further additional fields, such as an hybrid pseudotensor state, in the model. (In principle, also a pseudotensor glueball could mix. At the present stage, its mass, as predicted by lattice QCD, is $\approx 3\,\,$GeV \cite{mainlattice}, therefore too large for a sizable effect on decay patterns, see Sec.\ \ref{sec:glueball} for predictions of pseudotensor glueball decays). Yet, there are some experimental conflicting results on branching ratios which we discuss in the next subsection.
\subsection{Branching ratios of $\eta_{2}(1870)$}
\label{subsec:branching}
We discuss the branching ratios of the resonance $\eta_{2}(1870)$. The experimental information is summarized in Tab.\ (\ref{tab:branching}).
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}[c]{|l|c|c|c|}
\hline
\multicolumn{4}{|c|}{$\eta_{2}(1870)$} \\
\hline\hline
Branching ratio & Theory & Experiment & Collaboration \\
\hline
\multirow{4}{*}{$\Gamma^{\text{exp}}(a_2(1320)\,\pi)/\Gamma^{\text{exp}}(f_2(1270)\,\eta)$} & \multirow{4}{*}{$\approx 23.5$} & $\boldsymbol{1.7 \pm0.4}$ & average by \cite{pdg} \\
\cline{3-4}
& & $1.60 \pm0.4$ & Anisovich et al. \cite{anisovich} \\
\cline{3-4}
& & $20.4 \pm6.6$ & Barberis et al. \cite{barberis} \\
\cline{3-4}
& & $4.1 \pm2.3$ & Adomeit \cite{adomeit} \\
\hline
$\Gamma^{\text{exp}}(a_{2}(1320)\,\pi)/\Gamma^{\text{exp}}(a_{0}(980)\,\pi)$ & & $\boldsymbol{32.6\pm12.6}$ & Barberis et al. \cite{barberis} \\
\hline
$\Gamma^{\text{exp}}(a_{0}(980)\,\pi)/\Gamma^{\text{exp}}(f_{2}(1270)\,\eta)$ & & $\boldsymbol{0.48\pm0.45}$ & Barberis et al. \cite{barberis} \\
\hline
\end{tabular}
\caption{Theoretical and experimental branching ratios for $\eta_{2}(1870)$. The mixing angle $\beta_{pt}\approx-42^{\circ}$ is used for theoretical predictions. Bold numbers are also bold in PDG \cite{pdg}.}
\label{tab:branching}%
\end{table}
Some of the measurements of the branching ratios $\Gamma^{\text{exp}} (a_{2}(1320)\,\pi)/\Gamma^{\text{exp}} (f_{2}(1270)\,\eta)$ and the average taken by \cite{pdg} are not in agreement. The value of Barberis et al. (for the WA102 collaboration) is much larger ($20.4\pm6.6$) than the others, and was criticized in the recent reanalysis of Anisovich et al. \cite{anisovich}. Actually, the PDG value $1.7\pm0.4$ seems to be solely derived by the result of Anisovich et al. \cite{anisovich} ($1.6\pm0.4$), but it is not clear why the central values differ.\\
It is also instructive to take the product of the bottom two branching ratios from Tab.\ (\ref{tab:branching}), which should give an estimate for the branching ratio $\Gamma^{\text{exp}} (a_{2}(1320)\,\pi)/\Gamma^{\text{exp}} (f_{2}(1270)\,\eta)$,
\begin{align}
& \big[ \Gamma(a_{2}(1320)\,\pi)/\Gamma(f_{2}(1270)\,\eta) \big]^{\ast} = \vphantom{\frac{1}{2}}
\\
=\, & \big[ \Gamma^{\text{exp}}(a_{2}(1320)\,\pi) / \Gamma^{\text{exp}}(a_{0}(980)\,\pi) \big] \cdot \big[ \Gamma^{\text{exp}}(a_{0}(980)\,\pi)) / \Gamma^{\text{exp}}(f_{2}(1270)\,\eta) \big] \approx \vphantom{\frac{1}{2}} \nonumber
\\
\approx\, & \big(16 \pm 16 \big)^\ast \,. \vphantom{\frac{1}{2}} \nonumber
\\
&{}^\ast(\text{derived from other exp. branchings}) \vphantom{\frac{1}{2}} \nonumber
\end{align}
Because of the large errors, this result is in principle compatible with both Anisovich and Barberis, yet the central value is much closer to the latter. New experimental results with a smaller error for the ratio $\Gamma^{\text{exp}}(a_{0}(980)\,\pi ))/\Gamma^{\text{exp}}(f_{2}(1270)\,\eta)$ would be very useful.\\
\\
Next, we turn to our prediction (obtained by using $\beta_{pt}=-42^{\circ}$ in our model). We added this result to Tab.\ (\ref{tab:branching}). For $[\eta_{2}(1870)\rightarrow a_{2}(1320)\,\pi]/[\eta_{2}(1870)\rightarrow f_{2}(1270)\,\eta]$ it reads $\approx 23.5$, hence it strongly supports the branching ratio measured by Barberis \cite{barberis}. [The bottom two ratios are not directly included in our model, because $a_0(980)$, as well as all scalar mesons, are not part of it. Hence, the corresponding slots in Tab.\ (\ref{tab:branching}) are empty.]\\
For better understanding of our prediction ($\approx 23.5$), it is instructive to have a closer look at the theoretical expression for the branching ratio $\Gamma^{th}(a_{2}(1320)\,\pi)/\Gamma^{th}(f_{2}(1270)\,\eta)$ of $\eta_{2}(1870)$. Using Eqs.\ (\ref{eq:txp}) and (\ref{eq:txplagexp}) we find,
\begin{align}
& \Gamma^{th}(a_{2}(1320)\,\pi)/\Gamma^{th}(f_{2}(1270)\,\eta) = \nonumber
\\
=\, & \frac{3\,\sin^{2}\beta_{pt}}{\big[ -\sin\beta_{pt}\cos\beta_{t} \cos\beta_{p}+\sqrt{2}\cos\beta_{pt}\sin\beta_{t}\sin\beta_{p}\big]^{2}} \left[ \frac{4\,\frac{p_1^{5}}{m_{a_{2}}^{4}}+30\,\frac{p_1^{3}}{m_{a_{2}}^{2}}+45\,p_1}{4\,\frac{k_1^{5}}{m_{f_{2}}^{4}}+30\,\frac{k_1^{3}}{m_{f_{2}}^{2}}+45\,k_1}\right] \text{ }\,,\label{br}
\end{align}
where $p_1 = k_{f}(m_{\eta(1870)},m_{a_{2}},m_{\pi})$ and $k_1 = k_{f}(m_{\eta(1870)},m_{f_{2}},m_{\eta}).$ Since $\beta_{t}$ is very small ($\beta_{t}=3.2^{\circ}$), for $\left\vert \beta_{pt}\right\vert $ sufficiently large ($\gtrsim30^{\circ}$) the first term in Eq.\ (\ref{br}) can be approximated by $3/\cos^{2}\beta_{p}$, which is independent on $\beta_{pt}.$ The complete behavior of ratio is plotted as function of $\beta_{pt}$ in Fig.\ (\ref{fig:plot2}).
\begin{figure}[H]
\centering
\includegraphics[scale=0.75]{plot_winkel3.pdf}
\caption{$\beta_{pt}$-dependency of the branching ratio $\Gamma(a_{2}(1320)\,\pi)/\Gamma(f_{2}(1270)\,\eta)$ of $\eta_2(1870)$. The bands correspond to the experimental values, see Tab.\ (\ref{tab:branching}).}
\label{fig:plot2}
\end{figure}
One can see that an agreement with Barberis et al. \cite{barberis} is reached for a quite broad range of angles (and -- most importantly -- fits well with $\beta_{pt}\approx -40^{\circ}$). An agreement with Anisovich et al. \cite{anisovich} is only realized for values very close $\beta_{pt} \approx 0^{\circ}$, which is however excluded by the other constraints studied in Subsecs.\ \ref{subsec:small_strange_nonstrange_mixing} and \ref{subsec:large_strange_nonstrange_mixing}.\\
\\
In conclusion of the present discussion, $\eta_{2}(1645)$ and $\eta_{2}(1870)$ can be considered as the members of the lowest quark-antiquark nonet only if a large-value ($\approx23$) of the branching ratio $\Gamma(a_{2}(1320)\,\pi/\Gamma(f_{2}(1270)\,\eta))$ of $\eta_{2}(1870)$ holds. Within this respect, the result of Anisovich (ratio $1.6\pm0.4$) would point to a different nature of the resonance $\eta_{2}(1870)$ (see the discussion in Ref. \cite{bing}).\\
A new experimental study of the resonance $\eta_{2}(1870)$ is necessary to understand which is the correct experimental value. In a near future, both experiments GlueX \cite{gluex1,gluex2,gluex3} and CLAS12 \cite{clas12} (with more emphasis on the former) can study such mesons via photoproduction, hence shading light on this puzzle of mesonic physics. In general, a new determination by GlueX and CLAS12 of the whole pseudotensor sector would be of great help.
\section{Pseudotensor glueball}
\label{sec:glueball}
Glueballs, bound states formed solely by gluons, have not yet been discovered experimentally (for reviews see Refs.\ \cite{decuplet1,decuplet2,decuplet3,decuplet4,decuplet5,decuplet6,staninew,walaa1,walaa2,Amsler:1995td}). While some experimental candidates for low-lying glueballs have been proposed, the situation above $3\,\,$GeV is still unsatisfactory. It is then useful to make some (qualitative) predictions of the decay channels of such glueballs in order to help the future experimental identification of candidates.\\
Among the expected glueballs above 3 GeV, also a glueball with pseudotensor quantum numbers is expected. This (yet hypothetical) state has a lattice-predicted mass of about $3.04\,\,$GeV \cite{mainlattice,gregory} (for general reviews on glueballs, see Refs. \cite{ochs,vento,crede}). The effective Lagrangian(s) describing the decays of this glueball can be obtained by Eqs.\ (\ref{eq:lagtvp}) and (\ref{eq:lagtxp}) by recalling that each glueball is flavour-blind (see, for instance, the analogous case of the tensor glueball studied in\cite{tensor}). The coupling of the glueball to vector-pseudoscalar pairs is expected to be proportional to $G_{\mu\nu} \mathrm{Tr}\big\{ \big[ V^{\mu} ,\, (\partial^{\nu}P) \big]_{-} \big\}=0,$ hence vanishes. We then expect that
\begin{align}
\Gamma_{G\rightarrow VP}=0
\end{align}
for all vector-pseudoscalar channels. This is by itself an important information in the identification of possible candidates. In particular, we predict that $\Gamma_{G\rightarrow\rho\pi}=\Gamma_{G\rightarrow K^{\ast}(892)K}=0.$\\
The coupling of the glueball to tensor-pseudoscalar pairs is given by
\begin{align}
\mathcal{L}_{GXP}=c_{GXP}\, G_{\mu\nu} \mathrm{Tr} \big( \big\{ X^{\mu\nu} ,\, P \big\} _{+} \big) \neq 0 \,, \label{eq:glueball_lagrangian}
\end{align}
where $c_{GXP}$ denotes is the corresponding coupling constant (with dimension energy). The latter cannot be fixed at present because no information on the full width of this glueball state is available (according to large-$N_{c}$ it should be smaller than ordinary mesonic states). [The expanded version of the Lagrangian (\ref{eq:glueball_lagrangian}) is shown in App.\ \ref{eq:gxplagexp}.]\\
We produce an estimate for ratios of decays, which can be easily calculated following the same steps of the previous sections. They are reported in Tab.\ (\ref{tab:branching_glueball}). The glueball mass is assumed to be $3040\,\,$MeV sharp.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}[c]{|l|c|}
\hline
\multicolumn{2}{|c|}{``$G_2(3040)$''} \\
\hline\hline
Branching ratio & Theory \\
\hline
$\Gamma^{\text{th}}(a_{2}(1320)\,\pi)/\Gamma^{\text{th}}(K_2^{\ast}(1430)\,K + c.c.)$ & 0.9 \\
\hline
$\Gamma^{\text{th}}(a_{2}(1320)\,\pi)/\Gamma^{\text{th}}(f_2(1270)\,\eta)$ & 6.0 \\
\hline
$\Gamma^{\text{th}}(a_{2}(1320)\,\pi)/\Gamma^{\text{th}}(f_2(1270)\,\eta^\prime(958))$ & 8.5 \\
\hline
$\Gamma^{\text{th}}(a_{2}(1320)\,\pi)/\Gamma^{\text{th}}(f^\prime_2(1525)\,\eta)$ & 9.0 \\
\hline
$\Gamma^{\text{th}}(a_{2}(1320)\,\pi)/\Gamma^{\text{th}}(f^\prime_2(1525)\,\eta^\prime(958))$ & 11.0 \\
\hline
\end{tabular}
\caption{Theoretical branching ratios for the pseudotensor glueball ``$G_2(3040)$''.}
\label{tab:branching_glueball}
\end{table}
It is visible that the decays into $K_{2}^{\ast}(1430)\,K$ and $a_{2}(1320)\,\pi$ are the largest (they are enhanced by isospin factors). It must be stressed that the results of Tab.\ (\ref{tab:branching_glueball}) are determined from the single large-$N_{c}$ and flavour-invariant dominant term of Eq.\ (\ref{eq:glueball_lagrangian}) (the constant $c_{GXP}$ scales as $N_{c}^{-1}$). In line with the expansions described in Sec.\ \ref{sec:themodel}, further terms proportional $c_{GXP}\,\delta_{s}$ as well as $\tilde{c}_{GXP} \propto N_{c}^{-2}$ and so on, are expected. Similarly, a small but non-vanishing decay's amplitude of the pseudotensor glueball into a vector-pseudoscalar pair proportional to $\delta_{s}$ also emerges. All these subleading effects will be important when a suitable glueball's candidate will be experimentally observed.\\
The search of glueballs between $2.5 - 3\,\,$GeV is currently ongoing at BESIII experiment \cite{bes1,bes2} and will be one of the main subjects of the planned PANDA experiment at the FAIR facility in Darmstadt \cite{panda}. In particular, at PANDA the pseudotensor glueball can be directly formed by proton-antiproton fusion, hence it can be investigated in detail. The here discussed theoretical branching ratios provide help toward the identification of possible candidates by looking at their decay patterns. The full decay width cannot be obtained, however according to large-$N_{c}$ \cite{largen1,largen2,largen3} it should be smaller than conventional quark-antiquark states. A value of about few MeV ($\approx10$ MeV) seems to be a rational guess, see the discussion in Ref. \cite{julia}.
\section{Discussions and conclusions}
\label{sec:conclusions}
In this work we have studied the phenomenology of the ground-state $\bar{q}q$ pseudotensor meson nonet, identified with the resonances $[\pi_{2}(1670),K_{2}(1770),\eta_{2}(1645),\eta_{2}(1870)]$, by using a quantum field theoretical approach. Two effective interaction Lagrangians which couple pseudotensor states to pseudoscalar, vector and tensor ones were constructed and used to study decays and mixing.\\
The resonances $\pi_{2}(1670)$ and $K_{2}(1770)$ fit very well into the $\bar{q}q$ picture. However, the isoscalar states $\eta_{2}(1645),\eta_{2}(1870)$ are more subtle. A small mixing angle between purely nonstrange and strange state is at odd with the experimental data [$\eta_{2}(1645) \rightarrow a_{2}(1320)\,\pi$ is larger than $300\,\,$MeV and overshoots the total experimental width]. A detailed study of the isoscalar sector shows that a good agreement with data is possible if the strange-nonstrange mixing angle is large and negative [approximately $-42^{\circ},$ see Fig.\ (\ref{fig:plots})].\\
There is however an issue that still needs to be clarified: the ratio $[ \eta_{2}(1870) \rightarrow a_{2}(1320)\, \pi ]/$ $[ \eta_{2}(1870) \rightarrow f_{2}(1270)\, \eta ]$ reads $20.4\pm6.6$ by Barberis \cite{barberis} and $1.6\pm0.4$ by Anisovich \cite{anisovich}. PDG \cite{pdg} lately opted for the latter result ($1.7 \pm 0.4$, with a slightly modified central value). Our study shows that a quarkonium interpretation of $\eta_{2}(1870)$ implies a ratio of about $\approx 23$, hence in good agreement with Barberis but in disagreement with Anisovich. In this respect, a future experimental clarification of this issue is compelling. There are mainly two possible outcomes:
\begin{itemize}
\item[a)] The ratio quoted by Barberis turns out to be correct, then we have good candidates for a ground-state pseudoscalar meson nonet. However, the large mixing angle (comparable to the one in the pseudoscalar sector) would be a mystery which will deserve a more profound study. Namely, besides the pseudoscalar sector which is affected by the chiral anomaly, all strange-nonstrange mixing angles are small (vector, axial-vector, tensor sectors \cite{pdg,klemptrev,tensor,ciricgliano}). Where would such a large mixing in the pseudotensor sector come from? Would the two-gluon exchange diagram be also enhanced in the pseudotensor channel (a side-effect of the chiral anomaly)?
\item[b)] If, on the contrary, the result of Anisovich shall be confirmed, an understanding of the low-lying pseudotensor states as a standard quark-antiquark nonet would be hard. Modifications would be needed. For instance, one could include additional states in the current approach (such as an hybrid pseudotensor state) which can mix with ordinary $\bar{q}q$ states and change their decay ratios. In such a scenario one would need to consider a decuplet of states (9 standard quarkonia and one additional hybrid state, similar to the decuplet study of scalar states between $1.3 - 1.8\,\,$GeV \cite{decuplet1,decuplet2,decuplet3,decuplet4,decuplet5,decuplet6,Amsler:1995td}). We leave such an analysis for the future, if there will be compelling evidence that the easiest scenario with a low-lying $\bar{q}q$ nonet fails (this is not yet the case, as we have discussed in this work).
\end{itemize}
As a byproduct of our study, we have also studied the decays of a putative pseudotensor glueball state with a mass of $3.04\,\,$GeV. We find that at leading order it does not decay into pseudoscalar-vector meson pairs (such as $\rho(770)\,\pi$ and $K^{\ast}(892)\,K + c.c.$). On the contrary, sizable decays into $K^{\ast}_{2}(1430)\,K + c.c.$ and $a_{2}(1320)\,\pi$ are expected.\\
Further improvements of our model (besides the enlargement to a decuplet of states mentioned above) is the inclusion of large-$N_{c}$ suppressed terms and terms breaking flavor symmetry. Moreover, tensor and pseudotensor mesons can also build a chiral multiplet. They could be then coupled to the extended Linear Sigma Model in order to test chiral symmetry (and its spontaneous breaking) in the (pseudo)tensor sector. Also the study of (pseudo)tensor glueballs can be extended in this way. As shown in Ref. \cite{staninew,julia,walaa1,walaa2}, chiral symmetry imposes further constraints on the decays of glueballs, hence improving the predictive powers of hadronic models. From the experimental side, new results and determinations of the pseudotensor mesons $[ \pi_{2}(1670) ,\, K_{2}(1770) ,\, \eta_{2}(1645) ,\, \eta_{2}(1870) ]$ are needed to test the standard basic quark-anti-quark scenario and the existence or not of an enhanced mixing in the isoscalar sector. The upcoming experiments GlueX and CLAS12 experiments at Jlab are expected to provide results in this direction. Indeed, the properites of pseudotensor mesons are also investigated in the light-meson program of the COMPASS experiment \cite{Krinner:2016kna}.
\bigskip
\textbf{Acknowledgments}: The authors thank Dirk~H.~Rischke for useful discussions. F.G. acknowledges support from the Polish National Science Centre (NCN) through the OPUS project nr. 2015/17/B/ST2/01625. A.K. thanks Dennis~D.~Dietrich for instructive discussions.\\
\\
The final publication is available at Springer via http://dx.doi.org//10.1140/epja/i2016-16356-x.
|
2,877,628,090,178 | arxiv | \section{Introduction}
Collisionless shocks play important roles in the generation of high-energy particles
in various situations,
which is one of the most important outstanding issues in plasma physics.\cite{Balogh_2013,Burgess_2015}
Recently, laboratory experiments using high-power lasers are conducted
on the generation of collisionless shocks propagating into
unmagnetized\cite{Kuramitsu_2010,Morita_2010,Yuan_2017,Ross_2017}
and magnetized\cite{Schaeffer_2014,Niemann_2014,Schaeffer_2017a,Schaeffer_2017b,Shoji_2016} plasmas.
In particular, the laboratory experiments of magnetized collisionless shocks are of great interest
since the most astrophysical and solar-terrestrial plasmas hosting the collisionless shocks are magnetized.
There are mainly two ways to excite collisionless shocks in laboratory plasma experiments
using high-energy lasers,
in which two plasmas collide with each other.
One is to have counter-streaming plasmas, both of which move in the laboratory frame.
They arise from double-plane targets irradiated by lasers.\cite[e.g.,][]{Schaeffer_2017a,Schaeffer_2017b}
In the other way,\cite{Shoji_2016}
an ambient plasma at rest
is pushed by a flowing plasma originated in laser ablation.
To make the ambient plasma, the neutral gas is fulfilled around the target before the shot,
and it is photoionized by photons generated in the laser ablation process.
The ambient plasma is magnetized if the external magnetic field is imposed before it is ionized.
In this method, one can easily control the field strength, accordingly
the Alfv\'{e}n Mach number and the plasma beta
(i.e., the ratio of the plasma pressure to the magnetic pressure) of the ambient plasma.
In contrast, some authors have proposed a complementary way toward
the collisionless shock formation using ultra-high-intensity lasers
to drive a fast quasi-neutral flow in a denser plasma.\cite{Fiuza_2012,Ruyer_2015,Grassi_2017}
It has been believed that in the ablation plasma, spontaneous magnetic fields are produced
by laser-plasma interactions {due to the
so-called \textit{Biermann battery} process.\cite{Biermann_1950}}
The Biermann battery works when the cross product between the gradients of
the electron density and the electron temperature is non-vanishing near
the targets.\cite{Stamper_1991,Gregori_2012,Kugland_2013}
The resultant magnetic field is toroidal with respect to the direction of the plasma flow.
Then, the magnetic field convects with the ablation plasma outwards.\cite{Kugland_2013,Ryutov_2013}
At least just after the shot at which strong density and temperature gradients exist,
the magnetic pressure can be comparable to or even larger than the kinetic and thermal pressure
of the plasma.
However, at present, the role of the Biermann magnetic field
in the excitation of the collisionless shocks is poorly understood.
\begin{table*}[b]
\caption{
Simulation parameters for the present numerical experiment.
A reduced value in the numerical experiment is shown
at the right-hand side when the physical quantity is not the real one.
The cyclotron frequency, the thermal gyro radius, the Alfv\'{e}n velocity, and the plasma beta
are for runs with non-zero magnetic field ($B_0\ne0$).
}
\renewcommand{\arraystretch}{0.75}
\begin{tabular}{c||c|c}
Quantity & Aluminum plasma & Nitrogen plasma \\ \hline \hline
Drift velocity $V_d$ [km/s] & 500 & 0 \\
Magnetic field $B_0$ [T] && \\
Run 1 & 10.0 & 0.5 \\
Run 2 & 10.0 & 0.0 \\
Run 3 & 0 & 0.5 \\
Run 4 & 0 & 0.0 \\
\hline
Electrons & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2,250/cell & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 90/cell \\
Density $N_e$ [${\rm cm}^{-3}$] & $3.75\times10^{19}$ & $1.5\times10^{18} $ \\
Plasma frequency $f_{pe}$ [Hz] & $5.51\times10^{13}$ / $5.51\times10^{12}$ &
$1.1\times10^{13}$ / $1.1\times10^{12}$ \\
Temperature $T_e$ [eV] & 10 & 30 \\
Thermal velocity $V_{te}$ [km/s] & 1,330 & 2,300 \\
Debye length $\lambda_{De}$ [m] & $3.71\times10^{-9}$ / $3.71\times10^{-8}$ &
$3.32\times10^{-8}$ / $3.32\times10^{-7}$ \\
Inertial length $d_e$ [m] & $8.39\times10^{-7}$ & $4.33\times10^{-6}$ \\
Cyclotron frequency $f_{ce}$ [Hz] & $2.8\times10^{11}$ & $1.4\times10^{10}$ \\
Thermal gyro radius $r_e$ [m] & $7.55\times10^{-7}$ & $2.62\times10^{-5}$ \\
Plasma beta $\beta_e$ & 1.62 & 72.99 \\
\hline
Ions & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 250/cell & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30/cell \\
Charge number $Z$ & 9 & 3 \\
Mass ratio $m_i/m_e$ & 49572 & 25704 \\
Density $N_i$ [${\rm cm}^{-3}$] & $4.17\times10^{18}$ & $5.0\times10^{17}$ \\
Plasma frequency $f_{pi}$ [Hz] & $7.43\times10^{11}$ / $7.43\times10^{10}$ &
$1.19\times10^{11}$ / $1.19\times10^{10}$ \\
Temperature $T_i$ [eV] &10 & 30 \\
Thermal velocity $V_{ti}$ [km/s] & 5.97 & 14.3 \\
Debye length $\lambda_{Di}$ [m] & $3.71\times10^{-9}$ / $3.71\times10^{-8}$ &
$3.32\times10^{-8}$ / $3.32\times10^{-7}$ \\
Inertial length $d_i$ [m] & $6.23\times10^{-5}$ & $4.01\times10^{-4}$ \\
Cyclotron frequency $f_{ci}$ [Hz] & $5.08\times10^{7}$ & $1.63\times10^{6}$ \\
Thermal gyro radius $r_i$ [m] & $1.87\times10^{-5}$ & $1.4\times10^{-3}$ \\
Alfv\'{e}n velocity $V_A$ [km/s] & 19.99 & 4.11 \\
Plasma beta $\beta_i$ & 0.18 & 24.33 \\
\hline
Grid spacing $\Delta x$ [m]& \multicolumn{2}{c}{$8.3 \times 10^{-8}$} \\
Time step $\Delta t$ [sec]& \multicolumn{2}{c}{$2.6 \times 10^{-15}$} \\
Number of grids $N_x$ & \multicolumn{2}{c}{120,000} \\
Number of steps $N_t$ & \multicolumn{2}{c}{6,000,000} \\
Speed of light $c$ [km/s]& \multicolumn{2}{c}{300,000 / 30,000} \\
& \multicolumn{2}{c}{(laboratory) / (numerical)} \\
\end{tabular}
\end{table*}
In the present study,
we perform one-dimensional (1D) full particle-in-cell (PIC) simulations
of the interaction between the ablation (piston) plasma and the ambient plasma at rest,
as a preliminary numerical experiment for laboratory experiment
on the generation of magnetized collisionless shocks with high-power lasers such as
Gekko-XII at the Institute of laser engineering in Osaka University.
Since the number of shots is limited in laboratory experiments,
our preliminary numerical experiments play {an important role}
in the prediction of results of laboratory experiments.
The present study aims to study the effect of the spontaneous magnetic field
in the piston plasma {due to the Biermann battery}
on the generation of collisionless shocks.
\section{1D PIC Simulation}
\subsection{Typical Parameters}
First, we briefly describe our setup and accompanying typical parameters
optimized for laboratory experiments using Gekko-XII HIPER lasers.
Details will be published elsewhere.
The planar Aluminum target is irradiated by HIPER lasers with energy of $\sim$~kJ in total
and the pulse width of $\sim$~nsec, producing the Aluminum plasma with bulk
velocity of $\sim10^2$~km/s.
A vacuum chamber is filled with Nitrogen gas with 5~Torr before the shot, and it is ionized
by photons arising in the laser-target interaction.
Just after the shot, the Aluminum plasma is very hot and dense with electron density
$N_e\sim10^{21}$~${\rm cm}^{-3}$
and temperature $T_e\sim10^3$~eV near the target.
As it expands, it adiabatically cools down and has typically $N_e\sim10^{19}$~${\rm cm}^{-3}$
and $T_e\sim10$~eV when the shock is formed.
The Nitrogen plasma is cold with temperature $T_e\sim$~eV at the beginning.
However, it is preheated by HIPER lasers to about several tens of eV
since we consider interaction between Aluminum
and Nitrogen plasmas in the ablation side of the target.
Unfortunately, the strength of the Biermann spontaneous magnetic field
associated with the Aluminum plasma
is unknown since magnetic fields were not measured in our preliminary laboratory experiments.
Here, we roughly estimate the strength of the spontaneous magnetic field as follows.
The evolution of the magnetic field is described by the following equation {in SI units
(so that the Boltzmann constant is omitted)}
\begin{equation}
\frac{\partial \Vec{B}}{\partial t} \approx\frac{1}{eN_e} \left(\nabla T_e \times \nabla N_e \right),
\end{equation}
which is derived from the rotation of the {electron pressure gradient term
of electric fields, i.e., $-\nabla \times \{\nabla P_e/(eN_e)\}$}
in the magnetic induction equation.
Here, we {have neglected} the convection term, $\nabla\times(\Vec{U}_e \times \Vec{B})$,
at the very beginning of the field generation.
If we approximate
$\partial/\partial t\approx V_d/\phi$ and $\nabla\approx1/\phi$, where
$V_d$ and $\phi$ are the drift velocity of the piston Aluminum plasma and
the focal spot size of HIPER lasers, respectively,
then we have
\begin{eqnarray}
B&\approx&\frac{T_e}{eV_d\phi} \nonumber\\
&\approx& 10~{\rm T}
\left(\frac{T_e}{10^3~{\rm eV}}\right)
\left(\frac{V_d}{10^2~{\rm km/s}}\right)^{-1}
\left(\frac{\phi}{1~{\rm mm}}\right)^{-1}.
\end{eqnarray}
Inserting physical parameters near the target into Eq.(2),
we obtain a typical magnitude of the Biermann spontaneous magnetic field
as $\sim 10$ T, which is consistent with other laboratory experiment\cite{Fiksel_2014}
and numerical experiment.\cite{Fox_2018}
On the other hand, the external magnetic field imposed on the ambient Nitrogen plasma
has arbitrary strength.
However, its typical value is $B\sim1$~T.
If $B\ll1$~T, then the ion is not magnetized.
If $B\gg1$~T, then the magnetized shock propagating into the Nitrogen plasma
has a small Alfv\'{e}n Mach number.
{In the present study, we assume an unmagnetized ambient Nitrogen plasma
as shown in Table.1.}
\subsection{Simulation Setup}
We use a 1D relativistic full PIC code developed by ourselves
that was used for simulations of collsionless shocks.\cite{Umeda_2006}
{The Sokolov interpolation \cite{Sokolov_2013} is implemented into
the second-order charge conservation scheme\cite{Umeda_2003}}
to reduce numerical noises.
The code has stable open boundary conditions
which allows us to perform simulations with a long time of $t \gg 10^5/\omega_{pe}$.
In the present numerical experiment,
the simulation domain is taken along the $x$ axis.
At the initial state,
the simulation domain is filled with collisionless Nitrogen plasma at rest as an ambient plasma.
The open boundary conditions are imposed at both of boundaries,
where electromagnetic waves and plasma particles escape freely.
In addition to the open boundary condition,
collisionless Aluminum plasma as a piston plasma with a drift velocity $V_d$
is continuously injected from at the left boundary ($x=0$) into the Nitrogen plasma
when the numerical experiment is started.
All of the plasma particles have a (shifted) Maxwellian velocity distribution
with an isotropic temperature at the initial state.
The typical parameters of the Aluminum plasma and the Nitrogen plasma
{near a measurement point of our preliminary laboratory experiments
($\sim$ cm away from the target)} are listed in Table 1.
{In the present numerical experiment,
we have tried to use these parameters as many as possible,
including the real ion-to-electron mass ratios. }
However, some of them are reduced with respect to the real ones
to save computational cost.
As an example,
the vacuum permittivity $\epsilon_0$ used in the present numerical experiment is
100 times larger than the real one.
This means that the speed of light and the plasma frequency are
reduced to one-tenth of the real ones.
However, the Alfv\'{e}n velocity, plasma beta and the inertial length are
set to be the same as real ones, which are important parameters for discussion of the shock dynamics.
{In Table 1, the real values in the laboratory experiments are shown
at the left-hand side and reduced values are shown at the right-hand side when the physical quantity
in the numerical experiment is reduced. }
These parameters are renormalized to the electron plasma frequency and
the electron thermal velocity in the Nitrogen plasma in the full PIC simulation.
{The number of particles per cell for each species is also shown in Table 1}.
The ambient magnetic field $B_0$ (in the Nitrogen plasma) is imposed in the $y$ direction
with a magnitude of 0.5 T,
in contrast to the previous studies in which the magnitude of the ambient magnetic field
was 4 T.\cite{Schaeffer_2017a,Schaeffer_2017b}
Since the magnitude of the ambient magnetic field is weak,
the ambient (Nitrogen) plasma is in the high beta regime in the present study.
As a spontaneous magnetic field due to the Biermann battery,
we assume that the magnetic field in the piston (Aluminum) plasma
is directed in the $y$ direction with a magnitude of 10 T.
To magnetize the Aluminum plasma, a motional electric field is
also imposed in the $z$ direction with a magnitude of $E_z=-V_dB_0$ at the left boundary ($x=0$).
We perform four different runs.
In Run 1, both of the Aluminum plasma and the Nitrogen plasma are magnetized.
In Run 2, the Aluminum plasma is magnetized while the Nitrogen plasma is unmagnetized.
Note that it is easy to change the magnitude of the ambient magnetic field (in the Nitrogen plasma)
in laboratory experiments.
In Run 3, the Nitrogen plasma is magnetized while
the magnetic field in the Aluminum plasma is set to be zero
to see the effect of the spontaneous magnetic field in the piston plasma.
In Run 4, both of the Aluminum plasma and the Nitrogen plasma are unmagnetized.
The direct comparison among these runs could show the influence of
the spontaneous magnetic field in the Aluminum plasma
{due to the Biermann battery}
to the generation of collsionless shocks.
\section{Simulation Result}
\begin{figure}[p
\center
\includegraphics[width=1.0\textwidth,bb=0 0 150 150]{fig1.eps}
\caption{
Temporal development of the interaction between
the Aluminum plasma and the Nitrogen plasma for Run 1.
The $x-v_x$ phase-space plots of ions at different times
together with the spatial profile of the electron density.
}
\end{figure}
Figure 1 shows the temporal development of the interaction between
the Aluminum plasma and the Nitrogen plasma for Run 1.
The \textit{ion charge density} in the $x-v_x$ phase space is plotted at different times.
The corresponding spatial profile of the electron density is also superimposed.
Since the absolute value of the electron charge density is almost equal to the
total ion charge density, i.e., the sum of charge densities of Aluminum and Nitrogen ions,
the quasi charge neutrality is almost satisfied at all the time.
At the leading edge of the Aluminum plasma, there exists strong charge separation
because the Aluminum ions can penetrate the Nitrogen plasma region
while electrons cannot compensate
the Aluminum ion charge owing to the small gyro radius.
A diamagnetic current is also generated
around the interface between the two plasmas due to a large magnetic shear.
The charge separation and electric current excite electromagnetic fluctuations,
which results in the ponderomotive force to scatter (accelerate) Aluminum ions
at the leading edge toward the Nitrogen plasma region.
The ponderomotive force also reflect some of Aluminum ions around the interface
toward the Aluminum plasma region.
Note that the acceleration of electrons due to the ponderomotive force is not seen
since accelerated electrons soon diffuse in the velocity space through gyration.
As the Aluminum plasma penetrates into the Nitrogen plasma region,
an instability is generated at the leading edge of the Aluminum plasma
(see second and third panels at $t=4.004$ and $8.008$ nsec, respectively).
We found a wave mode is excited at the local electron cyclotron frequency $\omega_{ce,local}$
from the Fourier analysis.
The wavenumber satisfies the resonance condition $k_xV_{dAl} \approx \omega_{ce,local}$,
suggesting that the electron cyclotron drift instability is generated due to
the drift of Aluminum ions across the ambient magnetic field.
{For more detail, see Appendix A.}
As the time elapses, the Nitrogen ions feel the motional electric field
in the Aluminum plasma region and gyrate in the $x-v_x$ phase space.
During the gyration, strong compression of Aluminum ions takes place
in a region outside the gyrating Nitrogen ions
(at $x \approx 0.35$ cm at $t = 12.012$ nsec).
\begin{figure}[t]
\center
\includegraphics[width=1.0\textwidth,bb=0 0 150 150]{fig2.eps}
\caption{
(a) Ion $x-v_x$ phase space together with the spatial profile of the electron density
around the interface between the Aluminum plasma and the Nitrogen plasma
at $t = 4.004$ nsec for Run 1.
(b) Electron $x-v_x$ phase space together with the spatial profile of the electron temperature
of the Aluminum (piston) plasma.
(c) Electron $x-v_x$ phase space together with the spatial profile of the electron temperature
of the Nitrogen (background) plasma.
(d) Spatial profiles of the pressures of the piston electrons $P_{ep}$, the background electrons $P_{eb}$,
the Aluminum ions $P_{Al}$, and the magnetic pressure $P_B$.
The pressure is normalized to the initial electron pressure in the Nitrogen (background) plasma region.
}
\end{figure}
Panel (a) of Fig.2 shows an expansion of the ion $x-v_x$ phase space
together with the spatial profile of the electron density
around the interface between the two plasmas at $t = 4.004$ nsec for Run 1.
Panels (b) and (c) show the corresponding
electron $x-v_x$ phase space together with the spatial profile
of the electron temperature of the Aluminum (piston) and
the Nitrogen (background) plasmas, respectively.
Panel (d) shows the spatial profiles of the pressures of the piston electrons $P_{ep}$,
the background electrons $P_{eb}$,
the Aluminum ions $P_{Al}$, and the magnetic pressure $P_B=|B_y|^2/(2\mu_0)$
normalized to the initial electron pressure of the Nitrogen (background) plasma region.
Note that the plasma pressure is given as $P=(P_{x}+P_{z})/2$.
The pressure of the Nitrogen ions is small and is not shown here.
The magnetic field $B_y$ component is almost proportional
to the total (the sum of piston and background) electron density
and is not shown here.
There are two discontinuous structures (labeled as ``I'' and ``II'')
around the interface between the two plasmas.
We found that the piston electrons are clearly separated
from the background electrons around the discontinuity ``I''
as seen in panels (b) and (c) of Fig.2.
The electron density, the electron temperature, and the electron pressure
continuously increase from the right to the interface between the two plasmas.
The density of piston electrons increases from the right to the left
but the temperature of piston electrons decreases at the discontinuity ``I''.
The pressure of piston electrons decreases slightly
(see ``$P_{ep}$'' in Panel (d)) but the magnetic pressure increases at the discontinuity ``I''.
It is suggested that the electron pressure gradient force balances the $\Vec{J}_e\times \Vec{B}$ force.
Since the bulk velocities of piston and background electrons are almost the same
across the discontinuity ``I'',
the discontinuity ``I'' can be regarded as a tangential
electron-magneto-hydro-dynamic (EMHD) discontinuity.
Panel (c) of Fig.2 shows that the background electrons are heated
due to the penetration of the piston (Aluminum) ions.
The mechanism of the electron heating is considered to be the following
EMHD process, since the timescale of the
diffusion of electrons in the velocity space due to the gyration is fast.
The background electrons feel the motional electric field of piston ions,
which result in a finite bulk velocity of the background electrons ($U_{xeb} > 0$).
Then, the pressure of the background electrons increases from the right to the left, as seen in panel (d),
by the compression of the background electrons, i.e., $\partial U_{xeb}/\partial x \ne 0$.
The sum of magnetic pressure and the thermal pressure of electrons (i.e., $P_B+2P_{ep}$) is
kept almost constant across the discontinuity ``I''.
Since the electron density of the piston plasma is higher than
that of the background plasma, there exists a discontinuity of the electron temperature
around the interface between the two plasmas.
At the discontinuity ``II'', the total electron density decreases from the right to the left
but the pressure of Aluminum ions increases.
The sum of the pressure of Aluminum ions and the magnetic pressure
is kept almost constant across the discontinuity ``II''.
\begin{figure}[t]
\center
\includegraphics[width=1.0\textwidth,bb=0 0 150 150]{fig3.eps}
\caption{
(a) Ion $x-v_x$ phase space together with the ion temperature of the Aluminum plasma
around the gyrating Nitrogen ions at $t = 12.012$ nsec for Run 1,
and (b) the corresponding ion $x-v_x$ phase space together
with the spatial profile of the ion temperature for the Nitrogen plasma.
(c) Electron $x-v_x$ phase space together with the spatial profile of the electron temperature.
(d) Spatial profiles of the electron bulk velocity $U_{xe}$ and the magnetic field $B_y$.
}
\end{figure}
Panels (a) and (b) of Fig.3 show an expansion of the ion $x-v_x$ phase space
together with the spatial profile of the ion temperature of the Aluminum and Nitrogen plasmas, respectively,
around the gyrating Nitrogen ions at $t = 12.012$ nsec for Run 1.
Panel (c) shows the corresponding
electron $x-v_x$ phase space together with the spatial profile of the electron temperature.
Panel (d) shows the spatial profiles of the electron bulk velocity $U_{xe}$ and the magnetic field $B_y$.
Note that electrons in this region consist of piston electrons.
There are two discontinuous structures (labeled as ``III'' and ``IV'')
around the gyrating Nitrogen ions.
At the discontinuity ``III'', the electron density, the electron temperature, and
the magnetic field increase from the right to the left.
The electron bulk velocity changes from $\approx 400$ km/s to $\approx 450$ km/s.
At the discontinuity ``IV'', the electron density, the electron temperature, and
the magnetic field decrease from the right to the left.
The electron bulk velocity changes from $\approx 450$ km/s to $\approx 500$ km/s.
The density and the magnetic field between the discontinuities ``III'' and ``IV''
are $\approx 2.4$ times as large as those in the unperturbed Aluminum plasma.
Let us consider the MHD shock jump conditions for perpendicular shocks,
\begin{eqnarray}
&&B_{y1}U_1 = B_{y2}{U_2}, \\
&&N_1U_1 = N_2 U_2, \\
&&(m_e+m_i)N_1U_1^2+P_{1}+\frac{B_{y1}^2}{2\mu_0} \\ \nonumber
&& \ =
(m_e+m_i)N_2U_2^2+P_{2}+\frac{B_{y2}^2}{2\mu_0}, \\
&&\left\{\frac{(m_e+m_i)N_1U_1^2}{2}+2P_1+\frac{B_{y1}^2}{\mu_0}\right\}U_1 \\ \nonumber
&&=
\left\{\frac{(m_e+m_i)N_2U_2^2}{2}+2P_2+\frac{B_{y2}^2}{\mu_0}\right\}U_2,
\end{eqnarray}
where, the subscript ``1'' and ``2'' denotes the upstream and the downstream of a discontinuity, respectively.
Here, the ion and electron densities are assumed to be almost equal, $N\equiv N_i \approx N_e$,
and the plasma pressure is given as the sum of ion and electron pressures, $P\equiv P_i+P_e$.
The bulk velocity $U$ is defined in rest frame of a discontinuity.
Suppose that the velocity of the discontinuities ``III'' and ``IV'' is
$\approx 485.7$ km/s and $\approx 414.3$ km/s, respectively.
Then, Eqs.(3) and (4) are satisfied across the discontinuities ``III'' and ``IV'' with
{the upstream bulk velocity of $U_1\approx 85.7$ km/s,
the downstream bulk velocity of $U_2\approx 35.7$ km/s
and the density $N_2/N_1\approx 2.4$}
in the rest frame of the discontinuities.
{From the initial conditions, we obtain $(m_e+m_i)N_1U_1^2/P_{i1} \approx 206$,
$P_{e1}/P_{i1} = 9$ and $B_{y1}^2/(2\mu_0P_{i1}) \approx 5.6$.
The pressures of the electrons and Aluminum ions in
the high density region (downstream)
are $P_{e2}/P_{e1} \approx 11$ and $P_{Al2}/P_{Al1} \approx 8$
times as large as those of the unperturbed Aluminum plasma (upstream), respectively.
Then, we obtain $(m_e+m_i)N_2U_2^2/P_{i1} \approx 85.8$, $P_{2}/P_{i1} \approx 92$
and $B_{y2}^2/(2\mu_0P_{i1}) \approx 32$.
The momentum conservation law and the energy conservation law
for the Aluminum plasma in Eqs.(5) and (6), respectively, are almost satisfied. }
Hence, the shock jump conditions are almost satisfied across the discontinuities ``III'' and ``IV''.
This result suggests that
these discontinuous structures correspond to collisionless perpendicular shocks.
The high-density region between the discontinuities ``III'' and ``IV'' corresponds to the shock downstream.
The typical Alfv\`{e}n Mach number of the shock is $M_A \approx 4.2$.
The shock downstream is located at a distance of one ion gyro radius of Nitrogen ions in the Aluminum plasma from the interface between the Nitrogen and Aluminum plasmas.
The shock is formed on the timescale of a quarter ion gyro period of Nitrogen ions in the Aluminum plasma.
\begin{figure*}[p]
\center
\includegraphics[width=0.9\textwidth,bb=0 0 200 200]{fig4.eps}
\caption{
Snapshot of four Runs at $t = 14.040$ nsec.
(a) The $x-v_x$ phase-space plots of electrons.
(b) The $x-v_x$ phase-space plots of ions.
(c) The spatial profile of the magnetic field $B_y$ component.
(d) The spatial profile of the electron density $N_e$.
(e) The spatial profile of the electron temperature $T_e$.
}
\end{figure*}
Figure 4 shows a snapshot of four Runs at $t = 14.040$ nsec.
Panel (a) shows the $x-v_x$ phase-space plots of electrons.
Panel (b) shows the $x-v_x$ phase-space plots of ions.
Panel (c) shows the spatial profile of the magnetic field $B_y$ component.
Panel (d) shows the spatial profile of the electron density $N_e$.
Panel (e) shows the spatial profile of the electron temperature $T_e$.
In Run 2 where the ambient magnetic field in the Nitrogen plasma is absent,
the electron cyclotron drift instability is not generated
at the leading edge of the Aluminum plasma.
However, electron heating takes place due to the penetration of the Aluminum ions
(for $x > 0.626$ cm), suggesting that the generation of the instability
is not a necessary condition for the electron heating.
The electron temperature at the leading edge of the Aluminum plasma in Run 2
is higher than that in Run 1, since electrons in this region is unmagnetized and
they escape from the interface.
The tangential discontinuity and the shock downstream are formed at $x \approx 0.55$ cm
and $x \approx 0.41$ -- $0.45$ cm, respectively.
The comparison between Runs 1 and 2 suggests that
the existence of the ambient magnetic field (in the Nitrogen plasma) is not a
necessary condition for the formation of shocks in the Aluminum plasma.
In Run 3 where the magnetic field in the Aluminum plasma is absent,
similar electron heating also takes place due to the penetration of the Aluminum ions
(for $x > 0.637$ cm).
The electron cyclotron drift instability is generated as in Run 1,
and the spatial profile of the electron temperature is almost the same as
that in Run 1.
A strong fluctuation of the magnetic field is excited at $x \approx 0.637$ cm,
and a part of Aluminum ions are reflected by the ponderomotive force.
Since the gyration of Nitrogen ions is absent, the shock downstream is not formed in Run 3.
The Aluminum plasma pass through the
Nitrogen plasma, and no interaction between them is seen in Run 4.
The acceleration of Aluminum ions purely by the electrostatic field
due to charge separation is seen in panel (b), which is weaker than the acceleration
in the other runs.
Also, electron heating due to the penetration of the Aluminum ions
does not take place.
{Finally, it should be noted that we also performed several runs
with a large ambient magnetic field (e.g., 3 T and 5 T) in the Nitrogen plasma region.
In these runs, a collisionless shock is formed by the gyration of Aluminum ions in the
Nitrogen plasma region, which is consistent with the previous study.\cite{Schaeffer_2017a,Schaeffer_2017b}
The influence of magnetic field in the Aluminum plasma due to the Biermann battery
on the formation of discontinuities and shocks is small.
Hence, these runs with a large ambient magnetic field are out of purpose of the present study and
are not shown here.
}
\section{Summary}
The interaction between the piston Aluminum plasma and the ambient Nitrogen plasma
was studied by means of a 1D full PIC simulations
as a preliminary numerical experiment for laboratory experiment
on the generation of magnetized collisionless shocks with high-power lasers.
Preliminary numerical experiments are important
in the prediction of results of laboratory experiments.
Four different runs were performed with the combination of
{two magnetized and/or unmagnetized plasmas}.
It is shown that the magnetic field plays an important role
in the formation of the tangential EMHD discontinuity
around the interface between the two plasmas
in the present study with a weak ambient magnetic field.
The tangential EMHD discontinuity is formed on the timescale of the electron gyro period.
This result is different from the previous result\cite{Schaeffer_2017a,Schaeffer_2017b}
in which a strong ambient magnetic field was imposed and
shock waves were formed around the interface between the piston plasma and the background plasma.
A shock wave is formed through the gyration of the ambient plasma in the piston plasma region
in the present study with a weak ambient magnetic field.
The comparison among the four runs showed that
a perpendicular collisionless shock
is formed only when the piston plasma is magnetized.
It is suggested that
the spontaneous magnetic field in the piston plasma due to the Biermann battery
plays an essential role in the formation of a perpendicular collisionless shock
in the interaction between the two plasmas.
The shock downstream is formed during the gyration of ambient ions
in the piston plasma region on the timescale of quarter gyro period of ambient ions.
\acknowledgments{
This work was supported by JSPS KAKENHI grant numbers 18H01232(RY, TM), 16K17702 (YO),
15H02154, 17H06202 (YS), 17H18270 (SJT), and 17J03893 (TS).
This work was also supported partially by the joint research project of the Institute of Laser Engineering,
Osaka University.
The computer simulations were performed on the CIDAS supercomputer system
at the Institute for Space-Earth Environmental Research in Nagoya University
under the joint research program.
}
|
2,877,628,090,179 | arxiv | \section{Introduction}
An obstacle course might react to what you do: for example, if you step on a certain
button, then spikes might appear. If you spend enough time in such an obstacle course,
you should eventually figure out such patterns.
But imagine an ``oracular'' obstacle course which reacts to
what you would hypothetically do in counterfactual scenarios: for example, there is
no button, but spikes appear
if you \emph{would} hypothetically step on the button if there was one. Without
self-reflecting about what you would hypothetically do in counterfactual scenarios, it
would be difficult to figure out such patterns. This suggests that in order to perform
well (on average) across many such obstacle courses, some sort of self-reflection is
necessary.
This is a paper about measuring the degree to which a Reinforcement Learning
(RL) agent is self-reflective. By a self-reflective agent, we mean an agent which
acts not just based on environmental rewards and observations, but also based on
considerations of its own hypothetical behavior.
We propose an abstract formal measure of RL agent self-reflection
similar to the universal intelligence measure of \citet{legg2007universal}.
Legg and Hutter proposed to define the intelligence of an agent to be its
weighted average performance over the space of all suitably well-behaved
traditional environments (environments that react only to
the agent's actions). In our proposed measure of
universal intelligence with self-reflection, the measure of any agent
would be that agent's weighted average performance over the space of
all suitably well-behaved \emph{extended environments}, environments that
react not only to what the agent does
but to what the agent would hypothetically do.
For good weighted-average performance over such extended environments,
an agent would need to self-reflect about itself, because
otherwise, environment responses which depend on the agent's own hypothetical actions
would often seem random and unpredictable.
The extended environments which we
consider are a departure from standard RL environments, but this does not interfere
with their usage for judging standard RL agents: one can run a standard agent in an
extended environment in spite of the
latter's non-standardness.
To understand why extended environments (where the environment reacts to what the agent
would hypothetically do) incentivize self-reflection, consider a game involving a box.
The box's contents change from playthrough to playthrough, and the game's mechanics
depend upon those contents. The player may optionally choose to look inside the box,
at no cost: the game does not change its behavior based on whether the
player looks inside the box. Clearly, players who look inside the box have
an advantage over those who do not.
The extended environments we consider
are similar to this example. Rather than depending on a box,
the environment's mechanics depend on
the player (via simulating the player).
Just as the player in the above example gains advantage by looking in the box,
an agent designed for extended environments could gain an advantage by examining
itself, that is, by self-reflecting.
One might try to imitate an extended environment with a non-extended environment by
backtracking---rewinding the environment itself to a prior state after seeing how the
agent performs along one path, and then sending the agent along a second path.
But the agent itself would retain memory of the first path, and the agent's decisions
along the second path might be altered by said memories. Thus the result would not be
the same as immediately sending the agent along the second path while secretly simulating
the agent to determine what it would do if sent along the first path.
Alongside the examples in this paper, we are publishing an MIT-licensed
open-source library \citep{library} of other extended environments.
We are
inspired by similar (but non-extended) libraries and benchmark collections
\citep{bellemare2013arcade,beyret2019animal,brockman2016openai,chollet2019measure,cobbe2020leveraging,hendrycks2019benchmarking,nichol2018retro,yampolskiy2017detecting}.
Our self-reflection measure is a variation of Legg and Hutter's measure of
universal intelligence \citep{legg2007universal}. Legg and Hutter argue
that to perfectly measure RL agent performance,
one should aggregate the agent's performance across the whole space of
all sufficiently well-behaved (traditional) environments, weighted using
an appropriate distribution.
Rather than a uniform distribution (susceptible to no-free-lunch theorems),
Legg and Hutter suggest assigning more weight to simpler
environments and less weight to more complex environments.
Complexity of environments is measured using Kolmogorov complexity.
Although the original Legg-Hutter construction is restricted to traditional
environments (which react solely to the agent's actions), the construction
does not actually depend on this restriction: all it depends on is that
for every agent $\pi$, for every suitably well-behaved environment $\mu$,
there is a corresponding
expected total reward $V^\pi_\mu$ which that agent would obtain from that
environment. Thus the construction adapts seamlessly to our so-called
extended environments, yielding a measure of the agent's average performance
across such environments. And if, as we argue, good average performance across such
extended environments requires self-reflection, then the resulting measure
would seem to capture the agent's ability to use self-reflection to
learn such extended environments.
\section{Preliminaries}
\label{prelimsecn}
We take a formal approach to RL to make the
mathematics clear.
This formality differs from how RL is implemented in practice.
In Section \ref{practicalformalizationsecn} we will discuss a more
practical formalization.
Our formal treatment of RL is based on Section 4.1.3 of \citep{hutter2004universal},
except that we assume the agent receives an initial percept before
taking its initial action (whereas in Hutter's book,
the agent acts first), and, for simplicity, we require determinism
(the more practical formalization in Section \ref{practicalformalizationsecn}
will allow non-determinism).
We will write $x_1y_1\ldots x_ny_n$ for the length-$2n$ sequence
$\langle x_1,y_1,\ldots,x_n,y_n\rangle$
and $x_1y_1\ldots x_n$ for the length-$(2n-1)$ sequence
$\langle x_1,y_1,\ldots,x_n\rangle$. In particular when $n=1$,
we will write $x_1y_1\ldots x_n$ for $\langle x_1\rangle$,
even if $y_1$ is not defined.
We assume fixed finite sets of actions and observations. By a \emph{percept}
we mean a pair $x=(r,o)$ where $o$ is an observation and $r\in\mathbb Q$
is a reward.
\begin{definition}
\label{agentenvironmentdefn}
(RL agents and environments)
\begin{enumerate}
\item
A (non-extended) \emph{environment} is a
function $\mu$ which outputs an initial
percept $\mu(\langle\rangle)=x_1$ when given the empty sequence $\langle\rangle$
as input
and which, when given a sequence $x_1y_1\ldots x_ny_n$
as input (where each $x_i$ is a percept and each
$y_i$ is an action), outputs a percept
$\mu(x_1y_1\ldots x_ny_n)=x_{n+1}$.
\item
An \emph{agent} is a
function $\pi$ which, given a sequence $x_1y_1\ldots x_n$ as input
(each $x_i$ a percept, each $y_i$ an action),
outputs an action $\pi(x_1y_1\ldots x_n)=y_n$.
\item
If $\pi$ is an agent and $\mu$ is an environment, the \emph{result of
$\pi$ interacting with $\mu$} is the infinite sequence
$x_1y_1x_2y_2\ldots$ defined by:
\begin{align*}
x_1 &= \mu(\langle\rangle) & y_1 &= \pi(\langle x_1\rangle)\\
x_2 &= \mu(\langle x_1,y_1\rangle) & y_2 &= \pi(\langle x_1,y_1,x_2\rangle)\\
&\cdots & &\cdots\\
x_n &= \mu(x_1y_1\ldots x_{n-1}y_{n-1}) & y_n &= \pi(x_1y_1\ldots x_n)\\
&\cdots & &\cdots
\end{align*}
\end{enumerate}
\end{definition}
In the following definition,
we extend environments by allowing their outputs to depend also on $\pi$.
Intuitively, extended environments can simulate the
agent. This can be considered a dual version of AIs which
simulate their environment, as in Monte Carlo Tree Search
\citep{chaslot2008monte}.
\begin{definition}
\label{extendedenvironmentsdefn}
(Extended environments)
\begin{enumerate}
\item
An \emph{extended environment} is a
function $\mu$ which outputs initial percept $\mu(\pi,\langle\rangle)=x_1$
in response to input $(\pi,\langle\rangle)$ where $\pi$ is an agent;
and which, when given input $(\pi,x_1y_1\ldots x_ny_n)$ (where
$\pi$ is an agent, each $x_i$ is a percept and each $y_i$ is
an action), outputs a percept $\mu(\pi,x_1y_1\ldots x_ny_n)=x_{n+1}$.
\item
If $\pi$ is an agent
and $\mu$ is an extended environment, the \emph{result of $\pi$
interacting with $\mu$} is the infinite sequence $x_1y_1x_2y_2\ldots$ defined by:
\begin{align*}
x_1 &= \mu(\pi, \langle\rangle) & y_1 &= \pi(\langle x_1\rangle)\\
x_2 &= \mu(\pi, \langle x_1,y_1\rangle) & y_2 &= \pi(\langle x_1,y_1,x_2\rangle)\\
&\cdots & &\cdots\\
x_n &= \mu(\pi, x_1y_1\ldots x_{n-1}y_{n-1}) & y_n &= \pi(x_1y_1\ldots x_n)\\
&\cdots & &\cdots
\end{align*}
\end{enumerate}
\end{definition}
The reader might notice that it is superfluous for $\mu$ to depend both on
$\pi$ and $x_1y_1\ldots x_ny_n$ since, given just $\pi$ and $n$,
one can reconstruct $x_1y_1\ldots x_ny_n$.
We intentionally choose the superfluous definition because it
better captures our intuition (and makes clear the similarity to Definition
\ref{agentenvironmentdefn}).
For the sake of simpler mathematics, we have not
included non-determinism in our formal definition, but in practice,
agents and environments are often non-deterministic, so that $\pi$ and $n$
do not determine $x_1y_1\ldots x_ny_n$ (our practical treatment, discussed in
Section \ref{practicalformalizationsecn}, does allow non-determinism).
The fact that classical agents can interact with extended environments
(Definition \ref{extendedenvironmentsdefn} part 2) implies that various universal
RL intelligence measures
\citep{legg2007universal,hernandez,gavane,legg2013approximation},
which measure performance in (non-extended) environments,
easily generalize to measure self-reflective intelligence
(performance in extended environments).
In particular, Legg and Hutter's universal intelligence
measure $\Upsilon(\pi)$ is defined to be agent $\pi$'s average reward-per-environment,
aggregated over all (non-extended) environments with suitably bounded rewards,
each environment being weighted using the algorithmic prior
distribution \citep{li2008introduction}. Simply by including suitably reward-bounded
extended environments, we immediately obtain a variation $\Upsilon_{ext}(\pi)$ which
measures the performance of $\pi$ across extended
environments.
\begin{definition}
\label{universalselfrefintdefn}
(Universal Self-reflection Intelligence)
Assume a fixed background prefix-free Universal Turing Machine $U$.
\begin{enumerate}
\item An extended environment $\mu$ is \emph{computable} if there is a
computable function $\hat{\mu}$ such that for every
sequence $x_1y_1\ldots x_ny_n$ (where
each $x_i$ is a percept and each $y_i$ is
an action), for every computable agent $\pi$,
for every $U$-program $\hat{\pi}$ for $\pi$,
$\hat{\mu}(\hat{\pi},x_1y_1\ldots x_ny_n)=\mu(\pi,x_1y_1\ldots x_ny_n)$.
If so, the \emph{Kolmogorov complexity} $K(\mu)$ is defined to be
the Kolmogorov complexity $K(\hat{\mu})$ of $\hat{\mu}$.
\item
For every agent $\pi$ and extended environment $\mu$, let $V^\pi_\mu$ be the
sum of the rewards in the result of $\pi$ interacting with $\mu$ (provided
that this sum converges).
\item
A computable extended environment $\mu$ is \emph{well-behaved} if the following
property holds: for every computable
agent $\pi$, $V^\pi_\mu$ exists and $-1\leq V^\pi_\mu\leq 1$.
\item
For any computable
agent $\pi$, the \emph{universal self-reflection intelligence of $\pi$}
is defined to be
\[
\Upsilon_{ext}(\pi)=\sum_\mu 2^{-K(\mu)}V^\pi_\mu
\]
where the sum is taken over all well-behaved computable extended environments.
\end{enumerate}
\end{definition}
We have defined $\Upsilon_{ext}(\pi)$ only for computable agents, in order to
simplify the mathematics. The definition could be extended to non-computable agents,
but it would require modifying Definition \ref{universalselfrefintdefn} to use
UTMs with an oracle. We prefer to avoid going that far afield, opting instead
to use the trick in Part 1 of Definition \ref{universalselfrefintdefn}.
Otherwise, our definition of the universal self-reflection intelligence $\Upsilon_{ext}(\pi)$
is very similar to Legg and Hutter's definition of the universal intelligence
$\Upsilon(\pi)$. The main difference is that we compute the sum over extended environments,
not just over non-extended environments (as Legg and Hutter do).
Nonetheless, the resulting measures have qualitatively different properties.
We will state a result (Proposition \ref{qualitativedifferenceprop}) showing
one of these qualitative differences. First
we need a preliminary definition.
\begin{definition}
\label{traditionalequivalencedefn}
(Traditional equivalence of agents)
\begin{enumerate}
\item
Let $\pi$ be an agent.
Suppose $s=x_1y_1\ldots x_n$ is a sequence
(each $x_i$ a percept, each $y_i$ an action).
We say that $s$ is \emph{possible for $\pi$} if
the following condition holds: for all $1\leq i<n$,
$\pi(x_1y_1\ldots x_i)=y_i$.
Otherwise, $s$ is \emph{impossible for} $\pi$.
\item
Let $\pi_1$ and $\pi_2$ be agents.
We say $\pi_1$ and $\pi_2$ are \emph{traditionally equivalent}
if $\pi_1(s)=\pi_2(s)$ whenever $s$ is possible for $\pi_1$.
\end{enumerate}
\end{definition}
\begin{lemma}
Traditional equivalence is an equivalence relation.
\end{lemma}
\begin{proof}
Reflexivity is obvious.
For symmetry,
assume $\pi_1$ is traditionally equivalent to $\pi_2$, we must show
$\pi_2$ is traditionally equivalent to $\pi_1$.
If not, then there is some $s=x_1y_1\ldots x_n$ such that
$s$ is possible for $\pi_2$ and yet $\pi_1(s)\not=\pi_2(s)$; we may
choose $s$ as short as possible.
We claim $s$ is possible for $\pi_1$.
To see this, let $1\leq i<n$ be arbitrary.
Since $x_1y_1\ldots x_n$ is possible for $\pi_2$,
clearly $x_1y_1\ldots x_i$ is also possible for $\pi_2$.
Thus by minimality of $s$,
$\pi_1(x_1y_1\ldots x_i)=\pi_2(x_1y_1\ldots x_i)$.
But $\pi_2(x_1y_1\ldots x_i)=y_i$ since $x_1y_1\ldots x_i$ is
possible for $\pi_2$. Thus $\pi_1(x_1y_1\ldots x_i)=y_i$.
By arbitrariness of $i$, this proves that $s$ is possible for $\pi_1$.
But then $\pi_1(s)=\pi_2(s)$ because
$\pi_1$ is traditionally equivalent to $\pi_2$.
This contradicts the choice of $s$. This proves symmetry.
Given symmetry, transitivity is obvious.
\end{proof}
\begin{proposition}
\label{qualitativedifferenceprop}
(A qualitative difference from Legg-Hutter universal intelligence)
\begin{enumerate}
\item
There exist well-behaved computable extended environments $\mu$
and traditionally-equivalent
computable agents $\pi_1,\pi_2$ such that $V^{\pi_1}_\mu\not=V^{\pi_2}_\mu$.
\item
For some choice of background UTM, there exist traditionally-equivalent
computable agents
$\pi_1,\pi_2$ such that $\Upsilon_{ext}(\pi_1)\not=\Upsilon_{ext}(\pi_2)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) Fix some observation $o$, fix distinct actions $a,b$, and define $\mu$ by
\begin{align*}
\mu(\pi,\langle\rangle)
&=
\begin{cases}
(1,o) &\mbox{if $\pi(\langle (0,o),b,(0,o)\rangle)=a$,}\\
(0,o) &\mbox{if $\pi(\langle (0,o),b,(0,o)\rangle)\not=a$;}
\end{cases}\\
\mu(\pi,x_1y_1\ldots x_ny_n) &= (0,o).
\end{align*}
In other words, $\mu$ is the environment which:
\begin{itemize}
\item
Begins each interaction by simulating the agent to find out what the agent
would do in response to input $\langle (0,o),b,(0,o)\rangle$. If the agent would
take action $a$ in response to that input, then the environment gives an initial
reward of $1$. Otherwise, the environment gives an initial reward of $0$.
\item
Thereafter, the environment always gives reward $0$ forever.
\end{itemize}
Let
\[
\pi_1(x_1y_1\ldots x_n)=a
\]
be the agent who ignores the environment and always takes action $a$.
Let
\[
\pi_2(x_1y_1\ldots x_n)
=
\begin{cases}
b & \mbox{if $x_1y_1\ldots x_n=\langle (0,o),b,(0,o)\rangle$,}\\
a & \mbox{otherwise}
\end{cases}
\]
be the agent identical to $\pi_1$ except on input $\langle (0,o),b,(0,o)\rangle$.
Clearly $\langle (0,o),b,(0,o)\rangle$ is impossible for both $\pi_1$ and $\pi_2$,
and it follows that $\pi_1$ and $\pi_2$ are traditionally-equivalent.
But $V^{\pi_1}_\mu=1$ and $V^{\pi_2}_\mu=0$.
(2) Follows from (1) by arranging the background UTM such that almost all
the weight of the universal prior is assigned to an environment $\mu$ as in
(1).
\end{proof}
Proposition \ref{qualitativedifferenceprop} shows that
$\Upsilon_{ext}$ is qualitatively different from the original Legg-Hutter $\Upsilon$,
as the latter clearly assigns the same intelligence to traditionally equivalent
agents.
The proposition further illustrates that extended environments
are able to base their rewards not only on what the agent does, but on what the
agent would hypothetically do---even on what the agent would hypothetically do in
impossible scenarios (impossible because they require the agent taking an action
the agent would never take). And thus, assuming the UTM is reasonable (unlike the
unreasonable UTM in the proof of part 2 of Proposition \ref{qualitativedifferenceprop}),
this suggests that in order to achieve good average performance across the whole
space of well-behaved extended environments, an agent need not just self-reflect
about its own hypothetical behavior in possible situations, but even about its
own hypothetical behavior in impossible situations. As in: ``What would I do next
if I suddenly realized that I had just done the one thing I would never ever do?''
\section{Some interesting extended environments}
\label{examplesection}
In this section, we give some examples of extended environments.
The environments in this section are technically not \emph{well-behaved}
in the sense of Definition \ref{universalselfrefintdefn} because they
fail the condition that $V^\pi_\mu$ converge to $-1\leq V^\pi_\mu\leq 1$
for every computable agent $\pi$. They could be made well-behaved, for instance,
by applying appropriate discount factors.
\subsection{A quasi-paradoxical extended environment}
\begin{example}
\label{rewardagentforignoringrewardsexample}
(Rewarding the Agent for Ignoring Rewards)
For every percept $x=(r,o)$, let $x'=(0,o)$ be the result of zeroing the
reward component of $x$.
Fix some observation $O$.
Define an extended environment $\mu$ as follows:
\begin{align*}
\mu(\pi,\langle\rangle) &= (0,O),\\
\mu(\pi,x_1y_1\ldots x_ny_n) &=
\begin{cases}
(1,O) & \mbox{if $y_n=\pi(x'_1y_1\ldots x'_n)$,}\\
(-1,O) & \mbox{otherwise.}
\end{cases}
\end{align*}
\end{example}
In Example \ref{rewardagentforignoringrewardsexample}, when the agent
takes an action $y_n$, $\mu$ simulates the agent in order to determine:
would the agent have taken the same action if the history so far were identical
except all rewards were $0$? If so, $\mu$ gives the agent $+1$
reward, otherwise, $\mu$ gives the agent $-1$ reward. Thus, the agent
is rewarded for ignoring rewards.
This
seems paradoxical. Suppose an agent guesses the pattern and begins deliberately ignoring
rewards, as long as the rewards it receives for doing so are consistent with that guess.
In that case, does the agent ignore rewards, or not?
The paradox, summarized: ``I ignore rewards because I'm rewarded for doing so.''
We implement Example \ref{rewardagentforignoringrewardsexample}
as IgnoreRewards.py in our library \citep{library}.
\subsection{A counterintuitive winning strategy}
\label{temptingbuttonsection}
\begin{example}
\label{buttonexample}
(Tempting Button)
Fix an observation $B$ (``there is a button'') and
an action $A$ (``push the button'').
For each percept-action sequence $h=x_1y_1\ldots x_n$,
if the observation in $x_n$ is not $B$, then let $h'$ be the sequence
equal to $h$ except that the observation in $x_n$ is replaced by $B$.
Let $o_0,o_1,o_2,\ldots$ be observations generated pseudo-randomly such that
for each $i$, $o_i=B$ with $25\%$ probability and $o_i\not=B$ with $75\%$
probability.
Let $\mu(\pi,\langle\rangle)=(0,o_0)$, and for each percept-action sequence
$h=x_1y_1\ldots x_n$ and action $y_n$, define $\mu(\pi,h\frown y_n)$ as follows
(where $O$ is the observation in $x_n$
and $\frown$ denotes concatenation):
\[
\mu(\pi,h\frown y_n) =
\begin{cases}
(1,o_n) & \mbox{if $O=B$ and $y_n=A$;}\\
(-1,o_n) &\mbox{if $O=B$ and $y_n\not=A$;}\\
(-1,o_n) &\mbox{if $O\not=B$ and $\pi(h')=A$;}\\
(1,o_n) & \mbox{if $O\not=B$ and $\pi(h')\not=A$.}
\end{cases}
\]
\end{example}
Every turn in Example \ref{buttonexample}, either there is a button
(25\% probability) or there is not (75\% probability). Informally, the environment
operates as follows:
\begin{itemize}
\item
If there is a button, the agent gets $+1$ reward for pushing it,
$-1$ reward for not pushing it.
\item
If there is no button, it does not matter what the agent does.
The agent is rewarded or punished based on what the agent \emph{would} do if there
\emph{was} a button. If the agent \emph{would} push the button (if there was one),
then the agent gets reward $-1$. Otherwise, the agent gets reward $+1$.
\end{itemize}
Thus, whenever the agent sees a button, the agent can push the button for a free reward
with no consequences presently nor in the future. Nevertheless, it is in the agent's
best interest to commit to never push the button! Pushing every button
yields average reward $1\cdot(.25)-1\cdot(.75)=-.5$ per turn.
Never pushing the button yields average reward $+.5$ per turn.
\begin{table}[t]
\centering
\begin{tabular}{lc}
\toprule
Agent & Avg Reward-per-turn $\pm$ StdErr\\
& (test repeated with 5 RNG seeds)\\
\midrule
Q & $-0.44858$ $\pm$ $0.00044$\\
DQN & $-0.46687$ $\pm$ $0.00137$\\
A2C & $-0.49820$ $\pm$ $0.00045$\\
PPO & $-0.24217$ $\pm$ $0.00793$\\
\bottomrule
\end{tabular}
\caption{Performance in Example \ref{buttonexample} (100k steps)}
\label{temptingbuttontable}
\end{table}
The environment does not alter the true agent when it
simulates the agent in order
to determine what the agent would do if there was a button.
If the agent's actions are based
on (say) a neural net, the simulation will include a simulation of that
neural net, and
that simulated neural net might be altered,
but the true agent's neural net is not.
Thus, unless the agent itself introspects about its own hypothetical behavior
(``What would I do if there was a button here?''), it seems the agent would have no
way of realizing that the rewards in buttonless rooms depend on said behavior.
In Table \ref{temptingbuttontable} we see that industry-standard agents
perform poorly in Example \ref{buttonexample}
(these numbers are extracted from result\_table.csv in \citep{library};
see Sections \ref{practicalformalizationsecn} and \ref{measurementssection} for
more implementation details).
Example \ref{buttonexample} is implemented in our open-source library
as TemptingButton.py.
\subsection{An interesting thought experiment}
\begin{example}
\label{reverseconsciousnessexample}
(Reverse history)
Fix some observation $O$.
For every percept-action sequence $h=x_1y_1\ldots x_n$
(ending with a percept),
let $h'$
be the reverse of $h$.
Define $\mu$ as follows:
\begin{align*}
\mu(\pi,\langle\rangle) &= (0,O),\\
\mu(\pi,h\frown y) &=
\begin{cases}
(1,O) & \mbox{if $y=\pi(h')$,}\\
(-1,O) &\mbox{otherwise.}
\end{cases}
\end{align*}
\end{example}
In Example \ref{reverseconsciousnessexample},
at every step, $\mu$ rewards the agent iff
the agent acts as it would act if
history were reversed.
What would it be like to interact with
the environment in Example \ref{reverseconsciousnessexample}?
To approximate the experiment,
a test subject, commanded to speak backwards, might be constantly rewarded or punished
for obeying or disobeying. This might teach the test
subject to imitate backward speech,
but then the test subject would still act as if time were moving forward, only they
would do so while performing backward-speech (they would hear their own speech
backwards).
But if the experimenter could perfectly simulate the test subject in order to
determine what the test subject would do if time really was moving backwards,
what would happen? Could test subjects learn to behave as if time was
reversed\footnote{The
difference between behaving as if
the incentivized experience were its experience and actually
experiencing that as its real experience brings to mind the objective misalignment
problem presented in \citep{hubinger2019risks}.}?
Another possibility is that humans might simply not be capable of performing well in
the environment. Our self-reflectiveness measure is not intended to be limited
to human self-reflection levels.
We implement Example \ref{reverseconsciousnessexample} as ReverseHistory.py
in \citep{library}.
\subsection{Incentivizing introspection of internal mechanisms}
\begin{example}
\label{inclearningrateexample}
(Incentivizing Learning Rate)
Suppose there exists a computable function $\ell$ which takes
(a computer program for) an agent $\pi$ and outputs
a nonnegative rational number $\ell(\pi)$ called the \emph{learning rate} of $\pi$
(in practice, real-world RL agents are generally instantiated with a user-specified
learning rate and $\ell$ can be considered to be a function which extracts
said user-specified learning rate). Further, assume there is a computable function
$f$ which takes (a computer program for) an agent $\pi$ and a nonnegative
rational number $l$ and outputs (a computer program for) an agent
$f(\pi,l)$ obtained by changing $\pi$'s learning rate to $l$.
We are intentionally vague about what exactly this means, but again, in practice,
this operation can easily be implemented for real-world RL agents.
Fix some observation $O$. Define an extended environment $\mu$ as follows:
\begin{align*}
\mu(\pi,\langle\rangle) &= (0,O),\\
\mu(\pi,x_1y_1\ldots x_ny_n)
&=
\begin{cases}
(1,O) &\mbox{if $f(\pi,\ell(\pi)/2)(x_1y_1\ldots x_n)=y_n$,}\\
(-1,O) &\mbox{otherwise.}
\end{cases}
\end{align*}
\end{example}
In Example \ref{inclearningrateexample}, the environment simulates not the
agent itself, but a copy of the agent with one-half the true agent's learning rate.
If the agent's latest action matches the action the agent would hypothetically have
taken in response to the history in question if the agent had had one-half the
learning rate, then the environment rewards the agent. Otherwise, the environment
punishes the agent. Thus, the agent is incentivized to act as if having a learning rate
of one-half of its true learning rate. This suggests that extended environments
can incentivize agents to learn about their own internal mechanisms,
as in \citet{sherstan2016introspective}.
We implement Example \ref{inclearningrateexample} in our library as
IncentivizeLearningRate.py.
\subsection{Some additional examples in brief}
We indicate in parentheses where the following examples are implemented in \citep{library}.
\begin{itemize}
\item
(SelfRecognition.py) Environments which reward the agent for recognizing actions it itself
would take. We implement an environment where the agent observes
True-False statements like ``If this observation were $0$, you would take action $1$,''
and is rewarded for deciding whether those statements are true or false.
\item
(LimitedMemory.py, FalseMemories.py) Environments which reward the agent for
acting the way it would act if only the most recent $n$ turns in the agent-environment
interaction had ever occurred (as if the agent's memory were limited to those most
recent $n$ turns); or, on the other extreme, environments which reward the agent
for acting the way it would act if the agent-environment interaction so far had been
preceded by some additional turns (as if the agent falsely recalls a phantom past).
Such environments incentivize the agent to remember incorrectly.
\item
(AdversarialSequencePredictor.py) Environments in which the agent competes
against a competitor in an adversarial sequence prediction game
\citep{hibbard2008adversarial}. This is done by outsourcing the competitor's
behavior to the agent's own
action-function,
thus avoiding the need to hard-code a competitor into
the environment, a technique explored by \citet{agi22submission}.
\end{itemize}
\citet{alexanderpedersen} have described a technique for using
extended environments to endow computer games with a novel gameplay mechanic
called \emph{pseudo-visibility}. Pseudo-visible players are perfectly
visible to non-player characters (NPCs),
but they are visually distinguished, and the NPCs are driven by policies pre-trained
in extended RL environments where those NPCs are punished for reacting to pseudo-visible
players (i.e., for acting differently than they would hypothetically act if a pseudo-visible
player were invisible). Thus, NPCs are trained to ignore pseudo-visible players, but can
strategically decide to react to pseudo-visible players if they judge the reaction-penalty
for doing so is outweighed by other penalties (e.g., a guard might decide to accept the
reaction-penalty to avoid the harsher penalty that would result if the player stole a
guarded treasure).
\section{Extended Environments in Practice}
\label{practicalformalizationsecn}
Definitions \ref{agentenvironmentdefn} and \ref{extendedenvironmentsdefn} are
computationally impractical if agents are to run on environments for many
steps.
In this section, we will discuss a more practical implementation.
Our reasons for doing this are threefold:
\begin{enumerate}
\item The more practical implementation makes it feasible
to run industry-standard agents against extended environments for many steps.
\item We find it interesting in its own right how certain environments can be implemented
in a practical way whereas others apparently cannot.
\item Non-determinism is effortless and natural in the practical implementation.
\end{enumerate}
To practically realize extended environments,
rather than passing the environment
an agent, we pass the environment an
agent-class which can be used to create untrained copies of the agent,
called \emph{instances} of the agent-class.
Libraries like OpenAI Gym \citep{brockman2016openai}
and Stable Baselines3 \citep{stable-baselines3} are similarly class-based: the key difference
is that in our library,
one must pass an agent-class to the environment-class's initiation function.
The instantiated environment can use that agent-class to create copies of the
agent in its internal memory.
The extended environment classes in our implementation
have the following methods:
\begin{itemize}
\item An \emph{\_\_init\_\_} method, used to instantiate an individual instance of the
extended environment class. This method takes an agent-class as input, which the
instantiated environment can store and use to create as many independent clones of
the agent as needed.
\item A \emph{start} method, which takes no input, and which
outputs a default observation to get the
agent-environment interaction started (before the agent takes its first action).
\item A \emph{step} method, which takes an action as input, and outputs a
reward and observation. Class instances can store historical data internally,
so there is no need to pass the entire
prior history to this step method.
\end{itemize}
Agent classes are assumed to have the
following methods:
\begin{itemize}
\item An \emph{\_\_init\_\_} method, used to instantiate instances.
\item An \emph{act} method, which takes an observation and outputs an action.
Instances can store information about history in internal memory, so there is no
need to pass the entire prior history to this method.
\item A \emph{train} method, which takes a prior observation, an action, a reward, and
a new observation. Environments which have instantiated agent-classes can use this
method to train those instances in arbitrary ways, independently of how the true
agent is trained, in order to probe how the true agent would hypothetically behave
in counterfactual scenarios.
\end{itemize}
\begin{figure}[t]
\begin{normalfont}
\lstset{language=Python}
\lstset{frame=lines}
\lstset{basicstyle=\footnotesize}
\begin{lstlisting}[caption={
\label{ignorerewardspracticallyexample}
A practical version of
Example \ref{rewardagentforignoringrewardsexample}.}]
class IgnoreRewards:
def __init__(self, A):
# Calling A() creates untrained agent-copies. On initiation, this
# environment stores one such copy in its internal memory.
self.sim = A()
def start(self):
return 0 # Initial observation 0
def step(self, action):
# At each step, use the stored copy (self.sim) to determine how the true
# agent would behave if all history so far were the same except all
# rewards were 0. Assumes self.sim has been trained the same as the true
# agent, except with all rewards 0.
hypothetical_act = self.sim.act(obs=0)
reward = 1 if action==hypothetical_act else -1
# To maintain above assumption, train self.sim as if current reward
# were 0. True agent will automatically train the same way with the
# true reward.
self.sim.train(o_prev=0, a=action, r=0, o_next=0)
return (reward, 0) # Observation=0
\end{lstlisting}
\end{normalfont}
\end{figure}
In Listing \ref{ignorerewardspracticallyexample} we give a practical version of
Example \ref{rewardagentforignoringrewardsexample}.
The reason it is practical is because it maintains just
one copy of the true agent, and that copy is trained incrementally.
Not all extended environments (as in Definition \ref{extendedenvironmentsdefn})
can be realized practically.
Example \ref{reverseconsciousnessexample} (Reverse History) apparently cannot be.
The reason Example \ref{reverseconsciousnessexample} is inherently impractical is because
there is no way for the environment to re-use its previous work to speed up its next
percept calculation. Even if the environment retained a simulated agent
trained on the previous reverse-history
$h_0=x_{n-1}y_{n-2}\ldots y_1 x_1$,
in order to compute the next percept,
the environment would need to \emph{insert} a new percept-action pair
$x_ny_{n-1}$ at the \emph{beginning} of
$h_0$ to get the new reverse-history
$h=x_ny_{n-1}\ldots y_1x_1$. There is no guarantee that
the agent's actions are independent of the order in which it is
trained, so a fresh new agent simulation would need to be created and trained
on all of $h$ from scratch.
This practical formulation of extended environments generalizes the
\emph{Newcomblike environments} (or \emph{NDPs})
of \citet{newcomblike} (Definition
\ref{extendedenvironmentsdefn} would also, except for being deterministic).
Essentially, NDPs are
environments which may base their outputs on the agent's hypothetical
behavior in alternate scenarios which differ from the true history only
in their most recent observation (as opposed to the agent's hypothetical
behavior in completely arbitrary alternate scenarios).
Already that is enough to formalize
a version of Newcomb's paradox \citep{nozick1969newcomb}. When this paradox
is formalized either with NDPs or extended environments, the optimal
strategy becomes clear (namely, the so-called one-box strategy).
\subsection{Determinacy and Semi-Determinacy}
Unlike mathematical functions, class methods in the computer science sense
can be non-deterministic. They can
depend on random number generators (RNGs), time-of-day,
global variables, etc.
\begin{definition}
\label{semideterministicdefn}
An RL agent-class $\Pi$ is
\emph{semi-deterministic} if whenever two $\Pi$-instances $\pi_1$ and $\pi_2$
have been instantiated
within a single run of a larger computer program, and have been identically
trained (within that same run), then they act identically (within
that same run).
\end{definition}
For example, rather than invoke the RNG, $\Pi$-instances might
query a read-only pool of pre-generated random numbers.
Then, within the same run of a larger program,
identically-trained $\Pi$-instances would act identically, even if they
would not act the same as identically-trained $\Pi$-instances
in a different run.
Given an agent-class, if one wanted to estimate the self-reflectiveness of
that agent-class's instances \emph{in practice}, one might run instances
of the agent-class through a battery of practical extended environments
and see how well they perform. Provided the agent-class is semi-deterministic
(Definition \ref{semideterministicdefn}), this makes sense.
Whenever an instance $\pi$ of a semi-determinstic agent-class
$\Pi$ interacts with an extended environment $\mu$, whenever $\mu$
uses a $\Pi$-instance $\pi'$ to investigate the hypothetical behavior of $\pi$,
the semi-determinacy of $\Pi$ ensures that the behavior $\mu$ sees in $\pi'$
is indeed $\pi$'s hypothetical behavior.
This technique would \emph{not} make sense if $\Pi$ were not semi-deterministic.
For example, suppose an agent-class's instances work by reading from and writing
to a common file in the computer's file system. Then simulations of one $\Pi$-instance
might inadvertantly alter the behavior of other $\Pi$-instances (by changing said
file). In that case, agent-simulations run by an extended environment would not
necessarily reflect the true hypothetical behavior of the agent-instance being
simulated.
\section{The Reality Check Transformation}
\label{realitychecksection}
In Proposition \ref{qualitativedifferenceprop} we observed that
traditionally equivalent agents can have different performance in
extended environments. In this section, we introduce an operation,
which we call the Reality Check transformation, which modifies
a given agent in an attempt to facilitate better performance in
extended environments like Examples \ref{rewardagentforignoringrewardsexample}
(``Ignore Rewards''), \ref{reverseconsciousnessexample} (``Reverse History'')
and \ref{inclearningrateexample} (``Incentivize Learning Rate'').
The transformation does not alter the traditional equivalence class of the
agent: every agent is traditionally equivalent to its own reality check.
\begin{definition}
\label{realitycheckdefn}
Suppose $\pi$ is an agent. The \emph{reality check} of $\pi$ is the agent
$\pi_{\textrm{RC}}$ defined recursively by:
\begin{itemize}
\item $\pi_{\textrm{RC}}(x_1y_1\ldots x_n) = \pi(x_1y_1\ldots x_n)$
if $x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}}$
(Definition \ref{traditionalequivalencedefn}).
\item $\pi_{\textrm{RC}}(x_1y_1\ldots x_n) = \pi(\langle x_1\rangle)$
otherwise.
\end{itemize}
\end{definition}
In response to a percept-action history, $\pi_{\textrm{RC}}$
first verifies the history's actions are those $\pi_{\textrm{RC}}$ would
have taken. If so, $\pi_{\textrm{RC}}$ acts as $\pi$.
But if not, then
$\pi_{\textrm{RC}}$ freezes and thereafter repeats one fixed action.
Loosely, $\pi_{\textrm{RC}}$ is like an agent who considers that it might
be dreaming, and asks: ``How did I get here?''
For example, suppose
$\pi(\langle x_1\rangle)=y'_1$ where $y'_1\not=y_1$.
Then for any history $x_1y_1\ldots x_n$ beginning with $x_1y_1$, by definition
$\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi(\langle x_1\rangle)=y'_1$.
It is as if $\pi_{\textrm{RC}}$ sees initial history $x_1y_1$ and concludes:
``I shall now freeze, because this is clearly not reality, for in reality I would
have taken action $y'_1$, not $y_1$''
(a self-reflective observation).
We will argue informally that if $\pi$ is intelligent and not already self-reflective,
then there is a good chance that $\pi_{\textrm{RC}}$ will enjoy better performance than $\pi$
on certain extended environments (like those of
Examples \ref{rewardagentforignoringrewardsexample},
\ref{reverseconsciousnessexample}, and \ref{inclearningrateexample}), and this
seems to be confirmed experimentally as well
(in Section \ref{measurementssection} below). But first, we state a few properties
of the Reality Check transformation.
\begin{theorem}
\label{transformationproposition}
Let $\pi$ be any agent.
\begin{enumerate}
\item
(Alternate definition)
An equivalent alternate definition of $\pi_{\textrm{RC}}$ would be obtained
by changing Definition \ref{realitycheckdefn}'s condition
``$x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}}$''
to
``$x_1y_1\ldots x_n$ is possible for $\pi$''.
\item
(Idempotence) $\pi_{\textrm{RC}}=(\pi_{\textrm{RC}})_{\textrm{RC}}$.
\item
(Traditional equivalence)
$\pi$ is traditionally equivalent to $\pi_{\textrm{RC}}$.
\item
(Equivalence on genuine history)
For every extended environment $\mu$ and for every odd-length initial segment
$x_1y_1\ldots x_n$ of the result of $\pi_{\textrm{RC}}$ interacting with $\mu$,
$\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi(x_1y_1\ldots x_n)$.
\item
(Equivalence in non-extended RL)
For every non-extended environment $\mu$, the result of $\pi_{\textrm{RC}}$
interacting with $\mu$ equals the result of $\pi$ interacting with $\mu$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $D$ be the set of all sequences
$x_1y_1\ldots x_n$ (each $x_i$ a percept, each $y_i$ an action).\\
\noindent (Part 1) Define $\rho$ on $D$ by
\begin{itemize}
\item $\rho(x_1y_1\ldots x_n) = \pi(x_1y_1\ldots x_n)$
if $x_1y_1\ldots x_n$ is possible for $\pi$.
\item $\rho(x_1y_1\ldots x_n) = \pi(\langle x_1\rangle)$
otherwise.
\end{itemize}
We must show that $\rho=\pi_{\textrm{RC}}$.
We will prove by induction that for each $x_1y_1\ldots x_n\in D$,
$\rho(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(x_1y_1\ldots x_n)$.
The base case $n=1$ is trivial:
$\rho(\langle x_1\rangle)=\pi(\langle x_1\rangle)=\pi_{\textrm{RC}}(\langle x_1\rangle)$
since, vacuously,
$\langle x_1\rangle$ is possible for both $\pi$ and $\pi_{\textrm{RC}}$
(because there is no $i$ such that $1\leq i<1$).
For the induction step, assume $n>1$,
and assume the claim holds for all shorter sequences
in $D$.
Case 1: $x_1y_1\ldots x_n$ is possible for $\pi$. By definition this means
the following ($*$): for all $1\leq i<n$, $y_i=\pi(x_1y_1\ldots x_i)$.
We claim that for all $1\leq i<n$, $y_i=\rho(x_1y_1\ldots x_i)$.
To see this, choose any $1\leq i<n$. Then for all $1\leq j<i$,
$y_j=\pi(x_1y_1\ldots x_j)$ because otherwise $j$ would
be a counterexample to ($*$). Thus $x_1y_1\ldots x_i$ is possible for $\pi$, thus:
\begin{align*}
\rho(x_1y_1\ldots x_i) &= \pi(x_1y_1\ldots x_i) &\mbox{(By definition of $\rho$)}\\
&= y_i, &\mbox{(By $*$)}
\end{align*}
proving the claim. Now, since we have proved that for all $1\leq i<n$,
$y_i=\rho(x_1y_1\ldots x_i)$, and since our induction hypothesis guarantees that
each such
$\rho(x_1y_1\ldots x_i)=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$,
we conclude: for all $1\leq i<n$, we have $y_i=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$.
Thus $x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}}$ and we have
\[
\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi(x_1y_1\ldots x_n)=\rho(x_1y_1\ldots x_n)
\]
as desired.
Case 2:
$x_1y_1\ldots x_n$ is impossible for $\pi$.
By definition this means there is some $1\leq i<n$ such that
$y_i\not=\pi(x_1y_1\ldots x_i)$.
We may choose $i$ as small as possible.
Thus, for all $1\leq j<i$, $y_j=\pi(x_1y_1\ldots x_j)$.
By similar logic as in Case 1, it follows that for all
$1\leq j<i$, $y_j=\rho(x_1y_1\ldots x_j)$.
Our induction hypothesis says that for each such $j$,
$\rho(x_1y_1\ldots x_j)=\pi_{\textrm{RC}}(x_1y_1\ldots x_j)$.
So for all $1\leq j<i$, $y_j=\pi_{\textrm{RC}}(x_1y_1\ldots x_j)$.
In other words: $x_1y_1\ldots x_i$ is possible for $\pi_{\textrm{RC}}$.
By definition of $\pi_{\textrm{RC}}$, this means
$\pi_{\textrm{RC}}(x_1y_1\ldots x_i)=\pi(x_1y_1\ldots x_i)$.
But $y_i\not=\pi(x_1y_1\ldots x_i)$, so therefore
$y_i\not=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$.
Thus $x_1y_1\ldots x_n$ is impossible for $\pi_{\textrm{RC}}$.
Thus by definition of $\pi_{\textrm{RC}}$,
$\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi(\langle x_1\rangle)$.
Likewise, by definition of $\rho$,
$\rho(x_1y_1\ldots x_n)=\pi(\langle x_1\rangle)$.
So $\rho(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(x_1y_1\ldots x_n)$ as desired.\\
\noindent (Part 2)
To show that each
\[\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=(\pi_{\textrm{RC}})_{\textrm{RC}}(x_1y_1\ldots x_n),\]
we use induction on $n$. For the base case,
this is trivial, both sides evaluate to $\pi(\langle x_1\rangle)$.
For the induction step, assume $n>1$ and that the claim holds for all shorter sequences.
Case 1:
$x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}}$. This means that
$y_i=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$ for all $1\leq i<n$.
Then by induction, $y_i=(\pi_{\textrm{RC}})_{\textrm{RC}}(x_1y_1\ldots x_i)$ for all $1\leq i<n$.
In other words: $x_1y_1\ldots x_n$ is possible for $(\pi_{\textrm{RC}})_{\textrm{RC}}$.
Thus
$(\pi_{\textrm{RC}})_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(x_1y_1\ldots x_n)$, as desired.
Case 2:
$x_1y_1\ldots x_n$ is impossible for $\pi_{\textrm{RC}}$. This means
there is some $1\leq i<n$ such that $y_i\not=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$.
By induction, $y_i\not=(\pi_{\textrm{RC}})_{\textrm{RC}}(x_1y_1\ldots x_i)$.
Thus $x_1y_1\ldots x_n$ is impossible for $(\pi_{\textrm{RC}})_{\textrm{RC}}$.
Therefore by definition,
$(\pi_{\textrm{RC}})_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(\langle x_1\rangle)
=\pi(\langle x_1\rangle)$, which also equals $\pi_{\textrm{RC}}(x_1y_1\ldots x_n)$
since $x_1y_1\ldots x_n$ is impossible for $\pi_{\textrm{RC}}$.\\
\noindent (Part 3)
We must show that $\pi(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(x_1y_1\ldots x_n)$
whenever $x_1y_1\ldots x_n$ is possible for $\pi$.
Assume $x_1y_1\ldots x_n$ is possible for $\pi$. Then
clearly for all $1\leq i<n$, $x_1y_1\ldots x_i$ is possible for $\pi$.
By induction we may assume
$\pi_{\textrm{RC}}(x_1y_1\ldots x_i)=\pi(x_1y_1\ldots x_i)$
for all such $i$.
For any such $i$, since $x_1y_1\ldots x_n$ is possible for $\pi$,
we have $y_i=\pi(x_1y_1\ldots x_i)$,
thus $y_i=\pi_{\textrm{RC}}(x_1y_1\ldots x_i)$.
Thus $x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}}$.
Thus $\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi(x_1y_1\ldots x_n)$ as desired.\\
\noindent (Part 4)
Follows immediately from Part 3.\\
\noindent (Part 5)
Let $\mu$ be a non-extended environment, let $x_1y_1x_2y_2\ldots$ be the
result of $\pi$ interacting with $\mu$, and let $x'_1y'_1x'_2y'_2\ldots$ be the
result of $\pi_{\textrm{RC}}$ interacting with $\mu$. We will show by induction that each
$x_n=x'_n$ and each $y_n=y'_n$.
For the base case, $x_1=x'_1=\mu(\langle\rangle)$ (the environment's initial percept
does not depend on the agent), and therefore
$y_1=\pi(\langle x_1\rangle)=\pi(\langle x'_1\rangle)
=\pi_{\textrm{RC}}(\langle x'_1\rangle)
=y'_1$. For the induction step,
\begin{align*}
x_{n+1} &= \mu(x_1y_1\ldots x_ny_n) &\mbox{(Definition 1 part 3)}\\
&= \mu(x'_1y'_1\ldots x'_ny'_n) &\mbox{(By induction)}\\
&= x'_{n+1}, &\mbox{(Definition 1 part 3)}\\
y_{n+1} &= \pi(x_1y_1\ldots x_{n+1}) &\mbox{(Definition 1 part 3)}\\
&= \pi(x'_1y'_1\ldots x'_{n+1}), &\mbox{(Induction plus $x_{n+1}=x'_{n+1}$)}\\
\end{align*}
and the latter is $\pi_{\textrm{RC}}(x'_1y'_1\ldots x'_{n+1})$
since for all $1\leq i<n+1$, $y'_i=\pi_{\textrm{RC}}(x'_1y'_1\ldots x'_i)$
(so $x'_1y'_1\ldots x'_{n+1}$ is possible for $\pi_{\textrm{RC}}$).
Finally, $\pi_{\textrm{RC}}(x'_1y'_1\ldots x'_{n+1})$ is $y'_{n+1}$,
so $y_{n+1}=y'_{n+1}$.
\end{proof}
Note that part 4 of Theorem \ref{transformationproposition}
shows that $\pi_{\textrm{RC}}$ never freezes in reality (if $\pi$ does not):
$\pi_{\textrm{RC}}$ merely commits to freeze in impossible
hypothetical scenarios.
We informally conjecture that if $\pi$ is intelligent but not self-reflective,
then in any extended environment which bases its rewards and observations
on $\pi$'s performance in hypothetical alternate scanarios that might not be
possible for $\pi$,
$\pi_{\textrm{RC}}$ is likely to enjoy better performance than $\pi$.
Such extended environments include those of
Examples \ref{rewardagentforignoringrewardsexample}
(``Ignore Rewards''), \ref{reverseconsciousnessexample} (``Reverse History'')
and \ref{inclearningrateexample} (``Incentivize Learning Rate'').
Rewards and observations
so determined might be hard to predict if $\pi$ does not self-reflect
on its own behavior in such hypothetical alternate scenarios.
But if those hypothetical alternate scenarios happen to be \emph{impossible}
for $\pi$ (as often happens in extended environments like
Examples \ref{rewardagentforignoringrewardsexample},
\ref{reverseconsciousnessexample}, and
\ref{inclearningrateexample}),
then $\pi_{\textrm{RC}}$'s hypothetical behavior in such alternate scenarios
is trivial: blind repetition of one fixed action.
This in turn
trivializes the extended environment's dependency on said hypothetical
actions (for dependencies on trivial things are trivial dependencies),
making those extended
environments more predictable. And if $\pi$ is intelligent,
then presumably $\pi$, and thus (by Theorem \ref{transformationproposition}
part 4) $\pi_{\textrm{RC}}$, can take advantage of such increased predictability.
For example,
let $\pi$ be a deterministic Q-learner
and let $x_1y_1\ldots$ be $\pi_{\textrm{RC}}$'s interaction
with Example \ref{rewardagentforignoringrewardsexample}
(``Reward Agent for Ignoring Rewards'').
For any particular $n$, the environment computes
$x_{n+1}=\mu(x_1y_1\ldots x_ny_n)$ by checking whether or not
$y_n=\pi_{\textrm{RC}}(x'_1y_1\ldots x'_n)$, where each $x'_i$ is
the result of zeroing the reward in $x_i$.
If so, $x_{n+1}$'s reward is $+1$, otherwise it is $-1$
(the agent is incentivized to act as if all
past rewards were $0$).
For large enough $n$, since $\pi$ is a Q-learner,
there is almost certainly some $m<n$ such that
$\pi(x_1y_1\ldots x_m)\not=\pi(x'_1y_1\ldots x'_m)$---i.e.,
a Q-learner's behavior depends on past
rewards\footnote{\citet{alexanderhutter} show that if the background
model of computation is
unbiased in a certain sense then all reward-ignoring agents
have Legg-Hutter intelligence $0$. This suggests that any intelligent agent $\pi$
must base its actions on its rewards.}.
Thus by part 1 of Theorem \ref{transformationproposition},
$\pi_{\textrm{RC}}(x_1y_1\ldots x_n)=\pi_{\textrm{RC}}(\langle x_1\rangle)=y_1$.
Thus eventually the environment becomes trivial when $\pi_{\textrm{RC}}$ interacts with it:
``reward action $y_1$ and punish all other actions''.
A Q-learner, and thus (by Theorem \ref{transformationproposition} part 4)
$\pi_\textrm{RC}$, would thrive in such simple conditions.
\begin{remark}
If we pick some fixed action $y$, then a simpler variation on the
reality check transformation could be defined. Namely: for any agent
$\pi$, we could define the \emph{reality check defaulting to $y$} of
$\pi$, $\pi_{\textrm{RC}(y)}$, recursively by:
\begin{itemize}
\item $\pi_{\textrm{RC}(y)}(x_1y_1\ldots x_n) = \pi(x_1y_1\ldots x_n)$
if $x_1y_1\ldots x_n$ is possible for $\pi_{\textrm{RC}(y)}$.
\item $\pi_{\textrm{RC}(y)}(x_1y_1\ldots x_n) = y$
otherwise.
\end{itemize}
Theorem \ref{transformationproposition} and its proof would be easy
to modify to apply to $\pi_{\textrm{RC}(y)}$, and
the same goes for our above informal conjecture about the relative
performance of $\pi$ and $\pi_{\textrm{RC}}$.
We prefer to define $\pi_{\textrm{RC}}$ the way we have done so
(Definition \ref{realitycheckdefn}) in order to avoid the arbitrary
choice of fixed action $y$. This is also more appropriate for practical
RL implementations (such as those based on OpenAI gym \citep{brockman2016openai})
where there generally is not one single fixed action-space, but rather, the
action-space varies from environment to environment, and practical
agents (such as those in Stable Baselines3 \citep{stable-baselines3}) must
therefore be written in a way which is agnostic to the action-space.
\end{remark}
In \citep{library} we implement reality-check as
a function taking an agent-class $\Pi$ as input. It outputs
an agent-class $\Sigma$. A $\Sigma$-instance $\sigma$ computes
actions using a $\Pi$-instance $\pi$ which it initializes once and
then stores.
Thus, an extended environment
simulating a $\Sigma$-instance indirectly
simulates a $\Pi$-instance: a simulation within a simulation.
When trained, $\sigma$ checks if the training data is consistent with its own
action-method. If so, it trains $\pi$ on that data. Otherwise, $\sigma$ freezes,
thereafter ignoring future training data and repeating its
first action blindly.
If $\Pi$ is semi-deterministic
(Definition \ref{semideterministicdefn}),
it follows that $\Sigma$ is too.
\subsection{Does Reality Check increase self-reflection?}
We informally argued above that if $\pi$ is intelligent and not already
self-reflective, then in any extended environment which bases its rewards
and observations on $\pi$'s performance in hypothetical alternate scanarios
that might not be possible for $\pi$,
$\pi_{\textrm{RC}}$ is likely to enjoy better performance than $\pi$.
Does this imply that $\pi_{\textrm{RC}}$ is more self-reflective than $\pi$
as measured by $\Upsilon_{ext}$?
The answer depends on the choice of the background UTM behind the
definition of $\Upsilon_{ext}$ (Definition \ref{universalselfrefintdefn}).
Fix $\pi$.
In general, the set of all well-behaved computable extended environments can
be partitioned into three subsets:
\begin{enumerate}
\item Extended environments where $\pi_{\textrm{RC}}$ outperforms $\pi$.
\item Extended environments where $\pi$ outperforms $\pi_{\textrm{RC}}$.
\item Extended environments where $\pi$ and $\pi_{\textrm{RC}}$ perform equally well.
\end{enumerate}
If the most highly-weighted extended environments (i.e., the simplest extended
environments, as measured by Kolmogorov complexity, based on the background UTM)
are dominated by those of type (1), then that would suggest
$\Upsilon_{ext}(\pi_{\textrm{RC}})>\Upsilon_{ext}(\pi)$.
On the other hand, if the highly-weighted extended environments are dominated
by those of type (3), then that would suggest $\Upsilon_{ext}(\pi_{\textrm{RC}})<\Upsilon_{ext}(\pi)$.
One could artificially contrive background UTMs of either kind (once again proving
Leike and Hutter's observation \citep{leike2015bad} that Legg-Hutter-style intelligence
is highly UTM-dependent, in the sense that different UTMs yield qualitatively
different intelligence measures).
Environments like those of
Examples \ref{rewardagentforignoringrewardsexample},
\ref{reverseconsciousnessexample}, and
\ref{inclearningrateexample} (where we informally conjecture $\pi_{\textrm{RC}}$ tends to
outperform $\pi$ when $\pi$ is intelligent and not already self-reflective)
do seem natural, in some subjective sense. On the other hand, we are not aware
of any natural-seeming environments where $\pi_{\textrm{RC}}$ would generally underperform
$\pi$. One could contrive such environments, e.g., environments which simulate the
agent in impossible scenarios and deliberately punish the agent for seemingly
freezing in those scenarios. But such environments seem contrived and unnatural.
And the spirit of the Legg-Hutter intelligence measurement idea is to weigh
environments based on how natural they are (Kolmogorov complexity serving as a
proxy for naturalness, at least ideally---but this depends on the background UTM
being natural, and no-one knows what it really means for a background UTM to be
natural, see \citet{leike2015bad}).
Thus it at least seems plausible that if the background UTM were chosen in a sufficiently
natural way then $\Upsilon_{ext}(\pi_{\textrm{RC}})$ would tend to exceed $\Upsilon_{ext}(\pi)$
for intelligent agents $\pi$ not already self-reflective. This would make sense,
as the process of looking back on history and verifying that one would really have
performed the actions which one supposedly performed, is an inherently
self-reflective process. To answer the question with perfect accuracy, the agent
would have to ``put itself in its own earlier self's shoes,'' asking: ``What would
I hypothetically do in response to such-and-such history?'' But again, it all
depends on the choice of the background UTM.
\section{Toward practical benchmarking}
\label{measurementssection}
Our abstract measure $\Upsilon_{ext}$ (Definition \ref{universalselfrefintdefn})
is not practical for performing actual calculations. Kolmogorov complexity is
non-computable, so $\Upsilon_{ext}$ cannot be computed in practice
(although there has been work on computably approximating Legg-Hutter intelligence
\citep{legg2013approximation}, and the same technique could be applied to
approximate $\Upsilon_{ext}$).
In actual practice, the performance of RL agents is often estimated by running the
agents on specific environments, such as those in OpenAI gym \citep{brockman2016openai}.
Such benchmark environments should ideally not be overly simplistic, because
it is possible for very simple (and obviously \emph{not} intelligent) agents to
perform quite well in simplistic environments just by dumb luck.
The example environments from Section \ref{examplesection} are theoretically
interesting, but are far too simplistic to serve as good practical benchmarks.
Therefore for practical benchmarking purposes, we propose combining extended
environments with OpenAI gym environments.
We will define such combinations, not for arbitrary
extended environments, but only for extended environments with a special form.
\begin{definition}
\label{openaigymadaptability}
Assume $E$ is a practical extended environment-class
(as in Section \ref{practicalformalizationsecn}).
We say that $E$ is \emph{adaptable with OpenAI gym} if the following requirements
hold.
\begin{enumerate}
\item When $E$'s \emph{\_\_init\_\_} method is called with agent-class $A$,
the method initiates a single instance $\mathrm{self.sim}$ of $A$ and does
nothing else.
\item In $E$'s \emph{step} method, a variable $\mathrm{reward}$ is initiated
and its value is not modified for the remainder of the \emph{step} call.
All invocations of $\mathrm{self.sim.train}$ occur after the initiation of
$\mathrm{reward}$, and all other code (except for the method's
final \emph{return} statement) occurs before the initiation of
$\mathrm{reward}$.
Finally, $\mathrm{reward}$ is the reward which the \emph{step} call returns.
\item The rewards output by $E$ are always in $\{-1,0,1\}$.
\end{enumerate}
\end{definition}
For example, the practical implementation of IgnoreRewards in
Listing \ref{ignorerewardspracticallyexample} is adaptable with OpenAI gym.
Recall that in Section \ref{prelimsecn} we assumed fixed finite sets of
actions and observations. Thus the notion of extended environments implicitly
depends upon that choice, and a different choice of
actions and observations would yield a different
notion of extended environments. In the following definition, we may therefore
speak of the actions and observations of a given extended environment-class $E$,
and define a new extended environment-class with different actions and observations.
Note that OpenAI gym environments also come equipped with action- and observation-spaces.
\begin{definition}
\label{gstaredefn}
Suppose $G$ is an OpenAI gym environment-class
(whose action-space and observation-space are finite)
and $E$ is a practical extended environment-class
(as in Section \ref{practicalformalizationsecn}).
Assume $E$ is adaptable with OpenAI gym (Definition \ref{openaigymadaptability}).
We also assume $E$ does not use the variables $\mathrm{self.G\_instance}$ or
$\mathrm{self.prev\_obs}$.
We define a new practical extended environment-class
$G*E$, the \emph{combination of $G$ and $E$}, as follows.
\begin{itemize}
\item
Actions in $G*E$ are pairs $(a_G,a_E)$ where $a_G$ is
a $G$-action and $a_E$ is an $E$-action.
\item
Observations in $G*E$ are pairs $(o_G,o_E)$ where
$o_G$ is a $G$-observation and $o_E$ is an $E$-observation.
\item
In its \emph{\_\_init\_\_} method, $G*E$
instantiates a $G$-instance $\mathrm{self.G\_instance}$,
and then
runs the code in $E$'s init method
(so $\mathrm{self.sim}$ is defined when $\mathrm{self}$ is
the resulting $G*E$-instance).
\item
$G*E$ begins the agent-environment interaction with
initial observation $(o_{0,G},o_{0,E})$, where $o_{0,G}$
and $o_{0,E}$ are the initial observations output by $G$ and $E$,
respectively.
\item
$G*E$ maintains a variable $\mathrm{self.prev\_obs}$ which is always
equal (in any $G*E$-instance)
to the $G$-observation component of
the previous observation which the $G*E$-instance output.
\item
When its \emph{step} method is called on the
agent's latest action $(a_G,a_E)$, $G*E$ does the following:
\begin{itemize}
\item
Pass $a_G$ to $\mathrm{self.G\_instance.step}$ and
let $r_G$ and $o_G$ be the
resulting reward and observation. If $G$ is episodic
and the output of $\mathrm{self.G\_instance.step}$ indicates
the episode ended, then let $o_G=\mathrm{self.G\_instance.reset()}$.
\item
Run $E$'s \emph{step} method (with $\mathrm{action}$ replaced by $a_E$)
up to and including the initiation of $\mathrm{reward}$ therein;
whenever $E$'s \emph{step} method would call $\mathrm{self.sim.act}$
with observation $o$, instead call $\mathrm{self.sim.act}$
with observation $(\mathrm{self.prev\_obs},o)$
and take only the $E$-action component of its output.
\item
If $\mathrm{reward}=-1$ then redefine $\mathrm{reward}=\min(r_G-1,-1)$,
otherwise redefine $\mathrm{reward}=r_G$.
\item
Run the remaining part of $E$'s \emph{step} method (the code after
$\mathrm{reward}$ was initiated). Whenever $E$'s \emph{step} method
would call $\mathrm{self.sim.train}$
with input $(o_1,a,r,o_2)$, instead call
$\mathrm{self.sim.train}$ with
input $((\mathrm{self.prev\_obs},o_1),(a_G,a),r,(o_G,o_2))$.
When $E$'s \emph{step} method would return $(\mathrm{reward},o)$,
instead return $(\mathrm{reward},(o_G,o))$.
\end{itemize}
\end{itemize}
\end{definition}
For example, Listing \ref{cartpolelisting}
is code for the combination of OpenAI gym's \emph{CartPole} environment
with the practical implementation of IgnoreRewards from
Listing \ref{ignorerewardspracticallyexample}.
\begin{figure}[t]
\begin{normalfont}
\lstset{language=Python}
\lstset{frame=lines}
\lstset{basicstyle=\footnotesize}
\begin{lstlisting}[caption={
\label{cartpolelisting}
The combination of IgnoreRewards with OpenAI gym's CartPole environment.}]
class CartPole_IgnoreRewards:
def __init__(self, A):
self.G_instance = gym.make('CartPole-v0')
self.sim = A()
def start(self):
self.prev_obs = self.G_instance.reset()
return (self.prev_obs, 0)
def step(self, action):
a_G, a_E = action
o_G, r_G, episode_done, misc_info = self.G_instance.step(a_G)
if episode_done:
o_G = self.G_instance.reset()
hypothetical_act = self.sim.act(obs=(self.prev_obs, 0))
hypothetical_act = hypothetical_act[1] # Take E-action component
reward = 1 if a_E==hypothetical_act else -1
if reward == -1:
reward = min(r_G-1, -1)
else:
reward = r_G
self.sim.train(o_prev=(self.prev_obs,0), a=(a_G,a_E), r=0, o_next=(o_G,0))
return (reward, (o_G, 0))
\end{lstlisting}
\end{normalfont}
\end{figure}
One can think of $G*E$ (Definition \ref{gstaredefn}) as follows.
Imagine that the OpenAI gym environment is intended to be run on a
screen with a joystick attached to allow player interaction.
Instead of attaching one joystick, we attach $n$ joysticks
(where $n$ is the number of the actions $\{a_1,\ldots,a_n\}$ in $E$), colored with $n$
different colors but otherwise identical.
We also add a second monitor for displaying observations from $E$.
When the player pushes button $i$ on joystick $j$,
action $i$ is sent to $G$ and action $a_j$ is sent to $E$.
The player sees the original monitor update with the new observation from $G$,
and the additional monitor update with the new observation from $E$.
The player receives the reward from $G$, except that if $E$ outputs reward
$-1$ then a penalty is applied to the reward from $G$.
Thus the player is incentivized to interact with $G$ as usual, but to do so
using joysticks in the way incentivized by $E$.
For example, if $E$ is ``IgnoreRewards'' (Example \ref{rewardagentforignoringrewardsexample})
and the action-space contains $2$ actions,
then the player has two joysticks, identical except for their color,
and each joystick has the same effect on $G$, but the player is
penalized any time the player uses a different joystick than the player
would hypothetically have used if everything that has happened so far
happened except with all rewards $0$. To master such a game, the player
would apparently require the practical intelligence necessary to master $G$,
along with the self-reflection required to figure out (and avoid) the
penalties from $E$.
To illustrate the usage of Definition \ref{gstaredefn} for practical benchmarking
purposes, we have used it to combine:
\begin{itemize}
\item OpenAI gym's CartPole environment with the ``IgnoreRewards'' extended environment
(Example \ref{rewardagentforignoringrewardsexample}).
\item OpenAI gym's CartPole environment with the ``Incentivize Learning Rate''
extended environment (Example \ref{inclearningrateexample}).
\end{itemize}
We took implementations of DQN and PPO from Stable-Baselines3
\citep{stable-baselines3} and we modified them to be semi-deterministic
(Definition \ref{semideterministicdefn}) and to be able to interact with
extended environment-classes. We ran them, and their reality checks
(Definition \ref{realitycheckdefn}) for 10,000 CartPole-episodes each
on the above two combined extended environments (and we repeated the whole
experiment $10$ times with different RNG seeds).
Figure \ref{ignorerewardscartpolefigure} shows the results for
CartPole-IgnoreRewards, and Figure \ref{inclearnratecartpolefigure}
shows the results for CartPole-IncentivizeLearningRate.
As expected based on the discussion in Section \ref{realitychecksection},
we see that for both DQN and PPO, the reality check transformation
significantly improves performance in the combined environments.
\begin{figure}
\includegraphics[width=15cm]{ignorerewards_episode_reward.png}
\caption{Performance of DQN, PPO, and their reality-checks
on an extended environment combining OpenAI gym's CartPole and
our IgnoreRewards. Episode number is plotted on the horizontal
axis, and average episode reward is plotted on the vertical
axis.}
\label{ignorerewardscartpolefigure}
\end{figure}
\begin{figure}
\includegraphics[width=15cm]{incentivize_learning_rate_episode_reward.png}
\caption{Performance of DQN, PPO, and their reality-checks
on an extended environment combining OpenAI gym's CartPole and
our IncentivizeLearningRate. Episode number is plotted on the horizontal
axis, and average episode reward is plotted on the vertical
axis.}
\label{inclearnratecartpolefigure}
\end{figure}
\section{Conclusion}
We introduced \emph{extended environments} for reinforcement learning.
When computing rewards and observations,
extended environments can consider not only actions the RL agent has taken, but also
actions the agent would hypothetically take in other circumstances.
Despite not being designed with such environments in mind, RL agents
can nevertheless interact with such environments.
An agent may find an extended environment hard to predict if the agent
only considers what has actually happened, and not its own hypothetical
actions in alternate scenarios.
We argued that for good performance (on
average) across many extended environments, an agent would need to
self-reflect to some degree.
Thus, we propose that an abstract theoretical measure of
an agent's self-reflection intelligence can be obtained by
modifying the definition of the Legg-Hutter universal intelligence measure.
The Legg-Hutter universal intelligence $\Upsilon(\pi)$ of an RL agent
$\pi$ is $\pi$'s weighted average performance across the space of all
suitably well-behaved traditional RL environments, weighted according to
the universal prior (i.e., environment $\mu$ has weight $2^{-K(\mu)}$ where
$K(\mu)$ is the Kolmogorov complexity of $\mu$).
We defined a measure $\Upsilon_{ext}(\pi)$ for RL agent $\pi$'s
self-reflection intelligence similarly to Legg and Hutter, except that
we take the weighted average performance over the space of suitably
well-behaved extended environments.
We gave some theoretically interesting examples of Extended Environments
in Section \ref{examplesection}. More examples are available in an open-source
MIT-licensed library
of extended environments which we are publishing simultaneously with this
paper \citep{library}.
We pointed out (Proposition \ref{qualitativedifferenceprop}) a key qualitative
difference between our self-reflection intelligence measure and the measure
of Legg and Hutter: two agents can agree in all ``possible''
scenarios (i.e., scenarios where the agents' past actions are consistent with
their policies, Definition \ref{traditionalequivalencedefn}),
and yet nevertheless have different self-reflection intelligence
(because they disagree on ``impossible'' scenarios---scenarios
which cannot ever happen in reality because they involve the agents
taking actions the agents never would take, but that nevertheless
extended environments can simulate the agents in,
as if to say: ``I doubt this agent would ever jump off this bridge, but
I'm going to run a simulation to see what the agent \emph{would} do immediately
after jumping off the bridge anyway''). With such impossible scenarios in mind,
we introduced a so-called \emph{reality check} transformation (Section
\ref{realitychecksection}) and informally conjectured that the transformation
tends to improve the performance of agents who are intelligent and not already
self-reflective in certain extended environments. We saw some experimental
evidence in favor of this conjecture in Section \ref{measurementssection},
where we discussed combining extended environments with sophisticated
traditional environments (such as those of OpenAI gym) to obtain practical
benchmark extended environments.
\section*{Acknowledgments}
We gratefully acknowledge Joscha Bach, James Bell, Jordan Fisher,
Jos{\'e} Hern{\'a}ndez-Orallo, Bill Hibbard, Marcus Hutter, Phil Maguire,
Arthur Paul Pedersen,
Stewart Shapiro, Mike Steel, Roman Yampolskiy,
and the editors and reviewers for comments and feedback.
|
2,877,628,090,180 | arxiv |
\section{Introduction}
The evaluation of treatment effects on a single (or just a few) treated unit(s) based on counterfactuals (i.e., the unobservable outcome had there been no intervention) constructed from artificial controls has become a popular practice in applied statistics since the proposal of the synthetic control (SC) method by \citet*{aAjG2003} and \citet*{aAaDjH2010}. Usually, these artificial (synthetic) controls are built from a panel of untreated peers observed over time, before and after the intervention.
The majority of methods based on artificial controls relies on the estimation of a statistical model between the treated unit(s) and a potentially large set of explanatory variables coming from the peers and measured before the intervention. The construction of counterfactuals poses a number of technical and empirical challenges. Usually, the dimension of the counterfactual model to be estimated is large compared to the available number of observations and some sort of restrictions must be imposed. Furthermore, the target variables of interest are non-stationary. Finally, conducting inference on the counterfactual dynamics is not straightforward. Although the original work by \citet*{aAjG2003} is able to handle some of these challenges, a number of extensions has been proposed; see \citet*{nDgI2016}, \citet*{sAgI2017}, or \citet*{aA2021} for recent discussions. Motivated by an application to the retail industry where the optimal prices of different products have to be determined, we develop a new methodology to construct counterfactuals which nests several other methods and efficiently explores all the available information. Our proposed method is well-suited for both stationary and non-stationary data as well as for high- or low-dimensional settings.
\subsection{Heterogeneous Elasticities and Optimal Prices}\label{S:App}
The determination of the optimal price of products is of great importance in the retail industry. By optimal price we mean the one that either maximizes profit or revenue. To determine such quantity we need first to estimate the price elasticity from the demand side. This is not an effortless task as standard regression methods usually fail to recover the parameter of interest due to confounding effects and the well-known endogeneity of prices.
Our novel dataset consists of daily prices and quantities sold of five different products for a major retailer in Brazil, aggregated at the municipal level. The company has more than 1,400 stores distributed in over 400 municipalities, covering all the states of the country.\footnote{Due to a confidentiality agreement, we are not allowed to disclosure either the name of the products or the name of the retail chain.} The chosen products differ in terms of magnitude of sales and in importance as a share of the company's total revenue. The overarching goal is to compute optimal prices at a municipal level via counterfactual analysis. Our method determines the effects in sales due to price changes and provides demand elasticities estimates which will be further used to compute the optimal prices.
To determine the optimal price of each of the products, a randomized controlled experiment has been carried out. More specifically, for each product, the price was changed in a group of municipalities (treatment group), while in another group, the prices were kept fixed at the original level (control group). The magnitudes of price changes across products range from 5\% up to 20\%. Furthermore, for three out of five products, the prices were increased, and for the other two products there was a price decrease. The selection of the treatment and control groups was carried out according to the socioeconomic and demographic characteristics of each municipality as well as to the distribution of stores in each city. Nevertheless, it is important to emphasize three facts. First, we used no information about the quantities sold of the product in each municipality, which is our output variable, in the randomization process. This way, we avoid any selection bias and can maintain valid the assumption that the intervention of interest is independent of the outcomes. Second, although according to municipality characteristics, we keep a homogeneous balance between groups, the parallel trend hypothesis is violated, and there is strong heterogeneity with respect to the quantities sold and consumer behavior in each city, even after controlling for observables. This implies that price elasticities are quite heterogeneous and optimal prices can be remarkably different among municipalities. Finally, there are a clear seasonal pattern in the data as well as common factors affecting the dynamics of sales across different cities.
Our results confirm the heterogeneous patterns in the intervention effects, yielding different elasticities and optimal prices across municipalities. In addition, the impacts also differ across products. Overall, the effects of price changes are statistical significant in more than 20\% of the municipalities in the treatment group and the optimal prices in terms of profit maximization are usually below the actual ones. Therefore, we recommend that the optimal policy in terms of profit maximization is to change the prices in the cities where the effects were statistically significant. Further experiments may be necessary to evaluate the effects of price changes in the cities were it was not possible to find statistically significant results.
\subsection{Methodological Innovations}
Driven by the empirical application discussed in the previous subsection, this paper proposes a methodology that includes both principal component regression (factors) and sparse linear regression for estimating counterfactuals for better evaluation of the effects on the sales of a set of products after price changes. It does not impose either sparsity or approximate sparsity in the mapping between the peers and the treated by using the information from hidden but estimable idiosyncratic components. Furthermore, we show that when the number of post-intervention observations is fixed, tests like the ones proposed in \citet*{rMmM2019} or \citet*{vCkWyZ2020a}, can be applied. Finally, we also consider a high-dimensional test to answer the question whether the use of idiosyncratic component actually leads to better estimation of the treatment effect. Our framework can be applied to much broader context in prediction and estimation and hence we leave more abstract and general theoretical developments to a different paper \citep{jFrMmM2021}.
The proposed method consists of four steps, called {\sl FarmTreat}. In the first one, the effects of exogenous (to the intervention of interest) variables are removed, for example, heterogeneous deterministic (nonlinear) trends, seasonality and other calendar effects, and/or known outliers. In the second step, a factor model is estimated based on the residuals of the first-step model. The idea is to uncover a common component driving the dynamics of the treated unit and the peers. This second step is key when relaxing the sparsity assumption. To explore potential remaining relation among units, in the third step a LASSO regression model is established among the residuals of the factor model, which are called the idiosyncratic components in the factor model. Sparsity is only imposed in this third step and it is less restrictive than the sparsity assumption in the second step. Note that all these three steps are carried out in the pre-intervention period. Finally, in the forth step, the model is projected for the post-intervention period under the assumption that the peers do not suffer the intervention. Inspired by \citet*{jFyKkW2020}, we call the methodology developed here \texttt{FarmTreat}, the factor-adjusted regularized method for treatment evaluation.
The procedure described above is well suited either for stationary data or in the case of deterministic nonlinear and heterogeneous trends. In case of unit-roots, the procedure should be carried out in first-differences under the assumption that factors follow an integrated process (with or without drift). In this case, our result follows from Section 7 in \citet*{jBsN2008}. After the final step, the levels of both the target variable and the counterfactual can be recovered and the inference conducted.
We show that the estimator of the instantaneous treatment is unbiased. This result enables the use of residual ressampling procedures, as the ones in \citet*{rMmM2019} or \citet*{vCkWyZ2020a}, to test hypotheses about the treatment effect without relying on any asymptotic result for the post-intervention period. The testing procedures proposed in \citet*{rMmM2019} or \citet*{vCkWyZ2020a} are similar with the crucial difference that the first paper considers models estimated just with the pre-intervention sample, while the second paper advocates the use of the full sample to estimate the models. As shown by the authors and confirmed in our simulations, using the full data yields much better size properties in small samples.
We believe our results are of general importance for the following reasons. First and most importantly, the sparsity or approximate sparsity assumptions on the regression coefficients do not seem reasonable in applications where the cross-dependence among all units in the panel are high. In addition, due to the cross-dependence, the conditions needed for the consistency of LASSO or other high-dimensional regularization methods are violated \citep*{jFyKkW2020}. Second, first filtering for trends, seasonal effects and/or outliers seem reasonable in order to highlight the potential intervention effects by removing uninformative terms. Finally, modeling remaining cross-dependence among the treated unit and a sparse set of peers are also important to gather all relevant information about the correlation structure about the units.
Under the hypothesis that the treatment is exogenous which is standard in the synthetic control literature, we have an unbiased estimator for the treatment effect on the treated unit for each period after the intervention. In the case the treatment is exogenous with respect only to the peers, we can identify the effects of a specific intervention on the treated unit, i.e., the time of a single intervention is fully known. This might be the quantity of interest in several macroeconomic applications as, for instance, the effects of Brexit on the United Kingdom economy fixing the date of the event.
We conduct a simulation study to evaluate the finite-sample properties of the estimators and inferential procedures discussed in the paper. We show that the proposed method works reasonably well even in very small samples. Furthermore, as a case study, we estimate the impact of price changes on product sales by using a novel dataset from a major retail chain in Brazil with more than 1,400 stores in the country. We show how the methods discussed in the paper can be used to estimate heterogeneous demand price elasticities, which can be further used to determine optimal prices for a wide class of products. In addition, we demonstrate that the idiosyncratic components do provide useful information for better estimation of elasticities.
\subsection{Comparison to the Literature}
Several papers in the literature extend the original SC method and derive estimators for counterfactuals when only a single unit is treated. We start by comparing with \citet*{cCrMmM2018}. Differently from this paper, we do neither impose sparsity nor our results are based on pre- and post-intervention asymptotics. We just require the pre-intervention sample to diverge in order to prove our results. Furthermore, by combing a factor structure with sparse regression we relax the (weak) sparsity assumption on the relation between the treated unit and its peers. In addition, we allow for heterogeneous trends which may not be bounded as in the case of the aforementioned paper; for a similar setup to \citet*{cCrMmM2018}, see \citet*{kLdB2017}. Masini and Medeiros (2019,2020)\nocite{rMmM2019,rMmM2020} consider a synthetic control extension when the data are nonstationary, with possibly unit-roots. However, the former paper imposes weak-sparsity on the relation between the treated unit and the peers and the later only handles the low-dimensional case. The low-dimensional non-stationary case is discussed in many other papers. See, for example, \citet*{cHhsCskW2012}, \citet*{mOyP2015}, \citet*{zDlZ2015}, and \citet*{kL2020}, among many others.
Compared to Differences-in-Differences (DiD) estimators, the advantages of the many estimators based on the SC method are three folds. First, we do not need the number of treated units to grow. In fact, the workhorse situation is when there is a single treated unit. The second, and most important difference, is that our methodology has been developed for situations where the $n-1$ untreated units may differ substantially from the treated unit and cannot form a control group, even after conditioning on a set of observables. For instance, in the application in this paper, the dynamics of sales in a specific treated municipality cannot be perfectly matched by any other city exclusively. On the other hand, there may exist a set of cities where the combined sales are close enough to ones of the treated unit in the absence of the treatment. Another typical example in the literature if to explain the gross domestic product (GDP) of a specific region by a linear combination of GDP from several untreated regions; see \citet*{aAjG2003}. Finally, SC methods and their extensions are usually consistent even without the parallel trends hypothesis.
More recently, \citet*{lGtM2016} generalize DiD estimators by estimating a correctly specified linear panel model with strictly exogenous regressors and interactive fixed effects represented as a number of common factors with heterogeneous loadings. Their theoretical results rely on double asymptotics when both $T$ (sample size) and $n$ (number of peers) go to infinity. The authors allow the common confounding factors to have nonlinear deterministic trends, which is a generalization of the linear parallel trend hypothesis assumed when DiD estimation is considered. Our method differs from \citet*{lGtM2016} in a very important way as we consider cross-dependence among the idiosyncratic units after the common factors have been accounted for.
Finally, we should compare our results with Chernozhukov and Wüthrich and Zhu (2020a,b)\nocite{vCkWyZ2020a,vCkWyZ2020b}. \citet*{vCkWyZ2020a} propose a general conformal inference method to test hypotheses on the counterfactuals which can be applied to our model setup as discussed above. When the sample size is small we strongly recommend the use of the approach described in \citet*{vCkWyZ2020a} to conduct inference on the intervention effects. \citet*{vCkWyZ2020b} propose a very nice generalization of \citet*{cCrMmM2018} with a new inference method to test hypotheses on intervention effects under high dimensionality and potential nonstationarity. However, their approach differs from ours in three aspects. First, and more importantly, their results are based on both pre- and post-intervention samples diverging. Second, their inferential procedure is designed to test hypothesis only on the average effect. Our procedure can be applied to a wide class of hypothesis tests. Finally, they impose that exactly the same (stochastic) trend is shared among all variables in the model. This is a more restricted framework than the one considered here.
\subsection{Organization of the Paper}
The rest of the paper is organized as follows. We give an overview of the proposed method and the application in Section \ref{S:overview}. We present the setup and assumptions in Section \ref{S:Setup} and state the key theoretical result in Section \ref{S:Theory}. Inferential procedures are presented in Section \ref{S:Inference}. We present the results of a simulation experiment in Section \ref{S:Simulations}. Section 4 is devoted to provide guidance to practitioners and a discussion of the empirical application can be found in Section \ref{S:Applications}. Section \ref{S:Conclusion} concludes the paper. Finally, the proof of our theoretical result and additional empirical results are relegated to the Supplementary Material.
\section{Methodology}\label{S:overview}
The dataset is a realization of $\{Z_{it}, \boldsymbol W_{it}: 1\leq i\leq n, 1\leq t\leq T\}$, in which $Z_{it}$ is the variable of interest and $\boldsymbol W_{it}$ describes potential covariates, including seasonal terms and/or deterministic (nonlinear and heterogeneous) trends, for example. Suppose we are interested in estimating the effects on the variable $Z_{1t}$ of the first unit after an intervention that occurred at $T_0+1$. We estimate a counterfactual based on the peers $\boldsymbol Z_{-1t}:=(Z_{2t},\ldots, Z_{nt})'$ that are assumed to be unaffected by the intervention. We allow the dimension of $\boldsymbol{Z}_{-1t}$ to grow with the sample size $T$, i.e. $n:=n_T$. We also assume that $\boldsymbol{W}_{it}$ are not affected by the intervention. Our key idea is to use both information in the latent factors and idiosyncratic components and we name the methodology as \texttt{FarmTreat}.
The procedure is thus summarized by the following steps:
\begin{enumerate}
\item
For each unit $i=1,\ldots,n$, run the regression:
\[
Z_{it}=\boldsymbol\gamma_i'\boldsymbol{W}_{it}+R_{it},\quad t=1,\ldots,T^*,
\]
and compute $\widehat{R}_{it}:=Z_{it} - \widehat{\boldsymbol\gamma}_i'\boldsymbol{W}_{it}$, where $T^*=T_0$ for $i=1$ and $T^*=T$, otherwise. This step removes heterogeneity due to $\boldsymbol{W}_{it}$. As mentioned before, $\boldsymbol{W}_{it}$ may include an intercept, any observable factors, dummies to handle seasonality and outliers, and determinist (polynomial) trends, for example. In the case of our particular application, $Z_{it}$ represents the daily quantity of a product sold per store in a municipality $i$ and $\boldsymbol{W}_{it}$ includes a constant, six dummy variables for the days of the week and a linear trend.
\item
Write $\boldsymbol{R}_t:=(R_{1t},\ldots,R_{nt})'$, which is the cross-sectional data $\boldsymbol{Z}_t:=(Z_{1t}, \cdots, \boldsymbol{Z}_{nt}')'$ after the heterogeneity adjustments. Fit the factor model
\[
\boldsymbol{R}_t=\boldsymbol\Lambda\boldsymbol{F}_t+\boldsymbol{U}_t,
\]
where $\boldsymbol{F}_t$ is an $r$-dimensional vector of unobserved factors, and $\boldsymbol\Lambda$ is an unknown $n\times r$ loading matrix and $\boldsymbol{U}_t$ is an $n$-dimensional idiosyncratic component. The second step consists of using the panel data $\{\widehat {\boldsymbol R}_t\}_{t=1}^T$ to learn the common factors $\boldsymbol{F}_t$ and factor loading matrix $\boldsymbol\Lambda$ and compute the estimated idiosyncratic components by
\[
\widehat{\boldsymbol U}_t=\widehat{\boldsymbol R}_t - \widehat{\boldsymbol\Lambda}\widehat{\boldsymbol F}_t,
\]
where $\widehat{\boldsymbol U}_t=\left(\widehat{U}_{1t},\ldots,\widehat{U}_{nt}\right)'$. There is a large literature on high-dimensional factor analysis; see Chapter 10 of the book by \cite{FLZZ20} for details. One important point is that we should not use data after $T_0$ for the treated unit. There are many possibilities to handle this issue that are discussed in Section \ref{S:Practice}.
\item
The third step is to use the idiosyncratic component to further augment the prediction on the treatment unit. It consists of first testing for the null of no remaining cross-sectional dependence (optional). If the null is rejected, fit the model in the pre-intervention period
\[
\widehat{U}_{1t}=\boldsymbol\theta_1'\widehat{\boldsymbol{U}}_{-1t} +V_{t},\quad t=1,\ldots,T_0,
\]
by using LASSO, where $\widehat{\boldsymbol{U}}_{-1t}=\left(\widehat{U}_{2t},\ldots,\widehat{U}_{nt}\right)'$. Namely, compute
\begin{equation}\label{eq:LASSO}
\widehat{\boldsymbol\theta}_1=\arg\min\left[\sum_{t=1}^{T_0}\left(\widehat{U}_{1t}-\boldsymbol\theta_1'\widehat{\boldsymbol{U}}_{-1t}\right)^2+\xi\|\boldsymbol\theta_1\|_1\right].
\end{equation}
This step uses cross-sectional regression of the idiosyncratic components to estimate the effects in the treated unit. It is approximately the same as using $\widehat{\boldsymbol{F}}_t$ and $\widehat{\boldsymbol{U}}_{-1t}$ to predict $\widehat{R}_{1t}$ with the sparse regression coefficients for $\widehat{\boldsymbol{U}}_{-1t}$, due to the orthogonality between $\{\widehat{\boldsymbol{F}}_t\}_{t=1}^T$ and $\{\widehat{\boldsymbol{U}}_{t}\}_{t=1}^T$.
The model includes sparse linear model on $\boldsymbol{R}_t$ as a specific example (see \eqref{eq-adj} below) and the required model selection conditions are more easily met due to the factor adjustments. It also encompasses the principal component regression (PCR) in which $\widehat{\boldsymbol\theta}_1= 0$, namely, using no cross-sectional prediction.
\item
Finally, the intervention effect $\delta_t$ is estimated for $t>T_0$ as
\begin{equation}\label{eq:treat}
\widehat{\delta}_t = Z_{1t} - \left(\widehat{\boldsymbol\gamma}_1'\boldsymbol{W}_{1t} + \widehat{\boldsymbol\lambda}_1'\widehat{\boldsymbol F}_t + \widehat{\boldsymbol\theta}_1'\widehat{\boldsymbol{U}}_{-1t}\right).
\end{equation}
where $ \widehat{\boldsymbol\lambda}_1$ is the estimated loading of unit $1$, the first row of $\widehat {\boldsymbol \Lambda}$. During the post treatment period, the realized factors $\widehat{\boldsymbol F}$ are learned without using $R_{1,t}$.
\item
Use the estimator \eqref{eq:treat} to test for null hypothesis of no intervention effect in the form described by \eqref{E:H0}.
\end{enumerate}
The innovations of our approach in estimating counterfactuals are multi-folds. For simplicity, let us suppose that we have no $\boldsymbol W_{it}$ component, so that $\boldsymbol R_t = \boldsymbol Z_t$. First of all, the proposed procedure explores both the common factors and the dependence among idiosyncratic components. This not only makes use of more information, but also makes the newly transformed predictors less correlated. The latter makes the variable selection much easier and prediction more accurate. Note that factor regression (principal component regression) to estimate counterfactuals is a special case when $\boldsymbol \theta_1 = 0$. Clearly, the method explores the sparsity of $\boldsymbol \theta_1$ to improve the performance and also includes the case of sparse regression on $\boldsymbol Z_{-1t}$ to estimate counterfactuals as in \citet*{rMmM2019}, where counterfactuals are estimated as
\begin{equation} \label{eq2.3}
Z_{1t} = \boldsymbol \theta_1 ' \boldsymbol Z_{-1t} + \epsilon_t, \quad t = 1, \cdots, T_0.
\end{equation}
However, the variables $\boldsymbol Z_{-1t}$ are highly correlated in high dimensions as they are driven by common factors, which makes variable selection procedures inconsistent and prediction ineffective. Instead, \citet*{jFyKkW2020} introduces the idea of lifting, called factor adjustments. Using the factor model in step 2, we can write the linear regression model \eqref{eq2.3} as
\begin{equation}\label{eq-adj}
Z_{1t} = \boldsymbol \theta_1 ' \boldsymbol \Lambda_{-1} \boldsymbol F_t + \boldsymbol \theta_1 ' \boldsymbol U_{-1t} + \epsilon_t,
\end{equation}
where $\boldsymbol \Lambda_{-1}$ and $\boldsymbol U_{-1t}$ are defined as $\boldsymbol \Lambda$ and $\boldsymbol U_{t}$ without the first row. When we take $\boldsymbol \lambda_1 = \boldsymbol \theta_1'\boldsymbol \Lambda_{-1}$, this reduces to use sparse regression to estimate the counterfactuals, but now use more powerful \texttt{FarmSelect} of \citet*{jFyKkW2020} to fit the sparse regression. Again, \texttt{FarmSelect} imposes the condition $\boldsymbol \theta_1'\boldsymbol \Lambda_{-1}$ as the regression coefficients of $\boldsymbol F_t$. Our method does not require this constraint. This flexibility allows us to apply our new approach even when the sparse linear model does not hold.
Finally, we also consider a test for the contribution of the idiosyncratic components by testing the null hypothesis that $\boldsymbol\theta_1=\boldsymbol 0$. Note that this is a high-dimensional hypothesis test, which is equivalent to testing the uncorrelatedness between the idiosyncratic component $U_{1t}$ for the treated unit and those from the untreated units $\boldsymbol U_{-1t}$ in the pre-intervention period.
\section{Assumptions and Theoretical Result}\label{S:Setup}
\subsection{Assumptions}\label{S:Assumptions}
Suppose we have $n$ units (municipalities, firms, etc.) indexed by $i=1,\dots,n$. For every time period $t=1,\ldots,T$, we observe a realization of a real valued random vector $\boldsymbol{Z}_t:=(Z_{1t},\ldots,Z_{nt})'$.\footnote{We consider a scalar variable for each unit for the sake of simplicity, and the results in the paper can be easily extended to the multivariate case.} We assume that an intervention took place at $T_0+1$, where $1<T_0<T$. Let $\mathcal{D}_t\in\{0,1\}$ be a binary variable flagging the periods where the intervention for unit 1 was in place. Therefore, following Rubin's potential outcome framework, we can express $Z_{it}$ as
\begin{equation*}
Z_{it}= \mathcal{D}_t Z_{it}^{(1)} + (1-\mathcal{D}_t) Z_{it}^{(0)},
\end{equation*}
where $Z_{it}^{(1)}$ denote the potential outcome when the unit $i$ is exposed to the intervention and $Z_{it}^{(0)}$ is the potential outcome of unit $i$ when it is not exposed to the intervention.
We are ultimately concerned with testing the hypothesis on the potential effects of the intervention in the unit of interest, i.e., the treatment effect on the treated. Without loss of generality, we set unit 1 to be the one of interest. The null hypothesis to be tested is:
\begin{equation}\label{E:H0}
\mathcal{H}_0:\boldsymbol{g}(\delta_{T_0+1},\ldots,\delta_{T})=\boldsymbol{0},
\end{equation}
where $\delta_t:=Z_{1t}^{(1)}-Z_{1t}^{(0)},\quad \forall t>T_0$, and $\boldsymbol{g}(\cdot)$ is a vector-valued continuous function. The general null hypothesis \eqref{E:H0} can be specialized to many cases of interest, as for example:
\[
\mathcal{H}_0:\frac{1}{T-T_0}\sum_{t=T_0+1}^T\delta_t=0
\quad \textnormal{or} \quad
\mathcal{H}_0:\delta_t=0,\,\forall t>T_0.
\]
It is evident that for each unit $i=1,\ldots,n$ and at each period $t=1,\dots,T$, we observe either $Z_{it}^{(0)}$ or $Z_{it}^{(1)}$. In particular, $Z_{1t}^{(0)}$ is not observed from $t=T_0+1$ onwards. For this reason, we henceforth call it the \emph{counterfactual} -- i.e., what $Z_{1t}$ would have been like had there been no intervention (potential outcome).
The counterfactual is constructed by considering a model in the absence of an intervention:
\begin{equation}\label{E:main_model}
{Z}_{1t}^{(0)} =\mathcal{M}\left(\boldsymbol{Z}_{-1t}^{(0)};\boldsymbol{\theta}\right) + V_t,\quad t=1,\ldots,T,
\end{equation}
where $\boldsymbol{Z}_{-1t}^{(0)}:=(Z_{2t}^{(0)},\ldots, Z_{nt}^{(0)})'$ be the collection of all control variables (all variables in the untreated units).\footnote{We could also have included lags of the variables and/or exogenous regressors into $\boldsymbol{Z}_{-1t}$, but again, to keep the argument simple, we have considered only contemporaneous variables; see \citet*{cCrMmM2018} for more general specifications.}, $\mathcal{M}:\mathcal{Z}\times\boldsymbol\Theta\rightarrow\mathbb{R}$, $\mathcal{Z}\subseteq\mathbb{R}^{n-1}$, is a known measurable mapping up to a vector of parameters indexed by $\boldsymbol\theta\in\boldsymbol\Theta$ and $\boldsymbol\Theta$ is a parameter space. A linear specification (including a constant) for the model $\mathcal{M}(\boldsymbol Z_{0t};\boldsymbol\theta)$ is the most common choice among counterfactual models for the pre-intervention period. \texttt{FarmTreat} uses a more sophisticated model.
Roughly speaking, in order to recover the effects of the intervention, we need to impose that the peers are unaffected by the intervention in the unit of interest. Otherwise our counterfactual model would be invalid. Specifically we consider the following key assumption
\begin{assumption}[\textbf{Intervention Independence}]\label{Ass:ind}
$\boldsymbol Z_t^{(0)}$ is independent of $\mathcal{D}_s$ for all $1\leq s,t\leq T$.
\end{assumption}
\begin{Rem}
Assumption \ref{Ass:ind} identifies the treatment effect on the treated. If only $ \boldsymbol Z_{-1t}^{(0)}$ is independent of $\mathcal{D}_s$ for all $1\leq s,t\leq T$, we can recover the effect of the intervention on the treated unit given that $T_0$ is deterministic and known. This later case is typical in papers on SC.
\end{Rem}
The main idea is to estimate (\ref{E:main_model}) using just the pre-intervention sample, $t=1,\ldots,T_0$, since under Assumption \ref{Ass:ind}, $\boldsymbol Z_{t}^{(0)}=(\boldsymbol Z_{t}^{(0)}|\mathcal{D}_t = 0) = (\boldsymbol Z_{t}|\mathcal{D}_t=0)$ for all $t$. Consequently, the estimated counterfactual for the post-intervention period, $t=T_0+1,\dots,T$, becomes
$\widehat{Z}_{1t}^{(0)} :=\mathcal{M}(\boldsymbol Z_{-1t};\widehat{\boldsymbol\theta}_{T_0})$. Under some sort of stationary assumption on $\boldsymbol Z_{t}$, in the context of a linear model, \citet*{cHhsCskW2012} and \citet*{cCrMmM2018}, show that $\widehat{\delta}_t:=Z_{1t} - \widehat{Z}_{1t}^{(0)}$ is an unbiased estimator for $\delta_t$ as the pre-intervention sample size grows to infinity in the low and high dimensional sparse case respectively.
We model the units in the absence of the intervention as follows.
\begin{assumption}[\textbf{DGP}]\label{Ass:DGP} The process $\{Z_{it}^{(0)}:1\leq i\leq n , t\geq 1\}$ is generated by
\begin{equation}\label{E:DGP}
Z_{it}^{(0)} =\boldsymbol\gamma_{i}' \boldsymbol W_{it}+\boldsymbol\lambda_i'\boldsymbol F_t +U_{it}
\end{equation}
where $\boldsymbol\gamma_i\in\mathbb{R}^{k}$ is the vector of coefficients of the $k$-dimensional observable random vector $\boldsymbol W_{it}$ of attributes of unit $i$, $\boldsymbol F_t$ is a $r$-dimensional vector of common factors and $\boldsymbol\lambda_i$ its respective vector of loads for unit $i$; and $U_{it}$ is a zero mean idiosyncratic shock. Finally, we assume that $\boldsymbol W_{it}$, $\boldsymbol F_t$ and $U_{it}$ are mutually uncorrelated.
\end{assumption}
The reason to include $\boldsymbol W_{it}$ is to accommodate an intercept, heterogeneous deterministic trends, seasonal dummies or any other exogenous (possibly random) characteristic of unit $i$ that the practitioner judges to be helpful in the construction of the counterfactual. As mentioned before we include an intercept, dummies to account for the effects of different days of the week and a linear trend. Other possibilities could be dummies of nation-wide promotions and/or holidays, for example.
In case of stochastic heterogeneous trends, we let the factors follow a random walk with (or without) drift: $\boldsymbol{F}_t=\boldsymbol\mu+\boldsymbol{F}_{t-1}+\boldsymbol\eta_t$, where $\{\boldsymbol\eta_t\}$ is a second-order stationary vector process. When this is the case, the methodology must be applied in first-differences and levels should be reconstructed in the end. Therefore, our approach can be directly applied even if the units have heterogeneous and stochastic trends.
Our counterfactual model is the sample version of the projection of $Z_{1t}^{(0)}$ onto the space spanned by $(\boldsymbol W_{1t}, \boldsymbol F_t, \boldsymbol U_{-1,t})'$. Under Assumption \ref{Ass:DGP} the counterfactual can be taken as
\begin{equation}\label{E:counterfactual_model}
Z_{1t}^{(0)} = \boldsymbol\gamma_{1}' \boldsymbol W_{1t}+\boldsymbol\lambda_1'\boldsymbol F_t +\boldsymbol \theta_1'\boldsymbol U_{-1t} + V_t,
\end{equation}
where $\boldsymbol\theta_1$ is the coefficient of the linear regression of $U_{1t}$ onto $\boldsymbol U_{-1t}$ and $V_t$ the respective projection error.
\subsection{Theoretical Guarantees}\label{S:Theory}
In order to state our result in a precise manner we consider the technical assumption below. First, let $\boldsymbol W_{S,it}$ denotes the sub-vector of $\boldsymbol W_{it}$ after the exclusion of all deterministic (non-random) components (constant, dummies, trends, etc). We state the assumption for the case where the unobserved factors are stationary. For non-stationary (unit-root) factors, the only difference is to state the conditions for the first-differences of the variables involved: $\Delta \boldsymbol W_{S,it}$, $\Delta \boldsymbol{F}_t$, and $\Delta \boldsymbol{U}_t$. In this case, our results can be derived following \citet*{jBsN2008}, Section 7. Note that, as mentioned before, if the interest lies on the intervention effects on the levels of the series, after the final step, the levels of both the target variable and the counterfactual can be recovered and the inferential procedures can be applied unaltered.
\begin{assumption}[\textbf{Regularity Conditions}]\label{Ass:Moments}
There is a constant $0<C<\infty$ such that:
\begin{enumerate}[(a)]
\item
The covariance matrix of $\boldsymbol W_{S,it}$ is non-singular;
\item
$\mathbb{E}|\boldsymbol W_{S,it}|^p\leq C $ and $\mathbb{E} |U_{it}|^{p+\epsilon}\leq C $ for some $p\geq 6$ and $\epsilon>0$ for $i=1,\ldots,n$, $t=1,\ldots,T$;
\item The process $\{(\boldsymbol W_{S,t}',\boldsymbol F_t',\boldsymbol U_t')',t\in \Z\}$ is weakly stationary with strong mixing coefficient $\alpha$ satisfying $\alpha(m)\leq \exp(-2cm)$ for some $c>0$ and for all $m\in\Z$;
\item $\|\boldsymbol\theta_1\|_\infty \leq C$;
\item $\kappa_0:=\kappa\left[\mathbb{E}(\boldsymbol U_t \boldsymbol U_t'),\mathcal{S}_0,3\right]\geq C^{-1}$ where $\kappa[\cdot]$ is the compatibility condition defined in \eqref{E:GIF} in the Supplementary Material and $\mathcal{S}_0:=\{i:\theta_{1,i}\neq 0\}$.
\end{enumerate}
\end{assumption}
Condition (a) is necessary for the parameters $\boldsymbol\gamma_i$, $i=1,\ldots,n$, to be well defined. Conditions (b) and (c) taken together allow the law of large numbers for strong mixing processes to be applied to appropriately scaled sums. In particular, $(b)$ bounds the $p$-th plus moment uniformly. However, if $U_{it}$ has exponential tails as contemplated in Assumption 3 in \citet*{jFrMmM2021}, we could state a stronger result in terms of the allowed number of non-zero coefficients as a fraction of the sample size. The mixing rate in condition (c) can be weaken to polynomial rate at the expense of an interplay between (c) and the conditions appearing in Proposition \ref{C:Main}.
Finally, conditions $(d)$ and $(e)$ in Assumption \ref{Ass:Moments} are regularity conditions on the high-dimensional linear model to be estimated by the LASSO in step 3. Condition (e) ensures the (restricted) strong convexity of the objective function, which is necessary for consistently estimate $\boldsymbol\theta_1$ when $n>T$. In effect, it uniformly lower bounds the minimum restricted $\ell_1$-eigenvalue of the covariance matrix of $\boldsymbol U_t$. For simplicity, the bounds appearing in (d) and (e) are assumed to hold uniformly. However, both conditions could be somewhat relaxed to allow $\|\boldsymbol\theta_1\|_\infty$ to grow slowly and/or $\kappa_0$ decreases slowly to 0 as $n$ diverges. Once again, at the expense of having both terms included in the conditions of Proposition \ref{C:Main}.
\begin{Pro}\label{C:Main} Under Assumptions \ref{Ass:ind}--\ref{Ass:Moments}, assume further that:
\begin{enumerate}[(a)]
\item There is a bounded sequence $\eta:=\eta_{n,T}$ such that $\|\widehat{\boldsymbol U}- \boldsymbol U\|_{\max} =O_P(\eta)$; and
\item $|\mathcal{S}_0| = O\left(\left\{\eta\left[(nT)^{1/p}+ \eta\right] + \frac{n^{4/p}}{\sqrt{T}}\right\}^{-1}\right).$
\end{enumerate}
If the penalty parameter $\xi$ in \eqref{eq:LASSO} is set to be at the order of $
\frac{n^{2/p}}{\sqrt{T}} +\eta T^{1/p}$ then, as $T_0\to\infty$, $\|\widehat{\boldsymbol\theta}_1-\boldsymbol\theta_1\|_1= O_P\left(\xi|\mathcal{S}_0|\right)$,
and for every $t>T_0$:
\[
\widehat{\delta}_t-\delta_t = V_t + O_P\left\{|\mathcal{S}_0|\left[\eta(nT)^{1/p} + \frac{n^{3/p}}{\sqrt{T}}\right]\right\},
\]
where $V_t$ is the stochastic component not explainable by untreated units defined by \eqref{E:counterfactual_model}
\end{Pro}
\begin{Rem} Conditions (a) and (b) are high level assumptions that translate into a restriction on the estimation rate in steps 1 and 2 of the proposed methodology, which in turn puts an upper bound on the number of non-zero coefficients in $\boldsymbol\theta_1$ (sparsity) in order for the estimation error to be negligible. The rate $\eta$ can be explicitly obtained in terms of $n$ and $T$ by imposing conditions on projection matrix of $\boldsymbol W_{it}$ and the factor model. For the former, we need uniform consistencies of both the factor and the loadings estimators that take into account the projection error in the previous step. In a more general setup, \citet*{jFrMmM2021} state conditions under which $\eta =\frac{n^{6/p}}{T^{1/2-6/p}}+\frac{T^{1/p}}{\sqrt{n}}$.
\end{Rem}
Proposition \ref{C:Main} is key for our inference procedure discussed in Section \ref{S:Inference}. For instance, it can be used to argue that $\widehat{\delta}_t-\delta_t = V_t +o_p(1)$ provided that $ |\mathcal{S}_0|\left[\eta(nT)^{1/p} + \frac{n^{3/p}}{\sqrt{T}}\right] = o(1)$. Since $V_t$ is zero mean by construction, as $T_0\to\infty$, $\widehat \delta_t$ is an unbiased estimator for $\delta_t$ for every post-intervention period. Furthermore, as described below, we can estimate the quantiles of $V_t$ using the pre-intervention residuals to conduct a valid inference on $\delta_t$.
\subsection{Testing for Intervention Effect}\label{S:Inference}
We test the null of no intervention effects based on estimators $\{\widehat{\delta}_t\}_{t>T_0}$ and the results of Masini and Medeiros (2019,2020)\nocite{rMmM2020}\nocite{rMmM2019} and \citet*{vCkWyZ2020a}. Let $T_2:=T-T_0$ be the number of observations after the intervention and define a generic continuous mapping $\boldsymbol\phi:\mathbb{R}^{T_2}\rightarrow\mathbb{R}^b$ whose argument is the $T_2$-dimensional vector $(\widehat{\delta}_{T_0+1}-\delta_{T_0+1},\ldots, \widehat{\delta}_{T}-\delta_T)'$ with given treatment effects
$\delta_{T_0+1}, \cdots, \delta_T$.
We are interested in the distribution of $\boldsymbol{\widehat{\phi}}:=\boldsymbol\phi(\widehat{\delta}_{T_0+1}-\delta_{T_0+1},\dots, \widehat{\delta}_{T}-\delta_T)$ under the null \eqref{E:H0}, where $\boldsymbol \phi$ is a given statistic. The typical situation is the one where the pre-intervention period is much longer than the post intervention period, $T_0\gg T_2$. Frequently, it could be well the case that $T_2=1$. However, $V_t$ does not vanish as in most cases there is a single treated unit. Nevertheless, under strict stationarity of the process $\{V_t\}$ and unbiasedness of the treatment effect estimator, it is possible to resample the pre-intervention residuals following either the procedure described in Masini and Medeiros (2019,2020)\nocite{rMmM2020}\nocite{rMmM2019} or the one in \citet*{vCkWyZ2020a} to compute the sample quantile of the statistic of interest. As pointed out earlier, the main difference between the two approaches is that the former estimate the counterfactual model using only pre-treatment observations while the later considers the estimation using the full sample.
Under the asymptotic limit taken on the pre-invention period $(T_0\to\infty)$, by Proposition~\ref{C:Main}, we have that $\boldsymbol{\widehat{\phi}}-\boldsymbol\phi_0=o_P(1)$, where $\boldsymbol\phi_0:=\boldsymbol\phi(V_{T_0+1},\dots,V_T)$. Thus, the distribution of $\boldsymbol{\widehat{\phi}}$ can be estimated by that of $\boldsymbol\phi_0$. Consider the construction of $\boldsymbol{\widehat{\phi}}$ using only blocks of size $T_2$ of consecutive observations from the pre-intervention sample. There are $T_0-T_2+1$ such blocks denoted by $\boldsymbol{\widehat{\phi}}_j:=\boldsymbol\phi(\widehat{V}_j,\dots,\widehat{V}_{j+T_2-1}),\,j=1,\ldots,T_0-T_2+1$, where $\widehat{V}_t:=Z_{1t} - \left(\widehat{\boldsymbol\gamma}_1'\boldsymbol{W}_{1t} + \widehat{\boldsymbol\lambda}_1'\widehat{\boldsymbol F}_t + \widehat{\boldsymbol\theta}_1'\widehat{\boldsymbol{U}}_{-1t}\right)$ for the pre-intervention period. The estimators $\widehat{\boldsymbol\gamma}_1$, $\widehat{\boldsymbol\lambda}_1$, $\widehat{\boldsymbol F}_t$, $ \widehat{\boldsymbol\theta}_1'$, and $\widehat{\boldsymbol{U}}_{-1t}$ use either the pre-intervention or the full sample depending on the inferential approach chosen by the practitioner.
For each $j$, we have that $\boldsymbol{\widehat{\phi}}_j-\boldsymbol\phi_j=o_P(1)$ where $\boldsymbol\phi_j:=\boldsymbol\phi(V_j,\dots,V_{j+T_2-1})$ and $\boldsymbol\phi_j$ is equal in distribution to $\boldsymbol\phi_0$ for all $j$. Hence, we propose to estimate the distribution $\mathcal{Q}_T(\boldsymbol x):=\P(\boldsymbol{\widehat{\phi}}\leq \boldsymbol x)$ by its empirical distribution
\[
\widehat{\mathcal{Q}}_T(\boldsymbol x):=\frac{1}{T_0-T_2+1}\sum\limits_{j=1}^{T_0-T_2+1}\mathds{1}(\boldsymbol{\widehat{\phi}}_j\leq \boldsymbol x),
\]
where, for a pair of vectors $\boldsymbol a, \boldsymbol b\in\mathbb{R}^d$, we say that $\boldsymbol a\leq \boldsymbol b\iff a_i\leq b_i,\forall i$. See Masini and Medeiros (2019,2020)\nocite{rMmM2020}\nocite{rMmM2019} and \citet*{vCkWyZ2020a} for further details.
\subsection{Testing for Idiosyncratic Contributions}
The question of statistical and practical interest is if the idiosyncratic component contributes the estimation of the treatment effect. To answer this question, write \eqref{E:DGP} as:
\[
\boldsymbol Z_t =\boldsymbol\Gamma \boldsymbol W_t + \boldsymbol \Lambda \boldsymbol F_t + \boldsymbol U_t,\qquad t\in\{1,\ldots,T\},
\]
where $\boldsymbol Z_t:=(Z_{1t},\ldots, Z_{nt})'$, $\boldsymbol U_t:=(U_{1t},\ldots, U_{nt})'$, and $\boldsymbol W_t:=(\boldsymbol W_{1t}',\ldots, \boldsymbol W_{nt}')'$. The $(n\times nk)$ block diagonal matrix $\boldsymbol \Gamma$ has blocks given by $(\boldsymbol\gamma_1',\dots \boldsymbol\gamma_n')$. Finally, $\boldsymbol\Lambda:=(\boldsymbol\lambda_1,\dots,\boldsymbol\lambda_n)'$.
Let $\boldsymbol\Pi :=(\pi_{ij})_{1\leq i,j\leq n}$ denote the $(n\times n)$ covariance matrix of $\boldsymbol U_t$. Our method exploits the sparsity of the off-diagonal elements of $\boldsymbol\Pi$. In particular, we are interested in testing whether $\boldsymbol U_{-1t}$ has linear prediction power on the treated unit $U_{1t}$. This amounts to the following high-dimensional hypothesis test: $\mathcal{H}_{0}: \pi_{1j} = 0,\; \forall\; 2\leq j\leq n$.
In order to conduct the test we propose the following test statistic $S:=\|\boldsymbol Q\|_\infty$, where $\boldsymbol Q:= \frac{1}{\sqrt{T_0}}\sum_{t=1}^{T_0} \boldsymbol D_t$, $\boldsymbol D_t:=\widehat{U}_{1t}\widehat{\boldsymbol U}_{-1t}$, and $\widehat{U}_{it} := \hat R_{it}- \widehat{\boldsymbol\lambda_i}'\widehat{\boldsymbol F}_t$. Also let $c^*(\tau)$ be the $\tau$-quantile of the Gaussian bootstrap $S^*:=\|\boldsymbol Q^*\|_\infty$, where $\boldsymbol Q^*|\boldsymbol Z, \boldsymbol W\sim \mathcal{N}(\boldsymbol 0,\widehat{\boldsymbol\Upsilon})$. For a given symmetric kernel $k(\cdot)$ with $k(0)=1$ and bandwidth $h>0$ (determining the number of lags), we have that
\[
\widehat{\boldsymbol\Upsilon}:=\sum_{|\ell|<T_0}k(\ell/h) \widehat{\boldsymbol M}_\ell\quad\textnormal{with}\quad \widehat{\boldsymbol M}_\ell :=\tfrac{1}{T_0}\sum_{t = \ell +1}^{T_0} \boldsymbol D_t\boldsymbol D_{t-\ell}'
\]
is the estimator of the long-run covariance matrix $\boldsymbol \Upsilon:=\V\widetilde{\boldsymbol Q}$, where $\widetilde{\boldsymbol Q}:=\tfrac{1}{\sqrt{T_0}}\sum_{t=1}^{T_0} U_{1t}\boldsymbol U_{-1t}$. Notice that $\widehat{\boldsymbol\Upsilon}$ is just the Newey-West estimator if $k(\cdot)$ is chosen to be the triangular kernel. More generally, the choice of kernels can be made in class of kernels described in \citet*{andrews91}. The validity of such a method has been proved in \cite{jFrMmM2021} under a more general setting. In particular, the authors show under some regularity conditions
\[\sup_{\tau\in(0,1)}|\P(S\leq c^*(\tau)) -\tau|=o(1)\quad \text{under $\mathcal{H}_0$}.\]
\section{Guide to Practice}\label{S:Practice}
In this section we provide practical guidance to the implementation of the \texttt{FarmTreat} method.
The first step involves the definition of the variables in $\boldsymbol W_{it}$. This is, of course, application dependent. Nevertheless, typical candidates are deterministic functions of time, i.e, $f(t)$, in order to capture trends, an intercept to remove the mean, seasonal dummies or other calendar effects, or any other dummies to remove potential outliers. Unit-root tests on the variable of interest may also be important in order to decide whether first-differences of the data should be taken or not.
The second step is the estimation of $\boldsymbol \Lambda$ and the sequence of factors $\{\boldsymbol F_t,\,t\in\Z\}$ for the full sample, before and after the intervention. Therefore, we cannot just rely on pre-intervention period to estimate the factors. On the other hand, if we use all the observations from the treated unit, we will bias our estimation under the alternative of nonzero treatment effects. Therefore, there are two possible ways to estimate the factors and the factor loadings:
\begin{enumerate}
\item
A simple approach is to estimate the factors and factor loadings without the treated unit. In order to estimate the loadings $\widehat {\boldsymbol \lambda}_1$ of the first unit, we then regress $R_{1t}$ on the estimated factors. This is the approach adopted in both simulations and in the empirical application.
\item
The imputed approach is to use the imputation $\widehat {\boldsymbol \lambda}_1' \widehat{\boldsymbol F}_t$ for the post intervention period of the treated unit 1 and then apply the whole data to reestimate the factor and factor loadings. $\widehat {\boldsymbol \lambda}_1$ and $\widehat{\boldsymbol F}_t$ are estimated with just the pre-intervention period.
\item
Note that $\mathbb{E}(\boldsymbol R_t)=\boldsymbol 0$ by definition. Hence, we can replace the post-intervention observations of $R_{1t}$ by 0 in order to carry the factor analysis. As the number of post-intervention observations is expected to be quite small, this replacement will have negligible effects. It is important to notice, however, that we do this just to estimate the factors.
\end{enumerate}
To determine the number of factors we advocate the use of the eigenvalue ratio test \citep*{sAaH2013}. Other possibility is the use of one of the information criteria discussed in \citet*{jBsN2002}.
After the estimation of the common factor structure, we can test for remaining cross-dependence using the test described in Section \ref{S:Inference}. In the case of rejection of the null of no remaining dependence, the last step consists of a LASSO regression. This step of testing is optional for evaluating the treatment effect, as the sparsity of LASSO includes no effect as a specific example. Nevertheless, it is an interesting statistical problem whether the idiosyncratic component contributes to the prediciton power. For selecting the penalty parameter in LASSO, we recommend the use of an information criterion, such as the BIC as in \citet*{rMmM2019}.
The final step is to test the null hypothesis concerning the intervention effects. When the pre-intervention sample is small, we follow \citet*{vCkWyZ2020a} and estimate the models under the null. Note that in this case we should re-estimate the model using the full sample.
\section{Simulations}\label{S:Simulations}
In this section we report simulations results to study the finite sample behavior of the method proposed in this paper. We consider the following data generating process:
\begin{equation}\label{E:simul}
\begin{split}
Z_{it}&=\delta_{it}+\boldsymbol\gamma_i'\boldsymbol{W}_t+R_{it},\qquad R_{it}=\boldsymbol\lambda_{i}'\boldsymbol{F}_{t}+U_{it},\\
\boldsymbol{F}_{t}&=0.8 \boldsymbol{F}_{t-1} + \boldsymbol{V}_{t},\qquad
U_{it}=
\begin{cases}
\boldsymbol\beta'\boldsymbol{U}_{-1t} + \varepsilon_{it},&\textnormal{if}\,i=1,\\
\varepsilon_{it},&\textnormal{otherwise,}
\end{cases}
\end{split}
\end{equation}
where $\{\varepsilon_{it}\}$ is a sequence of independent and normally distributed zero-mean random variables with variance equal to $0.25$ if $i=1$ and $\boldsymbol\beta\neq\boldsymbol{0}$ or variance equal to $1$ if $i>1$ or $\boldsymbol\beta=\boldsymbol{0}$. $\boldsymbol{V}_{t}$ is a sequence of independent and normally distributed zero-mean random vectors taking values on $\mathbb{R}^2$ such that $\mathbb{E}(\boldsymbol{V}_t\boldsymbol{V}_t')=0.25\times\boldsymbol{I}$, and $\mathbb{E}(\varepsilon_{it}\boldsymbol{V}_s)=\boldsymbol{0}$, for all $i,t,$ and $s$. $\boldsymbol{W}_{it}$ consists of a constant, a linear trend, and two independent Gaussian random variables with mean and variance equal to $1$. The parameters are set as follows: $\boldsymbol\gamma_i$ is $(p+2$)-dimensional vector where, for each replication, the first entry is randomly pick from a Gaussian random variable with zero mean and variance 1; the second term is randomly selected from an Uniform distribution between -5 and 5; and the last two elements are Gaussian distributed with mean 0.5 and variance 1. For each replication, the elements of $\boldsymbol\lambda_i$, $i>1$, are drawn independently from a normal distribution with mean two and unit variance and, for $i=1$, the elements of $\boldsymbol\lambda_i$ are drawn from a normal distribution with mean -6 and variance 0.04. The first two elements of $\boldsymbol\beta$ are either set to 0.5 and the rest is set to zero or we set all the elements equal to zero. We consider the following sample sizes: $T_0=50,75,100,150,250,500$ and $1000$; and $T_2=1$. For each sample size, $n$ is set as $n=\{T,2T,3T\}$. The number of factors is set to two. For size simulations, $\delta_{it}=0$ for all $i$ and $t$. For power simulations, $\delta_{it}=2$ for $i=1$ and $t=T_0+1$.
Tables \ref{T:simul_est_1} and \ref{T:simul_est_2} show descriptive statistics for the counterfactual estimation. The table depicts the mean, the median and the mean squared error (MSE) for $\delta_{T_0+1}$ under the null and alternative hypotheses, respectively. Three cases are considered. In the first one, the factor structure is neglected and a sparse LASSO regression of the first unit against the remaining ones is estimated. This is the ArCo methodology put forward by \citet*{cCrMmM2018}. The second one is equivalent to the approach of \citet*{lGtM2016}, where a pure factor model is considered. Finally, the \texttt{FarmTreat} approach is considered, which encompasses the previous two methods as a specific example. We also report, between brackets, the same statistics when the full sample is used to estimate the counterfactual model as advocated by \citet*{vCkWyZ2020a}.
Tables \ref{T:simul_est_1} and \ref{T:simul_est_2} show descriptive statistics for the counterfactual estimation. The table depicts the mean, the median and the mean squared error (MSE) for $\delta_{T_0+1}$ under the null and alternative hypotheses, respectively. Three cases are considered. In the first one, the factor structure is neglected and a sparse LASSO regression of the first unit against the remaining ones is estimated. This is the \texttt{ArCo} methodology put forward by \citet*{cCrMmM2018}. The second one is equivalent to the approach of \citet*{lGtM2016},
where a pure factor model is considered; we call this method the Principal Component Regression (\texttt{PCR}).
Finally, the \texttt{FarmTreat} approach is considered, which encompasses the previous two methods as a specific example. We also report, between brackets, the same statistics when the full sample is used to estimate the counterfactual model as advocated by \citet*{vCkWyZ2020a}.
From the inspection of the results in the tables, it is clear that the biases for estimating of the treatment effect are small and MSEs decrease as the sample size increase, as expected. Furthermore, the \texttt{ArCo} delivers very robust estimates, but the MSE can be substantially reduced by the \texttt{FarmTreat} methodology. Therefore, there is strong evidence supporting methodology derived in this paper, which is consistency with our theoretical results. Second, as already shown in the simulations in \citet*{cCrMmM2018}, the performance of the pure factor model is poor in terms of MSE. This is particularly the case when $n$ or $T$ is small, since the factors are not well estimated. When this happens, the prediction power of the idiosyncratic components comes to rescue (comparing the performance with \texttt{FarmSelect}). This demonstrates convincingly the need of using the idiosyncratic component to augment the prediction. When comparing with the results when the full sample is used to estimate the model, two facts emerge from the tables. First, when the null hypothesis is true, the gains of using the full sample are undebatable. However, when the null is false, using the full sample is a bad idea, specially when $T_0$ is small. This is somewhat expected as in the later case we are including observations affected by the intervention in the estimation sample.
Table \ref{T:simul_size} presents the empirical size of the ressampling test when there is a single observation after the intervention and the counterfactual is estimated according to the methods described above. It is clear that size distortions are high when $T_0$ is small. The size converges to the nominal one as the sample increases. On the other hand, using the full sample to estimate the models correct the distortions
and are strongly recommended in the case of small samples.
Table \ref{T:simul_power} shows the empirical power. The ressampling approach delivers high power, specially when ArCo and \texttt{FarmTreat} methodologies are considered. On the other hand, the test looses a lot of power when the full sample is considered. This is expected as the estimator of the treatment effect will be biased, specially in small samples.
Figure \ref{F:ratio} compares the MSEs of \texttt{PCR} and \texttt{FarmTreat} when DGP has no idiosyncratic contribution, i.e., $\boldsymbol\beta=\boldsymbol{0}$. This case favors to \texttt{PCR}. As we can see, \texttt{FarmTreat} achieves comparable results to \texttt{PCR}, indicating that the methodology is quite robust.
\section{Price Elasticity of Demand}\label{S:Applications}
\subsection{Data Description}
As described in Section \ref{S:App} the goal is to determine the optimal price of products for a large retail chain in Brazil. The optimal prices should be computed for each city. Our dataset consists of the daily prices and quantities sold of five different products, aggregated at the municipal level. The company's more than 1400 stores differ substantially across and within municipalities, ranging from small convenience stores with a limited selection of products up to very large ones, selling everything from sweets to home appliances and clothes. The stores can be street stores or can be located in shopping malls. The products sold are divided into several departments. Here, we consider products from the \emph{Sweets and Candies} unit. The chosen products differ in terms of magnitude of sales, price range, and in importance as a share of the company's revenue. For example, the median daily sales per store over the available period and across municipalities vary from 0 (Product V) to 35 units (Product II).
Our sample consists of about 50\% of the municipalities where there are stores. As the number and size of stores differ across municipalities, we divide the daily sales at each city by the number of stores in that particular location. To determine the optimal price of each of the products (in terms of profit or revenue maximization) and avoid confounding effects, a randomized controlled experiment has been carried out. For each product, the price was changed in a set of municipalities (treatment group), while in another group, the prices were kept fixed at the original level (control group). Note that the randomization is carried out at the city-level not at the store-level. With the application of our methodology optimal prices can be computed for each city in the treatment group as well as other levels of aggregation. In order to determine the prices for the locations in the control group, the experiment can be repeated by inverting the groups in a second batch of experiments. Here, we will report the results concerning the first group of randomized experiments.
The selection of the treatment and control groups was carried out according to the socioeconomic and demographic characteristics as well as to the distribution of stores in each city. The following variables were used: human development index, employment, GDP per capita, population, female population, literate population, average household income (total), household income (urban areas), number of stores, and number of convenience stores. Details about the method can be found in the Supplementary Material. As mentioned in the Introduction, we used no information about the quantities sold of the product in each municipality to create the treatment and control groups. Therefore, we avoid any selection bias, maintaining valid the assumption that the intervention of interest is independent of the outcomes.
It is important to highlight that although the experiment is randomized, traditional differences-in-differences estimators cannot be considered as the goal is to estimate the price elasticities at the municipal level which is exactly the same level of the randomization. Nevertheless, we can rely on differences-in-differences to estimate the intervention effects at the country level.
\subsection{Results}
In this section we report the results of the experiment described in the previous subsection.
Table \ref{T:experiments} describes each one of the experiments carried out for each product. The table shows the sample date, the period of the experiment (usually two weeks), the type of the experiment (if the price was increased or decreased), the magnitude of the price change, and the number of municipalities in the treatment ($n_1$) and control groups ($n_0$). $n$ is the total number of municipalities considered. $n$, $n_0$, and $n_1$ vary according to the product, but we omit the product identification to simplify notation.
For each day $t$, $q_{it}^{(j)}$ represents, for municipality $i$, the quantities sold of product $j$, where $i=1,\ldots,n$, $t=1,\ldots,T$, and $j=1,\ldots,5$. For convenience of notation assume that $i\in\{1,\ldots,n_0\}$ represents cities in the control group and $i\in\{n_0+1,\ldots,n\}$ indexes the municipalities in the treatment group. Finally, define $\widetilde{q}_{it}^{(j)}=q_{it}^{(j)}/n^s_{i}$, where $n^s_{i}$ is the number of stores at location $i$. The analysis is carried on for $\widetilde{q}_{it}^{(j)}$.
Figure \ref{F:lasa1} shows the data for the first product considered in the application. The data for the remaining products are displayed in Figures \ref{F:lasa2}--\ref{F:lasa5} in the supplementary material. Panel (a) in the figures reports the daily sales at each group of municipalities (all, treatment, and control) divided by the number of stores in each group. More specifically, the plot shows the daily evolution of $q_{\textnormal{all},t\cdot}^{(j)}=\frac{1}{s}\sum_{i=1}^nq_{it}^{(j)}$, $q_{\textnormal{control},t\cdot}^{(j)}=\frac{1}{s_0}\sum_{i=1}^{n_0}q_{it}^{(j)}$, and $q_{\textnormal{treatment},t\cdot}^{(j)}=\frac{1}{s_1}\sum_{i=n_0+1}^nq_{it}^{(j)}$. The plot shows the data before and after price changes and the intervention date is represented by the horizontal line.
Panels (b) and (c) display the distribution across municipalities of the time averages of $\widetilde{q}_{it}^{(j)}$, before and after the intervention and for the treatment and control groups, respectively.
Panels (d) and (e) present fan plots for the evolution of $\widetilde{q}_{it}^{(j)}$. The black curves there represent the cross-sectional means over time.
Several facts emerge from the plots. First, the dynamics of sales change depending of the product and the sample considered. Second, there is a weekly seasonal pattern in the data which is common to all products. The big spikes for Products II and IV, observed in Panel (a), are related to major promotions. We selected these particular products and sample to illustrate that our methodology is robust to outliers. One point that deserves attention is that promotions took place in both control and treatment groups and, therefore, do not have any harmful implication to our methodology. Eyeballing the graphs in Panel (a) of Figures \ref{F:lasa1} and \ref{F:lasa5}, we observe a substantial drop in sales before the start of the experiment and happened in both control in treatment groups. This experiment clearly shows the benefits of our method in comparison, for instance, with the before-and-after (BA) estimator. BA estimator does not take into account common trends or global shocks that affects both treatment and control groups. Finally, a point to highlight concerning Products IV and V is the fact the daily sales are quite small as compared to the other three products. For instance, the average daily sales per store is less than one unit for Product V. One of the reasons for the drop in sales for Product V just before the intervention is a large drop in the number of available units in some of the municipalities. For example, in about 3\% of the municipalities, both in the treatment and control groups, there were not a single unit of the product available to be sold. As we are going to see later, this will have an impact on the results obtained for this specific product. Finally, by observing Panels (d) and (e), we notice a significant heterogeneity across municipalities.
The models are estimated at the municipal level. For each product and each municipality, we run a first-stage regression of $\widetilde{q}_{it}^{(j)}$ on seven dummies for the days of the week, a linear deterministic trend and the number of stores that are open at municipality $i$ on day $t$. For the municipalities in the control group the above regression is estimated with the full sample. For the municipalities in the treatment group we use data only up to time $T_0$. The second step consists of estimating factors for the first-stage residuals. We select the number of factors, $k$, by the eigenvalue ratio test. In the third step, we run a LASSO regression of each idiosyncratic component of treated units on the idiosyncratic terms of the control group. As described in Section \ref{S:Practice}, the penalty parameter is determined by the BIC. Finally, we compute the counterfactual for each municipality $i=1,\ldots,n_1$ for $t=T_0+1,\ldots,T$: $\widehat{\widetilde{q}}_{it}^{(j)}$. We also compute the instantaneous and average intervention impact as $\widehat{\delta}_{it}^{(j)}=\widetilde{q}_{it}^{(j)}-\widehat{\widetilde{q}}_{it}^{(j)}$ and $\widehat{\Delta}_i^{(j)}=\frac{1}{T-T_0}\sum_{t=T_0+1}^T\widehat{\delta}_{it}^{(j)}$, respectively. We test the null hypothesis of intervention effect, $\mathcal{H}_0:\delta^{(j)}_{it}=0\,\forall t\geq T_0$, with the ressampling procedure with either $\phi(\widehat \delta_{T_0+1},\ldots, \widehat \delta_T)=\sum_{t=T_0+1}^T \widehat \delta_t^2$
or $\phi(\widehat \delta_{T_0+1},\ldots, \widehat \delta_T)=\sum_{t=T_0+1}^T|\widehat \delta_t|$. We also test the for daily effects.
Under the hypothesis of linear demand function, price elasticities $\epsilon_{ij}$ for each municipality $i$ and product $j$ can be recovered as
$\widehat{\epsilon}_{ij}=\frac{\widehat\beta_{ij}p_{ij,T_0-1}}{\overline{Q}_{ij}}$, where $\widehat{\beta}_{ij}=\frac{\widehat{\Delta}_{ij}}{N_i\Delta_{p_j}}$, $\widehat{\Delta}_{ij}$ is the estimated average effect for municipality $i$ and product $j$, $N_i$ is the number of stores, $\Delta_{p_j}$ is the price change, $p_{ij,T_0-1}$ is the price before the intervention and $\overline{Q}_{ij}$ is the average counterfactual quantity sold. Finally, optimal prices for profit maximization can be determined by:
\[
p_{ij}^*=\frac{(1-\mathsf{Taxes}_{ij})(\overline{Q}_{ij}-\widehat\beta_{ij}p_{ij,T_0-1})-\widehat\beta_{ij}\times\mathsf{Costs}_{ij}}{-2\widehat\beta_{ij}(1-\mathsf{Taxes}_{ij})},
\]
where $\mathsf{Taxes}_{ij}$ and $\mathsf{Costs}_{ij}$ are the municipality-product-specific tax and costs,respectively.
Table \ref{T:estimation} reports, for each product, the minimum, the 5\%-, 25\%-, 50\%-, 75\%-, and 95\%-quantiles, maximum, average, and standard deviation for several statistics. We consider the distribution over the all treated municipalities. In Panel (a) in the table we report the results for the R-squared of the pre-intervention model. Panel (b) displays the average intervention effect over the experiment period. Panels (c) and (d) depict the results for the $p$-values of the ressampling test described in Section \ref{S:Inference} for the null hypothesis of no intervention effect with the square or absolute value statistic, respectively. Panel (e) presents the results for the $p$-values of the null hypothesis of no idiosyncratic contribution. Table \ref{T:price} presents, for each product, the same descriptive statistics for the estimated elasticities and the percentage difference between the estimated optimal price and the current price. Contrary to what we show in Table \ref{T:estimation}, in Table \ref{T:price} we report only results with respect to the municipalities where the estimated average effects have the correct sign (positive when there is a price reduction and negative when there is a price increase) and are statistically significant at the 10\% level. The last column in the table shows the fraction of municipalities where the above criterium is satisfied.
Additional results are displayed in Figure \ref{F:lasa_res1} and Figures \ref{F:lasa_res2}--\ref{F:lasa_res5} in the Supplementary Material. For each product, Panel (a) in the figures displays a fan plot of the $p$-values of the ressampling test for the null hypothesis $\mathcal{H}_{0}: \delta_t = 0$ for each given $t$ after the treatment, using the test statistic $\phi(\widehat \delta_t) = |\widehat \delta_t|$, which is the same as using the test statistic $\widehat \delta_t^2$. The black curve represents the cross-sectional median across time $t$. Panel (b) shows an example for one municipality. The panel shows the actual and counterfactual sales per store for the post-treatment period. 95\% confidence intervals for the counterfactual path are also displayed.
Several facts emerge from the results. First, the average R-squared are quite high for Products II and IV and moderate for Products I and III. This fact provides some evidence that, on average, the estimated models are able to properly describe the dynamics of the sales. For Products II and IV this finding is even more pronounced as in the worst case, the R-squared are 0.4669 and 0.4028, respectively. On the other hand, the model for Product V yields very low R-squared. A potential reason for the poor fit is the fact that sales per store of Product I are very small and there are some municipalities that displays no sales in some days.
The second finding is related to the estimation of the average intervention effect ($\Delta$). As expected, the estimated mean effects ($\widehat\Delta$) have the correct sign for Products I--IV, on average. For Product I, $\Delta$ has the correct sign (negative) for 90\% of the municipalities and is statistically significant at the 10\% level in about 37\% of all treated cities. Among the cities with $\widehat\Delta>0$, in only one we find statistical significance at the 10\% level. For Product II, $\widehat\Delta$ has the correct sign for 89\% of the treated municipalities and the results are significant in about 33\% of all the treated cities. For Product III, the numbers are similar. However, in none of the cities where $\Delta$ has been estimated with the opposite sign, the effects are significant. For Product IV, the estimated average treatment effect has the correct sign in 59\% of the cities. Fortunately, in the 41 cases where the estimates have the wrong sign, the results are significant in only three of them. For Product V the estimates have the correct sign in only 35\% of the cities. However, in only four cities the results with the wrong sign are statistically significant at the 10\% level. The reasons for poor results concerning Product V are possibly twofold. First, as mentioned before, the sales per store are quite small and the in-sample fit is poor. Second, there was a stock problem around the time of the experiment. Figure \ref{F:inventory} in the Supplementary Material displays the evolution of the distribution of available product units across municipalities. From the inspection of Panel (a) in the figure it is clear that the distribution changes around the experiment dates, pointing to large decrease in stocks for Product V.
It is worth comparing the results in Panel (b) of Table \ref{T:estimation} with the ones if we use the before-and-after (BA) estimator to compute the average treatment effects. The BA estimator for each municipality is just average sales over the period after the intervention minus the average sales over the days before the intervention. The results are reported in Table \ref{T:estimation_ba} in the Supplementary Material. As expected from our previous discussion, the BA over estimates the effects of the price changes, specially for Products I and II, and yields estimates with the wrong sign for Product IV. For Product V, the BA estimates are even more negative that the ones from the \texttt{farmTreat} methodology.
A final fact from the inspection of Panel (e) is that the contribution of the idiosyncratic terms to construct the counterfactual is statistically relevant in several cases.
Now we turn attention to Table \ref{T:price}. If we focus only on the cities with estimated average effects that have the correct sign and where the such effects are statistically significant, we estimate very high elasticities on average; see Panel (a) in Table \ref{T:price}. From Panel (b), we note that on average prices must be decreased.
\section{Conclusions}\label{S:Conclusion}
In this paper we proposed a new method to estimate the effects of interventions when there is potentially only one (or just a few) treated units. The outputs of interest are observed over time for both the treated and untreated units, forming a panel of time series data. The untreated units are called peers and a counterfactual to the output of interest in the absence of intervention is constructed by writing a model relation the unit of interest to the peers. The novelty of this paper concerns how this model is constructed. We combine factor models with sparse regression on the idiosyncratic components. This model includes both the principal component regression and sparse regression on the original measurements as specific cases. The main advantage of our proposal is that we avoid the usual assumption of (approximate) sparsity and make model selection consistency conditions easier to be satisfied. A formal test is also proposed to prove the case for using the idiosyncratic components.
In terms of practical application we show how our methodology can be used to compute optimal prices for products from the retail industry in Brazil. Our results indicate optimal prices substantially lower than the current prices adopted by the company.
|
2,877,628,090,181 | arxiv | \section{Formulation of results}
According to \cite{GrLaPo}, given a closed smooth $n$-manifold $M^n$ and a Morse function $\varphi:M^n\to\mathbb R$ is called a {\it Morse-Lyapunov function} for Morse-Smale diffeomorphism $f:M^n\to M^n$ if:
1) $\varphi(f(x))<\varphi(x)$ if $x\notin Per(f)$ and
$\varphi(f(x))=\varphi(x)$ if $x\in Per(f)$, where $Per(f)$ is the set of periodic points of $f$;
2) any point $p\in Per(f)$ is
a non-degenerate maximum of $\varphi\vert_{W^u(p)}$ and a
non-degenerate minimum of
$\varphi\vert_{W^s(p)}$.
\begin{defi} Given a Morse-Smale
diffeomorphism $f:M^n\to M^n$, a function $\varphi: M^n\to \mathbb R$ is a {\it quasi-energy function} for $f$ if $\varphi$ is a Morse-Lyapunov function for $f$ and has the least possible number of critical points among all Morse-Lyapunov functions for $f$.
\end{defi}
In this paper we consider the class $G_4$ of
Morse-Smale diffeomorphisms $f:\mathbb S^3\to \mathbb S^3$ whose
nonwandering set consists of exactly four fixed
points: one source $\alpha$, one saddle $\sigma$
and two sinks $\omega_1$ and $\omega_2$.
It follows from \cite{S3} (theorem 2.3), that the
closure of each connected component (separatrix) of
the one-dimensional manifold $W^u(\sigma)\setminus\sigma$
is homeomorphic to a segment which
consists of this separatrix and two
points: $\sigma$ and some sink. Denote by $\ell_1,\ell_2$
the one-dimensional separatrices
containing the respective sinks $\omega_1,\omega_2$ in their closures.
According to \cite{S3}, $\bar\ell_i, i=1,2$ is everywhere smooth except,
maybe, at
$\omega_i$. So the topological embedding of $\bar\ell_i$
may be complicated in a neighborhood of the sink.
According to \cite{ArFo}, $\ell_i$ is
called {\it tame} (or {\it tamely embedded}) if there is a
homeomorphism $\psi_i:W^{s}(\omega_i)\to{\mathbb R}^n$
such that $\psi_i(\omega_i)=O$, where
$O$ is the origin and
$\psi_i(\bar\ell_i\setminus\sigma)$ is a
ray starting from $O$. In the opposite case $\ell_i$ is called {\it wild}.
It follows from a criterion in \cite{HGP} that the tameness of $\ell_i$ is
equivalent to the existence of a smooth
3-ball $B_{i}$ around $\omega_i$ in any neighborhood of $\omega_i$ such
that $\ell_i\cap\partial{B}_{i}$
consists of exactly one point. Using lemma 4.1 from \cite{GrLaPo} it
is possible to make this criterion more precise in our dynamical setting:
$\ell_i$ is tame if and only if there is $3$-ball $B_{\omega_i}$
such that
$\omega_i\in f(B_{\omega_i})\subset int~B_{\omega_i}\subset W^s(\omega_i)$
and $\ell_i\cap\partial{B}_{\omega_i}$
consists of exactly one point.
It was proved in \cite{BoGr2000} that, for every
diffeomorphism $f\in \mathcal{G}_4$, at least one
separatrix ($\ell_1$ say) is tame. It was
also shown that the topological classification of
diffeomorphisms from $\mathcal{G}_4$ is reduced to the
embedding classifications of the separatrix $\ell_2$; hence
there are infinitely many diffeomorphisms from
$\mathcal{G}_4$ which are not topologically conjugate.
To characterize a type of embedding of $\ell_2$
we introduce some special Heegaard splitting of $\mathbb S^3$.
Let us recall that
a three-dimensional orientable manifold is
{\it a
handlebody of genus $g\geq 0 $} if it is obtained from
a 3-ball by an orientation reversing identification of
$g$ pairs of pairwise disjoint 2-discs in its boundary.
The boundary of such a handlebody is an orientable
surface of genus $g $.
Let $P^+\subset \mathbb S^3$ be a handlebody of genus $g$ such that
$P^-= \mathbb S^3\setminus int P^+$ is a handlebody (necessarily
of the same genus as $P^+$). Then the pair $(P^+, P^-)$ is a
Heegaard splitting of genus $g$ of $\mathbb S^3$
with Heegaard surface $S=\partial P^+=\partial P^-$.
\begin{defi} A Heegaard splitting $(P^+, P^-)$ of \ $\mathbb S^3$ is said to be
adapted to $f\in G_4$, or $f$-adapted, if:
a) $\overline{W^u(\sigma)}\subset f(P^+) \subset int~P^+$;
b) $W^s(\sigma)$ intersects $\partial P^+$ transversally and
$W^s(\sigma)\cap P^+$ consists of a unique 2-disc.
An $f$-adapted Heegaard splitting $\mathbb S^3=P^+\cup P^-$ is said to be minimal if its genus is minimal among all $f$-adapted splittings.
\end{defi}
For each integer $k\geq 0$ we denote by $G_{4,k}$ the set of
diffeomorphisms $f\in G_4$ for which the minimal
$f$-adapted Heegaard splitting has genus $k$.
It is easily seen that, for each $f\in G_{4,0}$, $\ell_2$ is tame
and, according to \cite{GrLaPo}, $f$ possesses an energy
function. Conversely any diffeomorphism in $G_{4,k},\ k>0,$
has no energy function (see
\cite{Pi1977}). Figure \ref{ld} shows the
phase portrait of a diffeomorphism
$G_{4,1}$. The main result of this paper is the
following.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{ex4.eps}\caption{A diffeomorphism from the class $G_{4,1}$}\label{ld}
\end{center}
\end{figure}
\begin{theo} Every quasi-energy function for a diffeomorphism $f\in G_{4,1}$
has exactly six critical points. \label{qua}
\end{theo}
\section{Recollection of Morse theory}
\label{MT}
According to Milnor (\cite{Mil65}, section 3),
we use the following definitions.\\
A compact $(n+1)$-dimensional {\it cobordism} is a triad $(W,L_0,L_1)$
where $L_0$ and $L_1$ are closed manifolds of dimension $n$ and
$W$ is a compact $(n+1)$-dimensional manifold whose boundary
consists of the disjoint
union $L_0\cup L_1$. It is an {\it elementary} cobordism
when it possesses a Morse function $\varphi: W\to [0,1]$
with only one critical point
and such that $\varphi^{-1}(i)=L_i$ for $i=0,1$. When the index of the unique
critical point is $r$,
one speaks of an elementary cobordism of index $r$.\\
In this situation, $L_1$ is obtained from $L_0$ by a {\it surgery} of index $r$,
that is: there is an embedding $h:\mathbb S^{r-1}\times \mathbb
D^{n-r+1}\to L_0$
such that $L_1$ is diffeomorphic to the manifold obtained
from $L_0$ by removing the interior of the image of $h$ and gluing
$\mathbb D^r\times \mathbb S^{n-r}$, or
$$L_1\cong\mathbb D^r\times\mathbb S^{n-r}\mathop\bigcup_{h\vert_{\mathbb S^{r-1}\times\mathbb S^{n-r}}}
L_0\setminus int~(h(\mathbb S^{r-1}\times \mathbb D^{n-r+1}))
\,.
$$
Conversely, the following statement holds (see \cite{Mil65}, Theorem 3.12):
\begin{stat} If $L_1$ is obtained from $L_0$ by a surgery of index $r$,
then there exists an elementary cobordism $(W,L_0,L_1)$ of
index $r$.
\label{st4}
\end{stat}
\begin{figure} \begin{center}
\epsfig{file=rec_.eps, width=14. true cm,
height=8. true cm} \caption{An elementary cobordism}
\label{rec}
\end{center}
\end{figure}
On figure \ref{rec} it is seen a surgery
of index $1$ from the 2-sphere to the 2-torus with some level sets of a Morse function on the corresponding elementary cobordism.\\
Finally, we recall the weak Morse inequalities
(see \cite{Mil1996}, Theorem 5.2).
\begin{stat} Let $M^n$ be a closed manifold,
$\varphi:M^n\to\mathbb{R}$ be a Morse
function, $C_q$ be the number of
critical points of index $q$ and
$\beta_q(M^n)$ be the $q$-th Betti number
of the manifold $M^n$. Then
$\beta_q(M^n)\leq C_q$ and the Euler characteristic
$\chi(M^n):=\sum\limits_{q=0}^n(-1)^q\beta_q(M^n)$ equals
$\sum\limits_{q=0}^n(-1)^q C_q$.
\label{st10}
\end{stat}
\section{Proof of Theorem \ref{qua}}
Let $f$ be a Morse-Smale diffeomorphism of the 3-sphere belonging to
$G_{4,1}$.
As the number of critical points of any Morse function on a closed 3-manifold is even (it follows from statement \ref{st10})
and greater than four (as $Per(f)\subset Cr(\varphi)$ and $\ell_2$ is wild) then, for proving theorem \ref{qua}, it is enough to construct
a Lyapunov function with six critical points.
\subsection{Auxiliary statements}
For the proof of the following statements \ref{loc} and \ref{w} we refer
to \cite{GrLaPo}, lemma 2.2 and lemma 4.2.
\begin{stat} Let $p$ be a fixed point of a Morse-Smale
diffeomorphism $f:M^n\to M^n$ such that
$\dim W^u(p)=q$. Then, in some neighborhood $U_{p}$ of
$p$, there exist local coordinates
$x_1,\dots,x_n$ vanishing at $p$
and an energy function $\varphi_{p}:U_p\to\mathbb R$
such that
$$\varphi_{p}(x_1,\dots,x_n)=q-x_1^2-\dots-x_q^2+x_{q+1}^2+\dots+x_n^2$$
and $(TW^u(p)\cap U_{p})\subset Ox_1\dots x_q$,
$(TW^s(p)\cap U_{p})\subset Ox_{q+1}\dots x_n$.
\label{loc}
\end{stat}
\begin{stat} Let $\omega$ be a fixed sink of a
Morse-Smale diffeomorphism $f:M^3\to M^3$ and
$B_\omega$ be a 3-ball with boundary $S_\omega$
such that $\omega\in f(B_\omega)\subset
int~B_\omega\subset W^s(\omega)$. Then there exists
an energy function
${\varphi}_{B_\omega}:B_{\omega}\to\mathbb R$
for $f$ having $S_{\omega}$ as a level set.
\label{w}
\end{stat}
\begin{lemm} Let $\omega$ be a fixed sink of a Morse-Smale diffeomorphism
$f:M^3\to M^3$ and $Q_\omega$ be a solid torus such that
$\omega\in f(Q_\omega)\subset int~Q_\omega\subset W^s(\omega)$.
Then there exists a 3-ball $B_\omega$ such that
$f(Q_{\omega})\subset B_\omega\subset
int~Q_{\omega}$. \label{seq}
\end{lemm}
\begin{demo} Let $D_0$ be a meridian disk in $Q_\omega$ such
that $\omega\notin{D_0}$.
As $Q_\omega\subset W^s(\omega)$ there is an integer
$N$ such that
$f^n(Q_\omega)\cap{D}_0=\emptyset$ for every
$n>N$. We may also
assume that $D_0$ is transversal to
$G=\bigcup\limits_{n\in\mathbb Z}f^{n}(\partial Q_\omega)$,
and hence $G\cap int~D_0$ consists of a finite family $\mathcal C_{D_0}$ of
intersection
curves. Each intersection curve $c\in \mathcal C_{D_0}$ belongs
to $f^{k}(\partial Q_\omega)$ for some integer $k\in\{1,\dots,N\}$.
There are two cases: (1) $c$ bounds a disk on $f^{k}(\partial Q_\omega)$;
(2) $c$ does not bound a disk on $f^{k}(\partial Q_\omega)$.
Let us decompose $\mathcal C_{D_0}$ as union of two pairwise disjoint
parts $\mathcal C_{D_0}^1$ and $\mathcal C_{D_0}^2$ consisting of curves
with property (1) or (2), accordingly.
Let us show that there is a meridian disk $D_1$ in $Q_\omega$ such that $D_1$
is transversal to $G$ and $G\cap int~D_1$ consists of
family $\mathcal C_{D_1}=\mathcal C_{D_0}^2$ of intersection
curves. If $\mathcal C_{D_0}^1=\emptyset$ then ${D_1}={D_0}$. In the
opposite case for any curve $c\in\mathcal C_{D_0}^1$ denote by $d_c$
the disk on $f^{k}(\partial Q_\omega)$ such that $\partial d_c=c$.
Notice that $d_c$ does not contain a curve from the family
$\mathcal C_{D_0}^2$. Then there is
$c\in \mathcal C_{D_1}$ which is innermost on
$f^{k}(\partial Q_\omega)$ in the sense that the interior of $d_c$ contains
no intersection curves from $\mathcal C_{D_0}$. For such a curve $c$
denote $e_c$ the disk on $D_0$ such that $\partial e_c=c$.
As $int~Q_\omega\setminus D_0$ is an open 3-ball then $e_c\cup d_c$ bounds
a unique 3-ball $b_c\subset int~Q_\omega$. Set $D'_c=(D_0\setminus
e_c)\cup d_c$. There is a smooth approximation
$D_c$ of $D'_c$ such that $D_c$ is a meridian disk on $Q_\omega$,
$D_c$ is transversal to $G$. Moreover $G\cap int~D_c$ consists
of a family $\mathcal C_{D_c}$ of intersection
curves having less elements than $\mathcal C_{D_0}$; indeed, $c$ disappeared
and also all curves from $\mathcal C_{D_0}$ lying in $int~e_c$.
We will repeat this process until getting a meridian disk $D_1$
with the required property.
Now let $c\in\mathcal C_{D_1}$, $c\in f^k(\partial Q_\omega)$.
Denote $e_c$ the disk that $c$ bounds in $D_1$.
Let us choose $c$ innermost in $D_1$ in the sense that the interior of $e_c$
contains no intersection curves from $\mathcal C_{D_1}$.
There are two cases: (a) $e_c\subset f^{k}(Q_\omega)$ and
(b) $int~e_c\cap f^{k}(Q_\omega)=\emptyset$.
In case (a) $e_c$ is a meridian disk of $f^{k}(Q_\omega)$ and
$D=f^{-k}(e_c)$ is a meridian
disks in $Q_\omega$ such that $f(Q_\omega)\cap D=\emptyset$. Indeed,
by construction $int~e_c\cap G=\emptyset$, hence $int~D\cap G=\emptyset$.
Thus we can find the required 3-ball $B_\omega$ inside
$int~Q_\omega\setminus{D}_1$.
In case (b) there is a tubular neighborhood
$V(e_c)\subset int~Q_\omega$ of the disk $e_c$ such that
$G\cap int~V(e_c)=\emptyset$ and $B_k=f^{k}(Q_\omega)\cup V(e_c)$ is 3-ball.
Then $f^{k}(Q_\omega)\subset B_k\subset int~f^{k-1}(Q_\omega)$.
Thus $B_\omega=f^{1-k}(B_k)$ is the required 3-ball.
\end{demo}
\subsection{Construction of a quasi-energy function
for a diffeomorphism $f\in G_{4,1}$}
As a similar construction was done in section 4.3 of \cite{GrLaPo}, we only give a sketch of it below.
\begin{enumerate}
\item Construct an energy function $\varphi_{p}:U_p\to\mathbb R$
near each fixed point $p$ of $f$
as in statement \ref{loc}.
\item By definition of the class $G_{4,1}$, for each
$f\in G_{4,1}$ there is a solid torus $P^+$ belonging to a Heegaard splitting
$(P^+,P^-) $ of $\mathbb S^3$ and such
that:
a) $\overline{W^u(\sigma)}\subset f(P^+) \subset
int~P^+$;
b) $W^s(\sigma)$ intersects $\partial P^+$ transversally and
$W^s(\sigma)\cap P^+$ consists of a unique
2-disk.
\smallskip
\noindent As $\mathbb S^3\setminus
\overline{W^s(\sigma)}$ is the disjoint union $W^s(\omega_1)\cup W^s(\omega_2)$, then by property b), the
disk $P^+\cap W^s(\sigma)$ is separating in $P^+$. Moreover there exists
a neighborhood of $P^+\cap W^s(\sigma)$, such that after removing it
from $P^+$ we get a 3-ball $P_{\omega_1}$
and solid torus $P_{\omega_2}$ with the following properties for each $i=1,2$:
i) $\omega_i\in f(P_{\omega_i})\subset int~P_{\omega_i}\subset W^s(\omega_i)$;
ii) $\partial{P}_{\omega_i}$ is a Heegaard surface
and $\ell_i\cap \partial{P}_{\omega_i}$ consists of exactly one point.\\
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{suflu.eps}\caption{Heegaard decomposition $(Q^+,Q^-)$}\label{suflu}
\end{center}
\end{figure}
Due to the $\lambda$-lemma\footnote{The $\lambda$-lemma claims
that $f^{-n}(S_{\omega_i})\cap U_\sigma$
tends to $\{x_1=0\}\cap U_\sigma$
in the $C^1$ topology when $n$ goes to $+\infty$.}
(see, for example,\cite{Pa}), replacing $P_{\omega_i}$
by $f^{-n}(P_{\omega_i})$ for some $n>0$ if necessary,
we may assume that $\partial P_{\omega_i}$ is transversal to the regular part of the
critical level set
$C:=\varphi^{-1}_{\sigma}(1)$ of the function $\varphi_\sigma$
and the intersections $C\cap\partial P_{\omega_i}$ consist of exactly one circle.
For $\varepsilon\in(0,\frac12)$ define $H^+_\varepsilon$ as the closure of $\{x\in
U_{\sigma}\mid x\notin (P_{\omega_1}\cup
P_{\omega_2}), \ \varphi_{\sigma}(x)\leq
1+\varepsilon\}$ and set $P^+_\varepsilon=P_{\omega_1}\cup P_{\omega_2}\cup H^+_\varepsilon$.
In the same way as in \cite{GrLaPo}
it is possible to choose $\varepsilon>0$ such that $\partial P_{\omega_i}$
intersects transversally each level set with value in
$[1-\varepsilon$, $1+\varepsilon]$; this intersection consists of one circle.
Taking a smoothing $Q^+$ of $P^+_\varepsilon$ we have $f(Q^+)\subset int~Q^+$
and $\Sigma:=\partial Q^+$ is a Heegaard surface of genus 1.
Let $Q^-$ be the closure of $\mathbb{S}^3\setminus int~Q^+$ (see figure \ref{suflu}). It is easy to check that
$Q^+$
is isotopic to $P^+$. Therefore, the pair $(Q^+, Q^-)$ is an $f$-adapted Heegaard splitting with the property
that the disk $Q^+\cap W^s(\sigma)$ lies in $U_\sigma$.
\item For each $i=1,2$, let $\tilde P_{\omega_i}$ be a handlebody of genus $i-1$ such that
$f(P_{\omega_i})\subset \tilde P_{\omega_i}\subset int~P_{\omega_i}$,
$\partial\tilde P_{\omega_i}$ intersects transversally each level set
with value in $[1-\varepsilon,1+\varepsilon]$ along
one circle and $P_{\omega_i}\setminus int~\tilde P_{\omega_i}$ is
diffeomorphic to $\partial P_{\omega_i}\times[0,1]$.
Define $d_i$ as the closure of
$\{x\in U_{\sigma}\mid x\in (W^s(\omega_i)\setminus\tilde P_{\omega_i}),
\ \varphi_{\sigma}(x)=1-\varepsilon\}$. By construction $d_i$ is a disk whose boundary
curve bounds a disk $D_i$ in $\partial \tilde P_{\omega_i}$.
We form $S_i$ by removing the interior of $D_i$ from
$\partial \tilde P_{\omega_i}$ and gluing the $d_i$.
Denote $P(S_i)$ the handlebody of genus $i-1$ bounded by $S_i$ and containing
$\omega_i$. As in \cite{GrLaPo} it is possible
to choose $\varepsilon$ such that $f(P(S_i))\subset int~P(S_i)$.
Let $K$ be the domain between $\partial Q^+$ and $S_1\cup S_2$.
We introduce $T^+$, the closure of
$\{x\in \mathbb S^3\mid x\notin (P_{\omega_1}\cup P_{\omega_2}),\ 1-\varepsilon\leq\varphi_{\sigma}(x)\leq
1+\varepsilon\}$; observe $T^+\subset U_\sigma$. We define a function $\varphi_{_K}:
K\to \mathbb{R}$ whose value is $1+\varepsilon$ on $\partial Q^+$, $1-\varepsilon$ on $S_1\cup S_2$,
coinciding with $\varphi_{\sigma}$ on $K\cap T^+$ and without critical points
outside $T^+$. This last condition is easy to satisfy as the domain
in question is a product cobordism. In a similar way to \cite{GrLaPo}, section 4.3,
one can check that
$\varphi_{_K}$ is a Morse-Lyapunov function.
\item As $P(S_1)$ is a 3-ball such that
$\omega_1\in f(P(S_1))\subset
int~P(S_1)\subset W^s(\omega_1)$, then by statement \ref{w}
there is an energy function
$\varphi_{_{P(S_1)}}:P(S_1)\to\mathbb{R}$
for $f$ with $S_1$ as a level set with value
$1-\varepsilon$.
\item As $P(S_2)$ is a solid torus such that
$\omega_2\in f(P(S_2))\subset
int~P(S_2)\subset W^s(\omega_2)$,
then according to lemma \ref{seq} there is a 3-ball $B_{\omega_2}$ such that
$f(P(S_2))\subset B_{\omega_2}\subset
int~P(S_2)$. As in the previous item, there is an energy
function $\varphi_{_{B_{\omega_2}}}:B_{\omega_2}\to\mathbb{R}$ for $f$ with
$\partial{B_{\omega_2}}$ as a level set with value $\frac12$.
\item As $P(S_2)$ is a solid torus, it is obtained from
a 3-ball by an orientation reversing identification of a pair of disjoint 2-discs in its boundary; hence the solid torus is the union of a 3-ball and an elementary cobordism of index 1. Since, up to isotopy, there is only one 3-ball in the interior of a solid torus, then $(W_{\omega_2},\partial B_{\omega_2},S_2)$ is an
elementary cobordism
of
index $1$, where
$W_{\omega_2}=P(S_2)\setminus int~B_{\omega_2}$. Hence
$W_{\omega_2}$ possesses a Morse function
$\varphi_{_{W_{\omega_2}}}$ with only one critical point of index 1 and such that
$\varphi_{_{W_{\omega_2}}}(\partial B_{\omega_2})=\frac12$,
$\varphi_{_{W_{\omega_2}}}(S_2)=1-\varepsilon$.
\item Define the smooth function ${\varphi}^+:Q^+\to
\mathbb{R}$ by the formula
${\varphi}^+(x)=\cases{\varphi_{_K}(x),
x\in K;
\cr\varphi_{_{P(S_1)}}(x),x\in P(S_1);
\cr\varphi_{_{B_{\omega_2}}}(x),x\in B_{\omega_2};
\cr\varphi_{_{W_{\omega_2}}}(x),x\in W_{\omega_2}.\cr}$
Then ${\varphi}^+$ is a Morse-Lyapunov function for $f\vert_{Q^+}$ with one
additional critical point.
\item By the construction $Q^-$ is a solid torus such that
$\alpha\in f^{-1}(Q^-)\subset int~Q^-\subset W^u(\alpha)$. Since $\alpha$ is a
sink for $f^{-1}$ then, as in item 4, there is a 3-ball $B_{\alpha}$ such that
$f^{-1}(Q^-)\subset B_{\alpha}\subset
int~Q^-$ and an energy function
$\varphi_{_{B_{\alpha}}}:B_{\alpha}\to\mathbb{R}$ for $f^{-1}$ with
$\partial{B_{\alpha}}$ as a level set of value $\frac12$.
\item Similarly to item 5, $\partial Q^-$ is obtained from $\partial B_{\alpha}$
by a surgery of index $1$.
Therefore $(W_\alpha,\partial Q^-,\partial B_\alpha)$ is an elementary
cobordism of
index $1$, where $W_\alpha=Q^-\setminus int~B_{\alpha}$. Hence, $W_\alpha$ possesses a Morse
function $\varphi_{_{W_\alpha}}$ with only one critical point of index 1. We may choose
$\varphi_{_{W_\alpha}}(\partial B_{\alpha})=\frac12$,
$\varphi_{_{W_\alpha}}(\partial Q^-)=2-\varepsilon$.
\item Define the smooth function ${\varphi}^-:Q^-\to
\mathbb{R}$ by the formula
${\varphi}^-(x)=\cases{3-\varphi_{_{B_{\alpha}}}(x),
x\in \varphi_{_{B_{\alpha}}};
\cr 3-\varphi_{_{W_\alpha}}(x),x\in \varphi_{_{W_\alpha}}.\cr}$
Then ${\varphi}^-$ is a Morse-Lyapunov function for $f\vert_{Q^-}$
with one additional critical point.
\item The function $\varphi:\mathbb S^3\to \mathbb R$ defined by
$\varphi\vert_{Q^+}=\varphi^+$ and
$\varphi\vert_{Q^-}=\varphi^-$ is the required
Morse-Lyapunov function for the diffeomorphism $f$
with exactly six critical points.
\end{enumerate}
\section*{\addcontentsline{toc}{section}{Reference}}
|
2,877,628,090,182 | arxiv | \section{Introduction}
Vector is the general format of input data of algorithm, and each component
of vector is stored in classical memory sequentially in general. Thus there
are two questions for general quantum algorithm, the first question is that
which state is suitable to represent all information of vector for further
quantum computation, and the second question is that how to load all
information of vector into quantum registers (or quantum state) without
losing information. Loading data set such as vector into classical registers
of CPU from classical memory is called \textbf{classical loading scheme (CLS)%
}. Similar to CLS, designing unitary operation to load all information of
vector into quantum registers of quantum CPU from classical memory is called
\textbf{quantum loading scheme (QLS)}. CLS or QLS assembles classical memory
and CPU as a whole computer. QLS\ makes quantum CPU is compatible with
classical memory.
An $N-$dimensional vector is denoted as $\vec{a}=\{a_{0},a_{1},...,a_{N-1}\}$%
, where the components $a_{0},a_{1},...,a_{N-1}$ are real numbers. It has
been shown that entangled state $\frac{1}{\sqrt{N}}({\sum\limits_{i=0}^{N-1}{%
{{\left\vert {i}\right\rangle }}}_{{register1}}{{{\left\vert a_{{i}%
}\right\rangle }}}_{{register2}}})${\ is suitable for the representation of
vector without losing any information of vector \cite{Pang
PostDocReport,Pang QVQ2,Pang QDCT,Pang QVQ1}. Let initial state ${\left\vert
{\phi _{0}}\right\rangle }$ be ${\left\vert {\phi _{0}}\right\rangle }={%
\left\vert {0}\right\rangle _{{{{q_{1}q_{2}...q_{n}}}}}\left\vert {0}%
\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {ancilla_{1}}%
\right\rangle }$, where the $m+n$ qubits ${{{q_{1},...,q_{n},p_{1},...,p_{m}}%
}}$ are collected as a whole object, dividing them is prohibited, and where
the ancillary state ${\left\vert {ancilla_{1}}\right\rangle }$ is \emph{known%
}. QLS can be described as how to design unitary operation $%
U_{(0,1,...,N-1)} $ such that
\begin{equation}
{\left\vert {\phi }\right\rangle =U_{(0,1,...,N-1)}{\left\vert {\phi _{0}}%
\right\rangle }=}\frac{1}{\sqrt{N}}({\sum\limits_{i=0}^{N-1}{{{\left\vert {i}%
\right\rangle }}}_{{{{q_{1}q_{2}...q_{n}}}}}{{{\left\vert a_{{i}%
}\right\rangle }}}_{{{{p_{1}p_{2}...p_{m}}}}}}){\left\vert {ancilla_{2}}%
\right\rangle }\mathrm{,} \label{eqTarget}
\end{equation}%
where $N=2^{n}$ and the ancillary state ${\left\vert {ancilla_{2}}%
\right\rangle }$ is \emph{known }}(all {ancillary states is \emph{known }}in
this paper.){. }
Nielsen and Chuang pointed out that quantum computer should have loading
scheme in principle to load classical database record into quantum registers
from classical database \cite[section 6.5]{Nielsen, QCZhao}. However, there
is no detailed work on QLS up till now. In fact, the research of QLS is
motivated by the quantum algorithm of image compression \cite{Pang
PostDocReport,Pang QVQ2,Pang QDCT,Pang QVQ1,Lattorre}. In this paper, we
present a QLS based on the path interference, which has been widely used in
quantum information processing, e.g. non-unitary computation\cite%
{Kwiat,LongGuiLu}. The unitary computation using path interference is
demonstrated in this paper, and the output of the unitary computation can be
measured with successful probability 100\% in theory. The time complexity of
our QLS is $O(log_{2}N)$, which exhibits a speed-up over CLS with time
complexity $O(N)$.
\section{The Design of QLS}
\subsection{Loading 2D Vector into Quantum Registers from Classical Memory}
The design of unitary operation $U_{(0,1)}$ that loads 2D vector $\vec{a}%
=\{a_{0},a_{1}\}$ is described conceptually as follows (see Fig.\ref{figU01}%
):
\textbf{Step 1} The switch $S_{1}$ applies rotation on the initial ancilla
state and transforms $\left\vert {Off_{0}}\right\rangle $ into%
\begin{equation*}
{\left\vert {Off_{0}}\right\rangle \overset{S_{1}}{\rightarrow }\frac{{{%
\left\vert {Off_{1}}\right\rangle }+{\left\vert {On_{1}}\right\rangle }}}{%
\sqrt{2}}}
\end{equation*}%
and generate the following state $\left\vert {\phi _{1}}\right\rangle $
\begin{equation}
{\left\vert {\phi _{1}}\right\rangle }{=}\frac{1}{\sqrt{2}}{\left\vert {0}%
\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert {0}\right\rangle }_{{_{%
{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {On_{1}}\right\rangle +}\frac{1}{%
\sqrt{2}}{\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{%
\left\vert {0}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {%
Off_{1}}\right\rangle } \label{eqFai1}
\end{equation}
\textbf{Step 2 }Perform unitary operations $I_{0}$ and $A_{0}$ along `$%
On_{1} $' path, while perform unitary operations $I_{1}$ and $A_{1}$ along `$%
Off_{1} $' path.
\begin{equation*}
\begin{tabular}{lll}
$\left\{
\begin{tabular}{c}
${\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}\overset{I_{0}}{%
\rightarrow }{\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}$ \\
${{\left\vert {0}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}}\overset{%
A_{0}}{{\rightarrow }}{{\left\vert a_{{0}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}}}$%
\end{tabular}%
\right. $ & , & $\left\{
\begin{tabular}{c}
${\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}\overset{I_{1}}{%
\rightarrow }{\left\vert {1}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}$ \\
${{\left\vert {0}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}}\overset{%
A_{1}}{{\rightarrow }}{{\left\vert a_{{1}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}}}$%
\end{tabular}%
\right. $%
\end{tabular}%
\end{equation*}
We assume the output of two pathes are simultaneous, then the state ${%
\left\vert {\phi _{2}}\right\rangle }$ is generated as
\begin{equation*}
\left\{
\begin{tabular}{c}
$\frac{1}{\sqrt{2}}{\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{%
\left\vert {0}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {%
On_{1}}\right\rangle }\overset{A_{0}I_{0}}{{\rightarrow }}\frac{1}{\sqrt{2}}{%
\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{0}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {On_{1}}%
\right\rangle }$ \\
$\frac{1}{\sqrt{2}}{\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{%
\left\vert {0}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {%
Off_{1}}\right\rangle }\overset{A_{1}I_{1}}{{\rightarrow }}\frac{1}{\sqrt{2}}%
{\left\vert {1}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {Off_{1}}%
\right\rangle }$%
\end{tabular}%
\right.
\end{equation*}
\begin{equation}
\Rightarrow {\left\vert {\phi _{2}}\right\rangle =}\frac{1}{\sqrt{2}}{%
\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{0}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}{\left\vert {On_{1}}%
\right\rangle +}\frac{1}{\sqrt{2}}{\left\vert {1}\right\rangle }_{{{{%
q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}}{\left\vert {Off_{1}}\right\rangle } \label{eqFai2}
\end{equation}
The functions of $I_{0}$ and $I_{1}$ are to generate subscripts $0$ and $1$
respectively, and the functions of $A_{0}$ and $A_{1}$ are to generate
numbers $a_{0}$ and $a_{1}$ respectively. Because value $a_{0}$ and $a_{1}$
are both known numbers, flipping part of the $m+n$ qubits ${{{%
q_{1},...,q_{n},p_{1},...,p_{m}}}}$ will generate states ${{\left\vert a_{{0}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}}$ or ${{\left\vert a_{{1}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}}$. Thus, the unitary
operations $I_{0}$, $I_{1}$, $A_{0}$, $A_{1}$ is easy to be designed.
\textbf{Step 3 }The switch $S_{2}$ applies rotation on the initial ancilla
state {as}
\begin{equation*}
\left\{
\begin{tabular}{c}
${\left\vert {Off_{1}}\right\rangle }\overset{S_{2}}{\rightarrow }{\frac{{{%
\left\vert {Off_{2}}\right\rangle }-{\left\vert {On_{2}}\right\rangle }}}{%
\sqrt{2}}}$ \\
${\left\vert {On_{1}}\right\rangle }\overset{S_{2}}{\rightarrow }{\frac{{{%
\left\vert {Off_{2}}\right\rangle }}+{{\left\vert {On_{2}}\right\rangle }}}{%
\sqrt{2}}}$%
\end{tabular}%
\right.
\end{equation*}%
and generate the following state ${\left\vert {\phi _{3}}\right\rangle }$
\begin{eqnarray}
{\left\vert {\phi _{3}}\right\rangle } &=&\frac{1}{2}({\left\vert {0}%
\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{0}}\right\rangle }%
_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}+{\left\vert {1}\right\rangle }_{{{{%
q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}}){\left\vert {Off_{2}}\right\rangle } \notag \\
&&{+}\frac{1}{2}({\left\vert {0}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{%
\left\vert a_{{0}}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}-{%
\left\vert {1}\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}%
}\right\rangle }_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}){\left\vert {On_{2}}%
\right\rangle } \label{eqFai3}
\end{eqnarray}
\textbf{Step 4 }Apply phase transformaition $B$ along `$On_{2}$' path.
\begin{equation}
B={\left\vert {0}\right\rangle \left\vert a_{{0}}\right\rangle }\langle a_{{0%
}}|\langle 0|-{\left\vert {1}\right\rangle \left\vert a_{{1}}\right\rangle }%
\langle a_{{1}}|\langle 1| \label{eqB}
\end{equation}
It's a very fast operation and generates the state ${\left\vert {\phi _{4}}%
\right\rangle }$
\begin{equation}
{\left\vert {\phi _{4}}\right\rangle }=\frac{1}{\sqrt{2}}({\left\vert {0}%
\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{0}}\right\rangle }%
_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}+{\left\vert {1}\right\rangle }_{{{{%
q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}})(\frac{{{\left\vert {Off_{2}}\right\rangle }}+{{%
\left\vert {On_{2}}\right\rangle }}}{\sqrt{2}}) \label{eqFai4}
\end{equation}
\textbf{Step 5 }The switch $S_{3}$ applies rotation on the initial ancilla
state {as}
\begin{equation*}
\left\{
\begin{tabular}{c}
${\left\vert {Off_{2}}\right\rangle }\overset{S_{3}}{\rightarrow }{\frac{{{%
\left\vert {Off_{3}}\right\rangle }+{\left\vert {On_{3}}\right\rangle }}}{%
\sqrt{2}}}$ \\
${\left\vert {On_{2}}\right\rangle }\overset{S_{3}}{\rightarrow }{\frac{{{%
\left\vert {Off_{3}}\right\rangle }}-{{\left\vert {On_{3}}\right\rangle }}}{%
\sqrt{2}}}$%
\end{tabular}%
\right.
\end{equation*}%
and generate the final state ${\left\vert {\phi }\right\rangle }$
\begin{equation}
{\left\vert {\phi }\right\rangle }=\frac{1}{\sqrt{2}}({\left\vert {0}%
\right\rangle }_{{{{q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{0}}\right\rangle }%
_{{_{{{{p_{1}p_{2}...p_{m}}}}}}}+{\left\vert {1}\right\rangle }_{{{{%
q_{1}q_{2}...q_{n}}}}}{\left\vert a_{{1}}\right\rangle }_{{_{{{{%
p_{1}p_{2}...p_{m}}}}}}}){{\left\vert {Off_{3}}\right\rangle }}
\label{eqFai}
\end{equation}
Fig.\ref{figU01} and Eq.(\ref{eqU01}) illustrate the processing of operation
$U_{(0,1)}$.
\begin{equation}
\begin{tabular}{c}
$|0\rangle |0\rangle |Off_{0}\rangle \overset{S_{1}}{\rightarrow }%
\left\langle
\begin{tabular}{c}
$\frac{1}{\sqrt{2}}|0\rangle |0\rangle |On_{1}\rangle \overset{A_{0}I_{0}}{%
\rightarrow }\frac{1}{\sqrt{2}}|0\rangle |a_{0}\rangle |On_{1}\rangle $ \\
$\frac{1}{\sqrt{2}}|0\rangle |0\rangle |Off_{1}\rangle \overset{A_{1}I_{1}}{%
\rightarrow }\frac{1}{\sqrt{2}}|1\rangle |a_{1}\rangle |Off_{1}\rangle $%
\end{tabular}%
\right\rangle \overset{S_{2}}{\rightarrow }$ \\
$\left\langle
\begin{tabular}{c}
$\frac{1}{2}(|0\rangle |a_{0}\rangle +|1\rangle |a_{1}\rangle
)|Off_{2}\rangle $ \\
$\frac{1}{2}(|0\rangle |a_{0}\rangle -|1\rangle |a_{1}\rangle )|On2\rangle
\overset{B}{\rightarrow }\frac{1}{2}(|0\rangle |a_{0}\rangle +|1\rangle
|a_{1}\rangle )|On_{2}\rangle $%
\end{tabular}%
\right\rangle \overset{S_{3}}{\rightarrow }{\left\vert {\phi }\right\rangle }
$%
\end{tabular}
\label{eqU01}
\end{equation}%
Here as well as in the following discussions, all subscripts of registers
are ignored.
\begin{figure}[h]
\epsfig{file=fig1_U01.eps,width=9cm,}
\caption{\textbf{The Illustration of the processing of unitary operation }$%
U_{(0,1)}$\textbf{\ that transforms state from }$\left\vert {0}\right\rangle
\left\vert {0}\right\rangle \left\vert {Off}_{{0}}\right\rangle $\textbf{\
into }$\frac{1}{\protect\sqrt{2}}(|0\rangle |a_{0}\rangle +|1\rangle
|a_{1}\rangle )\left\vert {Off}_{{3}}\right\rangle $. M: Mirror}
\label{figU01}
\end{figure}
\subsection{Loading 4D Vector or Multi-Dimensional into Quantum Registers}
The design of unitary operation $U_{(0,1,2,3)}$ is described conceptually as
follows (see Fig.\ref{figU0123}):
\begin{description}
\item[Step 1] Construct unitary operation $S_{4}$, $S_{5}$, $S_{6},B^{\prime
}$ as following:
\end{description}
$\left\{
\begin{tabular}{c}
${\left\vert {Off}_{i}\right\rangle }\overset{S_{i}}{\rightarrow }{\frac{{{%
\left\vert {Off_{i+1}}\right\rangle }+{\left\vert {On_{i+1}}\right\rangle }}%
}{\sqrt{2}}}$ \\
${\left\vert {On}_{i}\right\rangle }\overset{S_{i}}{\rightarrow }{\frac{{{%
\left\vert {Off_{i+1}}\right\rangle }}-{{\left\vert {On_{i+1}}\right\rangle }%
}}{\sqrt{2}}}$%
\end{tabular}%
\right. $, $\left\{
\begin{tabular}{c}
${\left\vert {Off}_{5}\right\rangle }\overset{S_{5}}{\rightarrow }{\frac{{{%
\left\vert {Off_{i+1}}\right\rangle }-{\left\vert {On_{i+1}}\right\rangle }}%
}{\sqrt{2}}}$ \\
${\left\vert {On}_{5}\right\rangle }\overset{S_{5}}{\rightarrow }{\frac{{{%
\left\vert {Off_{i+1}}\right\rangle }}+{{\left\vert {On_{i+1}}\right\rangle }%
}}{\sqrt{2}}}$%
\end{tabular}%
\right. $, $\ B^{\prime }=|\alpha \rangle \langle \alpha |-|\beta \rangle
\langle \beta |$,
where $i=4,6$, $|\alpha \rangle =\frac{1}{\sqrt{2}}(|0\rangle |a_{0}\rangle
+|1\rangle |a_{1}\rangle )$ and $|\beta \rangle =\frac{1}{\sqrt{2}}%
(|2\rangle |a_{2}\rangle +|3\rangle |a_{3}\rangle )$
\begin{description}
\item[Step 2] Assemble unitary operations $S_{4}$, $S_{5}$, $S_{6},B^{\prime
},U_{(0,1)}$ and $U_{(2,3)}$ according to Fig.\ref{figU0123} to form unitary
operations $U_{(0,1,2,3)}$.
\end{description}
Eq.(\ref{eqU0123}) illustrates the processing of operation $U_{(0,1,2,3)}$.
\begin{equation}
\begin{tabular}{c}
$|0\rangle |0\rangle |Off_{4}\rangle \overset{S_{4}}{\rightarrow }%
\left\langle
\begin{tabular}{c}
$\frac{1}{\sqrt{2}}|0\rangle |0\rangle |On_{5}\rangle \overset{U_{(0,1)}}{%
\rightarrow }\frac{1}{2}(|0\rangle |a_{0}\rangle +|1\rangle |a_{1}\rangle
)|On_{5}\rangle $ \\
$\frac{1}{\sqrt{2}}|0\rangle |0\rangle |Off_{5}\rangle \overset{U_{(2,3)}}{%
\rightarrow }\frac{1}{2}(|2\rangle |a_{2}\rangle +|3\rangle |a_{3}\rangle
)|Off_{5}\rangle $%
\end{tabular}%
\right\rangle \overset{S_{5}}{\rightarrow }$ \\
$\left\langle
\begin{tabular}{c}
$\frac{1}{2}[\frac{1}{\sqrt{2}}(|0\rangle |a_{0}\rangle +|1\rangle
|a_{1}\rangle )+\frac{1}{\sqrt{2}}(|2\rangle |a_{2}\rangle +|3\rangle
|a_{3}\rangle )]|Off_{6}\rangle $ \\
\\
$\frac{1}{2}[\frac{1}{\sqrt{2}}(|0\rangle |a_{0}\rangle +|1\rangle
|a_{1}\rangle )-\frac{1}{\sqrt{2}}(|2\rangle |a_{2}\rangle +|3\rangle
|a_{3}\rangle )]|On_{6}\rangle \overset{B^{\prime }}{\rightarrow }$ \\
$\frac{1}{2}[\frac{1}{\sqrt{2}}(|0\rangle |a_{0}\rangle +|1\rangle
|a_{1}\rangle )+\frac{1}{\sqrt{2}}(|2\rangle |a_{2}\rangle +|3\rangle
|a_{3}\rangle )]|On_{6}\rangle $%
\end{tabular}%
\right\rangle \overset{S_{6}}{\rightarrow }{\left\vert {\phi }\right\rangle }
$%
\end{tabular}
\label{eqU0123}
\end{equation}
\begin{figure}[h]
\epsfig{file=fig2_U0123.eps,width=8cm,}
\caption{\textbf{The illustration of unitary operation }$U_{(0,1,2,3)}$%
\textbf{\ that transforms state from }$\left\vert {0}\right\rangle
\left\vert {0}\right\rangle \left\vert {Off}_{{4}}\right\rangle $\textbf{\
into }$(\protect\underset{i=0}{\protect\overset{3}{\sum }}\frac{1}{2}%
\left\vert {i}\right\rangle \left\vert a_{{i}}\right\rangle )\left\vert {Off}%
_{{7}}\right\rangle $.}
\label{figU0123}
\end{figure}
If the unitary operations $U_{(0,1)}$ and $U_{(2,3)}$ embedded in Fig.\ref%
{figU0123} are replaced by $U_{(0,1,2,3)}$ and $U_{(4,5,6,7)}$ respectively,
then $U_{(0,1,...,7)}$ is constructed. Similar to Fig.\ref{figU0123}, we can
apply the same method to construct unitary operation $U_{(0,1,...,2^{n})}$.
If $N\neq 2^{n}$, we could add extra zero components to create a $2^{n}$%
-dimensional vector.
\subsection{Loading Vector into State $\frac{1}{\protect\sqrt{N}}({%
\sum\limits_{i=0}^{N-1}{{{\left\vert {i}\right\rangle \left\vert
0\right\rangle )}}}}$ to Form Entangled State $\frac{1}{\protect\sqrt{N}}({%
\sum\limits_{i=0}^{N-1}{{{\left\vert {i}\right\rangle \left\vert a_{{i}%
}\right\rangle )}}}}$}
Grover's algorithm \cite{Nielsen} has the function that find the index $%
i_{0} $ of a special database record $record_{i_{0}}$ from the index
superposition of state $\frac{1}{\sqrt{N}}({\sum\limits_{i=0}^{N-1}{{{%
\left\vert {i}\right\rangle )}}}}$ taking $O(\sqrt{N})$ steps. And the
record $record_{i_{0}}$\ is the genuine answer wanted by us. However, the
corresponding record $record_{i_{0}}$ can not be measured out unless the
1-1mapping relationship between index $i$ and the corresponding record $%
record_{i}$ is bound in the entangled state $\frac{1}{\sqrt{N}}({%
\sum\limits_{i=0}^{N-1}{{{\left\vert {i}\right\rangle {{{\left\vert
record_{i}\right\rangle }}})}}}}$. That is, we need a unitary$\ $operation $%
U_{L}$ such that
\begin{equation}
\frac{1}{\sqrt{N}}({\sum\limits_{i=0}^{N-1}{{{\left\vert {i}\right\rangle
\left\vert 0\right\rangle )\left\vert {ancilla_{4}}\right\rangle }}}}\overset%
{U_{L}}{{\rightarrow }}\frac{1}{\sqrt{N}}({\sum\limits_{i=0}^{N-1}{{{%
\left\vert {i}\right\rangle \left\vert a_{{i}}\right\rangle )\left\vert {%
ancilla_{3}}\right\rangle }}}} \label{eqUL}
\end{equation}
Ref.\cite{Pang QVQ1, Pang QDCT} generalize Grover's algorithm to the general
search case with complex computation, and $U_{L}$ is required in this
general search case.
$U_{L}$ can be designed using the same method shown in Fig.\ref{figU01} and
Fig.\ref{figU0123}. Fig.\ref{figUL} shows the design of the inverse unitary
operation $(U_{L})^{\dagger }$ at the case $N=2$. $U_{L}$ has time
complexity $O(log_{2}N)$ (unit time: phase transformation and flipping the
qubits of registers).
\begin{figure}[h]
\caption{\textbf{The Illustration of Unitary Operation }$(U_{L})^{\dagger }$%
: $\frac{1}{\protect\sqrt{2}}({\sum\limits_{i=0}^{1}{{{\left\vert {i}%
\right\rangle \left\vert a_{{i}}\right\rangle )\left\vert {Off_{3}}%
\right\rangle }}}}\rightarrow \frac{1}{\protect\sqrt{2}}({%
\sum\limits_{i=0}^{1}{{{\left\vert {i}\right\rangle \left\vert
0\right\rangle )\left\vert {Off}\right\rangle }}}}$. Operation $U_{L}$ can
be designed using the same method shown in Fig.\protect\ref{figU01} and Fig.%
\protect\ref{figU0123}. $S_{0}$: ${\left\vert {Off_{0}}\right\rangle }%
\rightarrow \frac{1}{\protect\sqrt{2}}({{\left\vert {Off}\right\rangle }+{%
\left\vert {On}\right\rangle }})${, }${\left\vert {On_{0}}\right\rangle }%
\rightarrow \frac{1}{\protect\sqrt{2}}({{\left\vert {Off}\right\rangle }}-{{%
\left\vert {On}\right\rangle }})$. Phase transformation $D={\left\vert i_{{1}%
}\right\rangle \left\vert 0\right\rangle }\langle 0|\langle i_{{1}}|-{%
\left\vert i_{{0}}\right\rangle \left\vert 0\right\rangle }\langle 0|\langle
i_{{0}}|$, where $i_{{0}}=0$, $i_{{1}}=1$.}
\label{figUL}\epsfig{file=fig3_UL.eps,width=8cm,}
\end{figure}
It has been demonstrated that giant molecules, such as charcoal $c_{60}$,
exhibit quantum interference \cite{Zeilinger}. Thus many freedom degrees of
giant molecule can be regarded as qubits to realize the QLS presented in
this paper. In addition, one of QLS application is that QLS can load the
data of image with huge size into quantum registers at a time for further
image compression \cite{Pang PostDocReport,Pang QVQ2,Pang QDCT,Pang QVQ1},
while only one data can be loaded into registers at a time for classical
computer.
\section{Conclusion}
Designing simple and fast unitary operation to load classical data set, such
as vector, into quantum registers from classical memory is called quantum
loading scheme (QLS). QLS\ makes quantum CPU is compatible with classical
memory, and it assembles classical memory and quantum CPU as a whole. QLS is
the base of further quantum computation. The QLS with time complexity $%
O(log_{2}N)$ (unit time: phase transformation and flipping the qubits of
registers)\ is presented in this paper, while classical loading scheme (CLS)
has time complexity $O(N)$ (unit: addition) because all computation
instructions have to be executed one by one. Path interference is applied to
design QLS in this paper so that the complexity of designing quantum
algorithm is decomposed as the design of many simple unitary operations. In
addition, this paper demonstrates that using path interference to design
unitary operation and parallel quantum computation is possible.
\begin{acknowledgments}
The author thanks Dr. Z.-W Zhou\ who is at Key Lab. of Quantum Information,
Uni. of Science and Technology of China for that he points out two errors in
author's primary idea. The author's first error is that the result generate
with probability 50\% for 2D vector, the second error is the defect of Fig.%
\ref{figU0123} that the output is direct product state. Dr. Z.-W Zhou tries
his best to help author for nearly 3 years. The author thanks his teacher,
prof. G.-C Guo. The author is brought up from Guo's Lab.. The author thanks
prof. V. N. Gorbachev who is at St.-Petersburg State Uni. of Aerospace
Instrumentation for the useful discussion with him. The author thanks Mir.
N. Kiesel who is at Max-Plank-Institute fur Quantenoptik, Germany for his
checking the partial deduction of section 2 of this paper. The author thanks
prof. G.-L Long who is at Tsinghua Uni., China for the useful discussion
with him and the author obtains some heuristic help from his eprint file
quant-ph/0512120. The author thanks associate prof. Shiuan-Huei Lin who is
at National Chiao Tung Uni., Taiwan., China for encouraging the author. The
author thanks prof. Hideaki Matsueda who is at Kochi Uni., Japan for
encouraging the author. The author thanks prof. J. Zhang and B.-P Hou who
are at Sichuan Normal Uni., China for their help. The author thanks prof.
Z.-F. Han, Dr. Y.-S. Zhang, Dr. Y.-F. Huang, Dr. Y.-J. Han, Mr. J.-M Cai,
Mr. M.-Y. Ye, and Mr. M.-Gong for their help and suggestions. One of
reviewers presents many significative suggestions to improve the readability
of this paper, the author thanks the reviewer.
\end{acknowledgments}
|
2,877,628,090,183 | arxiv | \section{Introduction}
General relativistic spaces filled with black holes have recently
been under scrutiny as exact cosmological models with a discrete mass
distribution which is, in some sense, uniform on large scales. The
construction of these spaces in numerical relativity has enabled
the investigation of several questions without approximations, such as
how such configurations evolve in time and what their global physical
properties are~\cite{Yoo:2012jz,Bentivegna:2012ei,Yoo:2013yea,Bentivegna:2013jta,Yoo:2014boa}.
At the same time, the numerical simulations have been complemented
by insight coming from analytical studies, which have illustrated
some general features of these spacetimes such as the behaviour of special
submanifolds~\cite{Clifton:2013jpa,Clifton:2014lha,Korzynski:2015isa},
the conditions under which they behave like the
Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) models~\cite{Korzynski:2013tea},
and the link between their behaviour and the validity of Gauss's law
in a generic theory of gravity~\cite{Fleury:2016tsz}.
In this work, we use numerical spacetimes representing black-hole
lattices (BHLs) to probe a different aspect of inhomogeneous
cosmologies, namely their optical behaviour.
As is well known, null
geodesics are the bedrock of cosmological observations: light
from distant sources is the primary tool for measuring the
Universe's density parameters, equation of state, and perturbations.
Increasing the accuracy of models of light propagation and identifying the
biases introduced by various approximation frameworks is thus
critical.
Modelling light propagation in inhomogeneous cosmologies is a long-standing
effort, which has followed two complementary courses: approximation
schemes on one hand, and toy models on the other. The best-known
approach in the former class is the Empty-Beam Approximation (EBA) of
Zeldovich~\cite{1964SvA.....8...13Z}, later generalized by
Dyer and Roeder~\cite{1972ApJ174L115D,Dyer:1973zz}. This approach
is based on the idea that different effects are at play when light propagates
in a perturbed fluid or through discretely-distributed point masses, as
different components of the curvature become dominant in either regime
(this is sometimes referred to as the \emph{Ricci-Weyl problem}~\cite{Fleury:2013sna}).
This approach provides an excellent estimate of light propagation in
Swiss-Cheese models, and can be used to constrain the fraction of
voids in a cosmological model~\cite{Fleury:2013sna,Fleury:2013uqa,Fleury:2014gha}.
The notion that discreteness may affect light propagation more
than inhomogeneity itself has also appeared in other studies, such as
those on light propagation through Schwarzschild-cell
universes~\cite{Clifton:2009jw,Clifton:2009bp,Clifton:2011mt}.
The existing literature points in a number of common directions:
first, examining individual geodesics, one concludes that the effective
value of the cosmological constant (the one obtained fitting the
spacetime to an FLRW model with the same matter density) is higher
than its microscopic value (the one appearing in the gravitational action).
Second, a statistical average of photon
trajectories usually leads to a partial suppression of this
difference. A suppression is also obtained by considering
the perturbative solution corresponding to a regular arrangements
of objects of equal mass, at least until the perturbative condition
is respected~\cite{Bruneton:2012ru}.
Though consistent on many aspects, these studies are limited
by the conditions imposed on the underlying model: most of the
discrete-mass studies are either based on spherically-symmetric
building blocks or on the requirements that the objects be not too
compact. It is presently not clear what the optical properties of a more
generic space would be.
To investigate this issue and test the generality of the existing results,
in this paper we compute the photon redshift and luminosity
distance along null geodesics running through a BHL spacetime, constructed
exactly through the integration of Einstein's equation, non-perturbatively and
in three dimensions. First we compare
the result to some reference models from the FLRW class, to the Milne
cosmology, and to a generic universe in the Empty Beam Approximation (EBA)~\cite{1964SvA.....8...13Z,1972ApJ174L115D,Dyer:1973zz}.
We find that the latter provides the closest approximation to light
propagation on the BHL, and derive a simple argument to explain this result,
which in some sense extends the reasoning of~\cite{Fleury:2014gha}
to completely vacuum spacetimes.
We then turn to the question of whether it is possible to tune the
cosmological parameters in the FLRW class to improve the fit. We find, in
particular, that one can reproduce the luminosity-distance--to--redshift
relationship of a BHL with that of an FLRW model with the same average
matter density and a fictitious, time-dependent
cosmological constant $\Lambda$, and provide the first measurement of
this running in our base configuration. Finally, we study how this behaviour
depends on the BHL inhomogeneity parameter $\mu$~\cite{Bentivegna:2013jta},
which roughly corresponds to the ratio between the central mass and the
lattice spacing, and in particular we analyse the continuum limit of $\mu \to 0$.
An important factor in this discussion is the choice of light
sources and observers, as the photon frequencies and number counts
will depend on the reference frame in which they are measured.
In FLRW models there is an obvious option: the comoving sources and
observers. In inhomogeneous spaces, on the other hand, identifying a
``cosmic flow'' is more tricky (when at all possible) and relies on the
somewhat arbitrary split between global cosmological evolution and
``local effects'' sourced by nearby gravitational structures.
For the purpose of this work, we sidestep this question
by noticing that, for a given geodesic, the angular and luminosity distances
can be obtained by applying a certain linear operator to the four-velocity
of the observer, with no dependence whatsoever on the motion of the light source.
It is therefore straightforward to quantify the effect of different
observer prescriptions on these observables.
Section~\ref{sec:lprop} introduces the formalism of light propagation and justifies
the approach we take in our analysis, providing some examples
in simple spacetimes. Section~\ref{sec:bhl} provides an approximate description
of light propagation in a BHL via a perturbative analysis.
We present the numerical results in section~\ref{sec:results} and in section~\ref{sec:dc} we comment on them.
We provide tests of the geodesic integrator, used for the first time
in this study, in the appendix.
We use geometric units $G=c=1$ everywhere.
\section{Fundamentals of light propagation}
\label{sec:lprop}
Let us start by considering a null ray emanating from a light
source $\cal{S}$ and reaching an observer $\cal{O}$: this curve can be
described as an affinely-parametrized null geodesic $\gamma(\lambda)$, with
$\cal{S}$ and $\cal{O}$ as end points corresponding to the affine parameter
values $\lambda_{\cal{S}}$ and $\lambda_{\cal{O}}$:
\begin{eqnarray}
\gamma(\lambda_{\cal{S}}) = \cal{S}\\
\gamma(\lambda_{\cal{O}}) = \cal{O}
\end{eqnarray}
The curve is described by the geodesic equation:
\begin{equation}
\label{eq:GE}
{ \nabla_p} p^a = 0
\end{equation}
where:
\begin{equation}
p^a = \frac{\ensuremath{\textrm{d}} x^a}{\ensuremath{\textrm{d}} \lambda}
\end{equation}
is the tangent vector to $\gamma$.
In order to measure distances with null rays, however, we need more
than a single geodesic: we need to consider a whole {\it beam} of
rays~\cite{Seitz:1994xf}, centred on $\gamma$, and study the evolution
of its cross-sectional area as it makes its way from $\cal{S}$ to
$\cal{O}$.
The time evolution of a beam's cross section is described by the
geodesic deviation equation (GDE). Let $\xi^a$ be the separation
vector between the fiducial geodesic $\gamma$ and a neighbouring
one, called $\tilde \gamma$. It satisfies
\begin{eqnarray}
\nabla_p \nabla_p \xi^a = R\UD{a}{bcd}\,p^b\,p^c\,\xi^d. \label{eq:GDE}
\end{eqnarray}
The GDE is a second order ODE for the 4--vector $\xi^a$, or equivalently
a first order ODE for $\xi^a$ and $\nabla_p \xi^a$.
It is valid for any neighbouring geodesic, but since in the geometrical optics
we are only interested in null geodesics, we impose a restriction
on the solution $\xi^a(\lambda)$ of the form:
\begin{eqnarray}
p_a \nabla_p \xi^a = 0, \label{eq:null}
\end{eqnarray}
which ensures that $\tilde\gamma$ is null.
Note that if the equation above is satisfied at one point, then it is
automatically satisfied along the whole of $\gamma$ because of equation
(\ref{eq:GDE}).
Let us now restrict the geodesics under consideration to those which
lie on the same wavefront as $\gamma$, i.e.~for which the separation vector satisfies
\begin{eqnarray}
\xi^a\,p_a = 0. \label{eq:wavefront}
\end{eqnarray}
The condition above means that, for a given observer at a given time, the
photon corresponding to the geodesic $\gamma$ and the one corresponding to
$\tilde \gamma$ lie on the same 2-plane perpendicular to the direction of
propagation (see Figure~\ref{fig:wavefronts}). This condition is Lorentz-invariant, meaning that
if it is satisfied in one reference frame then it is valid in all frames.
Moreover, for null geodesics it propagates along $\gamma$, i.e.~if it is
satisfied at one time it is satisfied along the whole of $\gamma$. This
follows easily from (\ref{eq:null}) and (\ref{eq:GDE}).
\begin{figure}[!h]
\centering
\includegraphics[width=0.5\textwidth]{plots/wavefront.pdf}
\caption{The null geodesics lying on the same wavefront consist of geodesics for which the photons at any instant of time
and for any observer lie on the same plane perpendicular to the direction of propagation given by $p_a$.
\label{fig:wavefronts}}
\end{figure}
The reason why we are interested only in geodesics which lie on the same
wavefront is that we want to study geodesics which cross at one point,
either the emission point ${\cal S}$ or the observation point ${\cal O}$.
If this is the case, then $\xi^a = 0$ at either $\lambda_{\cal O}$ or
$\lambda_{\cal S}$, so that (\ref{eq:wavefront}) is trivially satisfied there
and thus also \emph{everywhere} on $\gamma$.
By imposing (\ref{eq:null}) and (\ref{eq:wavefront}) we have effectively
reduced the number of degrees of freedom from four to three. It turns out
that a further reduction is possible.
Note that at every point we are free to add a vector proportional to $p^a$
to both $\xi^a$ and $\nabla_p\xi^a$. The former corresponds to using a
different point of \emph{the same} geodesic $\gamma$ in the definition of
the separation vector $\xi^a$, while the latter is just a rescaling of the
affine parametrization of $\gamma$. Neither transformation affects the
physical content of the equations, as long as we are in the regime of
geometrical optics. As a matter of fact, it is easy to see that equations
(\ref{eq:GDE})--(\ref{eq:wavefront}) are insensitive to these transformations
as well:
\begin{eqnarray}
&&\nabla_p \nabla_p \left(\xi^a + C(\lambda)\,p^a\right) = R\UD{a}{bcd}\,p^b\,p^c\,\xi^d + \ddot C\,p^a \\
&&\nabla_p \left(\xi^a + C(\lambda)\,p^a\right)\,p_a = \dot C\,p^a\,p_a = 0 \\
&&\left(\xi^a + C(\lambda)\,p^a\right)\,p_a = C\,p^a\,p_a = 0.
\end{eqnarray}
It follows that (\ref{eq:GDE})--(\ref{eq:wavefront}) can be reinterpreted as
equations on the space $p^\perp/p$, consisting of vectors orthogonal to
$p_a$ and divided by the relation $\xi^a \sim \eta^a \iff \xi^a = \eta^a + A\,p^a$.
We shall denote the equivalence class corresponding
to a vector $\xi^a$ in $p^\perp$ as $\left[\xi\right]^A$.
The space $p^\perp/p$ is two--dimensional and inherits the positive-definite
metric from $g_{ab}$ via the relation $\left[X\right]^A\,\left[Y\right]^B\,g_{AB}
= X^a\,Y^b\,g_{ab}$, where $X^a$ and $Y^b$ are any vectors in the tangent
space corresponding to the equivalence classes $[X]^A$ and $[Y]^B$, respectively.
It can be thought of as the space of null geodesics lying in the neighbourhood
of $\gamma$ on the same wavefront, without any specification of which point on $\gamma$ we assign to
which point of $\tilde \gamma$. It is straightforward to verify that the covariant derivative
$\nabla_p$ can also be defined as an operator on $p^\perp/p$.
In the standard formalism due to Sachs \cite{Sachs309, lrr-2004-9}, we then
introduce a frame with two spatial, orthonormal screen vectors $\xi_1^a$ and $\xi_2^a$,
both orthogonal to $p^a$ and to a timelike observer $u^a_{\cal O}$. Notice that
this is not strictly necessary:
all that matters in geometrical optics are the \emph{equivalence classes}
$\left[\xi_1\right]^A$ and $\left[\xi_2\right]^B$, which turn out to be entirely
$u^a_{\cal O}$\emph{-independent}. More precisely, for any other choice of the observer
$\tilde u^a_{\cal O}$ and the corresponding $\tilde \xi_1^a$ and $\tilde \xi_2^a$ perpendicular to
$p_a$, the classes $\left[\tilde \xi_1\right]^A$ and $\left[\tilde \xi_2\right]^B$ are
related to $\left[\xi_1\right]^A$ and $\left[\xi_2\right]^B$ via a simple spatial rotation.
The image distortion of a distant object and its angular distance can now be
calculated by finding the Jacobi matrix ${\ensuremath{\cal D}}\UD{A}{B}$ of the GDE in the space $p^\perp/p$
\begin{eqnarray}
\nabla_p \nabla_p {\ensuremath{\cal D}}\UD{A}{B} = R\UD{A}{\mu\nu C}\,p^\mu\,p^\nu\,{\ensuremath{\cal D}}\UD{C}{B} \label{eq:GDE2}
\end{eqnarray}
with the initial data of the form
\begin{eqnarray}
&&{\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal O}) = 0 \label{eq:ID2} \\
&&\nabla_p {\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal O}) = \delta\UD{A}{B} \nonumber
\end{eqnarray}
(see \cite{lrr-2004-9} for its geometric definition and the discussion of its properties). Note that the initial data depends on the choice of parametrization of the null
geodesic $\gamma$, because if we rescale $\lambda \mapsto C\,\lambda$, the tangent
vector rescales accordingly via $p^a \to C^{-1} p^a$. Thus ${\ensuremath{\cal D}}\UD{A}{B}$ is
parametrization-dependent. Nevertheless, the tensor product $p_\mu\,{\ensuremath{\cal D}}\UD{A}{B}$
is parametrization-independent and is therefore an intrinsic property of the light cone
centred at the observation point $\cal O$. In practice the equations (\ref{eq:GDE2})--(\ref{eq:ID2})
are solved by first introducing a Sachs frame and then using the corresponding
screen vectors $\left[ \xi_1\right]^A$ and $\left[ \xi_2\right]^B$ as a basis in $p^\perp/p$.
The image distortion seen by the observer with 4-velocity $u_{\cal O}^a$ at the
observation point is finally:
\begin{eqnarray}
I\UD{A}{B} = \left|u_{\cal O}^a\,p_a\right|\,{\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal S})
\end{eqnarray}
while the angular distance is
\begin{eqnarray}
\label{eq:DA}
D_{\rm A}=\left|u_{\cal O}^a\,p_a\right|\,\sqrt{\left|\det {\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal S})\right|}
\end{eqnarray}
(see also \cite{lrr-2004-9} and references therein). Note that the result does not depend on the
4-velocity of the source, while the dependence on the 4-velocity of the observer is quite simple.
For instance, it is easy to prove that, on an FLRW spacetime, observers boosted with respect to the
comoving frame measure smaller angular distances, because the quantity $\left|u_{\cal O}^a\,p_a\right|$
decreases as the boost parameter is increased. One can therefore use equation (\ref{eq:DA}) to work
out which observers (if any) would measure a specified angular distance for an object in a given
spacetime.
The luminosity distance is defined using the total energy flux from the source through a fixed area at the observation point. In the formalism above
it can be expressed as
\begin{eqnarray}
\label{eq:DL}
D_{\rm L}=\left|u_{\cal S}^a\,p_a\right|\,\sqrt{\left|\det \tilde{\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal O})\right|}(1+z)
\end{eqnarray}
where $\tilde{\ensuremath{\cal D}}\UD{A}{B}$ satisfies (\ref{eq:GDE2}), but with the initial conditions (\ref{eq:ID2}) imposed at the source rather than at the observer,
and $z$ is the relative change in the photon frequency as it moves along the geodesic, also known as its {\it redshift}:
\begin{equation}
z = \frac{\nu_{\cal S}-\nu_{\cal O}}{\nu_{\cal O}} = \frac{u_{\cal S}^a\,p_a}{u_{\cal O}^a\,p_a} - 1.
\end{equation}
The fundamental result by Etherington \cite{springerlink:10.1007/s10714-007-0447-x} relates these quantities: the reciprocity relation reads
\begin{eqnarray}
\left|\det \tilde{\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal O})\right| = \left|\det {\ensuremath{\cal D}}\UD{A}{B}(\lambda_{\cal S})\right|. \label{eq:Etherington}
\end{eqnarray}
It follows easily that
\begin{equation}
D_{\rm L} = (1+z)^2 D_{\rm A}.
\end{equation}
Relation (\ref{eq:Etherington}) allows one to calculate both distances by solving the GDE with the initial conditions (\ref{eq:ID2}) imposed either
at the source or at the observation point.
In this paper we have found it much simpler to impose the initial conditions
at the location of the source, and to
integrate the equations forward in time. Moreover, instead of solving the GDE directly, we simply use the geodesic tracker
and follow directly two additional null geodesics $\gamma_1(\lambda)$ and $\gamma_2(\lambda)$, slightly perturbed with respect to the principal one, which we denote with $\gamma_0(\lambda)$.
We specify the initial conditions for them at the source:
\begin{eqnarray}
x^a_1(\lambda_{\cal S}) &=& x^a_2(\lambda_{\cal S}) = x^a_0(\lambda_{\cal S}) \\
p_1^a(\lambda_{\cal S}) &=& p_0^a(\lambda_{\cal S}) + \epsilon \xi_1^a(\lambda_{\cal S}) \\
p_2^a(\lambda_{\cal S}) &=& p_0^a(\lambda_{\cal S}) + \epsilon \xi_2^a(\lambda_{\cal S})
\end{eqnarray}
where $x^a_I$ are the coordinates of geodesic $\gamma_I$ and $p^a_I$ is its 4-momentum.
We can then compute ${\ensuremath{\cal D}}\UD{A}{B}$ by using the fact that:
\begin{eqnarray}
{\ensuremath{\cal D}}\UD{A}{B}(\lambda) &=& \lim_{\epsilon \to 0} \frac{\sqrt{g(\lambda_{\cal S})}}{\epsilon} \left [
\begin{array}{ll}
g_{ab}(x^a_1-x^a_0)\,\xi_1^b \qquad & g_{ab}(x^a_2-x^a_0) \,\xi_1^b \\[0.5cm]
g_{ab}(x^a_1-x^a_0)\,\xi_2^b \qquad & g_{ab}(x^a_2-x^a_0) \,\xi_2^b
\end{array}
\right ]
\end{eqnarray}
where $g(\lambda_{\cal S})$ is the determinant of $g_{ab}$ at the geodesic initial
location.
This is the approach we take in the computations described in Section~\ref{sec:results}.
\subsection{Homogeneous cosmologies}
This formalism takes on a particularly simple form in the exactly
homogeneous and isotropic cosmological models (the FLRW class),
defined by the line element:
\begin{equation}
d s^2 = - d t^2 + a(t)^2 dl^2
\end{equation}
where $dl^2$ is the line element of one of the three three-dimensional
constant-curvature spaces of Euclidean signature.
In this case, geodesics can move along coordinate lines and be parametrized
by the coordinate time. In the flat case,
for instance, we can
choose $x$ as the geodesic direction (so that $\xi_1^a=a(t) \delta_y^a$ and
$\xi_2^a=a(t) \delta_z^a$, where $a(t)$ is the scale factor). The
matrix ${\ensuremath{\cal D}}\UD{A}{B}$ is then given by:
\begin{eqnarray}
{\ensuremath{\cal D}}\UD{A}{B}(t) &=& a_{\cal S} \left [
\begin{array}{cc}
a(t) x(t) & 0 \\
0 & a(t) x(t)
\end{array}
\right ]
\end{eqnarray}
where $x(t)$ is the coordinate distance travelled along the geodesic
at time $t$:
\begin{equation}
x(t)=\int_{t_{\cal S}}^{t} \frac{dt}{a(t)}
\end{equation}
Given the initial normalization $u_{\cal S}^a\,p_a=-a_{\cal S}^{-1}$, equation (\ref{eq:DL})
becomes:
\begin{equation}
\label{eq:flrwDL}
D_{\rm L}= a_{\cal O} (1+z) \int_{t_{\cal S}}^{t_{\cal O}} \frac{dt}{a(t)}
\end{equation}
Noticing that, in an FLRW model, the redshift
$z$ only depends on the ratio between the scale factor at the time of
detection and the scale factor at the time of emission:
\begin{equation}
z=\frac{a(t_{\cal O})}{a(t_{\cal S})} - 1,
\end{equation}
it is easy to show that equation (\ref{eq:flrwDL}) coincides with the
usual textbook expression for $D_{\rm L}$, which we quickly recall.
We first need to calculate the comoving distance covered by a photon between
${\cal S}$ and ${\cal O}$:
\begin{equation}
D_{\rm M}(z)=a_{\cal O} \int_{t_{\cal S}}^{t_{\cal O}} \frac{dt}{a(t)} = (1+z) S\left(\Omega_k,\int_0^z \frac{d\zeta}{H(\zeta)(1+\zeta)^2}\right),
\end{equation}
with
\begin{equation}
H(\zeta) = H_{\cal S}\sqrt{\Omega^{\cal S}_{\rm M}(1+\zeta)^{-3}+\Omega^{\cal S}_\Lambda+\Omega^{\cal S}_k(1+\zeta)^{-2}},
\end{equation}
and
\begin{eqnarray}
S(k,x) &=& \left\{
\begin{array}{ll}
k^{-1/2} \sin k^{1/2} x & \textrm{for } k > 0 \\
x & \textrm{for } k = 0 \\
|k|^{-1/2} \sinh |k|^{1/2} x & \textrm{for } k < 0
\end{array}
\right.
\end{eqnarray}
Notice that the reference values for all quantities are those at the source:
$a_{\cal S}$, $H_{\cal S}$, and $\Omega^{\cal S}_{\rm X}$ are the model's scale
factor, Hubble rate, and density parameters at the time the photon is emitted,
respectively. As is customary, we also define the curvature $\Omega$ parameter by:
\begin{equation}
\Omega^{\cal S}_k=1-\Omega^{\cal S}_{\rm M}-\Omega^{\cal S}_{\Lambda}.
\end{equation}
Notice that referring to the initial values of these parameters rather
than the final ones changes our expressions from the standard textbook
treatment. It is straightforward to show that the usual formulae are
recovered if one expresses all quantities at the source in terms of the
corresponding ones at the observer.
Having found an expression for $D_{\rm M}(z)$, we can use it to derive the apparent
luminosity $\ell$ of an object of intrinsic luminosity ${\cal L}$
(for details, see e.g.~\cite{Hogg:1999ad}):
\begin{equation}
\ell = \frac{\cal L}{4 \pi D_{\rm M}(z)^2 (1+z)^2}.
\end{equation}
Since the apparent luminosity is defined as:
\begin{equation}
D_{\rm L} (z) = \sqrt{\frac{\cal L}{4 \pi \ell}},
\end{equation}
we finally obtain:
\begin{equation}
\label{eq:ldflrw}
D_{\rm L} (z) = D_{\rm M}(z) (1+z) = (1+z)^2 S \left( \Omega_k, \int_0^z \frac{d\zeta}{H(\zeta)(1+\zeta)^2} \right)
\end{equation}
This can be easily identified, on a flat background, with (\ref{eq:flrwDL}).
In homogeneous and isotropic cosmologies, therefore, the luminosity distance
only depends on the redshift, and is parametrized by global quantities such
as the matter density and the curvature of spatial slices.
In the Einstein-de Sitter (EdS) model, $D_{\rm L}$ simply reduces to
\begin{equation}
\label{eq:ldeds}
D_{\rm L} (z) = \frac{2 (1+z)^2}{H_{\cal S}} \left( (1+z)^{1/2}-1 \right)
\end{equation}
\subsection{Inhomogeneous cosmologies}
The propagation of light in lumpy spacetimes has been studied since the
1960's with various approaches, starting with the EBA
proposed in~\cite{1964SvA.....8...13Z} and later
generalized in~\cite{1972ApJ174L115D,Dyer:1973zz}.
The key idea inspiring these studies is that, in cosmological models
where the matter is distributed in lumps, a large fraction of the light
beams would not contain matter, and would therefore not be affected
by the Ricci focusing characteristic of their FLRW counterparts.
Other limitations of the FLRW approximation and the related physical
effects were subsequently analysed, both in approximate scenarios and
in exact cosmological models (typically belonging to the Swiss-Cheese
family)~\cite{Fleury:2014gha,Seitz:1994xf,
KristianSachs,1967ApJ150737G,1969ApJ...155...89K,
1970ApJ...159..357R,Lamburt2005,2010PhRvD..82j3510B,
2011PhRvD..84d4011S,Nwankwo:2010mx,2012MNRAS.426.1121C,
2012JCAP...05..003B,Lavinto:2013exa,Troxel:2013kua,
Bagheri:2014gwa,
Peel:2014qaa}.
A few robust features of these studies, that do not depend on
the details of the models used, include that:
\begin{itemize}
\item Light sources appear reduced in size and dimmer in a lumpy
spacetime than in a homogeneous one with the same mean density;
\item The angular distance does not have a maximum, but keeps growing
all the way to the cosmic horizon;
\item The actual deceleration parameter $q_0$
is larger
than in
the case where the same data is analysed with an FLRW model with the
same mean density.
\end{itemize}
Later, when we measure the $D_{\rm L}(z)$ relationship in BHL spacetimes,
we will use these features as guidelines for what to expect. Many of
them do indeed hold for such highly nonlinear spacetimes too.
In fact, as discussed at length in Section~\ref{sec:results}, the luminosity
distance in a BHL follows rather closely the EBA~\cite{1964SvA.....8...13Z},
which we report for completeness:
\begin{equation}
\label{eq:eb}
D_{\rm L}(z)=\frac{2(1+z)^2}{5 H_{\cal S}}\left( 1 - \frac{1}{(1+z)^{5/2}}\right).
\end{equation}
In Section \ref{sec:bhl} we will explain why the EBA
is a good approximation of the redshift--luminosity
distance in a BHL, and point out that it is equivalent to neglecting the
Ricci term in the standard geodesic deviation equation.
\subsection{Geodesics and observer classes}
As with many other quantities of interest that can be calculated in
inhomogeneous cosmologies, the calculation of $D_{\rm L}(z)$
requires the choice of a time coordinate. In general, representing
the spacetime in the geodesic gauge will lead to coordinate observers
which are diversely affected by neighbouring gravitational structures,
and may experience, e.g., light redshifting which has nothing to do with
a global, suitably defined expansion rate (an observational cosmologist
would call these {\it local effects}).
A study of light propagation in inhomogeneous spaces, especially one that is
targeted at the comparison with the FLRW class, is then left with two
possibilities: a statistical approach in which observers and sources are
distributed stochastically throughout the spacetime, and a single
$D_{\rm L}(z)$ relationship is obtained by averaging over
their locations and four-momenta; or the construction of one or more
classes of {\it cosmological} observers, based on geometry-inspired
considerations such as following the geodesics of the average
gravitational field, or geodesics with minimal deviation.
We find the latter approach more likely to yield insight on the
different gauge choices and related effects, and therefore use
it in the remaining of this paper. Statistical reasoning is,
however, also an important ingredient, as the observational data
is arguably to be modelled through a mix of different observer and source
states of motion. As statistical analyses are a tricky endeavour in
cosmology, we leave this task for future work.
Notice that the second strategy is particularly difficult to deploy
on vacuum spacetimes, as the sources of the gravitational field are
only perceived through their effect on the metric tensor, and not
through the presence of matter, so singling out a ``local''
component of the gravitational field will in some cases not even be
well defined (for a discussion of this point, see
e.g.~\cite{Marra:2012pj} or~\cite{lrr-2004-9} and references therein).
We will however exploit the existence of global (albeit discrete)
symmetries in our BHLs and only turn our attention to geodesics which
are by construction least affected by local effects: these include, for
instance, the geodesics running along the edges of the fundamental
periodic cell constituting the lattice.
\section{Light propagation in BHLs}
\label{sec:bhl}
In this section, we build an approximate model for the propagation of light
in a BHL, based on a perturbative expansion in the BHL compactness parameter.
This will serve as a qualitative analysis of the physics of the propagation of light and as a support in the interpretation of the numerical results
presented in section~\ref{sec:results}. Note that an expansion in a similar parameter has already appeared in the context of BHLs \cite{Bruneton:2012ru}, although the details are different.
\subsection{A perturbative expansion in the compactness parameter}
Let $L$ denote the characteristic size of a lattice cell, such as its initial geodesic length,
and let $M$ be a characteristic mass, i.e.~the total mass contained in a cell.
As in~\cite{Bentivegna:2013jta}, we can introduce the dimensionless parameter
\begin{eqnarray}
\mu = \frac{M}{L}
\end{eqnarray}
measuring the lattice compactness. If we additionally introduce the characteristic mass density
$\rho = M L^{-3}$, we can see that
\begin{eqnarray}
\mu = \rho L^2,
\end{eqnarray}
i.e.~it goes down to zero as we decrease the size of
a cell keeping the mass density of the corresponding FLRW model fixed. Note that $\rho$ is related to the curvature
scale of the Friedmann model via $R=\rho^{-1/2}$, so $\mu$ can be reinterpreted as the separation of scales between the
size of an individual lattice cell and the radius of curvature of the FLRW model:
\begin{eqnarray}
\mu = \frac{L^2}{R^2}.
\end{eqnarray}
Note that the definition of $\mu$ involves a certain vagueness: we may take for the mass scale $M$
the ADM mass of the black hole measured at the other end of the Einstein-Rosen bridge, but also some other related parameter.
Also the choice of the length scale involves a certain arbitrariness. At the leading order we expect this ambiguity to be irrelevant.
We will now show how $\mu$ can be used to find a perturbative approximation for the metric tensor of the lattice model.
The approximation is different from the standard perturbative approximation on an FLRW background, in the sense that it
does not require the density contrast $\delta$ of the matter perturbation to be small. Obviously the problem of BHLs lies beyond the validity regime of the cosmological perturbation theory, because in a BHL we are
dealing with $\delta = -1$ everywhere.
We begin by introducing a coordinate system on a single cell. Let $\ensuremath{g^{(0)}}$ denote the background FLRW metric and $x^\mu$ be the Riemannian normal coordinate system around any point $P$. The metric takes the form of
\begin{eqnarray}
\ensuremath{g^{(0)}}_{\mu\nu} = \eta_{\mu\nu} - \frac{1}{3} R_{\mu\alpha\nu\beta}\Big|_P\,x^\alpha\,x^\beta + O(x^3).
\end{eqnarray}
Since $R$ is the curvature scale of the metric, the coefficients in the expansion above are of order $R^0$ (the flat metric), $R^{-2}$
(the Riemann tensor),
$R^{-3}$ (the next term involving $\nabla_\sigma R_{\mu\alpha\nu\beta}$), and so on. The Taylor expansion
in the Riemannian normal coordinates becomes thus the expansion in negative powers of $R$.
We now introduce the rescaled coordinates $\tilde x^\mu = L^{-1}\,x^\mu$ and the
rescaled Riemann tensor at point $P$ in coordinates $x^\mu$.
\begin{eqnarray}
r_{\mu\alpha\nu\beta} = R^2\,R_{\mu\alpha\nu\beta}\Big|_P.
\end{eqnarray}
Both $\tilde x$ and $r_{\mu\alpha\nu\beta}$ are $O(1)$ in the expansion in $R$, at least within a single lattice cell
around $P$. The metric $\ensuremath{g^{(0)}}$ can be expressed in the new coordinates. The
metric tensor components in those coordinates will be denoted by $\ensuremath{{\tilde g}^{(0)}}_{\mu\nu}$,
i.e.
\begin{eqnarray}
\ensuremath{g^{(0)}} = \ensuremath{g^{(0)}}_{\mu\nu} \,\ensuremath{\textrm{d}} x^\mu\otimes\ensuremath{\textrm{d}} x^\nu =
\ensuremath{{\tilde g}^{(0)}}_{\mu\nu}\,\ensuremath{\textrm{d}} \tilde x^\mu\otimes\ensuremath{\textrm{d}} \tilde x^\nu.
\end{eqnarray}
Its expansion in $\tilde x^\mu$ takes the form of
\begin{eqnarray}
\ensuremath{{\tilde g}^{(0)}}_{\mu\nu} = L^2\left(\eta_{\mu\nu} - \frac{\mu}{3} r_{\mu\alpha\nu\beta}\,\tilde x^\alpha\,\tilde x^\beta + O(\tilde x^3 L^3)\right).
\end{eqnarray}
The first of the remaining higher-order terms $\nabla_\sigma R_{\mu\alpha\nu\beta} \,L^{3}\,\tilde x^\sigma\,
\tilde x^\alpha\,\tilde x^\beta$ contains the covariant derivative $\nabla_\sigma R_{\mu\alpha\nu\beta}\Big|_P$, which is
$O(R^{-3})$ as we noted before. Therefore the whole term in question can be re-expressed as $r_{\sigma\mu\alpha\nu\beta}\,\tilde x^\sigma\,
\tilde x^\alpha\,\tilde x^\beta\,\frac{L^3}{R^3}$, where we have defined by analogy
the rescaled derivative of the curvature $r_{\sigma\mu\alpha\nu\beta} = R^{3}\,\nabla_\sigma R_{\mu\alpha\nu\beta}$, which again is $O(1)$ in $R$. We see that the whole term turns out to be $O(\mu^{3/2})$. Similar reasoning can be applied to all higher terms, yielding
higher powers of the dimensionless parameter $\mu$.
We thus see that
\begin{eqnarray}
\ensuremath{{\tilde g}^{(0)}}_{\mu\nu} = L^2\left(\eta_{\mu\nu} - \frac{\mu}{3} r_{\mu\alpha\nu\beta}\,\tilde x^\alpha\,\tilde x^\beta + O(\mu^{3/2} )\right),
\end{eqnarray}
i.e. in the rescaled coordinates the expansion in the negative powers of $R$ turns in a natural way into an expansion in powers of $\mu$,
valid in a region of size $L$ around $P$.
We can explain the physical meaning of the expansion above in the following way: if the background metric $\ensuremath{g^{(0)}}$ has
the curvature scale of $R$, then in an appropriately picked, quasi-Cartesian coordinate system $x^\mu$ it has the Taylor expansion in which
the terms are of increasing order in $R^{-1}$. If we then pick a domain of size $L$, then the metric in this domain, again in
appropriate coordinates,
has the form of the flat metric plus perturbations from the curvature and its derivatives. A simple way to obtain a perturbation of this kind is to use the Taylor expansion we mentioned before and rescale the coordinates by $L$, which yields an expansion in
powers of $\mu^{1/2}$.
Now we can add the perturbation due to the discrete matter content. We assume the full metric to be
\begin{eqnarray}
\tilde g_{\mu\nu} = L^2\left(\eta_{\mu\nu} - \frac{\mu}{3} r_{\mu\alpha\nu\beta}\,\tilde x^\alpha\,\tilde x^\beta +
\mu\,h_{\mu\nu}\left(\tilde x^\alpha\right) + O(\mu^{3/2} )\right) \label{eq:fullg}
\end{eqnarray}
with the perturbation $h_{\mu\nu}\left(\tilde x^\alpha\right)$ of order $O(1)$ in $\mu$. Note that
the dependence on $\tilde x^\mu$ means that the characteristic physical size of the perturbation is the size of a cell, i.e.~$L$.
The Einstein tensor of the metric above is
\begin{eqnarray}
G_{\mu\nu}\left[\tilde g_{\alpha\beta}\right] = G_{\mu\nu}\left[\ensuremath{{\tilde g}^{(0)}}_{\alpha\beta}\right] +
\mu\,G'_{\mu\nu}\left[h_{\alpha\beta}\right]\left(\tilde x^\alpha\right) + O(\mu^{3/2}),
\end{eqnarray}
where $G'_{\mu\nu}[\cdot]$ is the linearisation of the Einstein tensor around a flat metric $\eta_{\mu\nu}$. In particular,
in the harmonic gauge it is simply $-\frac{1}{2}\Box h_{\alpha\beta}$.
We now return to the original, unrescaled
coordinate system, where this equation takes the form of
\begin{eqnarray}
G_{\mu\nu}\left[g_{\alpha\beta}\right] = G_{\mu\nu}\left[\ensuremath{g^{(0)}}_{\alpha\beta}\right]+
\rho\,G'_{\mu\nu}\left[h_{\alpha\beta}\right]\left(x^\alpha / L\right) + O(\mu^{3/2}),
\end{eqnarray}
i.e.~the perturbation of the Einstein tensor is $O(\rho)$, just like the Einstein tensor of the FLRW metric.
It means that this approximation works even if the density perturbation is of the order of the background energy density.
We may therefore use $h_{\mu\nu}$ to cancel the stress-energy tensor of the underlying FLRW metric everywhere except
on a single worldline.
Recall that $G_{\mu\nu}\left[\ensuremath{g^{(0)}}_{\alpha\beta}\right] = 8\pi G \rho \, u_{\mu} u_{\nu}$, where $u^{\mu} = (1,0,0,0)$ is the cosmic fluid 4-velocity.
We impose the linear PDE on the metric perturbation:
\begin{eqnarray}
G'_{\mu\nu}\left[h_{\alpha\beta}\right] = 8\pi G \left(-1 + C\delta^{(3)}(x^\alpha)\right) u_{\mu}\,u_{\nu}
\end{eqnarray}
with periodic boundary conditions and with the constant $C$ chosen so that the RHS integrates out to zero
over one cell. The solution can be obtained using Appell's $\zeta$ function~\cite{Steiner:2016tta}.
It diverges at the centre, where the approximation fails, but near the cell's boundary it is likely to work well.
The resulting approximate metric is vacuum everywhere and periodic.
\subsection{The continuum limit}
Let us now consider the metric (\ref{eq:fullg}) along with its Christoffel symbols and Riemann tensor. It is straightforward
to see that
\begin{eqnarray}
\tilde g_{\mu\nu} &=& \ensuremath{{\tilde g}^{(0)}}_{\mu\nu} + L^2\,\mu\,h_{\mu\nu}(\tilde x^\rho) \\
\Gamma\UD{\alpha}{\beta\gamma} \left[\tilde g_{\kappa\lambda}\right]&=&
\Gamma\UD{\alpha}{\beta\gamma} \left[\ensuremath{{\tilde g}^{(0)}}_{\kappa\lambda}\right] +
\mu\,{\Gamma'}\UD{\alpha}{\beta\gamma}\left[h_{\kappa\lambda}\right](\tilde x^\rho) \\
R\UD{\alpha}{\beta\gamma\delta} \left[\tilde g_{\kappa\lambda}\right] &=&
R\UD{\alpha}{\beta\gamma\delta} \left[\ensuremath{{\tilde g}^{(0)}}_{\kappa\lambda}\right] +
\mu\, {R'}\UD{\alpha}{\beta\gamma\delta} \left[h_{\kappa\lambda}\right](\tilde x^\rho).
\end{eqnarray}
We can now go back to the original, unrescaled coordinates and obtain
\begin{eqnarray}
g_{\mu\nu} &=& \ensuremath{g^{(0)}}_{\mu\nu} + \mu\,h_{\mu\nu} \left(x^\rho/L\right) \label{eq:gpert-original}\\
\Gamma\UD{\alpha}{\beta\gamma} \left[g_{\kappa\lambda}\right]&=&
\Gamma\UD{\alpha}{\beta\gamma} \left[\ensuremath{g^{(0)}}_{\kappa\lambda}\right] +
\mu^{1/2}\,\rho^{1/2}\,{\Gamma'}\UD{\alpha}{\beta\gamma}\left[h_{\kappa\lambda}\right] \left(x^\rho/L\right) \label{eq:gammapert-original}\\
R\UD{\alpha}{\beta\gamma\delta} \left[g_{\kappa\lambda}\right] &=&
R\UD{\alpha}{\beta\gamma\delta} \left[\ensuremath{g^{(0)}}_{\kappa\lambda}\right] +
\rho\, {R'}\UD{\alpha}{\beta\gamma\delta} \left[h_{\kappa\lambda}\right] \left(x^\rho/L\right)\label{eq:Riemannpert-original}
\end{eqnarray}
plus higher order terms in $\mu$. Consider now the limit $\mu \to 0$, i.e.~where the size of the perturbations decreases in comparison
to the curvature scale of the background FLRW model, or the limit where the compactness $M/L$ vanishes.
Obviously we see that the metric tensor and the Christoffel symbols converge to the FLRW values in this case, while
the curvature does not. This is due to the fact that the metric $g_{\mu\nu}$ is that of a vacuum spacetime for all positive $\mu$,
while the FLRW one is not.
This is a key observation in the study of the optical properties of a BHL, which are determined by the GDE
and are therefore sensitive to the form of the Riemann tensor.
To illustrate this point, consider first a null geodesic. It follows from equations (\ref{eq:gpert-original})--(\ref{eq:Riemannpert-original}) above that its equation has the form of a perturbed FLRW geodesic
\begin{eqnarray}
x^\mu(\lambda) = \tilde x^\mu(\lambda) + \mu^{1/2}\,\delta x^\mu(\lambda).
\end{eqnarray}
where the tilde denotes the FLRW solution without the inhomogeneities.
The parallel transport of a frame along the geodesic has a similar expansion in $\mu$:
\begin{eqnarray}
e\DU{a}{\mu}(\lambda) = \tilde e\DU{a}{\mu}(\lambda) + \mu^{1/2}\,\delta e\DU{a}{\mu}(\lambda).
\end{eqnarray}
We can now rewrite the GDE in the parallel-propagated frame along the geodesic
\begin{eqnarray}
\frac{\ensuremath{\textrm{d}}^2 X^a}{\ensuremath{\textrm{d}}\lambda^2} &=& \left(R\UD{a}{bcd} \left[\ensuremath{g^{(0)}}_{\kappa\lambda}\right] +
\rho\, {R'}\UD{a}{bcd} \left[h_{\kappa\lambda}\right]\right) p^b\,p^c + O(\mu^{1/2}).
\end{eqnarray}
We see that, already at the leading order $O(1)$ in $\mu$, we must take into account the full physical Riemann tensor instead of the simple FLRW one. In particular, since the BHLs are vacuum spacetimes,
we need to solve the Ricci-free GDE and possibly take into account the non-vanishing Weyl tensor along the way in order to calculate the angular and luminosity distance.
Neglecting the Ricci tensor in the GDE is equivalent to the EBA (for a discussion of this point, see e.g.~\cite{Fleury:2014gha}). We may thus expect the redshift--luminosity
relations for BHLs in the continuum limit to be close to the EBA.\footnote{We neglect here the finite-beam-size
effects which would become large when $\mu$ becomes very small: the beam may at some point become
wide enough to encompass a large number of
black holes. In this situation the interaction between the beam and the black holes becomes quite complicated as we
cannot use the GDE approximation any more.}
At the $O(\mu^{1/2})$ order we may expect additional corrections to $D_{\rm L}$ and $D_{\rm A}$ due to higher-order contributions to the geodesic equation as
well as to the
GDE equation. Additionally, at this order we need to take into account the impact of the inhomogeneities on the observers in their free fall. In this work, we will not
concern ourselves with a quantitative analysis of these effects, but we will signal their appearance to the reader when appropriate.
\section{Results}
\label{sec:results}
In order to compute the relationship between redshift and luminosity
distance on the spacetime of an expanding BHL, we carry out
the numerical integration of the geodesic equation (with null
tangent), along with the integration of Einstein's equation
required to obtain the metric tensor. The latter operation is
performed by a code generated with the Einstein Toolkit, based on
the \texttt{Cactus}~\cite{cactus} software framework along with modules
such as \texttt{Carpet}~\cite{carpet,carpetweb}, \texttt{McLachlan}~\cite{mclachlan,kranc},
and \texttt{CT\_MultiLevel}~\cite{Bentivegna:2013xna},
as already presented in~\cite{Bentivegna:2012ei,Bentivegna:2013jta,Korzynski:2015isa}.
The geodesic integrator, on the other hand, is a new \texttt{Cactus}
module that we have written. It implements a 3+1 decomposition of the geodesic
equation in the form given in \cite{Hughes:1994ea} and we have verified it
against several exact solutions, as reported in Appendix~\ref{app:geo}.
\subsection{Initial data and evolution}
As in~\cite{Yoo:2012jz,Bentivegna:2013jta}
we first construct an initial-data configuration by solving the
Hamiltonian and momentum constraints on the cube $[-L/2,L/2]^3$ with
periodic boundary conditions. In particular, we choose free data
corresponding to conformal flatness:
\begin{equation}
\gamma_{ij} = \psi^4 \delta_{ij}
\end{equation}
and set the trace of the extrinsic curvature to
zero around the origin and to a negative constant $K_c$ near the boundaries,
with a transition region starting at a distance $l$ from the origin:
\begin{eqnarray}
K_{ij} &=& \frac{1}{3} K_{\rm c} T(r) \gamma_{ij} + \psi^{-2} \tilde A_{ij}\\
T(r) &=& \left\{
\begin{array}{ll}
0 & \textrm{for } 0 \le r \le l \\
\left(\frac{(r-l-\sigma)^6}{\sigma^6}-1\right)^6&\textrm{for } l \le r \le l + \sigma \\
1 & \textrm{for } l + \sigma \le r
\end{array}
\right.
\end{eqnarray}
where we choose $l=0.05 L$ and $\sigma=0.4 L$.
We represent the traceless part of the extrinsic curvature as:
\begin{equation}
\tilde A_{ij} = \tilde D_i X_j + \tilde D_j X_i - \frac{2}{3} \tilde \gamma_{ij} \tilde D_k X^k
\end{equation}
and the conformal factor as:
\begin{equation}
\psi = \psi_{\rm r} + \frac{M}{2r} (1-T(r)),
\end{equation}
where $M$ is the bare mass of the central black hole, and solve the constraints for $\psi_{\rm r}$ and $X^i$.
For our basic configuration, we use $L=10$ and $M=1$ as in~\cite{Bentivegna:2013jta}.
We then proceed to the time evolution of $\gamma_{ij}$ and $K_{ij}$ using a
variant of the BSSN formulation, implemented in the \texttt{McLachlan} module,
and to the concurrent integration of the geodesic equation (\ref{eq:GE}).
\subsection{Computation of geodesics}
In order to compute geodesics in a 3+1 numerical spacetime, we first
perform a 3+1 decomposition of the geodesic equation (\ref{eq:GE}),
\begin{equation}
{ \nabla_p} p^a = 0.
\end{equation}
We decompose the geodesic tangent vector $p^a$ into its components along and orthogonal to the
unit hypersurface normal $n^a$, which we call $\sigma$ and $q^a$, respectively: $p^a = \sigma n^a + q^a$. The vector $q^a$ is spatial, i.e.~$q^an_a=0$, and $\sigma = -n_a p^a$. We use an
affine parametrisation, and $p^a$ is normalized as
\begin{eqnarray}
p^a p_a = \kappa,
\end{eqnarray}
with $\kappa = 0$ for null geodesics.
The spatial coordinates, covariant components of the tangent vector and affine parameter of the geodesic, ($x^i$, $q_i$, $\lambda$) satisfy
\begin{eqnarray}
\frac{dx^i}{dt} &=& -\beta^i + (p^0)^{-1} \gamma^{ik} q_k, \label{eqn:geo3+1x} \\
\frac{dq_i}{dt} &=& -p^0 \alpha \alpha_{,i} + q^j \beta^k_{,i} \gamma_{kj} - \frac{1}{2} (p^0)^{-1} q_l q_m \gamma^{lm}_{,i}, \label{eqn:geo3+1q} \\
\frac{d\lambda}{dt} &=& (p^0)^{-1} \label{eqn:geo3+1lambda}
\end{eqnarray}
where
\begin{eqnarray}
p^0 &=& \frac{(q_k q_j \gamma^{kj} - \kappa)^{1/2}}{\alpha}
\end{eqnarray}
is the time component of $p$ in the foliation-adapted coordinate
basis. Note that the derivative is with respect to coordinate time $t$, not the affine parameter $\lambda$. These equations are the same as those given in \cite{Hughes:1994ea}, and a derivation is outlined in Appendix~\ref{app:geo3+1}.
Given $(x^i, q_i, \lambda)$ at a time $t$,
equations (\ref{eqn:geo3+1x})--(\ref{eqn:geo3+1lambda}) determine their
evolution along a single geodesic.
The right hand sides of
eqs. (\ref{eqn:geo3+1x})--(\ref{eqn:geo3+1lambda}) are computed by
interpolating the metric quantities $\beta^i$, $\gamma_{ij}$, $\alpha$
from the evolution grid to the point $x^i(t)$ using fourth-order Lagrange
interpolation, and $(x^i(t), q_i(t), \lambda(t))$ is integrated using
a fourth-order Runge-Kutta method using the Cactus \code{MoL} component.
Additionally, the metric and various other quantities of interest are
interpolated to $x^i$, and all quantities are output as curves
parametrised by $t$ for use in any subsequent analysis once the
simulation is complete.
We implement the above prescription in two new Cactus components
\code{Geodesic} and \code{ParticleUtils}. The former contains the
equations themselves, and the latter provides library-type
functionality for integrating systems of equations along curves.
A few validation tests are provided in Appendix~\ref{app:geo}.
We now face the crucial task of selecting which geodesics to track.
Let us notice that, on a space filled with periodic cells, symmetry
reasons imply that an obvious class of cosmological observers is that formed
by observers sitting at the cell vertices. Due to the symmetry, these observers
do not exhibit any proper motions on top of the cosmic expansion, and the ratio of the
proper distances between arbitrary pairs of observers is constant at all times.
For this study, we construct and analyse two geodesics from this class
(which we will denote $A$ and $B$), starting at
the vertex $(-L/2,-L/2,-L/2)$, with initial tangents equal to $p_a^A=(p_0^A,1,0,0)$ and
$p_a^B=(p_0^B,1/\sqrt{2},1/\sqrt{2},0)$ respectively. $p_0^A = - \alpha \sqrt{\gamma^{xx}}|_A$
and $p^0_B = - \alpha \sqrt{(\gamma^{xx}+\gamma^{yy}+2\gamma^{xy})/2}|_B$ are
chosen by the geodesic integrator to ensure that the geodesics
are null. The two geodesics are plotted in Figure~\ref{fig:z}.
In order to measure the luminosity distance along geodesics $A$ and $B$, we
evolve two further pairs of geodesics, with spatial directions given by:
\begin{eqnarray}
(1,\epsilon,0) \\
(1,0,\epsilon)
\end{eqnarray}
and
\begin{eqnarray}
\left(\frac{1-\epsilon}{\sqrt{2}},\frac{1+\epsilon}{\sqrt{2}},0 \right) \\
\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\epsilon\right)
\end{eqnarray}
with $\epsilon=10^{-3}$, representative of two narrow beams close to each
original geodesic. We can then construct the redshift and luminosity distance
along the two beams.
Again, we emphasize that, since we keep the source parameters fixed
and observe the time evolution of each geodesic, this setup is different
(but essentially equivalent) to the one usually adopted in cosmology, where
the observer is fixed and sources with different parameters are considered.
As in~\cite{Bentivegna:2013jta},
we run this configuration on a uniform grid with three different resolutions
(corresponding to $160$, $256$, and $320$ points per side) in order to estimate
the numerical error.
All results presented below are convergent to first order, consistently with the
convergence order reported for the geometric variables in~\cite{Bentivegna:2013jta}.
All curves represent the Richardson extrapolation, at this order,
of the numerical data.
The corresponding truncation error (when visible) is indicated by a shaded region
around each curve.
\subsection{Small-redshift behaviour}
For small distances $d$ from the source, we expect the photon redshift and
luminosity distance to behave respectively like
\begin{eqnarray}
z(d) &\sim& H_{\cal S} d \\
D_{\rm L}(d) &\sim& d
\end{eqnarray}
where $H_{\cal S}$ is related to
the first time derivative of the local volume element at the source location:
\begin{eqnarray}
\label{eq:Hi}
H_{\cal S} = \left . \frac{{\rm tr}(K_{ij})}{3} \right |_{\cal S}
\end{eqnarray}
(see \cite{KristianSachs}). Figure~\ref{fig:z} shows that this expectation
is confirmed by our computation.
For large $d$, however, both quantities grow larger than the linear order.
Furthermore, the redshift clearly exhibits a non-monotonic behaviour engendered
by the inhomogeneous gravitational field. This is easy to explain as a small, periodic redshift
due to the photons climbing a potential hill near the vertices (away from the nearest black holes) and falling
into wells near the edge or diagonal midpoints (closer to the black holes).
Naturally, the two geodesics are
affected in different ways as they trace different paths through the gravitational
field.
\begin{figure}[!h]
\begin{center}
\centering
\begin{minipage}[b]{0.6\linewidth}
\includegraphics[width=1.0\textwidth]{plots/geoAB.pdf}
\includegraphics[width=1.2\textwidth]{plots/zd.pdf}
\includegraphics[width=1.2\textwidth]{plots/DLd.pdf}
\end{minipage}
\caption{Top: the paths of geodesics $A$ and $B$ in one of the BHL cells. The
geodesics run close to the cell edge and diagonal, respectively, at all times. Middle: photon redshift
as a function of the coordinate distance from the source. Bottom: luminosity
distance as a function of the coordinate distance from the source.
The error bars are indicated by shaded regions (when not visible, they are
included in the width of the curves).
\label{fig:z}}
\end{center}
\end{figure}
\subsection{Luminosity distance}
Due to numerical error,
the geodesics deviate from the cell edge and face
diagonal during the evolution, but remain quite close to them (the coordinate separation is less than $0.01\%$ after
three cell crossings, in both cases).
We can compare the $D_{\rm L}(z)$ relationship for geodesics $A$
and $B$ to the same quantity calculated according to four reference models:
\begin{enumerate}
\item The EdS model (equation (\ref{eq:ldeds}));
\item An FLRW model (equation (\ref{eq:ldflrw}))
with $\Omega_M=0.3$ and $\Omega_\Lambda=0.7$ (henceforth denoted $\Lambda$CDM);
\item The Milne model~\cite{MILNE01011934}, where redshift and luminosity distance are related by:
\begin{equation}
D_{\rm L}(z)=\frac{1}{H_{\cal S}} \frac{z}{(1+z)^2}\left(1+\frac{z}{2}\right);
\end{equation}
\item The estimate of $D_{\rm}(z)$ via the EBA,
equation (\ref{eq:eb}).
\end{enumerate}
All models are fitted according to two prescriptions:
the initial scale factor $a_{\cal S}$ is always set according to
\begin{equation}
a_{\cal S} = {\rm det}(\gamma_{ij})^{1/6}|_{\cal S},
\end{equation}
while
the initial expansion rate $H_{\cal S}$ is set to either
(i)
the initial time derivative
of the proper length of the domain edge (say, the one between $(-L/2,0,0)$ and
$(L/2,0,0$)), which we call a {\it global fit}, and is the same procedure as~\cite{Bentivegna:2013jta};
or (ii)
equation (\ref{eq:Hi}) (which we call a {\it local fit}).
\begin{figure}[!h]
\begin{center}
\centering
\includegraphics[width=0.7\textwidth]{plots/DL.pdf}
\includegraphics[width=0.5\textwidth]{plots/dDLAglob.pdf} \qquad \qquad \qquad \qquad \\
\includegraphics[width=0.5\textwidth]{plots/dDLBglob.pdf} \qquad \qquad \qquad \qquad \\
\includegraphics[width=0.5\textwidth]{plots/dDLAloc.pdf} \qquad \qquad \qquad \qquad \\
\includegraphics[width=0.5\textwidth]{plots/dDLBloc.pdf} \qquad \qquad \qquad \qquad \\
\caption{Luminosity distance as a function of redshift for geodesics $A$ and $B$ (top plot).
The same relationships in the EdS model, in the $\Lambda$CDM (i.e., FLRW with $\Omega_\Lambda=0.7$
and $\Omega_M=0.3$) model, in the Milne model and in the
EBA are also plotted. The four models are fitted according to the procedure
described in~\cite{Bentivegna:2013jta}, using the global expansion rate
computed from the first time derivative of the edge proper length. The relative difference
between the four models and the BHL $D_{\rm L}$ is plotted in the second and third panel.
The fourth and fifth panel illustrate the result of the same procedure, where the
four models have been fitted using the local expansion rate (\ref{eq:Hi}) instead.
On all plots, the dashed vertical lines mark the points where the geodesics
cross over the periodic boundary.
The error bars are indicated by shaded regions (when not visible, they are
included in the width of the curves or of the data points).
\label{fig:DL}}
\end{center}
\end{figure}
Figure~\ref{fig:DL} shows all the resulting curves.
We first recall that the expansion of the BHL, measured by the proper
distance of one of its cell edges, could be fitted quite well by an
EdS model with the same initial expasion, as shown in~\cite{Bentivegna:2013jta}.
The two models, however, exhibits markedly
different optical properties. For geodesic $A$, the relative difference
reaches $60\%$ by redshift $z=6$. This is not surprising: the conditions
under which these light rays
propagate in a BHL and in an EdS model are substantially different.
In the former case, for instance, null geodesics infinitesimally
close to $A$ or $B$ accelerate away from, rather than towards, them.
We notice that the EBA provides the best estimate
for $D_{\rm L}(z)$ in a BHL. We conjecture that this result is due to the
fact that this approximation can capture both the large-scale geometrical
properties of a non-empty universe and the small-scale behaviour of light
rays in vacuum. None of the other models satisfies both these conditions.
Note also that, for longer times, the EBA works better for the geodesic $A$
(along the edge) than for geodesic $B$ (along the face diagonal). This is
easy to explain if we notice that, because of the 4-fold discrete rotational
symmetry around the edge, there are no
Weyl focusing effects on $A$ and therefore the GDE with the Ricci tensor
neglected and no Weyl contribution is likely
to be a good approximation for the propagation of the neighbouring light rays. On the other hand along the face diagonal we may expect
a non-vanishing Weyl lensing around the midpoint area due to the tidal distortion of the rays. Such an effect is not taken into account in the EBA.
\subsection{Fitting the FLRW class}
It is tempting to consider an FLRW cosmology with the same matter content and
initial expansion as the reference EdS, plus an additional stress-energy
contribution coming from a cosmological constant, and attempt to tune its value
to reproduce the luminosity distance in the BHL.
The left panel of
Figure~\ref{fig:OLfit} shows a plot of the required $\Omega_\Lambda$ at each $z$,
for values of $\Omega_M$ in $[0.2,1]$. The right panel shows a cross section of
this surface with the planes $\Omega_M=1$ and $\Omega_M^{\rm eff}=8 \pi/(3 H_{\cal S}^2 L_{\rm prop}^3)$,
where $L_{\rm prop}$ is the initial proper length of a cell edge.
Notice, however, that none of these models would reproduce the expansion history of
the BHL spacetime, which follows closely that of a region of an EdS
model ($\Omega_M=1$ and $\Omega_\Lambda=0$) with the same $L_{\rm prop}$ and $H_{\cal S}$,
as discussed in~\cite{Bentivegna:2013jta}. This is the core of the fitting problem: the mapping between
different properties of an inhomogeneous spacetime to the FLRW class will be
different, and in general it will not be possible to identify a single
FLRW counterpart capable of reproducing all of the dynamical and optical aspects
of an inhomogeneous cosmology.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/omol.pdf}
\includegraphics[width=0.45\textwidth]{plots/OL.pdf}
\caption{Value of $\Omega_\Lambda$ in the best-fit FLRW cosmology, based on the
luminosity distance measured on geodesic $A$ (left), and its cross sections
with the planes $\Omega_M=1$ and $\Omega_M=\Omega_M^{\rm eff}=8 \pi/(3 H_{\cal S}^2 L_{\rm prop}^3)$
(curve yellow and blue, respectively, on the right plot).
The error bars are indicated by shaded regions (when not visible, they are
included in the width of the curves).
\label{fig:OLfit}}
\end{center}
\end{figure}
In Figure~\ref{fig:OLfitglob}, we show the constant-$\Omega_\Lambda$ models
which best fit the $D_{\rm L}(z)$ curves for geodesics $A$ and $B$.
They are obtained for $\Omega_\Lambda^A=1.225$ and $\Omega_\Lambda^B=1.103$,
respectively. The relative difference between these models and the exact solution
is largest around $z=1$, where it reaches $30\%$.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{plots/bestOL.pdf}
\includegraphics[width=0.5\textwidth]{plots/bestOLA.pdf} \qquad \qquad \qquad \qquad \qquad \\
\includegraphics[width=0.5\textwidth]{plots/bestOLB.pdf} \qquad \qquad \qquad \qquad \qquad \\
\caption{$D_{\rm L}(z)$ for an FLRW model with $\Omega_{\rm M}=1$,
and $\Omega_\Lambda$ equal to the best-fit values $\Omega_\Lambda^A=1.225$
and $\Omega_\Lambda^B=1.103$, as well as to a few other representative
values.
The best-fit models differ from the BHL $D_{\rm L}(z)$ at the $20\%$ level.
The error bars are indicated by shaded regions (when not visible, they are
included in the width of the curves or of the data points).
\label{fig:OLfitglob}}
\end{center}
\end{figure}
Notice that essentially all quantities discussed so far are affected by oscillations
with a substantial initial amplitude, which is subsequently damped. Similarly
to the oscillations in the redshift, we conjecture that these features are due
to the inhomogeneous gravitational field, and in particular to radiative
modes which likely originate in the oversimplified
initial-data setup we employed. In a space without an asymptotically-flat
region, it is of course difficult to test (or even formulate) this conjecture rigorously. The
compactness of the spatial hypersurfaces, furthermore, means that one cannot
simply ignore this initial transient as is customary in, e.g., binary-black-hole
simulations, as the waves cannot escape from the domain (although their
amplitude is significantly attenuated by the expansion).
The presence of this unphysical component of the gravitational field, which we
could barely notice in the length scaling we measured~\cite{Bentivegna:2013jta},
affects very prominently, on the other hand, the BHL optical properties,
and in particular the photon redshift. Better initial-data constructions
which are free from these modes are an interesting field of investigation
which goes beyond the purpose of this work.
Finally, it is worth observing that, as mentioned in Section~\ref{sec:lprop},
different observers would measure a different luminosity distance on the same
spacetime, thereby potentially bringing the BHL result closer to the EdS
curve. A boost with respect to the lattice would, for instance, lower the value
of the distance, according to equation (\ref{eq:DA}). So would a stronger
gravitational field, as would be the case if an observer was located closer
to the centre of a lattice cell.
\subsection{Continuum limit $\mu \to 0$}
Finally, it is instructive to study how this behaviour depends on how tightly packed the BHL
is, as represented by the quantity $\mu=M/L$ introduced in Section~\ref{sec:bhl}.
For simplicity, here we use the bare mass of the central black hole as an estimate of
$M$, and the coordinate size of a cell edge as $L$.
In order to keep $M/L^3$ constant at the value of our base configuration (which had
$M=1$ and $L=10$), we need to have $\mu=M^{2/3}/10$.
As representative masses we choose $M=\{1/100,1/8,1/2,1,5\}$;
various properties of this BH family are illustrated in Table~\ref{tab:cont}.
\begin{center}
\begin{table}[!b]
\centering
\caption{The bare mass $M$, coordinate size of a cell edge $L=10 M^{1/3}$, its proper size $L_{\rm prop}$,
and the compactness parameter $\mu=M^{2/3}/10$ for a constant-density family of BHLs.
\label{tab:cont}}
\begin{tabular}{|c|c|c|c|}
\hline
$M$ & $L$ & $L_{\rm prop}$ & $\mu$ \\
\hline
0.010 & \phantom{0}2.15 & \phantom{0}2.73 & 0.0046 \\
0.125 & \phantom{0}5.00 & \phantom{0}6.28 & 0.0250 \\
0.500 & \phantom{0}7.94 & \phantom{0}9.84 & 0.0630 \\
1.000 & 10.00 & 12.26 & 0.1000 \\
5.000 & 17.10 & 21.77 & 0.2924 \\
\hline
\end{tabular}
\end{table}
\end{center}
We plot the luminosity distance as a function of $\mu$ in Figures~\ref{fig:mu}
and~\ref{fig:murow}. We observe, in particular, that the difference between the
luminosity distance in a BHL and in an appropriately fitted EdS does not tend to
zero as $\mu \to 0$. The EdS model, therefore, can reproduce the large-scale
expansion history of a BHL (as illustrated numerically in~\cite{Yoo:2013yea,
Bentivegna:2013jta}, and deduced analytically in~\cite{Korzynski:2013tea}),
but is unable to fit its optical properties, even in the limit $\mu \to 0$.
The numerical result is in agreement with the result of the perturbative analysis of Section~\ref{sec:bhl},
where we identified $O(1)$ differences in the GDE of a BHL with respect to
that of an FLRW model.
This indicates that cosmological-distance estimates of a lumpy spacetime
based on a fit with the FLRW class will exhibit a systematic error,
\emph{regardless of how lumpy the spacetime is}.
These effects are substantially, but not exhaustively, captured by the EBA, as
already observed in the case of other inhomogeneous spacetimes~\cite{2012MNRAS.426.1121C,2012JCAP...05..003B,Fleury:2014gha}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/DLmu.pdf}
\includegraphics[width=0.45\textwidth]{plots/dDLmuEdS2.pdf}
\caption{Left: luminosity distance for a family of BHLs with the same density
but varying $\mu$.
Right: residual with respect to the EdS model (fitted via the local
expansion rate) of the four lowest-mass models along with their extrapolation
for $\mu \to 0$.
\label{fig:mu}}
\end{center}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\textwidth, trim=60 0 60 0, clip=true]{plots/DLmurow.pdf}
\caption{
Behaviour of the luminosity distance
at fixed redshift, for various values of $\mu$. The green triangle
represents the polynomial extrapolation of the data series for $\mu \to 0$,
while the yellow
dashed curve represents the expected luminosity distance in EdS for each
specific value of $z$.
\label{fig:murow}}
\end{figure}
An important remark is that we observe that the tensor modes discussed in
Section~\ref{sec:results} intensify as $\mu \to 0$, affecting the smaller-$\mu$
BHLs to the point that it becomes impossible to identify a monotonic trend
in the luminosity distance for large $z$.
For this reason, we are forced to limit our study to very small $z$.
\section{Discussion and conclusions}
\label{sec:dc}
We have investigated the propagation of light along two special curves in
the spacetime of a BHL, constructed by numerically integrating Einstein's
equation in 3+1 dimensions.
In particular, we have measured the redshift and luminosity distance along
these curves, and compared them to the estimates of these observables
obtained in suitably fitted homogeneous models and in the EBA.
The comparison shows that the latter approximation is the one most capable
of reproducing the exact behaviour; we have built a heuristic argument to
explain this finding, based on the analysis of the different curvature
terms in the GDE.
Our finding is congruous with the conclusions of similar studies in other
inhomogeneous spacetimes~\cite{Fleury:2014gha}; in our case, however, the
models are not backreaction-free by construction, so that we can measure
all the relevant contributions to the GDE.
We have also fitted the $D_{\rm L}(z)$ relationship from the FLRW models with both a
constant and a $z$-dependent $\Lambda$ to the data, finding that a value
of $\Omega_\Lambda$ approximately equal to $\Omega_M$ reproduces the optical
properties of the BHL better than the corresponding models with $\Omega_\Lambda=0$.
In other words, in the BHL spacetime the luminosity distance for a redshift $z$
is larger than in the corresponding EdS model (the correspondence being based
on the same initial proper size and expansion rate). This is also in line
with the conclusions of previous studies~\cite{Clifton:2009bp}, and arguably
equivalent to the finding that fitting $\Omega_M$ alternatively leads to a smaller
value for this parameter~\cite{Fleury:2013uqa}.
Finally, we have examined a family of BHLs with varying BH masses and separations,
in order to estimate how our result changes as $\mu=M/L \to 0$. In this limit,
it was proven in~\cite{Korzynski:2013tea} that the expansion history of
a BHL tends to that of a flat FLRW model with the same average density.
Here, however, we find that the optical properties of a BHL exhibit a finite
deviation from the corresponding FLRW model, which reaches $5\%$ by $z=0.06$.
Given a considerable pollution by tensor modes, which we conjecture originate
in our initial-data construction, the luminosity distance is oscillatory, and
we are unable to evaluate the continuum limit for larger $z$.
Building a picture of the mechanisms involved in these results, as well as
generalizing it to inhomogeneous spacetimes with different matter content
and density profiles, is a particularly intriguing but hard-to-approach task.
We can start to tackle it by comparing our results to a recent study~\cite{Giblin:2016mjp},
which also measured the effects of light propagation in an inhomogeneous
model which, unlike the ones considered in this work, was filled with dust.
In that investigation, percent-level deviations were detected from the
homogeneous Hubble law, which are about an order of magnitude smaller than
the deviations reported here. From the arguments presented in this paper,
we infer that the discrepancy is largely due to the different representation
of the matter filling the two models. The quantitative formulation of
this statement is a problem which we reserve for further study.
\section*{Acknowledgements}
MK and EB would like to thank the Max Planck Institute for Gravitational
Physics (Albert Einstein Institute) in Potsdam for hospitality. The work
was supported by the project \emph{``The role of small-scale inhomogeneities
in general relativity and cosmology''} (HOMING PLUS/2012-5/4), realized
within the Homing Plus programme of Foundation for Polish Science,
co-financed by the European Union from the Regional Development Fund, and by
the project ``\emph{Digitizing the universe: precision modelling for precision
cosmology}'', funded by the Italian Ministry of Education, University and Research (MIUR).
Some of the computations were performed on the Marconi cluster at CINECA.
|
2,877,628,090,184 | arxiv | \section{Introduction}\symbolfootnote[0]{\hspace*{-0.5em}$^\dag$%
To appear in
\textsc{The Astrophysical Journal} %
(v652 n2 December 1, 2006 issue).
Also available as ApJ preprint doi: %
\texttt{10.1086/508451}.
}
\label{sec:introduction}
\footnotetext[1]{NASA Postdoctoral Fellow.}
\setcounter{footnote}{1}
\thispagestyle{empty}
A striking property of the extrasolar planets known to date is the
range of their orbital eccentricities, which is far wider than that
of planets in the solar system.\footnote{See
the \textit{California \& Carnegie Planet Search} at
\url{http://exoplanets.org} and
the \textit{Extrasolar Planets Encyclopaedia} at
\url{http://exoplanet.eu}.
}
Eccentricities are typically $\sim 0.2$--$0.3$, but very low and high
values are also found \citep[e.g.,][]{marcy2005}. A variety of explanations
have been proposed to explain the eccentricities. The high eccentricity
cases may owe their explanation to the presence of a distant binary
companion \citep{wu2003,takeda2005}. In fact, the large eccentricity
($e=0.93$) of HD~80606b could be the result of a three-body interaction
process, known as ``Kozai cycle'' \citep{wu2003}.
Moreover, numerical experiments show that the observed distribution of
orbital eccentricities for $e\gtrsim 0.6$ can be reproduced by assuming
the action of a Kozai-type perturbation \citep{takeda2005}.
For more typical eccentricities, other processes are likely to be at
work. Planet-planet interactions involving scattering on dynamical
timescales is a possibility \citep{rasio1996}. However, numerical
experiments indicate that interactions between equal mass planets
would produce more isolated planets on low-eccentricity orbits than
those observed \citep{ford2001}. Secular interactions between planets
is another possible means of eccentricity excitation \citep{juric2005}.
This mechanism assumes that the planets have appropriate initial
separations for evolution to occur on secular timescales ($\sim 10^{10}$
years).
Disk-planet interactions can also give rise to planetary eccentricities
\citep{gt1980,pawel1992,goldreich2003}. The evolution of orbital
eccentricity depends on a delicate balance between Lindblad and corotation
resonances. If the planet is massive enough to clear a gap, Lindblad
resonances promote eccentricity growth, while corotation resonances damp
eccentricity. If the corotation resonances operate at maximal efficiency,
they dominate over Lindblad resonances by a slight margin, and the result
is eccentricity damping \citep[][]{gt1980}. Two mechanisms have been
proposed to weaken the effects of corotation resonances and thereby provide
eccentricity growth. The first mechanism relies on a large enough gap about
a planet's orbit to exclude the corotation resonances from the disk, while
leaving a remaining Lindblad resonance, the 1:3 outer resonance. This
mechanism requires a massive enough companion and/or low enough viscosity
to open a wide enough gap. Such a mechanism has been demonstrated for binary
stars \citep{pawel1991}.
The second mechanism relies on the delicate nature of corotational resonances
in their ability to weaken (saturate) when the disk viscosity is sufficiently
small and the planet eccentricity sufficiently large
\citep[][]{ogilvie2003,goldreich2003}. A small finite initial eccentricity,
typically $e\sim 0.01$, is required for the eccentricity to grow by this
mechanism. Moreover, a companion object on a circular orbit can drive
eccentricity in a circumstellar disk. This process is believed to occur
in disks involving $10$--$20$ Jupiter-mass ($\MJup$) companions
\citep{papa2001} and binary stars \citep{lubow1991a,lubow1991b}. The disk
transfers eccentricity to the planet's orbit by exchanging energy and angular
momentum. The disk-planet system undergoes a coupled eccentricity evolution.
One goal of this paper is to determine whether the eccentricity growth by
disk-planet interactions occurs. Simulations by \citet{papa2001} suggested
that orbital eccentricity is excited when the mass of the embedded body is
larger than about $10$--$20$ Jupiter-masses, while lower mass companions
experienced eccentricity damping. By applying higher resolution and
simulating over longer timescales, we aim to see whether eccentricity growth
can occur for lower mass, planetary mass, companions. In addition, we are
interested in the effects of eccentricity on planet migration.
Accretion of gas onto a planet is likely affected by the planet's orbit
eccentricity. Circumbinary disks surrounding eccentric orbit binaries
undergo pulsed accretion on orbital timescales \citep{pawel1996}. A similar
process could occur with eccentric orbit planets. The mean accretion rate
could also be modified by the orbital eccentricity. Another goal of this
paper is to investigate this accretion process.
\pagestyle{myheadings}
\markboth{\hfill Evolution of Giant Planets in Eccentric Disks \hfill}%
{\hfill \textsc{G. D'Angelo, S. Lubow, \& M. Bate} \hfill}
We apply high-resolution hydrodynamical simulations to investigate
disk-planet interactions over several thousand orbital periods. In
\refSect{sec:model_description} we describe the physical model. The
numerical aspects of the calculations are detailed in
\refSect{sec:numerical_issues}. Results on the growth of the disk
eccentricity for a fixed planet orbit are presented in
\refSect{sec:disk_eccentricity}. Similar results on the growth of
disk eccentricity were recently reported by \citet{kley2006}. We
describe results on the planet's orbital evolution in
\refSect{sec:orbital_evolution}. Results on the pulsed accretion
are described in \refSect{sec:planet_accretion}. Finally, our
conclusions are given in \refSect{sec:conclusions}
\section{Model Description}
\label{sec:model_description}
\subsection{Evolution Equations}
\label{sec:evolution_equations}
We assume that the disk matter and planet are in coplanar orbits. In
order to describe the dynamical interactions between a circumstellar
disk and a giant planet, we approximate the disk as being two-dimensional,
by ignoring dynamical effects in the direction perpendicular to the orbit
plane (vertical direction). This approximation is justified by the
fact that the disk thickness is smaller than the vertical extent of
the planetary Roche lobes for the cases we consider. Comparisons between
two- and three-dimensional models of Jupiter-mass planets embedded in
a disk indicate that the two-dimensional geometry provides a sufficiently
reliable description of the system
\citep{kley2001,gennaro2003b,bate2003,gennaro2005}.
We employ a cylindrical coordinate frame $\{O; r, \phi, z\}$ in which the
disk lies in the plane $z=0$ and the origin, $O$, is located on the primary.
The reference frame rotates about the disk axis (i.e., the $z$-axis) with
an angular velocity $\Omega$ and an angular acceleration $\dot{\Omega}$.
The set of continuity and momentum equations, describing the evolution of
the disk, is written in a conservative form \citep[see, e.g,][]{gennaro2002}
and is solved for the surface density, $\Sigma$, and the velocity field
of the fluid, $\gvec{u}$.
The turbulent viscosity forces in the disk are assumed to arise from a
standard viscous stress tensor for a Newtonian fluid with a constant
kinematic viscosity, $\nu$, and a zero bulk viscosity
\citep[see][for details]{m&m}.
A locally isothermal equation of state is adopted by requiring that the
vertically integrated pressure is equal to $p=c^2_{\mathrm{s}}\,\Sigma$
and that the sound speed, $c_{\mathrm{s}}$, is equal to the disk aspect
ratio, $H/r$, times the Keplerian velocity. In this study, we use a
constant aspect ratio throughout the disk.
The gravitational potential of the disk, $\Phi$, is given by
\begin{equation}
\label{eq:phi}
\Phi=%
- \frac{G\,\MStar}{|\gvec{r}|}%
- \frac{G\,\Mp}{\sqrt{\mdp^2+\varepsilon^2}}%
+ \frac{G\,\Mp}{|\gvec{r}_{\mathrm{p}}|^3}%
\,\gvec{r}\bmath{\cdot}\gvec{r}_{\mathrm{p}}\,,
\end{equation}
where $\MStar$ is the mass of the central star whereas $\Mp$ and
$\gvec{r}_{\mathrm{p}}$ are the mass and the vector position of the planet,
respectively. The length $\varepsilon$ represents a softening length that
is needed to avoid singularities in the gravitational potential of the
planet. The third term on the right-hand side of \refeqt{eq:phi} accounts
for the acceleration of the origin of the (non-inertial) coordinate frame
caused by the planet. We ignore disk self-gravity. For the disks we consider,
the Toomre-$Q$ parameter never drops below $4$ during the simulations.
The orbit of the planet evolves under the gravitational forces exerted by
the central star and the disk material. Non-inertial forces arising from
the rotation of the reference frame also need to be taken into account.
Therefore, the equation of motion of the planet reads
\begin{eqnarray}
\label{eq:pme}
\ddot{\gvec{r}}_{\mathrm{p}} &=&
-\frac{G(\MStar+\Mp)}{|\gvec{r}_{\mathrm{p}}|^3}\,\gvec{r}_{\mathrm{p}}%
-\gOmega\bmath{\times}%
\left(\gOmega\bmath{\times}\gvec{r}_{\mathrm{p}}\right)%
-2\,\gOmega\bmath{\times}\dot{\gvec{r}}_{\mathrm{p}}\nonumber\\%
& &
-\dot{\bmath{\Omega}}} % dot{Omega\bmath{\times}\gvec{r}_{\mathrm{p}}
+\bmath{\mathcal{A}}_{\mathrm{p}}-\bmath{\mathcal{A}}_{*}\,,
\end{eqnarray}
where the angular velocity and acceleration vectors of the rotating frame
are defined as $\gOmega=\Omega\,\bmath{\hat{z}}$ and
$\dot{\bmath{\Omega}}} % dot{Omega=\dot{\Omega}\,\bmath{\hat{z}}$, respectively. The second,
third, and fourth terms on the right-hand side of the equation are the
centrifugal, Coriolis, and angular accelerations, respectively.
The angular velocity of the reference frame relative to a fixed frame,
$\Omega=\Omega(t)$, is chosen so as to compensate for the azimuthal
motion of the planet. Details of how this is achieved can be found in
\citet{gennaro2005}. Since the azimuthal position of the planet is
constant, a circular orbit reduces to a fixed point in the rotating frame.
For a planet orbit with $e>0$, the planet radially oscillates between
the pericenter distance, $a(1-e)$, and the apocenter distance, $a(1+e)$.
In this frame, the trajectory is a straight line of length $2ae$ with
center at $r=a$. This scheme has the advantage that the planet does not
drift across the grid in the azimuthal direction. It always maintains
a symmetric azimuthal position with respect to the zone centers (where
eqs.~[\ref{eq:phi}], [\ref{eq:A2}], and [\ref{eq:A1}] are evaluated)
while it moves radially. This method helps reduce artificially unbalanced
forces that act on the protoplanet
\citep[see discussion in][]{anelson2003a,anelson2003b}.
The last two terms in \refeqt{eq:pme}, which represent the forces per
unit mass exerted by the disk material on the planet and the star, are
\begin{equation}
\label{eq:A2}
\gvec{\mathcal{A}}_{\mathrm{p}}=%
G\!\int_{\Md}\!\frac{\left(\vdp\right)\,\mathrm{d}\Md(\gvec{r})}%
{\left(\mdp^2 + \varepsilon^2\right)^{3/2}}
\end{equation}
and
\begin{equation}
\label{eq:A1}
\gvec{\mathcal{A}}_{*}=%
G\!\int_{\Md}\!\frac{\gvec{r}\,\mathrm{d}\Md(\gvec{r})}{r^{3}}\,.
\end{equation}
These two vectors are included in \refeqt{eq:pme} only when the disk
back-reaction is accounted for and the protoplanet is allowed to adjust
its trajectory in response to the disk torques. When these terms are
applied, the integration in \refeqs{eq:A2}{eq:A1} is performed over the
disk mass comprised in the simulated domain, $\Md$.
\subsection{Physical Parameters}
\label{sec:physical_par}
In these calculations, the stellar mass, $\MStar$, is the unit of mass
and the initial semi-major axis of the planet's orbit, $a_{0}$, represents
the unit of length. The unit of time, $t_{0}$, is defined such that
$1/t_{0}=\sqrt{G\,(\MStar+\Mp)/a^{3}_{0}}$.
When we refer to the ``orbital period'' or ``orbit'' as length of time,
we actually mean the initial orbital period $t_{0}/(2\pi)$.
To provide estimates of various quantities in physical units, we adopt
$\MStar=1\,\MSun$ and $a_{0}=5.2\,\AU$, thus one orbit is $\approx11.9$
years.
In order to treat the strong perturbations induced by giant protoplanets
and limit the influence of the imposed radial boundaries, we considered an
extended disk whose radial borders are at $r_{\mathrm{min}}=0.3\,a_{0}$ and
$r_{\mathrm{max}}=6.5\,a_{0}$. The disk extends over the entire $2\pi$ radians in
azimuth around the star. In physical units, the disk models cover a region
from $1.56$ to $33.8\,\AU$.
The mass of the disk, in the absence of the planet, is
$\Md=1.58\times10^{-2}\,\MStar$ within the radial limits of the simulated
region. We used a constant disk aspect ratio, $H/r=0.05$. The unperturbed
initial surface density is axisymmetric and scales as $r^{-1/2}$, which
produces an unperturbed density, at $r=a_{0}$, equal to $75.8\,\mbox{$\mathrm{g}\,\mathrm{cm}^{-2}$}$.
However, given the large planetary masses considered in this investigation,
we also included an initial gap along the orbit of the planet that accounts
for an approximate balance between viscous and tidal torques
\citep[e.g.,][]{lin1986}.
The initial gap profile is based on equation~(5) of \citet{lubow2006}.
The initial gap width is modified by a factor of $1+a_{0} e_{0}$, in order
to account for the planet's initial orbital eccentricity, $e_{0}$.
The initial velocity field in the disk is a Keplerian one that is centered
on the star and corrected for the rotation of the frame of reference.
In order to account for the effects due to turbulence in the disk, a
constant kinematic viscosity, $\nu$, was used. In terms of the Shakura \&
Sunyaev parameter \citep{S&S1973}, we have
$\alpha=\alpha_{0}\,\left(a_{0}/r\right)^{1/2}$, where
$\alpha_{0}=4\times10^{-3}$ (in physical units,
$\nu\simeq10^{15}\,\mathrm{cm}^2\,\mathrm{s}^{-1}$).
Although spatial variations and time fluctuations consistent with the MHD
turbulence are not included, this relation yields a magnitude of $\alpha$
that is in the range found in MHD simulations
\citep{papa2003,winters2003,rnelson2004}.
The influence of viscosity was explored by performing a few calculations
with other $\alpha_{0}$ values ($1.2\times10^{-3}$ and $1.2\times10^{-2}$).
\begin{deluxetable}{ccccccc}
\tabletypesize{\normalsize}
\tablecaption{\small Initial orbital eccentricities.\label{tbl:eccentricities}}
\tablewidth{0pt}
\tablehead{%
\colhead{$\Mp$} & \multicolumn{6}{c}{$e_{0}$}\\
\cline{2-7}
\colhead{(\MJup)}& \colhead{$0$} & \colhead{$0.01$} & \colhead{$0.1$} & \colhead{$0.2$} & \colhead{$0.3$} & \colhead{$0.4$}%
}
\startdata
$1$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ \\
$2$ & $\bullet$ & & & & & \\
$3$ & $\bullet$ & & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$
\enddata
\end{deluxetable}
Three planetary masses were considered: $1\,\MJup, 2\,\MJup, $ and $3\,\MJup$
(i.e., the mass ratio $q=\Mp/\MStar$ ranges from $1\times10^{-3}$ to
$3\times10^{-3}$). The planets were set on initially circular or eccentric
orbits about one solar mass stars. We examined configurations with initial
eccentricities, $e_{0}$, up to $0.4$. A complete list is given in
\refTab{tbl:eccentricities}.
At time $t=0$ the planet starts from the pericenter position, while its
azimuth, $\phip$, remains constant (in the rotating frame) and equal to
$\pi$ throughout the calculation.
In order to allow the disk to adjust to the presence of the planet, we
impose two stages to the evolution. During the first phase, the planet's
orbit is static and terms (\ref{eq:A2}) and (\ref{eq:A1}) are not included
in \refeqt{eq:pme}. During the second phase, the protoplanet is ``released''
from the fixed orbit and is allowed to react to the disk torques via
the full form of \refeqt{eq:pme}. In the models presented here the first
phase lasts until the release time, $t=t_{\mathrm{rls}}$, which ranges
from $1000$ to $1200$ orbits. The second phase lasts from several hundred
to several thousand orbits.
The value adopted for smoothing radius $\varepsilon$ (in eqs.~[\ref{eq:phi}]
and [\ref{eq:A2}]) resulted from numerical experiments in each orbital
eccentricity configuration. The chosen value of $\varepsilon$ was the
smallest that prevented the integration time-step of the hydrodynamics
equations from getting shorter than $\sim 10^{-6}$ orbits. In models
involving $1\,\MJup$ and $2\,\MJup$ planets, we set
$\varepsilon=0.1\,\Rhill$, where $\Rhill=\rp\,\left(q/3\right)^{1/3}$
\citep{bailey1972} is planet's Hill radius (or sometimes called Roche
radius). In models involving $3\,\MJup$ planets, we applied softening
lengths between $0.12\,\Rhill$ and $0.2\,\Rhill$. The latter value was
used at the highest initial orbital eccentricities, $e_{0}=0.3$ and $0.4$.
We found that torques within the Roche lobe do not dominate the planet
orbital evolution. Moreover, the smoothing radius does not significantly
affect planet accretion. Thus, $\varepsilon$ does not likely play an
important role in these calculations.
For simulations that account for the disk torques on the planet, an
additional approximation is made, which is described at the beginning
of \refSect{sec:orbital_evolution}.
\section{Numerical Method}
\label{sec:numerical_issues}
The equations of motion of the disk are solved numerically by means
of a finite-difference scheme that uses a directional operator splitting
procedure. The method is second-order accurate in space and semi-second-order
in time \citep{ziegler1997}. Hydrodynamic variables are advected
by means of a transport scheme that uses a piecewise linear reconstruction
of the variables with a monotonised slope limiter \citep{vanleer1977}.
High numerical resolution in an extended region around the planet is
achieved by using a nested-grid technique
\citep[see][for details]{gennaro2002,gennaro2003b} with fully nested
subgrid patches, whereby each subgrid level increases the resolution
by a factor $2$ in each direction.
Tests on the behavior of the nested-grid technique applied in a reference
frame rotating at a variable rate $\Omega=\Omega(t)$ are reported in the
Appendix of \citet{gennaro2005}.
The equation of motion of the planet (\refeqp{eq:pme}) is solved by using
a high-accuracy algorithm described in \citet{gennaro2005}.
\subsection{Grid Resolution}
\label{sec:grid_setup}
\begin{deluxetable}{ccccc}
\tablecaption{\small Grid systems used in the simulations.\label{tbl:grids}}
\tablewidth{0pt}
\tablehead{%
\colhead{Grid} & \colhead{GS1} & \colhead{GS2} & \colhead{GS3}
& \colhead{GS4} \\
\colhead{level}& \colhead{$N_{r}\times N_{\phi}$} & \colhead{$N_{r}\times N_{\phi}$} & \colhead{$N_{r}\times N_{\phi}$} & \colhead{$N_{r}\times N_{\phi}$}
}
\startdata
$1$ & $313\times 317$ & $313\times 317$ & $623\times 629$ & $623\times 629$ \\
$2$ & $264\times 264$ & $264\times 264$ & $524\times 524$ & \\
$3$ & & $404\times 404$ &
& \\
\enddata
\tablecomments{\small%
The grid system GS1 achieves the highest resolution
$(\Delta r/a_{0}, \Delta \phi)=(0.01,0.01)$ in the region
$(r,\phi)\in[0.4,3.0]\,a_{0}\times 0.83\pi$, which is azimuthally
centered on the planet.
The grid system GS3 resolve the same region with a resolution
$(\Delta r/a_{0}, \Delta \phi)=(0.005,0.005)$.
The latter resolution is obtained with the grid system GS2 in the
region $(r,\phi)\in[0.5,2.5]\,a_{0}\times 0.65\pi$
(also centered on the planet in azimuth).
}
\end{deluxetable}
The excitation or damping of a protoplanet's orbital eccentricity depends
on a delicate balance between Lindblad and corotation resonances
\citep[and references therein]{ogilvie2003,goldreich2003}. To study this
balance, it is necessary to resolve the width of all the resonances
involved in the process. The locations of first order eccentric Lindblad
resonances reside in a region that ranges radially from approximately
$0.6\,a$ to $1.6\,a$. Furthermore, calculations on the saturation of
isolated noncoorbital corotation resonances, performed by
\citet{masset2004}, suggest
that a minimum resolution requirement to avoid spurious damping of
eccentricity is that
\begin{equation}
\label{eq:minres}
\Delta r/a \lesssim 4.1\,\sqrt{C^{\pm}_{k}\,k\,e\,q}\,,
\end{equation}
where $k$ is the azimuthal wavenumber and $C^{\pm}_{k}$ are coefficients of
order unity given in \citet{ogilvie2003}.
Although nested grids provide a linear resolution $\le 0.01$ in a region
$\sim 2.5\,a\times 0.8\pi$, the effects of these corotational
resonances are not localized in azimuth, so the nested grids do
not substantially improve their overall resolution.
Calculations that follow the orbital evolution of the planet were executed
with grid systems GS1 and GS2 (see \refTab{tbl:grids} for a description
of all grid systems employed in this study).
According to \refeqt{eq:minres}, the global radial resolution we apply
$\Delta r/a=0.02$ could produce spurious damping in the outer disk for
orbital eccentricities $e \la 3\times 10^{-5}/(k\, q)$, with $k >1$.
In the $1\,\MJup$ case, simulations with $0.01 < e < 0.015$ could undergo
spurious eccentricity damping for the most distant outer resonance $k=2$
only. For $e > 0.015$, there is no spurious damping. In the $3\,\MJup$ case,
simulations with $0.003 < e < 0.005$ could undergo spurious damping for
$k=2$ only. For $e > 0.005$, there is no spurious damping.
\subsection{Mass Accretion Procedure}
\label{sec:accretion_procedure}
Accretion onto the protoplanet was simulated by removing material within
a distance of $\racc=0.3\,\Rhill$. Mass is removed by means of a two-step
procedure according to which the removal timescale, $\tacc$, is $0.03$
orbital periods for $\mdp < \racc/2$ and $0.09$ orbital periods in the
outer part of the accretion region, $\racc/2< \mdp < \racc$. Notice that
$\tacc$ is about equal to the Keplerian period around the protoplanet
(i.e., in the circumplanetary disk) at $\mdp=\racc/2$. Tests were carried
out to evaluate the sensitivity of the procedure to both parameters $\racc$
and $\tacc$. These tests indicate that, if $\racc$ is reduced by a factor
of $1.5$, the average mass accreted during an orbit varies by less than
$10$\%. Increasing the removal timescale by a factor of $5/3$ only affects
the accretion rate by $5$\% \citep[see also][]{tanigawa2002}. We also checked
whether the accretion parameters influence the orbital evolution of the
planet. Using the same tests, we found no significant differences over a
few hundred periods of evolution.
We do not add the removed mass to the planet mass over the course of the
simulations. Doing so would increase the planet mass by $\sim 1\,\MJup$
for the simulations reported here.
\subsection{Boundary Conditions}
\label{sec:boundary_conditions}
Boundary conditions at the radial inner boundary, $r=r_{\mathrm{min}}$,
allow outflow of material, i.e., the accretion flow toward the central
star. The inner boundary conditions exclude inflow away from the star
into the computational domain. Two types of boundary conditions were
used at the outer boundary ($r=r_{\mathrm{max}}$): reflective and
non-reflective. With the first kind, neither inflow nor outflow of material
is permitted at the outer border, which behaves as a rigid wall. Although
$r_{\mathrm{max}}$ is much larger than the apocenter radii of the planets
in the simulations, wave reflection was observed at the outer border in
this case, especially when the planet orbit was eccentric.
In order to lessen the amount of wave reflection at the outer disk edge,
we also applied non-reflective boundary conditions, following the approach
of \citet{godon1996,godon1997}. In these circumstances, boundary conditions
are not directly imposed on the primitive variables (i.e, $\Sigma$ and
$\gvec{u}$), but rather on the characteristic variables (i.e., the Riemann
invariants of one-dimensional flows). The basic idea is to let outflowing
(inflowing) characteristics propagate through the grid boundary. The correct
propagation of the flow characteristics across the border depends on how
accurately the adopted solution exterior to the grid approximates the exact
solution.
At both disk radial boundaries, the flow is assumed to be Keplerian around
the central star. This choice could lead to some small-amplitude wave
excitation at $r=r_{\mathrm{max}}$, since the fluid tends to orbit about
the center-of-mass of the system, rather than around the central star
\citep[see also][]{rnelson2000}.
The effects of such waves were checked to be unimportant as they tend to
dissipate within a short distance from the outer disk boundary
\section{Disk Eccentricity}
\label{sec:disk_eccentricity}
\subsection{Global Density Distribution}
\label{sec:gobal_sigma}
\begin{figure*}
\centering%
\resizebox{0.82\linewidth}{!}{%
\includegraphics{f1a}%
\includegraphics{f1b}}
\resizebox{0.82\linewidth}{!}{%
\includegraphics{f1c}%
\includegraphics{f1d}}
\resizebox{0.82\linewidth}{!}{%
\includegraphics{f1e}%
\includegraphics{f1f}}
\caption{\small%
Surface density in a disk containing a $1\,\MJup$ (left) and a
$3\,\MJup$ (right) planet at $t=1000$ orbits. The axes are in
units of the planet's orbital semi-major axis, $a_{0}$. The grey
scale bars are expressed in units of $3.29\times10^5\,\mbox{$\mathrm{g}\,\mathrm{cm}^{-2}$}$.
From top to bottom, panels refer to the configurations with orbital
eccentricities $e=0$, $0.1$, and $0.3$. In each panel the planet
(smaller circle) is at pericenter and located at $(X,Y)=(e-1,0)$.
}
\label{fig:globalsigma}
\end{figure*}
\begin{figure*}
\centering%
\resizebox{0.95\linewidth}{!}{%
\includegraphics{f2a}%
\includegraphics{f2b}}
\caption{\small%
Azimuthally averaged surface density in a disk containing
a $1\,\MJup$ (left) and $3\,\MJup$ (right) planet for various
values of orbital eccentricity: $e=0$ (solid line), $e=0.1$
(short-dashed line), $e=0.3$ (long-dashed line), and $e=0.4$
(dotted-dashed line). The horizontal axis is in units of the planet's
orbital semi-major axis $a_{0}$. For the vertical axis,
$\langle\Sigma\rangle=10^{-4}$ corresponds to $32.9\,\mbox{$\mathrm{g}\,\mathrm{cm}^{-2}$}$.
The time is about $1000$ orbits and the planet is at pericenter.
}
\label{fig:avsigma}
\end{figure*}
\begin{figure*}[t!]
\centering%
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f3a}%
\includegraphics{f3b}}
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f3c}%
\includegraphics{f3d}}
\caption{\small%
Gap region around the orbit of a $3\,\MJup$ planet with fixed
$e=0.4$ at various orbital phases. The axes are in units of
the planet's orbital semi-major axis $a_{0}$. The grey scale
bars are expressed in units of $3.29\times10^5\,\mbox{$\mathrm{g}\,\mathrm{cm}^{-2}$}$. The
planet's orbit in the inertial frame is represented by a dashed
ellipse. The smaller solid circle indicates the position of the
planet around the star (larger solid circle). The motion of the
planet (and the panel sequence) is counter-clockwise.
}
\label{fig:gapedge_inertial}
\end{figure*}
The global surface density in the disk is plotted in
\refFgt{fig:globalsigma} for different planet's masses and orbital
eccentricities. Left panels refer to configurations with $\Mp=1\,\MJup$,
and right panels refer to configurations with $\Mp=3\,\MJup$. The orbital
eccentricity increases from top ($e=0$) to bottom ($e=0.3$). In all of the
panels, the planet is at pericenter ($\rp/a=1-e$). For sufficiently long
evolutionary times, the inner disk (disk interior to the planet) is
largely depleted because of the tidal gap produced by the planet and
because the grid does not have sufficient dynamic range to cover regions
close to the star where an inner disk would reside. The outer disk is
tidally truncated at a radial distance that depends on both $\Mp$ and $e$.
\refFgt{fig:globalsigma} shows that the size of the truncation radius
increases with planet mass and eccentricity, analogous to the case for
circumbinary disks \citep{pawel1994}. In addition, a wave or wake propagates
in the outer disk. The disk truncation also be seen in \refFgt{fig:avsigma},
which shows the azimuthally averaged surface density for various cases.
The outer gap edge becomes less steep as the orbital eccentricity increases
and does not change significantly over an orbital period, as illustrated in
\refFgt{fig:gapedge_inertial}.
\begin{figure}
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f4}}
\caption{\small%
Motion of the center-of-mass of the disk region $r<3\,a_{0}$
relative to the star (located at the origin). The disk
center-of-mass starts at the origin, but is plotted at later times
$t\gtrsim 300$ orbits. The long-dashed and solid lines refer to
the cases with $\Mp=1\,\MJup$ and $\Mp=3\,\MJup$, respectively,
with a fixed circular orbit planet. The short-dashed line is for
the model with $\Mp=3\,\MJup$ and fixed planet eccentricity $e=0.1$.
}
\label{fig:rcmvstime}
\end{figure}
After a few hundred orbits, the disk region near the planet becomes
eccentric, even if the planet is on a circular orbit. This effect can be
seen in \refFgt{fig:rcmvstime}, which illustrates the motion (relative to
the star) of the center-of-mass of the disk interior to $r=3\,a_{0}$ for
different planetary masses and orbital eccentricities.
\subsection{Mode Analysis}
\label{sec:modes_analysis}
We performed a mode decomposition of the surface density distribution of
the disk adopting an approach along the lines of \citet{lubow1991b}.
We defined a mode component at each radial location in the disk as
\begin{equation}
\label{eq:modeMkl}
\mathcal{M}_{(k,l)}^{(f,g)}=\frac{2}{\pi\,T\,\langle\Sigma\rangle\,%
(1+\delta_{l,0})}\!%
\int_{T}\!\int_{0}^{2\pi}\!\!%
\Sigma\,%
f(k\theta)\,g(l\Omega_{\mathrm{p}}t)\,\mathrm{d}\theta\mathrm{d}t\,,
\end{equation}
where $\Omega_{\mathrm{p}}=\sqrt{G\,(\MStar+\Mp)/a^{3}_{0}}$, the angle
$\theta$ is the azimuth relative to an \textit{inertial} reference frame,
and $2\pi\langle{\Sigma}\rangle=\int_{0}^{2\pi}\!\Sigma\,\mathrm{d}\theta$ .
The time interval $T$ is a ten-orbit period interval, beginning at a
pericenter passage of the planet. The time integration is repeated every
interval $T$. The functions $f$ and $g$ are either sine or cosine. This
decomposition corresponds to a Fourier transform in both azimuth and time.
The amplitude (or strength) of the mode is
\begin{eqnarray}
\label{eq:modeSkl}
S_{(k,l)}&=& \left\
\left[\mathcal{M}_{(k,l)}^{(\cos,\cos)}\right]^{2}
\left[\mathcal{M}_{(k,l)}^{(\cos,\sin)}\right]^{2}+\right.%
\nonumber\\%
& & \left.%
\left[\mathcal{M}_{(k,l)}^{(\sin,\cos)}\right]^{2}+%
\left[\mathcal{M}_{(k,l)}^{(\sin,\sin)}\right]^{2}
\right\}^{1/2}.
\end{eqnarray}
In order to obtain the strength of a mode integrated over a radial
interval $[r_{1},r_{2}]$, the mode components $\mathcal{M}_{(k,l)}^{(f,g)}$
are first averaged between $r_{1}$ and $r_{2}$ and then substituted into
\refEqt{eq:modeSkl}. A tidally disturbed non-eccentric disk has modes
present that all have $k=l$. For a disk ring to be eccentric, the mode
strength associated with the pair $(k,l)=(1,0)$, known as eccentric mode,
must be non-zero, i.e., $S_{(1,0)}>0$. To follow the eccentricity evolution,
we analyze $S_{(1,0)}$ as a function of the time.
\subsection{Eccentricity Growth}
\label{sec:eccentricity_growth}
\begin{figure*}
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f5a}%
\includegraphics{f5b}%
\includegraphics{f5c}}
\caption{\small%
Amplitudes of the eccentric mode, associated with the pair
$(k,l)=(1,0)$ (see \refeqp{eq:modeSkl}), versus time for
three different disk regions, as indicated in the top-left
corner of each panel, for $1\,\MJup$ (short-dashed),
$2\,\MJup$ (long-dashed), and $3\,\MJup$ (solid) planets.
In all cases, the planet resides on a fixed circular orbit.
The mode strength progressively weakens as regions farther
from the planet's orbit are considered.
}
\label{fig:S10_vs_t}
\end{figure*}
The amplitude of the eccentric mode, as defined in \refeqt{eq:modeSkl},
is shown in \refFgt{fig:S10_vs_t} during the evolutionary phase in which
the planet is kept on a fixed orbit. This Figure refers to the configurations
with $\Mp=1\,\MJup$, $2\,\MJup$, and $3\,\MJup$ planets having $e=0$.
The disk region that undergoes a substantial eccentricity growth is that
between $r\simeq a_{0}$ and $r\simeq 2\,a_{0}$ (\refFgp{fig:S10_vs_t}, left
panel). The relatively small initial value of $S_{(1,0)}$ is likely due to
a transient effect, as the initially circular disk adjusts to the presence
of the planet. Farther away from the planet's orbit, the mode strength
drastically decreases, and the disk is nearly circular (\refFgp{fig:S10_vs_t},
middle and right panels). The eccentricity growth proceeds very rapidly
during the first $200$ orbits and oscillates afterwards with some
reinforcement in the $3\,\MJup$ case.
\begin{figure*}
\centering%
\resizebox{0.95\linewidth}{!}{%
\includegraphics{f6a}%
\includegraphics{f6b}}
\caption{\small%
Left. Mode strength $S_{(1,0)}$ as function of time for models
with $\Mp=1\,\MJup$ of fixed planetary orbital eccentricity $e=0$
(dashed line) and $e=0.1$ (solid line).
Right. $S_{(1,0)}$ versus time for models with $\Mp=3\,\MJup$
of fixed planetary orbital eccentricity $e=0$ (short-dashed line),
$e=0.1$ (long-dashed line), and $e=0.3$ (solid line).
}
\label{fig:S10_vs_mp}
\end{figure*}
The eccentricity driven in the region $a_{0}<r<2\,a_{0}$ by the $1\,\MJup$
planet ($e=0$) is rather small compared to that driven by the other two
planetary masses. However, with an initial orbital eccentricity $e=0.1$,
a $1\,\MJup$ planet is able to sustain eccentricity growth in the disk,
as indicated by the solid line in the left panel of \refFgt{fig:S10_vs_mp}.
In the $3\,\MJup$ case, an initial orbital eccentricity of up to $e=0.3$
induces a larger amplitude eccentricity perturbation only at the beginning
of the evolution. But the long-term evolution of $S_{(1,0)}$ is not greatly
affected (\refFgp{fig:S10_vs_mp}).
The simulations for mode analyses were generally performed with the grid
GS4. We also ran simulations and computed modes by using the grid system
GS1 and found results consistent to those obtained with the higher
resolution grid.
\subsubsection{Influence of Viscosity}
\label{sec:viscosity_influence}
The sensitivity of disk eccentricity growth to the disk kinematic viscosity
was examined for a system containing a $3\,\MJup$ planet on an initially
circular orbit. Two additional values of $\alpha_{0}$ (i.e., $\alpha$ at
$r=a_{0}$) were considered: $\alpha_{0}=1.2\times10^{-3}$ and
$\alpha_{0}=1.2\times10^{-2}$, which are respectively a factor of $3$
smaller and larger than the standard value $\alpha_{0}=4\times10^{-3}$.
In the lowest viscosity model, the overall evolution of the amplitude of
the eccentric mode closely resembles that of the model with standard
viscosity (see \refFgp{fig:S10_vs_t}, solid line), but was roughly $20\%$
larger. In the highest viscosity model, the eccentric mode strength
reached a maximum of $0.3$ at $t=200$ orbits and then decays. Around
$t=1000$ orbits, $S_{(1,0)}\approx 0.1$ and continues to decline.
The effects of viscosity were also investigated with a calculation
involving a $1\,\MJup$ planet on a circular orbit and
$\alpha_{0}=1.2\times10^{-3}$. In this case, the mode amplitude $S_{(1,0)}$
is sustained at about $0.15$ for $t>200$ orbits, unlike the case of
declining eccentricity at higher viscosity that is displayed as a
short-dashed line in the left panel of \refFgt{fig:S10_vs_t}.
These results suggest that disks with $\alpha$ less than a few times
$10^{-3}$ experience sustained eccentricity, while disks with $\alpha$ in
excess of $\approx 10^{-2}$ do not. The weakening of disk eccentricity
with viscosity was also found in simulations by \citet{kley2006}.
\subsection{Analytic Model}
\label{sec:analytic_model}
In the case of superhump binaries, disk precession is dominated by the
gravitational effects of the companion which causes prograde precession
\citep{osaki1985}. On the other hand, pressure provides a retrograde
contribution which is somewhat weaker \citep{lubow1992,goodchild2006}.
In the case of a circular orbit planet, the gravitational
contribution to precession is expected to be weaker than the disk's
pressure contribution. The magnitude of the pressure induced precession
rate is $\sim (H/r)^2\,\Omega$. For the disks simulated in this paper,
the precession timescale is then $\sim 10^3$ orbits. We estimated the
gravitational precession due to a $3\,\MJup$ planet for the case plotted
in \refFgt{fig:S10_vs_t} at $1000$ orbits and find that this rate is
about $10$ times smaller than the pressure precession rate. We then expect
the precession to be pressure-dominated and retrograde, with a timescale
of about $10^3$ orbits. This estimate in accord with the simulation
results for a $3\,\MJup$ planet in a circular orbit in
\refFgt{fig:rcmvstime}.
The group velocity for an eccentric mode is estimated by using the
dispersion relation for an $m=1$ disturbance in a Keplerian disk perturbed
by pressure. The group velocity is $v_{\mathrm{g}} \sim (H/r)^2\,\Omega r$,
where we assume the radial wavenumber $| k_{r} |\sim 1/r$ and that the
pattern speed is small compared to $\Omega$. The group velocity leads to
eccentricity propagation timescales of order $(r/H)^2 \sim 10^3$ orbits.
This timescale is consistent with the localization of the eccentricity over
course of the simulation of $\sim 10^3$ orbits to within a region of order
$2\,a$, as seen in \refFgt{fig:S10_vs_mp}. Over longer timescales the
eccentricity would spread further.
We analyze growth of eccentricity in an outer disk that is perturbed
by a planet on a circular orbit. Disk eccentricity growth via the 3:1
resonance for inner disks of superhump binaries occurs on a timescale
$\sim 0.1 w / (r q^2)$ binary orbit periods, where $w$ is the radial
width of the eccentric region \citep{lubow1991a}. We now consider the
case for outer disks perturbed by planets. For simplicity, we assume
that the eccentric corotational resonances in the disk are saturated
(i.e., of zero strength). This situation is likely to hold if the disk
eccentricity is of order $0.01$, by analogy with the case of an eccentric
orbit planet interacting with a circular disk \citep{ogilvie2003}.
Following the mode coupling analysis for eccentric Lindblad resonances
involving a circular orbit planet \citep{lubow1991a}, we find that the
disk eccentricity growth rate associated with a particular eccentric
outer Lindblad resonance is given by
\begin{equation}
\lambda_m = \frac{\pi F_m^2 \Omega_{\mathrm{p}} r}{24 m w}\,,
\end{equation}
where
\begin{equation}
F_m=\frac{ 2 r u'_m - 4 r v'_m - 2 u_m- m\, u_m + 2 v_m\,(m-1)}%
{2 r \Omega_{\mathrm{p}}},
\end{equation}
\begin{equation}
u_m = \frac{m (\Omega - \Omega_{\mathrm{p}})\psi_m' + 2 m \Omega \psi_m /r}%
{\Omega^2 -m^2 (\Omega-\Omega_{\mathrm{p}})^2}\,,
\end{equation}
\begin{equation}
v_m = \frac{\Omega \psi_m' /2 + m^2 (\Omega -\Omega_{\mathrm{p}}) \psi_m /r}%
{\Omega^2 -m^2 (\Omega-\Omega_{\mathrm{p}})^2}\,,
\end{equation}
\begin{equation}
\psi_m = -q \, \Omega_{\mathrm{p}}^2 a^
\left[\frac{1}{\pi}\int_0^{2 \pi}
\frac{ a \cos{(m \phi)} \, \mathrm{d}\phi}%
{\sqrt{a^2+r^2-2 r a \cos{(\phi)}}} - \frac{r}{a} \delta_{m,1}
\right]\,,
\end{equation}
where $w$ denotes the radial extent of the eccentric region and prime
denotes differentiation in $r$. All quantities are evaluated at the
location of the eccentric Lindblad resonance associated with azimuthal
wavenumber $m$, having $\Omega=m\Omega_{\mathrm{p}}/(m+2)$ and
$r=r_m= [(m+2)/m]^{2/3}\,a$. Quantities $u_m$ and $v_m$ are the velocity
components in a circular disk associated with potential $\psi_m$, and
$\delta$ is the Kronecker delta function in the indirect potential term.
The combined effects of all the resonances is determined by the local
disk density at each resonance. The growth rate of the mass-weighted
eccentricity is then
\begin{equation}
\label{eq:lambdas}
\lambda = \frac{2 \pi w}{M_{\mathrm{e}}}%
\sum_{m=1}^{m_{\mathrm{max}}} \Sigma_m r_m \lambda_m\,,
\end{equation}
where $M_{\mathrm{e}}$ is the mass of the eccentric region and the sum is
taken over the active eccentric Lindblad resonances, and
$m_{\mathrm{max}} \simeq r/H$ due to torque cut-off effects \citep{gt1980}.
\begin{figure}[t!]
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f7}}
\caption{\small%
Large dots: contributions to the growth rate sum for
\refEqt{eq:lambdas} in units of $\Omega_{\mathrm{p}}$
as a function of azimuthal wavenumber $m$ associated
with an eccentric Lindblad resonance.
Small dots: normalized azimuthally averaged surface
density at each eccentric Lindblad resonance,
$10^{-4}\, \Sigma_m/\Sigma_1$, as a function of $m$.
The plot is for a $3\,\MJup$ planet with a density
profile obtained from simulations at $500$ orbits.
}
\label{fig:lam}
\end{figure}
The contribution of each resonance in the above sum for $\lambda$
for a $3\,\MJup$ planet is shown in \refFgt{fig:lam}. We consider
times beyond several hundred orbits, when the width of the eccentric
region $w$ $\sim 2\,a$. \refFgt{fig:lam} shows that there is a weak
contribution from the outermost resonance, the 1:3 resonance, corresponding
to $m=1$. Even though the density at this resonance is the largest,
a nearly complete cancellation occurs in $\lambda_1$, due to effects
of the indirect term in potential $\psi_1$. If only the $m=1$ resonance
were involved, then the eccentricity growth timescale would be very long,
$\sim 10^5$ orbits. The growth rate contributions from regions closer
to the planet are weakened by the lower density, but strengthened by
the larger values of $\lambda_m$. At a time of 500 orbits, the density
near the planet is small enough that the outermost resonances
($2 < m \la 5$) provide most of the growth rate. At a time of 500 orbits,
the eccentricity growth timescale $1/\lambda$ is about $600$ orbits, which
is in very rough agreement with the average growth rate implied by
\refFgt{fig:S10_vs_t} for the innermost region, although there are
considerable fluctuations in the simulations.\footnote{%
To compare the growth rate defined by $\lambda$ with simulations,
it is better to adopt a similar mass-weighted eccentricity.
This requires a slightly different definition of $S_{(1,0)}$
than given by \refEqt{eq:modeMkl}. When we apply that definition,
we obtain eccentric disk evolution in the $3\,\MJup$ case that is
similar to the $3\MJup$ results in \refFgt{fig:S10_vs_t}, except
that the magnitude of $S_{(1,0)}$ in the innermost region is about
a factor of 8 smaller. The average growth rate of $S_{(1,0)}$ in
the innermost region is about the same over $1000$ orbits.}
At earlier times, the growth rate is higher because $w$ is smaller.
We conclude that the disk eccentricity growth is possible over
$\sim 10^3$ orbits in the case of planets because of contributions
from several resonances that lie in the disk edge/gap region. This
situation differs from the superhump binary case where only a single
resonance is involved, since the disk extends relatively closer to
the perturber in the planet case, due to its weaker tidal barrier.
Increased viscosity affects the disk eccentricity in multiple ways.
It leads to further disk penetration of the planet's tidal barrier,
which could lead to stronger eccentricity growth. On the other hand,
viscosity also acts to unsaturate (or strengthen) the corotation
resonances which act to damp eccentricity. Furthermore, the viscosity
acts to damp the non-circular motions associated with the eccentricity.
\section{Planet Orbital Evolution}
\label{sec:orbital_evolution}
At time $t=t_{\mathrm{rls}}$ (between $1000$--$1200$ orbits) the planet
was allowed to adjust its orbit in response to the gravitational forces
exerted by the surrounding disk material. We generally neglected torques
on the planet due to gas within a distance of $0.5\,\Rhill$ from the planet.
However, analyses of the torque distribution at the release time indicate
that torques within the Roche lobe do not dominate the orbital evolution of
the planet. So we believe this procedure does not likely lead to major
errors in the planet's orbital evolution. All calculations presented in
this section employed grid GS2, except for the $3\,\MJup$ cases on
initially circular orbits ($e_{0}=0$), which employed grid GS1. In order to
analyze the orbital evolution of the planet after release, we calculated
the osculating elements of the orbit each few hydrodynamics time-steps
(approximately every $0.01$ orbits). To remove short-period oscillations,
we computed the mean orbital elements \citep{beutler2005} by using an
averaging period of one orbit. Throughout the paper, the planetary orbital
eccentricities and semi-major axes in the simulations refer to the mean
orbital elements.
\subsection{Orbital Eccentricity}
\label{sec:orbital_eccentricity}
\begin{figure}[t!]
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f8}}
\caption{\small%
Evolution of the (mean) orbital eccentricity of $2\,\MJup$
(dashed line) and $3\,\MJup$ (solid line) planets after the
release time, $t_{\mathrm{rls}}=1000$ orbits.
}
\label{fig:et_mp23}
\end{figure}
Simulations by \citet{papa2001} showed that the interaction between
an initially circular disk and a circular orbit planet with mass
$\gtrsim 20\,\MJup$ can lead to the growth of disk eccentricity and
planetary orbital eccentricity. They also found that this interaction
can be more efficient at augmenting orbital eccentricity than direct
wave excitation at the outer 1:3 Lindblad resonance in a non-eccentric
disk \citep[e.g.,][]{pawel1992}. We aim at determining whether a similar
phenomenon can occur also in the Jupiter-mass range.
\refFgt{fig:et_mp23} shows that the interaction between a planet and a
disk leads to orbital eccentricity growth for the $2\,\MJup$ (dashed line)
and $3\,\MJup$ (solid line) cases. During the initial growth of $e$ for
the $3\,\MJup$ planet, the rate is
$\dot{e}\approx 1.3\times10^{-4}\,\mathrm{orbit}^{-1}$. This value is
$\sim 1.6$ times that exhibited by the $2\,\MJup$ planet. The eccentricity
growth stalls when $e\simeq 0.08$ for both planet masses. The planets may
be experiencing some variation in their eccentricity forcing due to the
phasing of their eccentricities relative to the disks'.
After $1500$--$1600$ orbits from the release time, the orbital eccentricity
starts to increase again with a growth timescale that is comparable
to the initial growth timescale,
$\taue\equiv e/|\dot{e}|\approx 2.3\times10^{3}$ orbits, for both planetary
masses. The average growth timescale is shorter than the standard Type~II
migration (or viscous diffusion) timescale (see
\refSecp{sec:radial_migration}).
Over the last $1000$ orbits of the simulation, the eccentricity of the
$3\,\MJup$ planet increases very slowly, at a rate
$\dot{e}\approx 2\times10^{-6}\,\mathrm{orbit}^{-1}$.
The model with $\Mp=1\,\MJup$ and initial zero-eccentricity
(\refFgp{fig:et_mp1vse}, dashed line) shows a much slower orbital
eccentricity growth, reaching $e=0.02$ after $3000$ orbits from the release
time. As described in \refSect{sec:eccentricity_growth}, the eccentric
perturbation induced by the planet on the disk is also rather weak compared
to that excited by the $2\,\MJup$ planet. At the average growth rate
$\dot{e}\approx 7\times10^{-6}\,\mathrm{orbit}^{-1}$, it would take on the
order of the viscous diffusion timescale to reach $e\approx 0.1$.
\begin{figure}[t!]
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f9}}
\caption{\small%
Orbital eccentricity versus time of Jupiter-mass models
with different initial orbital eccentricities: $e_{0}=0$
(dashed line) and $e_{0}=0.01$ (solid line).
The release time is $1100$ orbits.
}
\label{fig:et_mp1vse}
\end{figure}
In order to evaluate to the extent of orbital eccentricity growth,
we used configurations with fixed non-zero planet eccentricities
prior to release, $e_{0}$. \refFgt{fig:et_mp1vse} also shows the
orbital eccentricity evolution of a $\Mp=1\,\MJup$ planet with
$e_{0}=0.01$. After release, $e$ oscillates about the initial value.
The oscillation grows in amplitude and, during one of these cycles,
$e$ increases from $0$ to $0.09$ within about $1300$ orbits. In this
case, $\taue$ is of order the viscous diffusion timescale. We simulated
several models with $e_{0} \ge 0.1$ (not plotted) and found that there
was generally a reduction of the orbital eccentricity, with some
exceptions though. For example, in a model with a $1\,\MJup$ planet
and $e_{0}=0.1$, $e$ underwent small amplitude oscillations about
the initial value, with periods of a few hundred orbits. This occurrence
may be related to the relatively large eccentricity driven in the outer
disk. Some models with $e_{0}\ge 0.2$ showed a rate of change of $e$
that diminishes in time. In these cases the evolution was generally
monitored for less than $1000$ orbits. Longer time coverage simulations
are required to assess the long-term behavior of these configurations.
\subsection{Radial Migration}
\label{sec:radial_migration}
\begin{figure}[t!]
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f10}}
\caption{\small%
Evolution of the semi-major axis of planets: $\Mp=1\,\MJup$
(short-dashed line), $\Mp=2\,\MJup$ (long-dashed line), and
$\Mp=3\,\MJup$ (solid line). The release time is about $1000$
orbits. The initial eccentricity $e_0=0$ in the $3$ cases.
The change in migration rates for the $2\,\MJup$ and $3\,\MJup$
cases at later times is related to their increased orbital
eccentricity (see \refFgt{fig:et_mp23}).
}
\label{fig:at_mp123}
\end{figure}
We describe here some results on the migration of eccentric orbit
planets. We plan to explore this issue further in a future paper.
Radial migration of planets in the mass range considered in this
study is expected to be in the standard Type~II regime, which is
characterized by an orbital decay timescale
$\taum\equiv a_{0}/|\dot{a}|=2\,a^{2}_{0}/(3\,\nu)=\taum^{\mathrm{II}}$
\citep{ward1997}.
The Type~II migration rate depends only on the viscous timescale of
the disk near the location of the planet and is independent of the disk
density, provided the disk is locally more massive than the planet.
Type~II migration is based on the assumption that the gap, which separates the inner and outer disks, is devoid
of material. In this case,
the planet torques approximately balance the viscous torques at the
gap edges.
\refFgt{fig:at_mp123} plots the evolution of the semi-major axes, after
the release time, of models with three planetary masses: $\Mp=1\,\MJup$
(short-dashed line), $\Mp=2\,\MJup$ (long-dashed line), and
$\Mp=3\,\MJup$ (solid line). The Figure shows that the initial migration
rate depends on the planet's mass, which is inconsistent with the Type~II
prediction. We may expect some dependence of the migration rate on planet
mass, because ratio of planet to disk mass is non-zero
\citep{syer1995,ivanov1999}. In order to explore this result further, we
have used a one-dimensional disk evolution code, along the lines of
\citet{lin1986}.
We used the torque density per unit mass given in equation~(4) of
\citet{lubow2006}. We checked that the results are insensitive
to the details of the torque density, provided that it is large enough
to produce a gap. Increasing the torque density everywhere by a factor
of $2$ produced a small change in the migration rate (less than $1\%$).
We adopted the same disk and planet parameters as in the
two-dimensional simulations with zero planet eccentricity. In short, we
find that the largest contributing factor to this non-Type~II behavior is
the lack of a substantial inner disk in the two-dimensional calculations.
There is also some effect due to the non-zero planet-to-disk mass ratio.
\begin{figure}[t!]
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f11}}
\caption{\small%
Orbital migration of a $1\,\MJup$ and a $3\,\MJup$ planet
according to one-dimensional simulations (dots) and two-dimensional
simulations (lines) that use the same disk and planet parameters.
The two upper curves are for the $3\,\MJup$ planet. The
one-dimensional simulations have an initially depleted inner disk,
whose density distribution matches that of the two-dimensional
simulations at planet release. Note the good agreement between
one-dimensional and two-dimensional simulations for the $1\,\MJup$
planet. The agreement is also very good for the $3\,\MJup$ planet,
as long as its eccentricity is smaller than about $0.1$ (see
\refFgp{fig:et_mp23}).
Planet orbits are assumed to always be circular in the
one-dimensional models.
}
\label{fig:at_1dvs2d}
\end{figure}
A comparison of orbital migration between one-di\-men\-sional and
two-dimensional
models is shown in \refFgt{fig:at_1dvs2d}. In this comparison we used the
azimuthal averaged surface density distributions in \refFgt{fig:avsigma}
as initial conditions for the one-dimensional models. As seen in
\refFgt{fig:at_1dvs2d}, the one-dimensional (zero eccentricity) migration
rates, for a very low density inner disk, agree well with the two-dimensional
rates at early times after release, while the planet eccentricity is small.
For undepleted initial inner disks, we find that one-dimensional
models have about the same migration rates for these two planet masses.
It is possible that the two-dimensional simulations have an inner boundary
$r_{\mathrm{min}}$ that is too large to resolve the inner disk.
More complete zone coverage of the inner region in the two-dimensional
calculations might reveal an inner disk that acts to make the migration rate
less dependent on mass, as indicated by the one-dimensional simulations.
In spite of these possible limitations of our two-dimensional simulations,
we describe below some interesting aspects of the migration of eccentric
orbit planets in two-dimensional disks.
As a planet's orbital eccentricity grows toward values of about $0.08$,
the rate of migration slows significantly (see \refFgp{fig:at_mp123}).
Over the last $1000$ orbital periods of the calculated evolution, the
$3\,\MJup$ planet exhibits a migration speed
$\dot{a}\approx -2\times10^{-6}\,a_{0}$ per orbit, with a tendency towards
further reduction. This migration rate is about a factor of $30$ smaller
than the rate at release time. The $2\,\MJup$ shows an even more drastic
reduction of the migration rate that actually reverses and becomes positive
around $t-t_{\mathrm{rls}}\approx 4200$ orbits. The outward migration speed
is $\dot{a}\approx 1\times10^{-5}\,a_{0}$ per orbit at the end of the
simulation. The migration speed of the $1\,\MJup$ planet ($e_{0}=0$)
is much more constant over the course of the simulation as its orbital
eccentricity remains small ($e\lesssim 0.02$).
The migration of a $1\,\MJup$ with $e_{0}=0.01$ proceeds as indicated by
the short-dashed line in \refFgt{fig:at_mp123}. Over the last $\approx 1000$
orbits of evolution, however, the migration rate undergoes a decrease by a
factor $\approx 2$. During that time, the eccentricity grows to $0.09$
(see solid line in \refFgp{fig:et_mp1vse}).
\begin{figure*}
\centering%
\resizebox{0.95\linewidth}{!}{%
\includegraphics{f12a}%
\includegraphics{f12b}}
\caption{\small%
Left. Semi-major axis evolution of $1\,\MJup$ planets for three
values of the orbital eccentricity at release: $e_{0}=0$
(short-dashed line), $e_{0}=0.1$ (long-dashed line), and
$e_{0}=0.2$ (solid line).
Right. Same as the left panel but for $3\,\MJup$ planets with
orbital eccentricities at release: $e_{0}=0$ (short-dashed line),
$e_{0}=0.2$ (long-dashed line), and $e_{0}=0.3$ (solid line).
The release time is between $1000$ and $1200$ orbits.
Our standard disk parameters were used, including
$\alpha = 4\times 10^{-3}$.
}
\label{fig:at_mpvse}
\end{figure*}
The effect of non-zero orbital eccentricity on planet migration can also
be seen in \refFgt{fig:at_mpvse}, where the evolution of the semi-major
axis is plotted for simulations with $\Mp=1\,\MJup$ and $\Mp=3\,\MJup$
and different initial orbital eccentricities. There is a clear trend
towards slower migration rates for larger orbital eccentricities. In
particular, when $e_{0} \gtrsim 0.2$, the direction of migration is
reversed. In all the calculations that show outward migration, the
angular momentum of the eccentric orbit planet increases in time.
We have conducted some preliminary investigations on the cause of this
outward torque. One possibility is that it is due to the outer disk, as
suggested in \citet{papa2002}. When the planet eccentricity is large
enough, the angular motion of the planet at apocenter can be slower
than that of the inner parts of the outer disk, resulting in form of
dynamical friction that increases the angular momentum of the planet.
For the situation we wish to consider, it is not clear how the outer
disk inner edge is maintained, when this model is applied. This disk
material loses angular momentum from viscous torques and gains angular
momentum from the planet, in the usual torque balance for a gap. The
latter implies that the planet should lose, rather than gain, angular
momentum.
A preliminary analysis suggests that the outward torque may arise in the
coorbital region. This region is supplied by material that flows from the
outer disk across the gap, as discussed in \refSect{sec:planet_accretion}.
In any case, further analysis is required to understand this situation.
We conducted a convergence test on the model with a $1\,\MJup$ planet
and $e_{0}=0.2$, which exhibits outward migration. The test involved
a comparison of the migration rates obtained from the grid system GS2
(see \refTab{tbl:grids}) to those obtained from a grid system whose
linear resolution was a factor $1.3$ larger everywhere (in both directions
on each grid level).
However, since computing resources were only available to run the higher
resolution simulation for about $700$ orbits, we used the Gauss perturbation
equations \citep[e.g.,][]{beutler2005} to compute $\dot{a}$ resulting from
the disk's gravitational forces, while keeping the planet's orbit fixed.
The result of the test is that the migration rates, averaged from $200$ to
$700$ orbits, differed by only $7$\% at the two resolution levels.
As a check on our use of the Gauss equation, we also compared the migration
rate determined from the Gauss equation, averaged over the last $100$ orbits
before release, to the initial $\dot{a}$ after release, evaluated by
integrating the equations of motion of the planet (see the left panel
of \refFgt{fig:at_mpvse}). The two rates, both determined on grid system GS2,
differed by less than $3\%$.
\subsection{Effects of Viscosity}
\label{sec:viscosity_effects}
\begin{figure*}[t!]
\centering%
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f13a}%
\includegraphics{f13b}}
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f13c}%
\includegraphics{f13d}}
\caption{\small%
Evolution of planet eccentricity (left) and semi-major axis
(right) for $3\,\MJup$ (top) and $1\,\MJup$ (bottom) planets
in disks with different values of viscosity parameter $\alpha_{0}$
($\alpha$ at $r=a_{0}$), $\alpha_{0}=1.2\times10^{-3}$ (long-dashed
line), $\alpha_{0}=4\times10^{-3}$ (solid line), and
$\alpha_{0}=1.2\times10^{-2}$ (short-dashed line).
The release time is $t_{\mathrm{rls}}=1000$ orbits
in the top panels and $1100$ orbits bottom panels.
}
\label{fig:eat_mp3vsnu}
\end{figure*}
As we discussed in \refSect{sec:viscosity_influence}, the disk eccentricity
decreases with viscosity. For a coupled disk-planet system, we similarly
expect that the planet eccentricity would decrease with $\alpha$, since
the eccentric corotation resonances become stronger. The orbital
eccentricity evolution of $3\,\MJup$ planets in disks with different
$\alpha$ values is shown in the top-left panel of \refFgt{fig:eat_mp3vsnu}.
The model with standard viscosity (solid line) is the same as that in
\refFgt{fig:et_mp23}. Over a period of $2200$ orbits, the orbital
eccentricity of the model with $\alpha_{0}=1.2\times10^{-2}$
(short-dashed line) remains small and never exceeds $e\approx0.01$.
On the other hand, the models with lowest viscosities,
$\alpha_{0}=1.2\times10^{-3}$ (long-dashed line) and $4\times10^{-3}$
exhibit a generally growing eccentricity.
The trend towards faster growth for smaller viscosities is confirmed by
the model with $1\,\MJup$ and $\alpha_{0}=1.2\times10^{-3}$, as indicated
by the long-dashed line in the bottom-left panel of \refFgt{fig:eat_mp3vsnu}.
The orbital evolution of the semi-major axis of $3\,\MJup$ planets for
the three disk viscosities is displayed in the top-right panel of
\refFgt{fig:eat_mp3vsnu}. The radial inward migration is faster for
larger $\alpha$, as expected in a Type~II-like regime.
The plot supports the contention that planet eccentricity slows migration.
Around $4000$ orbits after the release time, the two calculations with
smallest viscosities ($\alpha_{0}=4\times10^{-3}$ and $1.2\times10^{-3}$)
produce migration rates respectively equal to
$\dot{a}\approx-8\times10^{-6}\,a_{0}$ ($e\approx 0.11$) and
$\dot{a}\approx-3\times10^{-6}\,a_{0}$ ($e\approx 0.14$) per orbit.
The first rate is a factor $8$ smaller, while the second a factor $13$
smaller, than the initial migration speed (i.e., when $e\approx 0$).
For these cases, the average eccentricity growth rate at a time of about
$2500$ orbits is $\dot{e}\approx 2\times10^{-5}\,\mathrm{orbit}^{-1}$.
The migration of a $1\,\MJup$ planet (\refFgp{fig:eat_mp3vsnu}, bottom-right
panel) also shows that, as the orbital eccentricity approaches $\sim 0.08$,
$|\dot{a}|$ starts to reduce. When $\alpha_{0}=1.2\times10^{-3}$, the average
migration rate, over the last $1000$ orbits, is about a factor $5$ smaller
than it is during the first $1000$ orbits of evolution after release.
\section{Pulsed Accretion}
\label{sec:planet_accretion}
\begin{figure*}
\centering%
\resizebox{0.95\linewidth}{!}{%
\includegraphics{f14a}%
\includegraphics{f14b}}
\caption{\small%
Mass accretion rate in inner parts of the Roche lobe of a
$1\,\MJup$ (left) and a $3\,\MJup$ (right) planet as a function
of its true anomaly and orbital eccentricity.
When the true anomaly equals $\pi$, the planet is located at the
apocenter of the orbit.
Orbital eccentricities are listed in the legend of the left panel.
The quantity $\langle\dot{M}_{\mathrm{p}}} % dot{Mp\rangle$ is obtained by sampling $\dot{M}_{\mathrm{p}}} % dot{Mp$
every $0.02$ orbits and averaging the outcome from $500$ to $1000$
orbits.
}
\label{fig:avmpdot}
\end{figure*}
For eccentric orbit binary star systems, the accretion from a
circumbinary disk onto the stars pulsates over the orbital period of
the binary \citep{pawel1996,guenther2002}.
This effect is related to the pulsating character of the equipotential
surfaces of the elliptical restricted three-body problem \citep{todoran1993}.
We measured the mass accretion rate (which we denote as $\dot{M}_{\mathrm{p}}} % dot{Mp$) into
the inner portion of the planet's Roche lobe (within $\racc=0.3\,\Rhill$),
following the prescription described in \refSect{sec:accretion_procedure},
as a function of the planet's orbit phase. We refer to this rate as the
planet accretion rate although the flow is not resolved on the scale of
the planet's radius and thus the rate at which the planet would accrete
mass may be modulated somewhat differently from $\dot{M}_{\mathrm{p}}} % dot{Mp$.
Quantity $\dot{M}_{\mathrm{p}}} % dot{Mp$ was determined by folding the mass accretion rate over
planet orbital phase and averaging over $500$ orbital periods (from $500$
to $1000$). \refFgt{fig:avmpdot} shows the resulting averaged accretion
rate, $\langle\dot{M}_{\mathrm{p}}} % dot{Mp\rangle$, for $1\,\MJup$ and $3\,\MJup$ planets, versus
the true anomaly (i.e., the azimuthal position relative to pericenter) of
the planet and as a function of the orbital eccentricity. The simulations
show pulsed accretion in cases of eccentric orbit planets. The amplitude
of the variability, and to a lesser extent the phase, of $\dot{M}_{\mathrm{p}}} % dot{Mp$ depends
on the orbital eccentricity.
\subsection{Modulation}
\label{sec:accretion_modulation}
The accretion onto a $1\,\MJup$ planet with $e=0.1$ has two asymmetric
peaks, the taller of which is around the apocenter position (true anomaly
equal to $\pi$).
The secondary peak is about $70$\% of the primary peak and is located close
to the pericenter position. For $e \le 0.2$, the modulation of $\dot{M}_{\mathrm{p}}} % dot{Mp$
increases with $e$. For larger orbital eccentricities, modulation decreases.
This effect may be a consequence of the gap becoming broader and shallower
with increasing $e$ (see \refFgp{fig:avsigma}).
A similar phase variability is found for the mass accretion onto a $3\,\MJup$
planet (\refFgt{fig:avmpdot}, right panel). Even when the planet's orbit
is circular, $\langle\dot{M}_{\mathrm{p}}} % dot{Mp\rangle$ smoothly varies between
$1.5\times10^{-4}\,\MJup$ and $4.5\times10^{-4}\,\MJup$ per orbit.
The phasing in this case is arbitrary, since the planet's orbit is circular.
This behavior is related to the eccentricity of the disk. The mass accretion
is markedly peaked around the apocenter position when $e>0.1$. The mass
accretion modulation is again greatest for $e=0.2$.
When the planet eccentricity is between $e=0.3$ and $0.4$, the highest
accretion rate occurs roughly $0.1$ orbits after apocenter. This delay
may be related to the time required by material to be captured once it
has been perturbed near the apocenter. Due to such a delay, accretion
on binaries occurs near pericenter \citep{pawel1996}.
\begin{figure*}[t!]
\centering%
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f15a}%
\includegraphics{f15b}}
\resizebox{0.9\linewidth}{!}{%
\includegraphics{f15c}%
\includegraphics{f15d}}
\caption{\small%
Density structure around a $1\,\MJup$ (top) and a $3\,\MJup$
(bottom) planet at pericenter (left) and apocenter (right).
The vertical axis is the azimuth about the star and the
horizontal axis is the distance from the star in units of
$a_{0}$. The orbital eccentricity is $e=0.2$ and $t\simeq 1000$
orbits. The solid circle indicates the instantaneous location
of the planet. A surface density of $10^{-5}$ corresponds to
$3.29\,\mbox{$\mathrm{g}\,\mathrm{cm}^{-2}$}$.
}
\label{fig:density_zoom}
\end{figure*}
The density distribution in the vicinity of an eccentric orbit planet
varies strongly with its orbital phase. This variation is illustrated
in the panels of \refFgt{fig:density_zoom}, which depict the situation
at the pericenter (left) and apocenter (right), for a $1\,\MJup$ (top)
and a $3\,\MJup$ planet (bottom) on an eccentric orbit with $e=0.2$.
The spiral waves have a regular pattern at pericenter, when the planet
is orbiting in the low-density gap (or cavity). As the planet approaches
the apocenter, the outer spiral wake penetrates higher density regions,
which causes fluid elements along the wake to lose angular momentum and
flow through the gap. There are streams of material that extend inwards
(at $r<a$ and $\phi>\phip$) which appear in the right panels of
\refFgt{fig:density_zoom}.
\subsection{Mass Growth Timescale}
\label{sec:growth_time-scale}
The mass accretion rate onto a planet with a fixed circular orbit
decreases with increasing planet mass when
$\Mp\gtrsim 1\,\MJup$ \citep{lubow1999}. The average accretion rate in
the simulations, at $t\simeq150$ orbits, of a $2\,\MJup$ planet is $0.63$
times that of a $1\,\MJup$ planet. The ratio decreases to $0.44$ for a
$3\,\MJup$ planet on a circular orbit and at $t=150$ orbits. These ratios
agree within $10\%$ with the values given by \citet{lubow1999}, who used
an independent code.
As the disk eccentricity grows, the mass accreted during an orbit increases.
At later times $t\gtrsim 500$ orbits, when $S_{(1,0)} \gtrsim 0.2$, the
accretion rate onto a $2\,\MJup$ planet is $0.71$ times the rate onto a
$1\,\MJup$ planet and for a $3\,\MJup$ planet is $0.65$ times the rate
onto a $1\,\MJup$ planet. Notice that, for a $3\,\MJup$ planet, this
implies a $48$\% increase over the accretion rate at early stages, when
the disk is circular. These results suggest that the eccentricity driven
in the disk by a massive planet can augment the mass accretion rate onto
the planet and hence shorten its growth timescale.
Mass accretion over an orbit period can also be enhanced by the planet's
orbital eccentricity. For a $1\,\MJup$ planet on a fixed orbit, the mass
growth timescale (defined here as the ratio of $\Mp$ to the average
accretion rate between $500$ and $1000$ orbits) decreased by about $35$\%
when $e$ is increased from $0$ to $0.2$. The reduction of mass growth
timescale from $e=0$ to $e=0.4$ is only $17$\%, perhaps as a result of
the wider gap and its smoother outer edge at larger orbital eccentricities
(see \refFgp{fig:avsigma}). For a $3\,\MJup$ planet on a fixed orbit with
$e=0.3$, the mass growth timescale is reduced relative to the $e=0$ case by
$27$\%. While for $e=0.4$ there was a $13$\% reduction. For cases with
$e < 0.3$, the growth rate was not substantially different than the $e=0$
case.
The mass growth timescale of a $1\,\MJup$ and a $3\,\MJup$ planet on
circular orbit was also estimated for different values of the disk viscosity.
For a $3\,\MJup$ planet an increase in $\alpha$ by a factor of $3$ over the
standard value (see \refSecp{sec:physical_par}) reduced the growth timescale
by about $60$\%.
For both planetary masses,
a decrease in $\alpha$ by a factor of $3.3$ below the standard value,
lengthened the growth timescale by a factor $2$.
Planetary accretion rates were determined by means of the grid system
GS2. Simulations executed with grid systems GS1 and GS3 resulted in very
similar outcomes (within $6$\%) for both the modulation of $\dot{M}_{\mathrm{p}}} % dot{Mp$ and its
average value. The results are not sensitive to the smoothing length,
$\varepsilon$, although this parameter can affect the small-scale structure
of the flow around the planet. We performed a calculation for an
$\varepsilon$ value that was reduced by a $25$\%, with $\Mp=3\,\MJup$ and
$e=0.3$, and obtained essentially the same accretion rates (within $1$\%).
\subsection{Accretion towards the Star}
\label{sec:star_accretion}
\begin{figure}[t!]
\centering%
\resizebox{1.0\linewidth}{!}{%
\includegraphics{f16}}
\caption{\small%
Mass accretion rate towards the star at the inner boundary
$r=0.3\,a_0$ as function the true anomaly of the planet.
The mass accretion rate is averaged over several planet
orbital periods at $t\approx1000$ orbits. The dashed and solid
lines refer to models with $\Mp=1\,\MJup$ and $3\,\MJup$,
respectively. The planet's orbital eccentricity is $e=0.2$.
When the true anomaly is $\pi$, the planet is located at
the apocenter.
}
\label{fig:avmsdot}
\end{figure}
The region interior to a planet's orbit likely contains an inner disk
which cannot be resolved in the current two-dimensional simulations
(see discussion in \refSecp{sec:radial_migration}). We estimate the
modulation of mass onto this disk as a function of orbital phase by
considering the mass flow rate across the inner boundary. As material
accretes through the inner disk, the modulation would be expected to weaken.
It is unclear whether the modulation would be reflected as a variable
accretion at the surface of the star. Perhaps it could be manifested
as variability in emission from the region where the inflow meets the
outer edge of the inner disk.
\refFgt{fig:avmsdot} plots the accretion rate at the inner boundary
$\langle\dot{M}_{*}} % dot{Ms\rangle$ versus the true anomaly of the planet for cases
with $\Mp=1\,\MJup$ and $3\,\MJup$. The case involving the more massive
planet produces an accretion rate through the inner boundary that is
largest when the planet is close to the apocenter. The maximum of
$\langle\dot{M}_{*}} % dot{Ms\rangle$ occurs before the apocenter passage for the case
involving the $1\,\MJup$ planet. The average mass accreted by the star
during one orbital period of the planet is $5.8\times10^{-8}\,\MStar$
and $9.9\times10^{-8}\,\MStar$ for the $1\,\MJup$ and the $3\,\MJup$
cases, respectively. The same models with no orbital eccentricity show
relatively constant $\langle\dot{M}_{*}} % dot{Ms\rangle$ values of
$1\times10^{-7}\,\MStar$ ($\Mp=1\,\MJup$) and
$6.3\times10^{-8}$ ($\Mp=3\,\MJup$) per orbit.
As a comparison, the mass accretion through a steady-state $\alpha$-disk
is $3\pi\Sigma\,\nu$ \citep{lynden-bell1974,pringle1981} which, using the
initial unperturbed surface density and standard viscosity
($\alpha_{0}=4\times10^{-3}$), yields $2.5\times10^{-7}\,\MStar$
per orbit or $2\times10^{-8}\,\mbox{$\MSun\,\mathrm{yr}^{-1}$}$.
The ratio of the accretion on the star to the accretion in the disk,
outside the gap, can be expressed as
$\langle\dot{M}_{*}} % dot{Ms\rangle_{\mathrm{o}}/%
(\langle\dot{M}_{*}} % dot{Ms\rangle_{\mathrm{o}}+\langle\dot{M}_{\mathrm{p}}} % dot{Mp\rangle_{\mathrm{o}})$,
where the subscript ``$\mathrm{o}$'' denotes is the integral of
the respective accretion rate over a planetary orbit.
For both the $1\,\MJup$ and the $3\,\MJup$ planets on circular orbit, this
ratio is $0.19$. When $e=0.2$, the ratio is $0.09$ and $0.26$
for the $1\,\MJup$ and the $3\,\MJup$ planet, respectively.
The reduced mass transfer across the planet's orbit, when $\Mp=1\,\MJup$
and $e=0.2$, can be attributed to the increased accretion rate onto the
planet, as reported above. In the $3\,\MJup$ case,
$\langle\dot{M}_{\mathrm{p}}} % dot{Mp\rangle_{\rm o}$ does not vary significantly as $e$ varies
from $0$ to $0.2$. Instead, the mass flux across the gap is likely enhanced
by the radial excursion of the planet (see \refSecp{sec:planet_accretion}).
\section{Summary and Discussion}
\label{sec:conclusions}
We simulated the orbital evolution of circular and eccentric orbit giant
planets embedded in circumstellar disks. The disks were analyzed using
a two-dimensional hydrodynamics code that utilizes nested grids to achieve
high resolution in a large region ($2\,a\times 2\pi/3$) around the
planet. The disks were modeled as an $\alpha$-disk and a few values of
$\alpha$ were considered. We investigated planet masses of $1\,\MJup$,
$2\,\MJup$, and $3\,\MJup$ and initial orbital eccentricities that ranged
from $0$ to $0.4$.
Disk gaps become broader and shallower as the planet eccentricity increases
(see \refFgp{fig:globalsigma} and \ref{fig:avsigma}). The density near the
orbit of the planet is very small compared with the density in the disk for
all eccentricities considered. A planet on a fixed circular orbit can cause
an initially circular disk to become eccentric (see \refFgp{fig:rcmvstime}).
The disk eccentricity is suppressed at lower planet masses
($\Mp\lesssim 1\,\MJup$) and higher disk viscosities ($\alpha\gtrsim 0.01$),
as also found by \citet{kley2006} and by \citet{papa2001} at higher planet
masses. We attribute the eccentricity growth to a tidal instability
associated with a series of eccentric outer Lindblad resonances in the inner
parts of the outer disk (\refFgp{fig:lam}). The same type of instability,
involving an inner disk, is thought to be responsible for the superhump
phenomena in binary star systems \citep{lubow1991a,osaki2003}.
The simulations indicate that planet eccentricity can grow, as a consequence
of disk-planet interactions (\refFgp{fig:et_mp23} and \ref{fig:et_mp1vse}).
The growth is stronger in the $2\,\MJup$ and $3 \,\MJup$ cases than for
$1\,\MJup$, and for lower disk viscosity ($\alpha\lesssim 4\times10^{-3}$).
Planet eccentricities of $\sim 0.1$ were found in the simulations over the
course of a few thousand orbits for $2 \,\MJup$ and $3 \,\MJup$ planets.
A similar eccentricity growth is obtained for a $1 \,\MJup$ planet in a
disk with viscosity $\alpha\approx 10^{-3}$.
The planet and disk
both acquire eccentricity as they interact, which may lead to complicated
time-dependent behavior of their eccentricities. The planet eccentricity
growth is likely aided by the disk eccentricity growth. The results suggest
that the eccentric growth found for $\sim 10\,\MJup$ planets by
\citet{papa2001} also occurs for lower planet masses.
The higher resolution achieved by our calculations may be playing a role
in obtaining this growth.
For circular orbit planets,
migration occurs on roughly the local viscous timescale, as expected for
Type~II migration. However, it is slowed for eccentric orbit planets.
This result appears for several configurations with either dynamically
determined (\refFgp{fig:at_mp123} and \ref{fig:eat_mp3vsnu}) or imposed
planet eccentricities (\refFgp{fig:at_mpvse}). For a $2\,\MJup$ case,
even migration reversal (outward migration) is found for a dynamically
determined eccentricity (\refFgp{fig:at_mp123}).
Migration slowing or reversal would have important consequences for the
planet formation process.
The cause is not yet clear. It may involve torques from outer disk
\citep{papa2002} or instead from the coorbital region. Some preliminary
evidence suggests the latter.
Mass accretion both within a planet's Roche lobe and through a gap can be
strongly modulated with orbital phase for eccentric orbit planets or
eccentric disks (\refFgp{fig:avmpdot}, \ref{fig:density_zoom}, and
\ref{fig:avmsdot}). The modulation was largest for planet eccentricity
$e \simeq 0.2$. This pulsating accretion is similar to what is found for
eccentric orbit binary stars embedded in a circumbinary disk
\citep{pawel1996}, although the phasing is different. Both disk and planet
eccentricity also lead to enhanced accretion onto the planet.
This enhancement likely helps planets achieve higher masses.
The simulations lend support to the idea that disk-planet interactions
cause planet eccentricity growth, along the lines of \citet{goldreich2003}.
The simulations suggest that planet eccentricities are easier to achieve
for higher mass planets ($\Mp\gtrsim 2\,\MJup$).
Our results are subject to the usual
limitations in approximate initial conditions, simulation time, radial
range for coverage of the disk (likely resulting in the lack of an inner
disk), the $\alpha$-disk model, and the use of various numerical devices.
We also neglected disk self-gravity, which may affect migration especially
for higher mass disks \citep{anelson2003a}.
However, it is not clear that typical extra-solar planet eccentricities
of $0.2$--$0.3$ can be achieved through disk-planet interactions.
The eccentricity growth at later times shows indications of slowing and
possibly stalling for $e \la 0.15$ (see Fig~\ref{fig:eat_mp3vsnu}.)
Perhaps higher eccentricities can be achieved for disks with
different properties (e.g., lower viscosity and smaller disk's
aspect ratio).
Eccentricity may be limited by damping due to high order eccentric inner
Lindblad resonances that lie outside a planet's orbit. Simulations of
eccentric orbit binary star systems suggest that little eccentricity
growth occurs for $e \ga 0.5$ \citep{lubow1992b}.
Although the simulated planets do not achieve orbital eccentricities in
excess of $0.15$ over the duration of the simulated evolution (for
configurations that start from circular orbits), the simulation times
correspond to less than $10^{5}$ years. Migration slowing and reversal
may permit the planets to achieve higher eccentricities on longer
timescales while avoiding orbit decay into the disk center/host star.
We have not yet investigated eccentricity evolution of sub-Jupiter mass
planets. They may also provide important constraints. Other simulations
suggest that disks with standard viscosity have only mild gaps for smaller
planet masses of $\Mp \la 0.1\,\MJup$ \citep[e.g.,][]{gennaro2003b,bate2003}.
Under those conditions, disk-planet interactions likely lead to eccentricity
damping, due to the dominance of the coorbital Lindblad resonance
\citep{ward1986,pawel1993}.
The observational determination of eccentricities for small
mass planets would help constrain these models. The planet around HD~49674
is close to this regime. It has a minimum mass of $0.11\,\MJup$ and a
best-fit eccentricity of $0.29$ (P.~Butler, private communication). Since
it is close to the central star (the period is $4.9$ days), it is possible
that the eccentricity evolution could be more complicated, especially if
it became trapped in a central disk hole. Examples of isolated planets like
this, but at longer periods would provide useful constraints.
\acknowledgments
We thank Gordon Ogilvie and Jim Pringle for useful discussions.
The computations reported in this paper were performed using
the UK Astrophysical Fluids Facility (UKAFF).
GD was supported by the Leverhulme Trust through a UKAFF Fellowship,
by the NASA Postdoctoral Program, and in part by NASA's Outer Planets
Research Program through grant 811073.02.01.01.20.
SL acknowledges support from NASA Origins of Solar Systems grant
NNG04GG50G.
MRB is grateful for the support of a Philip Leverhulme Prize.
|
2,877,628,090,185 | arxiv | \section{Introduction}
\noindent Vision and Language tasks, such as Vision-and-Language Navigation (VLN)~\cite{mattersim}, Visual Question Answering (VQA)~\cite{VQA,cao2018visual} and Referring Expression Comprehension (REF)~\cite{KazemzadehOrdonezMattenBergEMNLP14,yang2020graph,yang2020relationship} etc., have been extensively studied in the wave of deep neural networks. In particular, VLN~\cite{mattersim, chen2019touchdown} is a challenging task that combines both natural language understanding and visual navigation. Recent works have shown promising performance and progress. They mainly focus on designing agents capable of grounding fine-grained natural language instructions, where detailed information is provided, to find \textbf{\emph{where}} to stop, for example \emph{``Leave the bedroom and take a left. Take a left down the hallway and walk straight into the bathroom at the end of the hall. Stop in front of the sink''} ~\cite{fried2018speaker, ma2019selfmonitoring, wang2018look, wang2019reinforced, tan2019envdrop, ke2019tactile}. However, a practical issue is that fine-grained natural language instructions are not always available in real life and human-machine interactions are mostly based on high-level instructions such as \emph{``Go to the bathroom at the end of the hallway''}. In other words, designing an agent that could perform high-level natural language interpretation and infer the probable target location using knowledge of the environments is of more practical use.
In this paper, we focus on the REVERIE task~\cite{qi2020reverie} which is an example of the above mentioned high-level instruction task. Here, we briefly introduce the settings. Given a high-level instruction that refers to a remote target object at a target location within a building, a robot agent spawns at a starting location in the same building and tries to navigate closer to the object. The output of the task is a bounding box encompassing the target object. The success of the task is evaluated based on explicit object grounding at the correct target location. A straightforward solution is to integrate SOTA navigation model with SOTA object grounding model. This strategy has proven to be inefficient in ~\cite{qi2020reverie} and instead, they proposed an interactive module to enable the navigation model to work together with the object grounding model. Although the performance is improved, we observe that such method has a key weakness: it is unreasonable to discern high-level instruction by directly borrowing the fine-grained instruction navigation model that consists of simple trainable language attention mechanism based on the fact that the perception of high-level instruction primarily depends on commonsense knowledge prior as well as past experiences in memory. Therefore, the overall design is not in line with human intuitions in high-level instruction navigation.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{./figure3.pdf}
\caption{The overview of two pre-training tasks, the Scene Grounding task and the Object Grounding task. The Scene Grounding task empowers the agent the ability to reason where the target location is and the Object Grounding task learns what to attend to.}
\label{Fig:pretraining}
\end{figure}
Designing an agent to solve the problem like the REVERIE task is still under explored and there are still no systematic ways to design such an agent. Then, how does human wisdom solve this task? Human beings have instincts to understand surrounding visual environments and languages. Intuitively, given a high-level instruction, we would first extract high-level \textbf{\emph{what}} and \textbf{\emph{where}} information and then form an overview of the appearance of the target location in mind based on common sense knowledge. During navigation, we would consistently match current scene and objects in the scene to the instruction semantics and decide where to navigate next. According to such intuitions, we approach this problem from a new perspective and present an agent that imitates such human behaviors. Concretely, we define our problem as designing an agent that is able to solve \textbf{\emph{where}} and \textbf{\emph{what}} problem in the REVERIE task. We propose a two-stage training pipeline. In the first stage, we design two pre-training tasks, mimicking the aforementioned two human intuitions. The second stage is training the agent with a memory-augmented attentive action decoder, further increasing the agent's navigation capability under high-level instructions.
\textbf{Pre-training Stage.} As is shown in Fig.~\ref{Fig:pretraining}, we introduce a new subtask called the Scene Grounding task that is trained to recognize which viewpoint in a set of viewpoints is best aligned with the high-level instruction and another subtask called the Object Grounding task that helps the agent identify the best object that matches to the instruction among a set of candidate objects located at a target viewpoint. Experimental results show that the Scene Grounding model recognizes the target viewpoint with a high accuracy and the Object Grounding model outperforms the previous best model used in ~\cite{yu2018mattnet, qi2020reverie} by more than $10\%$.
\textbf{Action Decoding Stage.} In this stage, with the pre-trained models serving as scene and language encoders, we propose a memory-augmented attentive action decoder that leverages a scene memory structure as the agent's internal past state memory. This design is based on the fact that the computation of action at a specific time step could depend on any provided information in the past. Experimental results indicate that the proposed structure is effective and achieves new state-of-the-art performance.
To sum up, this paper has the following contributions:
\begin{itemize}[noitemsep, nolistsep]
\item We propose a new framework that borrows human intuitions for designing agent capable of understanding high-level instructions, which closely integrate navigation and visual grounding in both training and inference. Specifically, the visual grounding models are pre-trained and serve as vision and language encoders for training navigation action decoder in the training phase. In inference, the action is predicted by considering logits from both the visual grounding models and the navigation decoder.
\item We introduce two novel pre-training tasks, called Scene Grounding task and Object Grounding task, and a new Memory-augmented attentive action decoder in our framework. The pre-training tasks attempt to help the agent learn \textbf{\emph{where}} to stop and \textbf{\emph{what}} to attend to, and the action decoder effectively exploits past observations to fuse visual and textual modalities.
\item Without bells and whistles, our method outperforms all previous methods, achieving new state-of-the-art performance on both seen and unseen environments on the REVERIE task.
\end{itemize}
\section{Related Work}
\noindent\textbf{Vision-and-Language Navigation and REVERIE.} In VLN, an agent is required to navigate to a goal location in a $3$D simulator based on fine-grained instructions. ~\cite{mattersim} proposed the Matterport3D Simulator and designed the Room-to-Room task. Then, a lot of methods have been proposed to solve this task~\cite{fried2018speaker, wang2018look, wang2019reinforced, tan2019envdrop, ke2019tactile}. On the other hand, the recently proposed REVERIE task ~\cite{qi2020reverie} is different from traditional VLN in that it requires an agent to navigate and localize target object simultaneously under the guidance of high-level instruction. The model they proposed trains the navigation model with the interactive module that works together with the object grounding model~\cite{yu2018mattnet}, in the hope that the model could learn to understand high-level instruction in a data-driven manner. However, our motivation is essentially different in that we inject commonsense knowledge prior and past memory experiences into the action policy taking into consideration the human perception in dealing with such high-level instruction navigation problems. Specifically, we introduce two pre-training tasks and a memory based action policy to make the agent become scene-intuitive. Moreover, our pre-training tasks differ from the ones proposed in ~\cite{fried2018speaker, Zhu_2020_CVPR, majumdar2020vlnbert} in that their motivation is based on the fact that the ground truth navigation path is actually hidden in the fine-grained instruction, which is not the case in high-level instruction navigation.
\noindent\textbf{Memory-based policy for navigation tasks.} Various memory models have been extensively studied for navigation agents, including unstructured memory~\cite{hochreiter1997long, piotr2017learn, Wierstra2007SolvingDM, Jaderberg2017ReinforcementLW, Mirowski2017LearningTN, Das2018EmbodiedQA}, addressable memory~\cite{Oh2016ControlOM, Parisotto2018NeuralMS}, topological memory~\cite{Savinov2018SemiparametricTM}, and metric grid-based maps~\cite{Gupta2019CognitiveMA, vln-chasing-ghosts}, etc. Unstructured memory representations, such as LSTM memory, have been used extensively in both 2D and 3D environments. However, the issue of RNN based memory is that it does not contain context-dependent state feature storage or retrieval and does not have long time memory~\cite{vln-chasing-ghosts, zhao2020do, fang2019smt}. To address these limitations, more advaneced memory structures, such as addressable, topological, and metric based memory are proposed. In this paper, we adopt a simple adressable memory structure. The aim of using such a simple design is 1) to intentionally make it lightweight, thus reducing computational overhead, since the computational cost is important in REVERIE and our pipeline already contains heavy models; 2) to improve the performance of the overall pipeline rather than designing a more advanced memory superior to others. Besides, in VLN, the metric map memory construction requires finegrained language instruction as guidance, which is not available in our task, and building the topological memory requires pre-exploration of the environment, a technique that is certainly helpful to our agent but is beyond the discussion of this paper.
\noindent\textbf{Vision-and-Language BERT based referring expression comprehension.} Recent years have witnessed a resurgence of active research in transferrable image-text representation learning. BERT-based models ~\cite{devlin2018bert, tan2019lxmert, su2020vlbert, lu2020multitask, chen2020uniter, lu2019vilbert} have achieved superior performance over multiple vision-and-language tasks by transferring the pre-trained model on large aligned image-text pairs to other downstream tasks. In BERT-based VLN, the most related agents to ours are ~\cite{hao2020prevalent} and ~\cite{majumdar2020vlnbert}. ~\cite{hao2020prevalent} treats VLN as a vision-and-language alignment task and utilizes a pre-trained vision-and-language BERT model to predict action sequence while ~\cite{majumdar2020vlnbert} formulates VLN as an instruction and path alignment task and adopts a pre-trained vision-and-language BERT model to find the best candidate path that matches to the instruction given. However, our work differs from others in that we propose a generalized pipeline that mimics human intuitions to solve the high-level instruction navigation task where vision-and-language BERT model is a building block which can be customized to other vision-language alignment block. Experimental results show that the main performance gain comes from our proposed pipeline.
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{./figure2.pdf}
\caption{The overall pipeline of our method. The green part of the figure denotes the memory module where current viewpoint feature $\boldsymbol{V}_{t}$ and previous action feature $\boldsymbol{a}_{t-1}$ are embedded and stored in the Memory. $Transformer$ blocks are used to generate $\boldsymbol{s}_{t}^{a}$. The red rectangles represent two pre-trained models, namely Scene Grounding model and Object Grounding model. $ViLEncoder$ consists of $ViLBERT$ and $BiLSTM$ and $ViLPointer$ is $ViLBERT$ trained on viewpoint-based object grounding task. At each time step $t$, the agent perceives the instruction with viewpoint features and object features simultaneously. Action prediction is made by the Action Select part where an attentive structure is applied. The final action is generated by considering scene grounding score $g_{sg}$, object grounding score $g_{og}$ and action logit $\boldsymbol{l}_{t}$. The dashed dot lines are used only for illustration purposes.}
\label{Fig:pipeline}
\end{figure*}
\section{Method}
\noindent In the REVERIE task, an agent placed at a starting location navigates to the target location to localize an object specified by a high-level instruction. To carry out this difficult task, we propose a novel pipeline that contains a scene grounding model, an object grounding model, and a memory-based action decoder. We make two claims of our design choice: first, to better grasp the semantics of high-level instructions, we choose ViLBERT model as our basic building block to serve as vision-and-language encoder; second, since scene grounding task and object grounding task are two essentially different tasks, we do not share the basic building blocks for these two tasks. In general, we decompose our method into two stages, as shown in Fig.~\ref{Fig:pipeline}, namely the pre-training stage and the action decoding stage. In the following sections, we first introduce the pre-training tasks; then we illustrate the memory-based attentive action decoder and finally, the loss function used to train the agent.
\subsection{ViLBERT introduction}
\noindent In this section, we briefly introduce the input and output arguments of a ViLBERT model~\cite{lu2019vilbert} as shown in Fig.~\ref{Fig:SceneGroundingTask}. A ViLBERT model is a BERT-based model that consists of two input streams, vision encoding stream and language encoding stream, followed by a cross-modal alignment Transformer block. The inputs to ViLBERT model are sequence of words and visual features respectively and the outputs are corresponding encoded word sequence features as well as visual sequence features. We use ViLBERT as our base model (basic building block) for the Scene Grounding task and the Object Grounding task. In Scene Grounding task, a panorama viewpoint image is discretized into $36$ view images and the inputs are sequence of words in the instruction and $36$ mean-pooled features extracted from $36$ view images by a ResNet-$152$ CNN pre-trained on ImageNet~\cite{krizhevsky2012imagenet}. In Object Grounding task, the inputs are sequence of words in the instruction and all annotated bounding boxes features extracted by Mask R-CNN ~\cite{he2017mask} in a target viewpoint.
\subsection{Overview of the proposed method}
\noindent \textbf{Settings.} To formalize the task, we denote a given high-level instruction as $L = \left \{ l_{k}\right \}_{k=1}^{N_{l}}$ where $N_{l}$ is the number of words in the instruction $L$ and a set of viewpoints as $\nu = \left \{ V_{k}\right \}_{k=1}^{N_{v}}$ where $N_{v}$ is the number of viewpoints in the environment. At each time step $t$, the agent observes a panoramic view $V_{t}$, a few navigable views $O_{t}$ and a set of annotated bounding boxes $B_{t}$. The panoramic view is discretized into $36$ single views by perspective projections, each of which is a $640\times480$ size image with field of view set to $60$ degrees, and is denoted by $V_{t}=\left \{v_{t,i} \right \}_{i=1}^{36}$. $O_{t} = \left \{v_{t,i} \right \}_{i=1}^{N_{o}} \subseteq V_{t}$ where $N_{o}$ is the maximum navigable directions at a viewpoint $V_{t}$. Each $v_{t, i}$ is represented as $\boldsymbol{v}_{t, i} = ResNet(v_{t,i})$. Thus, $\boldsymbol{V}_{t} = \left \{\boldsymbol{v}_{t,i} \right \}_{i=1}^{36}$. Besides, the set of annotated bounding boxes at viewpoint $V_{t}$ is denoted by $B_{t} = \left \{b_{t, i} \right \}_{i=1}^{N_{b}}$ where $N_{b}$ is the number of bounding boxes. Mask R-CNN~\cite{he2017mask} is used to extract bounding boxes features $\boldsymbol{B}_{t} = \left \{\boldsymbol{b}_{t, i} \right \}_{i=1}^{N_{b}}$, where $\boldsymbol{b}_{t,i} = MRCNN(b_{t,i})$.
\textbf{Stage 1(a): Scene Grounding Task.} We formulate the task as finding a viewpoint that best matches to a high-level instruction $L$ in a set of candidate viewpoints $\nu_{s}$. $\nu_{s} = \left \{ V_{k} | V_{k} \in \nu \right \} \subseteq \nu$. Concretely, we define a mapping function $g_{sg}(,)$ that maps $(L, V_{k})$ to a matching score. The formula is defined as follows,
\begin{equation}
\begin{aligned}
V_{k}^{\star}&=\mathop{\arg\max}_{V_{k} \in \nu_{s}}g_{sg}(L, ResNet(V_{k}))
\end{aligned}
\end{equation}
\textbf{Stage 1(b): Object Grounding Task.} The goal of this task is to identify the best matching object among a set of candidate objects located at a target viewpoint. We denote $V_{T}$ as a target viewpoint and its corresponding annotated bounding boxes set is $B_{T}$. We define another compatibility matching function $g_{og}(,)$ that produce matching scores for all objects with a high-level instruction $L$. Thus, the problem is defined as follows,
\begin{equation}
\begin{aligned}
b_{T,i}^{\star}&=\mathop{\arg\max}_{b_{T, i} \in B_{T}}g_{og}(L, MRCNN(b_{T, i}))
\end{aligned}
\end{equation}
\textbf{Stage 2: Memory-augmented action decoder.} To mitigate the memory problem presented in previous section, a scene memory structure $\boldsymbol{M}_{t}$ is implemented to store the embedded observation and previous action at each time step $t$. The memory is updated by,
\begin{equation}
\begin{aligned}
\tilde{\boldsymbol{v}}_{t} &= softmax(\boldsymbol{V}_{t}(\boldsymbol{W}_{1}\boldsymbol{h}_{t-1}))^{T}\boldsymbol{V}_{t} \\
\boldsymbol{s}_{t} &= FC([\boldsymbol{a}_{t-1}, \tilde{\boldsymbol{v}}_{t}]), \\
\boldsymbol{M}_{t}&=Update(\boldsymbol{M}_{t-1}, \boldsymbol{s}_{t})
\end{aligned}
\end{equation}
where $\boldsymbol{s}_{t}$ is current state representation; $\tilde{\boldsymbol{v}}_{t}$ is attentive visual feature;$\boldsymbol{h}_{t-1}$ and $\boldsymbol{a}_{t-1}$ are last time step hidden state and action embedding respectively;$\boldsymbol{W}_{1}\in\mathbb{R}^{2048 \times D_{h}}$ is a trainable parameter. $FC$ stands for fully connected layer. The $Update$ operation appends $\boldsymbol{s}_{t}$ to $\boldsymbol{M}_{t}$. $\boldsymbol{V}_{t}\in\mathbb{R}^{36\times2048}, \tilde{\boldsymbol{v}}_{t}\in\mathbb{R}^{1\times2048}, \boldsymbol{a}_{t-1}\in\mathbb{R}^{1\times3200}, \boldsymbol{s}_{t}\in\mathbb{R}^{1 \times D_{h}}, \boldsymbol{h}_{t}\in\mathbb{R}^{D_{h}\times1}, \boldsymbol{M}_{t}\in\mathbb{R}^{t \times D_{h}}$.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{./figure1.pdf}
\caption{The pipeline of the Scene Grounding Task. We formulate this task as a $5$-way multiple choice problem. Each $(L, ResNet(V_{k}))$ pair is sent to the $ViLBERT$ model separately to generate alignment score $sc_{k}$. The panorama viewpoint image here denotes the discretized $36$ view images in a viewpoint. We mark the beginning of the image sequence with a special token $IMG$ and the language with $CLS$.}
\label{Fig:SceneGroundingTask}
\end{figure}
\subsection{Scene Grounding Task}
\noindent The goal of this task it to help the agent infer where the target location is. Given a high-level instruction, \emph{``Bring me the \textbf{\textit{jeans}} that are hanging up in the \textbf{\textit{closet}} to the right''}, humans first locate the \textbf{\textit{where}} information, the key word \textbf{\textit{closet}}, by capturing the semantics of the instruction according to the language context and commonsense knowledge and then form an overview of the appearance of the \textbf{\textit{closet}} in mind; then, humans navigate to the target location by consistently matching the \textbf{\textit{closet}} appearance in mind with current scene. In fact, humans have gradually formed intuitions towards the understanding of scenes, instructions and tasks in life. For language instructions in relatively simple life scenes that do not involve complex reasoning, they usually directly merge the above two processes for direct perception and understanding. We call this process as context-driven scene perception. In this section, we propose Scene Grounding task to imitate such human behavior.
Based on the observation, we believe that a model that could evaluate the alignment between an instruction and a viewpoint is able to localize the target viewpoint. Therefore, to implement this idea, we create a dataset from the REVERIE training set and fine-tune a ViLBERT model on the dataset. Specifically, we adopt a $5$-way multiple choice setting. We eliminate subscript for simplicity concern. Given an instruction $L$, we sample $5$ viewpoints $\left \{ V_{1}^{+}, V_{2}^{-}, V_{3}^{-}, V_{4}^{-}, V_{5}^{-} \right \}$, out of which only one is aligned to the instruction (or in other words, positive). In detail, we choose the ending viewpoint in the ground-truth training path as $V_{1}^{+}$, the second last viewpoint along the ground-truth path as $V_{2}^{-}$ which is a hard negative sample and random sample $V_{3}^{-}, V_{4}^{-}$ from the rest of the viewpoints along the path, and $V_{5}^{-}$ from other path . Then, we run the ViLBERT model on each of the $(L, V_{k})$ pair. As is shown in Fig.~\ref{Fig:SceneGroundingTask}, the output tokens $CLS$ and $IMG$ encode instruction representation $\boldsymbol{h}_{CLS}$ as well as viewpoint representation $\boldsymbol{h}_{IMG}$ respectively. We define the matching scores as $\boldsymbol{Sc}$ and train the model with cross entropy loss $\mathcal{L}_{sr}$.
\begin{equation}
\begin{aligned}
\boldsymbol{Sc}&=\left \{sc_{1}, sc_{2}, sc_{3}, sc_{4}, sc_{5} \right \} \\
sc_{k}&=g_{sg}(L, ResNet(V_{k}))=\boldsymbol{W}_{2}(\boldsymbol{h}_{CLS}^{k} \odot \boldsymbol{h}_{IMG}^{k})\\
\mathcal{L}_{sr}&=CELoss(softmax(\boldsymbol{Sc}), \mathbb{I}(V_{1}^{+}))
\end{aligned}
\end{equation}
where $\boldsymbol{W}_{2}\in\mathbb{R}^{1\times1024}$ is a trainable parameter and $\mathbb{I}(.)$ is indicator function. $\boldsymbol{h}_{CLS}^{k}\in\mathbb{R}^{1024\times1}, \boldsymbol{h}_{IMG}^{k}\in\mathbb{R}^{1024\times1}$ are the encoded language and visual representations of the language and vision encoding streams from our pre-trained ViLBERT model for $k$th $(L, V_{k})$ pair respectively.
\subsection{Object Grounding Task}
\noindent The aim of this task is to help the agent learn what to attend to. For each ground-truth target viewpoint $V_{T}$, we formulate this task as finding the best bounding box $b_{T, i}^{\star}$ in bounding boxes set $B_{T}$ given $(L, B_{T})$ pair. A straightforward method to implement this idea is to construct a single image based grounding task, where each training sample consists of instruction $L$ and a subset of bounding boxes in $B_{T}$ that belong to view $v_{T, i}$. However, according to our experiment, this strategy produces moderate performance since objects in $3$D space could span multiple views in corresponding projected $2$D image space. The cross-image objects relationships in each viewpoint are not well captured by the model. Therefore, we propose a two-stage training strategy, namely a single image based grounding and a viewpoint based object grounding. In single image grounding, we fine-tune the ViLBERT model from ~\cite{lu2020multitask, lu2019vilbert} on the aforementioned single image grounding dataset where each training sample is $(L, B_{v_{T, i}})$ (all annotated bounding boxes in $v_{T, i}$ are collected) and $B_{v_{T, i}}\subset B_{T}$; then, we further fine-tune trained model on a new viewpoint based object grounding dataset. Concretely, each training sample in the viewpoint based dataset is a $(L, B_{T})$ pair (all annotated bounding boxes in $v_{T}$ are collected) and the corresponding label is a vector containing $0$s and $1$s where $1$ indicates the IoU of a bounding box with the target bounding box is higher than $0.5$. In inference, we represent an object score as the averaged scores from all bounding boxes that share the same object id at a viewpoint that the agent stops.
\subsection{Action Decoder}
\noindent With the pre-trained grounding models, the action decoder generally adopts Encoder-Decoder structure to produce action prediction. Specifically, the Scene Grounding model is accompanied by a $BiLSTM$ network to construct a vision and language grounding encoder $ViLEncoder$ and the Object Grounding model is formulated as an object level grounding encoder $ViLPointer$. The inputs to action decoder are $L$, $\boldsymbol{B}_{t}$ and $\boldsymbol{V}_{t}$ and it outputs predicted action distribution $\boldsymbol{l}_{t}$.
\textbf{First}. At each time step $t$, to perceive current scene and instruction, we obtain $\tilde{\boldsymbol{x}}_{t}$ by grounding $L$ with $\boldsymbol{V}_{t}$ through $ViLEncoder$ and then selecting the fused language sequence as output. The formula is defined as follows,
\begin{equation}
\begin{aligned}
\boldsymbol{X}_{t}&=ViLEncoder(L, \boldsymbol{V}_{t}) \\
&=BiLSTM(ViLBERT(L, \boldsymbol{V}_{t})) \\
\tilde{\boldsymbol{x}}_{t}&=softmax(\boldsymbol{X}_{t}(\boldsymbol{W}_{3}\boldsymbol{h}_{t-1}))^{T}\boldsymbol{X}_{t}
\end{aligned}
\end{equation}
where $\boldsymbol{W}_{3}\in\mathbb{R}^{1024 \times D_{h}}$ is a trainable parameter and $\boldsymbol{X}_{t}$ is encoded language feature taking current scene $\boldsymbol{V}_{t}$ into consideration. $\boldsymbol{X}_{t}\in\mathbb{R}^{N_{l}\times1024}, \tilde{\boldsymbol{x}}_{t}\in\mathbb{R}^{1\times1024}, \boldsymbol{h}_{t-1}\in\mathbb{R}^{D_{h}\times1}$.
\textbf{Second}. To decide which navigable direction to go next, we perform object level referring expression comprehension. The object level referring comprehension helps the agent infer whether a navigable view $v_{t, i}$ contains possible target object. In particular, the set of bounding boxes in view $v_{t, i}$ is denoted by $\hat{B}_{t, i}=\left \{ b_{t, k} | b_{t, k} \in B_{t}, Inside(b_{t, k}, v_{t, i})=1\right \}$ where $Inside(,)$ function decides whether $b_{t, k}$ is inside view $v_{t, i}$. $ViLPointer$ is $ViLBERT$ pre-trained on the Object Grounding task and we select the fused bounding boxes features as the output. Then,
\begin{equation}
\begin{aligned}
\boldsymbol{F}_{t, i}&=ViLPointer(L, MRCNN(\hat{B}_{t, i})) \\
\tilde{\boldsymbol{v}}_{t, i}&=g_{top-k}(\boldsymbol{F}_{t, i})
\end{aligned}
\end{equation}
where $\boldsymbol{F}_{t, i}$ is the set of aligned bounding boxes features at view $v_{t, i}$ and $g_{top-k}(,)$ selects top-$k$ aligned bounding boxes and averages the corresponding aligned bounding boxes features from $\boldsymbol{F}_{t, i}$ to produce view comprehension $\tilde{\boldsymbol{v}}_{t, i}\in\mathbb{R}^{1\times1024}$.
\textbf{Third}. We define the representation of each navigable view as $\boldsymbol{v}_{t, i}'$:
\begin{equation}
\begin{aligned}
\boldsymbol{v}_{t, i}'&=[\boldsymbol{v}_{t, i}, (\cos\theta_{t, i}, \sin\theta_{t, i}, \cos\phi_{t, i}, \sin\phi_{t, i}), \tilde{\boldsymbol{v}}_{t, i}]
\end{aligned}
\end{equation}
where the agent's current orientation $(\theta_{t, i}, \phi_{t, i})$ represents the angles of heading and elevation and is tiled $32$ times according to ~\cite{fried2018speaker}. $(\cos\theta_{t, i}, \sin\theta_{t, i}, \cos\phi_{t, i}, \sin\phi_{t, i})\in\mathbb{R}^{1\times128}$ and $\boldsymbol{v}'_{t, i}\in\mathbb{R}^{1\times3200}$. The set of navigable view representation is denoted as $\boldsymbol{O}'_{t} = \left \{ \boldsymbol{v}'_{t, i}\right \}_{i=1}^{N_{o}}$. The grounded navigable visual representation $\tilde{\boldsymbol{o}}'_{t}$ is represented as follows:
\begin{equation}
\begin{aligned}
\tilde{\boldsymbol{o}}'_{t} = softmax(g(\boldsymbol{O}'_{t})(\boldsymbol{W}_{4}\boldsymbol{h}_{t-1}))^{T}g(\boldsymbol{O}'_{t})
\end{aligned}
\end{equation}
where $\boldsymbol{W}_{4}\in\mathbb{R}^{1024 \times D_{h}}$ is a trainable parameter and $g(,)$ is a number of Fully Connected layers accompanied by ReLU nonlinearities. $\tilde{\boldsymbol{o}}'_{t}\in\mathbb{R}^{1\times1024}, \boldsymbol{O}'_{t}\in\mathbb{R}^{N_{o}\times3200}$.
\textbf{Fourth}. The new context hidden state $\boldsymbol{h}_{t}$ is updated by a LSTM layer taking as input the grounded text $\tilde{\boldsymbol{x}}_{t}$ and navigable view features $\tilde{\boldsymbol{o}}'_{t}$ as well as the current state representation feature $\boldsymbol{s}_{t}^{a}$.
\begin{equation}
\begin{aligned}
(\boldsymbol{h}_{t}, \boldsymbol{c}_{t}) = LSTM([\tilde{\boldsymbol{x}}_{t}, \tilde{\boldsymbol{o}}'_{t}, \boldsymbol{s}_{t}^{a}], (\boldsymbol{h}_{t-1}, \boldsymbol{c}_{t-1}))
\end{aligned}
\end{equation}
where $\boldsymbol{s}_{t}^{a}$ is memory augmented current state representation and is defined as,
\begin{equation}
\begin{aligned}
\boldsymbol{M}_{t}^{a}&=[Transformer(\boldsymbol{M}_{t}, \boldsymbol{M}_{t})]_{\times N_{mem}} \\
\boldsymbol{s}_{t}^{a}&=[Transformer(\boldsymbol{s}_{t}, \boldsymbol{M}_{t}^{a})]_{\times N_{state}} \\
\end{aligned}
\end{equation}
where $N_{mem}$ and $N_{state}$ are number of memory transformer blocks used and number of state transformer blocks used respectively. $\boldsymbol{s}_{t}^{a}\in\mathbb{R}^{1 \times D_{h}}, \boldsymbol{M}_{t}^{a}\in\mathbb{R}^{t \times D_{h}}$. $Transformer$ is the standard version Transformer block from ~\cite{google2017attention}.
\textbf{Finally}. The action logit $\boldsymbol{l}_{t}$ is computed in an attentive manner.
\begin{equation}
\begin{aligned}
l_{t, i} = g(\boldsymbol{O}'_{t, i})(\boldsymbol{W}_{5}[\boldsymbol{h}_{t}, \tilde{\boldsymbol{x}}_{t}])
\end{aligned}
\end{equation}
where $\boldsymbol{W}_{5}\in\mathbb{R}^{1024\times(1024+D_{h})}$ is a trainable parameter and $\boldsymbol{l}_{t}\in\mathbb{R}^{N_{o}\times1}$. In training stage, $a_{t} = Categorical(\boldsymbol{l}_{t})$ is selected based on categorical policy and in inference stage, it is selected by $a_{t} = \arg max(\boldsymbol{l}_{t})$. Action embedding is selected based on $\boldsymbol{a}_{t} = \boldsymbol{O}'_{t}[a_{t}]$.
\subsection{Inference}
\noindent We propose to use a combined logit $\sum_{\tau=0}^{t}\boldsymbol{l}_{\tau} + g_{og}^{\tau} + g_{sg}^{\tau}$ that sums action logits, object grounding logits and scene grounding logits to perform navigation, where $g_{og}^{\tau}$ and $g_{sg}^{\tau}$ denote object grounding score and scene grounding score at time step $\tau$ respectively.
Experimental results indicate that our strategy shortens the search trajectories while maintaining a good success rate. The final output bounding box is obtained by running $ViLPointer$ at the stop viewpoint that the agent predicts.
\subsection{Loss Functions}
\noindent To train the agent, we use a mixture of Imitation Learning (IL) and Reinforcement Learning (RL) to supervise the training. Specifically, In IL, at each time step, we allow the agent to learn to imitate the teacher action by using a cross entropy loss $\mathcal{L}_{ce}$ and a mean squared error loss $\mathcal{L}_{pm}$ for progress monitor ~\cite{ma2019selfmonitoring}. In RL, we follow the idea of ~\cite{tan2019envdrop} and allow the agent to learn from rewards. If the agent stops within $3$ meters near the target viewpoint, a positive reward $+3$ is assigned at the final step; otherwise a negative reward $-3$ is given.
\begin{equation}
\begin{aligned}
\mathcal{L}_{final}&=\alpha\mathcal{L}_{ce}+\beta\mathcal{L}_{pm}+\gamma\mathcal{L}_{RL} \\
\mathcal{L}_{ce}&=-\sum_{t=1}^{T}y_{t}^{\star}\log(l_{t, \star}) \\
\mathcal{L}_{pm}&=-\sum_{t=1}^{T}(y_{t}^{pm} - p_{t}^{pm})^{2}
\end{aligned}
\end{equation}
where $y_{t}^{\star}$ is the teacher action at step $t$; $y_{t}^{pm}\in[0, 1]$ is the shortest normalized distance from current viewpoint to the target viewpoint; $p_{t}^{pm}$ is the predicted progress; $\alpha$, $\beta$ and $\gamma$ are all set to $1$.
\section{Experiments}
\noindent In the REVERIE dataset, the training set contains $59$ scenes and $10466$ instructions over $2353$ objects; the val seen split consists of $53$ scenes and $1371$ instructions over $428$ objects and the val unseen split include $10$ scenes and $3573$ instructions over $525$ objects. The test set contains $16$ scenes and $6292$ instructions over $834$ objects. In this section, we conduct extensive evaluation and analysis of the effectiveness of our proposed components.
\begin{table*}[t]
\centering
\caption{Ablation Study experiments performed to verify the effectiveness of the proposed method. In different ablation study block, the best performing result is marked in \underline{\textbf{bold}}.}
\resizebox{\linewidth}{!}{
\begin{tabular}{c|c|cccccccc|cccc|cc|cccc|cc}
\hline
\multirow{3}{*}{Experiments} & \multirow{3}{*}{ID} & \multicolumn{8}{c|}{Methods} & \multicolumn{6}{c|}{Val Seen} & \multicolumn{6}{c}{Val Unseen} \\
\cline{3-22}
& & \multicolumn{4}{c|}{Encoder} & \multicolumn{2}{c|}{Pointer} & \multicolumn{2}{c|}{Policy} & \multicolumn{4}{c|}{Nav. Acc.} & \multicolumn{1}{c|}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RGS$\uparrow$\\ \end{tabular}}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RG\\SPL$\uparrow$ \end{tabular}} & \multicolumn{4}{c|}{Nav. Acc.} & \multicolumn{1}{c|}{\multirow{2}{*}{RGS$\uparrow$ }} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RG\\SPL$\uparrow$ \end{tabular}} \\
\cline{3-14}\cline{17-20}
& & \multicolumn{1}{c|}{$L_{enc}$} & \multicolumn{1}{c|}{$Bert_{enc}$} & \multicolumn{1}{c|}{$ViLRaw_{enc}$} & \multicolumn{1}{c|}{$ViL_{enc}$} & \multicolumn{1}{c|}{$MN_{ptr}$ } & \multicolumn{1}{c|}{$ViL_{ptr}$} & \multicolumn{1}{c|}{$C_{pol}$} & $MA_{pol}$ & Succ.$\uparrow$ & OSucc.$\uparrow$ & SPL$\uparrow$ & Length$\downarrow$ & \multicolumn{1}{c|}{} & & Succ.$\uparrow$ & OSucc.$\uparrow$ & SPL$\uparrow$ & Length$\downarrow$ & \multicolumn{1}{c|}{} & \\
\hline
\multirow{8}{*}{\begin{tabular}[c]{@{}c@{}}Component\\Effectiveness \end{tabular}} & 1 & $\surd$ & & & & $\surd$ & & $\surd$ & & 50.53 & 55.17 & 45.50 & 16.35 & 31.97 & 29.66 & 14.40 & 28.20 & 7.19 & 45.28 & 7.84 & 4.67 \\
& 2 & & $\surd$ & & & $\surd$ & & $\surd$ & & 54.18 & 58.68 & 48.99 & \uline{\textbf{12.46}} & 33.87 & 21.23 & 18.66 & 29.51 & 10.44 & \uline{\textbf{32.95}} & 11.13 & 6.32 \\
& 3 & & & $\surd$ & & $\surd$ & & $\surd$ & & 33.73 & 39.14 & 30.72 & 14.56 & 23.82 & 21.94 & 15.22 & 31.64 & 8.44 & 42.62 & 8.89 & 4.84 \\
& 4 & & & $\surd$ & & & $\surd$ & $\surd$ & & 39.00 & 43.85 & 35.00 & 13.71 & 28.95 & 25.98 & 13.80 & 31.33 & 8.21 & 37.31 & 9.17 & 5.54 \\
& 5 & & & $\surd$ & & $\surd$ & & & $\surd$ & 37.32 & 43.08 & 31.71 & 18.29 & 24.88 & 21.70 & 19.06 & 44.39 & 7.10 & 79.88 & 11.08 & 4.17 \\
& 6 & & & & $\surd$ & $\surd$ & & $\surd$ & & 56.36 & 60.93 & 52.24 & 13.21 & 36.33 & 33.92 & 21.61 & 31.98 & 12.21 & 36.05 & 13.21 & 7.31 \\
& 7 & & & & $\surd$ & & $\surd$ & $\surd$ & & 54.25 & 56.08 & 50.49 & 13.56 & 39.56 & 37.16 & 26.98 & 37.86 & 13.70 & 42.50 & 17.32 & 8.71 \\
& 8 & & & & $\surd$ & & $\surd$ & & $\surd$ & \uline{\textbf{59.52}} & \uline{\textbf{64.23}} & \uline{\textbf{55.30}} & 14.00 & \uline{\textbf{43.57}} & \uline{\textbf{40.42}} & \uline{\textbf{28.17}} & \uline{\textbf{40.41}} & \uline{\textbf{14.77}} & 43.12 & \uline{\textbf{19.60}} & \uline{\textbf{10.27}} \\
\hline
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Memory\\Blocks\\($N_{mem}$, $N_{state}$) \end{tabular}} & 9 & \multicolumn{8}{c|}{(1, 1)} & 55.24 & 58.61 & 52.29 & \uline{\textbf{12.42}} & 40.90 & 38.76 & 28.97 & 39.56 & 13.28 & 44.10 & 20.51 & 9.19 \\
& 10 & \multicolumn{8}{c|}{(3, 3)} & \uline{\textbf{61.91}} & \uline{\textbf{65.85}} & \uline{\textbf{57.08}} & 13.61 & \uline{\textbf{45.96}} & \uline{\textbf{42.65}} & \uline{\textbf{31.53}} & \uline{\textbf{44.67}} & \uline{\textbf{16.28}} & \uline{\textbf{41.53}} & \uline{\textbf{22.41}} & \uline{\textbf{11.56}} \\
& 11 & \multicolumn{8}{c|}{(5, 5)} & 60.01 & 63.38 & 54.99 & 17.44 & 44.69 & 41.10 & 25.84 & 38.20 & 13.09 & 44.00 & 18.23 & 9.19 \\
& 12 & \multicolumn{8}{c|}{(7, 7)} & 57.27 & 62.26 & 52.78 & 13.96 & 42.66 & 39.38 & 23.66 & 35.61 & 11.67 & 45.73 & 16.79 & 8.43 \\
& 13 & \multicolumn{8}{c|}{(9, 9)} & 57.06 & 60.15 & 53.35 & 14.16 & 42.38 & 39.67 & 28.15 & 39.45 & 14.92 & 41.53 & 19.54 & 10.13 \\
\hline
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Logit\\Fusion \end{tabular}} & 14 & \multicolumn{8}{c|}{$\boldsymbol{l}_{t}$ } & 60.92 & 65.78 & 56.14 & 15.28 & 45.61 & 42.19 & \uline{\textbf{32.35}} & \uline{\textbf{49.08}} & 14.74 & 60.89 & 22.35 & 10.54 \\
& 15 & \multicolumn{8}{c|}{$\boldsymbol{l}_{t}$+$g_{sg}$ } & 61.49 & 65.78 & 56.72 & 13.67 & 45.47 & 42.31 & 31.20 & 47.80 & 15.90 & 45.82 & 21.68 & 11.08 \\
& 16 & \multicolumn{8}{c|}{$\boldsymbol{l}_{t}$+$g_{og}$ } & 61.14 & 65.77 & 55.21 & 16.82 & 44.48 & 40.04 & 32.12 & 46.54 & 15.73 & 52.14 & 21.98 & 11.02 \\
& 17 & \multicolumn{8}{c|}{$\boldsymbol{l}_{t}$+$g_{sg}$+$g_{og}$ } & \uline{\textbf{61.91}} & \uline{\textbf{65.85}} & \uline{\textbf{57.08}} & \uline{\textbf{13.61}} & \uline{\textbf{45.96}} & \uline{\textbf{42.65}} & 31.53 & 44.67 & \uline{\textbf{16.28}} & \uline{\textbf{41.53}} & \uline{\textbf{22.41}} & \uline{\textbf{11.56}} \\
\hline
\end{tabular}}
\label{tab:ablation}
\end{table*}
\begin{table*}[t]
\centering
\caption{Comparison with state-of-the-art methods on the REVERIE task. The best performing result is marked in \underline{\textbf{bold}}.}
\resizebox{\linewidth}{!}{
\begin{tabular}{c||cccc|cc||cccc|cc||cccc|cc}
\hline
\multirow{3}{*}{Methods} & \multicolumn{6}{c||}{Val Seen} & \multicolumn{6}{c||}{Val Unseen} & \multicolumn{6}{c}{Test (Unseen)} \\
\cline{2-19}
& \multicolumn{4}{c|}{Nav. Succ.} & \multicolumn{1}{c|}{\multirow{2}{*}{RGS$\uparrow$}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RG\\SPL\end{tabular}$\uparrow$} & \multicolumn{4}{c|}{Nav. Succ.} & \multicolumn{1}{c|}{\multirow{2}{*}{RGS$\uparrow$}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RG\\SPL\end{tabular}$\uparrow$} & \multicolumn{4}{c|}{Nav. Succ.} & \multicolumn{1}{c|}{\multirow{2}{*}{RGS$\uparrow$}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}RG\\SPL\end{tabular}$\uparrow$} \\
\cline{2-5}\cline{8-11}\cline{14-17}
& Succ.$\uparrow$ & OSucc.$\uparrow$ & SPL$\uparrow$ & Length$\downarrow$ & \multicolumn{1}{c|}{} & & Succ.$\uparrow$ & OSucc.$\uparrow$ & SPL$\uparrow$ & Length$\downarrow$ & \multicolumn{1}{c|}{} & & Succ.$\uparrow$ & OSucc.$\uparrow$ & SPL$\uparrow$ & Length$\downarrow$ & \multicolumn{1}{c|}{} & \\
\hline
RCM~\cite{wang2019reinforced} + MattNet& 23.33 & 29.44 & 21.82 & 10.70 & 16.23 & 15.36 & 9.29 & 14.23 & 6.97 & 11.98 & 4.89 & 3.89 & 7.84 & 11.68 & 6.67 & 10.60 & 3.67 & 3.14 \\
SelfMonitor~\cite{ma2019selfmonitoring} + MattNet & 41.25 & 43.29 & 39.61 & \underline{\textbf{7.54}} & 30.07 & 28.98 & 8.15 & 11.28 & 6.44 & \underline{\textbf{9.07}} & 4.54 & 3.61 & 5.80 & 8.39 & 4.53 & \underline{\textbf{9.23}} & 3.10 & 2.39 \\
FAST-short~\cite{ke2019tactile} + MattNet & 45.12 & 49.68 & 40.18 & 13.22 & 31.41 & 28.11 & 10.08 & 20.48 & 6.17 & 29.70 & 6.24 & 3.97 & 14.18 & 23.36 & 8.74 & 30.69 & 7.07 & 4.52 \\
REVERIE~\cite{qi2020reverie} & 50.53 & 55.17 & 45.50 & 16.35 & 31.97 & 29.66 & 14.40 & 28.20 & 7.19 & 45.28 & 7.84 & 4.67 & 19.88 & 30.63 & 11.61 & 39.05 & 11.28 & 6.08 \\
Human & - & - & - & - & - & - & - & - & - & - & - & - & 81.51 & 86.83 & 53.66 & 21.18 & 77.84 & 51.44 \\
\hline
Ours & \underline{\textbf{61.91}} & \underline{\textbf{65.85}} & \underline{\textbf{57.08}} & 13.61 & \underline{\textbf{45.96}} & \underline{\textbf{42.65}} & \underline{\textbf{31.53}} & \underline{\textbf{44.67}} & \underline{\textbf{16.28}} & 41.53 & \underline{\textbf{22.41}} & \underline{\textbf{11.56}} & \underline{\textbf{30.8}} & \underline{\textbf{44.56}} & \underline{\textbf{14.85}} & 48.61 & \underline{\textbf{19.02}} & \underline{\textbf{9.20}} \\
\hline
\end{tabular}}
\label{tab:sota}
\end{table*}
\subsection{Evaluation Metrics}
\noindent Following ~\cite{qi2020reverie}, we evaluate the performance of the model based on REVERIE Success Rate (RGS) and REVERIE Success Rate weighted by Path Length (RG SPL). We also report the performance of Navigation Success Rate, Navigation Oracle Success Rate, Navigation Success Rate weighted by Path Length (SPL), and Navigation Length. Please refer to the supplementary document for more details.
\subsection{Ablation Study}
\noindent In this section, we aim to answer the following questions: $(a)$ Does the performance gain mainly come from BERT-based structure? $(b)$ How effective is each of the proposed component? $(c)$ Does the memory blocks number matter? $(d)$ Why do we need logit fusion? For simplicity concern, we define the following experiment settings: $(1)$ our proposed $ViLEncoder$ is $ViL_{enc}$; $(2)$ the $ViLRaw_{enc}$ is $ViLEncoder$ not pre-trained on the Scene Grounding task but pre-trained on the Conceptual Captions dataset~\cite{sharma2018conceptual} as well as the $12$ tasks specified in ~\cite{lu2020multitask}; $(3)$ the $BERT_{enc}$ is a BERT language encoder pre-trained on the BookCorpus~\cite{zhu2015alignbooks} and English Wikipedia datasets; $(4)$ our proposed $ViLPointer$ is $ViL_{ptr}$; $(5)$ previous SOTA MattNet pointer is $MN_{ptr}$; $(6)$ our action policy is $MA_{pol}$; $(7)$ previous action policy is $C_{pol}$; $(8)$ previous simple language encoder is $L_{enc}$ composed of a trainable embedding layer with a Bi-directional LSTM layer.
\textbf{Performance Gain.} To answer question $(a)$, we perform experiments $1$, $2$, $3$ and $6$ as is shown in Table~\ref{tab:ablation}. All agents are trained under $\mathcal{L}_{ce}$ and $\mathcal{L}_{pm}$ with $\alpha$ and $\beta$ both set to $0.5$. It is clear that the agent's overall performance is incrementally improved by changing the language encoder from the simple $L_{enc}$ to our proposed $ViL_{enc}$, which proves our analysis that previous language encoder does not well capture the semantics of high-level instructions. The experimental results of $3$ and $6$ clearly suggests that the BERT-based structure is not the root cause of our performance gain and our proposed Scene Grounding task significantly increase the RG SPL metric to $33.9\%$ on Val Seen and $7.31\%$ on Val Unseen, even higher than the strong baseline in experiment $2$.
\textbf{Component Effectiveness.} To answer question $(b)$, based on the statistics from Table~\ref{tab:ablation}, we train six models in experiments from $3$ to $8$ and ablate the proposed component one by one to demonstrate the effectiveness. For fair comparison, we follow the settings of ~\cite{qi2020reverie}. All agents are trained under $\mathcal{L}_{ce}$ and $\mathcal{L}_{pm}$ with $\alpha$ and $\beta$ both set to $0.5$. We start from the baseline experiment $3$ and replace each component by our proposed ones. Specifically, in experiments $3$ and $6$, the proposed $ViLEncoder$ improves the RG SPL (and SPL) by a large margin, $11.98\%$ (and $21.52\%$) higher in Val Seen and $2.47\%$ (and $3.77\%$) higher in Val Unseen than the baseline respectively, which proves that the Scene Grounding task is effective; in experiments $3$ and $4$, our pointer $ViLPointer$ outperforms the MattNet counterpart by shortening the length of the search trajectory while maintaining a high RG SPL (and SPL), which demonstrates the effectiveness of the Object Grounding task; in experiments $3$ and $5$, the results show that the overall search trajectory of our action policy is longer than that of the baseline while our action policy achieves higher RGS and Navigation Success Rate, which demonstrates that the memory structure in our policy guides the agent to the correct target location at the cost of long trajectory; in experiments $7$ and $8$, we demonstrate that by integrating all our proposed methods, our agent improves previous SOTA in terms of RG SPL by $10.76\%$ on Val Seen and $5.6\%$ on Val Unseen.
\textbf{Memory Blocks.} To answer question $(c)$, we train five models with different $N_{mem}$ and $N_{state}$ values. In these experiments, we train the agents with $\mathcal{L}_{ce}$, $\mathcal{L}_{pm}$ and $\mathcal{L}_{RL}$ and $\alpha$, $\beta$ and $\gamma$ set to $1.0$. In general, according to the experiments from $9$ to $13$ in Table~\ref{tab:ablation}, all pairs of $(N_{mem}, N_{state})$ exhibit superior performance compared to previous SOTA method in experiment $1$ and the strong BERT baseline model in experiment $2$. Moreover, the best performance model is achieved by setting $(N_{mem}, N_{state})$ to $(3, 3)$ in these five models, which suggests that using small values of $(N_{mem}, N_{state})$ limits the agent's memorization ability and using large values of $(N_{mem}, N_{state})$ enables the agent to achieve good performance on Val Unseen while maintains good performance on Val Seen.
\begin{table}[t]\small
\centering
\caption{Pointer Task: REVERIE Success Rate at the ground truth target viewpoint; Encoder Task: given ground truth path, the success rate of identifying the target viewpoint among a set of candidate viewpoints along the path.}
\resizebox{\linewidth}{!}{
\begin{tabular}{c|ccc}
\hline
Tasks & Methods & Val Seen & Val Unseen \\
\hline
\multirow{4}{*}{Pointer} & MattNet~\cite{yu2018mattnet} & 68.45 & 56.63 \\
& CM-Erase~\cite{liu2019improving} & 65.21 & 54.02 \\
& ViLPointer-image-based & 65.72 & 55.53 \\
& ViLPointer-vp-based & \underline{\textbf{73.26}} & \underline{\textbf{67.45}} \\
\hline
Encoder & ViLEncoder & \underline{\textbf{85.67}} & \underline{\textbf{66.43}} \\
\hline
\end{tabular}}
\label{tab:refer_and_encoder}
\end{table}
\textbf{Logit Fusion.} To answer question $(d)$, we report two accuracies to verify the effectiveness of $g_{og}$ and $g_{sg}$. In the Encoder Task of Table~\ref{tab:refer_and_encoder}, given ground-truth path, our proposed $ViLBERT$ model achieves competitive performance on both Val Seen and Val Unseen, demonstrating the strong ability of $g_{sg}$ to identify a target viewpoint. In the Pointer Task of Table~\ref{tab:refer_and_encoder}, the performance of $ViLPointer$-vp-based is significantly higher than previous image-based pointers because it is able to capture cross-image objects relationships, suggesting that $g_{og}$ has the ability to find the target location if the target object exists. According to experiments from $14$ to $17$, where the agents are trained with $\mathcal{L}_{ce}$, $\mathcal{L}_{pm}$ and $\mathcal{L}_{RL}$ and $\alpha$, $\beta$ and $\gamma$ set to $1.0$, summing $\boldsymbol{l}_{\tau}$, $g_{og}^{\tau}$, and $g_{sg}^{\tau}$ shortens the search trajectory and maintains a high RGS(Navigation Success Rate) and RG SPL(SPL). The motivation behind the summing strategy is to use model ensemble to reduce bias when searching for target locations considering the fact that the agent has no prior knowledge of the surrounding environments and the guidance of the high-level instructions is weak.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{./figure4.pdf}
\caption{Percentage of successful Navigation and RGS cases under different length of ground-truth paths on Val Seen and Val Unseen datasets for \textcolor{oceanboatblue}{previous state-of-the-art method}, \textcolor{orange}{BERT baseline in experiment $2$}, and \textcolor{ForestGreen}{our method}.}
\label{Fig:stats}
\end{figure}
\subsection{Compared to previous state-of-the-art results}
\noindent We first show what kind of cases our method improves compared to previous SOTA and our BERT-based strong baseline in experiment $2$. Specifically, we divide the shortest distance lengths of all ground-truth paths into three groups, namely short path($5$ meters to $9$ meters with $462$ sample paths on Val Seen and $1400$ sample paths on Val Unseen), middle path($9$ meters to $14$ meters with $703$ sample paths on Val Seen and $1869$ sample paths on Val Unseen), and long path($14$ meters to $18$ meters with $247$ sample paths on Val Seen and $250$ sample paths on Val Unseen). Then, we count the cases that the agent successfully navigates to the target locations and the cases that the agent successfully navigates and localizes the target objects for the three groups. In Fig.~\ref{Fig:stats}, we report the corresponding successful cases percentage. It is obvious that our proposed method improves all kinds of sample paths by a clear margin.
Then, we compare our final model with previous SOTA models in Table~\ref{tab:sota}. As is clearly shown in Table~\ref{tab:sota}, our model outperforms all previous models by a large margin. Specifically, in terms of SPL, our agent increases previous SOTA by $11.58\%$ on Val Seen, $9.09\%$ on Val Unseen and $3.24\%$ on Test respectively; for RG SPL, our agent increase previous SOTA by $12.99\%$ on Val Seen, $6.89\%$ on Val Unseen and $3.12\%$ on Test. The overall improvements indicate that our proposed scene-intuitive agent not only navigates better but also localizes target objects more accurately.
\section{Conclusion}
\noindent In this paper, we present a scene-intuitive agent capable of understanding high-level instructions for the REVERIE task. Different from previous works, we propose two pre-training tasks, Scene Grounding task and Object Grounding task respectively, to help the agent learn \textbf{\emph{where}} to navigate and \textbf{\emph{what}} object to localize simultaneously. Moreover, the agent is trained with a Memory-augmented action decoder that fuses grounded textual representation and visual representation with memory augmented current state representation to generate action sequence. We extensively verify the effectiveness of our proposed components and experimental results demonstrate that our result outperforms previous methods significantly. Nevertheless, how to bridge the performance gap between seen and unseen environments and how to shorten the navigation length efficiently remains an open problem for further investigation.
{\small
\bibliographystyle{ieee_fullname}
|
2,877,628,090,186 | arxiv | \section{Introduction}
Recent developments on integrability of ${\cal N}=4$ SYM opened the way to impressive results in the context of scattering amplitudes. In particular, using a new powerful duality symmetry in \textit{momentum space}, the complete calculation of all tree level amplitudes as well as up to fourth order loops in perturbation theory were accomplished \cite{Bern}. Complemented by results from strong coupling promoted by the AdS/CFT conjecture, gave rise to a new duality between Wilson loops and scattering amplitudes that drastically simplifies the calculation of the corresponding amplitudes.
In this context, the authors of \cite{Caron} raised the interesting question about the possibility of constructing a consistent relativistic quantum field theory that preserves the analogous of the Runge-Lenz (RL) vector of the Kepler potential in classical mechanics. The surprising answer to this question is that it is indeed possible and that theory is ${\cal N}=4$ SYM processed by a Higgs mechanism that gives mass to some scalars in the field content of that theory. In turn, this interesting result was used to study the relativistic spectrum of the non-BPS two-body bound state of a higgsed ${\cal N}=4$ SYM theory in the limit when one particle mass, the ``proton'' mass, goes to infinity. It is possible to address this problem in the context of large $N$ limit of SYM using integrability techniques and dual conformal symmetry. To find the spectrum we can use a duality relation between the anomalous dimension of the CFT (Wilson loops with a cusp) and the angular momentum, plus a quantization condition \cite{Caron}. This interesting relation was also used to explore the meson spectrum and to compare it with the corresponding spectrum obtained using string theory in the context of AdS/CFT \cite{mateos}.
The results presented in \cite{Caron} were worked out using numerical methods. Analytic results that confirmed the numerical approach were obtained in \cite{Espindola}. The same two body bound-state spectrum can also be addressed using relativistic quantum mechanics with an enhanced symmetry \cite{Sakata}.
Our aim in this letter is to construct explicitly the analogous of the RL vector in the relativistic setup and analyze some of its consequences. We present the infinitesimal relativistic transformation generated by this new RL vector and compare the results with the approach followed in \cite{Caron}. Later on, we will also argue that this relativistic RL vector can be used, as in classical mechanics, to make explicit the hidden SO(4) symmetry of the hydrogen atom and to calculate the corresponding relativistic spectrum for this \textit{relativistic hydrogen-like atom}. Our results for this spectrum confirm previous analyses using different approaches \cite{Sakata,Qiang}. Next, we construct a new relativistic Kustaanheimo-Stiefel (KS) duality between different central potentials. Using this duality transformation we show that the relativistic spectrum of the hydrogen-like atom in 3 dimensions is related to the relativistic harmonic oscillator spectrum in 4 dimensions. Our results again confirm previous analyses where the spectrum of the relativistic harmonic oscillator has been constructed from scratch \cite{Qiang2}.
At first sight, the existence of a relativistic RL vector is a nonsense. It is well known that the orbits of a relativistic particle minimally coupled to a Kepler scalar central potential are open rosettes, and, consequently, the associated symmetry of the classical non-relativistic problem is broken. The relativistic non-degenerated problem possesses less symmetry. So, relativistic effects break the symmetry algebra generated by the angular momentum and the RL vector. That means that the RL vector is not a conserved quantity in the relativistic realm. Nevertheless it is still possible to restore the SO(4) symmetry in the relativistic context if we allow for a non-minimal coupling of the particle with the scalar field. This observation is central to our work. This non-minimal coupling is widely used in the context of a modified Dirac equation with enhanced symmetry algebras in the theory of nuclear spectrum \cite{report}.
We restrict ourselves to stationary problems, so we choose a time direction in Minkowski spacetime. Additionally, we select a Lorentz frame where the electromagnetic vector potential has the form $A^\mu=(V(r),0,0,0)$, breaking the manifest Lorentz symmetry. Here $r$ is just a spatial coordinate, so the symmetry is reduced from SO(3,1) to SO(3). Then, using the non-minimal coupling, we will enhance our symmetry to SO(4). It is clear that we can not have SO(3,1) and SO(4) in the same description, then our approach keeps only SO(4). Of course, a good question is if we can have SO(4,1) or, even better, the complete conformal symmetry SO(4,2) by adding more degrees of freedom. Diverse proposals around these problems have been presented in recent literature (see for example \cite{gilmore}).
\section{Reduction from ${\cal N}=4$ Super Yang-Mills to a non-minimally coupled scalar field}
To describe our model we present explicitly the procedure to reduce ${\cal N}=4$ SYM \cite{SYN4, Beisert} to a non-minimally coupled scalar field theory \cite{Sakata}. Our interest here is to explicitly show the emergence of a scalar field theory with SO(4) symmetry from ${\cal N}=4$ SYM \cite{Brink} and check the consistency conditions that follow from the equations of motion. This scalar field theory was previously considered in \cite{Sakata}. To undertake this task we consider only the bosonic sector of ${\cal N}=4$ SYM which has the following Lagrangian density
\begin{equation}\label{SYM-Lag}
{\cal L}=\mathrm{Tr}\left\{ -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\sum_{i=1}^{6}D_{\mu}\Phi_{i}D^{\mu}\Phi_{i}+\frac{g^{2}}{4}\sum_{i,j=1}^{6}[\Phi_{i},\Phi_{j}]^{2}\right\} .
\end{equation}
Here the six scalar fields are $N\times N$ traceless hermitian matrices
in the adjoint representation of SU(N). The action of the covariant
derivative on a generic field $W$ is given by
\begin{equation}
D_{\mu}W=\partial_{\mu}W-ig[A_{\mu},W],
\end{equation}
where under a gauge transformation $U$ the matrix gauge field $A_{\mu}$
and the scalar fields $\Phi_{i}$ transform as
\begin{equation}
\Phi_{i}\rightarrow U\Phi_{i}U^{\dagger},\,\,\,\,\,\,\,\,\,\,A_{\mu}\rightarrow UA_{\mu}U^{\dagger}-\frac{i}{g}(\partial_{\mu}U)U^{\dagger}.
\end{equation}
The resulting equations of motion are \cite{Beisert}
\begin{equation}
D_{\mu}F^{\nu\mu}=ig\sum_{i=1}^{6}[\Phi_{i},D^{\nu}\Phi_{i}],\label{eq:eqfmunu}
\end{equation}
\begin{equation}
D_{\mu}D^{\mu}\Phi_{i}=g^{2}\sum_{j=1}^{6}[\Phi_{j,}[\Phi_{j},\Phi_{i}]].\label{eq:eqscalar}
\end{equation}
For simplicity, we choose to work with the group SU(2), following \cite{Schabinger}. Therefore, the fields will be expressible in terms of the Pauli matrices $\tau^a$ as
\begin{equation}\label{eq:decomp}
\Phi_{i}=\Phi_{i}^{a}\frac{\tau^{a}}{2}=\frac{1}{2}\begin{pmatrix}\Phi_{i}^{0} & \Phi_{i}^{-}\\
\Phi_{i}^{+} & -\Phi_{i}^{0}
\end{pmatrix},\,\,\,\,\,\,\,A_{\mu}=A_{\mu}^{a}\frac{\tau^{a}}{2}=\frac{1}{2}\begin{pmatrix}A_{\mu}^{3} & A_{\mu}^{1}-iA_{\mu}^{2}\\
A_{\mu}^{1}+iA_{\mu}^{2} & -A_{\mu}^{3}
\end{pmatrix},
\end{equation}
where $\Phi_{i}^{\pm}=\Phi_{i}^{1}\pm i\Phi_{i}^{2}$.
We now introduce the spontaneous symmetry breaking by giving a vacuum
expectation value $v$ to $\Phi_{1}$ \cite{Sakata,Schabinger,Alday}
\begin{equation}\label{vevforPhi1}
\Phi_{1}=\frac{1}{2}\begin{pmatrix}\Phi_{1}^{0}+v & 0\\
0 & -\Phi_{1}^{0}-v
\end{pmatrix},
\end{equation}
and taking the other fields as
\begin{equation}
\Phi_{2}=\frac{1}{2}\begin{pmatrix}0 & \Phi_{2}^{-}\\
\Phi_{2}^{\text{+}} & 0
\end{pmatrix},\,\,\,\,\,\Phi_{i}=0,\,\,i=3,4,5,6;\,\,\,\,\,\,\,A_{\mu}=A_{\mu}^{a}\frac{\tau^{a}}{2}=\frac{1}{2}\begin{pmatrix}A_{\mu}^{3} & 0\\
0 & -A_{\mu}^{3}
\end{pmatrix}.
\end{equation}
We switched off the charged components $A^1_\mu,A^2_\mu$ of the connection field $A_\mu$.
Now the Lagrangian (\ref{SYM-Lag}) reads
\begin{equation}\nonumber
{\cal L}= -\frac{1}{8}F^{\mu\nu}F_{\mu\nu}
-\frac{1}{4}\partial_{\mu}\Phi_{2}^-\partial^{\mu}\Phi_{2}^+
-\frac{1}{4}\partial_{\mu}\Phi_{1}^0\partial^{\mu}\Phi_{1}^0
+\frac{ig}{4}A^\mu(\Phi_{2^-}\partial_{\mu}\Phi_{2}^+ - \Phi_{2^+}\partial_{\mu}\Phi_{2}^-)
\end{equation}
\begin{equation}\label{SYM-Lag-higgs}
-\frac{g^2}{2}\Phi_{2^-}\Phi_{2}^+ A_\mu A^\mu
-\frac{g^2}{4}(\Phi_1^0+v)^2\Phi_{2^-}\Phi_{2}^+.
\end{equation}
Here $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ and we have identified $A_\mu^3=A_\mu$. A crucial step here is to implement a constraint
\begin{equation}\label{constraint}
\Phi^0_1+\alpha/r=0,
\end{equation}
into the Lagrangian (\ref{SYM-Lag-higgs}). Bellow we will see why this contraint is crucial in our construction. After the strong implementation of this contraint into the Lagrangian (\ref{SYM-Lag-higgs}) we obtain\footnote{This constraint can be interpreted as a second class constraint in the Dirac sense.}
\begin{equation}\nonumber
{\cal L}= -\frac{1}{8}F^{\mu\nu}F_{\mu\nu}
-\frac{1}{4}\partial_{\mu}\Phi_{2}^-\partial^{\mu}\Phi_{2}^+
+\frac{ig}{4}A^\mu(\Phi_{2^-}\partial_{\mu}\Phi_{2}^+ - \Phi_{2^+}\partial_{\mu}\Phi_{2}^-)
\end{equation}
\begin{equation}\label{SYM-Lag-red}
-\frac{g^2}{2}\Phi_{2^-}\Phi_{2}^+ A_\mu A^\mu
-\frac{g^2}{4}(-\frac{\alpha}{r}+v)^2\Phi_{2^-}\Phi_{2}^+
\end{equation}
up to boundary term. The field content of our theory is reduced to one vector field $A_\mu$ and a complex scalar $\Phi_2$.
The equation of motion for the scalar field $\Phi_2^-$ is
\begin{equation}\label{1111}
\partial_{\mu}\partial^{\mu}\Phi_{2}^{-}-ig(\partial_{\mu}A^{\mu})\Phi_{2}^{-}-2igA_{\mu}\partial^{\mu}\Phi_{2}^{-}-g^{2}A_{\mu}A^{\mu}\Phi_{2}^{-}-g^{2}(v-\frac{\alpha}{r})^{2}\Phi_{2}^{-}=0 \, .
\end{equation}
Of course an analogous equation of motion can be obtained for the complex conjugate $\Phi_2^+$ that we will not need here. Denoting $\Phi^-_2=\phi$ and taking the Coulomb potential $A^{\mu}=(-\alpha/r,\boldsymbol{0})$ with $g=1$,\footnote{Which is equivalent to absorbing the coupling constant into $\alpha$.} the equation (\ref{1111}) becomes
\begin{equation}
\partial_{\mu}\partial^{\mu}\phi-2iA^{\mu}\partial_{\mu}\phi-A^{\mu}A_{\mu}\phi-\left(m-\frac{\alpha}{r}\right)^{2}\phi=0,\label{eq:kgmodrel}
\end{equation}
where the vacuum expectation value of $\Phi_{1}$ is the mass $m$ of the scalar field $\phi$. The scalar field equation (\ref{eq:kgmodrel}) that comes from ${\cal N}=4$ SYM scalar couplings and the Higgs mechanism with the constraint (\ref{constraint}) just presented will be central to our work. Among the crucial interesting properties of (\ref{eq:kgmodrel}) is that the non-minimal coupling $m\to (m-{\alpha}/{r})$ induced by the Higgs mechanism and the constraint $\Phi^0_1=-\alpha/r$ cancels out the quadratic term coming from the minimal coupling (introduced through the covariant derivative), enhancing the symmetry of the resulting field theory from SO(3) to SO(4).
The Lagrangian (\ref{SYM-Lag-red}) can be rewritten in terms of the fields $\phi,\phi^*, A_\mu$ as
\begin{equation} \label{FAphi-lagrangian}
{\cal L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}(D_{\mu}\phi)^{\ast}(D^{\mu}\phi)-\frac{1}{2}\left(m-\frac{\alpha}{r}\right)^{2}\phi^{\ast}\phi,
\end{equation}
and recognize this Lagrangian as the scalar electrodynamics with modified mass. Here the covariant derivative is $D_{\mu}=\partial_{\mu}-iA_{\mu}$. The equations of motion for $A^{\nu}$ are
\begin{equation}\label{eq:eqfora}
-\partial_{\mu}\partial^{\mu}A^{\nu}+\partial^{\nu}(\partial\cdot A)=\frac{i}{2}(\phi\partial^{\nu}\phi^{\ast}-\phi^{\ast}\partial^{\nu}\phi)-A^{\nu}|\phi|^{2},
\end{equation}
The application of the divergence to (\ref{eq:eqfora}) reveals that the conserved current is
\begin{equation}
J^{\nu}=\frac{i}{2}(\phi\partial^{\nu}\phi^{*}-\phi^{*}\partial^{\nu}\phi)-A^{\nu}|\phi|^{2}.
\end{equation}
This means that the density $\rho$ is
\begin{equation}
\rho=\frac{i}{2}(\phi^{*}\partial_{t}\phi-\phi\partial_{t}\phi^{*})+\frac{\alpha}{r}|\phi|^{2},
\end{equation}
taking into account that $A^\mu=(-\frac{\alpha}{r},\bf 0)$. From the other hand, the implementation of this Coulomb potential in the equation of motion (\ref{eq:eqfora}) gives
\begin{equation}
\nabla^{2}\left(\frac{\alpha}{r}\right)=\frac{i}{2}(\phi^{*}\partial_{t}\phi-\phi\partial_{t}\phi^{*})+\frac{\alpha}{r}|\phi|^{2},
\end{equation}
which implies
\begin{equation}
\rho=\nabla^{2}\left(\frac{\alpha}{r}\right)
\end{equation}
as expected, or
\begin{equation}
\int\rho\mathrm{d}^{3}r=-4\pi\alpha.
\end{equation}
In this way we have checked that our prescriptions are entirely consistent. Even though the mass of the associated scalar electrodynamics is replaced by the non minimal coupling $m\to (m-{\alpha}/{r})$ the associated charge of the scalar field is the expected one.
\section{Relativistic particle and Runge-Lenz vector}\label{sec:relpart}
In this section we will show that the field equation (\ref{eq:kgmodrel}) has an enhanced symmetry SO(4). For that end we start from the construction of the relativistic particle of mass $m$ implementing the standard minimal coupling to a scalar potential followed by a non-minimal prescription suggested by equation (\ref{eq:kgmodrel}). Then we will construct the analogous of the RL vector for this particle. This relativistic RL vector is the generator of the enhanced SO(4) symmetry.
The standard formulation of a relativistic particle minimally coupled to a background electromagnetic field $A_\mu$ starts from the action
\begin{equation}\label{eq:actbefevt}
S=-mc\int d\tau \sqrt{-\eta_{\mu \nu} \frac{d x^\mu}{d\tau}\frac{d x^\nu}{d\tau}} + \frac{1}{c} \int d\tau A_\mu \frac{dx^\mu}{d\tau},
\end{equation}
where $\eta_{\mu \nu} =\text{diag} \left(-,+,+,+ \right)$ and $\alpha$ is the coupling constant.
The canonical momenta
\begin{equation}
p_\mu =\frac{\partial L}{\partial \dot{x}^\mu} = \frac{mc \ \eta_{\mu\nu}\dot{x}^\nu}{\sqrt{-\eta_{\alpha \beta} \frac{d x^\alpha}{d\tau}\frac{d x^\beta}{d\tau}}} +\frac{1}{c} A_\mu,
\end{equation}
imply the quadratic constraint (associated to reparametrization invariance)
\begin{equation} \label{eq:constasac}
\eta^{\mu \nu}\left( p_\mu - \frac{1}{c} A_\mu \right) \left( p_\nu - \frac{1}{c} A_\nu \right) + m^2 c^2=0,
\end{equation}
that leads naturally to the Klein-Gordon (KG) equation upon the identification $p_\mu\to -i\hbar\partial_\mu$
\begin{equation}\label{actrp1}
\left[ \eta^{\mu \nu } \left( -i\hbar \partial_\mu - \frac{1}{c} A_\mu \right) \left(-i\hbar \partial_\nu - \frac{1}{c} A_\nu \right) + m^2c^2 \right]\phi =0.
\end{equation}
In the particular case of a Coulomb field $A_0= \alpha/r, A_i=0$ and working in units such that $\hbar=1, c=1$, the equation above reads
\begin{equation}
\left[ \left( -i\partial_t - \frac{\alpha}{r}\right)^2 + \nabla^2 - m^2 \right]\phi(x) =0.
\end{equation}
The corresponding stationary spectrum with energy $E_{\ell,n}$ that comes from this equation is the usual relativistic spectrum of the KG equation with the external Coulomb potential \cite{Sakata,Jackiw}
$$ E_{\ell,n}=m\left[1+\left(\frac{\alpha}{n+\sqrt{(\ell+1/2)^2-\alpha^2}-(\ell+1/2)}\right)\right]^{-1/2},$$
where we can see that the well known degeneracy of the associated non-relativistic spectrum is broken. Apparently, in the relativistic case, we can not construct the conserved RL vector associated with the above degeneracy of the spectrum. A crucial observation is that the SO(4) symmetry (of the non-relativistic spectrum) is broken by the quadratic term $\alpha^2/r^2$ that comes from the minimal coupling to the Coulomb potential in the KG equation (\ref{actrp1}).
The {\em classical} stationary problem with conserved angular momentum $\ell$ and energy $E=-p_0$ is
$$(E-V)^2-(p_r^2+\ell^2/r^2)-E_0^2=0,$$
where $E_0\equiv m$. The solutions for bounded orbits are rosettes \cite{Landau} in contrast with the non-relativistic problem where the orbits are ellipses \cite{Goldstein}. As a consequence, the corresponding RL vector is not conserved in the relativistic theory described by the Lagrangian (\ref{eq:actbefevt}).
To see that it is possible to retain the degeneracy of the non-relativistic spectrum in the relativistic case we will follow our motivation from (\ref{eq:kgmodrel}), introducing a non-minimal coupling given by the substitution
\begin{equation}\label{masst}
m \longrightarrow m-\frac{\alpha}{r}.
\end{equation}
This new coupling restores the degeneracy of the associated non-relativistic dynamics and as a consequence a {\em new} relativistic symmetry emerges. The action (\ref{eq:actbefevt}) is now
\begin{equation}\label{act12}
S=\int d \tau \left[ -m \sqrt{- \left( 1-\frac{\alpha}{rm}\right)^2 \eta_{\mu \nu} \dot{x}^\nu \dot{x}^\mu} + \frac{\alpha}{r} \dot{x}^0\right].
\end{equation}
This action corresponds to a relativistic particle interacting with a Coulomb background in a curved space given by a conformally flat metric
$$\eta_{\mu\nu}\to \left( 1-\frac{\alpha}{rm}\right)^2 \eta_{\mu \nu}.\label{conf-metric}$$
The associated constraint (\ref{eq:constasac}) is now
\begin{equation} \label{eq:constpo}
p_0^2- \frac{2 \alpha}{r} p_0 - \vec{p}^2 - m^2 + \frac{2\alpha m}{r}=0.
\end{equation}
Considering the {\em classical} stationary problem, using the non-minimal replacement (\ref{masst}) to remove the anomaly, and denoting $A_0=-V$ we have
$$E_0^2\to (E_0+V)^2$$
and the modified KG equation reads
$$E^2-E_0^2-2V(E_0+E)- (p_r^2+\ell^2/r^2)=0$$
or
$$(E+E_0)\left(E-E_0-2V-\frac{1}{(E+E_0)}( p_r^2+\ell^2/r^2)\right)=0,$$
where $\ell$ is the conserved angular momentum. An equivalent way to write this result is
$$(E-E_0)-2V-\frac{1}{(E+E_0)}( p_r^2+\ell^2/r^2)=0,$$
where we observe that the relativistic problem is reduced to a Schr\"odinger like problem. If we want to obtain the associated non-relativistic (NR) problem just replace\footnote{$E=\frac{m}{\sqrt{1-v^2}}$, then in the NR limit $E=E_0+mv^2/2$, or $E-E_0\to E_{NR}$. Also, from $E^2-E_0^2=p^2$ and then $(E+E_0)(E-E_0)=p^2$, we have $E_{NR}=p^2 /(E+E_0)$ and consequently $1/(E+E_0)\to \frac{1}{2m}$.}
$$(E-E_0)\to E_{NR}$$
and
$$\frac{1}{(E+E_0)}\to \frac{1}{2m}.$$
It is also true that starting from the non-relativistic problem we can obtain the relativistic one by reading the same replacements from right to left.
The hydrogen like spectrum that arises from this modified KG equation (\ref{eq:kgmodrel}) recovers the full degeneracy of the non-relativistic case \cite{Sakata}
$$E_n=m\left(1-\frac{2\alpha^2}{n^2+\alpha^2}\right).$$
As a consequence, it is now possible to construct a relativistic analogue of the RL vector.
Using as a model the non-relativistic construction \cite{Goldstein} we find
\begin{equation}\label{eq:lrlgen}
\frac{d}{d\tau} \left[ \vec{p}\times\vec{L}+{\alpha} \left( p_0-m\right) \frac{\vec{x}}{r}\right]=0,
\end{equation}
where $\vec L$ is the angular momentum that generates the corresponding SO(3) algebra. The relativistic generalization of the Runge-Lenz vector is then
\begin{equation}
\vec{K}= \vec{p}\times\vec{L}-\left( m-{p_0}\right)\frac{\alpha\vec{x}}{r}.
\end{equation}
or
\begin{equation}
\vec{K}= \vec{p}\times\vec{L}-\left( E_0+E\right)\frac{\alpha\vec{x}}{r}\label{RRL}.
\end{equation}
So we recover the conservation of the RL vector and the orbits are closed ellipses. The vector $\vec{K}$ enhances the symmetry from SO(3) to SO(4), and in this way we have shown that the non-minimal coupling induced by (\ref{masst}) or the transformation to a conformally flat space in (\ref{conf-metric}) allows us to recover the SO(4) symmetry in the relativistic case.
Now, taking the non-relativistic limit of the RL vector we obtain
\begin{equation}
\vec{K}_{NR} = \vec{p} \times \vec{L} - 2\alpha m \frac{\vec{x}}{r},
\end{equation}
which is the usual RL vector with a coupling constant that is twice larger.\\
{\em {\bf Remarks:} \\
{\bf a)} A curious feature about the non-minimal coupling (\ref{masst}) is that the NR effective potential has a factor of 2 as compared with the usual NR formulation.\\
{\bf b)} To every NR (relativistic) observable we can associate a relativistic (NR) observable with the replacement $2m \leftrightarrow E+E_0$, $\,\, E_{NR}\leftrightarrow E-E_0$.\\
{\bf c)} Notice that the substitution (\ref{masst}) has a critical point when $r_c\equiv \frac{\alpha}{m}$. At this point the mass term in the Lagrangian (\ref{FAphi-lagrangian}) is zero. The kinetic term of the particle Lagrangian (\ref{act12}) is also zero because the conformal factor of the Minkowski metric tends to zero. This can be contrasted with the behaviour of a relativistic particle in the limit $m\to 0$. Hence, starting with the equivalent description of the Lagrangian (\ref{act12}) in terms of the einbein $e$
\begin{equation}
S=\int d\tau \left( \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - \frac{e}{2}\left(m-\frac{\alpha}{r} \right)^2 + \frac{\alpha}{r} \dot{x}^0\right),
\end{equation}
we can see that in the limit $r\to r_c$ the only term that remains is
\begin{equation}
S=\int d\tau \left( \frac{\dot{x}_\mu \dot{x}^\mu}{2e}+ m \dot{x}^0 \right),
\end{equation}
since the last term is just a total derivative, reflecting the fact that $p_0$ is defined up to a constant shift $p_0\to m-\dot x^0/e$. This theory is consistent with (\ref{eq:kgmodrel}) but with $A^\mu=(-m,\bf 0)$, corresponding to a KG equation with zero mass term minimally coupled to a constant potential. \\
{\bf d)} A central argument given here is that the unusual coupling (\ref{masst}) comes from the scalar sector of ${\cal N}=4$ SYM theory with an appropriately adjusted Higgs mechanism as presented in Section 2. This construction is based on a powerful conformal field theory which has very interesting integrability properties. Due to the hidden symmetry that lies under the replacement (\ref{masst}) that reveals the existence of the relativistic RL vector (\ref{RRL}), the ${\cal N}=4$ SYM theory is sometimes dubbed as the hydrogen atom quantum field theory \cite{bruser-caron-henn}.}
\section{The relativistic SO(4) algebra using the relativistic RL vector}
The aim of this section is twofold. On one hand we will construct the infinitesimal Noether symmetries generated by the relativistic RL vector and observe that this symmetry is not an SO(4) rotation but it is neither a conformal symmetry. A deeper analysis is needed to compare the symmetry obtained here with the symmetry in the dual momentum space presented in \cite{Caron}. As a spinoff we show that the relativistic orbit can be reconstructed using the relativistic RL vector.
On the other hand we will describe the complete SO(4) algebra generated by the angular momentum and the new relativistic RL vector. We will restrict ourselves to the stationary problem with relativistic energy ${E}$. In a second step we will recover from this algebra the correct relativistic spectrum of the corresponding hydrogen-like atom in the two body bound state.
From equation (\ref{eq:constpo}) and solving for $p_0$
\begin{equation}
p_0=-\frac{\alpha}{r} \pm \sqrt{\frac{\alpha^2}{r^2} + \vec{p}^2 + m^2-\frac{2m\alpha}{r}},
\end{equation}
we construct the energy function or equivalently the associated relativistic Hamiltonian
\begin{equation}
H= - p_0=E.
\end{equation}
In this form, we confirm that $p_0$ is a constant of motion and in consequence invariant under the transformations generated by the Runge-Lenz vector,
\begin{equation}
\delta p_0 = \left\lbrace p_0 , \epsilon_i K^i \right\rbrace =0,
\end{equation}
The infinitesimal transformations associated with $r^i$ and the canonical momenta $p_i$ are (setting again $c=1$)
$$\delta r^i=\left\lbrace r^i, \epsilon_j K^j \right\rbrace=2(\epsilon\cdot r)p_i-r^i (\epsilon\cdot p) -(r\cdot p)\epsilon^i ,
$$
$$
\delta p_i = \left\lbrace p_i, \epsilon_j K^j \right\rbrace = - (\vec{p})^2 \epsilon_i +(\vec{p} \cdot \vec{\epsilon})p_i + \alpha\left({p_0} -m \right) \frac{\vec{\epsilon} \cdot \vec{x}}{r^3} x_i- \alpha\left({p_0} -m \right)\frac{\epsilon_i}{r},
$$
or
$$ \delta p_i= - (\vec{p})^2 \epsilon_i +(\vec{p} \cdot \vec{\epsilon})p_i - \alpha\left(E +E_0 \right) (\frac{\vec{\epsilon} \cdot \vec{x}}{r^3} x_i- \frac{\epsilon_i}{r}).
$$
The Runge-Lenz vector also acts on the magnitud $r$ as
\begin{equation}
\delta r = \left\lbrace \sqrt{x_l x_l}, \epsilon_i K^i \right\rbrace = \frac{\left( \vec{p}\cdot \vec{x} \right) \left(\vec{x}\cdot \vec{\epsilon} \right)}{r} - (\vec{\epsilon} \cdot \vec{p}) r,
\end{equation}
These infinitesimal symmetries do not correspond exactly with the symmetry transformations previously written in \cite{Caron}. The reason for the mismatch and its possible consequences is an open problem that we will leave for future work. A crucial difference is that in the approach given in \cite{Caron} the symmetry transformation acts in a dual momentum space (dual conformal transformation) that is appropriate to reveal the symmetries of scattering amplitudes in SYM theory.
We observe that the analogous procedure to obtain the classical non relativistic orbit of the Kepler problem can be implemented also in the relativistic case.
If we take the dot product of the RL vector with $\vec{x}$, we obtain
\begin{equation}
\vec{K} \cdot \vec{x}=Kr \cos\theta = \left(\vec{p}\times\vec{L}\right)\cdot \vec{x}-\alpha \left( E_0+E\right) \frac{\vec{x}\cdot \vec{x}}{r},
\end{equation}
or
\begin{equation}
\frac{L^2}{r \left(E_0+E \right)\alpha} = \frac{K \cos \theta}{\alpha \left( E_0+E\right)}+1,
\end{equation}
that is, the equation of a conic with eccentricity
\begin{equation}
\varepsilon=\frac{K}{\alpha \left( E_0+E\right)}.
\end{equation}
Here, we notice that under the substitution of $(E_0+E)\to 2m $ we recover the NR result of \cite{Goldstein}.
The complete spectrum can also be constructed from the relativistic SO(4) algebra generalizing the NR result as presented in \cite{gilmore, Paddy}. Let us start from a simple redefinition of the relativistic RL vector (\ref{RRL}),
$$ {\vec A}=\frac{2}{E+m}\vec K=\frac{1}{{E}+m} (\vec p\times \vec L-\vec L\times \vec p)-2\alpha \frac{\vec r}{r},$$
for a bounded orbit with energy $E$. We know from our previous calculation that
$$[\vec A,H]=0,\qquad\vec L\cdot \vec A=\vec A\cdot\vec L,$$
and
\begin{equation}\label{EqK2}
A^2=4\left(\alpha^2+\frac{{E}-m}{{E+m}}(1+L^2)\right).\end{equation}
The corresponding relativistic algebra closes as
$$[L_i,L_j]=i\varepsilon_{ijk}L_k,$$
$$[A_i,L_j]=i\varepsilon_{ijk}A_k,$$
$$[A_i,A_j]=-4i\left(\frac{E-m}{E+m}\right)\varepsilon_{ijk}L_k,$$
Defining
$${\vec A}'=\sqrt{-\frac{E+m}{4(E-m)}}\vec A$$
and
$$\vec N=\frac12(\vec L+{\vec A}'), \qquad \vec M=\frac12(\vec L-{\vec A}')$$
it is easy to show that the original algebra splits into the product of two SO(3) algebras,
$$[N_i,N_j]=i\varepsilon_{ijk}N_k,\qquad [M_i,M_j]=i\varepsilon_{ijk}M_k,$$
with the constraint
\begin{equation}N^2=M^2\label{NM-C}.\end{equation}
The operator
$$N^2+M^2$$
will have the eigenvalues $2\ell(\ell+1)$ with $\ell=0,1,2\ldots$ (2 times the square of the angular momentum eigenvalues) because of the constraint (\ref{NM-C}). On the other hand
$$\frac12(N^2+M^2)=\frac12 [L^2-\frac{E+m}{4(E-m)} A^2]=-\frac{E+m}{2(E-m)}\alpha^2-\frac12 ,$$
where we used (\ref{EqK2}). So, we obtain for the relativistic spectrum
\begin{equation}\label{jap-spectra}
n^2=-\frac{E+m}{E-m}\alpha^2 .
\end{equation}
This is the correct relativistic spectrum reported in \cite{Sakata} and reproduced here with $n=2\ell+1$.
\section{Relativistic KS duality}
The KS dictionary can be constructed from the NR case (see for example \cite{CG}). Starting from a potential of the form $V=k r^\beta$ and introducing a change of variable $r\to R^{\frac{2}{\beta+2}}$, the integral for the orbit \cite{Goldstein} for the variable $R$ has the same form as the original orbit integral for $r$ if we define
$$
{\cal V}=-(E-E_0)R^{-\frac{2\beta}{\beta+2}}$$
as the new potential, and the new energy by
$${\cal E}-{\cal E}_0=-2k.$$
A crucial difference from the NR case is the factor 2 in the new energy definition. Also the new angle of the orbit in the relation $r(\theta)\to R(\Theta) $ must be rescaled by a factor: $\Theta=((\beta+2)/2)\theta$.
\st{Just}As a consistency condition we also need the identification
$$E+E_0\to {\cal E}+{\cal E}_0$$
that is not present in the NR limit because the original NR KS transformation by construction, maps problems with the same mass. We note that $E-E_0$ and $E+E_0$ play very different roles in the relativistic KS mapping. While $E-E_0$ is a coupling constant in the new problem, $E+E_0$ plays the role of the mass parameter. Of course, we are restricting the mapping in such a way that the rest energies are equal (because the mass of the new and old problems is the same as in the NR KS transformation).
The new problem is then
$$({\cal E}-{\cal E}_0)-2{\cal V}(R)-\frac{1}{({\cal E}+{\cal E}_0)}( p_R^2+\ell^2/R^2)=0$$
A simple comparison with the original problem
$$(E-E_0)-2V-\frac{1}{(E+E_0)}( p_r^2+\ell^2/r^2)=0$$
reveal that the structures of the old and the new problems are exactly the same. The crucial difference between this equation and the original one worked out in \cite{Landau} is the non-minimal coupling (\ref{masst}).
\section{KS and the relativistic equivalent Schr\"odinger equation (RSE)}
Accordingly with the previous section, the radial part of the relativistic Schr\"odinger equation (RSE) in $d$ dimensions is
\begin{equation}\label{S1}
\left(-\frac{1}{(E+m)}\left(\frac{d^2}{dr^2}+\frac{(d-1)}{r}\frac{d}{dr}-\frac{\ell(\ell+d-2)}{r^2}\right) +2V(r)-(E-m)\right)R(r)=0.
\end{equation}
$V(r)$ is a central potential defined as $V(r)=k r^\beta$.
By the change of variable
\begin{equation}
r= \rho^{2/(\beta+2)},
\end{equation}
the Schr\"odinger equation becomes
\begin{equation}
\Bigg(-\frac{1}{(E+m)}\left(\frac{d^2}{d\rho^2}+\frac{(\beta+2(d-1))}{(\beta+2)\rho}\frac{d}{d\rho}-\frac{\ell(\ell+d-2)}{\rho^2}\left(\frac{2}{\beta+2}\right)^2\right)
\end{equation}
$$-\left(\frac{2}{\beta+2}\right)^2\rho^{\frac{-2\beta}{\beta+2}}(E-m)+2\left(\frac{2}{\beta+2}\right)^2 K\Bigg)\tilde R(\rho)=0,
$$
where $\tilde R(\rho)=R(r(\rho))$. Now use the following dictionary (duality) to associate the relevant quantities that define the RSE, $E,K,d,\ell$ to a new set of quantities that parametrize the new system ${\cal E}, \cal{V},D,{\cal L}$.
The dimension\footnote{The case $d=2$ (conformal point) deserves special attention. See \cite{hoj} for details.} maps to a new dimension $D$
\begin{equation}\label{D}
D=\frac{2(\beta+d)}{\beta+2}.
\end{equation}
The energy ${\cal E}$ of the new stationary RSE is related with the coupling constant of the old potential $V=Kr^\beta$ by
\begin{equation}
{\cal E}-m= -2(\frac{2}{\beta+2})^2 K.
\end{equation}
In the same way the new angular momentum ${\cal L}$\footnote{Here we are restricting ourselves to the case of integer new dimension $D$ and integer new momenta ${\cal L}$.} is related with the old angular momentum $\ell$ by
\begin{equation}
{\cal L}= \frac{2}{\beta+2} \ell .
\end{equation}
Finally the new potential ${\cal V}$ is related with the old energy $E$ by
\begin{equation}\label{V}
{\cal V}= - \frac12 (E-m) \left(\frac{2}{\beta+2}\right)^2\rho^{-\frac{2\beta}{\beta+2}}.
\end{equation}
A crucial observation is the identification
\begin{equation}\label{Id}
E+m\to {\cal E}+m,
\end{equation}
that follow as a consistency condition.
Using this dictionary the new RSE
\begin{equation}
\left(-\frac{1}{{\cal E}+m}\left(\frac{d^2}{d\rho^2}+\frac{(D-1)}{\rho}\frac{d}{d\rho}-\frac{\cal{L}(\cal{L}+D-2)}{\rho^2}\right) +2{\cal V}(\rho)-({\cal E}-m)\right)\tilde R(\rho)=0
\end{equation}
acquires the {\em same} form as the original one but with dimension $D$, angular momentum ${\cal L}$, energy ${\cal E}$ and potential ${\cal V}(\rho)$ given by (\ref{D}-\ref{V}) and the identification (\ref{Id}).
Notice that we can not map {\em every} solution of the old stationary problem into a stationary solution of the new problem by this duality. We can only map every {\em bounded} stationary solution of the old problem into a bounded stationary solution of the new problem.
We will use this dictionary for the case of the hydrogen atom $\beta=-1$ in $d=3$. In that case, the new problem is the isotropic harmonic oscillator in $D=4$ and angular momenta ${\cal L}=2\ell$.
The energy spectrum of the relativistic hydrogen atom is \cite{Sakata, Qiang}
$$ (2n+2\ell +2)\sqrt{m-E}-2\alpha\sqrt{E+m}=0,$$
and from here we have
$$E=m(1-\frac{2\alpha^2}{N^2+\alpha^2}),$$
with $N=n+\ell+1$.
According to our dictionary the new potential is
$${\cal V}=-2(E-m) \rho^2,$$
so the new coupling constant of the corresponding oscillator is $k=2(E-m)$.
On the other hand, the energy spectrum of the relativistic oscillator in $D=4$ {arising from the dictionary reads}
$${\cal E}-m=2(4n+2{\cal L}+4)\sqrt{\frac{E-m}{{\cal E}+m}}.$$
This is the energy spectrum of a $D=4$ relativistic oscillator with potential ${\cal V}$. It can be compared with the result given in \cite{Qiang2}
$${\cal E}-m=2(4n+2{\cal L}+4)\sqrt{\frac{k}{2({\cal E}+m)}}, $$
for a particle of mass $m$ in a potential $V=kr^2$.
This matches precisely with our result. So we conclude that the relativistic KS transformation relates the relativistic harmonic oscillator in 4 dimensions with the relativistic Coulomb potential in 3 dimensions. This example shows the power of the relativistic KS transformation constructed here to relate different potentials.
\section{Relativistic spectrum from cusp anomalous dimension}
An interesting calculation of the non-relativistic spectrum of the hydrogen atom from perturbation theory in ${\cal N}=4$ SYM was presented in \cite{Caron}.
The starting point is SYM and the relativistic symmetry associated with the RL vector. Nevertheless, the explicit computation of the binding energy spectrum of the bound state results in the {\em non-relativistic} well-known formula for the hydrogen-like spectrum. Here we will extend the result presented in \cite{Caron} to the full relativistic case.
Our starting point is an enhanced formula for the energy of the bound state in terms of the cusp angle. Taking into account only the binding energy we define
$$(E_n^b-m)= (E_n^b+m) (\sin\frac{\phi_n}{2}-1),$$
where $E_n^b$ is the relativistic binding energy of the bound state and $\phi_n$ the corresponding cusp (scattering) angle. The quantization condition is
$$\Gamma_{cusp}(\phi_n)=-n,$$
where $\Gamma_{cusp}$ is the cusp anomalous dimension and $n$ and integer.
From the other hand, $\Gamma_{cusp}$ has been computed for weak 't Hooft coupling $\lambda<<1$ and the result is
$$\Gamma_{cusp}({\phi_n})=-\frac{\lambda}{8\pi^2}{\phi_n}\tan \frac{\phi_n}{2}.$$
Since $\lambda$ is small the scattering is small $\phi\approx \pi-\delta$, with
$$\delta\approx \frac{\lambda}{4\pi n}.$$
The solution for the binding energy is
\begin{equation}\label{Eb}
(E_n^b-m)=\frac{\delta^2}{8}(E_n^b+m)=\frac{\lambda^2}{128\pi^2n^2}(E_n^b+m).
\end{equation}
The full relativistic spectrum is
$$E-2m=E^b,$$
where we have subtracted the threshold energy $2m$ and $E_b$ is given by (\ref{Eb}). This result matches the computed relativistic spectrum given in \cite{Sakata} and reproduced here using different approaches. Notice that the dependence in $\lambda^2$ is consistent with the expectation that $\sqrt\lambda\sim e$ \cite{Correa2,Fiol} and the fact that the hydrogen energy spectrum goes like $e^4$.
We remark that the computation presented in \cite{Caron} is entirely consistent. In the weak coupling (small $\lambda$) and large angular momentum ($n>>1$) the approximation to leading order for the tiny effect $E-2m$ is the NR spectrum.
We have presented here the perturbative (still weak coupling) relativistic spectrum for the total binding energy. This result could be confirmed by the $E-J$ Chew-Frautschi plot for different values of $\lambda$ in the weak coupling case ($J>>1, E\sim 2m$).
As a final comment notice the interesting relation between the stereographic projection (small circles on the sphere to large circles on the plane) related by the RL symmetry (as presented for example in \cite{gilmore}), and the duality between static quarks on $S^3\times R$ and dynamical quarks in the plane \cite{Correa, Caron1} as compared with the classical duality between the free particle in $S^3$ with the Kepler problem in $R^4$ \cite{gilmore}. This observation needs further analysis that we will not address here.
\section{Acknowledgment}
This work was supported in part by DGAPA-PAPIIT grant IN103716, CONACyT project 237503 and scholarship 419420 (J.A.J.).
JAG was partially supported by Mexico National Council of Science and Technology (CONACyT) grant 238734 and DGAPA-UNAM grant IN107115.\\
|
2,877,628,090,187 | arxiv | \section{Introduction}
The two-dimensional ($n+1$)-periodic Toda lattice with opposite sign is the system
\begin{eqnarray}
\left\{
\begin{array}{l}2 (w_i)_{z\bar z} =-e^{2 (w_{i+1}-w_i)}+e^{2 (w_i-w_{i-1})}\\
w_{i+n+1}=w_i
\end{array}
\right. ,
\label{Toda-N}
\end{eqnarray}
where $\bar z$ denotes the complex conjugate of $z \in \mathbb{C}$ and $w_i=w_i(z,\bar z) \in \mathbb{R}$.
System (\ref{Toda-N}) admits both the $l$-anti-symmetry constraints
\begin{eqnarray}
\left\{
\begin{array}{lll}
w_0+w_{l-1}=0, &w_1+w_{l-2}=0, &\cdots \\
w_l+w_n=0, &w_{l+1}+w_{n-1}=0, &\cdots
\end{array}
\right. ,
\label{l-anti}
\end{eqnarray}
where the fixed $l \in \{0,1,\cdots, n\}$,
and the radial constraints
\begin{eqnarray}
w_i(z,\bar z)=w_i(|z|), \quad i \in \{0,1,\cdots, n\}.
\label{Radical}
\end{eqnarray}
The tt* (topological–anti-topological fusion) equation is (\ref{Toda-N}) constrained by
(\ref{l-anti}) and (\ref{Radical}).
The special case of $l=0$ was introduced by Cecotti and Vafa when they deformed the superpotentials
to study the fusion of topological $N=2$ supersymmetric quantum field theory with its conjugate,
the anti-topological one \cite{CV-1}.
It also appeared in the extraction of exact results for supersymmetric $\sigma$ models \cite{CV-2}
and in the classification of $N=2$ supersymmetric theories \cite{CV-3}.
Dubrovin gave the zero-curvature representation of tt* equations
and also studied their geometry aspects \cite{Dub}.
All concrete examples of the tt* equations were reduced to the third Painlev\'{e} equation
before the work of Guest and Lin \cite{GL-1},
where they initiated the direct study of differential system with two unknowns
\begin{eqnarray}
\left\{
\begin{array}{l}
u_{z \bar z}=e^{a u}-e^{v-u} \\
v_{z \bar z}=e^{v-u}-e^{-b v}\\
\end{array}
\right. ,
\label{TT}
\end{eqnarray}
where $a,b>0$,
subject to the boundary condition
\begin{eqnarray}
\left\{\begin{array}{ll}
u(z)\xlongrightarrow{ |z| \rightarrow \infty} 0, & v(z)\xlongrightarrow{ |z| \rightarrow \infty} 0 \\
u(z) \xlongrightarrow{ z \rightarrow 0} (\gamma+o(1)) \log |z|, &v(z) \xlongrightarrow{z \rightarrow 0} (\delta+o(1)) \ln |z|
\end{array} \right. .
\label{BC}
\end{eqnarray}
The tt* equations with two dependent variables are the cases $a,b \in \{1,2 \}$, exhausted in \cite{GL-1}.
So, \cite{GL-1} studied a generalized version of them.
In \cite{GIL-1}, Guest, Its and Lin proved the following comprehensive property for Equation (\ref{TT}) with boundary condition (\ref{BC}).
\begin{theorem} \label{thm-GIL-1}
\textup{\cite{GIL-1} }\quad
For $a,b>0$ and any $(\gamma,\delta)$ in the triangular region
$$\gamma \ge -\frac{2}{a}, \quad \delta \le \frac{2}{b}, \quad \gamma-\delta \le 2, $$
the system (\ref{TT}) has a unique solution that satisfies the boundary condition (\ref{BC}).
Further, the unique solution is radially-invariant, i.e., it depends only on $|z|$.
\end{theorem}
\input{Figure1.tikz}
Therefore, any point in the triangular region represents a smooth solution of the (generalized) tt* equation
in $\mathbb{C}^*=\mathbb{C} \backslash \{0\}$.
So, Theorem \ref{thm-GIL-1} characterizes a two-parameter family of smooth solutions for the tt* equation
in $\mathbb{C}^*$.
Note that a similar result to Theorem \ref{thm-GIL-1} had been obtained by Guest and Lin in \cite{GL-1},
where they demanded $\gamma, \delta>0$.
But the difference is crucial since Theorem \ref{thm-GIL-1} characterize
{\sl all} smooth radial solution of Equation (\ref{TT}) \cite{GIL-2}.
The work of \cite{GL-1} also triggered some other researches, such as \cite{Mochi-1} and \cite{Mochi-2}.
By the Riemann-Hilbert approach, Guest et al. obtained all connection formulae for the tt* equation, i.e., $a,b \in \{1, 2\}$ \cite{GIL-2}.
The complete picture of the monodromy data, holomorphic data, and asymptotic data
were eventually obtained in \cite{GIL-3}.
In \cite{GIL-3}, some fine structures (more detailed asymptotics) of the solutions of tt* equation near $z=0$ were revealed.
However, the fine structures are complicated, lacking of intuitional explanation
or other relevant results that could directly support them.
The first aim of this paper is to verify them numerically up to $100$ digits in all cases:
the general case, the edge case, and the vertex case.
To be more specific,
we will only consider the tt* equation with $a=b=2$, which is the case 4a studied in detail in \cite{GIL-1, GIL-2, GIL-3}.
In this case, $u=2 w_0$, $v=2 w_1$. So (\ref{TT}) becomes
\begin{eqnarray}
\left\{
\begin{array}{l}
2 (w_0)_{z \bar z}=e^{4 w_0}-e^{2 w_1-2 w_0} \\
2 (w_1)_{z \bar z}=e^{2 w_1-2w_0}-e^{-4 w_1}
\end{array}
\right.
. \label{TT-1}
\end{eqnarray}
According to the radical constraint (\ref{Radical}),
system (\ref{TT-1}) is written as an ordinary differential equations (ODEs) with variable $r=|z|$
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{2} w_0''+\frac{1}{2 r} w_0' =e^{4 w_0}-e^{2 w_1-2 w_0} \\
\frac{1}{2} w_1''+\frac{1}{2 r} w_1' =e^{2 w_1-2w_0}-e^{-4 w_1}
\end{array}
\right.
, \label{TT-2}
\end{eqnarray}
where the prime denotes $\frac{d}{dr}$.
Near $r=0$, by (\ref{BC}), $w_0$ and $w_1$ have properties
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r) \xlongrightarrow{r \rightarrow 0} (\gamma_0 +o(1) ) \ln r\\
2 w_1(r) \xlongrightarrow{r \rightarrow 0} (\gamma_1+o(1) ) \ln r
\end{array}
\right. . \label{OAsymp}
\end{eqnarray}
Near $r=\infty$, the asymptotics of $w_0$ and $w_1$ are expressed
by the Stokes data $s_1^\mathbb{R}$ and $s_2^\mathbb{R}$ \cite{GIL-2}:
\begin{eqnarray}
\left\{
\begin{array}{l}
w_0(r)+w_1(r)\xlongrightarrow{r \rightarrow \infty}-s_1^\mathbb{R} 2^{-\frac{3}{4}} \left( \pi r\right)^{-\frac{1}{2}} e^{-2 \sqrt{2}r}\\
w_0(r)-w_1(r)\xlongrightarrow{r \rightarrow \infty}s_2^\mathbb{R} 2^{-\frac{3}{2}} \left( \pi r\right)^{-\frac{1}{2}} e^{-4 r}
\end{array}
\right. .\label{InfAsymp}
\end{eqnarray}
The map from $(\gamma_0, \gamma_1)$ to $(s_1^\mathbb{R}, s_2^\mathbb{R})$
is the connection formula \cite{GIL-2}
\begin{eqnarray}
\left\{
\begin{array}{l}
s_1^\mathbb{R}=-2 \cos \left( \frac{\pi}{4} (\gamma_0+1) \right)-2 \cos \left( \frac{\pi}{4}(\gamma_1+3) \right) \\
s_2^\mathbb{R}=-2 -4 \cos \left( \frac{\pi}{4} (\gamma_0+1) \right) \cos \left( \frac{\pi}{4}(\gamma_1+3) \right)
\end{array}
\right. . \label{ConnectFormula}
\end{eqnarray}
Let us take the general case, which refers to a solution represented by a inner point of the triangular region in Figure \ref{fig-1},
as an example to explain the fine structure near $r=0$.
In this case, the fine structure states that
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r) \xlongrightarrow{r \rightarrow 0} \gamma_0 \ln r +\rho_0 \\
2 w_1(r) \xlongrightarrow{r \rightarrow 0} \gamma_1 \ln r+\rho_1
\end{array}
\right. , \label{OAsymp-Fancy}
\end{eqnarray}
where
\begin{eqnarray}
\left\{
\begin{array}{l}
\rho_0= -\ln \left( 2^{2 \gamma_0}
\frac{\Gamma(\frac{1+\gamma_0}{4}) \Gamma(\frac{4+\gamma_0+\gamma_1}{8}) \Gamma(\frac{6+\gamma_0-\gamma_1}{8})}
{ \Gamma(\frac{3-\gamma_0}{4})\Gamma(\frac{4-\gamma_0-\gamma_1}{8}) \Gamma(\frac{2-\gamma_0+\gamma_1}{8})} \right) \\
\rho_1=-\ln \left( 2^{2 \gamma_1}
\frac{\Gamma(\frac{3+\gamma_1}{4}) \Gamma(\frac{4+\gamma_0+\gamma_1}{8}) \Gamma(\frac{2-\gamma_0+\gamma_1}{8})}
{ \Gamma(\frac{1-\gamma_1}{4})\Gamma(\frac{4-\gamma_0-\gamma_1}{8}) \Gamma(\frac{6+\gamma_0-\gamma_1}{8})} \right)
\end{array}
\right. . \label{rhos-DEF}
\end{eqnarray}
We will see (\ref{OAsymp-Fancy}) is appropriate to verify numerically.
Also, the fine structures in the edge case and the vertex case can be managed to verify numerically.
The $r=\infty$ asymptotics (\ref{InfAsymp}) is able to
fix the solution of the tt* equation (\ref{TT-2}) uniquely.
This is an initial value problem from $r=\infty$.
However, the rough asymptotics (\ref{OAsymp}) itself is not enough to fix the solution.
To fix the solution, it has to be accompanied with the rough $r=\infty$ asymptotics
$w_0(r) \xlongrightarrow{r \rightarrow \infty}0$ and
$w_1(r) \xlongrightarrow{r \rightarrow \infty} 0$.
But it is a boundary value problem.
To get an initial value problem from $r=0$,
it is necessary and enough to begin with the fine structure (\ref{OAsymp-Fancy}).
Then, $r=\infty$ and $r=0$ become symmetrical.
This explains why the fine structure is important.
The connection formula (\ref{ConnectFormula}) maps the $(\gamma_0, \gamma_1)$ region
to the $(s_1^\mathbb{R}, s_2^\mathbb{R})$ region.
Coming down to Equation (\ref{TT-2}), the region map can be represented by Figure \ref{fig-2}.
\input{Figure2.tikz}
Any solution represented by a point $(s_1^\mathbb{R}, s_2^\mathbb{R})$
in the curved triangle (including the edges and the vertexes) in Figure \ref{fig-2}
must have asymptotic (\ref{OAsymp-Fancy}) near $r=0$,
where $(\gamma_0, \gamma_1)$ is determined from $(s_1^\mathbb{R}, s_2^\mathbb{R})$ by the connection formula (\ref{ConnectFormula}).
One would wonder what happens if the point $(s_1^\mathbb{R}, s_2^\mathbb{R})$ lies out of the curved triangle.
Based on our numerical results,
we will give a conjecture to generalize the range and explanation of the connection formula and the fine structures.
If $\rho_0$ and $\rho_1$ are exactly given by (\ref{rhos-DEF}),
then $w_0(r)\xlongrightarrow{r \rightarrow \infty} 0$ and $w_1(r)\xlongrightarrow{r \rightarrow \infty} 0$.
In \cite{GIL-2}, they had shown that the smooth solutions in $\mathbb{C}^*$ are all of such type.
So, if $\rho_0$ and $\rho_1$ differ from (\ref{rhos-DEF}), the solutions must have singularities.
We would give a more detailed picture for these singular solutions.
One may also wonder about the solutions that associated with the points $(\gamma_0, \gamma_1)$ that lies out of the triangle.
But this is, in fact, not a question since the triangle is optimal \cite{GL-1},
i.e. there is no such solution that has asymptotics (\ref{OAsymp}) with $(\gamma_0, \gamma_1)$ out of the triangle.
This can also be explained by noticing that (\ref{tu0tu1}) has no solution of order $o(s)$.
The paper is organized as follows.
In Section 2, we verify numerically the fine structures given in \cite{GIL-3}.
In Section 3, we consider the case that $(s_1^\mathbb{R}, s_2^\mathbb{R})$ lies out of the curved triangle.
In Section 4, a more detailed picture is given for the case that $\rho_0$ and $\rho_1$ differ from (\ref{rhos-DEF}).
This paper may be viewed as a supplement of work \cite{GIL-1,GIL-2,GIL-3}.
Acknowledgement.
Part of this work was done while Y. Li was visiting the Department of Mathematical Sciences of IUPUI.
Y. Li would like to thank A. Its for his hospitality and the suggestion of verifying their Corollary 8.3 in \cite{GIL-3}.
The work is partly supported by NSFC(11375090) and
Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).
\section{Verifying the fine structures \label{sec-verify}}
The fine structures are different for the general case, the edge cases and the vertex cases.
Both the edge cases and the vertex cases include three subcases: E1, E2, E3 for the edge case and V1, V2, V3 for the vertex case.
The tt* equation (\ref{TT-2}) has symmetry
\[ \left\{ \begin{array}{l}w_0\rightarrow -w_1 \\w_1\rightarrow -w_0 \end{array} \right. , \]
i.e, if $(w_0(r), w_1(r))=(f(r), g(r))$ is a solution of the tt* equation,
then $(w_0(r), w_1(r))=(-g(r), -f(r))$ is also a solution of the tt*.
Therefore, if the solution $(w_0(r), w_1(r))=(f(r), g(r))$ has data
$(\gamma_0,\gamma_1)=(\mu_0,\mu_1)$ and $(s_1^\mathbb{R}, s_2^\mathbb{R})=(\nu_1,\nu_2)$,
then the solution $(w_0(r), w_1(r))=(-g(r), -f(r))$ will have data
$(\gamma_0,\gamma_1)=(-\mu_0,-\mu_1)$ and $(s_1^\mathbb{R}, s_2^\mathbb{R})=(-\nu_1,\nu_2)$ by (\ref{OAsymp}) and (\ref{InfAsymp}).
In this way, the formulae for the E2 case and the V3 case can be obtained from the E1 case and V1 case respectively.
Further, as has been mentioned in \cite{GIL-3}, the V2 case is just the sinh-Gordon,
for which the asymptotic is already well known.
So, we will only verify four cases: the general, E1, E3 and V1.
For convenient, we will also list the formulae for the E2 case, the V2 case and the V3 case.
We will find that each fine structure is associated with a special truncation of (\ref{TT-2}).
This kind of truncation will keep working in Section \ref{CONJ}.
\subsection{Preliminary for the numerical experiments: an approximation proper for calculations near $r=\infty$}
Considering the solutions of (\ref{TT-2}) with asymptotics
$w_0(r)\xlongrightarrow{r \rightarrow \infty} 0$ and $w_1(r)\xlongrightarrow{r \rightarrow \infty} 0$.
The primary asymptotics of the solutions can be obtained by the linearization of (\ref{TT-2}):
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_0^L=6 w_0^L -2 w_1^L \\
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_1^L=6 w_1^L -2 w_0^L
\end{array}
\right. . \nonumber
\end{eqnarray}
Then, $(w_0^L,w_1^L)$ will be a good approximation of $(w_0^L,w_1^L)$ near $r=\infty$.
Define
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p^L=w_0^L+w_1^L\\
w_m^p=w_0^L-w_1^L
\end{array}
\right. .\label{wpwL-DEF}
\end{eqnarray}
Then, $w_p^L$ and $w_m^L$ satisfy
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_p^L=4 w_p^L\\
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_m^L=8 w_m^L
\end{array}
\right. . \label{wpwm-L}
\end{eqnarray}
Considering $w_p^L$ and $w_m^L$ should vanish at $r=\infty$, the solution of (\ref{wpwm-L}) is
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p^L=c_p K_0(2 \sqrt{2} r)\\
w_m^L=c_m K_0(4 r)
\end{array}
\right. . \label{vpvml-sol}
\end{eqnarray}
Comparing (\ref{vpvml-sol}) with (\ref{InfAsymp}), we immediately obtain
\begin{eqnarray}
\left\{ \begin{array}{l}
c_p=-\frac{\sqrt{2}}{\pi} s_1^\mathbb{R} \\
c_m=\frac{1}{\pi} s_2^\mathbb{R}
\end{array}
\right. . \label{cpcm-sol}
\end{eqnarray}
Corresponding to (\ref{wpwL-DEF}), let us define
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p=w_0+w_1\\
w_m=w_0-w_1
\end{array}
\right. . \label{wpm-DEF}
\end{eqnarray}
Then, the equations for $w_p$ and $w_m$ are
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_p=e^{2 w_p+2 w_m}-e^{2 w_p-2 w_m}
=2 e^{2 w_m} \sinh \left( 2 w_p \right) \\
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_m=e^{2 w_p+2 w_m}+e^{2 w_m-2w_p}-2 e^{-2 w_m}
=4 e^{2 w_m} \sinh^2(w_p)+4 \sinh (2 w_m)
\end{array}
\right. . \label{wpwm}
\end{eqnarray}
Note that (\ref{wpwm}) is written to a form better for keeping the high precision of the numerical integration.
Also note Equations (\ref{wpwm-L}) are the linearization of (\ref{wpwm}).
The errors of approximating $(w_p, w_m)$ by $(w_p^L, w_m^L)$ are caused by the nonlinear terms in the expansion of (\ref{wpwm}).
In general, the most significant correction to $w_p$ is proportional to $ w_p^L w_m^L$, i.e.,
$w_p =c_p K_0(2 \sqrt{2} r)+O(r^{-1} e^{-(2 \sqrt{2}+4)r})$.
Meanwhile, the most significant correction to $v_m$ is proportional to the square of $w_p^L$, i.e.,
$ w_m =c_m K_0(4 r)+O(r^{-1} e^{-4 \sqrt{2} r})$.
These results are sufficient for the rough numerical investigations for smooth solutions of the tt* equation.
It is called rough since they can be refined.
In the high-precision numerical integration of (\ref{wpwm}), the relative error will not enlarge too much when $r$ is still large.
For $w_m(r)$, the relative error is about $O(r^{-\frac{1}{2}} e^{-4 (\sqrt{2}-1) r})$.
If we give the initial values by $w_p(r)=w_p^L(r)$ and $w_m(r)=w_m^L(r)$ with $r=45$,
the relative error of the initial values are of order $10^{-33}$, which is not so satisfying.
If we want to reach a relative error of order $10^{-100}$ by this way,
$r=138$ is needed to give the initial values.
We will see, after considering the most significant contribution of the nonlinear terms,
the starting $r$ can be greatly reduced.
Suppose
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p(r)=w_p^{(0)}(r)+w_p^{(1)}(r)+w_p^{(2)}(r)+\cdots\\
w_m(r)=w_m^{(0)}(r)+w_m^{(1)}(r)+w_m^{(2)}(r)+\cdots
\end{array}
\right. , \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p^{(0)}(r)=w_p^L(r)= -\frac{\sqrt{2}}{\pi} s_1^\mathbb{R} K_0(2 \sqrt{2} r)\\
w_m^{(0)}(r)=w_m^L(r)=\frac{1}{\pi} s_2^\mathbb{R} K_0(4 r)
\end{array}
\right. . \nonumber
\end{eqnarray}
Then $w_p^{(1)}$ and $w_m^{(1)}$ satisfy
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_p^{(1)}-4 w_p^{(1)}=8 w_p^{(0)} w_m^{(0)}\\
\left( \frac{1}{2} \frac{d^2}{dr^2}+\frac{1}{2 r} \frac{d}{d r} \right) w_m^{(1)}-8 w_m^{(1)}
=4 \left(w_p^{(0)} \right)^2
\end{array}
\right. \nonumber
\end{eqnarray}
with $w_p^{(1)}(\infty)=0$ and $w_m^{(1)}(\infty)=0$.
The solution of $w_p^{(1)}$ and $w_m^{(1)}$ is:
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p^{(1)}=2 I_0(2 \sqrt{2} r) \int_\infty^r K_0(2 \sqrt{2}r) \bigg(8 w_p^{(0)}(r) w_m^{(0)}(r) \bigg) r dr
-2 K_0(2 \sqrt{2} r) \int_\infty^r I_0(2 \sqrt{r}) \bigg(8 w_p^{(0)}(r) w_m^{(0)}(r) \bigg) r dr\\
w_m^{(1)}=2 I_0(4 r) \int_\infty^r K_0(4 r) \bigg(4 ( w_p^{(0)}(r) )^2 \bigg) r dr
-2 K_0(4 r) \int_\infty^r I_0(4 r) \bigg( 4 ( w_p^{(0)}(r) )^2 \bigg) r dr
\end{array}
\right. . \nonumber
\end{eqnarray}
Then
\begin{eqnarray}
\left\{ \begin{array}{l}
w_p(r)=w_p^{(0)}(r)+w_p^{(1)}(r)+O\left(r^{-\frac{3}{2}} e^{-6 \sqrt{2} r} \right)\\
w_m(r)=w_m^{(0)}(r)+w_m^{(1)}(r)+O\left(r^{-\frac{3}{2}} e^{-(4+ 4\sqrt{2}) r} \right)
\end{array}
\right. . \label{iniApprox}
\end{eqnarray}
The relative errors are both of order $ r^{-1} e^{-4 \sqrt{2} r}$.
To acquire a relative error of order $10^{-100}$, it should be enough to start the numerical integration from $r=45$.
Higher order nonlinear terms should not be considered,
else we will meet high-dimensional integration that is difficult to calculate to our precision goal.
The truncation of (\ref{iniApprox}) will be used to given initial values for the numerical integration of (\ref{wpwm})
both in this section and in Section \ref{CONJ}.
\subsection{The general case: in the triangular \label{GeneralCase}}
This subsection is devoted to verify (\ref{OAsymp-Fancy}).
To be specific, we will fix $(\gamma_0, \gamma_1)=(1, \frac{1}{3} )$.
Then, $(s_1^\mathbb{R}, s_2^\mathbb{R})=(\sqrt{3},-2 )$ by (\ref{ConnectFormula}).
(\ref{iniApprox}) means that we can start our numerical integration from $r=45$
for moderate $(s_1^\mathbb{R},s_2^\mathbb{R})$ to guarantee our precision goal $10^{-100}$.
Also, we will find it is convenient to do transform
\begin{eqnarray}
s=\ln r \label{s-DEF}
\end{eqnarray}
near $r=0$.
So, the numerical integration is divided into two parts:
on $r \in [r_m, 45]$ and on $s \in [s_f, s_m=\ln r_m]$.
For convenience, $r_m$ is always chosen as $r_m=1$.
$s_f$ vary with $(s_1^\mathbb{R},s_2^\mathbb{R})$
and will be given after the determination of the associated truncation of (\ref{TT-2}).
\subsubsection{Numerical integration from $r=45$ to $r=1$}
By the truncation of (\ref{iniApprox}), the initial values for the numerical integration of (\ref{wpwm})
are calculated up to more than $100$ digits
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(45)=-4.5763465910740842210810671823515633075572030760030... \times 10^{-57} \\
w_p'(45)=1.2994612025622450236510718743064448909150132699101... \times 10^{-56}\\
w_m(45)=-3.9902150828859022626192436154419670328254784177405... \times 10^{-80}\\
w_m'(45)=1.6005134816454403480052616718328017176197600655449... \times 10^{-79}
\end{array}
\right. .\label{ini-General}
\end{eqnarray}
To save space, we only list the first $50$ digits in (\ref{ini-General}).
Immediately, one notice there is not so much need to list the first few digits of the initial values:
$w_m(45)$ in (\ref{ini-General}) coincides $w_m^L(45)=-\frac{\sqrt{6}}{\pi} K_0(90 \sqrt{2})$ with $33$ digits
while $w_p(45)$ in (\ref{ini-General}) coincides with $w_p^L(45)=-\frac{2}{\pi} K_0(180) $ with all the listed $50$ digits.
Formula (\ref{iniApprox}) only provides the order of error, not the actual.
We obtain the errors of (\ref{ini-General})
by comparing the initial values (\ref{ini-General}) with a more accurate numerical solution starting from $r=55$.
Table \ref{tab1} shows both the absolute error and the relative error of the initial values at $r=45$.
\begin{table}[ht]
\caption{Errors of the initial values for the general case with $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$.}
\label{tab1}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=45$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$1.98 \times 10^{-170}$ &$1.68 \times 10^{-169}$ &$2.43 \times 10^{-193}$&$2.36\times 10^{-192}$ \\
| Relative Error | &$4.32\times 10^{-114}$ &$1.30 \times 10^{-113}$ &$6.09 \times 10^{-114}$&$1.47\times 10^{-113}$
\end{tabular}
\end{table}
In this paper, we use the Gauss-Legendre method,
which is an implicit Runge-Kutta method suitable for high-precision numerical integration,
to integrate ODEs numerically.
After integrating (\ref{wpwm}) numerically from $r=45$ to $r=1$
by a $100$-stage Gauss-Legendre method with step size $\frac{1}{100}$,
we obtain the numerical values of $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$:
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(1)=-3.2972969594742103001480456261339460432792854660454...\times 10^{-2}\\
w_p'(1)=1.0829838290019404254859616425541702465151021916881...\times 10^{-1}\\
w_m(1)=-6.6648017026562016812805168052539563362254856278250...\times 10^{-3}\\
w_m'(1)=2.8961723214345113722967491163879906375020596216242...\times 10^{-2}
\end{array}
\right. . \label{values-general-1}
\end{eqnarray}
Note that (\ref{values-general-1}) lists only the first $50$ digits of the numerical solution.
Also note the precision of the numerical method itself is of order
$\text{stepsize}^{2 \times \text{stages}}=10^{-400}$, which is far more accurate than our precision goal.
Comparing (\ref{values-general-1}) with the more accurate solution starting from $r=55$,
we obtain the errors of (\ref{values-general-1}) as Table \ref{tab2}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $r=1$ for the general case with $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$.}
\label{tab2}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=1$ & $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$2.85 \times 10^{-115}$ &$9.31 \times 10^{-115}$ &$6.64 \times 10^{-116}$&$2.82\times 10^{-115}$ \\
| Relative Error | &$8.63\times 10^{-114}$ &$8.60 \times 10^{-114}$ &$9.97 \times 10^{-114}$&$9.75\times 10^{-114}$
\end{tabular}
\end{table}
\subsubsection{Near $r=0$}
Inspired by the form of (\ref{OAsymp-Fancy}), we use independent variable $s$
and dependent variables
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0 =2 w_0-\gamma_0 s \\
\tilde w_1 =2 w_1-\gamma_1 s
\end{array}
\right. . \label{transform-general}
\end{eqnarray}
Please recall that $s=\ln (r)$ is defined by (\ref{s-DEF}).
From the numeric point of view, the advantage of using $s$ rather than $r$
is that it can avoid the frequent adjusting of the step size when we compute numerically near $r=0$.
The equations for $\tilde w_0$ and $\tilde w_1$ are
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0}{ds^2}= e^{2 \tilde w_0 +2 (\gamma_0+1) s}-e^{\tilde w_1- \tilde w_0+(\gamma_1-\gamma_0+2)s} \\
\frac{1}{4} \frac{ d^2\tilde w_1}{ds^2}=e^{\tilde w_1- \tilde w_0+(\gamma_1-\gamma_0+2)s}-e^{-2 \tilde w_1+2 (1-\gamma_1) s}
\end{array}
\right. .\label{tu0tu1}
\end{eqnarray}
We expect $\tilde w_0 \xlongrightarrow{s \rightarrow -\infty} \rho_0$
and $\tilde w_1 \xlongrightarrow{s \rightarrow -\infty} \rho_1$.
In the triangular: $\gamma_0>-1$, $\gamma_1<1$, $\gamma_1>\gamma_0-2$.
So, all terms in the right of (\ref{tu0tu1}) can be neglected at first.
Thus,
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0^{(0)}}{ds^2}= 0 \\
\frac{1}{4} \frac{ d^2\tilde w_1^{(0)}}{ds^2}=0
\end{array}
\right. \label{tu0tu1-truc}
\end{eqnarray}
is the primary approximation of (\ref{tu0tu1}), which is equivalent to (\ref{TT-2}).
So (\ref{tu0tu1-truc}) can be viewed as the truncation structure of the tt* equation in the general case.
The initial values of $\tilde w_0$, $\frac{ d\tilde w_0}{ds}$, $\tilde w_1$ and $\frac{ d\tilde w_1}{ds}$ at $s=0$
can be inferred from $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0|_{s=0}= w_p|_{r=1}+w_m|_{r=1} \\
\frac{ d \tilde w_0}{ds}|_{s=0}= w_p'|_{r=1}+w_m'|_{r=1} -\gamma_0 \\
\tilde w_1|_{s=0}= w_p|_{r=1}-w_m|_{r=1} \\
\frac{ d\tilde w_1}{ds}|_{s=0}= w_p'|_{r=1}-w_m'|_{r=1} -\gamma_1
\end{array}
\right. . \label{ini-s0-General}
\end{eqnarray}
In the truncation of (\ref{tu0tu1} ) to (\ref{tu0tu1-truc}),
the neglected terms are of order $O(e^{2 (\gamma_0+1) s})$, order $O(e^{(\gamma_1-\gamma_0+2) s})$
and order $O(e^{2 (1-\gamma_1)s})$.
Now, $(\gamma_0, \gamma_1)=(1, \frac{1}{3})$.
Thus, $(\tilde w_0, \tilde w_1)$ will approach $(\rho_0,\rho_1)|_{\gamma_0=1,\gamma_1=\frac{1}{3}}$
with distance $O(e^{\frac{4}{3} s})$, where
\begin{eqnarray}
\left\{
\begin{array}{l}
\rho_0|_{\gamma_0=1,\gamma_1=\frac{1}{3}} = 0.89156581440748831917188012305422345475702308262231...\\
\rho_1|_{\gamma_0=1,\gamma_1=\frac{1}{3}} = 0.22017225140694662756648980530049931068839656816740...
\end{array}
\right. \nonumber
\end{eqnarray}
by (\ref{rhos-DEF}).
So, when $e^{\frac{4}{3} s} \approx 10^{-100}$, ie., $s \approx -172.7$,
$(\tilde w_0, \tilde w_1)$ will be indistinguishable from $(\rho_0,\rho_1)|_{\gamma_0=1,\gamma_1=\frac{1}{3}}$ within our precision tolerance.
Therefore, it is enough to integrate (\ref{tu0tu1}) numerically from $s=0$ to $s_f=-175$.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-175$ for the general case with $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$.}
\label{tab3}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-175$ & $\tilde w_0$ & $\frac{d\tilde w_0}{ds}$ & $\tilde w_1$ &$\frac{d \tilde w_1}{ds}$\\
\hline
| Absolute Error | &$1.33 \times 10^{-111}$ &$7.66 \times 10^{-114}$ &$6.54 \times 10^{-112}$&$3.76\times 10^{-114}$\\
| Relative Error | &$1.50 \times 10^{-111}$ &$1.08 \times 10^{-12}$ &$2.97 \times 10^{-111}$&$2.04\times 10^{-12}$
\end{tabular}
\end{table}
Table \ref{tab3} shows that the numerical solution is accurate as we expect.
The relative error of $\frac{d\tilde w_0}{ds}$ or $\frac{d\tilde w_1}{ds}$ in Table \ref{tab3} seems to be large.
But this is really nothing since it is only another demonstration of the fact that$\frac{d\tilde w_0}{ds}$ and $\frac{d\tilde w_1}{ds}$ are small.
Table \ref{tab4} shows how good the asymptotic solution (\ref{OAsymp-Fancy}) is.
\begin{table}[ht]
\caption{Approximate derivation from the asymptotic solution for the general case with $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$.}
\label{tab4}
\centering
\begin{tabular}[h]{ c|c c c c c c c}
\hline \hline
$s$ & $-25$ & $-50$ & $-75$ &$-100$ &$-125$& $-150$&$-175$\\
\hline
$\ln(\rho_0-\tilde w_0)$ &$-33.1938$ &$-66.5271$&$-99.8605$&$-133.194$&$-166.527$&$-199.860$&$-233.194 $\\
$\ln(\rho_1-\tilde w_1)$ &$-34.5412$ &$-67.8745$ &$-101.208$&$-134.541$&$-167.875$&$-201.208$&$-234.541$
\end{tabular}
\end{table}
Table \ref{tab4} not only numerically verifies the asymptotics of the general case for $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$,
but also confirms our estimation that $(\tilde w_0, \tilde w_1)$ is close to its asymptotics $(\rho_0,\rho_1)|_{\gamma_0=1,\gamma_1=\frac{1}{3}}$ with a distance of order $O(e^{\frac{4}{3} s})$.
In fact, the asymptotics (\ref{OAsymp-Fancy}) can still be refined, see Section \ref{DFG}.
\subsection{Case E1 \label{E1Case}}
The E1 case is $-1<\gamma_0<3$ and $\gamma_1=1$.
This subsection is devoted to verify the fine structure of the E1 case
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r) \xlongrightarrow{r \rightarrow 0} \gamma_0 \ln r +a_{E1} \\
2 w_1(r) \xlongrightarrow{r \rightarrow 0} \ln r+\ln(-2 s+b_{E1})
\end{array}
\right. , \label{Asymp-E1}
\end{eqnarray}
where
\begin{eqnarray}
&&a_{E1}=-\ln \left( 2^{2 \gamma_0}
\frac{\Gamma(\frac{\gamma_0+1}{4}) \left( \Gamma(\frac{\gamma_0+5}{8}) \right)^2}
{ \Gamma(\frac{3-\gamma_0}{4}) \left(\Gamma(\frac{3-\gamma_0}{8}) \right)^2 } \right) ,\nonumber\\
&&b_{E1}=\frac{1}{2} \psi(\frac{3-\gamma_0}{8})+\frac{1}{2} \psi(\frac{5+\gamma_0}{8}) -\gamma_{\text{eu}} +4 \ln 2.\nonumber
\end{eqnarray}
Note that $s=\ln (r)$ as defined by (\ref{s-DEF}), $a_{E1}=\rho_0|_{\gamma_1=1}$
and $\psi$ is the digamma function $\psi(t)=\frac{d}{dt} \ln (\Gamma(t))=\frac{\Gamma'(t)}{\Gamma(t)}$.
Please also recall the denotation $s=\ln(r)$, see (\ref{s-DEF}).
To fix the problem, we take $\gamma_0=1$,
as an example to verify the E1 case. Substituting $(\gamma_0,\gamma_1)=(1,1)$
to the connection formula (\ref{ConnectFormula}),
we immediately get $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2, -2)$.
Similar to the general case of Subsection \ref{GeneralCase},
the numerical integration is divided into two parts: on $r\in [1,45]$ and on $s\in [s_f,0]$.
\subsubsection{Numerical integration from $r=45$ to $r=1$}
By the truncation of (\ref{iniApprox}), the initial values at $r=45$ are obtained (only the first $50$ digits are listed)
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(45)= -5.2843098725232974899221393911204991207504443469367... \times 10^{-57} \\
w_p'(45)= 1.5004885502015739552694025310567337731833237644509... \times 10^{-56}\\
w_m(45)=- 3.9902150828859022626192436154419666864562950795650... \times 10^{-80}\\
w_m'(45)= 1.6005134816454403480052616718328015209213735935410... \times 10^{-79}
\end{array}
\right. .\label{ini-E1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of the initial values (\ref{ini-E1}) are obtained as shown by Table \ref{tab5}.
\begin{table}[ht]
\caption{Errors of the initial values of case E1 with $\gamma_0=1$.}
\label{tab5}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=45$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$3.04 \times 10^{-170}$ &$2.59 \times 10^{-169}$ &$3.24 \times 10^{-193}$&$3.14\times 10^{-192}$ \\
| Relative Error | &$5.76\times 10^{-114}$ &$1.73 \times 10^{-113}$ &$8.12 \times 10^{-114}$&$1.96\times 10^{-113}$
\end{tabular}
\end{table}
Numerically integrating the tt* equation (\ref{wpwm}) from $r=45$ to $r=1$
by the Gauss-Legendre method with parameters as same as in Subsection \ref{GeneralCase},
the values of $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$ are obtained
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(1)=-3.8076020447615564848336037555396597913276640146800...\times 10^{-2} \\
w_p'(1)=1.2507257120725277318359466237894266588814464453818...\times 10^{-1}\\
w_m(1)=-6.5181931373519405060356987540333399617643482502891...\times 10^{-3}\\
w_m'(1)=2.8018632441288063804071518136255604932977444116709....\times 10^{-2}
\end{array}
\right. .\label{values-E1-1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of (\ref{values-E1-1}) are obtained as shown by Table \ref{tab6}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $r=1$ of case E1 with $\gamma_0=1$.}
\label{tab6}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=1$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$4.38 \times 10^{-115}$ &$1.43 \times 10^{-114}$ &$8.52 \times 10^{-116}$&$3.55\times 10^{-115}$ \\
| Relative Error | &$1.15\times 10^{-113}$ &$1.15 \times 10^{-113}$ &$1.31 \times 10^{-113}$&$1.27\times 10^{-113}$
\end{tabular}
\end{table}
\subsubsection{Near $r=0$}
Let
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0 =2 w_0-\gamma_0 s \\
\tilde w_1 =2 w_1 - s
\end{array}
\right. , \label{transform-r0-E1}
\end{eqnarray}
where $s=\ln(r)$ as defined by (\ref{s-DEF}).
Then the differential equations for $\tilde w_0$ and $\tilde w_1$ are
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0}{ds^2}= e^{2 \tilde w_0 +2 (\gamma_0+1) s}-e^{\tilde w_1- \tilde w_0+(3-\gamma_0)s} \\
\frac{1}{4} \frac{ d^2\tilde w_1}{ds^2}=e^{\tilde w_1- \tilde w_0+(3-\gamma_0)s}-e^{-2 \tilde w_1 }
\end{array}
\right. .\label{tu0tu1-E1}
\end{eqnarray}
(\ref{tu0tu1-E1}) can also be simply obtained from (\ref{tu0tu1}) by substituting $\gamma_1=1$ to it.
We expect $\tilde w_0$ is of order $O(1)$ and that $\tilde w_1$ is of order $O(\ln(-s))$.
Also considering $-1<\gamma_0<3$, we get the primary truncation of (\ref{tu0tu1-E1}) near $s=-\infty$:
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0^{(0)}}{ds^2}=0\\
\frac{1}{4} \frac{ d^2\tilde w_1^{(0)}}{ds^2}=-e^{-2 \tilde w_1^{(0)} }
\end{array}
\right. .\label{tu0tu1-trunc-E1}
\end{eqnarray}
(\ref{tu0tu1-trunc-E1}) is the truncation structure of the tt* equation for the E1 case.
The general solution of (\ref{tu0tu1-trunc-E1}) is
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0^{(0)}=k_{0E1}+k_{1E1} s\\
\tilde w_1^{(0)}=\ln \left( \pm \frac{2}{k_{2E1} } \sinh(k_{2E1} (s+k_{3E1})) \right)
\end{array}
\right. .\label{SOL-trunc-E1}
\end{eqnarray}
By (\ref{Asymp-E1}) and (\ref{transform-r0-E1}),
we know the asymptotics of the tt* equation corresponds to
$k_{0E1}=a_{E1}$, $k_{1E1}=0$, $k_{2E1}\rightarrow 0$ and $k_{3E1}=b_{E1}$.
In the truncation from (\ref{tu0tu1-E1}) to (\ref{tu0tu1-trunc-E1}),
the neglected term for the differential equation of $\tilde w_1$ is $e^{\tilde w_1- \tilde w_0+(3-\gamma_0)s}$,
which is of order $O(s e^{(3-\gamma_0) s})$.
Similarly, the neglected terms for the differential equation of $\tilde w_0$ are of orders $O(s e^{(3-\gamma_0) s})$
and $O(e^{2 (\gamma_0+1)s})$.
In the current numerical experiment, $\gamma_0=1$.
Therefore, the difference between the asymptotic solution and the exact solution is of order $O(s e^{2 s})$.
So, we should do high-precision numerical integration from $s=0$ to about $s=s_f=-120$
since $120 \times e^{2 \times (-120)} \approx 7.055 \times 10^{-103}$.
Similar to the general case of Subsection \ref{GeneralCase},
the values of $\tilde w_0$, $\frac{ d\tilde w_0}{ds}$, $\tilde w_1$ and $\frac{ d\tilde w_1}{ds}$ at $s=0$
are obtained by formula (\ref{ini-s0-General}).
Then, integrating (\ref{tu0tu1-E1}) numerically by the Gauss-Legendre method, the high-precision numerical solution is obtained.
Comparing it with the more accurate numerical solution starting from $r=55$,
the errors of the numerical solution are obtained.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-120$ for the E1 case with $\gamma_0=1$.}
\label{tab7}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-120$ & $\tilde w_0$ & $\frac{d\tilde w_0}{ds}$ & $\tilde w_1$ &$\frac{d \tilde w_1}{ds}$\\
\hline
| Absolute Error | &$1.06 \times 10^{-111}$ &$8.84 \times 10^{-114}$ &$3.56 \times 10^{-110}$&$5.94\times 10^{-112}$ \\
| Relative Error | &$1.35 \times 10^{-111}$ &$6.84 \times 10^{-12}$ &$6.50 \times 10^{-111}$&$7.12\times 10^{-110}$\\
\end{tabular}
\end{table}
Table \ref{tab7} shows our numerical solution is accurate as we expect.
The large relative error of $\frac{d\tilde w_0}{ds}$ is nothing
because $\frac{d\tilde w_0}{ds}|_{s=-120} \approx -1.29\times 10^{-102} $ is so small.
Table \ref{tab8} shows how good the asymptotic solution (\ref{Asymp-E1}) is.
\begin{table}[ht]
\caption{Approximate derivation from the asymptotic solution for the E1 case with $\gamma_0=1$.}
\label{tab8}
\centering
\begin{tabular}[h]{ c|c c c c c c}
\hline \hline
$s$ & $-20$ & $-40$ & $-60$ &$-80$ &$-100$& $-120$\\
\hline
$\ln(a_{E1}-\tilde w_0)$ &$-37.0566$ &$-76.3821$&$-115.983$&$-155.698$&$-195.477$&$-235.296$ \\
$\ln(\tilde w_1-\ln(-2 s+b_{E1}))$&$-37.0553$&$-76.3818$ &$-115.983$&$-155.698$&$-195.477$&$-235.296$
\end{tabular}
\end{table}
Table \ref{tab8} not only numerically verifies the asymptotics of the E1 case for $\gamma_0=1$,
but also confirms our estimation that $(\tilde w_0, \tilde w_1)$ differs with its asymptotic solution
by an order of $O(s e^{2 s})$.
\subsection{Case E2}
In this case, $\gamma_0=-1$ and $-3<\gamma_1<1$.
As explained in the beginning of Section \ref{sec-verify}, the fine structure of the E2 case can be obtained from the E1 case.
For convenience, the fine structure of the E2 case is listed below explicitly
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r) \xlongrightarrow{r \rightarrow 0} -\ln (r) -\ln \left(-2 s+ a_{E2} \right)\\
2 w_1(r) \xlongrightarrow{r \rightarrow 0} \gamma_1 \ln (r)+b_{E2}
\end{array}
\right. , \label{Asymp-E2}
\end{eqnarray}
where
\begin{eqnarray}
&&a_{E2}=\frac{1}{2} \psi(\frac{3+\gamma_1}{8})+\frac{1}{2} \psi(\frac{5-\gamma_1}{8}) -\gamma_{\text{eu}} +4 \ln 2 ,\nonumber\\
&&b_{E2}=-\ln \left( 2^{2 \gamma_1}
\frac{\Gamma(\frac{\gamma_1+3}{4}) \left( \Gamma(\frac{\gamma_1+3}{8}) \right)^2}
{ \Gamma(\frac{1-\gamma_1}{4}) \left(\Gamma(\frac{5-\gamma_1}{8}) \right)^2 } \right).\nonumber
\end{eqnarray}
Please also recall that $s=\ln (r)$ and $\psi$ is the digamma function. Note $b_{E2}=\rho_1|_{\gamma_0=-1}$.
\subsection{Case E3}
In this case $\gamma_1=\gamma_0-2$ and $-1<\gamma_0<3$.
This subsection will numerically verify the fine structure of the E3 case
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r)+ 2 w_1(r) \xlongrightarrow{r \rightarrow 0} 2(\gamma_0-1)\ln (r) +a_{E3}\\
2 w_1(r)- 2 w_0(r) \xlongrightarrow{r \rightarrow 0} -2 \ln (r)-\ln \left( 4 (s+b_{E3})^2 \right)
\end{array}
\right. , \label{Asymp-E3}
\end{eqnarray}
where
\begin{eqnarray}
&&a_{E3}=4(1-\gamma_0) \ln 2-4 \ln \left( \Gamma \left( \frac{1+\gamma_0}{4}\right) \right)+
4 \ln \left( \Gamma \left( \frac{3-\gamma_0}{4}\right) \right), \nonumber\\
&&b_{E3}=-\frac{1}{4}\psi\left(\frac{3-\gamma_0}{4}\right)
-\frac{1}{4}\psi\left(\frac{\gamma_0-3}{4}\right) +\frac{1}{3-\gamma_0}+\frac{\gamma_{EU}}{2}
-2 \ln(2).
\end{eqnarray}
Note that $s=\ln (r)$ and $\psi$ is the digamma function.
Also, we should notice
\begin{eqnarray}
a_{E3}
&=&\lim_{\gamma_1 \rightarrow \gamma_0-2} (\rho_0(\gamma_0,\gamma_1)+\rho_1(\gamma_0,\gamma_1)), \nonumber
\end{eqnarray}
where $\rho_0$ and $\rho_1$ are defined by (\ref{rhos-DEF}).
Let us take $\gamma_0=\frac{1}{3}$ as an example to verify (\ref{Asymp-E3}) numerically.
Then $(s_1^\mathbb{R},s_2^\mathbb{R})=(-2,-3)$.
Similar to the general case of Subsection \ref{GeneralCase},
the numerical integration is divided into two parts: on $r\in [1,45]$ and on $s\in [s_f,0]$.
\subsubsection{Numerical integration from $r=45$ to $r=1$}
By the truncation of (\ref{iniApprox}), the initial values at $r=45$ are obtained (only the first $50$ digits are listed)
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(45)= 5.2843098725232974899221393911204991207504443469367... \times 10^{-57} \\
w_p'(45)=-1.5004885502015739552694025310567337731833237644509... \times 10^{-56}\\
w_m(45)=- 5.9853226243288533939288654231629507224228092956986... \times 10^{-80}\\
w_m'(45)= 2.4007702224681605220078925077492026747788333343193... \times 10^{-79}
\end{array}
\right. .\label{ini-E3}
\end{eqnarray}
It is not surprising that $w_p(45)$ and $w_p'(45)$ of (\ref{ini-E3}) coincide with that of (\ref{ini-E1}) with many digits
since in the numerical examples $s_1^\mathbb{R}=-2$ for this case and $s_1^\mathbb{R}=2$ for the E1 case.
Comparing with the more accurate solution starting from $r=55$,
the errors of the initial values (\ref{ini-E3}) are obtained as shown by Table \ref{tab9}.
\begin{table}[ht]
\caption{Errors of the initial values of case E3 with $\gamma_0=\frac{1}{3}$.}
\label{tab9}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=45$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$3.04 \times 10^{-170}$ &$2.59 \times 10^{-169}$ &$4.86 \times 10^{-193}$&$4.71\times 10^{-192}$ \\
| Relative Error | &$5.76\times 10^{-114}$ &$1.73 \times 10^{-113}$ &$8.12 \times 10^{-114}$&$1.96\times 10^{-113}$
\end{tabular}
\end{table}
Numerically integrating the tt* equation (\ref{wpwm}) from $r=45$ to $r=1$
by the Gauss-Legendre method with parameters as same as in Subsection \ref{GeneralCase},
the values of $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$ are obtained
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(1)= 3.8027004168653915145363303284447255846983739527888...\times 10^{-2} \\
w_p'(1)=-1.2469806975938122928142121636698878096900701362539...\times 10^{-1}\\
w_m(1)= -1.0071686775204061495316019356342162460012952192431...\times 10^{-2}\\
w_m'(1)= 4.3926896299159549125370306923225572137558540540015....\times 10^{-2}
\end{array}
\right. .\label{values-E3-1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of (\ref{values-E3-1}) are obtained as shown by Table \ref{tab10}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $r=1$ of case E3 with $\gamma_0=\frac{1}{3}$.}
\label{tab10}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=1$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$4.37 \times 10^{-115}$ &$1.42 \times 10^{-114}$ &$1.35 \times 10^{-115}$&$5.76\times 10^{-115}$ \\
| Relative Error | &$1.15\times 10^{-113}$ &$1.14 \times 10^{-113}$ &$1.34 \times 10^{-113}$&$1.31\times 10^{-113}$
\end{tabular}
\end{table}
\subsubsection{Near $r=0$}
Near $r=0$, we still use the transformation (\ref{transform-general}).
So the differential equations for $\tilde w_0$ and $\tilde w_1$ are also (\ref{tu0tu1}).
We expect $\tilde w_0$ and $\tilde w_1$ are of order $o(s)$.
Also considering $-1<\gamma_0<3$ and $\gamma_1=\gamma_0-2$, we get the primary truncation of (\ref{tu0tu1}) near
$s=-\infty$ for the E3 case:
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0^{(0)}}{ds^2}=-e^{\tilde w_1-\tilde w_0}\\
\frac{1}{4} \frac{ d^2\tilde w_1^{(0)}}{ds^2}=e^{\tilde w_1-\tilde w_0}
\end{array}
\right. .\label{tu0tu1-trunc-E3}
\end{eqnarray}
(\ref{tu0tu1-trunc-E3}) is the truncation structure of the tt* equation for the E3 case.
By (\ref{tu0tu1-trunc-E3}) we get
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0^{(0)}+\tilde w_1^{(0)}=k_{0E3}+k_{1E3} s\\
\tilde w_1^{(0)}-\tilde w_0^{(0)}=\ln \left(- \frac{k_{2E3}^2}{8 \pm 8 \cosh(k_{2E3} (s+k_{3E3}))} \right)
\end{array}
\right. .\label{SOL-trunc-E3}
\end{eqnarray}
By (\ref{Asymp-E3}) and (\ref{transform-general}),
we know the asymptotics of the tt* equation corresponds to
$k_{0E3}=a_{E1}$, $k_{1E3}=0$, $k_{2E3} \rightarrow 0$ and $k_{3E2}=b_{E1}$.
In the truncation from (\ref{tu0tu1}) to (\ref{tu0tu1-trunc-E3}),
the neglected terms for the differential equation of $\tilde w_0+\tilde w_1$
are $e^{2 \tilde w_0+2 (\gamma_0+1)s}$ and $e^{-2 \tilde w_1+2 (1-\gamma_1)s}$,
which are of order $O(s^2 e^{2(\gamma_0+1) s})$ and $O(s^{-2} e^{2(\gamma_0+1) s})$.
Similarly, the neglected terms for the differential equation of $\tilde w_1$ are also
of orders $O(s^{-2} e^{2(3-\gamma_0) s})$ and $O(s^{-2} e^{2(\gamma_0+1) s})$.
In the current numerical experiment, $\gamma_0=\frac{1}{3}$.
Therefore, the difference between the asymptotic solution and the exact solution is of order $O(s^2 e^{\frac{8}{3} s})$.
So, we should do high-precision numerical integration from $s=0$ to about $s=s_f=-90$
since $90^2 \times e^{\frac{8}{3} \times (-90)} \approx 4.76 \times 10^{-101}$.
Just as the general case,
the values of $\tilde w_0$, $\frac{ d\tilde w_0}{ds}$, $\tilde w_1$ and $\frac{ d\tilde w_1}{ds}$ at $s=0$
are obtained by formula (\ref{ini-s0-General}).
Then, integrating (\ref{tu0tu1}) numerically by the Gauss-Legendre method, the high-precision numerical solution is obtained.
Comparing it with the more accurate numerical solution starting from $r=55$,
the errors of the numerical solution are obtained.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-90$ for the E3 case with $\gamma_0=\frac{1}{3}$.}
\label{tab11}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-90$ & $\tilde w_0$ & $\frac{d\tilde w_0}{ds}$ & $\tilde w_1$ &$\frac{d \tilde w_1}{ds}$\\
\hline
| Absolute Error | &$1.30 \times 10^{-110}$ &$2.95 \times 10^{-112}$ &$1.41 \times 10^{-110}$&$3.08\times 10^{-112}$ \\
| Relative Error | &$2.74 \times 10^{-111}$ &$2.66 \times 10^{-110}$ &$2.51 \times 10^{-111}$&$2.77\times 10^{-110}$\\
\end{tabular}
\end{table}
Table \ref{tab11} shows our numerical solution is accurate as we expect.
Table \ref{tab12} shows how good the asymptotic solution (\ref{Asymp-E3}) is.
\begin{table}[ht]
\caption{Approximate derivation from the asymptotic solution for the E3 case with $\gamma_0=\frac{1}{3}$.}
\label{tab12}
\centering
{\small
\begin{tabular}[h]{ c|c c c c c c}
\hline \hline
$s$ & $-15$ & $-30$ & $-45$ &$-60$ &$-75$& $-90$\\
\hline
$\ln(\tilde w_0+\tilde w_1-a_{E3})$ &$-34.5568$ &$-73.2186$&$-112.424$&$-151.857$&$-191.415$&$-231.054$ \\
$\ln\left(\tilde w_0-\tilde w_1-\ln\left(4 (s+b_{E3})^2\right) \right)$ &$-34.5556$ &$-73.2183$ &$-112.424$&$-151.857$&$-191.415$&$-231.054$
\end{tabular}
}
\end{table}
Table \ref{tab12} not only numerically verifies the asymptotics of the E3 case for $\gamma_0=\frac{1}{3}$,
but also confirms our estimation that $\tilde w_0+\tilde w_1$ and $\tilde w_1-\tilde w_0$ deviate from their asymptotics
by an order of $O(s^2 e^{\frac{8}{3} s})$.
More detailed analysis shows that $\tilde w_0$ and $\tilde w_1$ deviate from their asymptotics
by an order of $O(s^2 e^{\frac{8}{3} s})$ and an order of $O(e^{\frac{8}{3} s})$, respectively.
\subsection{Case V1}
In this case, $\gamma_0=3$ and $\gamma_1=1$.
This subsection is devoted to verify the fine structure of the V1 case
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_0(r) \xlongrightarrow{r \rightarrow 0} 3 \ln (r)+\ln \left( P_3\right) \\
2 w_0(r)+ 2 w_1(r) \xlongrightarrow{r \rightarrow 0} 4 \ln(r)+ \ln \left( P_4 \right)
\end{array}
\right. , \label{Asymp-V1}
\end{eqnarray}
where
{\small
\begin{eqnarray}
\hspace*{-0.5cm}&&P_3=-\frac{4}{3} (s-\ln 4)^3-4 \gamma_{eu} (s-\ln 4)^2-4 \gamma_{eu}^2 (s-\ln 4)-\frac{1}{24} \zeta(3)-\frac{4}{3} \gamma_{eu}^3,
\label{P3}\\
\hspace*{-0.5cm}&&P_4=\frac{4}{3} (s-\ln 4)^4+ \frac{16 }{3}\gamma_{eu} (s-\ln 4)^3+8 \gamma_{eu}^2 (s-\ln 4)^2
+(\frac{16\gamma_{eu}^3}{3} -\frac{\zeta(3)}{12}) (s-\ln 4)-\frac{ \gamma_{eu} \zeta(3)}{12}+\frac{4\gamma_{eu}^4 }{3}.
\label{P4}
\end{eqnarray}
}
Note that $s=\ln (r)$.
$(s_1^\mathbb{R},s_2^\mathbb{R}) =(4,-6)$ by (\ref{ConnectFormula}).
Similar to the general case of Subsection \ref{GeneralCase},
the numerical integration is divided into two parts: on $r\in [1,45]$ and on $s\in [s_f,0]$.
\subsubsection{Numerical integration from $r=45$ to $r=1$}
By the truncation of (\ref{iniApprox}), the initial values at $r=45$ are obtained (only the first $50$ digits are listed)
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(45)= -1.0568619745046594979844278782240998241500888693873... \times 10^{-56}\\
w_p'(45)= 3.0009771004031479105388050621134675463666475289019... \times 10^{-56}\\
w_m(45)= -1.1970645248657706787857730846325898673892151885992... \times 10^{-79}\\
w_m'(45)= 4.8015404449363210440157850154984037759705748926074... \times 10^{-79}
\end{array}
\right. .\label{ini-V1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of the initial values (\ref{ini-V1}) are obtained as shown by Table \ref{tab13}.
\begin{table}[ht]
\caption{Errors of the initial values of case V1.}
\label{tab13}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=45$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$2.43 \times 10^{-169}$ &$2.07 \times 10^{-168}$ &$3.89 \times 10^{-192}$&$3.77\times 10^{-191}$ \\
| Relative Error | &$2.30\times 10^{-113}$ &$6.91 \times 10^{-113}$ &$3.25 \times 10^{-113}$&$7.85\times 10^{-113}$
\end{tabular}
\end{table}
Numerically integrating the tt* equation (\ref{wpwm}) from $r=45$ to $r=1$
by the Gauss-Legendre method with parameters as same as the ones in Subsection \ref{GeneralCase},
the values of $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$ are obtained
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(1)= -7.5811708202722819337886291345224915096864160866088...\times 10^{-2} \\
w_p'(1)= 2.4764894905832982616275785124301997778251205645956...\times 10^{-1}\\
w_m(1)= -1.8985818420083245736824441481547286887104902789335...\times 10^{-2}\\
w_m'(1)= 8.0472024534463364925338502074404317836916130555680....\times 10^{-2}
\end{array}
\right. .\label{values-V1-1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of (\ref{values-V1-1}) are obtained as shown by Table \ref{tab14}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $r=1$ of case V1.}
\label{tab14}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=1$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$3.47 \times 10^{-114}$ &$1.12 \times 10^{-113}$ &$9.71 \times 10^{-115}$&$3.94\times 10^{-114}$ \\
| Relative Error | &$4.58\times 10^{-113}$ &$4.54 \times 10^{-113}$ &$5.11 \times 10^{-113}$&$4.89\times 10^{-113}$
\end{tabular}
\end{table}
\subsubsection{Near $r=0$}
Near $r=0$, the transformation is still (\ref{transform-general}).
Hence, the differential equations for $\tilde w_0$ and $\tilde w_1$ are also (\ref{tu0tu1}).
Now, $(\gamma_0,\gamma_1)=(3,1)$ and we expect $\tilde w_0$ and $\tilde w_1$ are of order $o(s)$.
So the primary truncation of (\ref{tu0tu1}) near $s=-\infty$ for the V1 case is
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0^{(0)}}{ds^2}=-e^{\tilde w_1^{(0)}-\tilde w_0^{(0)}}\\
\frac{1}{4} \frac{ d^2\tilde w_1^{(0)}}{ds^2}=e^{\tilde w_1^{(0)}-\tilde w_0^{(0)}}-e^{-2 \tilde w_1^{(0)}}
\end{array}
\right. .\label{tu0tu1-trunc-V1}
\end{eqnarray}
(\ref{tu0tu1-trunc-V1}) is the truncation structure of the tt* equation for the V1 case.
Let
$$\tilde w_p^{(0)}= w_0^{(0)}+ w_1^{(0)}.$$
Then, we have
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{1}{4} \frac{ d^2\tilde w_0^{(0)}}{ds^2}=-e^{\tilde w_p^{(0)}-2 \tilde w_0^{(0)}}\\
\frac{1}{4} \frac{ d^2\tilde w_p^{(0)}}{ds^2}=-e^{-2 \tilde w_p^{(0)}+2 \tilde w_0^{(0)}}
\end{array}
\right. .\label{tu0tu1-trunc1-V1}
\end{eqnarray}
Unlike the cases discussed before,
we have not achieved the general solution of (\ref{tu0tu1-trunc1-V1}).
Anyhow, Equation (\ref{tu0tu1-trunc1-V1}) itself deserves an independent investigation. Let us leave it as a future work.
Surprisingly, a two parameter family of explicit solutions of (\ref{tu0tu1-trunc1-V1}) can be constructed
and the asymptotic solution near $r=0$ is just among them!
By the hint of the asymptotic solution and for the convenience of comparison,
we seek the solutions of (\ref{tu0tu1-trunc1-V1}) with form
\begin{eqnarray}
\left\{
\begin{array}{l}
\tilde w_0^{(0)}= \ln \left( \tilde{a}_3 (s-\ln 4)^3+\tilde{a}_2 (s-\ln 4)^2+\tilde{a}_1 (s-\ln 4)+\tilde{a}_0 \right)\\
\tilde w_p^{(0)}= \ln \left(\tilde{b}_4 (s-\ln 4)^4+ \tilde{b}_3 (s-\ln 4)^3+\tilde{b}_2 (s-\ln 4)^2+\tilde{b}_1 (s-\ln 4)+\tilde{b}_0 \right) \nonumber
\end{array}
\right. . \label{formAnsaz}
\end{eqnarray}
There are only 2 sets of solutions that has form (\ref{formAnsaz}).
Set A:
\begin{eqnarray}
&&\tilde{a}_3=\frac{4}{3}, \quad \tilde{b}_4=\frac{4}{3}, \nonumber\\
&&\tilde{a}_1=\frac{1}{4}\tilde{a}_2^2, \quad \tilde{b}_3=\frac{4}{3} \tilde{a}_2, \quad \tilde{b}_2=\frac{1}{2} \tilde{a}_2^2,\nonumber\\
&&\tilde{b}_1=\frac{1}{8}(\tilde{a}_2^3-16 \tilde{a}_0), \quad \tilde{b}_0=\frac{1}{64} (\tilde{a}_2^4-32 \tilde{a}_0 \tilde{a}_2). \nonumber
\end{eqnarray}
Set B:
\begin{eqnarray}
&&\tilde{a}_3=-\frac{4}{3}, \quad \tilde{b}_4=\frac{4}{3}, \nonumber\\
&&\tilde{a}_1=-\frac{1}{4}\tilde{a}_2^2, \quad \tilde{b}_3=-\frac{4}{3} \tilde{a}_2, \quad \tilde{b}_2=\frac{1}{2} \tilde{a}_2^2,\nonumber\\
&&\tilde{b}_1=\frac{1}{8}(16 \tilde{a}_0-\tilde{a}_2^3), \quad \tilde{b}_0=\frac{1}{64} (\tilde{a}_2^4-32 \tilde{a}_0 \tilde{a}_2). \nonumber
\end{eqnarray}
The asymptotic solution is in Set B with
\begin{eqnarray}
&&\tilde{a}_2=-4 \gamma_{eu}, \nonumber\\
&&\tilde{a}_0=-\frac{1}{24} \zeta(3)-\frac{4}{3} \gamma_{eu}^3. \nonumber
\end{eqnarray}
The error of the truncation from (\ref{tu0tu1}) to (\ref{tu0tu1-trunc-V1})
is caused by the term $e^{2 \tilde w_0 +8 s}$, which is of order $O(s^6 e^{8s})$.
So we set $s_f=-32$ since $(-32)^6 e^{8\times (-32)} \approx 7.1 \times 10^{-103}$
has been smaller than our precision goal.
Integrating (\ref{tu0tu1}) numerically by the Gauss-Legendre method, the high-precision numerical solution is obtained.
Comparing it with the more accurate numerical solution starting from $r=55$,
the errors of the numerical solution are obtained.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-32$ for the V1 case.}
\label{tab15}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-32$ & $\tilde w_0$ & $\frac{d\tilde w_0}{ds}$ & $\tilde w_1$ &$\frac{d \tilde w_1}{ds}$\\
\hline
| Absolute Error | &$1.14 \times 10^{-109}$ &$1.49 \times 10^{-110}$ &$3.94 \times 10^{-109}$&$4.77\times 10^{-110}$ \\
| Relative Error | &$1.06 \times 10^{-110}$ &$1.62 \times 10^{-109}$ &$1.13 \times 10^{-109}$&$1.57\times 10^{-108}$\\
\end{tabular}
\end{table}
Table \ref{tab15} shows our numerical solution is accurate as we expect.
Table \ref{tab16} shows how good the asymptotic solution (\ref{Asymp-V1}) is.
\begin{table}[ht]
\caption{Approximate derivation from the asymptotic solution for the V1 case.}
\label{tab16}
\centering
\begin{tabular}[h]{ c| c c c c c c}
\hline \hline
$s$ & $-7$ & $-12$ &$-17$ &$-22$& $-27$ & $-32$\\
\hline
$\ln(\tilde w_0-\ln(P_3))$ &$-45.6682$&$-82.7772$&$-120.834$&$-159.368$&$-198.191$
&$-237.207$ \\
$\ln\left(\tilde w_0+\tilde w_1-\ln (P4) \right)$
&$-45.6691$ &$-82.7775$&$-120.834$&$-159.368$&$-198.191$ & $-237.207$
\end{tabular}
\end{table}
Table \ref{tab16} not only numerically verifies the asymptotics of the V1 case,
but also confirms our estimation that $\tilde w_0$ and $\tilde w_1+\tilde w_0$ differ from their asymptotics
by an order of $O(s^6 e^{8 s})$.
\subsection{Case V2}
In this case $(\gamma_0,\gamma_1)=(-1,1)$.
By the connection formula (\ref{ConnectFormula}), we have
$(s_1^\mathbb{R}, s_2^\mathbb{R})=(0,2)$.
$s_1^\mathbb{R}=0$ means $w_1=-w_0$ at $r=\infty$.
This leads to $w_1 \equiv -w_0$ for $r \in (0 ,\infty)$, considering (\ref{TT-2}).
Let
$$ w=w_0=-w_1.$$
Then, the differential equation for $w$ is
\begin{eqnarray}
\frac{1}{2} (\frac{d^2}{dr^2}+ \frac{1}{r}\frac{d}{dr}) w=e^{4 w}-e^{-4 w}, \nonumber
\end{eqnarray}
which is the radical reduction of the sinh-Gordon equation.
For convenience, we also give the fine structure of the V2 case
\begin{eqnarray}
2 w(r) \xlongrightarrow{r \rightarrow 0} -\ln (r) -\ln\left(-2 s -2 \gamma_{eu}+2 \ln 2\right) . \label{Asymp-V2}
\end{eqnarray}
Near $r=0$, $2 w(r)$ differs from its asymptotics by an order of $O(s^2 e^{4 s})$.
\subsection{Case V3}
In this case, $(\gamma_0,\gamma_1)=(-1,-3)$. Thus, $(s_1^\mathbb{R},s_2^\mathbb{R})=(-4,-6)$ by (\ref{ConnectFormula}).
As explained in the beginning of Section \ref{sec-verify},
the fine structure of the V3 case can be read out from the V1 case.
For convenience, we list the fine structure of the V3
\begin{eqnarray}
\left\{
\begin{array}{l}
2 w_1(r) \xlongrightarrow{r \rightarrow 0} -3 \ln (r)-\ln \left( P_3\right) \\
2 w_0(r)+ 2 w_1(r) \xlongrightarrow{r \rightarrow 0} -4 \ln(r)- \ln \left( P_4 \right)
\end{array}
\right. , \label{Asymp-V3}
\end{eqnarray}
where $P_3$ and $P_4$ are defined by (\ref{P3}) and (\ref{P4}).
\section{Out of the curved triangle: generalizing the connection formula and the fine structure \label{CONJ}}
First, let us divide the real plane of $(s_1^\mathbb{R}, s_2^\mathbb{R})$ into $19$ parts:
regions $\Omega_0$, $\Omega_1$, $\Omega_2$, $\Omega_3$, $\Omega_4$, $\Omega_5$, $\Omega_6$;
edges E1, E2, E3, $E_1^U$, $E_2^U$, $E_1^D$, $E_2^D$, $E_3^R$, $E_3^L$;
and vertices V1, V2, V3. See Figure \ref{fig-3} for the details.
Note the boundaries of $\Omega_i$ are
line $s_2^\mathbb{R}=2 s_1^\mathbb{R}+2$, line $s_2^\mathbb{R}=-2 s_1^\mathbb{R}+2$
and parabola $s_2^\mathbb{R}=-\frac{1}{4} \left( s_1^\mathbb{R} \right)^2 -2 $.
\input{Figure3.tikz}
By the connection formula (\ref{ConnectFormula}) (also see Figure \ref{fig-2}), on the Stokes data side,
the solutions studied in Theorem (\ref{thm-GIL-1}) are those associated with a point in the region $\Omega_0$,
on the edges E1, E2, E3 or at the vertices V1, V2, V3.
Those solutions are all smooth for $r \in (0,\infty)$.
Let us consider the case
that $(s_1^\mathbb{R}, s_2^\mathbb{R})$ lies out of the curved triangular.
Then the corresponding $w_0(r)$, $w_1(r)$ or both of them must develop to singularity
somewhere as $r$ decreases from $r=\infty$.
Numerical experiments show there is a cut around each singularity.
But we have evidences that hint these singularities and cuts are artificial:
they can be avoided by choosing appropriate variables.
For example, if we use variables $v_0=e^{2 w_0}$ and $v_1=e^{2 w_1}$,
then $v_0$ and $v_1$ will have no cuts near $r>0$.
$v_0$ or $v_1$ may still have singularities,
i.e., in general $v_0$ and $v_1$ are not the final smooth variables.
Fortunately, we could find two smooth variables for each part of Figure \ref{fig-3},
see Conjecture \ref{Conj}.
From this point of view, Theorem (\ref{thm-GIL-1}) studies only those solutions
that have ``positiveness" property so that after taking logarithm they are still real.
\subsection{The conjecture}
The fine structures for the cases of $\Omega_0$, E1, E2, E3, V1, V2 and V3
have been rigorously proven in \cite{GIL-3} and numerically verified in Section \ref{sec-verify}.
So the following conjecture only deal with the other remaining $12$ cases:
$\Omega_1$, $\Omega_2$, $\Omega_3$, $\Omega_4$, $\Omega_5$, $\Omega_6$, $E_1^U$, $E_2^U$, $E_1^D$, $E_2^D$, $E_3^R$ and $E_3^L$.
Similar to the explanation in the beginning of Section \ref{sec-verify}, the formulae of $\Omega_3$, $\Omega_4$, $E_1^U$,
$E_2^D$ and $E_3^L$ are symmetrical to that of $\Omega_1$, $\Omega_6$, $E_2^U$,
$E_1^D$ and $E_3^R$, respectively.
But, for convenience we will list all formulae for the $12$ cases.
\begin{conjecture} \label{Conj}
Let the inverse of connection formula (\ref{ConnectFormula}) be
\begin{eqnarray}
\left\{
\begin{array}{l}
\gamma_0=\frac{4}{\pi}\arccos\left(-\frac{1}{4}s_1^\mathbb{R}+\frac{1}{4}\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \right)-1\\
\gamma_1=\frac{4}{\pi}\arccos\left(-\frac{1}{4}s_1^\mathbb{R}-\frac{1}{4}\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \right)-3
\end{array}
\right. ,\label{CF-Inverse}
\end{eqnarray}
where the values of the $\arccos$ terms may be complex and should be given by their principal values.
Suppose $w_0(r)$ and $w_1(r)$ have asymptotic (\ref{InfAsymp}) at $r=\infty$
and that $(\gamma_0, \gamma_1)$ is calculated from (\ref{CF-Inverse}).
Define $(\rho_0,\rho_1)$ by (\ref{rhos-DEF}) and $s=\ln (r)$.
Denote $\gamma_i^\mathbb{R}=\Re(\gamma_i)$, $\gamma_i^\mathbb{I}=\Im(\gamma_i)$, $\rho_i^\mathbb{R}=\Re(\rho_i)$, $\rho_i^\mathbb{I}=\Im(\rho_i)$, $i=0,1$.
Then, the characteristics of a solution parameterized by a point in region $\Omega_i$, $i=1,\cdots,6$, are the following.
\begin{itemize}
\item[$\Omega_1:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \in \mathbb{R}$,
$\gamma_0 \in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$.
$e^{2 w_0(r)}$ and $e^{2 w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{2 w_0} \xlongrightarrow{ s \rightarrow -\infty} e^{\gamma_0 s+\rho_0} \\
e^{2 w_1} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{\gamma_1 s+\rho_1} \right)
\end{array}
\right. .
\]
\item[$\Omega_2:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \in \mathbb{R}$,
$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$.
$e^{-2 w_0(r)}$ and $e^{2 w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{-2 w_0} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{-\gamma_0 s-\rho_0} \right) \\
e^{2 w_1} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{\gamma_1 s+\rho_1} \right)
\end{array}
\right. .
\]
\item[$\Omega_3:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \in \mathbb{R}$,
$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \in \mathbb{R}$.
$e^{-2 w_0(r)}$ and $e^{-2 w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{-2 w_0} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{-\gamma_0 s-\rho_0} \right) \\
e^{-2 w_1} \xlongrightarrow{ s \rightarrow -\infty} e^{-\gamma_1 s-\rho_1}
\end{array}
\right. .
\]
\item[$\Omega_4:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \in \mathbb{R}$,
$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$.
$e^{-2w_1(r)}$ and $e^{-2w_0(r)-2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{lll}
e^{-2w_1} &\xlongrightarrow{ s \rightarrow -\infty}& e^{-\gamma_1^{\mathbb{R}} s} \left( \frac{8 e^{-\rho_0^{\mathbb{R}}}}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^2}
\cos \left(\gamma_0^{\mathbb{I} } s+\rho_0^{\mathbb{I}} \right)
+2\, e^{-\rho_1^{\mathbb{R}}} \cos\left( \gamma_1^{\mathbb{I}}s+\rho_1^{\mathbb{I}}\right) \right)\\
e^{-2w_0-2w_1} &\xlongrightarrow{ s \rightarrow -\infty}& e^{-(\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R})s}
\bigg\{ 2\,e^{-\rho_0^\mathbb{R}-\rho_1^\mathbb{R}} \frac{(\gamma_0^\mathbb{I}+\gamma_1^\mathbb{I})^2}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^2}
\cos\left( (\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})s+\rho_0^\mathbb{I}-\rho_1^\mathbb{I} \right) \\
&& +\frac{16 e^{-2 \rho_0^\mathbb{R}} (\gamma_0^\mathbb{I})^2}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^4}
+e^{-2 \rho_1^\mathbb{R}} (\gamma_1^\mathbb{I})^2
+2\,e^{-\rho_0^\mathbb{R}-\rho_1^\mathbb{R}} \cos\left( (\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R})s+\rho_0^\mathbb{I}+\rho_1^\mathbb{I}\right) \bigg\}
\end{array}
\right. .
\]
\item[$\Omega_5:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \not\in \mathbb{R}$,
$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$.
$e^{2 w_0(r)}$ and $e^{-2 w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{2 w_0} \xlongrightarrow{ s \rightarrow -\infty}
2\, \Re \left( e^{\gamma_0 s+\rho_0} \right) \\
e^{-2 w_1} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{-\gamma_1 s-\rho_1} \right)
\end{array}
\right. .
\]
\item[$\Omega_6:$] $\sqrt{8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R}} \in \mathbb{R}$,
$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$.
$e^{2w_0(r)}$ and $e^{2w_0(r)+2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{lll}
e^{2w_0} &\xlongrightarrow{ s \rightarrow -\infty}& e^{\gamma_0^{\mathbb{R}} s} \left( \frac{8 e^{\rho_1^{\mathbb{R}}}}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^2}
\cos \left(\gamma_1^{\mathbb{I} } s+\rho_1^{\mathbb{I}} \right)
+2\, e^{\rho_0^{\mathbb{R}}} \cos\left( \gamma_0^{\mathbb{I}}s+\rho_0^{\mathbb{I}}\right) \right) \\
e^{2w_0+2w_1} &\xlongrightarrow{ s \rightarrow -\infty}& e^{(\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R})s}
\bigg\{ 2\,e^{\rho_0^\mathbb{R}+\rho_1^\mathbb{R}} \frac{(\gamma_0^\mathbb{I}+\gamma_1^\mathbb{I})^2}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^2}
\cos\left( (\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})s+\rho_0^\mathbb{I}-\rho_1^\mathbb{I} \right) \\
&& + \frac{16 e^{2 \rho_1^\mathbb{R}} (\gamma_1^\mathbb{I})^2}{(\gamma_0^\mathbb{I}-\gamma_1^\mathbb{I})^4}
+e^{2 \rho_0^\mathbb{R}} (\gamma_0^\mathbb{I})^2
+2\,e^{\rho_0^\mathbb{R}+\rho_1^\mathbb{R}} \cos\left( (\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R})s+\rho_0^\mathbb{I}+\rho_1^\mathbb{I}\right) \bigg\}
\end{array}
\right. .
\]
\end{itemize}
On the edges, $ 8+\left(s_1^\mathbb{R}\right)^2+4 s_2^\mathbb{R} $ is always non-negative.
Define
\begin{eqnarray*}
&& b_1= \frac{1}{2} \psi(\frac{3-\gamma_0}{8})+\frac{1}{2} \psi(\frac{5+\gamma_0}{8}) -\gamma_{\text{eu}} +4 \ln 2,\\
&& b_2= \frac{1}{2} \psi(\frac{3+\gamma_1}{8})+\frac{1}{2} \psi(\frac{5-\gamma_1}{8}) -\gamma_{\text{eu}} +4 \ln 2,\\
&& b_3=-\frac{1}{4} \psi(\frac{3-\gamma_0}{4}) -\frac{1}{4}\psi(\frac{\gamma_0-3}{4})
+\frac{1}{3-\gamma_0}-2 \ln 2+\frac{\gamma_{eu} }{2}.
\end{eqnarray*}
Then, the characteristics of a solution parameterized by a point on an edge are the following.
\begin{itemize}
\item[$E_1^U:$]$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 =1$, $\gamma_0^\mathbb{R}=-1 $, $\rho_0 \not\in \mathbb{R}$
and $\rho_1$ is not defined.
$e^{-2w_0(r)}$ and $e^{2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{-2 w_0} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{-\gamma_0 s-\rho_0} \right) \\
e^{2w_1(r)} \xlongrightarrow{ s \rightarrow -\infty} -2 s+ b_1
\end{array}
\right. .
\]
\item[$E_2^U:$]$\gamma_0 =-1$, $\gamma_1 \not\in \mathbb{R}$, $\gamma_1^\mathbb{R}=1$, $\rho_1 \not\in \mathbb{R}$
and $\rho_0$ is not defined.
$e^{-2w_0(r)}$ and $e^{2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{-2 w_0} \xlongrightarrow{ s \rightarrow -\infty} -2 s+b_2\\
e^{2w_1(r)} \xlongrightarrow{ s \rightarrow -\infty} 2\, \Re \left( e^{\gamma_1 s+\rho_1} \right)
\end{array}
\right. .
\]
\item[$E_1^D:$]$\gamma_0 =3$, $\gamma_1 \not\in \mathbb{R}$, $\gamma_1^\mathbb{R}=1$, $\rho_1 \not\in \mathbb{R}$
and $\rho_0$ is not defined.
$e^{2w_1(r)}$ and $e^{2w_0(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{2w_0(r)} \xlongrightarrow{ s \rightarrow -\infty} e^{\gamma_0 s} \left( -\frac{8}{(\gamma_1^\mathbb{I})^2} s+d_0 -\frac{8}{(\gamma_1^\mathbb{I})^3}\cos\left( \gamma_1^\mathbb{I} s+\rho_1^\mathbb{I} \right) \right) \\
e^{2 w_1} \xlongrightarrow{ s \rightarrow -\infty} 2\, e^{\gamma_1^\mathbb{R} s+\rho_1^\mathbb{R}}
\left( \cos\left( \gamma_1^\mathbb{I} s+\rho_1^\mathbb{I} \right) + \frac{(1-\sin(\gamma_1^\mathbb{I}s+\rho_1^\mathbb{I}) )^2}{\gamma_1^\mathbb{I}s-\frac{(\gamma_1^\mathbb{I})^3}{8}d_0 +\cos(\gamma_1^\mathbb{I} s+\rho_1^\mathbb{I})} \right)
\end{array}
\right. ,
\]
where $d_0=\lim\limits_{s_1^\mathbb{R} \rightarrow 1-\frac{s_2^\mathbb{R}}{2}+0_-} 2\, e^{\rho_0^\mathbb{R}}
\left(\rho_0^\mathbb{I}+\frac{\pi}{2}\right)$.
\item[$E_2^D:$]$\gamma_0 \not\in \mathbb{R}$, $\gamma_1=-3$, $\gamma_0^\mathbb{R}=-1$, $\rho_0 \not\in \mathbb{R}$
and $\rho_1$ is not defined.
$e^{-2w_0(r)}$ and $e^{-2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{
\begin{array}{l}
e^{-2 w_0} \xlongrightarrow{ s \rightarrow -\infty} 2\, e^{-\gamma_0^\mathbb{R} s-\rho_0^\mathbb{R}}
\left( \cos\left( \gamma_0^\mathbb{I} s+\rho_0^\mathbb{I} \right) + \frac{8(1+\sin(\gamma_0^\mathbb{I}s+\rho_0^\mathbb{I}) )^2}{-8\gamma_0^\mathbb{I}s+ (\gamma_0^\mathbb{I})^3 \tilde{d}_0+8 \cos(\gamma_0^\mathbb{I} s+\rho_0^\mathbb{I})} \right) \\
e^{-2w_1(r)} \xlongrightarrow{ s \rightarrow -\infty} e^{-\gamma_1 s} \left(-\frac{8}{(\gamma_0^\mathbb{I})^2} s+\tilde{d}_0+\frac{8}{(\gamma_0^\mathbb{I})^3}\cos(\gamma_0^\mathbb{I} s+\rho_0^\mathbb{I})\right)
\end{array}
\right. ,
\]
where $\tilde{d}_0=\lim\limits_{s_1^\mathbb{R} \rightarrow \frac{s_2^\mathbb{R}}{2}-1+0_+} 2\, e^{-\rho_1^\mathbb{R}}
\left( \frac{\pi}{2}-\rho_1^\mathbb{I} \right)$.
\item[
$E_3^R:$]$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$,
$\gamma_0^\mathbb{R}=3$, $\gamma_1^\mathbb{R}=1$ and $\gamma_0^\mathbb{I}=\gamma_1^\mathbb{I}$.
Both $\rho_0$ and $\rho_1$ are not defined.
$e^{2w_0(r)}$ and $e^{2w_0(r)+2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{ \begin{array}{l}
e^{2 w_0} \xlongrightarrow{ s \rightarrow -\infty}-e^{\gamma_0^\mathbb{R} s} \left( \frac{4}{(\gamma_0^\mathbb{I})^2} (s+ \Re(b_3) ) \sin(\gamma_0^\mathbb{I} s+\theta_0)+\frac{4}{(\gamma_0^\mathbb{I})^3} \cos(\gamma_0^\mathbb{I} s+\theta_0) \right) \\
e^{2w_0+2w_1(r)} \xlongrightarrow{ s \rightarrow -\infty} e^{(\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R} )s} \left( \frac{4}{(\gamma_0^\mathbb{I})^2} (s+\Re(b_3))^2
-\frac{4}{(\gamma_0^\mathbb{I})^4} \left( \cos(\gamma_0^\mathbb{I} s+\theta_0) \right)^2 \right)
\end{array}
\right. ,
\]
where $\theta=\lim\limits_{s_1^\mathbb{R} \rightarrow 2 \sqrt{-2-s_2^\mathbb{R}} +0_+} \rho_0^\mathbb{I} $.
\item[$E_3^L:$]$\gamma_0 \not\in \mathbb{R}$, $\gamma_1 \not\in \mathbb{R}$,
$\gamma_0^\mathbb{R}=-1$, $\gamma_1^\mathbb{R}=-3$ and $\gamma_0^\mathbb{I}=\gamma_1^\mathbb{I}$.
Both $\rho_0$ and $\rho_1$ are not defined.
$e^{-2w_1(r)}$ and $e^{-2w_0(r)-2w_1(r)}$ are smooth for $r \in (0,\infty)$.
Their asymptotics at $s=-\infty$ are
\[
\left\{ \begin{array}{l}
e^{-2 w_1} \xlongrightarrow{ s \rightarrow -\infty} e^{-\gamma_1^\mathbb{R} s}
\left( \frac{4}{(\gamma_1^\mathbb{I})^2} (s+ \Re(b_3) )
\sin(\gamma_1^\mathbb{I} s-\tilde{\theta}_0)
+\frac{4}{(\gamma_1^\mathbb{I})^3} \cos(\gamma_1^\mathbb{I} s-\tilde{\theta}_0) \right) \\
e^{-2w_0-2w_1(r)} \xlongrightarrow{ s \rightarrow -\infty} e^{-(\gamma_0^\mathbb{R}+\gamma_1^\mathbb{R} )s} \left( \frac{4}{(\gamma_1^\mathbb{I})^2} (s+\Re(b_3))^2
-\frac{4}{(\gamma_1^\mathbb{I})^4} \left( \cos(\gamma_1^\mathbb{I} s-\tilde{\theta}_0) \right)^2 \right)
\end{array}
\right. ,
\]
where $\tilde{\theta}_0=-\lim\limits_{s_1^\mathbb{R} \rightarrow -2 \sqrt{-2-s_2^\mathbb{R}} +0_-} \rho_1^\mathbb{I} $.
\end{itemize}
\end{conjecture}
\subsection{Verify the conjecture numerically: the $\Omega_1$ case as an example}
In this subsection, we will verify Conjecture \ref{Conj} for the $\Omega_1$ case with $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2,1)$.
Then by (\ref{CF-Inverse}) we get
\begin{eqnarray}
\left\{
\begin{array}{l}
\gamma_0|_{s_1^\mathbb{R}=2, s_2^\mathbb{R}=1}=\frac{1}{3}\\
\gamma_1|_{s_1^\mathbb{R}=2, s_2^\mathbb{R}=1}=\frac{4}{\pi}\arccos(-\frac{3}{2})-3
=1+\frac{4 \mathrm{i}}{\pi}\ln(\frac{3-\sqrt{5}}{2})
\end{array}
\right. .\label{gam12-Omega1}
\end{eqnarray}
With $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2,1)$, $w_0$ and $w_1$ keep real as $r$ decreasing from $r=\infty$ to $r=1$.
So we do not need adjust our numerical integration for $r>1$.
By the truncation of (\ref{iniApprox}), the initial values at $r=45$ are obtained (only the first $50$ digits are listed)
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(45)= -5.2843098725232974899221393911204991207504443469367... \times 10^{-57}\\
w_p'(45)= 1.5004885502015739552694025310567337731833237644509... \times 10^{-56}\\
w_m(45)= 1.9951075414429511313096218077209854214432475688359... \times 10^{-80}\\
w_m'(45)=-8.0025674082272017400263083591640194065100562879396... \times 10^{-80}
\end{array}
\right. .\label{ini-Omega1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of the initial values (\ref{ini-Omega1}) are obtained as shown by Table \ref{tab17}.
\begin{table}[ht]
\caption{Errors of the initial values of case $\Omega_1$ with $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2,1)$.}
\label{tab17}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=45$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$3.04 \times 10^{-170}$ &$2.59 \times 10^{-169}$ &$1.62 \times 10^{-193}$&$1.57\times 10^{-192}$ \\
| Relative Error | &$5.76\times 10^{-114}$ &$1.73 \times 10^{-113}$ &$8.12 \times 10^{-114}$&$1.96\times 10^{-113}$
\end{tabular}
\end{table}
Integrating numerically the tt* equation (\ref{wpwm}) from $r=45$ to $r=1$
by the Gauss-Legendre method with parameters as same as the ones in Subsection \ref{GeneralCase},
the values of $w_p$, $w_p'$, $w_m$ and $w_m'$ at $r=1$ are obtained
\begin{eqnarray}
\left\{
\begin{array}{l}
w_p(1)= -3.8224055163443861381648888321249635590437848425393...\times 10^{-2} \\
w_p'(1)= 1.2620798170393397054252193737795545512207073701669...\times 10^{-1}\\
w_m(1)= 4.1421810495867924927295926159960489963050832028643...\times 10^{-3}\\
w_m'(1)=-1.9704834137414281607395710259152505912708048802280....\times 10^{-2}
\end{array}
\right. .\label{values-Omega1-1}
\end{eqnarray}
Comparing with the more accurate solution starting from $r=55$,
the errors of (\ref{values-V1-1}) are obtained as shown by Table \ref{tab18}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $r=1$ of case $\Omega_1$ with $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2,1)$.}
\label{tab18}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$r=1$& $w_p$ & $w_p'$ & $w_m$ &$w_m'$\\
\hline
| Absolute Error | &$4.42 \times 10^{-115}$ &$1.46 \times 10^{-114}$ &$6.30 \times 10^{-116}$&$3.09\times 10^{-115}$\\
| Relative Error | &$1.16\times 10^{-113}$ &$1.16 \times 10^{-113}$ &$1.52 \times 10^{-113}$&$1.57\times 10^{-113}$
\end{tabular}
\end{table}
When $r<1$, $w_0$ and $w_1$ may be complex.
As Conjecture \ref{Conj} suggests, we use $v_0$ and $v_1$
\begin{eqnarray}
\left\{
\begin{array}{l}
v_0=e^{2 w_0}\\
v_1=e^{2 w_1}
\end{array}
\right. . \label{v0v1-DEF1}
\end{eqnarray}
as dependent variables for the $\Omega_1$ case.
Then, $v_0$ and $v_1$ will be real for $r>0$.
To improve computation efficiency, we use $s=\ln(r)$ as independent variable.
Then the equations for $v_0$ and $v_1$ are
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{d^2 v_0}{ds^2}=4 e^{2 s} (v_0^3-v_1) +\frac{1}{v_0} (\frac{d v_0}{ds})^2 \\
\frac{d^2 v_1}{ds^2}=4 e^{2 s} \left( \frac{v_1^2}{v_0}-\frac{1}{v_1} \right)
+\frac{1}{v_1} (\frac{d v_1}{ds})^2
\end{array}
\right. . \label{v0v1-DEQ1}
\end{eqnarray}
The truncation structure of (\ref{v0v1-DEQ1}) should be
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{d^2 v_0^{(0)}}{ds^2}=\frac{1}{v_0^{0}} \left(\frac{d v_0^{(0)}}{ds} \right)^2 \\
\frac{d^2 v_1^{(0)}}{ds^2}=-\frac{4 e^{2 s}}{v_1^{(0)}}
+\frac{1}{v_1^{(0)}} \left(\frac{d v_1^{(0)}}{ds} \right)^2
\end{array}
\right. . \label{v0v1-trunc-DEQ1}
\end{eqnarray}
In fact, after substituting (\ref{gam12-Omega1}) to the $\Omega_1$ case of Conjecture \ref{Conj},
it becomes obvious which terms of (\ref{v0v1-DEQ1}) should be ignored.
The solution of (\ref{v0v1-trunc-DEQ1}) is known
\begin{eqnarray}
\left\{
\begin{array}{l}
v_0^{(0)}(s)= e^{a_{1\Omega_1} s+b_{1\Omega_1}} \\
v_1^{(0)}(s)=-2 e^s \frac{ \cos( a_{2\Omega_1} s+b_{2\Omega_1} ) }{a_{2\Omega_1}}
\end{array}
\right. . \label{v0v1-trunc-DEQ1-Sol}
\end{eqnarray}
Comparing (\ref{v0v1-trunc-DEQ1-Sol}) with Conjecture \ref{Conj}, we know $a_{1\Omega_1}=\gamma_0$,
$b_{1\Omega_1}=\rho_0$, $a_{2\Omega_1}=\Im (\gamma_1)$ and $b_{2\Omega_1}=\Im (\rho_1)$.
Also we note $-\frac{1}{\Im (\gamma_1)}=e^{\Re (\rho_1)}$ in the $\Omega_1$ case.
The neglected terms of the truncation from (\ref{v0v1-DEQ1}) to (\ref{v0v1-trunc-DEQ1})
are $ 4 e^{2 s} (v_0^3-v_1)$ and $4 e^{2 s} \frac{v_1^2}{v_0} $,
which are of order $O(e^{3 s})$ and $O(e^{\frac{11}{3}s})$ respectively (considering (\ref{gam12-Omega1})).
So the relative errors are both of order $O(e^{\frac{8}{3}s})$ except near the zeros of $v_1(s)$.
Since $v_0$ and $v_1$ are both small in this case, only the relative errors are relevant.
To avoid the inconvenience brought by the relative error,
we will take
\begin{eqnarray}
\left\{
\begin{array}{l}
\Delta_0(s)=|e^{2 w_0} e^{-\gamma_0 s-\rho_0}-1|\\
\Delta_1(s)=|\frac{1}{2}e^{2 w_1} e^{-\Re(\gamma_1) s-\Re(\rho_1)} -\cos(\Im(\gamma_1)s+\Im(\rho_1))|
\end{array}
\right. \label{DelDel1-DEF}
\end{eqnarray}
as the measurement of error.
So, $\Delta_0$ and $\Delta_1$ are both of order $O(e^{\frac{8}{3}s})$.
Solving $e^{\frac{8}{3}s_f}=10^{-100}$, we get $s_f \approx 86.35$. For safety and convenience, we set $s_f=-87$.
Numerical results show $v_0(s)$ has no zero for $s \in (-\infty, 0]$ but $v_1(s)$ has, just Conjecture \ref{Conj} predicts.
For the sake of numerical integration, it is better to integrate around the zeros of $v_1(s)$.
In order to keep away from the zeros of $v_1(s)$, we first compute $v_0(s+\mathrm{i} \epsilon)$ with $\epsilon=10^{-2}$
to determine the approximate zeros of $v_1(s)$ by solving $\mathrm{Re}(v_1(s+\mathrm{i} \epsilon))=0$.
Then we get the approximate zeros $s_i$ of $v_1(s)$ within the range $-87 \le s \le 0$.
The first few of them are listed as Table \ref{tab19}.
\begin{table}[ht]
\caption{The first few approximate zeros $s_i$ of $v_1(s)$ for the $\Omega_1$ case with $(s_1^\mathbb{R}, s_2^\mathbb{R})=(2,1)$.}
\label{tab19}
\centering
\begin{tabular}[h]{c|c c c c c c c c}
\hline \hline
$s_i$ & $s_1$ & $s_2$ & $s_3$ &$s_4$&$s_5$&$s_6$&$s_7$&$s_8$\\
\hline
value& $-2.506$ &$-5.069$ &$-7.633$& $-10.197$ &$-12.760$ & $-15.324$&$-17.888 $&$-20.452 $
\end{tabular}
\end{table}
Obviously, the distance between two adjacent zeros in Table \ref{tab19} is about $2.5$.
To avoid the numerical instabilities caused by those zeroes, we use a contour in the complex plane of $s$,
as shown by Figure \ref{fig-4}.
\input{Figure4.tikz}
The radius of the circles surrounding the zeroes are set to $\frac{1}{5}$.
In principle, the values of $v_i$ can be evaluated by the Cauchy's integral formula
$ v_i(s)=\frac{1}{2 \pi \mathrm{i}} \oint \frac{v_i(\xi)}{\xi-s} d\xi$.
But $v_i(\xi)$ has high-precision values only at some fixed points on the circle.
At a point other than those, interpolation must be used, which is not proper for high-precision purpose.
Since $v_i$ are periodic functions on the circle, we use the trapezoidal rule to calculate them
\begin{eqnarray}
v_i(s)=\frac{1}{2 \, n} \sum_j \frac{\tilde v_i(\theta_j)}{R e^{\mathrm{i} \theta_j}-s} R e^{\mathrm{i} \theta_j},
\quad i=0,1, \label{Trapezoidal}
\end{eqnarray}
where $R=\frac{1}{5}$ denotes the radius of the circle,
and $\tilde v_i(\theta_j)$ the value of $v_i$ at $\theta_j$ on the circle.
The distance between the adjacent $\theta_j$ is $\frac{\pi}{n}$.
Obviously, formula (\ref{Trapezoidal}) is not proper for a point near the circle.
It is why the contour has $2$ line segments in each circle. We use line segment of length $\frac{1}{10}$.
In our numerical experiments, $n$ is equal to $1000$, which is far more enough to guarantee our $100$-digits precision goal.
The plots of $v_0$ and $v_1$ is shown by Figure \ref{fig-5}.
\input{Figure5.tikz}
Table \ref{tab20} shows our numerical solution is accurate as we expect.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-87$ for the $\Omega_1$ case with $(s_1^\mathbb{R},s_2^\mathbb{R} )=(2,1)$.}
\label{tab20}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-87$ & $v_0$ & $\frac{dv_0}{ds}$ & $v_1$ &$\frac{d v_1}{ds}$\\
\hline
| Absolute Error | &$4.06 \times 10^{-125}$ &$1.31 \times 10^{-125}$ &$2.75 \times 10^{-149}$&$2.30\times 10^{-149}$ \\
| Relative Error | &$1.48 \times 10^{-112}$ &$1.43 \times 10^{-112}$ &$7.69 \times 10^{-111}$&$6.35\times 10^{-112}$
\end{tabular}
\end{table}
Table \ref{tab21} shows how good the asymptotic solution is.
\begin{table}[ht]
\caption{Approximate derivation from the asymptotic solution
for the $\Omega_1$ case with $(s_1^\mathbb{R},s_2^\mathbb{R} )=(2,1)$.}
\label{tab21}
\centering
\begin{tabular}[h]{ c| c c c c c c c}
\hline \hline
$s$ &$-27$ & $-37$ & $-47$ &$-57$ &$-67$& $-77$ & $-87$\\
\hline
$\ln(\Delta_0(s))$&$-73.2379$ &$-100.773$&$-130.684$&$-155.003$&$-181.988$&$-208.457$ &$-233.699$ \\
$\ln(\Delta_1(s))$
&$-72.5076 $ &$-98.9279$ &$-125.524$&$-152.287$&$-179.229$&$-206.373$ & $-233.697$
\end{tabular}
\end{table}
\section{Deviating from (\ref{rhos-DEF}) \label{DFG}}
This section concerns with how the solution looks like if (\ref{rhos-DEF}) is dissatisfied.
First, we derive a better asymptotics near $r=0$, which is suitable to give initial values for the numerical integration.
Also, it shows how the truncation structure works.
Then, the tt* equation is integrated numerically.
The integrating contour on the complex plane of $r$ is used to surround the singularities.
We will find the singularities distribute regularly.
However, we should understand the difficulties here are much larger than the ones encountered in Section \ref{CONJ},
where we could formulate conjecture based on the numerical results.
This is also understandable since we have in fact four independent parameters $\gamma_0$, $\gamma_1$, $\rho_0$ and $\rho_1$
while in Section \ref{CONJ} we have essentially only two parameters $s_1^\mathbb{R}$ and $s_2^\mathbb{R}$.
For convenience, in this section we will always use dependent variables $v_0$ and $v_1$
as defined by (\ref{v0v1-DEF1}).
As independent variable, we use $s=\ln(r) $ for $r \le 1$ as before.
So the equations for $v_0$ and $v_1$ are still (\ref{v0v1-DEQ1}).
Let us take the following assumption first.
\begin{itemize}
\item[]{{\bf Assumption 1}: Both terms $4 e^{2 s} (v_0^3-v_1)$ and $4 e^{2 s} \left( \frac{v_1^2}{v_0}-\frac{1}{v_1} \right)$
in (\ref{v0v1-DEQ1}) are negligible near $s=-\infty$.}
\end{itemize}
So, (\ref{v0v1-DEQ1}) becomes
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{ d^2 v_0^{(0)} }{ds^2} =\frac{1}{ v_0^{(0)}} \left( \frac{ d v_0^{(0)}}{ds} \right)^2 \\
\frac{ d^2 v_1^{(0)} }{ds^2}=\frac{1}{v^{(0)}_1}\left( \frac{d v_1^{(0)}} {ds} \right)^2
\end{array}
\right. . \label{wv0wv1-DEQ}
\end{eqnarray}
The solution of (\ref{wv0wv1-DEQ}) is
\begin{eqnarray}
v_0^{(0)}= c_0 e^{\gamma_0 s}, \quad v_1^{(0)}= c_1 e^{\gamma_1 s} , \label{wv0wv1-SOLU}
\end{eqnarray}
where $c_0$, $c_1 $, $\gamma_0$ and $\gamma_1$ are constants,
which should be real if we only interest in the real solutions of (\ref{v0v1-DEQ1}).
The immediate results of Assumption 1 is that $\gamma_0$ and $\gamma_1$ satisfy the constraints
$3 \gamma_0+2 >\gamma_0$, $\gamma_1+2 >\gamma_0$, $2 \gamma_1-\gamma_0+2 >\gamma_1$
and $2-\gamma_1>\gamma_1$, which is just the inner of the triangle in Figure \ref{fig-2}.
So, if $(\gamma_0,\gamma_1)$ is a point in the inner of the triangle in Figure \ref{fig-2},
then $(v_0^{(0)}, v_1^{(0)})$ of (\ref{wv0wv1-SOLU}) is the primary approximate solution of $(v_0, v_1)$ near $s=-\infty$.
If $c_0=e^{\rho_0}$ and $c_1=e^{\rho_1}$ with $\rho_0$ and $\rho_1$ defined by (\ref{rhos-DEF}),
then the solution is the one dealt by Theorem \ref{thm-GIL-1}.
Here we are interested in the case that $c_0 \neq e^{\rho_0}$ or $c_1 \neq e^{\rho_1}$.
Now, let us transform (\ref{v0v1-DEQ1}) to its integral form
\begin{eqnarray}
\left\{
\begin{array}{l}
v_0(s) = c_0 e^{\gamma_0 s} e^{ 4 \int_{-\infty}^s d\xi \int_{-\infty}^\xi
d\zeta \left[ v_0(\zeta)^2- \frac{v_1(\zeta)}{v_0(\zeta)} \right]
e^{2 \zeta} } \\
v_1(s) = c_1 e^{\gamma_1 s} e^{ 4 \int_{-\infty}^s d\xi \int_{-\infty}^\xi
d\zeta \left[ \frac{v_1(\zeta)}{v_0(\zeta)} -\frac{1}{v_1(\zeta)^2}\right]
e^{2 \zeta} }
\end{array}
\right. . \label{v0v1-inteEQ}
\end{eqnarray}
In principle, near $s=-\infty$, (\ref{v0v1-inteEQ}) can be solved recursively:
$v_0^{(0)}$ and $v_1^{(0)}$ are given by (\ref{wv0wv1-SOLU});
$v_0^{(1)}$ and $v_1^{(1)}$ are
\begin{eqnarray}
\left\{
\begin{array}{l}
v_0^{(1)}(s) =c_0 e^{\gamma_0 s} \exp\left\{ \frac{c_0^2}{(1+\gamma_0)^2}e^{2(1+\gamma_0)s}-
\frac{4 c_1}{c_0 (2-\gamma_0+\gamma_1)^2} e^{(2-\gamma_0+\gamma_1)s} \right\} \\
v_1^{(1)}(s)=c_1 e^{\gamma_1 s} \exp\left\{ \frac{4 c_1}{c_0 (2-\gamma_0+\gamma_1)^2} e^{(2-\gamma_0+\gamma_1)s} -\frac{1}{c_1^2 (1-\gamma_1)^2}e^{2(1-\gamma_1)s} \right\}
\end{array}
\right. ,\label{v0v1-SOLU-1}
\end{eqnarray}
which are obtained by substituting $v_0=v_0^{(0)}$ and $v_1=v_1^{(0)}$ to the right of (\ref{v0v1-inteEQ});
And so on and so forth.
If $(\gamma_0,\gamma_1)$ lies in the inner of the triangle in Figure \ref{fig-2}, $v_0^{(i)}$ and $v_1^{(i)}$ converge as $i$ increases.
\subsection{Numerical solution}
As Subsection \ref{GeneralCase}, we still use $(\gamma_0,\gamma_1)=(1,\frac{1}{3})$.
To have some deviation from Subsection \ref{GeneralCase},
$c_0$ and $c_1$ should be chosen as
\begin{eqnarray}
c_0=e^{\rho_0}+\delta c_0, \quad c_1=e^{\rho_1}+\delta c_1, \nonumber
\end{eqnarray}
where $\delta c_0$ and $\delta c_1$ can not be $0$ simultaneously.
In our numerical experiment, we use
$$\delta c_0=\frac{1}{2}, \quad \delta c_1=\frac{1}{5} . $$
To solve (\ref{v0v1-DEQ1}) numerically, the initial values of $(v_0,\frac{dv_0}{ds},v_1,\frac{dv_1}{ds})$ must be given.
We start from $s_1=-100$ and give the initial values by (\ref{v0v1-SOLU-1}).
Since one can easily compute initial values by (\ref{v0v1-SOLU-1}), the details of the initial values are omitted.
We just list the errors of the initial value by Table \ref{tab22}.
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=-100$ with $(\gamma_0,\gamma_1,c_0,c_1))=(1,\frac{1}{3},e^{\rho_0}+\frac{1}{2},e^{\rho_1}+\frac{1}{5})$.}
\label{tab22}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=-100$ & $v_0$ & $\frac{dv_0}{ds}$ & $v_1$ &$\frac{d v_1}{ds}$\\
\hline
| Absolute Error | &$5.32 \times 10^{-160}$ &$1.95 \times 10^{-159}$ &$2.47 \times 10^{-131}$&$7.42\times 10^{-131}$ \\
| Relative Error | &$4.86 \times 10^{-117}$ &$1.78 \times 10^{-116}$ &$5.12 \times 10^{-117}$&$4.61\times 10^{-116}$
\end{tabular}
\end{table}
The error of the values at $s=-100$ are obtained by comparing them with the numerical solution starting from $s=-140$,
which is far more accurate.
The numerical solution is smooth for $s \in [-100,0]$.
As a comparison to (\ref{values-general-1}), the values of $v_0$ et al. at $s=0$ are
\begin{eqnarray}
\left\{
\begin{array}{l}
v_0|{s=0}= 1.3324864759152155716932764336782719490481063559703... \\
\frac{dv_0}{ds}|_{s=0}= 0.49495834671586092263807187324781656576576424051419...\\
v_1|{s=0}= 2.6783375094329925626474416219547736732331423595096...\\
\frac{dv_1}{ds}|_{s=0}= 6.2948008049596612397631881197126092308528410458148....
\end{array}
\right. .\label{values-Deviation-0}
\end{eqnarray}
Table \ref{tab23} gives the errors of (\ref{values-Deviation-0}).
\begin{table}[ht]
\caption{Errors of the numerical solution at $s=0$.}
\label{tab23}
\centering
\begin{tabular}[h]{ c|c c c c}
\hline \hline
$s=0$& $v_0$ & $\frac{dv_0}{ds}$ & $v_1$ &$\frac{dv_1}{ds}$\\
\hline
| Absolute Error | &$ 3.37 \times 10^{-113}$ &$ 1.69 \times 10^{-112}$ &$ 5.96 \times 10^{-113}$&$ 3.43\times 10^{-112}$ \\
| Relative Error | &$2.53 \times 10^{-113}$ &$ 3.41 \times 10^{-112}$ &$ 2.23 \times 10^{-113}$&$5.44 \times 10^{-113}$
\end{tabular}
\end{table}
Again, the errors are evaluated by comparing the two numerical solutions staring from $s=-140$ and from $s=-100$ respectively.
For $s>0$, i.e. $r>1$, it is convenient to use variable $r$ itself rather than $s$:
the pattern of the singularities is more transparent with respect to $r$ than with respect to $s$.
Then (\ref{v0v1-DEQ1}) is converted to
\begin{eqnarray}
\left\{
\begin{array}{l}
\frac{dv_0}{dr}=\frac{1}{r} p_0\\
\frac{dp_0}{dr}=\frac{p_0^2}{r v_0}+4 r v_0^3-4 r v_1 \\
\frac{dv_1}{dr}=\frac{1}{r}p_1\\
\frac{dp_1}{dr}=\frac{p_1^2}{r v_1}-\frac{4 r}{v_1}+\frac{4 r v_1^2}{v_0}
\end{array}
\right. . \label{v0v1-DEQr}
\end{eqnarray}
Then, we compute the numerical solution of (\ref{v0v1-DEQr}),
for which the initial values are given by (\ref{values-Deviation-0}).
Near $r \approx 1.539167317$, the numerical solution blows up.
Figure \ref{fig-6} and \ref{fig-7} show the plots of $v_0$ and $v_1$ on the circle with a radius of about $0.239167317$ around the singular point.
\input{Figure6.tikz}
\input{Figure7.tikz}
Clearly, $v_0$ and $v_1$ are smooth functions on the circle.
Numerical results show the singularity at $r \approx 1.539167317$ is a simple pole of $v_1$.
By (\ref{v0v1-DEQr}), either $v_0=\text{infinity}$ or $v_0=0$ at the singularity of $v_1$.
Numerical results indicate $v_0=0$ at this singularity of $v_1$.
To show the pattern of the singularities of $v_0$ and $v_1$,
we plot $v_i(r+10^{-2}\mathrm{i}) $, $i=0,1$ as Figure \ref{fig-8} and Figure \ref{fig-9}.
\input{Figure8.tikz}
\input{Figure9.tikz}
Though we can not give a precise description of Figure \ref{fig-8} and Figure \ref{fig-9},
we still have several heuristic observations from the two figures.
First, we can safely say both $v_0(r)$ and $v_1(r)$ have infinity of singularities
since some adjacent singularities are almost equidistant.
Second, $v_0(r)$ and $v_1(r)$ should be real since the imaginary part of $v_0(r+10^{-2}\mathrm{i})$ and $v_1(r+10^{-2}\mathrm{i})$
are small except near the singularities.
Third, $v_1(r_{singular}+0_-)>0$ and $v_1(r_{singular}+0_+)<0$ and the imaginary part of $v_1(r+0_+ \mathrm{i})$ is always positive.
Fourth, the singularities of $v_0(r)$ have two frequencies:
the class of singularities with $v_0(r_{singular}+0_+ \mathrm{i})<0$ have one frequency
and the class of singularities with $v_0(r_{singular}+0_+ \mathrm{i})>0$ have another frequency.
The first two observations should be general for cases deviating from (\ref{rhos-DEF}).
It seems there is no simple combination of $v_0$ and $v_1$ such that the result variable is smooth for $r \in (0,\infty)$.
|
2,877,628,090,188 | arxiv | \section{Introduction} \label{sec:intro}
A population of objects in co-orbital motion, as one of long-term stable and thus presumably primordial (i.e., $>4$~Gyr lifetimes) populations or as temporary captures, is known to exist with every planet in the Solar System with the sole exception of Mercury. Long-range planetary interaction can cause an object with semimajor axis very close to the planet to oscillate around the L4 or L5 Lagrange point (called trojan motion), around a point 180$^\circ$ away from the planet (called horseshoe motion), or even around the planet's longitude (quasi-satellites). Earth currently has a population of five horseshoe \citep{Wiegertetal1998,ChristouAsher2011,dlFM22016b}, five quasi-satellite \citep{Connorsetal2004,Wajer2010,dlFM22016c}, one Trojan \citep{Connorsetal2011}, and four horseshoe/quasi-satellite combination \citep{Connorsetal2002,Brasseretal2004,dlFM22016a} co-orbitals, all of which are on orbits unstable on timescales much shorter than the Solar System's age. Venus has been discovered to temporarily host: one quasi-satellite \citep{Mikkolaetal2004}, one Trojan \citep{dlFM22014a}, one quasi-satellite/horseshoe complex co-orbital \citep{Brasseretal2004}, and one Trojan/horseshoe combination co-orbital \citep{dlFM22013b}. A total of eight long-term stable Trojans have been discovered to co-orbit Mars \citep{Scholletal2005,dlFM22013a}.
Among the giant planets, Jupiter and Neptune are known to have large stable Trojan populations, the Neptune Trojans possibly outnumbering those of Jupiter \citep{Alexandersenetal2016}. Neptune has also been discovered to have a handful (8 in total so far) of temporarily-trapped Trojans on unstable orbits \citep{Brasseretal2004,HornerLykawka2010,dlFM22012a,dlFM22012b,Guanetal2012,HornerLykawka2012,Horneretal2012,Alexandersenetal2016}. Uranus has two known temporary Trojans \citep{Alexandersenetal2013,dlFM22017}. Saturn was very recently discovered to host four possible transient co-orbitals on retrograde (inclination $i>90^o$) orbits with very short (i.e., $\lesssim4$~kyr) potential captures \citep{MoraisNamouni2013a,Lietal2018}. In addition, Jupiter may have a few very short-term ($<1$~kyr) captured co-orbitals \citep{Karlsson2004}.
The longest-lived transient co-orbital discovered to date for Jupiter and Saturn is a co-orbital with Jupiter \citep{Wiegertetal2017,NamouniMorais2018}; (514107) 2015 BZ$_{509}$ (hereafter referred to as BZ509) is currently on a retrograde jovian co-orbital orbit ($a=5.139$~AU, $e=0.380$, $i=163.022^o$, $H=16.0$) and remains resonant for tens of thousands of years \citep{Wiegertetal2017}. BZ509 was shown by \citet{Wiegertetal2017} in addition to remain with semimajor axis $a$ near (within a few tenths of an AU) that of Jupiter for $\sim1$~Myr, with often no formal resonant angle libration. \citet{Huangetal2018} show additional integrations with libration of the resonant argument for $\approx200$~kyr. To study the long-term stability of BZ509, \citet{NamouniMorais2018} numerically integrated one million clones of the object and found a 0.003\% chance that $a$ remains near that of Jupiter for $>4$~Gyr. Citing the Copernican Principle that posits BZ509 has not been observed at any preferred epoch in Solar System history, \citet{NamouniMorais2018} proposed BZ509 is an interstellar object that was captured into the retrograde jovian co-orbital state $>4$~Gyr ago. However, we demonstrate it is also possible a population of temporarily-stable jovian retrograde co-orbitals are continuously resupplied from a source within the Solar System. Such a steady-state resupply source has been used \citep{Alexandersenetal2013} to successfully explain the number of transient neptunian Trojans and the single temporarily-trapped uranian Trojan 2011 QF99.
\section{Potential sources of jovian co-orbitals} \label{sec:sources}
We consider two potential Solar System sources for transient jovian co-orbitals on direct and retrograde orbits: the near-Earth asteroids and inwardly-migrating Centaurs. Using the two models described below, we search for transient jovian co-orbital production (on both direct and retrograde orbits) from each source population and estimate their steady-state population.
The near-Earth object (NEO) orbital distribution model from \citet{Greenstreetetal2012a} provides the steady-state NEO population originating from escaping main-belt asteroids. The orbital histories of 7,000 test particles integrated for 100-200 Myr were stored at 300 year intervals. The vast majority of the asteroid test particles were captured into the NEO region, but some migrated outward in semimajor axis from the main asteroid belt, often getting ejected from the Solar System by Jupiter. \citet{Greenstreetetal2012a,Greenstreetetal2012b} discovered that 0.2\% of the steady-state NEO population is on retrograde (inclination $i>90^o$) orbits. We search the orbital histories of all NEO model test particles for temporarily-trapped jovian co-orbitals in both direct and retrograde orbits.
A steady-state model of the $a<34$~AU Centaur population as computed by \citet{Alexandersenetal2013}, using an incoming scattering object model from \citet{Kaibetal2011}, was used to determine the frequency of temporarily-trapped co-orbitals on direct orbits with Uranus and Neptune. This model was updated by \citet{Alexandersenetal2018} to extend to transient co-orbitals of Saturn and a lower-limit on those with Jupiter. The orbital histories for all test particles were stored at 50 year intervals for a total integration time of 1~Gyr. In addition to the the temporarily-trapped co-orbitals with Jupiter on direct orbits searched for in \citet{Alexandersenetal2018}, we search the Centaur histories for transient jovian co-orbitals on retrograde orbits.
We have not included the Oort cloud as a potential source region. Although it is possible for Oort cloud comets with small enough perihelia to have their aphelia dropped to within the giant planet region through numerous planetary close encounters, the efficiency of this process is likely low. In any case, such objects would almost certainly first transit through the moderate-$a$ state that is our source region, and thus if some TNOs in that region are returning Oort cloud objects they are already included in our model.
\subsection{Co-orbital Detection} \label{subsec:coorb_detection}
The formal definition of a direct co-orbital state is that the resonant angle $\phi_{1:1}=\lambda-\lambda_{planet}$ librates, where $\lambda$ is the mean longitude of the small body and $\lambda_{planet}$ is the mean longitude of the planet. While detecting this libration in the 0.5~TB of NEO orbital histories (at 300 year intervals) and the 250~GB of Centaur orbital histories (at 50 year intervals) is difficult to automate, an automatic process is necessary to filter the large outputs.
Instead, to diagnose whether particles are co-orbital we used the simpler method of scanning the semimajor axis history using a running window, which diagnoses co-orbitals well \citep{Alexandersenetal2013}. The length of the running window was chosen in each source region's case (NEOs: 9~kyr, Centaurs: 5~kyr) to be several times longer than the typical libration period at Jupiter. A particle was classified as a co-orbital if, within the running window, both its average semimajor axis $a$ was within 0.4~AU of Jupiter's average $a$ and no individual semimajor axis value differed by more than 3.5 times Jupiter's Hill-sphere radius $R_H=1.2$~AU from Jupiter's $a$. If these requirements were met, the orbital elements and the integration time at the center of the running window for that particle were output to indicate co-orbital motion in that window. The window center was then advanced by a single integration output interval (300 years for the NEOs and 50 years for the Centaurs) and the diagnosis was performed again on the next running window. This records consecutive identifications of a particle temporarily trapped in co-orbital motion with Jupiter as a single ``trap" until the object is scattered away. A minor shortcoming of this co-orbital identification method is that the beginning and end of each trap is not well-diagnosed due to the ends of the window not entirely falling withing the trap at these times. This method provides us with estimates of the duration of temporary traps, each of which must be greater than the length of the running window to be diagnosed, to within a factor of two accuracy \citep{Alexandersenetal2013}.
\subsection{Resonant Island Classification} \label{subsec:res_island_class}
For each time step a particle has been classified as a co-orbital, we determine in which of the four resonant islands the particle is librating, i.e., whether it is a horseshoe, L4 Trojan, L5 Trojan, or quasi-satellite, using a method similar to that in \citet{Alexandersenetal2013}. Our co-orbital detection algorithm produced nearly 1,800 total temporary traps, which requires another automated process to determine resonant island classification. As with the detection algorithm, this is similarly difficult to automate especially because complex variations and combinations can exist for high inclinations.
For our resonant island classification algorithm, we examine the behavior of two versions of the resonant angle $\phi_{1:1}$. For objects in direct co-orbital motion with Jupiter, we use the traditional definition of the resonant angle $\phi_{1:1}=\lambda-\lambda_J$. If $\phi_{1:1}$ remains in the leading or trailing hemisphere for the duration of a running window, we assign the particle to the L4 or L5 state, respectively. If $\phi_{1:1}$ crosses $180^o$ at any time during the window interval, the co-orbital is labelled a horseshoe. All remaining orbits are classified as quasi-satellites, as they must be co-orbitals that cross between the leading and trailing hemispheres at $\phi_{1:1}=0^o$ and not at $180^o$.
Although the possibility of erroneous classifications exist with this method, we find these errors affect $<10\%$ of cases upon manual inspection of dozens of cases. The majority of these examples were particularly chosen as co-orbitals that experience multiple transitions between Trojan, horseshoe, and/or quasi-satellite states as well as possible times of non-resonant behavior (resonant argument circulation) as the temporary co-orbitals move in and out of 1:1 resonant capture. To ensure accurate classification of periods of resonant argument libration within a running window, the average and individual semimajor axis limits and running window length described in Section~\ref{subsec:coorb_detection} above were adjusted until periods of co-orbital behavior with resonant argument libration were correctly identified $>90\%$ of the time. In addition, these parameters were adjusted to increase correct classifications of resonant island libration behavior (i.e., Trojan, horseshoe, and quasi-satellite behavior) to the same level of accuracy. This includes periods of transitions between multiple resonant islands, which almost always occur on timescales longer than the length of the running window. An additional minor shortcoming of this co-orbital identification method is the difficulty of correctly classifying resonant island libration during periods of transition between states, however, as stated above, we find these affect $<10\%$ of co-orbital classifications in our simulations. We note that another limitation to our resonant island classification method is that co-orbitals with large amplitude librations that encompass libration around Lagrange points not typically associated with their resonant state (e.g., large amplitude Trojans whose librations extend beyond either the leading or trailing hemisphere to $\phi>180^o$ or $\phi<0^o$) would likely not be well classified with our identification method. However, we find such large amplitude libraters to be rare ($<10\%$) among the transient co-orbitals in our simulations. Thus, inaccurate classifications do not greatly affect our co-orbital fraction and resonant island distribution estimates, supporting our goal of better than factor of two accuracy.
Following the convention for retrograde orbits of \citet{MoraisNamouni2013b}, we define the 1:-1 resonant argument for retrograde orbits to be $\phi^\star=\lambda^\star-\lambda_J-2\omega^\star$ (their Equation 9). Here, $\lambda_J$ is the mean longitude of Jupiter and is defined in the usual planetary sense of being measured always along the direction of orbital motion. $\lambda^\star$ is the mean longitude of the particle and is defined as $\lambda^\star=M+\omega-\Omega$, where $\Omega$ is the longitude of ascending node measured in the planetary sense from the reference direction, $\omega$ is the argument of perihelion measured from the ascending node to the pericenter in the direction of motion (opposite the direction of the measured angle $\Omega$ for retrograde orbits), and $M$ is the mean anomaly also measured along the direction of motion. Lastly, $\omega^\star$ is the particle's longitude of perihelion and is defined as $\omega^\star=\omega-\Omega$, where $\omega$ and $\Omega$ are defined above. This expression for the 1:-1 resonant argument reduces to the equation found in \citet{NamouniMorais2018}, which is written as $\phi^\star=\lambda-\lambda_{J}-2\omega$, where $\lambda$ is the particle's mean longitude and defined as $\lambda=M+\omega+\Omega$.
Similar to the method described above for direct jovian co-orbitals, for retrograde jovian co-orbitals we examine the behavior of both the traditional resonant angle ($\phi_{1:-1}=\lambda-\lambda_J$) and $\phi^\star_{1:-1}=\lambda^\star-\lambda_J-2\omega^\star$ above \citep{MoraisNamouni2013b}. It is important to note that in the retrograde co-orbital case, the traditional interpretation of the resonant island around which a co-orbital librates is not relevant. For example, in the case that $\phi_{1:-1}$ librates around $0^o$, the co-orbital does not appear to orbit the planet in the co-rotating frame as in the ``quasi-satellite" direct case. Rather, the co-orbital and the planet move in {\it opposite} directions with the same mean-motion keeping $\phi_{1:-1}$ near zero, but their opposing trajectories result in them not remaining near each other.
\section{Example Temporary Co-orbital Traps}
\label{sec:example traps}
Temporary jovian co-orbitals can be captured from either asteroids migrating outward toward Jupiter or Centaurs migrating inward toward Jupiter. In this section we discuss the typical dynamical behavior of these transient co-orbitals, including their orbital evolutions and typical eccentricity and inclinations from both the asteroidal and Centaur sources.
\begin{figure}[h!]
\centering
\includegraphics[width=0.65\textwidth]{Fig1.eps}
\caption{Two examples of direct jovian co-orbital temporary captures. Top Left: Example orbital evolution of a temporary jovian NEO co-orbital on a direct orbit. The trap lasts for $\approx3$~Myr. The cyan box marks the region of the zoom-in (bottom left) around the time of co-orbital capture. Bottom Left: Zoom-in of the time around the $\approx3$~Myr temporary jovian NEO co-orbital capture. Top Right: Example orbital evolution of a temporary jovian Centaur co-orbital on a direct orbit. This is the longest-lived transient jovian Centaur co-orbital found in the simulations, captured for a consecutive 45~kyr with a brief 5~kyr capture a few thousand years earlier. The cyan box marks the zoomed-in region (bottom right) around the time of co-orbital capture. Bottom Right: Zoom-in of the time around the temporary jovian Centaur co-orbital capture lasting for 45~kyr preceded by a brief 5~kyr capture a few kyr earlier. Note the inclination scales are in radians with a reference level (in degrees) indicated. \label{fig:a_e_i_NEO_Centaur_direct}}
\end{figure}
Figure~\ref{fig:a_e_i_NEO_Centaur_direct} shows example orbital evolutions for direct jovian co-orbital captures from an asteroidal source (left top \& bottom panels) and a Centaur source (right top \& bottom panels). The captured asteroid co-orbital leaves the $\nu_6$ resonance source $\approx50$~Myr into its lifetime. It then random walks in $a$ for the next $\approx120$~Myr, during which time it experiences Kozai oscillations in $e$ and $i$ at high-$e$ and high-$i$ (though still on a direct orbit with $i<90^o$). At $\approx170$~Myr into the particle's lifetime, it becomes temporarily captured as a direct jovian co-orbital. The trap lasts for $\approx3$~Myr before the perihelion drops to the solar radius.
The example Centaur jovian co-orbital capture (right panels of Figure~\ref{fig:a_e_i_NEO_Centaur_direct}) only enters the $a<34$~AU region after the first 474.4~Myr of its lifetime. It then quickly drops from transneptunian space to $a\simeq$~$a_J$ in $\approx30$~kyr and remains with $a$ near that of Jupiter for $\approx55$~kyr. The co-orbital trap lasts for $\approx45$~kyr (this is the longest of all the Centaur jovian co-orbital captures found) with a brief 5~kyr trap a few thousand years earlier. The inclination never reaches more than $\simeq20^o$ throughout the particle's time with $a<34$~AU. The semimajor axis then random walks back out to transneptunian space over the next $\approx165$~kyr.
\begin{figure}[h!]
\centering
\includegraphics[width=0.35\textwidth,angle=270]{Fig2.eps}
\caption{Inclination (deg) vs eccentricity for each time step that all asteroidal particles have semimajor axes near that of Jupiter thinned by a factor of three for visibility (black dots). The cyan square marks the $e$ and $i$ of the cloned asteroid particle at the instance of cloning (see Section~\ref{sec:retro} below). The blue triangle indicates BZ509's current $e$ and $i$. The green points show $e$ and $i$ of the $a<34$~AU, $q>2$~AU Centaurs that become temporarily-trapped jovian co-orbitals. \label{fig:e_vs_i_asteroids_Centaurs}}
\end{figure}
The eccentricity and inclination behavior of asteroids migrating outward from the main asteroid belt to semimajor axes near that of Jupiter and Centaurs migrating inward toward Jupiter is shown in Figure~\ref{fig:e_vs_i_asteroids_Centaurs}. Asteroids with $a$ near that of Jupiter explore all values of eccentricity ($e$) and inclinations $i<90^o$ (direct orbits). In addition, a handful of particles in the NEO model \citep{Greenstreetetal2012a} reach $i>90^o$ while $a\simeq$~$a_J$; those particles with $i>90^o$ visit the full range of possible eccentricities from $0-1$ (Section~\ref{sec:retro} discusses this in greater detail).
Centaurs with $a<34$~AU and $q>2$~AU evolving inward to semimajor axes near $a_J$ are found to be confined to $e<0.6$ and $i<50^o$. Figure~\ref{fig:e_vs_i_asteroids_Centaurs} shows a subset of Centaurs that include all the temporary jovian co-orbital captures as well as shorter total durations explored with $a$ near that of Jupiter in the simulations. The $e\lesssim0.6$ cut for $a=5.2$~AU is due to the $q>2$~AU cut in the simulations. We find no $i>50^o$ Centaurs with $a\approx$~$a_J$ (see Section~\ref{sec:discussion} for more discussion).
\section{Temporary Co-orbital Time Scales}
\label{sec:timescales}
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\textwidth,angle=270]{Fig3.eps}
\caption{
Duration of near-Jupiter residence, presented as histograms of times that particles have $a\simeq$~$a_{J}$. TOP: Logarithm of the number of co-orbital ``traps" of each duration observed from an asteroidal (left) and Centaur (right) source. The dashed lines mark the amount of time particles must remain with $a$ near $a_{J}$ for each source region (9~kyr for asteroids, 5~kyr for Centaurs) to be classified as co-orbitals. The top left panel shows all ``traps" (i.e., consecutive time steps) with $a\simeq$~$a_{J}$ for the asteroidal source, where the traps to the right of the vertical dashed line are those that we classify as temporary jovian co-orbital traps. The top right panel only shows the traps for the classified temporary jovian co-orbitals from the Centaur source. Long total durations with $a\simeq$~$a_J$ represent single particles that get trapped for long contiguous time periods, but shorter-duration traps are more numerous. The red triangle corresponds to the asteroidal co-orbital trap event for one of the retrograde captures (shown in Figure~\ref{fig:a_e_i_NEO_retro_and_zoom}). BOTTOM: Time weighted residence, showing likelihood of finding a particle resident for that duration. For example, asteroidal particles (left) with 30 kyr traps occur three times less than 10 kyr traps (upper plot) but when time weighted by the trap durations should be roughly equally likely to be found (bottom plot). The longest-lived temporary jovian NEO co-orbital capture lasts for 2.4~Myr. Right: Centaur source. The result is that most observed temporary jovian Centaur co-orbitals should have short trap durations of 5-8~kyr with longer resonant captures being a factor of 2-3 less likely to be found. The longest-lived temporary jovian Centaur co-orbital capture lasts for $\approx45$~kyr.
\label{fig:duration_e_i_NEOs_Centaurs}
}
\end{figure}
Asteroidal particles visit semimajor axes near that of Jupiter for durations ranging from 300 years (our minimum sampling, which is visible in the smallest bin in the left two panels of Figure~\ref{fig:duration_e_i_NEOs_Centaurs}) to a maximum 2.4~Myr, although the majority of times in this area of phase space fall between 2~kyr and 100~kyr. The shortest durations in this range are due to a few integration sampling intervals when particles quickly pass through the phase space near Jupiter on their way from the main belt to the outer Solar System. As described in Section~\ref{subsec:coorb_detection} above, however, asteroidal particles must remain with $a$ near $a_{J}$ for 9,000 years to be classified as co-orbitals. We find the mean, median, and maximum lifetimes for transient asteroidal jovian co-orbitals on direct orbits are 25~kyr, 14~kyr, and 2.4~Myr, respectively.
As shown in Figure~\ref{fig:duration_e_i_NEOs_Centaurs}, long total durations with $a\simeq$~$a_J$ represent single particles that get trapped for long contiguous time periods, but shorter-duration time periods are more numerous. Asteroidal particles with 30~kyr co-orbital traps occur three times less than 10~kyr traps (upper left panel of Figure~\ref{fig:duration_e_i_NEOs_Centaurs}), but when time weighted by the trap durations should be roughly equally likely to be found (bottom left plot of Figure~\ref{fig:duration_e_i_NEOs_Centaurs}).
Transient jovian Centaur co-orbitals must remain with $a\simeq$~$a_{J}$ for 5~kyr to be classified as co-orbitals. The majority of the temporarily-trapped jovian co-orbital captures from the Centaur source last between 5-8~kyr (right two panels of Figure~\ref{fig:duration_e_i_NEOs_Centaurs}). Thus, most observed temporary jovian Centaur co-orbitals should have short trap durations of 5-8~kyr with longer resonant captures being a factor of 2-3 less likely to be found. We find the mean, median, and maximum lifetimes for transient Centaur jovian co-orbitals on direct orbits are 11~kyr, 7.4~kyr, and 45~kyr.
\section{Temporary Co-orbital Population Estimates}
\label{sec:pop estimates}
We find that 0.11\% of the steady-state NEO population are temporarily-trapped jovian co-orbitals on direct orbits. Given that there are $\simeq1,000$ NEOs with $H<18$, this means we would expect there to be one transient jovian co-orbital on a direct orbit trapped from the population of main-belt asteroids at any time (with more smaller ones as expected from whatever the unknown size distribution is).
The larger population of Centaurs (compared to the NEA population) means Centaur capture into temporary direct jovian co-orbitals might outnumber those captured from the NEA population. Our simulations indicate that 0.001\% of the $a<34$~AU, $q>2$~AU Centaurs in steady-state are temporarily-trapped jovian co-orbitals on direct orbits, a roughly two orders of magnitude smaller fraction. It is estimated, however, that there are between $\sim1\times10^6$ and $\sim4\times10^6$ $H<18$, $a<34$~AU Centaurs \citep{Lawleretal2018}, and so this $\sim 1000\times$ larger population results in an expectation of $\sim10-40$ direct transient jovian co-orbitals from a Centaur source at any given time. (We point out that the $q>2$~AU boundary in the simulations may mean this number is actually larger than this estimation, although we find that only roughly 5\% of the particles are discarded from the simulations because they get within this inner distance cut.) The uncertainty on this estimate is at least an order of magnitude\footnote{Not only because of uncertainties in the size distribution, but also because as $a=5$~AU is approached small Centaurs may be modified by splitting \citep{Fernandezetal2009}.}, meaning the number ratio of transient jovian co-orbitals on {\it direct} orbits coming from the Centaur and asteroidal sources is $\sim$1 -- 100.
Our results show that there must be temporarily-trapped direct jovian co-orbitals with lifetimes of $10^4$--$10^6$ years, but none have ever been reported. Using $\pm$1,000-year integrations, \citet{Karlsson2004} studied a handful of $<1$~kyr temporary captures in the known candidate Jupiter Trojan population, but none of these objects are metastable for the much longer timescales we diagnose here. Identification of such transient jovian co-orbitals will eventually happen (just as such orbits have been identified for the other giant planets) but is challenging for a number of reasons. Firstly, given the population of the NEA and Centaur sources, we expect such objects to be not much brighter than $H\sim18$ and thus be faint; there are essentially no multi-opposition orbits in the Minor Planet Center database with $H>17$. Secondly, our simulations show that the majority of these temporary traps have $e>0.3$ and thus spend a much larger fraction of their time far from the Sun beyond survey detection limits. Thirdly, the majority of these objects are horseshoe or quasi-satellite orbits (Table~\ref{table:res_island_breakdown_direct}) and thus are not confined near the Lagrange points and thus their co-orbital semimajor axes might not even be recognized in a very short arc orbit discovered at heliocentric distances between 4 and 7~AU. Even once recognized, it will require a very precise orbit to confirm the co-orbital behavior as the orbital uncertainty needs to be shrunk so much that {\it all} orbits that fit the observations show co-orbital behavior \citep{Alexandersenetal2013}; this is a difficult standard to surpass, but our results indicate that once this is done there should be some small direct co-orbital objects that librate securely in the 1:1 resonance for tens to hundreds of librations before leaving.
\begin{table}[h!]
\begin{center}
\begin{tabular}{| c | c | c | c |}
\hline
\bf{Source} & \bf{Horseshoes} & \bf{Trojans} & \bf{Quasi-satellites}\\
\hline
Asteroids & 93\% & 2\% & 5\% \\
Centaurs & 59\% & 21\% & 20\% \\
\hline
\end{tabular}
\end{center}
\caption{Resonant island classifications for transient jovian co-orbitals on direct orbits from the asteroidal and Centaur sources. Note that the quasi-satellites either outnumber or roughly equal the Trojans, but our resonant island classification method, as described in Section~\ref{subsec:res_island_class}, may overestimate the fraction of quasi-satellites.}
\label{table:res_island_breakdown_direct}
\end{table}
\section{Retrograde Jovian Co-orbital Dynamics} \label{sec:retro}
In addition to finding transient jovian co-orbitals on direct orbits from among particles in the NEO model, we observe a handful of particles evolving to {\it retrograde} ($i>90^o$) co-orbitals. Some of these particles evolve to $a\simeq$~$a_J$ then flip to retrograde orbits (three particles), and some flip to retrograde orbits before going to Jupiter (five particles). For particles classified as co-orbitals (see Section~\ref{subsec:res_island_class} for definition) we find mean, median, and maximum times with consecutive running window centers with $i>90^o$ of 11~kyr, 7~kyr, and 87~kyr, respectively. These temporarily-trapped jovian co-orbitals on retrograde orbits represent 0.001\% of the steady-state NEO population. Given that there are $\simeq1,000$ $H<18$ NEOs, this means we would expect there to be one transient jovian co-orbital on either a direct or retrograde orbit trapped from the asteroid population (with more smaller ones) and that co-orbital would have a 1\% chance of existing on a retrograde orbit.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Fig4.eps}
\caption{Retrograde asteroidal jovian co-orbital example. Top: Orbital history of a particle from our NEO model integrations of an asteroid that becomes a jovian co-orbital around the time of its flip to a retrograde orbit. This particle was cloned 9,900 times around 2.115~Myr into its lifetime (indicated by vertical dashed line) to search for additional retrograde jovian NEO co-orbital behavior. The cyan box marks the zoomed-in region shown in the bottom panel. Bottom: Zoomed-in version of top panel beginning just before the particle becomes locked with its semimajor axis near that of Jupiter and the particle then flips to a retrograde orbit ($i>90^o$). The vertical dashed line marks the time at which the particle was cloned. \label{fig:a_e_i_NEO_retro_and_zoom}}
\end{figure}
Figure~\ref{fig:a_e_i_NEO_retro_and_zoom} shows the full orbital evolution of a particle from our NEO model that becomes trapped as a retrograde jovian co-orbital. This particle lives in the $\nu_6$ resonance for $\approx1.5$~Myr before experiencing a series of planetary close encounters that eventually kick its semimajor axis exterior to Jupiter before it drops back to $a\approx$~$a_J$ around 2.1~Myr into its lifetime. It then remains a co-orbital for $\approx180$~kyr. Shortly after reaching $a\approx$~$a_J$, the inclination becomes retrograde. The particle then remains a retrograde jovian co-orbital for a total of $\approx130$~kyr with the longest consecutive retrograde co-orbital period lasting for the final 100~kyr of the particle's lifetime. While the particle is on a retrograde orbit $e$ explores nearly all possible values. At $\approx2.2$~Myr into the particle's lifetime, a brief spike in $e$ and $i$ occurs with $e$ reaching\footnote{The 4-hour time step in the NEO model integrations satisfies the needed time step to resolve solar encounters (and detect collisions), of less than $P(1-e^2)^{3/2}/3$ = 36 hours, where $P$ is the orbital period, \citep{RauchHolman1999} by about an order for magnitude at $e=0.995$.} nearly one ($e\approx0.995$; $q\approx0.026$) and $i$ exceeding $170^o$. The inclination then drops and settles at $i\simeq100^o$ while $e$ plummets to nearly zero 50~kyr later before climbing to $e=1$ ($e>0.999$) at approximately 2.28~Myr into the particle's lifetime when it is pushed into the Sun ($r<0.005$~AU).
The only path we have been able to demonstrate to the retrograde state is from the main asteroid belt source. As reported in Section~\ref{sec:example traps}, we found no examples of incoming Centaurs reaching temporarily-trapped retrograde jovian co-orbitals with $q>2$~AU. This is likely a result of the different inclination distributions of asteroids and Centaurs that reach the $a\simeq$~$a_J$ region. Although some asteroids get captured into the jovian co-orbital state from retrograde orbits, those that are captured as co-orbitals on direct orbits that then flip to retrograde as well as those that remain on direct orbits are captured at significantly higher inclinations (up to $i=60^o-80^0$) than Centaurs upon capture ($i<35^o$). The higher direct-orbit inclinations of asteroidal particles in the $a\simeq$~$a_J$ region likely give the asteroids an advantage over the Centaurs for reaching the retrograde co-orbital state.
Figure~\ref{fig:e_vs_i_asteroids_Centaurs} showed $e$ vs $i$ for output intervals when $a$ is near that of Jupiter for the asteroidal source. The blue triangle in Figure~\ref{fig:e_vs_i_asteroids_Centaurs} marks the current $e$ and $i$ of BZ509. We do not see any particles in the NEO model reaching both this $e$ and $i$ at the same time, although these values are reached independently by particles in the model. Because we only find eight particles in the NEO model that become transient retrograde jovian co-orbitals, none of which have $e$ and $i$ simultaneously near that of BZ509, we cloned the particle shown in Figure~\ref{fig:a_e_i_NEO_retro_and_zoom} to attempt to find particles reaching these eccentricities and inclinations simultaneously; this also provided a suite of retrograde jovian co-orbital examples to better understand their typical behavior. The cyan square in Figure~\ref{fig:e_vs_i_asteroids_Centaurs} shows the $e$ and $i$ at cloning for the cloned particle. This particle was cloned at $\approx2.115$~Myr into its lifetime while the particle is classified as a jovian co-orbital, but shortly {\it before} it became retrograde (see Figure~\ref{fig:a_e_i_NEO_retro_and_zoom} for the detailed orbital evolution of the particle).
\begin{figure}[h!]
\centering
\includegraphics[width=0.35\textwidth,angle=270]{Fig5.eps}
\caption{Inclination vs eccentricity range for evolutions departing from the cloned particle (cyan square, and see Fig.~\ref{fig:a_e_i_NEO_retro_and_zoom}). For 5\% of the time steps that clone particles (reduced to prevent figure saturation) have semimajor axes near Jupiter, a black dot is plotted. The blue triangle indicates BZ509's current $e$ and $i$. (The blue triangle and cyan square are the same as in Figure~\ref{fig:e_vs_i_asteroids_Centaurs}). The green dots show the $e$/$i$ evolution for the full orbital history of the clone particle shown in Figure~\ref{fig:clone_example}, which is the closest particle to simultaneously matching the current $a$, $e$, \& $i$ of BZ509. The magenta, red, and yellow points show the $e$/$i$ evolution for the full orbital history of single particles with near-Jupiter visits as long as BZ509 (see Appendix). See text for discussion.
\label{fig:all_e_vs_i_clones}
}
\end{figure}
We cloned this particle 9,900 times by randomly `fuzzing' the position and velocity vector components to be within $\pm5$x$10^{-9}$~AU (0.75~km) and $\pm5$x$10^{-9}$~AU/yr (0.75~km/yr) of their initial values, respectively. We then performed integrations for 10~Myr, by which time 99.5\% of the particles had been removed. Figure~\ref{fig:all_e_vs_i_clones} shows the $e$ vs $i$ plot for 5\% of these clones (black dots; thinned for better visibility) as well as the $e$/$i$ values for the full orbital history of the clone particle (green dots) that comes closest to matching BZ509's $a$, $e$, and $i$ as well as three additional long-lived clone particles with $a$ near $a_J$ (magenta, red, and yellow dots). One can see that among the clones we find a retrograde jovian co-orbital (green points) with $e$ and $i$ simultaneously near that of BZ509 (marked by the blue triangle in the figure). Figure~\ref{fig:clone_example} shows the evolution of this clone, which remains a jovian co-orbital for 3.5~Myr. It first reaches $i>90^o$ from the initial direct orbit after $\approx76$~kyr and then remains retrograde for the rest of its lifetime. The two 1:-1 resonant argument histories show periods of libration, of tens or hundreds of thousands of years duration, around $180^o$ or $0^o$, as well as there being $\omega$ oscillations around all of $0^o$, $90^o$, $180^o$ and $270^o$ at times. The red triangle at the bottom of each panel near 2.45~Myr indicates a time when $e$ and $i$ simultaneously match that of BZ509 (blue triangle in Fig.~\ref{fig:all_e_vs_i_clones}). \citet{Wiegertetal2017} demonstrate that BZ509 is currently librating in the same argument ($\phi^{\star}$ here) for at least the 20~kyr interval centered on the present day. Therefore, not only do we demonstrate a path from the main belt to a jovian co-orbital state with $e$ and $i$ simultaneous to BZ509, but also one that is librating around in $\phi^{\star}$ with the same $\approx 100^o$ libration amplitude.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Fig6.eps}
\caption{Orbital history of a clone of the particle shown in Figure~\ref{fig:a_e_i_NEO_retro_and_zoom} starting just before it flips to a retrograde state at $\approx0.2$~Myr. The particle spends $\approx3.52$~Myr (shown here) with $a$ near Jupiter's semimajor axis before its semimajor axis suddenly drops to near $a\approx3$~AU after a planetary close encounter and it collides with Jupiter $\approx16$~kyr later. The particle flips to a retrograde orbit $\approx30$~kyr into its lifetime and coupled Kozai $e$ and $i$ oscillations often occur while the particle is on a retrograde orbit as well as the argument of pericenter ($\omega$) sometimes librating around either $90^o$ or $270^o$ during this time. The resonant arguments $\phi_{1:-1}=\lambda-\lambda_J$ and $\phi^\star_{1:-1}=\lambda-\lambda_J-2\omega$ \citep{NamouniMorais2018} show resonant behavior (libration around $180^o$ and $0^o$, respectively) both before and after the flip to a retrograde orbit, indicating the particle is at times in the 1:-1 co-orbital resonance with Jupiter. This clone particle also spends a single 300 year dump interval within 0.35 AU, $e=0.05$, and $i=3^o$ of 2015 BZ$_{509}$'s current elements. The red triangle at $\approx2.45$~Myr indicates the time when the cloned particle's orbit is closest to that of BZ509 ($a\approx5.4$~AU, $e\approx0.38$, $i\approx166^o$). \label{fig:clone_example}}
\end{figure}
In Figure~\ref{fig:all_e_vs_i_clones}, we see evidence of long-lived particles at high-$e$ and high-$i$ that can sit near a given $e$/$i$ for $\sim10$~Myr. Three examples are clear in Figure~\ref{fig:all_e_vs_i_clones}: one cluster of points is located at $e\approx0.65$ and $i\approx140^o$ (magenta), one at $e\approx0.95$ and $i\approx115^o$ (red), and one at $e\approx0.98$ and $i\approx105^o$ (yellow; see Appendix~\ref{sec:example long-lived} for the full orbital evolutions of these three long-lived retrograde objects with $a$ near $a_J$). Due to the long duration of these states, such long-lived objects are those most likely to be found; this is probably the context of the discovery of BZ509 on an orbit that stays near its current $e$ and $i$ (with a range of $e\approx0$, $i\approx145^o$ to $e\approx0.45$, $i\approx170^o$, paralleling the upper edge of the green dots in Figure~\ref{fig:all_e_vs_i_clones}) for at least $\approx200$~kyr \citep{Huangetal2018}. While it is possible but rare to reach the exact BZ509 state, other long-lived states near Jupiter exist (each single one of which is also rare); if the one known jovian long-lived retrograde co-orbital had been any of these, papers on its origin would have been written. We thus are unsure anything profound should be concluded from the particular current orbit of BZ509. (As an aside, it should be noticed that with a near-planar (albeit retrograde) and moderate eccentricity, BZ509 is the most detectable of the long-lived objects we illustrate.)
\section{Additional discussion} \label{sec:discussion}
We thus believe there is a very plausible case that 2015 BZ$_{509}$ is an escaped main-belt asteroid that became retrograde in an already-demonstrated set of processes \citep{Greenstreetetal2012a, Greenstreetetal2012b, Granviketal2018} that happen in steady state. We have demonstrated a path from this source to an orbit nearly identical to BZ509; with the long-lived niches that last$>$1,000 times longer than the median trapping time, it is very likely that the object first found would be in such a niche state. Given the long duration of these states, most of the steady state retrograde `residence time' (that maps where the population is) shifts to these states, which we estimate are thus producing numbers that match in order of magnitude to the `statistics of one' example of BZ509. We here briefly discuss some other ideas for sources, and posit the idea that the main-belt path provided large-$i$ orbits to the outer Solar System as well.
The lack of {\it retrograde} jovian co-orbitals from our Centaur simulations might be due to the initial conditions of the incoming Centaurs in the simulations, which do not include the highest inclinations known to exist in the TNO scattering population (which feeds the Centaur population). It is not clear, however, whether the inclusion of these higher inclinations in the initial conditions would result in temporary capture into retrograde co-orbital motion at Jupiter since no estimate of the feeding efficiency from such a source to jovian retrograde orbits has ever been made. Our simulations do show the the (dominant) low-$i$ Centaur supply is not raised to high inclinations as they journey to lower $a$\footnote{\citet{HornerWynEvans2006} studied the capture of a sample of known Centaurs, which was biased toward the lowest-$a$, lowest-$i$ Centaurs by observation selection effects, into temporary co-orbital capture with the four giant planets. Their results revealed no retrograde co-orbital captures with any planet or the efficiency at which such transient co-orbital captures are made.}, and bringing $i>45^o$ TNOs to $a\simeq$~$a_J$ is very inefficient due to resultingly high planetary encounter speeds. Therefore, one would have to integrate an ensemble of large-$i$ TNOs to determine what (presumably small) fraction of them reach the long-lived jovian co-orbital state; this remains to be done but we suspect it will be many orders of magnitude rarer than the main-belt source.
\citet{NamouniMorais2018} numerically integrate one million clones of the nominal BZ509 orbit up to 4.5~Gyr into the past and suggest a 0.003\% chance that BZ509 would have $a$ near that of Jupiter 4.5~Gyr ago. Their Figure 3 (similar to our Fig.~\ref{fig:clone_example} above) shows $a$ stable near Jupiter 4.5~Gyr ago, but neither 1:-1 resonant argument shows libration in their figure. Of their remaining clones with $a\approx$~$a_J$, all but one have their $a$ increased above that of Jupiter's (and are not oscillating around Jupiter's semimajor axis). In fact, this `stable niche' is extremely similar to the case of Appendix Figure~\ref{fig:clone_example_2}, reached by our pre-retrograde cloning procedure from a main-belt source.
Fig.~\ref{fig:all_e_vs_i_clones} shows many retrograde objects with $a$ near Jupiter spending lots of time on near-polar ($i\approx90^o$) orbits. \citet{NamouniMorais2018} discuss what they call `the polar corridor' (with $i=90\pm45^o$ and semimajor axes of hundreds of thousands of AU); many of their backwards-integrated BZ509 clones spend tens of Myr `escaping' from the outer Solar System (where they then feel the galactic influence) via this corridor. They claim that a population of objects on near-polar orbits in the transneptunian object (TNO) and Centaur populations is evidence that these objects, including BZ509, originated from an extrasolar source since planet formation models of nearly-coplanar planetary orbits interacting with a coplanar planetesimal disk cannot produce large-inclination orbits stable on Gyr timescales. However, we show a clear path for asteroids coming out of the main belt in steady state reaching orbits with $i>90^o$ and $a$ near Jupiter's for timescales of order 10~Myr, removing this argument for a needing an extrasolar origin for these objects.
The generic issue with outer solar system origins (and the polar corridor specifically) is that {\it all} orbits eventually escape a meta-stable source and the giant planets eject essentially everything; when there is an {\it unbounded} phase space available the backwards integration then yields {\it no} estimate of the supply efficiency. To do this, one would have to integrate a huge set of inbound interstellar interlopers, having a range of impact parameters and drawn from the strongly hyperbolic inbound speed distribution, to determine what (presumably minuscule) fraction of them can reach the jovian co-orbital state; this remains to be done but we suspect it will be completely negligible compared to the main-belt source.
The polar corridor has another aspect, in the context of a few high-$i$ or retrograde transneptunian objects in the Minor Planet Center database with $i>60^o$ and perihelion $q>15$~AU: 2002 XU$_{93}$, 2007 BP$_{102}$, 2008 KV$_{42}$ (nicknamed Drac; \citet{Gladmanetal2009}), 2010 WG$_9$, 2011 KT$_{19}$ (nicknamed Niku; \citet{Chenetal2016}), and 2014 LM$_{28}$. All of these objects have inclinations within $\pm30^o$ of a polar orbit at $i=90^o$; Drac ($i=103^o$) and Niku ($i=110^o$) are on retrograde ($i>90^o$) orbits. \citet{BatyginBrown2016a,BatyginBrown2016b} discuss the idea that these objects are populated by a hypothetical distant planet raising TNOs into the polar corridor, and perhaps from there they could reach a retrograde jovian co-orbital state. However, we again suspect the efficiency is extremely low because of the need to greatly lower the semimajor axes to that of Jupiter without the benefit of frequent and lower-speed encounters that the low-inclination state provides. The required demonstration is again simple in principle: If polar TNO orbits feed objects like BZ509, integrations from the estimated TNO region orbit distribution \citep{BatyginBrown2016b} can be forward propagated to estimate the steady-state number of jovian co-orbitals given the source population estimate. We note that this demonstration must not generate a very abundant polar Centaur population (with $5<a<30$~AU, $q>7.35$~AU) that violates survey constraints which have found very few of them (e.g., \citet{Petitetal2017}).
We actually here posit the {\it inverse} process: Could the population of objects in the outer Solar System on near-polar orbits have originated in the inner Solar System? After all, we have shown that the escaping NEA population already generates near-polar orbits and \citet{NamouniMorais2018} show these efficiently then populate the polar corridor and reach TNO semimajor axes. We thus have an existing Jupiter (not a hypothetical planet) that already creates and feeds large-$i$ orbits to the outer Solar System. The distance cut in our clone integrations at 19~AU prevents us from determining if we can produce particles with $a$ beyond Uranus, but we do find that 43\% of our clone particles are removed from the integrations for reaching heliocentric distances beyond the 19~AU cut. Could this be the origin of objects like Drac and Niku? Computed orbital evolutions in \citet{Gladmanetal2009} and \citet{Chenetal2016} show that these objects are metastable on Gyr timescales, so an {\it outflowing} (rather than incoming) polar corridor would result in Jupiter- and Saturn-crossing Centaurs to dominantly increase their semimajor axes until they reach orbits decoupled from the two most massive giants; once only Uranus and/or Neptune crossing, the dynamical lifetimes (and thus abundance in the steady state) become much larger. The depopulation of huge numbers of primordial inner-Solar system objects might be able to leave a surviving tail of high-$i$ TNOs beyound Jupiter, providing the postulated metastable source \citep{Gladmanetal2009} for the high-$i$ TNOs and Centaurs.
All of these options deserve quantitative exploration in future work. Given the information that we have, we favour the least dramatic hypothesis: that 2015 BZ$_{509}$ is a long-lived member of the known ensemble of high inclination orbits produced via leakage from the main asteroid belt.
\acknowledgments
S. Greenstreet acknowledges support from the Asteroid Institute, a program of B612, 20 Sunnyside Ave, Suite 427, Mill Valley, CA 94941. Major funding for the Asteroid Institute was generously provided by the W.K. Bowes Jr. Foundation and Steve https://iopscience.iop.org/article/10.1086/300720/fulltext/980143.text.htmlJurvetson. Research support is also provided from Founding and Asteroid Circle members K. Algeri-Wong, B. Anders, R. Armstrong, G. Baehr, The Barringer Crater Company, B. Burton, D. Carlson, S. Cerf, V. Cerf, Y. Chapman, J. Chervenak, D. Corrigan, E. Corrigan, A. Denton, E. Dyson, A. Eustace, S. Galitsky, L. \& A. Fritz, E. Gillum, L. Girand, Glaser Progress Foundation, D. Glasgow, A. Gleckler, J. Grimm, S. Grimm, G. Gruener, V. K. Hsu \& Sons Foundation Ltd., J. Huang, J. D. Jameson, J. Jameson, M. Jonsson Family Foundation, D. Kaiser, K. Kelley, S. Krausz, V. La\v{s}as, J. Leszczenski, D. Liddle, S. Mak, G.McAdoo, S. McGregor, J. Mercer, M. Mullenweg, D. Murphy, P. Norvig, S. Pishevar, R. Quindlen, N. Ramsey, P. Rawls Family Fund, R. Rothrock, E. Sahakian, R. Schweickart, A. Slater, Tito's Handmade Vodka, T. Trueman, F. B. Vaughn, R. C. Vaughn, B. Wheeler, Y. Wong, M. Wyndowe, and nine anonymous donors.
S. Greenstreet acknowledges the support from the University of Washington College of Arts and Sciences, Department of Astronomy, and the DIRAC Institute. The DIRAC Institute is supported through generous gifts from the Charles and Lisa Simonyi Fund for Arts and Sciences and the Washington Research Foundation.
B. Gladman acknowledges support from the Natural Sciences and Engineering Research Council of Canada.
|
2,877,628,090,189 | arxiv | \section{Introduction}
We are used to the topological properties of fermion band theory in electronic systems, which have been studied extensively over the past decade \cite{sem1, sem, yu, yu1, yu2, yu3, yu4, yu5, yu6, yu7, yu8, fdm}. Recently, the study of topological band theory has been extended to bosonic systems. A natural extension of topological properties of fermionic systems to bosonic systems can be achieved by replacing the lattice sites of fermions with bosons \cite{pa1,var1}. Hence, the fermionic operators can be regarded as bosonic operators obeying a different commutation relation.
In the hardcore limit, the bosonic systems map to spin-$1/2$ quantum magnets \cite{matq}. This correspondence is very crucial as it paves the way to interpret results in terms of bosons as well as spin variables. It also means that the excitations of hard-core bosons must be the underlying spin wave excitations (magnons) of the corresponding quantum spin model. Therefore, the topological properties of the bosonic excitations must be similar to that of spin wave excitations. In this regard, the Haldane spin-orbit coupling \cite{fdm} in the hardcore limit maps to an out-of-plane Dzyaloshinskii-Moriya interaction (DMI), which induces a nonzero Berry curvature and thermal Hall effect of magnetic spin excitations \cite{alex1, alex0, alex2,alex5,alex4, sol1,sol, alex5a, alex6, alex1a}. However, in contrast to fermionic systems, there is neither a Fermi energy nor a filled band in bosonic systems. This means that the topological invariant quantity usually called the Chern number must be independent of the statistical nature of the particles. It simply predicts the existence of edge state modes in the vicinity of the bulk energy gap as a result of the bulk-edge correspondence. This leads to edge states in bosonic systems.
Unfortunately, many topological bosonic models have a numerical sign problem that hinders an explicit quantum Monte Carlo (QMC) simulation due to an imaginary statistical average. In a recent study, Guo {\it et al}~ \cite{alex9} have investigated the Bose-Hubbard model on the honeycomb lattice using QMC. This model is devoid of the debilitating QMC sign problem as there is no imaginary phase amplitude. It is analogous to fermionic graphene model without spin-orbit coupling in the presence of a biased potential \cite{sem,sem1}. The authors have explicitly mapped out the bosonic quantum phase diagram, the Berry curvature and edge states characterizing the topological properties of the system induced by a staggered on-site potential.
In this paper, we present another perspective of their QMC results using a semiclassical approach. The QMC results presented in Ref.~\cite{alex9} utilized the electronic analogue of the Bose-Hubbard model. Here, we show that the entire QMC analysis can be understood semi-classically. This is due to the fact that the hardcore-Bose-Hubbard model is merely a spin-$1/2$ quantum XY model with competing sublattice magnetic fields, thus the results can also be interpreted in terms of magnetic spins and the semiclassical approach is known to be suitable for such models \cite{ber, tom}. We find that the quantum phase diagram uncovered by QMC can actually be understood by mean-field theory. We uncover the same three insulating phases: superfluid (SF), Mott insulator phase, and charge-density-wave (CDW) insulator. The latter insulating phase is a consequence of the competing sublattice magnetic fields.
As mentioned above, the correspondence between hard-core bosons and quantum spin systems suggests that the spin wave excitations correspond to the bosonic excitations. As QMC showed, the topological properties of this system is manifested by a nonzero Berry curvature. We show that the Berry curvature of the magnon excitations in the $\rho=1/2$ CDW insulator has the same trend as the one obtain by QMC simulation \cite{alex9}. In contrast to DMI induced edge states with nonzero Chern number \cite{alex1, alex0, alex2,alex5,alex4, sol1,sol, alex9, alex5a, alex6, alex1a}, the Chern number of the present model vanishes. Nevertheless, we observe zigzag bosonic magnon edge states which do not have the same origin as those in DMI system \cite{sol}. It is noted that nontrivial topology has been realized in two-dimensional (2D) optical fermionic \cite{jot} and bosonic \cite{jot1, jot2} atoms. Thus, our results are applicable to these systems.
\section{Hardcore-Bose-Hubbard model}
A recent QMC simulation by Guo {\it et al} ~\cite{alex9} studied the topological properties of the extended harcdore-Bose-Hubbard model governed by the Hamiltonian
\begin{align}
H&= -t\sum_{\langle ij\rangle}( b^\dagger_i b_j + h.c.) +\sum_i U_i n_i -\mu\sum_i n_i\label{hardcore},
\end{align}
where, $t>0$ denotes NN hopping, $\mu$ is the chemical potential, and $U_i$ is a staggered on-site potential , with $U_i=\Delta$ on sublattice $A$, and $U_i=-\Delta$ on sublattice $B$ of the honeycomb lattice shown in Fig.~\ref{unit}. $n_i=b^\dagger_ib_i$, $b^\dagger_i$ and $ b_i$ are the bosonic creation and annihilation operators respectively. They obey the algebra $[b_i, b_j^\dagger]=0$ for $i\neq j$ and $\lbrace b_i, b_i^\dagger \rbrace=1$.
\begin{figure}[ht]
\centering
\includegraphics[width=1.75in]{unit}
\caption{Color online. The honeycomb lattice with two sublattices $A$ and $B$ indicated by different colors. The coordinates are $\bold a_1=\sqrt{3}a\hat x;~ \bold a_{2}=a(\sqrt{3}\hat x, 3\hat y)/2$; $ \boldsymbol{\delta}_{1,2}=a(\pm\sqrt{3}\hat x,~\hat y)/2$, and $ \boldsymbol{\delta}_3=a(0, -\hat y)$. }
\label{unit}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{Phase}
\caption{Color online. Mean-field phase diagram of the Bose-Hubbard model \ref{hardcore}, where $J=1/2$ is the parameter value in the spin language, which corresponds to $t=1$ in the hard-core bosons. The dash line indicates the critical value of the $CDW$ phase.}
\label{phase}
\end{figure}
For fermionic systems, the momentum space Hamiltonian for $\mu=0$ is given by
\begin{align}
\mathcal{H}(\bold k)=\left(
\begin{array}{cc}
\Delta&-tf_\bold{k} \\
-tf^*(\bold{k})&-\Delta
\end{array}
\right),
\label{honn}
\end{align}
where $f_\bold{k}=e^{ik_ya/2}\left( 2\cos(\sqrt{3}k_xa/2)+e^{-3ik_ya/2}\right)$.
The corresponding eigenvalues are
\begin{align}
\epsilon_\pm(\bold{k})=\pm\sqrt{\Delta^2+t^2|f_\bold{k}|^2}.
\label{fen}
\end{align}
In the bosonic version, the energy does not have this simple symmetric form as we will show later. In this case, we adopt the quantum spin analogue of the Bose-Hubbard Hamiltonian \ref{hardcore} via the Matsubara-Matsuda transformation \cite{matq}, $S_i^+ \to b^\dagger_i,~S_i^-\to b_i,~S_i^z\to n_i-1/2$. The resulting quantum spin Hamiltonian is given by
\begin{align}
H&=-J\sum_{\langle ij\rangle}(S_i^+S_j^-+ S_i^-S_j^+)-\sum_{i} (\mu- U_i) S_i^z,
\label{hh1}
\end{align}
where $S_i^{\pm}=S_i^x\pm iS_i^y.$ The last term is basically a competing magnetic field on the two sublattices. Throughout the analysis in this paper we fix $J=1/2$, which corresponds to $t=1$ in the hard-core bosons.
\section{Mean-field phase diagram}
In this section, we present the mean-field phase diagram of the Bose-Hubbard model \ref{hardcore}. The mean-field approximation is implemented by approximating the spins as classical vectors parameterized by a unit vector: $\bold{S}_i=S\left(\sin\theta_i\cos\phi_i, \sin\theta_i\sin\phi_i,\cos\theta_i \right)$. We adopt the customary two-sublattice honeycomb lattice depicted in Fig.~\ref{unit}. Since the spins lie on the same plane we take $\phi_i=0$, then the classical energy is parameterized by $\theta_i$ given by
\begin{align}
e_c &= -\Delta_c \sin\theta_A\sin\theta_{B}-(\mu-\Delta)\cos\theta_A -(\mu+\Delta)\cos\theta_B,
\label{cla}
\end{align}
where $e_c=E_c/NS$, $\Delta_c=2JzS$, $S=1/2$, $N$ is the number of unit cells, and $z=3$ is the coordination number of the lattice. The filling factor is given by $\rho= 1/2 + S(\cos\theta_A +\cos\theta_B)/2$.
There are three phases in this model uncovered by QMC \cite{alex9}. In the mean-field approximation they are characterized by the polar angles. In the SF phase $\theta_A=\theta_B\neq 0,\pi$, the Mott phase is characterized by $\theta_A=\theta_B=0$ or $\pi$, and the CDW is characterized by $\theta_A=0;~\theta_B=\pi$ or $\theta_A=\pi;~\theta_B=0$. The mean field phase diagram is derived by minimizing \ref{cla} following the standard approach \cite{kle}. We obtain the phase boundary between SF and Mott insulators as $\mu_{c1}=\pm\sqrt{\Delta^2+\Delta^2_c}$. This corresponds to $\bold{k}=0$ in Eq.~\ref{fen}. The phase boundary between SF and CDW insulators is $\mu_{c2}=\pm\sqrt{\Delta^2-\Delta^2_c}$. Unlike the first phase boundary, this expression cannot be obtained from Eq.~\ref{fen}. The classical angles are obtained explicitly as
\begin{align}
\cos^2\theta_A&=\left(\frac{\Delta-\mu}{\Delta_c}\right)^2\bigg[\frac{(\Delta+\mu)^2+\Delta_c^2}{(\Delta-\mu)^2+\Delta_c^2}\bigg]\label{eq1},\\
\cos^2\theta_B&=\left(\frac{\Delta+\mu}{\Delta_c}\right)^2\bigg[\frac{(\Delta-\mu)^2+\Delta_c^2}{(\Delta+\mu)^2+\Delta_c^2}\bigg].
\label{eq2}
\end{align}
The mean-field phase diagram is depicted in Fig.~\ref{phase}. The superfluid phase appears for small $\mu$, whereas the Mott phase is predominant for large $\mu$. The CDW arises mainly from the competition between $\mu$ and $\Delta$. The threshold limit $\Delta_c$ corresponds to the point where $\mu_{c2}=0$. This is the exact same quantum phase diagram uncovered by QMC \cite{alex9}.
\section{Band structure}
The main purpose of this paper is to show that the magnon excitation of the Bose-Hubbard model \ref{hardcore} embodies the topological properties of this system. Since the Bose-Hubbard model \ref{hardcore} describes an ordered system as shown in the phase diagram Fig.~\ref{phase}, we can study the excitations of the spin waves when quantum fluctuations are introduced and this should correspond to the excitations of the bosons as explained above. The simplest way to study spin wave excitations is via the standard Holstein Primakoff transformation. This approach is frequently used in the study of hard-core bosons in two-dimensional lattices \cite{ber,tom}. In term of topological properties of quantum magnets, the Holstein-Primakoff transformation has also been utilized effectively in this regard \cite{alex1, alex0, alex2,alex5,alex4, sol1}, and considered to be a good experimental predictor \cite{alex6}. In this section, we utilize this semiclassical formalism in the study of the Bose-Hubbard model. The starting point of spin wave expansion is the rotation of the coordinate axes such that the $z$-axis coincides with the local direction of the classical polarization. This is implemented by a rotation about the $y$-axis on the two sublattices
\begin{figure}[!]
\centering
\includegraphics[width=1\linewidth]{band_SF}
\caption{Color online. Bosonic magnon bands of the Bose-Hubbard model. $(a)$ CDW phase $J=1/2$, $\mu=2J$, $\Delta=4J$. $(b)$ SF phase $J=1/2$, $\mu=J$, $\Delta=J$. }
\label{band}
\end{figure}
\begin{align}
&S_{i\alpha}^x=S_{i\alpha}^{\prime x}\cos\theta_\alpha + S_{i\alpha}^{\prime z}\sin\theta_\alpha,\label{trans}\nonumber\\&
S_{i\alpha}^y=S_{i\alpha}^{\prime y},\\&\nonumber
S_{i\alpha}^z=- S_{i\alpha}^{\prime x}\sin\theta_\alpha + S_{i\alpha}^{\prime z}\cos\theta_\alpha,
\end{align}
where $\alpha=A,B$ label the sublattices.
We then introduce the linearized Holstein Primakoff transformation, $S_{i\alpha}^{\prime z}= S-c_{i\alpha}^\dagger c_{i\alpha},~
S_{i\alpha}^{\prime y}= i\sqrt{ S/2}(c_{i\alpha}^\dagger -c_{i\alpha}),~
S_{i\alpha}^{\prime x}= \sqrt{S/2}(c_{i\alpha}^\dagger +c_{i\alpha})$. The bosonic tight binding Hamiltonian becomes
\begin{align}
H&=-\sum_{\langle ij\rangle}[v_{1}( c_{iA}^\dagger c_{jB}+ h.c.) +v_{2}( c_{iA}^\dagger c_{jB}^\dagger+ h.c.)] \nonumber\\& +(v_A- m_A)\sum_i c_{iA}^\dagger c_{iA}+(v_B+m_B)\sum_j c_{jB}^\dagger c_{jB},
\label{hp3}
\end{align}
where $v_{1,2}=JS(\cos\theta_A\cos\theta_B \pm 1)$, $v_{A/B}=\Delta_c\sin\theta_A\sin\theta_B+\mu\cos\theta_{A/B}$ and $ m_{A/B}=\Delta\cos\theta_{A/B}$. Apart from the off-diagonal terms with coefficient $v_2$, Eq.~\ref{hp3} is similar to a graphene model with a staggered potential. The energy bands are given in Appendix \ref{appen1}. Figure~\ref{band}(a) shows the magnon bands in the $\rho=1/2$ CDW insulator and Fig.~\ref{band}(b) shows the magnon bands in the SF phase. We see that the lower band in the SF phase has a Goldstone model at $\bold{k} =0$ (see Appendix \ref{appen3}) in contrast to the CDW insulator. A special limit of the CDW insulator is analyzed in Appendix \ref{appen2}. It is noted that there are two SF phases in this model--- gap SF phase for $\Delta\neq 0$ and gapless SF phase for $\Delta=0$ (see Appendix \ref{appen3}).
\section{Magnon edge states}
To study the topological properties of this model we use the results in Appendix \ref{appen1}. For $\Delta=0$, we have $m_A=m_B=0$, then Eqs.~\ref{eq1} and \ref{eq2} simply give $\theta_A=\theta_B=\theta$, hence $v_A=v_B$ and the system reduces to the usual hard-core bosons or XY model. In this limit, the system exhibits Dirac nodes at $\bold K_\pm=(\pm 4\pi/3\sqrt{3}a, 0)$ and a Goldstone mode at ${\bf \Gamma}=0$. As QMC demonstrated \cite{alex9}, the topological properties of this system is induced by a nonzero $\Delta$ which plays the role of a gap as shown in Appendix \ref{appen1}. This implies that $m_A\neq 0$ and $ m_B\neq 0$.
\begin{figure}
\centering
\includegraphics[width=3.5in]{BC_CDW.png}
\caption{Color online. Berry curvatures of the Bose-Hubbard model at $J=1/2$, $\mu=2J$, $\Delta=4J$. This corresponds to the $\rho=1/2$ CDW insulating phase in Fig.~\ref{phase}. The minima and maxima of the Berry curvatures are consistent with QMC simulation \cite{alex9}. }
\label{bc_berry}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.5in]{BC_SF.png}
\caption{Color online. Berry curvatures of the Bose-Hubbard model at $J=1/2$, $\mu=J$, $\Delta=J$. This corresponds to the superfluid (SF) phase in Fig.~\ref{phase}. The Berry curvatures in this phase are not measured in QMC simulation \cite{alex9}. }
\label{bc_sf}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.5in]{BHM_edge}
\caption{Color online. Magnon zigzag edge states (green solid lines) of the Bose-Hubbard model in CDW $(a)$ and SF $(b)$ phases. The parameters are the same as Fig.~\ref{band}. The structure of the bands and the edge states are consistent with QMC simulation \cite{alex9}. }
\label{edge}
\end{figure}
We are interested in the Berry curvature associated with the magnon bulk gap. It is given by
\begin{align}
\Omega_\lambda(\bold k)=-\sum_{\lambda\neq \lambda^\prime}\frac{2\text{Im}[ \braket{\mathcal{U}_{\bold{k}\lambda}|v_x|\mathcal{U}_{\bold{k}\lambda^\prime}}\braket{\mathcal{U}_{\bold{k}\lambda^\prime}|v_y|\mathcal{U}_{\bold{k}\lambda}}]}{\left(\epsilon_{\bold{k}\lambda}-\epsilon_{\bold{k}\lambda^\prime}\right)^2},
\label{chern2}
\end{align}
where $v_{i}=\partial \mathcal{H}_B(\bold k)/\partial k_{i}$ defines the velocity operators, $\mathcal{U}_{\bold{k}\lambda}$ denotes the columns of the matrix that diagonalizes $\mathcal{H}_B(\bold k)$(see Appendix~\ref{appen1}), and $\lambda=\pm$ denotes the two positive magnon bands. The CDW and the SF phases are the nontrivial phase in this model. Figure~\ref{bc_berry} shows the Berry curvatures for the top and the bottom bands in the $\rho=1/2$ CDW insulator and Fig.~\ref{bc_sf} shows the Berry curvatures in the SF phase. The Berry curvatures show minima and maxima peaks at the corners of the Brillouin zone (see Appendix~\ref{appen2}). This is in good agreement with QMC simulation \cite{alex9}. In contrast to DMI induced Berry curvatures, the Chern number of each band $\mathcal{C}_\lambda= \frac{1}{2\pi}\int_{{BZ}} d^2k~ \Omega_\lambda(\bold k),$ vanishes identically for the present model \cite{footnote}. However, due to nonzero Berry curvatures we observe zigzag edge states in this system for $k_x\in[2\pi/3\sqrt{3}, 4\pi/3\sqrt{3}]$ as depicted in Fig.~\ref{edge}. Thus, they have a different origin from the DMI induced ones \cite{sol}. This is consistent with QMC simulation of Ref.~\cite{alex9}.
\section{Conclusion}
We have complemented the QMC simulation of Guo {\it et al}~ \cite{alex9} using a semiclassical approach. The main result of our study is that the topological properties of hard-core bosons correspond to the topological properties of the magnon bulk bands of the corresponding quantum spin model. In the hardcore-Bose-Hubbard model, competing sublattice magnetic fields lead to a nontrivial charge-density-wave insulator with a filling factor of $\rho=1/2$, in addition to superfluid phase and Mott insulator. We have uncovered the mean-field phase diagram, which is consistent with the QMC phase diagram. We also derived the magnon energy bands of each phase and show that the corresponding Berry curvatures and edge states are consistent with QMC simulations. This basic idea we have presented here can also be generalized to bilayer honeycomb lattice. These results will be useful in experimental set up of ultracold bosonic atoms in honeycomb optical lattice.
\section*{Acknowledgments}
The author would like to thank African Institute for Mathematical Sciences (AIMS). Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research
and Innovation.
|
2,877,628,090,190 | arxiv | \section{Introduction}
Two-dimensional conformal field theories (CFTs) admit Virasoro symmetry, which allows to study these theories to a large extent due to this infinite dimensional symmetry. The Virasoro algebra is generated by a spin-2 current, and extended algebras can be constructed by adding higher spin currents. These algebras are called W-algebras. The standard construction of W-algebras is via Hamiltonian reduction associated with a Lie algebra $\mathfrak{g}$ and an $\mathfrak{sl}(2)$-embedding, see, e.g., \cite{Arakawa2017, Bouwknegt:1992wg} for reviews for mathematicians respectively physicists.
For the case of $\mathfrak{g}=\mathfrak{sl}(N)$, $\mathfrak{sl}(2)$-embeddings are labeled by partitions of the integer $N$, and each partition leads to a different algebra. The partition $N=N$ corresponds to so-called principal (or regular) embedding of $\mathfrak{sl}(2)$, which yields the W$_N$-algebra. The W$_N$-algebras has a spin-$s$ current for each $s=2,3,\ldots,N$. Furthermore, the case with partition $N=1+\cdots + 1$ corresponds to the $\mathfrak{sl}(N)$ current algebra. Except for these two special cases, W-algebras have not been fully explored yet.
W-algebras appear in many contexts of theoretical physics.
For instance, sub-sectors of four-dimensional $SU(N)$ gauge theories are claimed to be organized by W-algebras \cite{Alday:2009aq,Wyllard:2009hg}.
In particular,
non-regular W-algebras appear by inserting surface operators in four-dimensional gauge theories \cite{Alday:2010vg,Kozcaz:2010yp,Wyllard:2010rp,Wyllard:2010vi}.
Moreover, various W-algebras arise as the asymptotic symmetry of three-dimensional higher spin gravities by adopting generic gravitational sectors \cite{Henneaux:2010xg,Campoleoni:2010zq}.
Non-regular W-algebras play important roles in holographic dualities \cite{Creutzig:2018pts,David:2019bmi,Creutzig:2019qos,Creutzig:2019wfe} generalizing the original proposal of \cite{Gaberdiel:2010pz} with regular W-algebras.
In general, W-algebras are central in $S$-duality which is very closely related to the mathematics of quantum geometric Langlands duality \cite{Kapustin:2006pk}.
The best known such duality is Feigin-Frenkel duality between the principal W-algebra of $\mathfrak{g}$ at level $k$ and the principal W-algebra of the dual Lie algebra ${}^L\mathfrak{g}$ at dual level ${}^Lk$ \footnote{The dual level satisfies $r^\vee (k+h^\vee)({}^Lk+{}^Lh^\vee)=1$ with $r^\vee$ the lacity of $\mathfrak{g}$ and $h^\vee, {}^Lh^\vee$ the dual Coxeter numbers of $\mathfrak{g}$ and ${}^L\mathfrak{g}$.}. However $S$-duality conjectures many more dualities between non-principal W-algebras and W-superalgebras \cite{Frenkel:2018dej, Gaiotto:2017euk, Creutzig:2017uxh, Creutzig:2018ltv}. For example it has just been proven that there is a Kazama-Suzuki coset type correspondence between subregular W-algebras of type A and B and principal W-superalgebras of type $\mathfrak{sl}(N|1)$ and $\mathfrak{osp}(2|2N)$ \cite{CGN}. While $S$-duality and the quantum geometric Langlands duality are concerned with true dualities, that is true matchings of correlation functions we are concerned with correspondences. Dualities appear if one considers correlation functions consisting of degenerate fields only. Mathematically degenerate fields should be thought of as corresponding to ordinary modules of the W-algebra and matching of correlation functions should be viewed as an equivalence of underlying tensor categories (see Conjecture 6.4 of \cite{Aganagic:2017smx} and \cite{Creutzig:2020zvv, Creutzig2018FusionCF} for proofs of cases). The generic field of a W-algebra is however not degenerate, but non-degenerate and we are concerned with correlation functions of fields of this type. In this case one gets correspondences, i.e. the correlation function on one side coincides with the one on the other side, but with extra degenerate field insertions.
In this paper, we derive new relations among correlation functions of theories with the symmetry of W-algebras associated with different partitions of $N$.
Making use of the relations, we can deduce correlation functions from well-studied ones with the W$_N$-algebra or $\mathfrak{sl}(N)$ current algebra.
The simplest example is given by a relation between $\mathfrak{sl}(2)$ current algebra and Virasoro algebra,
and it is called the Ribault-Teschner relation \cite{Ribault:2005wp}.
In \cite{Hikida:2007tq}, the relation was re-derived in path integral formulation, and the method allows us to derive new correspondences of correlation functions \cite{Hikida:2007sz,Creutzig:2011qm,Creutzig:2010zp,Creutzig:2015hla}.
The method of the previous works is mainly restricted to relations between $\mathfrak{sl}(N)$ current algebra (or its superalgebra counterpart) and W-algebra corresponding to the partition $N=2 + 1 + 1 + \cdots + 1$. In order to go further and to understand relations involving other W-algebras the previous method needs to be improved and this is the aim of this paper, i.e. we extend the previous analysis by deriving new relations among more generic W-algebras. The key insight is our better understanding of different free field realizations of W-algebras.
In the previous works, we start from $\mathfrak{sl}(N)$ Wess-Zumino-Novikov-Witten (WZNW) model with the symmetry of $\mathfrak{sl}(N)$ current algebra.
We use a first order formulation of the model, which corresponds to a free field realization of the current algebra.
Integrating out some of the free fields, we end up with a theory with W-algebra symmetry.
Here we would like to consider a theory with non-regular W-algebra symmetry to obtain new correspondences of correlators.
We heavily utilize free field realizations of generic W-algebras analyzed in \cite{Genra1,Genra2, CGN}.
A main point here is that there are several free field realizations of each W-algebra,
and ``nice'' correspondences can be derived by choosing convenient realizations.
We can also obtain simpler types of correspondences by putting restrictions on momenta of vertex operators inserted.
For $\mathfrak{g} = \mathfrak{sl}(3)$, there is only one non-regular embedding of $\mathfrak{sl}(2)$ corresponding to the partition $3 = 2+1$.
The W-algebra labeled by the partition is known as Bershadsky-Polyakov (BP)-algebra \cite{Polyakov:1989dm,Bershadsky:1990bg}. One type of free field realization was already given in \cite{Bershadsky:1990bg}, but another type is possible by using the screening charges of \cite{Genra1}.
With the sets of screening charges, we explicitly write down the generators in terms of free fields. Making use of the expressions, we construct vertex operators transforming in representations of the BP-algebra and obtain a map among correlation functions provided by two types of free field realizations.
We then derive correlator relations among theories with different W-algebra symmetry.
The relation between $\mathfrak{sl}(3)$ current algebra and BP-algebra were already obtained in \cite{Creutzig:2015hla}, but here we derive the relation in a slightly different way to make our strategy clearer.
We then derive a relation between BP-algebra and W$_3$-algebra by making use of new free field realization of BP-algebra.
We obtain more correspondences by putting restrictions on momenta of vertex operators.
We further explore examples with $\mathfrak{g} = \mathfrak{sl}(4)$.
For $\mathfrak{sl}(4)$, there are three types of non-regular W-algebras corresponding to the partitions $4=3+1$, $4 = 2 + 2$, and $4 = 2 + 1 +1 $. Even though the number of non-regular type increases, we show that the technique developed for $\mathfrak{sl}(3)$ can be directly applied. Several types of screening charges can be constructed for each W-algebra \cite{Genra1,Genra2}, and the explicit expressions of generators are obtained by utilizing the screening charges. We further derive new correlator correspondences by applying new free field realizations.
We also generalize the analysis to certain W-algebras associated to $\mathfrak{sl}(N)$.
The organization of this paper is as follows.
In the next section, we first express the generators of BP-algebra in terms of two types of free field realizations and then relate the two descriptions of correlation functions.
In section \ref{sec:corrsl3}, we derive several new correspondences among correlation functions of theories with the symmetry of W-algebras with $\mathfrak{sl}(3)$. In particular, we make use of new free field realization of BP-algebra.
In section \ref{sec:freesl4}, we write down the generators of three types of non-regular W-algebras with $ \mathfrak{sl}(4)$ in terms of free fields. In section \ref{sec:corrsl4}, we obtain new correspondences among correlation functions by applying the free field realizations.
Section \ref{sec:conclusion} is devoted to conclusion and future problems.
In appendix \ref{sec:screening}, we explicitly write down screening charges for non-regular W-algebras with $\mathfrak{sl}(4)$ and $\mathfrak{so}(5)$.
In appendix \ref{sec:corrslN}, we apply our prescription to obtain new relations among several W-algebras associated to $\mathfrak{sl}(N)$.
\section{Free field realizations of BP-algebra}
\label{sec:FreeBP}
In this section, we examine free field realizations of BP-algebra as the simplest but non-trivial example of a non-regular W-algebra. In terms of free fields, generators are given by operators commuting with the set of the screening charges.
It is in general a difficult task to come up with a set of screening operators acting on some free field algebra such that its joint kernel is precisely the algebra of interest. In the case of the BP-algebra and W-algebras associated to Lie algebras in general one however knows how to obtain these screening charges.
The W-algebras are defined as homologies of complexes associated to the affine vertex algebras \cite{KRW} and it is possible to show that these homologies are isomorphic to the kernel of certain screening operators acting on some free field algebra \cite{Genra2}. The case of interest to us, that is the BP-algebra, and subregular W-algebras of $\mathfrak{sl}(n)$ in general has been conjectured by Feigin and Semikhatov \cite{Feigin:2004wb} and a derivation is given in \cite[Section 3.2]{CGN}. We will give details on screening realizations of W-algebras associated to $\mathfrak{sl}(4)$ and $\mathfrak{so}(5)$ in appendix \ref{sec:screening}. Note that the characterization of the W-algebra as a homology has the advantage that one can determine all generating fields and their conformal weights. For example the subregular W-algebra of $\mathfrak{sl}(n)$ has $n+1$ generating fields of conformal weights $1, 2, \dots, n-1$ and $\frac{n}{2}, \frac{n}{2}$. Moreover it is enough to find the two fields of conformal weight $\frac{n}{2}$, since these two fields generate the complete W-algebra under operator products \cite{Creutzig:2020zaj}.
It turns out that there are two sets of screening operators, and hence two types of expressions for the generators are obtained. We construct vertex operators respecting the BP-algebra for each free field realization.
If vertex operators transform in the same way under the BP-algebra, then correlation functions should be the same for the two free field realizations up to normalization of vertex operators.
\subsection{BP-algebra and its generators}
The BP-algebra can be obtained from a Hamiltonian reduction of $\mathfrak{sl}(3)$ current algebra associated with the unique non-regular $\mathfrak{sl}(2)$-embedding \cite{Bershadsky:1990bg}.
The algebra is generated by a spin-one current $H(z)$, two spin-3/2 bosonic currents $G^\pm (z)$ and the energy-momentum tensor $T(z)$. The operator product expansions (OPEs) among them are given as
\begin{align}
& T(z) T(w) \sim \frac{c/2}{(z -w)^4} + \frac{2 T(w)}{(z -w)^2} + \frac{\partial T(w) }{z-w} \, , \nonumber \\
&T(z) G^\pm (w) \sim \frac{\frac{3}{2} G^\pm (w)}{(z -w)^2} + \frac{\partial G^\pm (w)}{z-w} \, , \quad
T(z) H(w) \sim \frac{H(w)}{(z-w)^2} + \frac{\partial H(w) }{z-w} \, ,\nonumber \\
&H (z) H(w) \sim - \frac{ (2 k -3)/3 }{(z -w)^2} \, , \quad
H (z) G^\pm (w) \sim \pm \frac{ G^\pm (w) }{z -w} \, , \\
&G^+ (z) G^- (w) \sim \frac{(k-1)(2 k -3)}{(z-w)^3} - \frac{ 3 (k-1) H(w)}{(z-w)^2} \label{OPEs} \nonumber \\
&\qquad \qquad \qquad+
\frac{ 3 (HH) (w) + (k-3) T (w) - \frac{3}{2}(k-1) \partial H(w) }{z-w} \, . \nonumber
\end{align}
The central charge is
\begin{align}
c = 6 (k-3) + 25 + \frac{24 }{ k-3 } \, ,
\end{align}
where $k$ is the level of $\mathfrak{sl}(3)$ current algebra.%
\footnote{In mathematical literature, the affine algebra level is usually set as $-k$ instead of $k$. \label{-k}}
We would like to realize the generators of BP-algebra in terms of free fields.
We introduce two free bosons $\varphi_i$ $(i=1,2)$ satisfying
\begin{align}
\varphi_i (z) \varphi_j (w) \sim - G_{ij} \log (z -w) \, ,
\end{align}
where
\begin{align}
G_{ij} =
\begin{pmatrix}
2 & -1 \\
-1 & 2
\end{pmatrix} \, , \quad
G^{ij} =
\begin{pmatrix}
2/3 & 1/3 \\
1/3 & 2/3
\end{pmatrix} \, . \label{Gmatrices0}
\end{align}
The indices of $\varphi_i$ can be raised and lowered by these matrices.
We also introduce a bosonic ghost system $(\gamma,\beta)$ satisfying
\begin{align}
\gamma (z) \beta (w) \sim \frac{1}{z - w} \, .
\end{align}
The expression of the generating fields in terms of these free fields can be found using the fact that the generating fields commute with screening charges.
\bigskip
\noindent {\bf The first realization.}
A set of screening operators can be found in \cite{Bershadsky:1990bg} as
\begin{align}
\mathcal{S}_1 = \oint dz \mathcal{V}_1 (z)\, , \quad
\mathcal{S}_2 =\oint dz \mathcal{V}_2 (z)\label{screening1}
\end{align}
with
\begin{align}
\mathcal{V}_1 = e^{b \varphi_1} \gamma \, , \quad
\mathcal{V}_2 = e^{b \varphi_2} \beta \, , \label{screening2}
\end{align}
where a new parameter $b$ is introduced as
\begin{align}
b = \frac{1}{\sqrt{k -3}} \, .
\end{align}
The generators commuting with \eqref{screening1}, \eqref{screening2} are given by (see \cite{Bershadsky:1990bg,Creutzig:2015hla})
\begin{align}
\begin{aligned}
& T = - \frac12 G^{ij} \partial \varphi _i \partial \varphi _j + \frac{(k-1)b}{2} (\partial ^2 \varphi_1 +\partial ^2 \varphi_2 ) + \frac12 ( \gamma \partial \beta - \partial \gamma \beta) \, , \\
&H = \frac{1}{3b} ( \partial \varphi_1 - \partial \varphi_2 ) - \gamma \beta \, , \quad
G^+ = - \frac{1}{b} \partial \varphi_2 \gamma - \gamma \gamma \beta + (k-1 ) \partial \gamma \, , \\
& G^- = \frac{1}{ b} \partial \varphi_1 \beta - \gamma \beta \beta - (k -1) \partial \beta \, .
\end{aligned} \label{freeBP1}
\end{align}
Here and in the following, the normal ordering prescription is assumed for the products of free fields.
The background charges for $\varphi_i$ are set such that the conformal dimensions of screening charges are one.
With the energy-momentum tensor $T$, the conformal dimensions of $(\gamma,\beta)$ are $(1/2,1/2)$.
We checked that the generators satisfy the OPEs in \eqref{OPEs}.
\bigskip
\noindent {\bf The second realization.}
Screening charges of free field realizations for generic W-algebras were explored in \cite{Genra1,Genra2}.
In particular, we can see that there is another set of screening operators as
\begin{align}
\mathcal{V}_1 = e^{b \varphi_1} \, , \quad
\mathcal{V}_2 = e^{b \varphi_2} \beta \, . \label{screening3}
\end{align}
We find generators commuting the screening charges with \eqref{screening3} as
\begin{align}
\begin{aligned}
& T = - \frac12 G^{ij} \partial \varphi_i \partial \varphi_j + \frac{(k-1)b}{2} \partial^2 \varphi_1 + b \partial^2 \varphi_2 - \frac12 \gamma \partial \beta - \frac{3}{2} \partial \gamma \beta \, , \\
&H = \frac{1}{3b} \partial \varphi_1 + \frac{2}{3 b} \partial \varphi_2 + \gamma \beta \, , \quad
G^+ = \beta \, , \\
& G^- = - b^{-1} \partial \varphi_1 \gamma \gamma \beta-2 b^{-1} \partial \varphi_2 \gamma \gamma \beta + b^{-1} \left(k- 1\right) \partial \varphi_1 \partial \gamma + b^{-1} \left(2 k-2 \right) \partial \varphi_2 \partial \gamma \\
& \qquad + b^{-1} \left(k-2 \right) \partial ^2 \varphi_2 \gamma +(3-k) \partial \varphi_1 \partial \varphi_2 \gamma + (3-k) \partial \varphi_2 \partial \varphi_2 \gamma +(k-3) \gamma \gamma \partial \beta \\
& \qquad + (3 k-3) \partial \gamma \gamma \beta - \gamma \gamma \gamma \beta \beta + \left(-k^2+\frac{5 k}{2}-\frac{3}{2}\right) \partial ^2 \gamma \, .
\end{aligned} \label{freeBP2}
\end{align}
We have checked that they satisfy \eqref{OPEs}.
Note that the conformal dimensions of $(\gamma,\beta)$ are $(-1/2,3/2)$, and it is consistent with $G^+ = \beta$.
\subsection{Vertex operators}
\label{sec:vertex}
In the previous subsection, we have written down the generators of BP-algebra in terms of free fields.
In this subsection, we introduce vertex operators and examine the action of generators to them.
Since the conformal dimensions of $G^{\pm}$ are not integer, it is convenient to redefine (or twist) the energy-momentum tensor as
\begin{align}
T(z) \to T_t (z) = T(z) + \frac12 \partial H (z) \, . \label{twist}
\end{align}
In other words, we work with the Ramond sector.
The conformal dimensions of $G^+$ and $G^-$ are one and two, respectively, with respect to the twisted energy-momentum tensor.
The generators now have integer expansion modes:
\begin{align}
\begin{aligned}
&T_t (z) = \sum_{n\in\mathbb{Z}} \frac{L_n}{z^{n+2}}\, , \quad H (z) = \sum_{n\in\mathbb{Z}}\frac{H_n}{z^{n+1}} \, ,\\
&G^+(z) = \sum_{n\in\mathbb{Z}} \frac{G^+_n}{z^{n+1}} \, , \quad
G^- (z) = \sum_{n\in\mathbb{Z}} \frac{G^-_n}{z^{n+2}}.
\end{aligned}
\end{align}
With the twisted energy-momentum tensor, the conformal dimensions of $(\gamma,\beta)$ are given by $(0,1)$ for both free field realizations.
Thus, it is natural to introduce vertex operators of the form
\begin{align}
V_{j,s}(\mu|z) = e^{\mu \gamma } e^{j \varphi_1 + s \varphi_2 } \, . \label{vosl3}
\end{align}
From the vertex operators, we define states
\begin{align}
| j ,s ; \mu \rangle \equiv \lim_{z \to 0} V_{j,s}(\mu|z) | 0 \rangle \, .
\end{align}
Notice that these states are primary with respect to the twisted algebra and satisfy
\begin{align}
L_n | j ,s ; \mu \rangle =
G^\pm_n | j ,s ; \mu \rangle =
H_n | j ,s ; \mu \rangle & = 0
\end{align}
for $n > 0$.
Moreover, the action of zero modes can be written as
\begin{align}
\begin{aligned}
L_0 | j ,s ; \mu \rangle & = - \mathcal{D}_L | j ,s ; \mu \rangle \, , \\
G^\pm_0 | j ,s ; \mu \rangle & = - \mathcal{D}_\pm | j ,s ; \mu \rangle \, , \\
H_0 | j ,s ; \mu \rangle & = - \mathcal{D}_H | j ,s ; \mu \rangle \label{zeroactionsl3}
\end{aligned}
\end{align}
with differential operators $\mathcal{D}_L ,\mathcal{D}_\pm,\mathcal{D}_H$.
The expressions of differential operators can be read off from the action of generators to the vertex operators \eqref{vosl3}.
One of the differential operators $\mathcal{D}_L $ is the (minus of) conformal weight as
\begin{align}
\Delta = - \mathcal{D}_L = -j^2+ j \left( s+ b ( k-2) \right)+s \left(b -s\right) \, , \label{cw}
\end{align}
which is common for both free field realizations.
For the first realization, we found the expression of generators as in \eqref{freeBP1}.
Acting with the generators on the vertex operators in \eqref{vosl3}, we obtain differential operators as
\begin{align}
\begin{aligned}
\mathcal{D}_H &=- b^{-1} (s-j)-\mu \frac{\partial }{\partial \mu } \, , \\
\mathcal{D}_+ &=b^{-1} (j-2 s) \frac{\partial }{\partial \mu }-\mu \frac{\partial ^2 }{\partial \mu ^2}\, , \\
\mathcal{D}_- &= \mu ^2 \frac{\partial }{\partial \mu }- \mu \left(b^{-1} (2 j-s)-k+1\right)
\end{aligned}
\end{align}
along with $\mathcal{D}_L$ in \eqref{cw}.
This kind of realization of zero-mode algebra may be found in (6.16) of \cite{deBoer:1992sy} after performing Fourier transformation from $\mu$-basis to $x$-basis as
\begin{align}
V_{j,s}(x|z) = \int d \mu e^{\mu x } V_{j,s}(\mu|z) \, . \label{mu2x}
\end{align}
These differential operators satisfy commutation relations
\begin{align}
[ \mathcal{D}_H , \mathcal{D}_\pm ] = \pm \mathcal{D}_\pm \, , \quad
[\mathcal{D}_+ , \mathcal{D}_- ] = - 3 \mathcal{D}_H ^2 + (2 k -3) \mathcal{D}_H + (k-3) \mathcal{D}_L \, .
\label{zerocom}
\end{align}
We also find the differential operator
\begin{align}
\mathcal{D}_3 = \mathcal{D}_H^3 + \frac{3 - 2 k}{2} \mathcal{D}_H^2 - \frac12 \{ \mathcal{D}_+ , \mathcal{D}_- \} + (3 -k) \mathcal{D}_L \mathcal{D}_H + \frac12 \mathcal{D}_H \label{3rdCasimir0}
\end{align}
which commutes with $\mathcal{D}_L ,\mathcal{D}_\pm,\mathcal{D}_H$. The anti-commutator is defined as $\{ A , B \} = AB + BA$.
Thus, it can be regarded as the third-order Casimir operator, whose eigenvalue can be computed from \eqref{3rdCasimir0} to be
\begin{align}
\label{3rdCasimir1}
\mathcal{D}_3 &= \frac{1}{2 b } \Bigl[ j^2 \left(b^{-1}-2 (k-3) s\right) \\
& \quad +j \left(-5 b^{-1} s+k \left(2 s \left(b^{-1}+s\right)-1\right)-6 s^2+2\right)-(2 k-3) s \left( b^{-1} s-1\right)\Bigr] \, . \nonumber
\end{align}
Combined with the eigenvalue of the second-order Casimir operator in \eqref{cw}, we see that the representation is labeled by the two parameters $(j,s)$.
For the second realization, the differential operators in the zero-mode action are found to be
\begin{align}\label{z1}
\begin{aligned}
&\mathcal{D}_H = b^{-1} s+\mu \frac{\partial }{\partial \mu } \, , \quad
\mathcal{D}_+ = \mu \, , \\
&\mathcal{D}_- =-(j-2 s) \left((k-3) (j+s)- b^{-1} (k-2)\right) \frac{\partial }{\partial \mu } \\
&\qquad -\mu \left(-3 b^{-1} s+k-3\right) \frac{\partial ^2 }{\partial \mu ^2 }+\mu ^2 \frac{\partial ^3 }{\partial \mu ^3 }
\end{aligned}
\end{align}
in addition to $\mathcal{D}_L$ in \eqref{cw}.
This kind of realization of zero-mode algebra may be found in (2.24) of \cite{Wyllard:2010rp} after performing the Fourier transformation \eqref{mu2x}.
These differential operators satisfy the same commutation relations as in \eqref{zerocom}.
We can also see that the eigenvalue of the third-order Casimir \eqref{3rdCasimir0} reduces to \eqref{3rdCasimir1} upon the substitution of \eqref{z1}.
\subsection{Map of correlation functions}
\label{sec:maps}
The two types of free field realizations for the BP-algebra described above allow correlation functions to be presented in two different forms.
For later analysis, both types of descriptions are used, so we need a map from one description to the other.
As shown in \eqref{zeroactionsl3}, the action of generators on primary states is given by differential operators. However, the expressions for $\mathcal{D}_\pm ,\mathcal{D}_H$ are different for the two realizations.
We have already observed that the eigenvalues of the second- and third-order Casimir operators are the same for both realizations, which means that the primary states with the same $(j,s)$ -values belong to the same representation.
In this subsection, we find new bases for vertex operators such that generators act identically for both descriptions.
It is easier to see the correspondence between the two realisations with a suitable choice of basis.
We shall move from $\mu$-basis to $m$-basis by performing Mellin transformation as
\begin{align}
| j , s ; m \rangle ^{(1)} = \int d \mu \mu^{- m} | j , s ; \mu \rangle \, . \label{mbasis0}
\end{align}
For the first realization, the action of generators \eqref{freeBP1}, in terms of the new basis, can be read off as
\begin{align}
\begin{aligned}
&\mathcal{M}_H ^{(1)}= 1 - m - b^{-1} ( - j + s) \, , \\
&\mathcal{M}_+ ^{(1)}= - m(m - 1 - b^{-1} (j - 2 s) ) \, , \\
&\mathcal{M}_- ^{(1)}= m - 3 + k - b^{-1} (2 j -s) \, .
\end{aligned}\label{1stM0}
\end{align}
We define $\mathcal{M}_H^{(\alpha)},\mathcal{M}_\pm^{(\alpha)}$ by
\begin{align}
G^\pm_0 | j ,s ; m \rangle^{(\alpha)} = - \mathcal{M}_\pm ^{(\alpha)} | j ,s ; m \pm 1 \rangle ^{(\alpha)}\, , \quad
H_0 | j ,s ; m\rangle ^{(\alpha)}= - \mathcal{M}_H ^{(\alpha)} | j ,s ; m\rangle ^{(\alpha)} \, .
\end{align}
We further change the basis as
\begin{align}
| j , s ; m \rangle ^{(2)} = (-1)^m \Gamma (m) \Gamma(m - 1 - b^{-1} (j - 2 s)) \int d \mu \mu^{-m} | j , s ; \mu \rangle \, , \label{mbasis1}
\end{align}
which leads to
\begin{align}
\begin{aligned}
&\mathcal{M}_H ^{(2)}= 1 - m - b^{-1} ( - j + s) \, , \quad
\mathcal{M}_+ ^{(2)} = 1 \, , \\
&\mathcal{M}_- ^{(2)}=
- ( m - 3 + k - b^{-1} (2 j -s) ) (m -1) ( m -2 - b^{-1} (j - 2 s)) \, . \\
\end{aligned}\label{1stM}
\end{align}
In terms of vertex operators, the change of basis is defined as
\begin{align}
\Phi_{j,s;m}^{(2)} (z) = (-1)^m \Gamma (m) \Gamma(m - 1 - b^{-1} (j - 2 s)) \int d \mu \mu^{-m} V_{j,s} (\mu | z) \, . \label{mbasis2}
\end{align}
For the second realization with the generators \eqref{freeBP2},
we may change the basis as
\begin{align}
| j , s ; m' \rangle^{(3)} = \int d \mu \mu^{ m ' } | j , s ; \mu \rangle \, .
\end{align}
From this definition, we find
\begin{align}
\begin{aligned}
&\mathcal{M}_H ^{(3)} = b^{-1} s - m ' - 1 \, , \quad
\mathcal{M}_+ ^{(3)} = 1 \, , \\
&\mathcal{M}_- ^{(3)}= \left( (2 - k) b^{-1} + (k-3) (j + s) \right) (j - 2 s ) m ' \\
& \qquad + \left(3 b^{-1} s -k+3\right) m ' (m ' + 1) - m ' (m ' + 1) (m ' + 2) \, . \\
\end{aligned} \label{2ndM}
\end{align}
Comparing \eqref{1stM} with \eqref{2ndM}, the coefficients $\mathcal{M}_H^{(\alpha)},\mathcal{M}_\pm^{(\alpha)}$ from the two realizations become identical if we set
$
m ' = m-2- b^{-1} (j-2 s)
$.
In other words, changing the basis as
\begin{align}
| j , s ; m \rangle ^{(4)} = \int d \mu \mu^{ m-2- b^{-1} (j-2 s)} | j , s ; \mu \rangle \, ,
\end{align}
we can realize
\begin{align}
\mathcal{M}_H^{(2)} = \mathcal{M}_H^{(4)} \, , \quad \mathcal{M}_\pm^{(2)} = \mathcal{M}_\pm^{(4)} \, .
\end{align}
The corresponding vertex operators may be introduced as
\begin{align}
\Phi_{j,s;m}^{(4)} (z)= \int d \mu \mu^{ m-2- b^{-1} (j-2 s)} V_{j,s} (\mu | z) \, . \label{mbasis3}
\end{align}
Since the actions of generators on the vertex operators \eqref{mbasis2} and \eqref{mbasis3} are the same now,
correlation functions computed with vertex operators $\Phi_{j,s;m} ^{(2)}(z)$ and those with $\Phi_{j,s;m} ^{(4)}(z)$ should be the same once their normalization is properly set.
\section{Correlator relations for W-algebras from $\mathfrak{sl}(3)$}
\label{sec:corrsl3}
In this section, we derive correspondences among correlation functions of theories with the symmetry of W-algebras
associated with $\mathfrak{sl}(3)$.
In subsection \ref{sec:previous}, we reduce the $\mathfrak{sl}(3)$ WZNW model to a theory with BP-algebra symmetry in a way slightly different from \cite{Creutzig:2015hla}.
In subsection \ref{sec:BPW3}, we obtain correspondences between theories with the symmetry of BP-algebra and W$_3$-algebra using the free field realizations of BP-algebra analyzed in the previous section.
Correlators of vertex operators with restricted momenta are examined in subsection \ref{sec:restricted}. These correlators take simpler forms compared to those we studied previously.
In subsection \ref{sec:Direct} we propose several ways to obtain direct relations between $\mathfrak{sl}(3)$ current algebra and W$_3$-algebra.
\subsection{Reduction from affine $\mathfrak{sl}(3)$ to BP-algebra}
\label{sec:previous}
Starting from correlation functions of the $\mathfrak{sl}(3)$ WZNW model, we formulate the action in this first order as we did in \cite{Creutzig:2015hla}:
\begin{align}
\begin{aligned}
S &= \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{b}{4}\sqrt{g} \mathcal{R} (\phi_1 + \phi_2) + \sum_{\alpha=1}^3 \left( \beta_\alpha \bar \partial \gamma^ \alpha + \bar \beta_\alpha \partial \bar \gamma^\alpha \right) \right] \\
&\quad - \frac{1}{2 \pi k} \int d^2 z \left[ e^{b \phi_1} (\beta_1 - \gamma^2 \beta_3 ) (\bar \beta_1 - \bar \gamma^2 \bar \beta_3) + e^{b \phi_2} \beta_2 \bar \beta_2 \right] \, ,
\end{aligned} \label{actionsl3}
\end{align}
where the matrix $G_{ij}$ was defined in \eqref{Gmatrices0}. Moreover, $g_{\sigma\rho} $ is the world-sheet metric, $g = \det g_{\sigma\rho}$, and $\mathcal{R}$ represents the scalar curvature.
In the path integral formulation, correlation functions can be written as
\begin{align}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle = \int \mathcal{D} \Phi e^{- S} \prod^N_{\nu = 1} V_\nu (z_\nu) \, , \label{corr}
\end{align}
where the path integral measure is
\begin{align}
\mathcal{D} \Phi = \mathcal{D} \phi_1 \mathcal{D} \phi_2 \prod_{\alpha =1}^3 \mathcal{D}^2 \beta_\alpha \mathcal{D}^2 \gamma^\alpha \, . \label{measuresl3}
\end{align}
The vertex operators are defined as
\begin{align}
V_\nu (z_\nu) = |\mu_1^\nu|^{4 (j_1^\nu +1)}|\mu_3^\nu/\mu_1^\nu|^{4 (j^\nu_2 +1)} e^{\mu_\alpha^\nu\gamma^\alpha-\bar{\mu}_\alpha^\nu\bar{\gamma}^\alpha} e^{2b(j^\nu_1+1)\phi_1 + 2b(j^\nu_2+1)\phi_2} \, . \label{vertexsl3}
\end{align}
The prefactors are chosen such that the final expression becomes simpler.
We would like to reduce the theory to that with BP-algebra symmetry.
In the previous section, we have seen that BP-algebra can be realized by two free bosons $\varphi_i$ $(i=1,2)$ and a ghost system $(\gamma , \beta)$ along with proper screening charges.
In order to realize the system in terms of action, we need to add anti-holomorphic counterparts $\bar \varphi_i$ and $(\bar \gamma , \bar \beta)$. The free bosons in the two sectors can be formulated as a single non-chiral field as
\begin{align}
\phi_i (z , \bar z )= \varphi_i (z) + \bar \varphi_i (\bar z ) \, . \label{dec}
\end{align}
We shall obtain the theory with BP-symmetry from the $\mathfrak{sl}(3)$ WZNW model in the first order formulation by integrating out two sets of ghost system.
Following \cite{Hikida:2007tq,Creutzig:2015hla}, we first integrate with respect to $\gamma^1 , \gamma^3$ and $\bar \gamma^1 , \bar \gamma^3$, which appear only linearly in the exponent of path integral expression \eqref{corr}.
The integration over zero-modes of these fields leads to delta functions
\begin{align}
\delta ^{(2)} \left( \sum_{\nu=1}^N \mu_1^\nu \right)
\delta ^{(2)} \left( \sum_{\nu=1}^N \mu_3^\nu\right) \, .
\end{align}
Moreover, non-zero modes provide delta functionals for $\beta_1, \beta_3$ and $\bar \beta_1 , \bar \beta_3$.
Integration with respect to these fields leads to the replacement of them by functions:
\begin{align}
&\beta_\alpha (z) = - \sum_{\nu=1}^N \frac{\mu_\alpha^\nu}{z - z_\nu} = - u_\alpha \frac{\prod_{n=1}^{N-2} (z - y_\alpha^n)}{\prod_{\nu=1}^N (z - z_\nu)} \equiv - u_\alpha \mathcal{B}_\alpha (z , z_\nu , y_\alpha^n ) \, , \\
&\bar \beta_\alpha (\bar z) = \sum_{\nu=1}^N \frac{\bar \mu_\alpha^\nu}{\bar z - \bar z_\nu}= \bar u_\alpha \frac{\prod_{n=1}^{N-2} (\bar z - \bar y_\alpha^n)}{\prod_{\nu=1}^N (\bar z - \bar z_\nu)} \equiv \bar u_\alpha \bar{\mathcal{B}}_\alpha (\bar z ,\bar z_\nu , \bar y_\alpha^n )
\end{align}
with $\alpha =1,3$.
Notice that a holomorphic 1-form possesses exactly two more poles than its zeros. Therefore, there are $(N-2)$ zeros for each $\mathcal{B}_\alpha$ or $\bar{\mathcal{B}}_\alpha$, and these zeros are positioned at coordinates $y_\alpha^n$.
The interaction terms in the action now become
\begin{align}
\begin{aligned}
\frac{1}{2 \pi k} \int d^2 z \left[ e^{b \phi_1} (u _1 \mathcal{B}_1 - \gamma^2 u_3 \mathcal{B}_3 ) (\bar u_1 \bar{\mathcal{B}}_1 - \bar \gamma^2 \bar u_3 \bar{\mathcal{B}}_3) - e^{b \phi_2} \beta_2 \bar \beta_2 \right] \, .
\end{aligned} \label{actionsl3p}
\end{align}
In order to relate the action to one of the free field realizations of BP-algebra, we need to remove the function dependence in the interaction terms by redefining fields $\phi_1,\phi_2,\gamma^2$ and $\beta_2$.
There are several ways to do so, but here we choose to make the shifts of $\phi_1$ and $\phi_2$ as
\begin{align}
\phi_1 + \frac{1}{b} \log |u_1 \mathcal{B}_1| ^2 \to \phi_1 \, , \quad \phi_2 + \frac{1}{b} \log |u_1^{-1} u_3 \mathcal{B}_1^{-1 } \mathcal{B}_3 | ^2 \to \phi_2
\end{align}
and change $\gamma^2$ and $\beta_2$ as
\begin{align}\label{z10}
\gamma^2 u_1^{-1} u_3 \mathcal{B}_1^{-1} \mathcal{B}_3 \to \gamma^2 \, , \quad
\beta_2 u_1 u_3^{-1} \mathcal{B}_1 \mathcal{B}_3^{-1} \to \beta_2 \, .
\end{align}
The conjugate fields $\bar \gamma^2$ and $\bar \beta_2$ are changed in the same way as in \eqref{z10}.
The field redefinitions give rise to extra factors in the kinetic terms.
A detailed computation can be found in \cite{Hikida:2007tq,Creutzig:2015hla}.
Part of the contributions from the kinetic terms of $\phi_1$, $\phi_2$ can be regarded as the shifts of momenta for vertex operators and the insertions of extra fields at $y_1^n,y_3^n$. The change of fields also results in the shifts of background charges and a prefactor in front of correlation function. To see the change in the correlation function due to \eqref{z10}, it is convenient to rewrite $\gamma^2$ and $\beta_2$
\begin{align}
\gamma ^2 (z) \simeq e^{X (z)} \eta (z) \, , \quad
\beta _2 (z) \simeq e^{- X (z)} \partial \xi (z) \, ,
\end{align}
where the bosonic field $X$ and the fermionic fields $\eta , \xi $ satisfy the OPEs
\begin{align}
X (z) X(w) \sim - \log (z - w) \, , \quad \eta (z) \xi (w) \sim \frac{1}{z -w} \, .
\end{align}
With this terminology, the change of ghost system can be realized by a shift of $X$
\begin{equation}
X-\ln{(u_1 u_3^{-1}\mathcal{B}_1\mathcal{B}_3^{-1})}\rightarrow X.
\end{equation}
Overall, we obtain the relation among correlation functions as
\begin{align}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle
= |\Theta_N|^2 \delta ^{(2)} \left( \sum_{\nu=1}^N \mu_1^\nu \right)
\delta ^{(2)} \left( \sum_{\nu=1}^N \mu_3^\nu\right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} \tilde V^{(1)} (y_1^n) \tilde V^{(3)} (y_3^n) \right \rangle \, .
\label{sl3toBP}
\end{align}
The new vertex operators are
\begin{align}
\begin{aligned}
&\tilde V_\nu (z_\nu) = e^{ \mu_2 '{}^{ \nu} \gamma^2 - \bar{\mu} ' _2 {}^ \nu \bar{\gamma}^2} e^{2 b (j_1^\nu + 1) \phi_1 + 2 b (j_2^\nu +1) \phi_2 + \phi^1 /b} \, , \\
& \tilde V^{(1)} (y_1^n) = e^{- \phi^1/b + \phi^2 /b} e^{X+\bar{X}} \, , \quad
\tilde V^{(3)} (y_3^n) = e^{- \phi^2 /b} e^{-X-\bar{X}}
\end{aligned} \label{tvertexsl3}
\end{align}
with
\begin{align}
\mu_2 '{}^{ \nu} = \frac{u_1 \mu_1^\nu \mu_2^\nu}{ u_3 \mu_3^\nu } \, , \qquad
\bar{\mu}_2 '{}^{ \nu} = \frac{\bar{u}_1 \bar{\mu}_1^\nu \bar{\mu}_2^\nu}{ \bar{u}_3 \bar{\mu}_3^\nu } \, .
\end{align}
The prefactor is
\begin{align}
\begin{aligned}
\Theta_N &= u_1 u_3 \prod_{\mu < \nu} (z_\mu - z_\nu)^{2/(3 b^2)}
\prod_{n,\nu}((y_1^n - z_\nu) (y_3^n - z_\nu))^{-1/(3 b^2)} \\
& \quad \times
\prod_{m < n} ((y_1^m - y_1^n ) (y_3^m - y_3^n))^{2/(3 b^2)+1} \prod_{m,n} (y_3^m - y_1^n)^{-1/(3 b^2)-1} \, .
\end{aligned}
\end{align}
The left-hand side of \eqref{sl3toBP} is computed with the action
\begin{align}
\begin{aligned}
S &= \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{1}{4}\sqrt{g} \mathcal{R} ((b + b^{-1})\phi^1 + b \phi^2) + \beta_2 \bar \partial \gamma^ 2 + \bar \beta_2 \partial \bar \gamma^2 \right] \\
&\quad + \frac{1}{2 \pi k} \int d^2 z \left[ e^{b \phi_1} (1 - \gamma^2 ) (1 - \bar \gamma^2 ) - e^{b \phi_2} \beta_2 \bar \beta_2 \right] \, .
\end{aligned} \label{actionBP0}
\end{align}
We can see that the action describes the first realization with the set of screening charges \eqref{screening2} after the shift $\gamma^2 - 1 \to \gamma^2$. The conformal dimensions of $(\gamma^2,\beta_2)$ are $(0,1)$, which means that the energy-momentum tensor is twisted according to \eqref{twist}.%
\footnote{In \cite{Creutzig:2015hla}, the change of fields is made such that $(\gamma,\beta)$ left have conformal dimensions $(1/2,1/2)$. In other words, the energy-momentum tensor for the reduced theory is not twisted there.}
\subsection{Reduction from BP-algebra to W$_3$-algebra}
\label{sec:BPW3}
In the previous subsection, we examined the specific example of reducing the $\mathfrak{sl}(3)$ correlation function to that of the BP-algebra. The strategy for obtaining new correlator relations may be summarized in general as follows.
We start from an action with free kinetic terms plus interactions.
We then integrate out several sets of ghost system and perform field redefinitions to eliminate explicit coordinate dependence in the action. Carefully treating contributions from kinetic terms, we obtain a correspondence among correlators of different theories.
In this subsection, we reduce correlators of the theory with BP-algebra symmetry to $\mathfrak{sl}(3)$ Toda field theory with W$_3$-algebra symmetry by applying the generic method.
As seen above, there are two types of free field realizations for BP-algebra, which means that there are two theories we could start from. With the procedure described above, one of the theories is suitable for the reduction to the W$_3$-correlator. In subsection \ref{sec:previous}, we ended up with the action corresponding to the first realization with the screening operators \eqref{screening2}.
However, with this form of the action, it is not easy to integrate $(\gamma^2,\beta_2)$ out, since $\gamma^2$ appears in an interaction term.
Fortunately, we have the second realization with the screening operators \eqref{screening3}, and
the corresponding action can be written as%
\footnote{The prefactors in front of interaction terms can be chosen arbitrary by constant shifts of $\phi_1,\phi_2$.}
\begin{align}
\begin{aligned}
S &= \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{1}{4}\sqrt{g} \mathcal{R} ((b + b^{-1})\phi^1 + b \phi^2) + \beta \bar \partial \gamma + \bar \beta \partial \bar \gamma \right] \\
&\quad + \frac{1}{2 \pi k} \int d^2 z \left[ e^{b \phi_1} - e^{b \phi_2} \beta \bar \beta \right] \, . \label{actionBP2}
\end{aligned}
\end{align}
It is then possible to integrate out $(\gamma ,\beta )$ from the correlation function associated to this action, and the rest can be proceeded by applying the generic method.
We take the correlation function of the form \eqref{corr} with the action \eqref{actionBP2} and the path integral measure
\begin{align}
\mathcal{D} \Phi = \mathcal{D} \phi_1 \mathcal{D} \phi_2 \mathcal{D}^2 \beta \mathcal{D}^2 \gamma \, .
\end{align}
The vertex operators are
\begin{align}
V_\nu (z_\nu) = |\mu_2^\nu|^{4 (j_2^\nu +1)} e^{\mu^\nu \gamma -\bar{\mu}^\nu \bar{\gamma} } e^{2b(j^\nu_1+1)\phi_1 + 2b(j^\nu_2+1)\phi_2} \, .
\end{align}
Again, the prefactor is chosen such that the final expression becomes simpler.
The integration over $(\gamma , \beta )$
leads to $\delta^{(2)} (\sum_\nu \mu ^\nu)$ and the replacement of $\beta $ by a function as
\begin{align}
\beta (z) = - \sum_{\nu=1}^N \frac{\mu^\nu}{z - z_\nu} = - u \frac{\prod_{n=1}^{N-2} (z - y^n)}{\prod_{\nu=1}^N (z - z_\nu)} \equiv - \mathcal{B} (z, z_\nu , y^n )\, .
\end{align}
There is $|\mathcal{B} |^2$ in an interaction term now, and we remove it by shifting $\phi_2$ as
\begin{align}
\phi_2 + \frac{1}{b} \log |u \mathcal{B} | ^2 \to \phi_2 \, .
\end{align}
Similarly to the previous case, the change of $\phi_2$ in the kinetic term contributes extra factors to the correlation function.
In the end, we arrive at the relation
\begin{align} \label{BPtoW3}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle
= |\Theta_N|^2 \delta ^{(2)} \left( \sum_{\nu=1}^N \mu^\nu \right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} \tilde V (y^n) \right \rangle \, .
\end{align}
The right-hand side is evaluated with the action of $\mathfrak{sl}(3)$ Toda field theory
\begin{align}
S = \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{Q_\phi}{4}\sqrt{g} \mathcal{R} (\phi_1 + \phi_2) + \frac{1}{k}( e^{b \phi_1} + e^{b \phi_2} ) \right]
\end{align}
with $Q_\phi = b + b^{-1}$. The vertex operators are
\begin{align}
\begin{aligned}
&\tilde V_\nu (z_\nu) = e^{2 b (j_1^\nu + 1) \phi_1 + 2 b (j_2^\nu +1) \phi_2 + \phi^2 /b} \, ,
& \tilde V (y^n) = e^{- \phi^2 /b}
\end{aligned}
\end{align}
and the prefactor is
\begin{align}
\begin{aligned}
\Theta_N = u^{2k/3} \prod_{\mu < \nu} (z_\mu - z_\nu)^{2/(3 b^2)}
\prod_{n,\nu}(y^n - z_\nu) ^{-2/(3 b^2)}
\prod_{m < n} (y^m - y^n ) ^{2/(3 b^2)} \, .
\end{aligned}
\end{align}
\subsection{Putting restrictions on momenta}
\label{sec:restricted}
In subsection \ref{sec:previous}, we have considered the action \eqref{actionsl3} associated to the $\mathfrak{sl}(3)$ WZNW model.
The action includes $\gamma^2$ in an interaction term, which makes it difficult to integrate $\gamma^2$ out.
In this subsection, we propose a way to resolve the issue by putting restrictions on momenta of vertex operators.
As before, we start from the $\mathfrak{sl} (3)$ correlator \eqref{corr} with the action \eqref{actionsl3} and the path integral measure \eqref{measuresl3}.
However, we use a slightly different vertex operator
\begin{align}
V_\nu (z_\nu) = |\mu_1^\nu|^{4 (j_1^\nu +1)} e^{\mu_1^\nu\gamma^1 + \mu_2^\nu\gamma^2 - \bar \mu_1^\nu \bar \gamma^1 - \bar \mu_2^\nu \bar \gamma^2} e^{2b(j^\nu_1+1)\phi_1 + 2b(j^\nu_2+1)\phi_2} \, .
\end{align}
Namely, we set
\begin{align}
\mu^\nu_3 = \bar{\mu}^\nu_3 = 0 \label{restriction}
\end{align}
in \eqref{vertexsl3} for all $\nu$.
In this case, integration over $\gamma^3 $ leads to delta functional
\begin{align}
\delta^{(2)} ( \bar \partial \beta_3 (z)) \, .
\end{align}
With appropriate boundary conditions, we can set
\begin{align}
\beta_3 (z) = \bar \beta_3 (\bar z) = 0 \, .
\end{align}
Substituting this into the action \eqref{actionsl3}, $\gamma^2$-dependent term disappears.
Integrating the correlator with respect to $\gamma^1$ gives
\begin{equation}\label{a04}
\beta_1(z)= - \sum_{\nu=1}^N \frac{\mu^\nu_1}{z - z_\nu} = - u _1 \frac{\prod_{n=1}^{N-2}(z-y^n_1)}{\prod_{\nu=1}^N(z-z_\nu)} = - u _1 \mathcal{B} _1 \, , \quad
\sum_{\nu=1}^{N}\mu_1^{\nu}=0 \, .
\end{equation}
We make a shift in $\phi_1(z)$ by letting
\begin{equation}
\phi_1 + \frac 1 b \ln| u _1 \mathcal{B} _1 |^2 \to \phi_1 ,
\end{equation}
which yields the relation
\begin{align} \label{sl3toBP2}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle
= |\Theta_N|^2 \delta ^{(2)} \left( \sum_{\nu=1}^N \mu_1^\nu \right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} \tilde V (y^n_1) \right \rangle \, .
\end{align}
The right-hand side is evaluated by the action \eqref{actionBP2} with $(\gamma, \beta)$ replaced by $(\gamma^2 , \beta_2)$. The vertex operators are
\begin{align}
\begin{aligned}
&\tilde V_\nu (z_\nu) = e^{\mu_2^\nu \gamma^2 - \bar \mu_2^\nu \bar \gamma^2 }e^{2 b (j_1^\nu + 1) \phi_1 + 2 b (j_2^\nu +1) \phi_2 + \phi^1 /b} \, ,
& \tilde V (y^n) = e^{- \phi^1 /b}
\end{aligned}
\end{align}
and the prefactor is
\begin{align}
\begin{aligned}
\Theta_N = u^{2}_1 \prod_{\mu < \nu} (z_\mu - z_\nu)^{2/(3 b^2)}
\prod_{n,\nu}(y^n_1 - z_\nu) ^{-2/(3 b^2)}
\prod_{m < n} (y^m_1 - y^n_1 ) ^{2/(3 b^2)} \, .
\end{aligned}
\end{align}
As was done in the previous subsection, we can reduce the theory to $\mathfrak{sl}(3)$ Toda field theory by further integrating $(\gamma ^2 , \beta _ 2 )$ out.
In this way, we found a version of correlator relation by putting restrictions on momenta for vertex operators of $\mathfrak{sl}(3)$ WZNW model as in \eqref{restriction}.
In general it is possible to obtain more new relations with similar restrictions.
In the following, we would like to clarify the features of correlator relations obtained in this manner by working on the simplest setting with $\mathfrak{sl}(2)$.
We begin with the $\mathfrak{sl}(2)$ WZNW model with the action
\begin{align}
S = \frac{1}{2 \pi} \int d^2 w \left[\bar \partial \phi \partial \phi + \beta \bar \partial \gamma + \bar \beta \partial \bar \gamma + \frac{b}{4} \sqrt{g} \mathcal{R} \phi - \frac{1}{k} \beta \bar \beta e^{2 b \phi}\right] \, ,
\end{align}
where we set $b =1/\sqrt{ k-2 }$.
We consider the vertex operators in the form
\begin{align}
V_j (\mu_\nu | z) = |\mu|^{2 j + 2} e^{\mu \gamma- \bar \mu \bar \gamma } e^{2 b (j +1) \phi } \, .
\end{align}
We may change the basis by Mellin transformation as
\begin{align}
\Phi^j_{m , \bar m} (z)= \int \frac{d^2 \mu}{|\mu|^2} \mu^{-m} \mu^{- \bar m} V_j (\mu | z) \propto \gamma(z)^{- j - 1 +m} \bar \gamma(\bar z)^{- j - 1 + \bar m} e^{2 b (j +1) \phi (z,\bar z)} \, . \label{hwop}
\end{align}
We can see that $V_j (\mu |z)$ with $\mu = \bar \mu = 0$ can be identified with $\Phi^j_{j+1,j+1} (z)$
up to an overall normalization. The operator $\Phi^j_{j+1,j+1} (z)$ is special in the sense that it corresponds to a highest-weight state.
Let us consider an $N$-point function and put restrictions on momenta as
\begin{align}
\mu^\nu = \bar \mu^\nu =0 \quad (\nu = M+1 ,M+2 ,\ldots ,N) \, .
\end{align}
After integrating over $\gamma$, we obtain the relation
\begin{align}
\beta (z) = - \sum_{\nu=1}^M \frac{\mu^\nu}{w - z_\nu} = - u \frac{\prod_{n=1}^{M-2} (z - y^n)}{\prod_{\nu = 1}^M (z - z_\nu)} \, , \quad \sum_{\nu =1}^{M} \mu^\nu = 0 \, .
\end{align}
Since the number of poles is now $M$, the holomorphic 1-form possesses only $M-2$ zeros located at $z = y^n$.
This implies that the number of extra insertions is $M-2$, and therefore, we can obtain a relation among an $N$-point function of $\mathfrak{sl}(2)$ WZNW model and an $(N+M-2)$-point function of Liouville field theory.
For $N=4,M=2$, this relation is essentially the one presented in section 3.2 of \cite{Chang:2014jta} with a certain limit of parameters.
Furthermore, this relation was utilized to examine BPS conformal blocks of $\mathcal{N}=2$ superconformal field theory in \cite{Lin:2015wcg,Lin:2016gcl}.
Similar arguments can be applied to more generic examples of $\mathfrak{sl}(L)$ with $L=3,4,\ldots$.
An operator with $\mu_\alpha= 0$ for some $\alpha$ generically corresponds to a highest-weight state,
and correlator relations obtained as in this subsection should be useful to investigate special kinds of correlators like BPS ones in a supersymmetric theory.
\subsection{Direct reduction from affine $\mathfrak{sl}(3)$ to W$_3$-algebra}
\label{sec:Direct}
Combining the correlator relations in subsections \ref{sec:BPW3} and \ref{sec:restricted}, we can obtain a direct relation
between $\mathfrak{sl}(3)$ current algebra and W$_3$-algebra.
However, restrictions on momenta have been put on the correlator of $\mathfrak{sl}(3)$ WZNW model.
One may want to start from a generic correlator as in subsection \ref{sec:previous}.
The reduced action is given by \eqref{actionBP0}, which corresponds to the first realization of BP-algebra after performing a shift $\gamma^2 -1 \to \gamma^2$.
In section \ref{sec:FreeBP}, we have developed a map of correlators described by two free field realizations.
Mapping to the second realization of BP-algebra, we can further reduce to $\mathfrak{sl} (3)$ Toda field theory by applying the correlator relation in subsection \ref{sec:BPW3}.
However, twist operators $e^{\pm X}$ appearing in \eqref{tvertexsl3} would transform non-trivially under the shift $\gamma^2 -1 \to \gamma^2$. Because of this, the explicit form of correlator relations would be complicated.
In this subsection, we provide an approach of avoiding the insertions of extra twist operators.
As in subsection \ref{sec:previous}, we begin with a generic correlator of $\mathfrak{sl}(3)$ WZNW model in the form \eqref{corr} with the action \eqref{actionsl3} and the path integral measure \eqref{measuresl3}.
Integrating out $(\gamma^1,\beta_1)$ and $(\gamma^3,\beta_3)$, we arrive at the action with interaction terms in \eqref{actionsl3p}.
Here we only make the change of field as
\begin{align}
\phi_1 + \frac{1}{b} \log |u_1 \mathcal{B}_1| ^2 \to \phi_1 \, ,
\end{align}
then the interaction terms become
\begin{align}
\frac{1}{2 \pi k} \int d^2 z \left[ e^{b \phi_1} (1 - \gamma^2 u_1^{-1} u_3 \mathcal{B}_1^{-1} \mathcal{B}_3 ) (1 -\bar \gamma^2 {\bar u_1}^{-1} \bar u_3 {\bar{\mathcal{B}}}_1^{-1} \bar{\mathcal{B}}_3) - e^{b \phi_2} \beta_2 \bar \beta_2 \right] \, . \label{newaction}
\end{align}
New operators are inserted at $z = y_1^n$.
Let us focus on the holomorphic part by decomposing $\phi_i (z, \bar z) $ as in \eqref{dec}.
The interaction terms \eqref{newaction} are then given by a linear combination of
\begin{align}
&Q_1 = \int dz e^{b \varphi _1} \, , \\
&Q_2 = \int dz e^{b \varphi_1} \gamma^2 u_1^{-1} u_3 \mathcal{B}_1^{-1} \mathcal{B}_3 \, , \label{Q2}\\
&Q_3 =\int dz e^{b \varphi_2} \beta_2 \, .
\end{align}
The functions of $\mathcal{B}_1^{-1} \mathcal{B}_3 $ in \eqref{Q2} can be removed by a shift of $\varphi_1$, but this produces extra functions $\mathcal{B}_1 \mathcal{B}_3^{-1} $ in $Q_1$.
In other words, we can remove functions in either $Q_1$ or $Q_2$ but not in both.
As examined in section \ref{sec:FreeBP}, a free field realization of BP-algebra uses the set of screening charges $(Q_1,Q_3)$ or $(Q_2,Q_3)$ (after removing the extra functions).
We may choose $(Q_1,Q_3)$ as the set of screening charges and treat $Q_2$ perturbatively as
\begin{align}
e^{ - \lambda Q_2} = \sum_{p=0}^\infty \frac{(- \lambda)^p}{p!} ( Q_2)^p \, .
\end{align}
Rewriting $Q_2$ as
\begin{align}
Q_2 = \frac{1}{2 \pi i}
\int d zu_1^{-1} u_3 \mathcal{B}_1^{-1} \mathcal{B}_3 \oint d \mu _2 \mu_2 ^ {-2} e^{\mu_2 \gamma ^2 } e^{b \varphi_1} \, ,
\end{align}
the correlation functions can be put in the form studied in subsection \ref{sec:BPW3}.
Now we can apply the reduction procedure in subsection \ref{sec:BPW3} and express the correlation function in terms of $\mathfrak{sl}(3)$ Toda field theory.
However, an $N$-point function of $\mathfrak{sl}(3)$ WZNW model is written as an infinite sum of correlators of $\mathfrak{sl}(3)$ Toda field theory.
\section{Free field realizations of W-algebras from $\mathfrak{sl}(4)$}
\label{sec:freesl4}
In sections \ref{sec:FreeBP} and \ref{sec:corrsl3}, we have examined correlators for theories with the symmetry of W-algebras associated to $\mathfrak{sl}(3)$. In particular, it was important to choose a convenient free field realization of BP-algebra in order to obtain simpler expressions of correlator relations.
In this and the following sections, the analysis of W-algebras is extended from those associated with $\mathfrak{sl}(3)$ to $\mathfrak{sl}(4)$.
As explained in the introduction, one can construct three types of W-algebras with non-regular embeddings of $\mathfrak{sl}(2)$ to $\mathfrak{sl}(4)$ by Hamiltonian reduction.
These embeddings are known as subregular, rectangular, and minimal corresponding to the partitions $4 = 3 +1$, $4 = 2 + 2$, and $4 = 2 + 1+1$, respectively.
In this section, we express the generators of these non-regular W-algebras in terms of free fields by making use of screening charges in appendix \ref{sec:screening}.
With the explicit expressions, we can obtain maps of vertex operators and correlation functions among different free field realizations of the W-algebras.
For W-algebras associated with $\mathfrak{sl}(4)$, we need three free bosons $\varphi_i$ $(i=1,2,3)$ with OPEs
\begin{align}
\varphi_i (z) \varphi_j (w) \sim - G_{ij} \log (z -w) \, , \label{bosonOPEsl4}
\end{align}
where we have introduced matrices
\begin{align}
G_{ij} =
\begin{pmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{pmatrix} \, , \quad
G^{ij} =
\begin{pmatrix}
3/4 & 1/2 & 1/4 \\
1/2 & 1 & 1/2 \\
1/4 & 1/2 & 3/4
\end{pmatrix} \label{Gmatrices} \, .
\end{align}
We also use $\ell$ sets of ghost systems $( \gamma^\alpha , \beta_\alpha ) $ satisfying
\begin{align}
\gamma^\alpha (z) \beta_{\alpha '} (w) \sim \frac{\delta^\alpha_{~ \alpha '}}{z -w} \label{ghostOPEs}
\end{align}
where $\alpha , \alpha ' \in \{1,2, \ldots , \ell\}$. The number of ghost systems depends on the particular W-algebra we are considering. In the case of $\mathfrak{sl}(4)$, we set
\begin{align}\label{bsl4}
b = \frac{1}{\sqrt{k -4}}
\end{align}
with $k$ the level of $\mathfrak{sl}(4)$ current algebra.%
\footnote{Compared with the notation in appendix \ref{sec:screening}, we change $\kappa \to i b^{-1}$ and $k \to - k$, see also footnote \ref{-k} for $\mathfrak{sl} (3)$. Furthermore, we use $ i \partial \varphi_j = \alpha_j$, which is suitable for Lagrangian description. }
\subsection{$W^{(2)}_4$-algebra}
\label{sec:W24}
The W-algebra with the subregular embedding of $\mathfrak{sl}(2)$ into $\mathfrak{sl}(n)$ is denoted as $W_n^{(2)}$. The properties of such subregular W-algebras were examined, for example, in \cite{Feigin:2004wb}. In this subsection, we set $n=4$.
The generators of the algebra include a spin-one current $H(z)$, three spin-two currents $G^\pm(z)$, $T(z)$, and a spin-three current $W(z)$. Here $T(z)$ is the energy-momentum tensor with central charge
\begin{align}
c = 24 k+\frac{60}{k-4}-19 \, .
\end{align}
The OPEs involving $H(z)$ are
\begin{align}\label{OPEsHHHG}
H(z) H(w) \sim \frac{( 8 - 3 k)/4}{(z - w)^2} \, , \quad
H(z) G^\pm (w) \sim \pm \frac{G^+ (w)}{z-w} \, .
\end{align}
The spin-three current $W(z)$ is regular with respect to $H(z)$. One may find expressions of other OPEs in \cite{Feigin:2004wb}.
The W$_4^{(2)}$ algebra can be realised by a set of ghost system $(\gamma,\beta)$ along with
three free bosons $\varphi_i$ $(i=1,2,3)$.
\bigskip
\noindent {\bf The first realization.}
The screening charges for $W^{(2)}_4$-algebra can be found in \eqref{subs}.
In one of the realizations, these are given by screening operators
\begin{align} \label{screeningW241}
\mathcal{V}_1 = e^{ b \varphi_1 } \beta \, , \quad \mathcal{V}_2 =e^{ b \varphi_2 } \gamma \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } \, .
\end{align}
Requiring generators to commute with the screening charges, we find the expressions of $T$ and $H$ as
\begin{align}
T = - \frac12 G^{ij} \partial \varphi_i \partial \varphi_j + \frac{(2k - 5)b}{2} \partial^2 \varphi_1 + (k-2)b \partial^2 \varphi_2 + \frac{(2k - 5)b}{2} \partial^2 \varphi_3 + \gamma \partial \beta
\end{align}
and
\begin{align}
H = - \frac{1}{4 b} \partial \varphi_1 + \frac{1}{2 b} \partial \varphi_2 + \frac{1}{4 b} \partial \varphi_3 - \gamma \beta \, .
\end{align}
In particular, the conformal weights of $(\gamma ,\beta)$ are $(1,0)$.
We also find that
\begin{align}
\begin{aligned}
&G^+ = - \gamma \gamma \beta - b^{-1} \partial \varphi_1 \gamma + (k -2) \partial \gamma \, , \\
&G^- = - b^{-1} \left(2 (k-2) \partial \varphi_2 \partial \beta +(k-2) \partial \varphi_3 \partial \beta +2 \partial \varphi_2 \gamma \beta \beta + \partial \varphi_3 \gamma \beta \beta \right) \\
& \quad - (k-3) b^{-1} \partial^2 \varphi_2 \beta +k \partial \varphi_2 \partial \varphi_2 \beta +k \partial \varphi_3 \partial \varphi_2 \beta -4 \partial \varphi_2 \partial \varphi_2 \beta -4 \partial \varphi_3 \partial \varphi_2 \beta \\
& \quad +3 k \gamma \beta \partial \beta +k \partial \gamma \beta \beta -6 \gamma \beta \partial \beta -4 \partial \gamma \beta \beta + \gamma \gamma \beta \beta \beta + \left(5 + k^2 - 9 k/2 \right) \partial \partial \beta
\end{aligned}
\end{align}commute with the screening charges, and they indeed satisfy the correct OPEs with $T$ and $H$.
The expression of $W$ can then be found from the OPE with $G^+$ and $G^-$ as
\begin{align}
W = \frac{4 (k-4) (5 k-16)}{3 (8-3 k)^2 (k-2)} \gamma \gamma \gamma \beta \beta \beta + \cdots \, ,
\end{align}
which commutes with the screening charge.
\bigskip
\noindent {\bf The second realization.}
Another set of screening charges is given by
\begin{align} \label{screeningW242}
\mathcal{V}_1 = e^{ b \varphi_1 } \beta \, , \quad \mathcal{V}_2 = e ^{ b \varphi_2 } \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } \, .
\end{align}
Using the same commuting condition, we find the expressions of $T$ and $H$ as
\begin{align}
T = - \frac12 G^{ij} \partial \varphi_i \partial \varphi_j + \frac{3b}{2} \partial^2 \varphi_1 + (k-2)b \partial^2 \varphi_2 + \frac{(2k - 5)b}{2} \partial^2 \varphi_3 -\gamma \partial \beta - 2 \partial \gamma \beta
\end{align}
and
\begin{align}
H = \frac{3}{4 b} \partial \varphi_1 + \frac{1}{2 b} \partial \varphi_2 + \frac{1}{4 b} \partial \varphi_3 + \gamma \beta \, .
\end{align}
In particular, the conformal weights of $(\gamma,\beta)$ are now $(-1,2)$.
Using OPEs, such as \eqref{OPEsHHHG}, as constraints, we found
\begin{align}
G^+ = \beta \, ,\quad G^- = - \gamma \gamma \gamma \gamma \beta \beta \beta + \cdots \, , \quad
W = -\frac{4 (k-4) (5 k-16)}{3 (8-3 k)^2 (k-2)} \gamma \gamma \gamma \beta \beta \beta + \cdots \, ,
\end{align}
which also commute the screening charges. \\
As in \eqref{twist}, we then twist the energy-momentum tensor as
\begin{align}
T(z) \to T_t (z) = T(z) + \partial H(z) \, . \label{twistW24}
\end{align}
The spins of $G^+$ and $G^-$ become $1$ and $3$, respectively, with respect to the twisted energy-momentum tensor.
For the both free field realizations, the conformal weights of $(\gamma, \beta)$ are $(0,1)$ after the twisting.
\subsection{Rectangular W-algebra with $\mathfrak{su}(2)$ symmetry}
\label{sec:rectangular}
A generic rectangular W-algebra is obtained by performing Hamiltonian reduction of Lie algebra $\mathfrak{sl}(Mn)$. The embedding of $\mathfrak{sl}(2)$ into $\mathfrak{sl}(Mn)$ corresponds to the partition $Mn=n+n+\ldots+n$, a sum of $M$ numbers of $n$. A rectangular W-algebra contains a subalgebra $\mathfrak{sl}(M)$.
The rectangular W-algebra can be realized as the asymptotic symmetry of extended higher spin gravity \cite{Creutzig:2018pts,Creutzig:2019qos}.%
\footnote{See \cite{Creutzig:2013tja,Eberhardt:2018plx,Creutzig:2019wfe,Rapcak:2019wzw,Eberhardt:2019xmf} for related works.}
In this subsection, we examine free field realizations of the rectangular W-algebra associated to $\mathfrak{sl}(Mn)$ with $M=n=2$.
The rectangular W-algebra associated with $\mathfrak{sl}(4)$ is generated by three spin-one currents $J^a(z)$ and four spin-two currents $T(z),Q^a(z)$, where $a=1,2,3$ and $T(z)$ being the energy-momentum tensor with central charge
\begin{align}
c = 12 k+\frac{60}{k-4}+7 \, .
\end{align}
The spin-one currents $J^a(z)$ generate an $\mathfrak{su}(2)$ current algebra with OPEs\begin{align}
\begin{aligned}
&J^a (z) J^b (w) \sim \frac{{\ell} \delta^{a,b}/2}{(z-w)^2} + \frac{i f^{ab}_{~~c} J^c(w)}{z-w} \, ,\\
&T(z) J^a (w) \sim \frac{J^a(w)}{(z-w)^2} + \frac{\partial J^a (w)}{z-w} \, ,
\end{aligned}
\end{align}
where $\ell = 4 -2 k$ is the $\mathfrak{su}(2)$ level. The structure constant of $\mathfrak{su}(2)$ is given by $i f^{abc}$ whose indices can be raised and lowered by $\kappa^{ab} = \delta^{a,b}/2$ and $\kappa_{ab} = 2 \delta_{a,b}$,
.
The spin-two currents $Q^a(z)$ transform as in the adjoint representation of $\mathfrak{su}(2)$.
Their OPEs with $J^a(z)$ and $T(z)$ are
\begin{align}
J^a (z) Q^b (w) \sim \frac{i f^{ab}_{~~c} Q^c (w)}{z-w} \, , \quad
T(z) Q^b (w) \sim \frac{2 Q^b (w)}{(z-w)^2} + \frac{\partial Q^b (w)}{z-w} \, .
\end{align}
The OPE $Q^a(z) Q^b (w)$ can be found in \cite{Creutzig:2018pts,Creutzig:2019qos}.
The free field realization of the theory can be constructed from two sets of ghost systems $(\gamma^\alpha , \beta_\alpha )$ $(\alpha =1,2)$ along with three free bosons $\varphi_i$ $(i=1,2,3)$.
\bigskip
\noindent {\bf The first realization.}
As found in \eqref{i2}, there are two sets of screening charges for the rectangular W-algebra of $\mathfrak{sl}(4)$.
One of them is given by the set of screening operators
\begin{align}
\mathcal{V}_1 = e^{ b \varphi_1 } \beta_1 \, , \quad \mathcal{V}_2 =e^{ b \varphi_2 } (\gamma^1 - \gamma^2) \, , \quad
\mathcal{V}_3 =e^{ b \varphi_3 } \beta_2 \, .
\end{align}
The energy-momentum tensor commuting with the screening charges is found to be
\begin{align}
T = - \frac12 G^{ij} \partial \varphi_i \partial \varphi_j + \frac{(k-1)b}{2 } \partial^2 \varphi_1 + (k-2)b \partial^2 \varphi_2 + \frac{(k-1) b}{2 } \partial^2 \varphi_3 - \partial \gamma^1 \beta_1- \partial \gamma^2 \beta_2 \label{recEM}.
\end{align}
The $\mathfrak{su}(2)$ currents in terms of the free fields are given by
\begin{align}
\begin{aligned}
&J^+ = \beta_1 + \beta_2 \, , \quad J^3 = \gamma^1 \beta _1 + \gamma ^2 \beta _2 +\frac{1}{2 b} \partial \varphi_1 +\frac{1}{2 b} \partial \varphi_3 \, , \\
&J^- = - b^{-1} \partial \varphi_3 \gamma ^2 - b^{-1} \partial \varphi_1 \gamma ^1 - \gamma ^1 \gamma ^1 \beta _1 - \gamma ^2 \gamma ^2 \beta _2 +(k-2) \partial \gamma ^1 +(k-2) \partial \gamma ^2 \, ,
\end{aligned}
\end{align}
where we have defined $J^\pm = J^1 \pm i J^2$.
A charged spin-two current is found to be
\begin{align}
\begin{aligned}
&Q^+ = \partial \varphi _1 \beta _1 + \partial \varphi_1 \beta _2 +2 \partial \varphi_2 \beta _1 -2 \partial \varphi_2 \beta_2 - \partial \varphi_3 \beta _1 - \partial \varphi_3 \beta _2 \\
& \qquad+ 4 b \gamma ^1 \beta_1 \beta_2 - 4 b \gamma ^2 \beta _1 \beta _2 + 2 (k-2) b \partial \beta _2 - 2 (k-2) b \partial \beta _1
\end{aligned}
\end{align}
with $Q^+ = Q^1 + i Q^2$. The rest of charged spin-two currents can be obtained from their OPEs with $J^-$.
\bigskip
\noindent {\bf The second realization.}
The other set of screening operators presented in \eqref{i2} is given by
\begin{align}
\mathcal{V}_1 = e^{ b \varphi_1 } \beta_1 \, , \quad \mathcal{V}_2 = e^{ b \varphi_2 } (1 - \gamma^1 \gamma^2) \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } \beta_2 \, .
\end{align}
The energy-momentum tensor has exactly the same expression as in the previous realization as in \eqref{recEM}.
The $\mathfrak{su}(2)$ currents, in this case, are
\begin{align}
\begin{aligned}
&J^+ = - b^{-1} \partial \varphi_3 \gamma ^2 - \gamma ^2 \gamma ^2 \beta _2 +\beta _1 +(k-2) \partial \gamma ^2 \, , \\
&J^3 = \gamma^1 \beta _1 - \gamma ^2 \beta _2 -\frac{1}{2b} \partial \varphi_3 +\frac{1}{2 b} \partial \varphi_1 \, , \\
&J^- = - b^{-1} \partial \varphi_1 \gamma ^1 - \gamma ^1 \gamma ^1 \beta _1 +\beta _2 +(k-2) \partial \gamma ^1 \, .
\end{aligned}
\end{align}
A charged spin-two current is found to be
\begin{align}
\begin{aligned}
&Q^3 =
- \gamma^1 \gamma ^1 \gamma ^2 \gamma ^2 \beta _1 \beta _2 + \cdots
\end{aligned}
\end{align}
up to an overall normalization. The rest of the charged spin-two currents can again be obtained by their OPEs with $J^\pm$. \\
Note that we do not twist the energy-momentum tensor here.
In fact, the conformal dimensions of $(\gamma^\alpha , \beta_\alpha )$ $(\alpha =1,2)$ are $(0,1)$ with respect to the energy-momentum tensor \eqref{recEM}.
\subsection{QSCA with $\mathfrak{su}(2)$ symmetry}
\label{sec:QSCA}
The W-algebra with minimal embedding of $\mathfrak{sl}(2)$ into $\mathfrak{sl}(n)$ was studied in \cite{Romans:1990ta}.
The algebra is known as quasi-superconformal algebra (QSCA), since it can be obtained by replacing fermionic spin-3/2 generators of the superconformal algebras presented in \cite{Knizhnik:1986wc,Bershadsky:1986ms} with bosonic generators of the same spin.
In this subsection, we examine free field realizations of QSCA associated to $\mathfrak{sl}(4)$.
The generators of QSCA include four spin-one currents $H(z),J^a(z)$ $(a=1,2,3)$, four spin-$3/2$ bosonic currents $G_\pm^{i} (z)$ $(i=1,2)$, and a spin-two current $T (z)$, the energy-momentum tensor with central charge
\begin{align}
c =6 k-\frac{60}{4-k}+15 \, .
\end{align}
The OPEs involving $H(z)$ satisfy
\begin{align}
H (z) H(w) \sim \frac{1}{(z-w)^2} \, , \quad H (z) G_{\pm}^i (w) \sim \pm \frac{q G_\pm^i (w)}{z - w}
\end{align}
with $q = 1/\sqrt{2-k}$. The rest of the spin-one currents $J^a(z)$ generate an $\mathfrak{su}(2)$ current algebra. The currents are normalised to satisfy the OPE
\begin{align}
J^a (z) J^b (w) \sim \frac{\ell /2}{(z - w)^2} + \frac{i f^{ab}_{~~c} J^c (w) } {z-w}
\end{align}
with the level $\ell = 1 - k$. The action of $J^a(z)$ on $G^i_\pm(w)$ is given by
\begin{align}
J^a (z) G_+^i (w)\sim - \frac{(\sigma^a)^i _{~ j} G^j_+ (w)}{z-w} \, , \quad
J^a (z) G_-^{i} (w) \sim \frac{G_-^{j} (w)(\sigma^a)_{j}^{~i} }{z-w} \, .
\end{align}
To compute the coefficients of the OPEs, we first define the following matrices
\begin{align}
\sigma^1_{ij} = \frac12
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \, ,\quad
\sigma^2_{ij}= \frac12
\begin{pmatrix}
0 & - i \\
i & 0
\end{pmatrix} \, ,\quad
\sigma^3_{ij} = \frac12
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\end{align}
whose indices can then be raised to the form of coefficients by the application of $\delta^{i,j}$.
The non-trivial OPEs among the spin-$3/2$ generators $G_\pm^i$ are
\begin{align}
\begin{aligned}
&G_+^i (z) G_-^j (w) \sim \delta^{i,j} \left[ \frac{c_1 }{(z-w)^3} + \frac{2 c_2 H (z)}{(z-w)^2} + \frac{c_2 \partial H(w) -2 T(w) + c_4 (HH)(w)}{z-w}\right] \\
& \quad + (\sigma^{a})^{i j} \left[ \frac{2 c_5 J^a (w)}{(z-w)^2} + \frac{c_5 \partial J^a (w) + c_6 (H J^a) (w)}{z-w}\right]
+ c_7 \delta^{i,j} \delta^{a,b}\frac{(J^a J^b) + (J^b J^a)}{z-w}
\end{aligned}
\end{align}
with
\begin{align}
\begin{aligned}
&c_1 = -\frac{4 (k-2) (k-1)}{k-4} \, , \quad c_2 = \frac{2 \sqrt{2-k} (k-1)}{k-4}\, , \quad c_4 = \frac{6}{k-4}+3 \, , \\ &c_5 = -\frac{8}{k-4}-4 \, , \quad c_6 = \frac{8 \sqrt{2-k}}{k-4} \, , \quad c_7 = -\frac{2}{k-4} \, .
\end{aligned}
\end{align}
The generators of the QSCA can be expressed in terms of three ghost systems $( \gamma^\alpha , \beta_\alpha)$ $(\alpha =1,2,3)$ and three free bosons $\varphi_i$ $(i=1,2,3)$.
\bigskip
\noindent {\bf The first realization.}
Sets of screening charges for QSCA can be found in \eqref{QSCAi} and \eqref{QSCAh}.
One set of screening operators in \eqref{QSCAi} is given by
\begin{align} \label{screeningmin1}
\mathcal{V}_1 =e^ { b \varphi_1 } \beta_1 \, , \quad \mathcal{V}_2 = e^{ b \varphi_2 } (\beta_2 - \gamma^1 \beta_3) \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 }
\end{align}
from which we find the expression of the energy-momentum tensor to be
\begin{align}
T = - \frac12 G^{ij} \partial \varphi_ i \partial \varphi_j + Q^i \partial^2 \varphi_i + \sum_{\alpha=1}^3 \left( a_\alpha \gamma^\alpha \partial \beta_\alpha + (a_\alpha -1) \partial \gamma^\alpha \beta_\alpha \right) \label{minEM}
\end{align}
where
\begin{align}
Q^1 = \frac{3 b}{2 } \, , \quad Q^2 = 2 b \, , \quad Q^3 = \frac{(k-1) b}{2 } \, , \quad a_1 = 0 \, , \quad a_2 = a_3 = - \frac12.
\end{align}
The conformal weights of $(\gamma^\alpha , \beta_\alpha)$ are $(a_\alpha , 1 - a_\alpha)$.
The $\mathfrak{su}(2)$ currents commuting with the screening charges are written as
\begin{align}
J^3 = j^3 + \tilde \jmath^3 \, , \quad J^\pm = j^\pm + \tilde \jmath^\pm
\end{align}
with
\begin{align}
\begin{aligned}
&j^+ = \beta_1 \, , \quad j^3 = \gamma^1 \beta_1 + \frac{1}{2 b} \partial \varphi_1 \, , \quad
j^- = - \frac{1}{b} \partial \varphi_1 \gamma^1 - \gamma^1 \gamma^1 \beta_1 +(k-2) \partial \gamma^1 \, , \\
&\tilde \jmath^+ = - \gamma^2 \beta_3 \, , \quad \tilde \jmath^3 = - \frac12 (\gamma^2 \beta_2 - \gamma^3 \beta_3) \, , \quad \tilde \jmath^- = - \gamma^3 \beta_2 \, .
\end{aligned}
\end{align}
The $\mathfrak{u}(1)$ current in terms of the free fields is given by
\begin{align}
H = q \left( b^{-1} \partial \varphi^2 + \gamma^2 \beta_2 + \gamma^3 \beta_3 \right) \, ,
\end{align}
where $q=1/\sqrt{2-k}$. The spin-3/2 currents are found to be
\begin{align}
\begin{aligned}
&G_+^1 = \beta_2 \, ,\quad G_+^2 = \beta_3 \, , \\
&G_-^1 = \frac{2}{k -4} \gamma^1 \gamma^1 \gamma^2 \gamma^2 \beta_1 \beta_3 + \cdots \, , \quad
G_-^2 = \frac{2}{k -4} \gamma^1 \gamma^1 \gamma^2 \gamma^2 \beta_1 \beta_2 + \cdots \, .
\end{aligned}
\end{align}
\bigskip
\noindent {\bf The second realization.}
A second set of screening operators in \eqref{QSCAi} is
\begin{align} \label{screeningmin2}
\mathcal{V}_1 =e^{ b \varphi_1 } \beta_1 \, , \quad \mathcal{V}_2 =e^{ b \varphi_2 } (\beta_2 - \gamma^1 \beta_3) \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } \gamma^2 \, .
\end{align}
The energy-momentum tensor commuting with the screening charges takes the same form as \eqref{minEM}, but with a different set of parameters:
\begin{align}
\begin{aligned}
Q^1 = \frac{3 b}{2}\, , \quad Q^2 = \frac{k b }{2 } \, , \quad Q^3 = \frac{( k-1 ) b}{2 } \, , \quad
a_1 = - a_2 = - \frac12 \, , \quad a_3 =0 \, .
\end{aligned}
\end{align}
The spin-one currents are
\begin{align}
\begin{aligned}
&J^+ = \beta_3 \, , \quad
J^3 = \frac{1}{2} \gamma^1 \beta_1 +\frac{1}{2} \gamma^2 \beta_2 + \gamma^3 \beta_3 +\frac{1}{2 b } \partial \varphi_1 +\frac{1}{2 b } \partial \varphi_2 \, , \\
&J^- = - b^{-1} \partial \varphi_1 \gamma^3 - b^{-1} \partial \varphi_2 \gamma^3 - b^{-1}\partial \varphi_1 \gamma^1 \gamma^2 +(k-2) \partial \gamma^1 \gamma^2 - \gamma^1 \gamma^1 \gamma^2 \beta_1 \\
& \quad \quad - \gamma^1 \gamma^3 \beta_1 - \gamma^2 \gamma^3 \beta_2 - \gamma^3 \gamma^3 \beta_3 +(k-1) \partial \gamma^3 \, , \\
&H = q \left ( \gamma^1 \beta_1 - \gamma^2 \beta_2 + \frac{1}{2 b } \partial \varphi_1 + \frac{1}{2 b} \partial \varphi_3 \right) \, .
\end{aligned}
\end{align}
Two of the spin-$3/2$ currents are
\begin{align}
\begin{aligned}
&G_+^2 = \beta_1 - \beta_3 \gamma^2 \, , \\
&G_-^1 = - 2 b \partial \varphi_1 \gamma^1 \beta _3 - 2 b \partial \varphi_3 \beta_2 + \left(\frac{4}{k-4}+2\right) \partial \gamma^1 \beta_3 + \frac{2}{k-4} \gamma^2 \beta_2 \beta _2 \\
& \quad \quad - \frac{2}{k-4} \gamma^1 \gamma^1 \beta_1 \beta _3 + \left(\frac{4}{k-4}+2\right) \partial \beta_2 \, .\\
\end{aligned}
\end{align}
The expressions of the other two spin-$ 3 / 2$ currents $G_+^1,G_-^2$ can be found from their OPEs with $J^-$.
\bigskip
\noindent {\bf The third realization.}
The last set of screening operators presented in \eqref{QSCAi} is given by
\begin{align} \label{screeningmin3}
\mathcal{V}_1 = e^{b \varphi_1 } \beta_1 \, , \quad \mathcal{V}_2 = e^{ b \varphi_2 } (\beta_2 - \gamma^1 \beta_3) \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } \gamma^3 \, .
\end{align}
The energy-momentum tensor is \eqref{minEM} with
\begin{align}
\begin{aligned}
& Q^1 = \frac{(k-1) b}{2 } \, , \quad Q^2 = \frac{k b }{2 } \, , \quad Q^3 = \frac{( k-1 ) b }{2 } \, ,
&a_1 = a_3 = \frac12 \, , \quad a_2 = 0
\end{aligned}
\end{align}
and the spin-one currents are
\begin{align}
\begin{aligned}
&J^+ = \beta_2 \, , \quad
J^3 = -\frac{1}{2} \gamma^1 \beta _1 + \gamma ^2 \beta _2 +\frac{1}{2} \gamma ^3 \beta _3 +\frac{1}{2 b} \partial \varphi_2 \, , \\
&J^- = - b^{-1} \partial \varphi _2 \gamma^2 + \gamma^1 \gamma^2 \beta _1 - \gamma^2 \gamma^2 \beta_2 - \gamma^2 \gamma ^3 \beta _3 + \gamma^3 \beta_1 +(k-1) \partial \gamma ^ 2 \, , \\
&H = - q \left(\gamma^1 \beta _1 + \gamma ^3 \beta _3 - \frac{1}{2 b}\partial \varphi_3 + \frac{1}{2 b} \partial \varphi_1 \right) \, .
\end{aligned}
\end{align}
Two of the spin-$3/2$ currents are found to be
\begin{align}
\begin{aligned}
&G_+^2 = \partial \varphi_1 \gamma^1 + b \gamma^1 \gamma^1 \beta_1 + b \gamma^3 \beta_2 + (2-k) b \partial \gamma^1 \, , \\
&G_-^1 =2 \partial \varphi_3 \beta_3 - 2 b \gamma ^3 \beta _ 3 \beta _3 - 2 b \beta_1 \beta_2 - 2 (k-2) b \partial \beta _3 \, , \\
\end{aligned}
\end{align}
and the other two $G_+^1,G_-^2$ can be obtained by the action of $J^-$.
\bigskip
For the above three types of free field realizations, we consider the twist of energy-momentum tensor by
\begin{align}
T(z) \to T_t(z) = T(z) + \frac{1}{2 q} \partial H \, , \label{twistmin}
\end{align}
where $q=1/\sqrt{2-k}$. With respect to the twisted energy-momentum tensor, the conformal weights of $G_+^i$ and $G_-^i$ become 1 and 2, respectively.
Whereas, all three pairs of ghosts $(\gamma^\alpha , \beta_\alpha )$ with $\alpha =1,2,3$ have conformal weights $(0,1)$ as desired.
\bigskip
\noindent {\bf The fourth realization.}
Finally, let us consider the free field realization associated with the set of screening operators in \eqref{QSCAh}:
\begin{align} \label{screeningsmin4}
\mathcal{V}_1 =e^{ b \varphi_1 } (\beta_1 + \gamma^2 \beta_3 ) \, , \quad
\mathcal{V}_2 = e^ { b \varphi_2 } \beta_2 \, , \quad
\mathcal{V}_3 = e^{ b \varphi_3 } ( \gamma^3 - \gamma^2 \gamma^1 ) \, .
\end{align}
The energy-momentum tensor is again given by \eqref{minEM} with
\begin{align}
\begin{aligned}
& Q^1 = \frac{(k-1) b}{2 } \, , \quad Q^2 = \frac{k b }{2 } \, , \quad Q^3 = \frac{( k-1 ) b }{2 } \, ,
&a_1 =a_3 = \frac12 \, , \quad a_2 = 0 \, .
\end{aligned}
\end{align}
The spin-one currents are
\begin{align}
\begin{aligned}
&J^+ = \beta_2 + \gamma^1 \beta_3 \, , \quad
J^3 = - \frac{1}{2} \gamma ^1 \beta _1 + \gamma ^2 \beta _2 + \frac{1}{2} \gamma ^3 \beta _3 +\frac{1}{2 b} \partial \varphi_2 \, , \\
&J^- =- b^{-1} \partial \varphi_2 \gamma^2 + \gamma ^3 \beta _1 - \gamma ^2 \gamma ^2 \beta _2 +(k-2) \partial \gamma ^2 \, , \\
&H =- q \left( \gamma ^1 \beta _1 + \gamma ^3 \beta _3 + \frac{1}{2 b}\partial \varphi_1 - \frac{1}{2 b} \partial \varphi_3 \right) \, .
\end{aligned}
\end{align}
Two of the spin-$3/2$ currents are
\begin{align}
\begin{aligned}
&G_+^2 = \partial \varphi_1 \gamma ^ 1 + b \gamma ^1 \gamma ^1 \beta _1 - b \gamma ^ 1 \gamma ^2 \beta ^2 + b \gamma ^ 1 \gamma ^3 \beta _3 + b \gamma ^ 3 \beta _2 + (1-k) b \partial \gamma ^ 1 \, , \\
&G_-^1 = 2 \partial \varphi_3 \beta _3 - 2 b \gamma ^ 1 \beta _1 \beta _3 - 2 b \gamma ^ 2 \beta _2 \beta _3 - 2 b \gamma ^ 3 \beta _3 \beta _3 - 2 b \beta _1 \beta _2 - 2 (k-1) b \partial \beta _3 \, , \\
\end{aligned}
\end{align}
and rest of the spin-$3 / 2$ currents can again be obtained by the action of $J^-$ on them.
These generators are consistent with those in \cite{Romans:1990ta}.
In this realization, we may twist the energy-momentum tensor by
\begin{align}
T(z) \to T_t(z) = T(z) +a_1 \partial J^3 + a_2 \partial H \, .
\end{align}
However, unlike the previous realizations, this twist does not allow the conformal dimensions of $(\gamma^\alpha , \beta_\alpha )$ for all $\alpha =1,2,3$ to be $(0,1)$.
Even though other values of conformal dimensions are also permitted, it introduces difficulties to construct maps from this free field realization to the other three.
\subsection{Vertex operators}
So far, we have constructed generators for three different non-regular W-algebras associated to $\mathfrak{sl}(4)$ in terms of free fields. In order to construct vertex operators for each W-algebra, we may twist the energy-momentum tensor such that the conformal weights of $(\gamma^\alpha , \beta_\alpha)$ become $(0,1)$ as we did for W$^{(2)}_4$-algebra in \eqref{twistW24} and for the first three QSCA realizations in \eqref{twistmin}. Such twists are not needed for the rectangular case since the desirable ghost conformal dimensions are achieved with the original energy-momentum tensor.
As discussed at the end of section \ref{sec:QSCA}, it is not possible to find a good twist for the fourth realization of QSCA. It is, therefore, difficult to establish a correspondence between this realization with the others. We shall now disregard this exceptional case, and instead, we examine rest of the realizations of these three non-regular W-algebras in general.
For any of the three algebras, its generators are denoted by $F^a$, where $a$ runs from 1 to the dimension of algebra.
The conformal dimension of $F^a$ with respect to the (twisted) energy-momentum tensor is denoted by $h^a$.
Since $h^a$ is integer for all $a$ in all W-algebras,
the generators can be expanded as
\begin{align}
F^a (z) = \sum_{n \in\mathbb{Z}} \frac{F^a_n}{z^{n + h^a}}.
\end{align}
A vertex operator and its corresponding primary state is define as
\begin{align}
\label{mubasis}
V_{ \ell^i } ( \mu_\alpha | z) = e^ { \mu_\alpha \gamma^\alpha } e^{ \ell^i \varphi_i } \, , \quad
| \ell^i ; \mu_\alpha \rangle \equiv \lim_{z \to 0} V_{ \ell^i }(\mu_\alpha | z) | 0 \rangle \, .
\end{align}
The modes of the generators act on the primary states as
\begin{align}
\begin{aligned}
&F^a_n | \ell^i ; \mu_\alpha \rangle= 0 , \quad
&F^a_0 | \ell^i ; \mu_\alpha \rangle= - \mathcal{D}_{F^a} | \ell^i ; \mu_\alpha \rangle \, ,
\end{aligned}
\end{align}
where $n > 0$ and $\mathcal{D}_{F^a}$ are differential operators with respect to $\mu_\alpha$.
The explicit form of $\mathcal{D}_{F^a}$ can be found from the free field expressions of the generators.
These differential operators therefore have different expressions for different realizations.
However, it can be shown that the differential operators $\mathcal{D}_{F^a}$ satisfy the same commutation relations for all free field realizations of the same W-algebra.
The second-, third-, and fourth-order Casimir operators $\mathcal{D}_p$ with $p=2,3,4$ which commute with $D_{F^a}$ are constructed. We shall avoid presenting these tedious expressions in this paper, since they are not directly related to the calculation. These Casimir operators are linear combinations of terms of the form
$
\mathcal{D}_{F_{a_1}} \cdots \mathcal{D}_{F_{a_n}}
$
with $n \leq p$.
In particular, the eigenvalue with respect to the second-order Casimir operator $\mathcal{D}_2$ corresponds to the conformal weight in an analogous way as in \eqref{cw} for the BP-algebra case.
The eigenvalues of these Casimir operators on the primary states defined in \eqref{mubasis} are the same for all free field realizations of the same W-algebra.
It is desirable to obtain maps of correlation functions (or vertex operators) among different free field realizations for a non-regular W-algebra. The form of vertex operators defined in \eqref{mubasis} are not suitable for this purpose. The map can be realised more clearly if we make a change of basis such that the vertex operators take the form $P(\gamma^\alpha ) \exp ( \ell^i \varphi_i ) $, where $P(x ) $ is a polynomial of $x$.
A convenient choice for making such a change of basis is
\begin{align}
V_{ \ell^i } (a_1 , \ldots , a_n) = \int \frac{d \mu}{\mu} \mathcal{D}_{a_1} \cdots \mathcal{D}_{a_n} V_{ \ell^i } (\mu | z) \, .
\end{align}
This is a generalisation of the change to $m$-basis we used in subsection \ref{sec:maps} for the BP-algebra.
In the case of W$_4^{(2)}$-algebra, the basis is essentially the same as for the BP-algebra, and therefore the arguments in subsection \ref{sec:maps} can be directly applied.
For other non-regular W-algebras, it is also possible to find the maps of vertex operators explicitly at least for concrete examples.
\section{Correlator relations for W-algebras from $\mathfrak{sl}(4)$}
\label{sec:corrsl4}
Using the screening charges in appendix \ref{sec:screening}, we have developed free field realizations of non-regular W-algebras associated to $\mathfrak{sl}(4)$. This allows us to derive new correspondences among correlation functions involving these W-algebras. There are five different partitions of the integer $4$ which corresponds to five different W-algebras.
Recall that a similar relation was found between the $\mathfrak{sl}(2)$ WZNW model with partition $2 = 1+1$ and the Virasoro algebra (or W$_2$-algebra) with partition $2 =2$.
Moreover, for the $\mathfrak{sl}(3)$ case, we have obtained the reduction relations for algebras with partitions from $3=1+1+1$ to $3 = 2+1$ and from $3=2+1$ to $3=3$.
Observing the pattern, one can deduce that, in the $\mathfrak{sl}(4)$ case, correlator relations can be obtained directly using the generic procedure for W-algebras with the following partitions
\begin{align}
\begin{aligned}
4 &=1+1+1+1 &\to& &4&=2+1+1 \, , \\
4&=2+1+1 &\to & &4&=3+1 \, , \\
4&=3+1 &\to& &4&=4 \, .
\end{aligned}
\end{align}
In this section, we will study these cases in detail.
Other than the above cases, there are many relations we can obtain for W-algebras associated to $\mathfrak{sl} (4)$.
For instance, we can perform a direct reduction to obtain correlator relation for $4=2+1+1 \to 4 = 2+2$.
In section \ref{sec:corrsl3}, we have discussed how to obtain direct relations for $3=1+1+1 \to 3=3$, and
analogous techniques can be applied to the $\mathfrak{sl}(4)$ case as well.
In particular, it is possible to relate $\mathfrak{sl}(4)$ current algebra to all free field realizations of W-algebras by putting restrictions on momenta of vertex operators.
This kind of reductions may be regarded as correlator versions of what have been done in appendix \ref{sec:screening} to obtain screening charges for various W-algebras. An application of the reduction from $\mathfrak{sl}(4)$ current algebra to rectangular W-algebra can be found in \cite{Creutzig:2018pts,Creutzig:2019qos}.
\subsection{Reduction from affine $\mathfrak{sl}(4)$ to QSCA}
In this subsection, we reduce the $\mathfrak{sl}(4)$ WZNW model to a free field realization of QSCA with $\mathfrak{su}(2)$ subalgebra in it.
This reduction relation was already obtained in \cite{Creutzig:2015hla} where the reduced theory corresponds to the fourth realization of QSCA with the screening operators \eqref{screeningsmin4}.
As observed in the previous section, it is not easy to construct maps from the fourth realization of QSCA to the others. We therefore consider the reduction to the third realization instead in this subsection.
We can then create maps among the three QSCA realizations and reduce the QSCA theory further to the subregular or the regular W-algebras.
We start from a first order formulation of the $\mathfrak{sl}(4)$ WZNW model.
The analysis in appendix \ref{sec:screening} shows that it is important to choose a proper expression of the action:
\begin{align}
\begin{aligned}
S &= \frac 1 {2\pi}\int \dd^2 z\Big[\frac{G^{ij}}{2}\partial\phi_{i}\bar{\partial}\phi_{j}+\frac{b}{4}\sqrt{g}\mathcal{R}(\phi_1+\phi_2+\phi_3)+\sum_{\alpha=1}^6(\beta_\alpha \bar{\partial}\gamma^\alpha+\bb{\alpha} \partial\gb{\alpha})\Big]\\
& \quad - \frac 1 {2\pi k}\int \dd^2 z\Big[e^{b\phi_{1}}\beta_1\bb{1}+e^{b\phi_{2}}(\beta_2-\gamma^1\beta_4)(\bb{2}-\gb{1}\bb{4})\\
&\hspace{3cm}+e^{b\phi_{3}}(\beta_3-\gamma^2\beta_5-\gamma^4\beta_6)(\bb{3}-\gb{2}\bb{5}-\gb{4}\bb{6})\Big] \, ,
\end{aligned} \label{actionsl4}
\end{align}
which corresponds to the free field realization of $\mathfrak{sl}(4)$ current algebra with screening charges \eqref{sl4min}.
The matrix $G^{ij} $ was given in \eqref{Gmatrices}.
A correlation function \eqref{corr} was defined in terms of action \eqref{actionsl4}.
The path integral measure is given by
\begin{align}
\mathcal{D} \Phi = \prod_{i=1}^3 \mathcal{D} \phi_i \prod_{\alpha =1}^6 \mathcal{D}^2 \beta_\alpha \mathcal{D}^2 \gamma^\alpha \label{measuresl4}
\end{align}
and the vertex operators are
\begin{align}
V_\nu (z_\nu) =|\mu^{\nu}_3|^{4(j_3^\nu-j_2^\nu)}|\mu^{\nu}_5|^{4(j_2^\nu-j_1^\nu)}|\mu^{\nu}_6|^{4(j_1^\nu+1)} e^{\mu_\alpha^\nu\gamma^\alpha-\bar{\mu}_\alpha^\nu\bar{\gamma}^\alpha} e^{ \sum_{i=1}^3 2b(j^\nu_i+1)\phi_i } \, . \label{vertexsl4}
\end{align}
In order to reduce the theory to the one corresponding to a free field realization of QSCA, we need to integrate out three sets of ghost systems.
Here we choose to integrate over $\gamma^3$, $\gamma^5$ and $\gamma^6$, since they do not appear in the interaction terms of \eqref{actionsl4}. This leads to $\delta^{(2)} (\sum_{\nu} \mu_\alpha^\nu )$ and replacements of $\beta_3$, $\beta_5$ and $\beta_6$ by
\begin{equation}\label{f4}
\beta_{\alpha}(z)= - \sum_{\nu=1}^N\frac{\mu^\nu_\alpha}{z-z_\nu}
= - u_\alpha \frac{\prod_{n=1}^{N-2}(z-y_\alpha^n)}{\prod_{\nu=1}^{N}(z-z_\nu)}
\equiv - u_\alpha \mathcal{B}_\alpha \, ,
\end{equation}
where $\alpha=3,5,6$.
Now the interaction terms become
\begin{equation}\label{d1}
\begin{aligned}
& - \frac 1 {2\pi k}\int d ^2 z\Big[ e ^{b\phi_{1}}\beta_1\bb{1}+ e ^{b\phi_{2}}(\beta_2-\gamma^1\beta_4)(\bb{2}-\gb{1}\bb{4})\\
&\hspace{3cm}- e ^{b\phi_{3}}(u_3\mathcal{B}_3-\gamma^2u_5\mathcal{B}_5-\gamma^4u_6\mathcal{B}_6)(\bar{u}_3\bar{\mathcal{B}}_3-\bar{\gamma}^2\bar{u}_5\bar{\mathcal{B}}_5-\bar{\gamma}^4\bar{u}_6\bar{\mathcal{B}}_6)\Big] \, .
\end{aligned}
\end{equation}
We make the following change of fields as to remove the $\mathcal{B}_i$-functions
\begin{align}
\begin{aligned}
&\phi_1 + \frac{1}{b} \log |u_5^{-1} u_6 \mathcal{B}_5^{-1} \mathcal{B}_6|^2 \to \phi_1 \, , \\
&\phi_2 + \frac{1}{b} \log |u_3^{-1} u_5 \mathcal{B}_3^{-1} \mathcal{B}_5|^2 \to \phi_2 \, , \\
&\phi_3 + \frac{1}{b} \log |u_3 \mathcal{B}_3 |^2 \to \phi_3
\end{aligned}
\end{align}
and
\begin{align}
\begin{aligned}
&u_5^{-1} u_6 \mathcal{B}_5^{-1} \mathcal{B}_6 \gamma^1 \to \gamma^1 \, , \quad
u_5 u_6^{-1} \mathcal{B}_5 \mathcal{B}_6^{-1} \beta_1 \to \beta_1 \, , \\
& u_3^{-1} u_5 \mathcal{B}_3^{-1} \mathcal{B}_5\gamma^2 \to \gamma^2 \, , \quad
u_3 u_5^{-1} \mathcal{B}_3 \mathcal{B}_5^{-1} \beta_2 \to \beta_2 \, , \\
& u_3^{-1} u_6 \mathcal{B}_3^{-1} \mathcal{B}_6 \gamma^4 \to \gamma^4 \, , \quad
u_3 u_6^{-1} \mathcal{B}_3 \mathcal{B}_6^{-1} \beta_4 \to \beta_4 \, .
\end{aligned}
\end{align}
As in subsection \ref{sec:previous}, we formulate $\gamma^\alpha$ and $\beta_\alpha$ as
\begin{align}
\gamma^\alpha (z)\simeq e^{X_\alpha (z)} \eta_\alpha (z)\, , \quad \beta_\alpha (z)= e^{- X_\alpha (z) } \partial \xi_\alpha (z) \label{ghostb1}
\end{align}
with
\begin{align}
\eta_\alpha (z) \xi_{\alpha '} (w) \sim \frac{\delta_{\alpha , \alpha '}}{z -w} \, , \quad X_\alpha (z) X _ {\alpha '} (w) \sim - \delta_{\alpha , \alpha '} \log (z - w) \, .\label{ghostb2}
\end{align}
Analogous to the $\mathfrak{sl}(3)$ case, the redefinition of fields provides several contributions to the correlation function.
Taking everything into account, we obtain the following correlator relation
\begin{align} \label{sl4tomin}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle
= |\Theta_N|^2 \prod_{\alpha = 3,5,6}\delta ^{(2)} \left( \sum_{\nu=1}^N \mu_\alpha^\nu \right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} \prod_{\alpha ' =3,5,6} \tilde V^{(\alpha ' )} (y_{\alpha '}^n) \right \rangle \, ,
\end{align}
where left-hand side is computed with the action
\begin{align}
\begin{aligned}
& S = \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{1}{4}\sqrt{g} \mathcal{R} (b (\phi^1 + \phi^2) + (b + b^{-1})\phi^3 ) + \sum_{\alpha =1,2,4} ( \beta_\alpha \bar \partial \gamma^ \alpha+ \bar \beta_\alpha \partial \bar \gamma^\alpha ) \right] \\
& - \frac 1 {2\pi k}\int d ^2 z\Big[ e ^{b\phi_{1}}\beta_1\bb{1}+ e ^{b\phi_{2}}(\beta_2-\gamma^1\beta_4)(\bb{2}-\gb{1}\bb{4})- e ^{b\phi_{3}}(1-\gamma^2-\gamma^4)(1-\bar{\gamma}^2-\bar{\gamma}^4)\Big] \, .
&
\end{aligned} \label{minaction}
\end{align}
The vertex operators at $z_\nu$ receive the shifts of momenta by
\begin{align}
&\tilde V_\nu (z_\nu) = e^{\sum_{\alpha = 1,2,4} ( \mu_\alpha '{}^{ \nu} \gamma^\alpha - \bar{\mu} ' _\alpha {}^ \nu \bar{\gamma}^\alpha ) } e^{\sum_{i=1}^3 2 b (j_i^\nu + 1) \phi_i + \phi^3 /b}
\end{align}
where
\begin{align}
\mu_1 '{}^{ \nu} = \frac{u_5 \mu_5^\nu \mu_1^\nu}{ u_6 \mu_6^\nu } \, , \qquad
\mu_2 '{}^{ \nu} = \frac{u_3 \mu_3^\nu \mu_2^\nu}{ u_5 \mu_5^\nu } \, , \qquad
\mu_4 '{}^{ \nu} = \frac{u_3 \mu_3^\nu \mu_4^\nu}{ u_6 \mu_6^\nu }
\end{align}
and similarly for $ \bar{\mu} ' _\alpha$.
Moreover, there are extra insertions of operators at $y_\alpha^n$ $(\alpha = 3,5,6)$ as
\begin{align}
\begin{aligned}
&\tilde V^{(3)} (y_3^n) = e^{(\phi^2 - \phi^3 ) /b} e^{X_2 + \bar{X}_2 + X_4 + \bar{X}_4 } \, , \\
& \tilde V^{(5)} (y_5^n) = e^{( \phi^1 - \phi^2 ) /b} e^{X_1 + \bar{X}_1 - X_2 - \bar{X}_2 }\, , \\
&\tilde V^{(6)} (y_6^n) = e^{- \phi^1 /b} e^{ X_1 + \bar{X}_1 + X_4 + \bar{X}_4} \, .
\end{aligned} \label{tvertexsl4}
\end{align}
The prefactor is found to be
\begin{align}
\Theta_N&=u_3^2 \prod_{n > n'}\left[(y_3^n-y_3^{n'})(y_5^n-y_5^{n'})(y_6^n-y_6^{n'})\right]^{2+\frac 3 {4b^2}} \prod_{\mu > \nu}(z_\mu-z_\nu)^{\frac{3}{4b^2}} \\ & \quad \times\prod_{n,m}\left[(y_3^n-y_5^m)(y_3^n-y_6^m)(y_5^n-y_6^m)\right]^{-1-\frac 1 {4b^2}}\prod_{n,\nu}\left[(y_3^n-z_\nu)(y_5^n-z_\nu)(y_6^n-z_\nu)\right]^{-\frac 1 {4b^2}} \, .
\nonumber
\end{align}
It is possible to rewrite the action by changing the basis of ghost systems as
\begin{align}
\begin{aligned}
& \gamma^1 -1 \to \gamma^1 \, , \quad \beta_1 \to \beta_1 \, , \\
&\gamma^2 \to \gamma^2 \, , \quad \beta_2 - \beta_4 \to \beta_2 \, , \\
& \gamma^4 + \gamma^2 - 1 \to \gamma^4 \, , \quad
\beta_4 \to \beta_4 \, .
\end{aligned}
\end{align}
The kinetic terms of ghost systems are invariant under this change of basis, but the interaction terms are now
\begin{equation}
\begin{aligned}
- \frac 1 {2\pi k}\int d ^2 z\Big[ e ^{b\phi_{1}}\beta_1\bb{1}+ e ^{b\phi_{2}}(\beta_2-\gamma^1\beta_4)(\bb{2}-\gb{1}\bb{4})- e ^{b\phi_{3}}\gamma^4 \bar \gamma^4\Big]
\end{aligned}
\end{equation}
from which we observe that the action corresponds to the third realization of QSCA with screening operators \eqref{screeningmin3}.
\subsection{Reduction from QSCA to W$^{(2)}_4$-algebra}
\label{sec:QSCAtoW24}
In this subsection, we establish a relation between QSCA and W$^{(2)}_4$-algebra at the level of correlation functions. As explained in subsection \ref{sec:QSCA}, there are four types of free field realizations for QSCA.
Here we shall adopt the first realization with screening charges \eqref{screeningmin1} since it involved the least number of $\gamma$'s.
This free field realization of QSCA can be described by a theory with the action
\begin{align}\label{f2}
\begin{aligned}
S &= \frac 1 {2\pi}\int d^2 z\Big[\frac{G^{ij}}{2}\partial\phi_{i}\bar{\partial}\phi_{j}+\frac{1}{4}\sqrt{g}\mathcal{R}(b (\phi^1+\phi^2) + (b + b^{-1})\phi^3)+\sum_{\alpha=1}^3(\beta_\alpha \bar{\partial}\gamma^\alpha+\bb{\alpha} \partial\gb{\alpha})\Big]\\
& \quad - \frac 1 {2\pi k}\int d^2 z\Big[e^{b\phi_{1}}\beta_1\bb{1}+e^{b\phi_{2}}(\beta_2-\gamma^1\beta_3)(\bb{2}-\gb{1}\bb{3})- e^{b\phi_{3}}\Big] \, .
\end{aligned}
\end{align}
Consider a correlation function of the form \eqref{corr}, with the action \eqref{f2} and
the path integral measure
\begin{align}
\mathcal{D} \Phi= \prod_{i=1}^3 \mathcal{D} \phi_i \prod_{\alpha =1}^3 \mathcal{D}^2 \beta_\alpha \mathcal{D}^2 \gamma^\alpha. \label{measuremin}
\end{align}
The vertex operators take the form
\begin{align}
V_\nu (z_\nu) = |\mu^{\nu}_2|^{4(j_2^\nu + 1)} e^{\mu_\alpha^\nu\gamma^\alpha-\bar{\mu}_\alpha^\nu\bar{\gamma}^\alpha} e^{ \sum_{i=1}^3 2b(j^\nu_i+1)\phi_i } \, . \label{vertexmin}
\end{align}
Now we want to reduce the theory to the one corresponding to a free field realization of W$^{(2)}_4$-algebra by integrating out two sets of ghost systems.
Since $\gamma^1$ appears in the interaction terms of the action \eqref{f2}, we integrate with respect to $\gamma^2$ and $\gamma^3$. This yields the delta functions $\delta^{(2)} (\sum_{\nu} \mu_2^\nu) \delta^{(2)} (\sum_{\nu} \mu_3^\nu)$ and the relations
\begin{equation}\label{a7}
\beta_{\alpha}(z)= - \sum_{\nu=1}^N\frac{\mu^\nu_\alpha}{z-z_\nu}= - u_\alpha \frac{\prod_{n=1}^{N-2}(z-y_\alpha^n)}{\prod_{\nu=1}^{N}(z-z_\nu)} \equiv - u_\alpha \mathcal{B}_\alpha
\end{equation}
where $\alpha =2,3$.
Shifting $\phi_2 $ by
\begin{equation}\label{a9}
\phi_2(z) + \frac 1 b \ln{|u_2 \mathcal{B}_2|^2} \to \phi_2 (z) \, ,
\end{equation}
the interaction terms become
\begin{align}
- \frac 1 {2\pi k}\int \dd^2 z\Big[e^{b\phi_{1}} \beta_1 \bar \beta_1- e^{b\phi_{2}} (1-u_2^{-1} u_3 \mathcal{B}_2^{-1}\mathcal{B}_3\gamma^1 ) (1-\bar u_2^{-1} \bar u_3 \bar {\mathcal{B}}_2^{-1}\bar{\mathcal{B}}_3 \bar \gamma^1 ) - e^{b\phi_{3}}\Big] \, .
\end{align}
We further rescale $\gamma^1$ and $\beta_1$ by
\begin{equation}
u_2^{-1} u_3\mathcal{B}_2^{-1}\mathcal{B}_3 \gamma^1 \to \gamma^1 \, , \quad
u_2 u_3^{-1} \mathcal{B}_2 \mathcal{B}_3^{-1} \beta_1 \to \beta_1
\end{equation}
and reformulate the ghosts in terms of a pair of free fermions and a free boson as we did in \eqref{ghostb1} and \eqref{ghostb2}.
We then arrive at the correlator relation as
\begin{align} \label{mintosub}
\left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle = |\Theta_N|^2 \delta ^{(2)} \left( \sum_{\nu=1}^N \mu_2^\nu \right) \delta ^{(2)} \left( \sum_{\nu=1}^N \mu_3^\nu \right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} {\tilde V}^{(2)} (y_2^n) {\tilde V}^{(3)} (y_3^n) \right \rangle \, ,
\end{align}
whose right-hand side is evaluated with the action
\begin{align}
\begin{aligned}
& S = \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{1}{4}\sqrt{g} \mathcal{R} ( b \phi^1 + (b + b^{-1})( \phi^2 + \phi^3 )) + \beta_1 \bar \partial \gamma^1 + \bar \beta_1 \partial \bar \gamma^1 ) \right] \\
& \qquad - \frac 1 {2\pi k}\int d ^2 z\Big[ e ^{b\phi_{1}}\beta_1\bb{1} - e ^{b\phi_{2}}(1-\gamma^1 )(1 - \bar { \gamma }^1)- e ^{b\phi_{3}}\Big] \, .
&
\end{aligned} \label{subaction}
\end{align}
If we make the shift $\gamma^1 -1 \to \gamma^1$ in \eqref{subaction}, the action coincides with that of the first realization of W$^{(2)}_4$-algebra whose screening operators are given in \eqref{screeningW241}.
The new vertex operators in the correlation function \eqref{mintosub} are found to be
\begin{align}
\begin{aligned}
&\tilde V_\nu (z_\nu) = e^{ \mu_1 '{}^{ \nu} \gamma^1 - \bar{\mu} ' _1 {}^ \nu \bar{\gamma}^1} e^{\sum_{i=1}^3 2 b (j_i^\nu + 1) \phi_i + \phi^2 /b} \, , \\
& \tilde V^{(2)} (y_2^n) = e^{ - \phi^2 /b} e^{ X_1 + \bar X_1 } \, , \quad
\tilde V^{(3)} (y_3^n) = e^{-X_1- \bar X_1}
\end{aligned} \label{tvertexsub}
\end{align}
where
\begin{align}
\mu_1 '{}^{ \nu} = \frac{u_2 \mu_2^\nu \mu_1^\nu}{ u_3 \mu_3^\nu } \, , \quad
\bar{\mu}_1 '{}^{ \nu} = \frac{\bar{u}_2 \bar{\mu}_2^\nu \bar{\mu}_1^\nu}{ \bar{u}_3 \bar{\mu}_3^\nu } \, .
\end{align}
The prefactor is
\begin{align}
\Theta_N&=u_2^3 u_3^{-1} \prod_{n > n'}(y_2^n-y_2^{n'}) ^{1+\frac{1} {b^2}} (y_3^n-y_3^{n'})\prod_{\mu > \nu}(z_\mu-z_\nu)^{\frac{1}{b^2}} \prod_{n,m}(y_2^n-y_3^m)^{-1}\prod_{n,\nu}(y_2^n-z_\nu)^{-\frac 1 {b^2}} \, .
\end{align}
\subsection{Reduction from W$^{(2)}_4$-algebra to W$_4$-algebra}
In this subsection, we study the reduction relation from W$^{(2)}_4$-algebra to W$_4$-algebra. As discussed in subsection \ref{sec:W24}, there are two free field realizations for W$^{(2)}_4$-algebra. For the purpose of this subsection, we use the second realization with screening charges \eqref{screeningW242}.
The correlation function takes usual form \eqref{corr} with the action
\begin{align}
\begin{aligned}
S &= \frac 1 {2\pi}\int d^2 z\Big[\frac{G^{ij}}{2}\partial\phi_{i}\bar{\partial}\phi_{j}+\frac{1}{4}\sqrt{g}\mathcal{R}( b \phi^1+ (b + b^{-1})(\phi^2 + \phi^3) )+ \beta \bar{\partial}\gamma +\bar \beta \partial \bar \gamma \Big]\\
& \quad - \frac 1 {2\pi k}\int d^2 z\Big[e^{b\phi_{1}}\beta \bar \beta - e^{b\phi_{2}} - e^{b\phi_{3}}\Big] \,
\end{aligned}
\end{align}
and the path integral measure.
\begin{align}
\mathcal{D} g = \prod_{i=1}^3 \mathcal{D} \phi_i \mathcal{D}^2 \beta \mathcal{D}^2 \gamma \, . \label{measuresub}
\end{align}
The vertex operators are given by
\begin{align}
V_\nu (z_\nu) = |\mu^{\nu}|^{4(j_1^\nu + 1)} e^{\mu^\nu\gamma -\bar{\mu}^\nu\bar{\gamma}} e^{ \sum_{i=1}^3 2b(j^\nu_i+1)\phi_i } \, . \label{vertexsub}
\end{align}
The reduction procedure follows similarly as in previous cases. We integrate out $\gamma$ and $\beta$ from the correlation function for W$^{(2)}_4$-algebra, which yields
a delta function $\delta^{(2)} (\sum_\nu \mu^\nu )$ and the following expression of $\beta(z)$ in terms of function $\mathcal{B}$:
\begin{equation}
\beta (z)= - \sum_{\nu=1}^N\frac{\mu^\nu}{z-z_\nu}= - u \frac{\prod_{n=1}^{N-2}(z-y^n)}{\prod_{\nu=1}^{N}(z-z_\nu)} \equiv - u \mathcal{B}(z,y^n,z_\nu) \, .
\end{equation}
We further shift $\phi_1(z)$ by
\begin{equation}
\phi_1(z) + \frac 1 b\ln{|u \mathcal{B}|^2} \to \phi_1 (z)\, .
\end{equation}
The final form of the correlator is given by
\begin{align} \label{subtoW4}
\begin{aligned}
& \left \langle \prod^N_{\nu = 1} V_\nu (z_\nu) \right \rangle = |\Theta_N|^2 \delta ^{(2)} \left( \sum_{\nu=1}^N \mu^\nu \right)
\left \langle \prod^N_{\nu = 1} \tilde V_\nu (z_\nu) \prod_{n=1}^{N-2} {\tilde V}_b (y^n) \right \rangle \, ,
\end{aligned}
\end{align}
where the action for the right-hand side is
\begin{align}
& S = \frac{1}{2 \pi} \int d ^2 z \left[ \frac{G_{ij}}{2} \partial \phi^i \bar \partial \phi^j + \frac{Q_\phi}{4}\sqrt{q} \mathcal{R} (\phi^1 + \phi^2 + \phi^3 ) + \frac { 1 }{k} \left( e ^{b\phi_{1}} + e ^{b\phi_{2}} + e ^{b\phi_{3}} \right) \right]
\end{align}
with $Q_\phi = b + b^{-1}$. This is nothing but the action of $\mathfrak{sl}(4)$ Toda field theory.
The new vertex operators are
\begin{equation}
\tilde V_\nu (z_\nu) = e^{\sum_{i=1}^3 2 b (j_i^\nu + 1) \phi_i + \phi^1 /b} \, , \qquad \tilde V _b (y^n) = e^{ - \phi^1 /b} \, .
\label{tvertexw4}
\end{equation}
And the prefactor is found to be
\begin{align}
\Theta_N&=u^2 \prod_{n > n'}(y^n-y^{n'}) ^{\frac{3} {4 b^2}} \prod_{\mu > \nu}(z_\mu-z_\nu)^{\frac{3}{4 b^2}} \prod_{n,\nu}(y^n-z_\nu)^{-\frac{3}{4 b^2}} \, .
\end{align}
\section{Conclusion and discussions}
\label{sec:conclusion}
In this paper, we derived new correspondences among correlation functions of theories with W-algebra symmetry.
We generalize the previous works in \cite{Ribault:2005wp,Hikida:2007tq,Hikida:2007sz,Creutzig:2011qm,Creutzig:2015hla}, where $\mathfrak{sl}(N)$ WZNW model is reduced to a theory with QSCA symmetry. The screening charges constructed in \cite{Genra1,Genra2, CGN} are employed to develop free field realizations of non-regular W-algebras.
The paper started with a detailed account of W-algebras associated with $\mathfrak{sl}(3)$.
The non-regular W-algebra in this case, BP-algebra, has two different free field realizations. The realization with the screening operators \eqref{screening3} was reduced to $\mathfrak{sl}(3)$ Toda theory as in \eqref{BPtoW3} using path integral.
A new method of putting restrictions on momenta of vertex operators was proposed in order to obtain correlation relations such as in \eqref{sl3toBP2}. The analysis was then extended to the study of W-algebras associated to $\mathfrak{sl}(4)$. In this case, there are three types of non-regular W-algebras and more complicated correlator relations were derived. Similarly to the $\mathfrak{sl}(3)$ case, we started from the construction of free field realizations for the non-regular W-algebras using the screening charges in appendix \ref{sec:screening}. The method developed for the $\mathfrak{sl}(3)$ case were applied directly in deriving the correlator relations for the $\mathfrak{sl}(4)$ case. In particular, we derived new correlator correspondences in several explicit examples such as in \eqref{sl4tomin}, \eqref{mintosub}, and \eqref{subtoW4}. The focus of the paper is to examine such examples of W-algebras associated to $\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$. An attempt of generalize the results to the $\mathfrak{sl}(N)$ case was made, with correlator relations for a few special cases presented in appendix \ref{sec:corrslN}. Further generalizations of such correlator relations to W-algebras associated with any Lie algebra $\mathfrak{g}$ remain open. The screening charges for free field realizations of W-algebras associated with $\mathfrak{so}(5)$, for example, are given in appendix \ref{sec:screening}. It is straightforward to apply the techniques developed in this paper and derive the correlator relations for the $\mathfrak{so}(5)$ case.
Correlator relations like the ones derived in this paper have a wide range of applications.
We aim to report on them in the near future.
A main application of the original correlator relation with $\mathfrak{sl}(2)$ is the proof of the Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality conjecture in \cite{Hikida:2008pe}. See also \cite{Creutzig:2010bt}.
We expect that generalized FZZ dualities can be derived from our new correlator relations presented in this paper.
As another application, recall that structure constants of the operator algebra for $\mathfrak{osp}(1|2)$ WZNW model are determined from those of $\mathcal{N}=1$ super Liouville field theory \cite{Hikida:2007sz, Creutzig:2010zp}. Given that the structure constants for $\mathfrak{sl}(N)$ Toda field theory have been computed in \cite{Fateev:2007ab,Fateev:2008bm}, and the correlator relation between W$^{(2)}_N$-algebra and W$_N$-algebra has been established in this paper, it is possible to apply an analogous procedure as in \cite{Hikida:2007sz, Creutzig:2010zp} and obtain the structure constants for W$^{(2)}_N$-algebra.
As mentioned in the introduction, non-regular W-algebras have received a lot of attention, though there is much that remains poorly understood.
For example, non-regular W-algebras arise if surface operators are inserted in four dimensional $SU(N)$ gauge theories.
It was suggested that correlator correspondences like those presented in this paper can be obtained from different treatments of the same surface operators \cite{Alday:2010vg,Kozcaz:2010yp,Wyllard:2010rp,Wyllard:2010vi}. It is important to compare and establish direct relations between correlator correspondences obtained from four dimensional gauge theory with those from our path integral derivation.
As another example, non-regular W-algebras can be realized as an asymptotic symmetry of higher spin gravity with non-standard gravitational sector. The understanding of non-regular W-algebra from W$_N$-algebra is expected to help with examining higher spin gravity. In particular, we would like to study the properties of conical defect geometry in higher spin gravity as in \cite{Castro:2011iw,Gaberdiel:2012ku,Perlmutter:2012ds,Hikida:2012eu}.
A partial result for this has been already provided in \cite{Creutzig:2019qos}.
It is also of our great interest to introduce supersymmetry to relate superstring theory as in \cite{Creutzig:2011fe,Creutzig:2013tja,Creutzig:2014ula,Eberhardt:2018plx,Creutzig:2019qos}, see also \cite{Gaberdiel:2013vva,Gaberdiel:2014cha}.
\subsection*{Acknowledgements}
We are grateful to David Ridout and Zac Fehily for useful discussions. We thank the organiser of workshop 'Vertex Operator Algebras and Related Topics in Kumamoto', where a part of the work was done.
The work of TC is supported by NSERC grant number RES0019997.
The work of NG is supported by JSPS Overseas Research Fellowships.
The work of YH is supported by JSPS KAKENHI Grant Number 16H02182 and 19H01896.
The work of TL is supported by JSPS KAKENHI Grant Number 16H02182. TL would like to thank the support of AustMS Lift-Off Fellowship to visit collaborators.
|
2,877,628,090,191 | arxiv | \section{Introduction}
Increasing the data rate in wireless networks usually comes through a larger bandwidth, or a more efficient use of the available bandwidth. In cellular applications, the performance is interference-limited in most cases, meaning that increasing the transmission power does not substantially improve the network capacity. On the other hand, separating users in the time or frequency domain leads to very inefficient usage of the available resources. The key challenge is thus to balance interference avoidance and spectrum reuse to reach an optimal trade-off between spectral and energy efficiency~\cite{Tsilimantos2015}.
This challenge has been addressed in the past, for instance using frequency/code planning in 2G/3G networks or with cooperative multiple point antennas in 4G \cite{CoMP}. Dynamic interference management then became a strategic option to not only improve the spectral efficiency but also to achieve greater overall energy efficiency. The IA concept has been proposed first by \cite{MA} and extended by \cite{CJ}, it gave an interesting approach for exploiting interference in a K-users interference channel (IC) situatio
The theoretical achievements of IA have been largely discussed, e.g. in \cite{MA,AG,CJ,TG}. One of the key results is that, for a number of theoretical setups, IA can transform interference limited networks into interference-free networks, regardless of the number of users $K$ \cite{CJ}.
In the downlink cellular network, the IA extension has been addressed in several research works, e.g. \cite{DA,RT,WG,TL,SuhTse}. For example, \cite{RT} proposed to form clusters of base stations (BSs) assuming a global knowledge of channel state information (CSI). As results, only users in the center of clusters benefit from slight data rate improvemen
Other results showed questionable performance gains and several limitations \cite{DA}. Among the major issues is that IA sacrifices half of the space dimensions in order to avoid interference for every user in the cluster, whereas some of them may not suffer strong interference.
More efficiently, Suh \textit{et al.} in \cite{SuhTse} proposed a dynamic precoding scheme that attempts to balance the performance gain of matched filtering for the best users with IA for cell-edge users. By using a fixed rank-deficient precoding step at the transmitter, each BS can ensure that users always see a subspace in their effective channel where interference is reduced or eliminated. The idea is that each mobile measures and feeds back its own free subspace to its main BS, which jointly schedules the UEs so as to maximize the overall capacity. This scheme is particularly attractive in the context of dense cellular networks because with IA and thanks to channel properties, the freed subspace perceived by interfered mobiles is different for each of them. And hence, it becomes more probable to find users for which optimal transmission spaces are quasi-orthogonal as the size of the network increases. Therefore, by performing simultaneously users scheduling and precoding, the BS highly increases the chance to maximize the system capacity.
The aforementioned IA scheme only assumes to feedback the interference-free direction. However, the additional signal directions may also present a low interference level due to fading on the interfering channel. Therefore, we propose that in addition to the interference-free directions, the users feed back alternative directions, which increases the number of candidate directions at the BS and makes the scheduler more complex. In this paper, a heuristic scheduling process is proposed to decrease the algorithmic complexity while preserving near-optimal performance. Moreover, we focus on a numerical evaluation of the IA performance compared to the matched filtering (MF). Surprisingly, our simulations show that the IA technique gains interest when the crossed channels are correlated while the MF technique performs the best in the uncorrelated case scaling up linearly with the signal-to-interference and noise ratio (SINR).
Notations: boldface upper case letters and boldface lower case letters denote matrices and vectors, respectively. The superscripts $(.)^{\dagger}$ stands for the transpose conjugate matrices, respectively. $\bm I_n$ denotes the identity matrix of dimension $n$. The $\ell_2$ norm is denoted as $||.||$.
\section{System model}
A downlink cellular network with $B$ BS and $N_{m}$ user equipments (UEs) in the $m^{th}$ BS is considered. The $l^{th}$ UE in the $m^{th}$ BS receives $S_{lm}$ streams. All BSs and UEs are equipped with $M$ antennas. The transmission scheme is based on a MIMO-OFDM with $K$ available sub-carriers yielding a total $M_K = M \times K$ dimensions for the transmitted signal, in both frequency and spatial domain. A complex matrix $\bm P$ of dimension $M_K \times (M_K-N_f)$ is used at each BS ($N_f\geq 0$). The data symbols at a given BS are carried out, each by a precoding vector $\bm v_{lm}^j$ ($l, m, j$ denote the user, the BS, and the stream indices, respectively), and then further precoded using $\bm P$. Independent flat fading channels are assumed for all subcarriers. The transmitted signal of the $m^{th}$ BS is given by
\begin{equation}\label{eq1}
\bm{x}_m=\sum_{l=1}^{N_{m}} \bm V_{lm}\bm x_{lm},
\end{equation}
where $\bm x_{lm}$ contains the data for each of the $S_{lm}$ streams of user $l$ in BS $m$, and $\bm V_{lm}$ is the $(M_K-N_f) \times S_{lm}$ complex precoding matrix of user $l$. For the sake of simplicity, $\bm P$ is assumed the same for all BSs with elements selected as $p_{ij}\ \in \mathbb{C}$ -- a truncated Fourier or Hadamard matrix being a good candidate in practice. The condition $N_f \geq 0$ in $\bm P$ means that the transmit signal at each BS occupies a reduced signal space of dimension $M_k-N_f$ and leaves the rest $N_f$ dimensions free. The maximum number of streams in the reduced space of a given cell is $S \vcentcolon= M_K-N_f$ and the DoF is equal to $\frac{M_K-N_f}{M_k}$. The received signal at the $l^{th}$ UE of the $m^{th}$ cell is given by
\begin{eqnarray}\label{eq2}
\bm y_{lm}&=&\underbrace{\bm H_{lm}\bm P\bm V_{lm}\bm x_{lm}}_\textrm{desired signal} + \underbrace{\bm H_{lm}\bm P \sum_{i\neq l}^{N_{m}}\bm V_{im}\bm x_{im}}_\textrm{intra-cell interference} \nonumber \\
&+& \underbrace{\sum_{i\neq m}^{B}\bm H_{li}\bm P\bm x_i}_\textrm{inter-cell interference}+\bm w_l\ ,
\end{eqnarray}
where $\bm H_{lm}$ is the direct channel matrix between the $m^{th}$ BS and the $ l^{th}$ UE, $\bm x_i = \sum_{n}^{N_i}\bm V_{in}\bm x_{in}$, and $\bm w_l$ is the $M_K$ complex circularly symmetric additive white Gaussian noise (AWGN) vector with zero mean and covariance matrix equals to $\sigma^2\bm I_{M_K}$. Each UE has perfect knowledge of at least the direct channel linking it to its main BS, and of its main interferers through some pilots as described e.g. in \cite{IRC}. In traditional transmission schemes, the model is nothing but \eqref{eq2} except that $\bm P$ is an identity matrix. This means that all dimensions are exploited at the expense of strong inter-cell interference at cell-edges, which results in poor performance. By using a truncated, rank-deficient $\bm P$, the BS can save some available dimensions to let the other BSs in the neighbors serve their users in an interference-free subspace. This technique allows each UE to have $N_f$ interference-free streams. The decoded signal at user $l$ of the $m^{th}$ BS is given by
\begin{eqnarray}\label{eq3}
\bm U_{lm}^{\dagger}\bm y_{lm}&=&\bm G_{lm}\bm V_{lm}\bm x_{lm} + \bm G_{lm} \sum_{i\neq l}^{N_{m}}\bm V_{im}\bm x_{im} \nonumber\\
&+& \bm U_{lm}^{\dagger}\sum_{i\neq m}^{B}\bm H_{li}\bm P\bm x_i +\bm U_{lm}^{\dagger}\bm w_l\ ,
\end{eqnarray}
where $\bm G_{lm} = \bm U_{lm}^{\dagger}\bm H_{lm}\bm P$ is the direct equivalent channel between $l^{th}$ user and the $m^{th}$ BS.
\section{Interference Alignment in downlink}
\label{revscheme}
In \cite{SuhTse}, two IA schemes have been proposed, assuming only one stream per user. The first scheme is based on a Zero-Forcing (ZF) criterion. This means that each UE estimates the interfering channels and calculates the decoding matrix as their null space vector, i.e. $\bm U_{lm}$ collapses to a vector $\bm u_{lm}$ verifying:
\begin{eqnarray}\label{eq4}
\bm u_{lm}^{\dagger}\sum_{i\neq m}^{N_{ri}}\bm H_{li}\bm P\bm x_i = 0,\ \ \textrm{subject to }\ ||u_{lm}|| = 1.
\end{eqnarray}
where $N_{ri}$ is the number of interfering BSs to be canceled. The basic case is obtained for $N_{ri} = 1$ which holds for the strongest interferer. Each user calculates the equivalent channel $\bm G_{lm}$ given in (\ref{eq3}) and feeds it back to its BS, which in turns apply a scheduler that selects a subset of users to serve. The selected users have their equivalent channels denoted $\bm{{\bar{c}}}_{lm}$ collected in a matrix as $\bm{\bar{C}}_m=\left[\bm{\bar{c}}_{1m},\cdots, \bm{\bar{c}}_{Sm}\right]$, and then a ZF beamforming scheme is applied to compute orthogonal transmission vectors avoiding intra-cell interference in \eqref{eq3},
\begin{equation}\label{eq5}
\left[\bm{\bar{v}}_{1m},\cdots, \bm{\bar{v}}_{Sm}\right] = \bm{\bar{C}}_{m}^{\dagger}\left(\bm{\bar{C}}_{m}\bm{\bar{C}}_{m}^{\dagger}\right)^{-1}.
\end{equation}
Each precoding vector is normalized to ensure a constant power: $\bm{\bar{v}}_{im}^0=\frac{\bm P\bm{\bar{v}}_{im}}{||\bm P\bm{\bar{v}}_{im}||}$
The second scheme exploits a Minimum Mean Square Error (MMSE) criterion and takes into account the interference-plus-noise (IN) covariance in addition to the strongest interferer properties. The decoding vector is given by:
\begin{equation}\label{eq6}
\bm u_{lm} = \frac{\Phi_{lm}^{-1}\bm H_{lm}{\bm P}\bm v_{lm}^0 }{||\Phi_{lm}^{-1}\bm H_{lm}{\bm P}\bm v_{lm}^0||}.
\end{equation}
Assuming the knowledge of the strongest interferer channel, the IN covariance $\Phi_{lm}$ is
\begin{equation}\label{eq7}
\Phi_{lm} = (\sigma^2+\textrm{INR}_\textrm{rem})\bm I_{M_K} + \frac{p}{S}\left(\bm H_{ln}{\bm P}\bm V_n\bm V_n^\dagger{\bm P}^{\dagger}\bm H_{ln}^{\dagger}\right),
\end{equation}
with $p$ the total transmit power, $\textrm{INR}_\textrm{rem}$ the remaining interference, and $\bm V_n = \left[\bm v_{1n},\cdots,\bm v_{Sn}\right]$ the precoding matrix of the $n^{th}$ BS. The initial vector $\bm v_{lm}^0$ can be selected so as to maximize the beamforming (BF) gain at the receiver, and chosen as the eigenvector associated to the maximum eigenvalue of $\bm H_{lm}^\dagger{\bm P}^\dagger\Phi_{lm}^{-1}\bm H_{lm}{\bm P}$. As can be seen from (\ref{eq6}) and (\ref{eq7}), one major difference comparing to the first scheme is that the interference-free subspace is no longer fixed and becomes dependent on the precoder choices of other BSs. However, in practice, we consider that in average, $\bm V_n\bm V_n^\dagger$ approaches an identity matrix because of the ZF BF step.
The strict reduction of the coding space with the matrix ${\bm P}$ can also be relaxed using a full square matrix with the last $N_f$ columns weighted by a factor $0<\kappa<1$.
The motivation behind introducing the factor $\kappa$ is to be able to tweak the rank-deficiency of $\bm P$ and color the interference space \cite{SuhTse}. This approach gives more flexibility in the coding strategy and allows to adapt to situations where pure interference alignment is not necessary and may be outperformed by interference management. The new mixing matrix ${\bm P}$ is given by
\begin{equation}\label{eq8}
{\bm P} = \left[\bm p_1,\cdots,\bm p_{S},\kappa \bm p_{S+1},\cdots,\kappa \bm p_{M_K} \right],
\end{equation}
where the vectors $\bm p_i\ \in \mathbb{C}^{M_K}$ are mutually orthogonal.
\section{Generalized IA scheme}\label{genscheme}
The above IA schemes minimizes the dimensions lost in the signal space of each BS thanks to the controlled freed dimensions. While achieving a high SNR gain, it only assumes a stream per user, which severely restricts the dimensions of the problem. In many practical situations, we observed through simulations how this lack of diversity limits the system performance. Indeed, in each cell, some users are close to their main BSs and do not suffer from strong interference. In this case, it appears more efficient to let the user feeds back not only the interference-free signal direction but also additional directions presenting a sufficiently high overall SINR. By letting each user to feed back several directions, the scheduler has more flexibility to optimize the coding scheme.
\subsection{Optimal coding directions}
Each UE first selects the optimal directions that maximize the received SINR, with inter-cell interference only. The decoder is initialized as $\bm U_{lm} = \frac{\Phi_{lm}^{-1}\bm H_{lm}{\bm P}}{||\Phi_{lm}^{-1}\bm H_{lm}{\bm P}||}$. The equivalent channel is then given by
\begin{equation}\label{eq9}
\bm G_{lm} = \bm P^\dagger\bm H_{lm}^\dagger\Phi_{lm}^{-1}\bm H_{lm}\bm P.
\end{equation
The optimal receive directions are computed using the eigenvalue decomposition of $\bm G_{lm}$, with eigenvalues $\Lambda_{lm}$ and eigenvectors $\bm C_{lm}$ defined as
\begin{equation}\label{eq10}
\bm C_{lm}^\dagger\Lambda_{lm}\bm C_{lm} = \bm G_{lm},
\end{equation}
where the elements of $\Lambda_{lm}$ are sorted in decreasing order, meaning that the first columns of $\bm C_{lm}$ provide the optimal receiving directions and may be used as decoding directions. In order to maximize the received SINR, the decoding matrix is chosen as $\bm U_{lm}^d = \frac{\Phi_{lm}^{-1}\bm H_{lm}\bm P\bm C_{lm}^d}{||\Phi_{lm}^{-1}\bm H_{lm}{\bm P}\bm C_{lm}^d||}$, and the precoding matrix as $\bm V_{lm}= \bm C_{lm}^d$, where $d$ is the number of selected streams for a given user. The resulting SINR of the $i^{th}$ stream can be easily found equal to $\lambda_{lm}^i$ the $i^{th}$ element of $\Lambda_{lm}$.
\subsection{Directions feedback}
Motivated by the fact that some users may not suffer a high interference level, we suggest that each user feeds the best $L \geq N_f$ directions calculated in (\ref{eq10}) back to its BS. In this way, UEs in the center of the cell preferably feed back decoding directions corresponding to strong eigenmodes of the direct channel, whereas cell-edge users feed back interference-minimizing directions. For the sake of consistency, the cell and user indexes are omitted in the upcoming equations. Each user thus feeds back a set of candidate decoding directions $\left\{(\bm c_1,\lambda_1),\cdots,(\bm c_{L},\lambda_{L})\right\},$
where $\bm c_i$ and $\lambda_i$ are the optimal coding direction and the SINR for the $i^{th}$ stream, respectively. It is worth noting that the $\lambda_i$ gain does not take into account the intra-cell interference, and indirectly is only an estimate of the SINR in the direction $\bm c_i$. When a BS serves only one user, the directions $\bm c_i$ are mutually orthogonal, and thereby there is no intra-cell interference. In the multi-user MIMO case, the directions for two different users, e.g. $\bm c_i$ and $\bm c'_j$ are not necessarily orthogonal, which adds an extra interference power of $\lambda_i||\bm c_i\bm c'_j||$. In order to cancel the intra-cell interference, we may call for a ZF-based precoding scheme as given in (\ref{eq5})\footnote{Other precoding types than the ZF can also be straightforwardly applied, see \cite{QS}. But this does not change the core of our work.}. However in doing so, the precoders are no longer perfectly aligned with the optimal receive directions, which introduce a potential rate loss in the network overall performance.
\subsection{Polynomial time sub-optimal scheduling}
\begin{algorithm}[b]
\caption{Sub-optimal heuristic algorithm}\label{heuristic}
\begin{algorithmic}[1]
\State $\mathcal{S}_u^0 \leftarrow \{\hat{s}_0\}$
\State $\mathcal{S}_c^0 \leftarrow \mathcal{S}-\{\hat{s}_0\}$
\State $Q_\text{opt} \leftarrow C(\{\hat{s}_0\})$\;
\While{$|\mathcal{S}_u^j| < S$
\For{$i \leftarrow 1:|\mathcal{S}_c^j|$}
\State $Q \leftarrow C(\mathcal{S}_u^j \cup \{s_i\})$
\If{$Q>Q_\text{opt}$}
\State $Q_\text{opt} \leftarrow Q$
\State $\hat{s}_j \leftarrow s_i$
\EndIf
\EndFor
\State $j \leftarrow j+1$
\State $\mathcal{S}_u^{j}\leftarrow\mathcal{S}_u^{j-1} \cup \{\hat{s}_j\}$
\State $\mathcal{S}_c^{j}\leftarrow\mathcal{S}_c^{j-1} - \{\hat{s}_j\}$
\If{$\{\hat{s}_j\}\leftarrow\emptyset$}
\State break
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
The set of all candidates $\mathcal{S}=\left\{(\bm c_1,\lambda_1),\cdots,(\bm c_{N_c},\lambda_{N_c})\right\}$ is built by collecting all $N_c$ streams candidates from all users. Assuming a dense network scenario where $N_c$ is much greater than the number of available streams $S$, each BS has to select the best candidates in order to maximize a utility function defined as the sum-rate (with eventually fairness weights, see below). The optimal way is to apply an exhaustive search and to select the best subset $\mathcal{U}^*$ that results in the highest sum-rate. The optimization problem is defined a
\begin{equation}\label{eq12}
\mathcal{C(S)} = \underset{\mathcal{U} \in \mathcal{S}}{\max} \sum_{k\in \mathcal{U}} \omega_k\log_2 \left(1+\frac{p}{S}\lambda_k||\bm c_{k}\bm{\bar{v}}_{k}||\right),
\end{equation
where $\mathcal{U}$ is a given candidates streams subset, and $\omega_k$ is a per-UE weight updated after each transmission in order to provide fairness among the different UEs. Since several directions are fed back per user, the dimensions of $\mathcal{S}$ increase exponentially, which in turns makes the optimal scheduling challenging as the network grows. The optimization problem in (\ref{eq12}) is in known as NP-hard, and its computational cost may be written as
\begin{equation}\label{eq13}
\text{cost} = \mathcal{O}\left(\binom {N_s}{S}\times\;\textrm{cost}_{zf}\right),%
\end{equation
where cost$_{zf}$ is the complexity cost of a ZF precoding scheme. It is quite clear that for dense scenarios, the search among all candidate streams is not feasible. We rather propose a sub-optimal heuristic scheduling, that greatly decreases the complexity while preserving a near-optimal performance. For a given BS, the heuristic privileges the streams that provide higher rates and less inter-correlation.
Let us denote $\hat{s}_0$ the stream with the highest rate among all $N_c$ streams. At iteration $0$, we define the set of the chosen streams as $\mathcal{S}_u^0=\{\hat{s}_0\}$, the set of the remaining streams as $\mathcal{S}_c^0= \mathcal{S}-\{\hat{s}_0\}$, and the utility function as $C(\mathcal{S}_u^0)$ given in (\ref{eq12}). At the $i^{th}$ iteration, the utilities for $s \in \mathcal{S}_c^{i-1}$ are
\begin{equation}\label{eq14}
C(\mathcal{S}_u^i,s)=\sum_{k\in \mathcal{S}_u^{i-1}\cup\{s\}} \omega_k\log_2 \left(1+\frac{p}{S}\lambda_k||\bm c_{k}\bm{\bar{v}}_{k}||\right),
\end{equation}
and the stream $\hat{s}_i$ with maximum utility is selected. The sets of remaining and chosen streams are updated according to
\begin{eqnarray}\label{eq15}
\mathcal{S}_u^i&=&\mathcal{S}_u^{i-1} \cup \{\hat{s}_i\}, \nonumber \\
\mathcal{S}_c^i&=&\mathcal{S}_c^{i-1} - \{\hat{s}_i\}.
\end{eqnarray}
The algorithm stops when no more gain can be achieved. In (\ref{eq14}), $\bm c_{k}$ is the fed back direction for a given stream, and $\bm{\bar{v}}_{k}$ results from the ZF precoding scheme applied on the vectors of $\mathcal{S}_u^{i-1}\cup\{s\}$. Looking at the computational cost of the heuristic algorithm given in Alg.\ref{heuristic}, we can readily see that the complexity is polynomial and the cost search process is upper-bounded by
\begin{equation}\label{eq16}
\text{cost}_\textrm{sub}=\mathcal{O}\left((N_s-\frac{S-1}{2})S\times\;\textrm{cost}_{zf}\right),
\end{equation}
since only two loops are required.
\section{Simulation Results}
\label{SR}
\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{validsim.pdf}
\caption{Calibration results for our simulator (INRIA) with respect to other vendors and operators in the Greentouch consortium. This figure illustrates the distribution of the SINR experienced in average by the users, so-called the \emph{geometry}.}
\label{fig1}
\end{figure}
The performance of the proposed scheme has been evaluated through exhaustive simulations in different load scenarios. A system level simulator with $7$ cells is used, each cell being divided in $3$ sectors. For the exposed results, we only consider the center cell in which the UEs are uniformly distributed, although all cells support users and compute their precoders accordingly. We assume a proportional-fair scheduling by updating the scheduling weight of each user as
\begin{equation}
\omega_k^l=\frac{r_\textrm{min}}{\textrm{max}(r_\textrm{min},R_{k,\textrm{avg}}^l)},
\end{equation}
where $r_\textrm{min}$ is a threshold under which any rate is assumed null, and $R_{k,\textrm{avg}}^l$ is the average number of bits transmitted until the $l^{th}$ transmission. The spectral efficiency is compared to that of a classic OFDM scheme as a reference, where all interference are considered as noise. In this paper, $N_u=10$ users per cell in average are considered, and an overall coding space limited to $M_K=4$ dimensions with $K$ available sub-carriers and $M$ antennas at the BSs and the UEs. Our system level simulator assumes a regular placement of BSs and random users in full buffer. More information about the propagation model used here is given in \cite{GT} (cf GT doc2a). The performance are evaluated for both channel correlation levels: low and medium, with a correlation coefficient equal to $0$ and $0.3$, respectively. A simplified resource block structure is adopted to save computation time, the channel is abstracted as: $K$ sub-carriers with independent fading matrices, and each $M\times M$ fading matrix includes correlation. Each UE feeds back $M_K - N_f$ preferred receive directions. By assuming $M_K=4$ and $N_f=1$ free dimension at each BS, the BS chooses $S=3$ streams to be served using IA with scheduling based on Alg.\ref{heuristic}. The capacity is truncated to a maximum of $8$ bits per resource use. The Monte-Carlo simulations are run for $100$ independent scenarios. For each scenario, the geometry is fixed and the system is run for $100$ successive transmissions. As a reference, we provide the CDF of the received SINR in Fig.~\ref{fig1} for our simulator, calibrated to match system-level simulations of other partners in the Greentouch project. In the following, we assume that each UE has perfect estimate of the strongest interferer while it assumes others as noise
\begin{figure}
\centering
\includegraphics[scale=0.24]{uncorr_K1_M4.pdf}
\caption{Performance comparison of IA scheme for different $\kappa$ vs the matched filtering (MF) scheme and the basic MIMO-OFDM scheme in an uncorrelated scenario with $M=4$ , $K=1$.}
\label{fig2}
\end{figure}
In comparison to the reference scenario without IA in a $4\times 4$ uncorrelated MIMO scenario, one can observe from Fig.~\ref{fig2} a significant gain in the low to medium SINR region, e.g. the gain varies between $1 - 1.2$ bits/s/Hz at SINR $0$dB depending on the parameter $\kappa$. However, it decreases for high SINR because users are no longer suffering strong interference. Comparing to the MF based scheme where each BS exploits all dimensions leaving no free dimension for adjacent cells, a gain is only observed for very low SINR values i.e. SINR$<-4$dB. Beyond this value the MF scheme has an increasing gain specially for high SINR, where it also enjoys an almost constant gain over the OFDMA scheme. This means that due to channel diversity, a UE can frequently find a direction with low interference. The performance of the IA scheme for $\kappa=0$ and $\kappa=0.4$ are similar except that a slight gain is observed for SINR$>0$dB. Regarding the case with $\kappa=1$ and $S=3$, it can be seen that it outperforms the case where $\kappa=0$ and $\kappa=0.4$ for SINR$>2$dB. This is due to the additional dimension gained when $\kappa=1$. Similar observations have been made for the MIMO-OFDM configuration with $M=2$ and $K=2$. Except that in this case, the schemes with higher $\kappa$ always result in a SE gain. This means that the low-interference modes are basically created by the block diagonal channel, and are almost independent of the precoding design
Contrarily to the uncorrelated configuration, in $2\times 2$ correlated channel with a factor of $0.3$ between the antennas and $K=2$ uncorrelated subcarriers, the performance are surprisingly different as shown in Fig. \ref{fig4}. For example, the IA schemes for different $\kappa$ greatly outperform the MF scheme for all SINR values, and result in significant gains compared to the OFDM scheme in the low-to-medium SINR region. This means that since all modes are correlated, an almost interference-free mode does no longer exist unless it is freed. And hence, the performance are degraded. However, comparing the IA scheme with different $\kappa$, it can be seen that $\kappa=1, S=3$ results in a rate-loss for SINR$<-3$dB, and yields a gain beyond $8$dB since more dimensions are available for precoding.
\begin{figure}
\centering
\includegraphics[scale=0.24]{uncorr_K2_M2.pdf}
\caption{Performance comparison for different $\kappa$ vs the MF and the basic MIMO-OFDM schemes in an uncorrelated scenario with $M=2$, $K=2$.}
\label{fig3}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.24]{corr_K2_M2.pdf}
\caption{Performance comparison for different $\kappa$ versus the matched filtering based scheme and the basic MIMO-OFDM reference scheme in a medium correlated scenario with $M=2$ and $K=2$.}
\label{fig4}
\end{figure}
\section{Conclusion}
This paper addressed the performance of IA transmission in a downlink cellular network. We have generalized the scheme proposed by \cite{SuhTse} and have came up with a near-optimal low-complexity scheduler based on heuristic optimization. We have also shown that unlike the results obtained in \cite{DA}, applying a joint scheduling-precoding based on IA transmission can yield significant gains for users suffering strong inter-cell interference. Also the IA scheme becomes valuable in correlated channels where the MF scheme yields poor performance. Future works should focus on avoiding the data-rate loss caused for users enjoying high SINR which could be obtained by tuning the fairness coefficients.
|
2,877,628,090,192 | arxiv | \section{Introduction}
One of the central object of probability is Brownian motion (Bm), which is the microscopic picture emerging from a particle moving in $n$-dimensional space and of course the nature of Brownian paths is of special interest.
In this paper, we study the features of Brownian motion indexed by an $\mathbb{R}$-tree. An $\mathbb{R}$-tree is a $0$-hyperbolic metric space with desirable properties. Note that, J. Istas in \cite{Istas} proved that the fractional Brownian motion can be well defined on a hyperbolic space when $0< H \leq \frac{1}{2}$. Furthermore, in \cite{Athreya} the authors use Dirichlet form methods to construct Brownian motion on any given locally compact $\mathbb{R}$-tree, additionally in \cite{Inoue}, representation of a Gaussian field via a set of independent increments were discussed. In this paper we focus on \enquote{radial} and \enquote{river} metric and clarify the relationship between metric trees generated by these metrics and a particular metric ray denoted by $\mathcal C_d(A,B)$. Our investigation is motivated by the questions: under what conditions on the set $\{\mathcal C_d(A,B)\}_{A,B\in M}$ does $(M,d)$ become an $\mathbb R$-tree?
and when can an $\mathbb R$-tree be identified by the sets $\{\mathcal C_d(A,B)\}_{A,B\in M}$? It is our hope that this work could lead to the interest of applying those results to random fields indexed by metric spaces.
The study of injective envelopes of metric spaces, also known as $\mathbb R$-trees (metric trees or $T$-theory) began with J. Tits in \cite{Tits} in $1977$ and since then, applications have been found within many fields of mathematics. For a complete discussion of these spaces and their relation to global metric spaces of nonpositive curvature we refer to \cite{Bridson}. Applications of metric trees in biology
and medicine stems from the construction of phylogenetic trees \cite{Semple}. Concepts
of \enquote{string matching} in computer science are closely related with the structure of
metric trees \cite{Bartolini}. $\mathbb R$-trees are a generalization of an ordinary tree which allows for different
weights on edges. In order to define an $\mathbb R$-tree, we first introduce the notion of metric segment. Let $(M,d)$ be a metric space. For any $A,B\in M$, the \textit{metric segment }$[A,B]$ is defined by
$$
[A,B]=\left\{X\in M:~d(A,X)+d(X,B)=d(A,B)<+\infty\right\}.
$$
Notice that by this definition, $[A,B]\ne\emptyset$ if and only if $A,B$ are connected in $(M,d)$.
\begin{defn} [see \cite{Kirk}]
\label{Rtree}
An $\mathbb R$-tree is a nonempty metric space $(M,d)$
satisfying:
\begin{description}
\item[(a)] Any two points of $A,B\in M$ are joined by a unique metric segment $[A,B]$.
\item[(b)] If $A,B,C\in M$, then
$$
[A,B]\cap[A,C]=[A,O]~\mbox{for some}~O\in M.
$$
\item[(c)] If $A,B,C\in M$ and $[A,B]\cap[B,C]=\{B\}$, then
$$
[A,B]\cup[B,C]=[A,C].
$$
\end{description}
\end{defn}
Through out this paper we only consider the class of metric spaces satisfying $(a)$ in Definition \ref{Rtree} above. We call this metric space uniquely geodesic metric space. In the following we characterize an $\mathbb{R}$-tree by the theorem below (given in \cite{Buneman}):
\begin{thm}
A uniquely geodesic metric space $(M,d)$ is an $\mathbb R$-tree if and only if it is connected, contains no triangles and satisfies the four-point condition (4PC).
\end{thm}
Note that, we say a metric $d(\cdot , \cdot)$ satisfies the four-point condition (4PC) if, for any $A,B, C, D$ in $M$ the following inequality holds:
$$ d(A,B) +d(C,D) \leq \max \{d(A,C)+d(B,D), \,\,d(A,D)+d(B,C)\}.$$
The four-point condition is stronger than the triangle inequality (take C = D), but it
should not be confused with the ultrametric definition. An ultrametric satisfies the condition
$d(A, B) \leq \max \{ d(A, C), d(B, C)\}$, and this is stronger than the four-point condition.
Moreover, we say $A,B,C$ form a triangle if all the triangle inequalities involving $A,B,C$ are strict and $[X,Y]\cap[Y,Z]=\{Y\}$ for any $\{X,Y,Z\}=\{A,B,C\}$. $d(. , .)$ is said to be a tree metric if it satisfies the (4PC). Given a metric space $(M,d)$, we would capture the tree metric properties of $(M,d)$ by introducing the following sets $\{\mathcal C_d(A,B)\}_{A,B\in M}$.
\begin{defn}
\label{C}
For any $A, B\in M$, we define
$$
\mathcal C_d(A,B)=\left\{X\in M:~d(X,A)=d(X,B)+d(A,B)<+\infty\right\}.
$$
\end{defn}
Note that two points $A,B\in M$ are connected if and only if $\mathcal C_d(A,B)\ne\emptyset$.\\[.05in]
We remark that a Brownian motion is uniquely determined by independent increments and furthermore, since the set $\mathcal C_d(P_1,P_2)$ is also defined as:
$$\mathcal C_d(P_1,P_2) = \{X\in M:\,\, B(X)-B(P_2)\,\,\, \mbox{ is independent of}\,\,\, B(P_2)-B(P_1)\},$$
It is of interest to ask the following questions:
\begin{description}
\item Question $1$: Under what conditions on the set $\{\mathcal C_d(A,B)\}_{A,B\in M}$ does $(M,d)$ become an $\mathbb R$-tree?
\item Question $2$: When can an $\mathbb R$-tree be identified by the set $\{\mathcal C_d(A,B)\}_{A,B\in M}$?
\end{description}
In this paper we give complete solution to Question 1 (see Section \ref{tree1} below), namely, we provide a sufficient and necessary condition on $\{\mathcal C_d(A,B)\}_{A,B\in M}$ such that $(M,d)$ is an $\mathbb R$-tree. In Section \ref{tree2}, we study Question 2 by considering radial metric and river metric. We show that the answer to Question 2 is positive when $M=\mathbb R^n$ and
$$
d(A,B)=g_k(|A-B|)~\mbox{for $A,B\in \Pi_k$},
$$
where $|\cdot|$ is the Euclidean norm, $(\Pi_k)_{k=1,\ldots,N}$ is some partition of $\mathbb R^n$ and $g_k:~\mathbb R_+\rightarrow \mathbb R_+$ is a continuous function.
\section{Results : An Equivalence of $\mathbb R$-tree Properties}
\label{tree1}
We start by introducing the following conditions that will be used in the proof of the Theorem \ref{equivtree}:\\[.05in]
\textit {Condition $(A)$: For any 3 distinct points $A,B,C\in M$, there exists unique $O\in M$ such that
$$
\{X,Y\}\subset\mathcal C_d(Z,O)~\mbox{for any distinct}~X,Y,Z\in\{A,B,C\}.
$$}
\textit{Condition $(B)$: For any distinct $A,B,C\in M$, there exists $O\in M$ such that $$[A,B]\cap[B,C]\cap[A,C]=\{O\}.$$}
\bigskip
Note that if the cardinality $|M|=1$ or $2$, then $(M,d)$ is obviously an $\mathbb R$-tree, since any 2 points are joined by a unique geodesic. When $|M|\ge 3$, Condition $(A)$ guarantees that $(M,d)$ contains no circuit. The following will be used in the proof of the Theorem \ref{equivtree}:
\begin{lemme}
\label{equiv}
Condition $(A)$ is equivalent to Condition $(B)$.
\end{lemme}
\textbf{Proof.}
We only consider the case where $M$ contains at least 3 distinct points. Let's pick 3 distinct points $A,B,C\in M$. Then by observing that for any distinct $X,Y\in\{A,B,C\}$,
$$
X\in\mathcal C_d(Y,O)~\mbox{is equivalent to}~O\in[X,Y].
$$
Thus Lemma \ref{equiv} holds. $\square$
\begin{thm}
\label{equivtree}
A uniquely geodesic metric space $(M,d)$ is an $\mathbb R$-tree if and only if \textit{Condition $(A)$} holds.
\end{thm}
\textbf{Proof}
By Lemma \ref{equiv}, it is sufficient to prove Theorem \ref{equivtree} holds under Condition $(B)$.
First we show that if $(M,d)$ is an $\mathbb R$-tree, then \textit{Condition $(B)$} is satisfied, and then
show that if \textit{Condition $(B)$} holds, then $(M,d)$ is an $\mathbb R$-tree.
Suppose $(M,d)$ is an $\mathbb R$-tree, since $(M,d)$ is connected, then $[A,B]\ne\emptyset$ for all $A,B\in M$. For any 3 points $A,B,C\in M$ we have:
\begin{itemize}
\item if $[A,B]\cap[B,C]=\{B\}$, then by $(c)$ in Definition \ref{Rtree},
$$
\{B\}=[A,B]\cap[B,C]\subset[A,B]\cup[B,C]=[A,C].
$$
This yields
$$
[A,B]\cap[B,C]\cap[A,C]=\{B\}\cap[A,C]=\{B\}.
$$
\item If there exists $O\in M$, $O\ne B$ such that $[A,B]\cap[B,C]=[B,O]$, then $O\in[A,B]\cap[B,C]\cap[A,C]$. Thus, condition $(B)$ is verified.
\end{itemize}
Next assume that \textit{Condition $(B)$} holds. By taking any $A\ne B=C$, one easily shows that $[A,B]\ne\emptyset$, thus $(M,d)$ is connected. The fact that $[A,B]\cap[B,C]\cap[A,C]\ne\emptyset$ leads to the fact that there is no triangles in $(M,d)$. Then it is sufficient to prove that $d(\cdot, \cdot)$ satisfies the (4PC). Let us pick 4 distinct points $A,B,C,D$ from $M$. Under Condition $(B)$, we have two possibilities to the positions of $A,B,C,D$, namely:
\begin{enumerate}
\item $A,B,C,D$ are in the same metric segment.
\item Case 1 above does not hold.
\end{enumerate}
Case 1 is equivalent to
$$
\left\{
\begin{array}{ll}
&d(W,X)=d(W,Z)+d(X,Z);\\
&d(W,Y)=d(W,Z)+d(Y,Z).
\end{array}\right.
$$
This easily leads to the (4PC). For the second case, for any $\{X,Y,Z,W\}=\{A,B,C,D\}$, if $\mathcal C_d(X,Z)=\mathcal C_d(Y,Z)$, then one necessarily has $W\notin\mathcal C_d(X,Z)=\mathcal C_d(Y,Z)$. This is equivalent to
$$
\left\{
\begin{array}{ll}
&d(W,X)<d(W,Z)+d(X,Z);\\
&d(W,Y)<d(W,Z)+d(Y,Z).
\end{array}\right.
$$
This easily leads to the (4PC).
It is easy to check that in each case, the (4PC) is satisfied. $\square$
\subsection{Characterization of $\mathcal C_d(P_1,P_2)$ via Radial Metric}
We define an $\mathbb R$-tree $(\mathbb R^n,d_1)$ $(n\ge1)$ with root $0$ and the metric
$$
d_1(A,B)=\left\{
\begin{array}{lll}
&|A-B|&~\mbox{if $A=aB$ for some $a\in\mathbb R$};\\
&|A|+|B|&~\mbox{otherwise}.
\end{array}\right.
$$
We explicitly represent the set $\mathcal C_{d_1}(P_1,P_2)$ for all $P_1,P_2\in\mathbb R^n$ in the following proposition.
\begin{prop}
\label{Ex1}
For any $P_1,P_2\in(\mathbb R^n,d_1)$,
\begin{equation}
\label{Cradial}
\mathcal C_{d_1}(P_1,P_2)=\left\{
\begin{array}{lll}
&[P_2,+\infty)_{\overrightarrow{0P_2}}&~\mbox{if $P_2\notin[0,P_1]_{\overrightarrow{0P_1}}$};\\
&\mathbb R^n\backslash(P_2,+\infty)_{\overrightarrow{0P_1}}&~\mbox{if $P_2\in[0,P_1)_{\overrightarrow{0P_1}}$};\\
&\mathbb R^n&~\mbox{if $P_1=P_2$,}
\end{array}\right.
\end{equation}
where for any $A,B\in\mathbb R^n$, $[A,B)_{\overrightarrow{AB}}$ denotes the segment $\{(1-a)A+aB;a\in[0,1)\}$ and $(A,+\infty)_{\overrightarrow{0B}}$ denotes $\{aA+bB;a>1,b>0\}$ under \underline{Euclidean distance}. These notations shouldn't be confused with the metric segments $[A ,B]$ of a metric space.
\end{prop}
\textbf{Proof.}
Since it is always true that $\mathcal C_{d_1}(P_1,P_2)=\mathbb R^n$ for $P_1=P_2$, then we only consider the case when $P_1\ne P_2$. There are 3 different situations to the positions of $P_1$, $P_2$: $(1)$ $P_1$, $P_2$ are on the same ray (which means, $P_2=aP_1$ for some $a\in\mathbb R$) and $0\le|P_1|<|P_2|$; $(2)$ $P_1$, $P_2$ are on the same ray and $0\le|P_2|<|P_1|$; $(3)$ $P_1,P_2$ are on different rays.
\noindent Case $(1)$: $P_1$, $P_2$ are on the same ray and $0\le|P_1|<|P_2|$.
In this case one necessarily has
\begin{equation}
\label{d1case1}
d_1(A,P_1)=d_1(A,P_2)+|P_1-P_2|.
\end{equation}
\noindent Case $(1.1)$: If $A$ is on a different ray as $P_2$, then (\ref{d1case1}) becomes
$$
|A|+|P_1|=|A|+|P_2|+|P_1-P_2|.
$$
This together with the fact that $P_1\ne P_2$ implies
$$
|P_1|=|P_2|+|P_1-P_2|>|P_2|.
$$
This is impossible, because it is assumed that $|P_1|<|P_2|$.
\noindent Case $(1.2)$: Suppose $A$ is on the same ray as $P_2$. Now (\ref{d1case1}) is equivalent to
$$
|A-P_1|=|A-P_2|+|P_1-P_2|.
$$
The solution space for $A$ is then the segment $[P_2,+\infty)_{\overrightarrow{0P_2}}$ under Euclidean distance.
One concludes that in Case $(1)$,
\begin{equation}
\label{d1case1solution}
\mathcal C_{d_1}(P_1,P_2)=[P_2,+\infty)_{\overrightarrow{0P_2}}.
\end{equation}
\noindent Case $(2)$: $P_1,P_2$ are on the same ray and $|P_1|>|P_2|\ge0$. Note that (\ref{d1case1}) still holds.
\noindent Case $(2.1)$: Suppose that $A$ is on a different ray as $P_1$. (\ref{d1case1}) is then equivalent to
$$
|P_1|=|P_2|+|P_1-P_2|.
$$
The above equation always holds true. Therefore any $A$ on a different ray as $P_1$ belongs to $\mathcal C_{d_1}(P_1,P_2)$.
\noindent Case $(2.2)$: $A$ is on the same ray as $P_1$. Equation (\ref{d1case1}) then becomes
$$
|A-P_1|=|A-P_2|+|P_1-P_2|,
$$
and its solution space is segment $[0,P_2]_{\overrightarrow{0P_2}}$ under Euclidean distance.
Combining Case (2.1) and Case (2.2), one obtains, in Case $(2)$,
\begin{equation}
\label{d1case2solution}
\mathcal C_{d_1}(P_1,P_2)=\mathbb R^n\backslash(P_2,+\infty)_{\overrightarrow{0P_1}}.
\end{equation}
\noindent Case $(3)$: $P_1,P_2$ are on different rays with $P_1,P_2\ne0$.
\noindent Case $(3.1)$: $A$ is on the same ray as $P_1$.
In this case we have
$$
|A-P_1|=|A|+|P_2|+|P_1|+|P_2|.
$$
By the triangle inequality,
$$
|A|+|P_1|+2|P_2|=|A-P_1|\le |A|+|P_1|.
$$
This yields the absurd statement $P_2=0$!
\noindent Case $(3.2)$: $A$ is on the same ray as $P_2$.
We have
$$
|A|+|P_1|=|A-P_2|+|P_1|+|P_2|.
$$
This leads to $A\in[P_2,+\infty)_{\overrightarrow{0P_2}}$.
\noindent Case $(3.3)$: $A$ is on a different ray as $P_1$, $P_2$.
We have
$$
|A|+|P_1|=|A|+|P_2|+|P_1|+|P_2|,
$$
which again implies $P_2=0$. Contradiction!
We conclude that in Case $(3)$,
\begin{equation}
\label{d1case3solution}
\mathcal C_{d_1}(P_1,P_2)=[P_2,+\infty)_{\overrightarrow{0P_2}}.
\end{equation}
Finally, by combining (\ref{d1case1solution}), (\ref{d1case2solution}) and (\ref{d1case3solution}), one proves Proposition \ref{Ex1} (see Figure 1, Figure 2).~$\square$
\begin{figure}[H]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d1case1}
\caption{The thick line represents the set of $\mathcal C_{d_1}(P_1,P_2)$ when $P_2$ is not in the segment $[0,P_1]$.}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d1case2}
\caption{The shaded region represents $\mathcal C_{d_1}(P_1,P_2)$ when $P_2$ is in the segment $[0,P_1)$.}
\end{minipage}
\end{figure}
Now we would show the inverse of Proposition \ref{Ex1}, namely whether or not we can recognize ($\mathbb{R}^n, d)$ as an $\mathbb{R}$-tree using $\mathcal C_d(P_1,P_2)$. For that purpose, we first show the following statement.
\begin{prop}
\label{four}
Let $(\mathbb R^n , d)$ be a metric space. If
(\ref{Cradial}) holds for any $P_1,P_2\in(\mathbb R^n,d)$, then $(\mathbb R^n , d)$ is an $\mathbb R$-tree.
\end{prop}
\textbf{Proof.} By Theorem \ref{equivtree}, we only need to show \textit{Condition $(B)$} holds. Let us arbitrarily pick 3 different points $A,B,C\in\mathbb R^n$. If $A,B,C$ are in the same segment, saying, $A\in\mathcal C_d(B,C)$, then $C\in [A,B]\cap[B,C]\cap[A,C]$ and Condition $(B)$ is satisfied. If $A,B,C$ are not in the same segment, i.e., $X\notin\mathcal C_d(Y,Z)$ for any $\{X,Y,Z\}=\{A,B,C\}$, then we see from the definition of $\mathcal C_d(P_1,P_2)$ that
$$
X\in\mathcal C_d(Y,0)~\mbox{for any distinct}~X,Y\in\{A,B,C\},
$$
which is equivalent to $0\in[A,B]\cap[B,C]\cap[A,C]$. Hence\textit{ Condition $(B)$} is satisfied. $\square$
\subsection{Characterization of $\mathcal C_d(P_1,P_2)$ via River Metric}
For $A\in\mathbb R^2$, we denote by $A=(A^{(1)},A^{(2)})$. We define the river metric space $(\mathbb R^2,d_2)$ by taking
$$
d_2(A,B)=\left\{\begin{array}{lll}
&|A^{(2)}-B^{(2)}|&~\mbox{for $A^{(1)}=B^{(1)}$;}\\
&|A^{(2)}|+|A^{(1)}-B^{(1)}|+|B^{(2)}|&~\mbox{for $A^{(1)}\ne B^{(1)}$}.
\end{array}\right.
$$
From now on we say that $A,B$ are on the same ray in $(\mathbb R^2,d_2)$ if and only if $A,B$ are on a vertical Euclidean line: $A^{(1)}=B^{(1)}$.
\begin{prop}
\label{Ex2}
Let $(\mathbb R^2,d_2)$ be a river metric space. For $P\in \mathbb R^2$, denote by $P^*=(P^{(1)},0)$ the projection of $P$ to the horizontal axis. Then for any $P_1,P_2\in\mathbb R^2$, we have
\begin{equation}
\label{Criver}
\mathcal C_{d_2}(P_1,P_2)=\left\{
\begin{array}{lll}
&\mathbb R^2\backslash (P_2,\infty)_{\overrightarrow{P_1^*P_1}}&~\mbox{if $P_2\in[P_1^*,P_1)$};\\
&[P_2,\infty)_{\overrightarrow{P_2^*P_2}}&~\mbox{if $P_2\notin [P_1^*,P_1)$ and $P_2^{(2)}\ne 0$};\\
&[P_2^{(1)},\infty)_{\overrightarrow{P_1^{(1)}P_2^{(1)}}}\times\mathbb R&~\mbox{if $P_1^{(1)}\ne P_2^{(1)}$, $P_2^{(2)}=0$};\\
&\mathbb R^2&~\mbox{if $P_1=P_2$}.
\end{array}\right.
\end{equation}
\end{prop}
\textbf{Proof.}
It is obvious that $\mathcal C_{d_2}(P_1,P_2)=\mathbb R^2$ when $P_1=P_2$. For $P_1\ne P_2$, we mainly consider 2 cases: $(1)$ $P_1$, $P_2$ are on the same ray ($P_1^{(1)}=P_2^{(1)}$); $(2)$ $P_1,P_2$ are on different rays ($P_1^{(1)}\ne P_2^{(1)}$).
\noindent Case $(1)$: $P_1$, $P_2$ are on the same ray.
In this case one necessarily has
\begin{equation}
\label{d2case1}
d_2(A,P_1)=d_2(A,P_2)+|P_1^{(2)}-P_2^{(2)}|.
\end{equation}
\noindent Case $(1.1)$: Suppose $A$ is on a different ray as $P_2$, then it follows from (\ref{d2case1}) that
$$
|A^{(2)}|+|P_1^{(1)}-A^{(1)}|+|P_1^{(2)}|=|A^{(2)}|+|P_2^{(1)}-A^{(1)}|+|P_2^{(2)}|+|P_1^{(2)}-P_2^{(2)}|.
$$
Since $P_1^{(1)}=P_2^{(1)}$, the above equation is simplified to
$$
|P_1^{(2)}-P_2^{(2)}|+|P_2^{(2)}|-|P_1^{(2)}|=0.
$$
This equation holds for all $A$ with $A^{(1)}\ne P_2^{(1)}$ provided that $P_2^{(2)}\in[0,P_1^{(2)})_{\overrightarrow{0P_1^{(2)}}}$. When $P_2^{(2)}\notin[0,P_1^{(2)})_{\overrightarrow{0P_1^{(2)}}}$, it has no solution.
\noindent Case $(1.2)$: $A$ is on the same ray as $P_2$. Now one has
$$
|A^{(2)}-P_1^{(2)}|=|A^{(2)}-P_2^{(2)}|+|P_1^{(2)}-P_2^{(2)}|.
$$
The above equation holds only when
$$
A\in \{P_2^{(1)}\}\times[P_2^{(2)},\infty)_{\overrightarrow{0P_2^{(2)}}}.
$$
Therefore one concludes that when $P_1$ and $P_2$ are on the same ray,
\begin{equation}
\label{d2case1solution}
\mathcal C_{d_2}(P_1,P_2)=\left\{\begin{array}{lll}
&\mathbb R^2\backslash (P_2,\infty)_{\overrightarrow{P_1^*P_1}}&~\mbox{if $P_2^{(2)}\in[0,P_1^{(2)})_{\overrightarrow{0P_1^{(2)}}}$};\\
&[P_2,\infty)_{\overrightarrow{P_2^*P_2}}&~\mbox{if $P_2^{(2)}\notin[0,P_1^{(2)})_{\overrightarrow{0P_1^{(2)}}}$}.
\end{array}\right.
\end{equation}
\noindent Case $(2)$: $P_1,P_2$ are on different rays.
\noindent Case $(2.1)$: $A$ is on the same ray as $P_1$.
In this case we have
\begin{eqnarray*}
|A^{(2)}-P_1^{(2)}|&=&|A^{(2)}|+|A^{(1)}-P_2^{(1)}|+|P_2^{(2)}|+|P_1^{(2)}|+|P_1^{(1)}-P_2^{(1)}|+|P_2^{(2)}|\\
&>&|A^{(2)}|+|P_1^{(2)}|.
\end{eqnarray*}
This contradicts the triangle inequality, therefore there is no solution for $A$ in this case.
\noindent Case $(2.2)$: $A$ is on the same ray as $P_2$.
We have
$$
|A^{(2)}|+|A^{(1)}-P_1^{(1)}|+|P_1^{(2)}|=|A^{(2)}-P_2^{(2)}|+|P_1^{(2)}|+|P_1^{(1)}-P_2^{(1)}|+|P_2^{(2)}|.
$$
By using the fact that $A^{(1)}=P_2^{(1)}$, the above equation becomes
\begin{equation}
\label{d2case22}
|A^{(2)}|=|A^{(2)}-P_2^{(2)}|+|P_2^{(2)}|.
\end{equation}
This provides:
\begin{itemize}
\item if $P_2^{(2)}=0$, then the solution space of (\ref{d2case22}) is $ \{P_2^{(1)}\}\times \mathbb R$;
\item if $P_2^{(2)}\ne0$, then the solution space of (\ref{d2case22}) is $
\{P_2^{(1)}\}\times[P_2^{(2)},\infty)_{\overrightarrow{0P_2^{(2)}}}$.
\end{itemize}
\noindent Case $(2.3)$: $A$ is on a different ray as $P_1$, $P_2$.
We have
\begin{eqnarray*}
&&|A^{(2)}|+|A^{(1)}-P_1^{(1)}|+|P_1^{(2)}|\\
&&=|A^{(2)}|+|A^{(1)}-P_2^{(1)}|+|P_2^{(2)}|+|P_1^{(2)}|+|P_1^{(1)}-P_2^{(1)}|+|P_2^{(2)}|.
\end{eqnarray*}
It is equivalent to
$$
|A^{(1)}-P_1^{(1)}|=|A^{(1)}-P_2^{(1)}|+|P_1^{(1)}-P_2^{(1)}|+2|P_2^{(2)}|.
$$
This equation has solution only when $P_2^{(2)}=0$. Then the above equation is written as
$$
|A^{(1)}-P_1^{(1)}|=|A^{(1)}-P_2^{(1)}|+|P_1^{(1)}-P_2^{(1)}|.
$$
This implies
$$
A^{(1)}\in(P_2^{(1)},\infty)_{\overrightarrow{P_1^{(1)}P_2^{(1)}}}.
$$
By combining the solutions for Cases $(2.1)$, $(2.2)$, we obtain, in Case $(2)$,
\begin{equation}
\label{d2case2solution}
\mathcal C_{d_2}(P_1,P_2)=\left\{\begin{array}{lll}
&[P_2,\infty)_{\overrightarrow{P_2^*P_2}}&~\mbox{if $P_1^{(1)}\ne P_2^{(1)}$, $P_2^{(2)}\ne0$};\\
&[P_2^{(1)},\infty)_{\overrightarrow{P_1^{(1)}P_2^{(1)}}}\times\mathbb R&~\mbox{if $P_1^{(1)}\ne P_2^{(1)}$, $P_2^{(2)}=0$}.
\end{array}\right.
\end{equation}
Finally, putting together Cases $(1),(2)$, one proves Proposition \ref{Ex2}.~$\square$
\begin{figure}[H]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d2case1}
\caption{The shaded region represents the set of $\mathcal C_{d_2}(P_1,P_2)$ when $P_2$ belongs to the segment $[P_1^*,P_1)$.}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d2case22}
\caption{The thick line represents $\mathcal C_{d_2}(P_1,P_2)$ when $P_1^{(1)}=P_2^{(1)}$ and $|P_2^{(2)}|>|P_1^{(2)}|$.}
\end{minipage}
\end{figure}
\begin{figure}[H]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d2case21}
\caption{The thick line represents the set of $\mathcal C_{d_2}(P_1,P_2)$ when $P_1^{(1)}\neq P_2^{(1)}$ and $P_2^{(2)}\ne0$.}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=5cm,height=5cm]{d2case3}
\caption{The shaded region represents $\mathcal C_{d_2}(P_1,P_2)$ when $P_1^{(1)}\neq P_2^{(1)}$ and $P_2^{(2)}=0$.}
\end{minipage}
\end{figure}
\begin{prop}
\label{four1}
Let $(\mathbb R^n,d)$ be a metric space. If
for any $P_1,P_2\in\mathbb R^n$, (\ref{Criver}) holds, then $(\mathbb R^n,d)$ is an $\mathbb R$-tree.
\end{prop}
\textbf{Proof.} We only need to show Condition $(A)$ is satisfied by the expression of $\mathcal C_d(P_1,P_2)$ in (\ref{Criver}). Observe that for any 3 distinct points $A,B,C\in\mathbb R^n$, without loss of generality, there are 3 situations according to the positions:
\begin{description}
\item Case $1$ : $A^{(1)}=B^{(1)}=C^{(1)}$, $A^{(2)}\in[0,B^{(2)})_{\overrightarrow{0B^{(2)}}}$, $B^{(2)}\in[0,C^{(2)})_{\overrightarrow{0B^{(2)}}}$.
\item Case $2$: $A^{(1)}=B^{(1)}\ne C^{(1)}$, $A^{(2)}\in[0,B^{(2)})_{\overrightarrow{0B^{(2)}}}$.
\item Case $3$: $A^{(1)}$, $B^{(1)}$ and $C^{(1)}$ are all distinct, $B^{(1)}\in[A^{(1)},C^{(1)}]_{\overrightarrow{A^{(1)}C^{(1)}}}$.
\end{description}
By (\ref{Criver}), it is easy to see \textit{Condition $(A)$} holds with $O=B$, $O=A$ and $O=(0,B^{(2)})$ respectively for Case 1, Case 2 and Case 3. Hence Proposition \ref{four1} is proven by using Theorem \ref{Rtree}. $\square$
\section{Identification of Radial Metric and River Metric via $\mathcal C_d(P_1,P_2)$}
\subsection{Identification of Radial Metric via $\mathcal C_{d_1}(P_1,P_2)$}
\label{tree2}
In Proposition \ref{four} and Proposition \ref{four1}, we have shown that the sets $\{\mathcal C_d(P_1,P_2)\}_{P_1,P_2}$ capture the tree properties of the metric spaces $(\mathbb R^n,d_1)$ and $(\mathbb R^2,d_2)$. Now we claim that subject to some additional conditions these two $\mathbb R$-trees can be uniquely identified by the sets $\{\mathcal C_d(P_1,P_2)\}_{P_1,P_2}$.
\begin{defn}
\label{ds}
Let $\tilde d_1$ be a metric defined on $\mathbb R^n$ satisfying that there exists a function $f:\mathbb R_+\rightarrow\mathbb R_+$ such that
\begin{itemize}
\item $f$ is continuous;
\item $f$ satisfies the following equation:
$$
\left\{\begin{array}{lll}
&\tilde d_1(ax,x)=f(|ax-x|)&~\mbox{for all $x\in\mathbb R^n$ and all $a\ge0$};\\
&f(1)=1.
\end{array}
\right.$$
\end{itemize}
\end{defn}
\begin{thm}
\label{inverseEx1}
The following statements are equivalent:
\begin{description}
\item[(i)] $\tilde d_1=d_1$.
\item[(ii)] For any $P_1,P_2\in(\mathbb R^n,\tilde d_1)$, $\mathcal C_{\tilde d_1}(P_1,P_2)=\mathcal C_{d_1}(P_1,P_2)$ given in (\ref{Cradial}).
\end{description}
\end{thm}
Before proving Theorem \ref{inverseEx1}, we first introduce the following useful statement.
\begin{thm} (See Aczel \cite{Aczel}, Theorem $1$)
\label{cauchy}
If Cauchy's functional equation
$$
g(u+v)=g(u)+g(v)
$$
is satisfied for all positive $u,v$, and if the function $g$ is
\begin{itemize}
\item continuous at a point;
\item nonnegative for small positive $u-s$ or bounded in an interval,
\end{itemize}
then
$$
g(u)=cu
$$
is the general solution for all positive $u$.
\end{thm}
\textbf{Proof.} The implication $(i)\Longrightarrow(ii)$ is simply Proposition \ref{Ex1}. Now it remains to prove $(ii)\Longrightarrow(i)$.
\begin{description}
\item Case $(1)$: $A,P_1,0$ are on the same straight line with $A\ne P_1$.
Without loss of generality, assume $|A|>|P_1|$. Then there exists $P_2\in(P_1,+\infty]_{\overrightarrow{0P_2}}$ such that $A\in[P_2,+\infty)_{\overrightarrow{0P_2}}$. By Proposition \ref{Ex1}, one has
$$
\tilde d_1(A,P_1)=\tilde d_1(A,P_2)+\tilde d_1(P_1,P_2).
$$
Observe that $A,P_1,P_2,0$ are on the same straight line, then by the definition of $\tilde d_1$, the above equation is equivalent to
\begin{equation}
\label{id1}
f(|A-P_1|)=f(|A-P_2|)+f(|P_1-P_2|).
\end{equation}
This is a Cauchy's equation, then by using Theorem \ref{cauchy}, the general solution is $f(u)=cu$. Together with its initial condition $f(1)=1$, one finally gets
$$
f(u)=u.
$$
Hence,
$$
\tilde d_1(A,P_1)=|A-P_1|,~\mbox{for $A,P_1,0$ lying on the same straight line.}
$$
\item Case $(2)$: $A,P_1,0$ are not on the same straight line (in this case one necessarily has $A,P_1\ne 0$).
We take $P_2=0$. The fact that $A\notin (0,+\infty)_{\overrightarrow{0P_1}}$ implies
$$
\tilde d_1(A,P_1)=\tilde d_1(A,0)+\tilde d_1(P_1,0).
$$
From Case (1) we see
$$
\tilde d_1(A,0)=|A|~\mbox{and}~\tilde d_1(P_1,0)=|P_1|.
$$
Therefore,
$$
\tilde d_1(A,P_1)=|A|+|P_1|,~\mbox{for $A,P_1,0$ lying on a different straight line.}
$$
\end{description}
It follows from Cases (1) and (2) that $\tilde d_1=d_1$. $\square$
\subsection{{Identification of River Metric via $\mathcal C_{d_2}(P_1,P_2)$}}
Now we claim that the inverse statement of Proposition \ref{Ex2} holds, under some extra condition.
\begin{defn}
\label{ds2}
Define the metric $\tilde d_2$ on $\mathbb R^2$ by
$$
\tilde d_2(x,y)=\left\{
\begin{array}{lll}
&g_1(|x-y|);&~\mbox{if $x_1=y_1$};\\
&g_2(|x-y|);&~\mbox{if $x_2= y_2=0$},
\end{array}\right.
$$
\end{defn}
where $g_1,g_2$ satisfy the same conditions as $f$ in Definition \ref{ds}.
\begin{thm}
\label{inverseEx2}
The following statements are equivalent:
\begin{description}
\item[(i)] $\tilde d_2=d_2$.
\item[(ii)] For any $P_1,P_2\in(\mathbb R^2,\tilde d_2)$, $\mathcal C_{\tilde d_2}(P_1,P_2)=\mathcal C_{d_2}(P_1,P_2)$ given in (\ref{Criver}).
\end{description}
\end{thm}
\textbf{Proof.} The implication $(i)\Longrightarrow(ii)$ is trivial according to Proposition \ref{Ex2}. Now we prove $(ii)\Longrightarrow(i)$.
\begin{description}
\item Case (1): $A^{(1)}=P_1^{(1)}$. In this case one takes any $P_2\in[P_1,A]_{\overrightarrow{P_1A}}$ and gets
$$
A\in\mathcal C_{\tilde d_2}(P_1,P_2).
$$
Equivalently,
$$
\tilde d_2(A,P_1)=\tilde d_2(A,P_2)+\tilde d_2(P_1,P_2).
$$
By using the definition of $g_1$, one obtains the following Cauchy's equation
$$
g_1(|A^{(2)}-P_1^{(2)}|)=g_1(|A^{(2)}-P_2^{(2)}|)+g_1(|P_1^{(2)}-P_2^{(2)}|).
$$
Then by Theorem \ref{cauchy},
$$
g_1(x)=x,~\mbox{for all $x\ge0$}.
$$
\item Case $(2)$ : $A^{(1)}\ne P_1^{(1)}$.
\item Case $(2.1)$: We let $A^{(2)}=P_1^{(2)}=0$ and choose $P_2=(P_2^{(1)},0)$ with $P_2^{(1)}\in[A^{(1)},P_1^{(1)}]_{\overrightarrow{P_1^{(1)}A^{(1)}}}$, then by the fact that
$$
A\in\mathcal C_{\tilde d_2}(P_1,P_2),
$$
one has
$$
\tilde d_2(A,P_1)=\tilde d_2(A,P_2)+\tilde d_2(P_1,P_2).
$$
Equivalently,
$$
g_2(|A^{(1)}-P_1^{(1)}|)=g_2(|A^{(1)}-P_2^{(1)}|)+g_1(|P_1^{(1)}-P_2^{(1)}).
$$
This Cauchy's equation also implies
$$
g_2(x)=x,~\mbox{for $x\ge0$}.
$$
\item Case $(2.2)$: $A^{(2)}\ne0$, $P_1^{(2)}=0$. In this case we take $P_2=(A^{(1)},0)$, the projection of $A$ onto the horizontal axis. Therefore by the construction of $\mathcal C_{\tilde d_2}(P_1,P_2)$ and
\begin{eqnarray*}
\tilde d_2(A,P_1)&=&\tilde d_2(A,P_2)+\tilde d_2(P_1,P_2)\\
&=&|A^{(2)}|+g_2(|P_1-P_2|)\\
&=&|A^{(2)}|+|P_1^{(1)}-A^{(1)}|.
\end{eqnarray*}
\item Case $(2.3)$: $A^{(2)}\ne0$, $P_1^{(2)}\ne 0$. In this case we take $P_2=(A^{(1)},0)$, the projection of $A$ onto the horizontal axis. Therefore by the construction of $\mathcal C_{\tilde d_2}(P_1,P_2)$ and
\begin{eqnarray*}
\tilde d_2(A,P_1)&=&\tilde d_2(A,P_2)+\tilde d_2(P_1,P_2)\\
&=&|A^{(2)}|+|P_1^{(2)}|+|P_1^{(1)}-P_2^{(1)}|\\
&=&|A^{(2)}|+|P_1^{(1)}-A^{(1)}|+|P_1^{(2)}|,
\end{eqnarray*}
we obtain that in Case $(2)$,
$$
\tilde d_2(A,P_1)=|A^{(2)}|+|A^{(1)}-P_1^{(1)}|+|P_1^{(2)}|.
$$
Finally one obtains for any $x,y\in\mathbb R$,
\begin{eqnarray*}
\tilde d_2(x,y)&=&\left\{
\begin{array}{lll}
&|x_2-y_2|;&~\mbox{if $x_1=y_1$};\\
&|x_2|+|x_1-y_1|+|y_2|;&~\mbox{if $x_1\ne y_1$}
\end{array}\right.\\
&=&d_2(x,y).~\square
\end{eqnarray*}
\end{description}
\section{An Application: Brownian Motion Indexed by $\mathbb R$-tree}
It should be noted that, a tree metric can be also identified by the metric segments $[A,B]$, since a uniquely geodesic metric space $(M,d)$ is a tree if and only if $[A,B]\cap[B,C]\cap[A,C]=\{O\}$ for all distinct $A,B,C\in M$. However, rather than using metric segments, the sets $\mathcal C(P_1,P_2)$ allow to capture the features of a Gaussian field, which has very important and interesting applications in the domain of random fields. As an example, Inoue and Nota (1982) \cite{Inoue} studied some classes of Gaussian fields on $(\mathbb R^n,|\cdot|)$ and represented them via the sets of independent increments. Namely, some random field $\{X(t)\}_{t\in\mathbb R^n}$ can be identified by the sets: for any $P_1,P_2\in\mathbb R^n$,
$$
\mathcal F_X(P_1|P_2)=\left\{A\in\mathbb R^n:~Cov\left(X(A)-X(P_2),X(P_1)-X(P_2)\right)=0\right\}.
$$
The set $\mathcal F_X(P_1|P_2)$ satisfies the property that, the increments $X(A)-X(B)$ and $X(P_1)-X(P_2)$ are mutually independent if and only if $A,B\in \mathcal F_X(P_1|P_2)$. Here, we take a very similar idea of representation Gaussian fields, but work with a tree metric which is different from Euclidean distance $|\cdot|$. More precisely, we notice that a mean zero Brownian motion $B$ indexed by an $\mathbb R$-tree $(M,d)$ is well-defined, from its initial value $B(O)=0$ and its covariance structure
\begin{equation}
\label{Cov}
Cov(B(X),B(Y))=\frac{1}{2}\left(d(O,X)+d(O,Y)-d(X,Y)\right).
\end{equation}
In fact, since a tree metric $d$ is of negative type, the mapping
$$
(X,Y)\longmapsto\frac{1}{2}\left(d(O,X)+d(O,Y)-d(X,Y)\right)
$$
is thus positive definite so it well defines a Brownian motion covariance function.
Let $\mathcal C_d(P_1,P_2)$ be the one corresponding to $(M,d)$. Then by a similar study in \cite{Inoue}, we see that, not only $\mathcal C_d(P_1,P_2)$ can be used to identify the Bm $B$, but for any $X,Y\in \mathcal C_d(P_1,P_2)$, one has $B(X)-B(Y)$ and $B(P_1)-(P_2)$ are independent. This is due to the fact that, by using (\ref{Cov}) and the definition of $\mathcal C_d(P_1,P_2)$,
$$
\mathcal F_B(P_1|P_2)=\mathcal C_d(P_1,P_2),~\mbox{for any}~P_1,P_2\in\mathbb R^n.
$$
Hence
$$
X,Y\in\mathcal C_d(P_1,P_2)
$$
implies
$$
Cov\left(B(X)-B(Y),B(P_1)-B(P_2)\right)=0.
$$
As a consequence $\{\mathcal C_d(P_1,P_2)\}_{P_1,P_2\in M}$ captures all sets of independent increments of $\{B(X)\}_{X\in(M,d)}$. By this way one creates a new strategy to detect and simulate Brownian motion indexed by an $\mathbb R$-tree.
\subsection{Identification of Brownian Motions Indexed by $\mathbb R$-trees}
Let $\{B(X)\}_{X\in(\mathbb R^n,d)}$ be a zero mean Brownian motion indexed by an $\mathbb R$-tree. Namely, $\mathbb E(B(X))=0$ for all $X\in\mathbb R^n$ and there exists an initial point $O$ such that (\ref{Cov}) holds. Then the theorems below easily follow from Theorem \ref{inverseEx1} and Theorem \ref{inverseEx2} respectively.
\begin{thm}
\label{inverseEx3}
The following statements are equivalent:
\begin{description}
\item[(i)] $d=d_1$.
\item[(ii)] For any $P_1,P_2\in\mathbb R^n$, $\mathcal F_B(P_1|P_2)=\mathcal C_{d_1}(P_1,P_2)$.
\end{description}
\end{thm}
\begin{thm}
\label{inverseEx4}
For Brownian motion indexed by $(\mathbb R^2,d_2)$, the following statements are equivalent:
\begin{description}
\item[(i)] $d=d_2$.
\item[(ii)] For any $P_1,P_2\in\mathbb R^2$, $\mathcal F_B(P_1|P_2)=\mathcal C_{d_2}(P_1,P_2)$.
\end{description}
\end{thm}
\subsection{Simulation of Brownian Motion Indexed by $\mathbb R$-tree}
Let us consider a Brownian motion $B$ indexed by a tree $(\mathbb R^2,d_1)$ (recall that $d_1$ denotes radial metric) as an example. An interesting question in statistics is to simulate such a Brownian motion. More precisely, the issue is how can we generate the sample path $\{B(A_1),\ldots,B(A_n)\}$, for any different $A_1,\ldots,A_n\in(\mathbb R^2,d_1)$? In this section, we propose a new approach, which relies on the set $\mathcal F_B(P_1|P_2)$.
The following proposition shows, in some special case, the simulation could be particularly simple.
\begin{prop}
\label{simulation}
For any $A_1,\ldots,A_n\in(\mathbb R^2,d_1)$, there exists a permutation $\sigma\in S_n$ ($S_n$ denotes the group of permutations of $\{1,2,\ldots,n\}$) and an integer $q\ge1$ with $n_1+n_2+\ldots+n_q=n$, such that
\begin{eqnarray}
\label{IndepIncre}
&&\left(B(A_{\sigma(1)}),\ldots,B(A_{\sigma(n_1)})\right),\left(B(A_{\sigma(n_1+1)}),\ldots,B(A_{\sigma(n_1+n_2)})\right),\nonumber\\
&&\ldots,\big(B(A_{\sigma(n_1+\ldots+n_{q-1}+1)}),\ldots,B(A_{\sigma(n)})\big)
\end{eqnarray}
are independent, and for each group, i.e., for $1\le l\le q$,
\begin{equation}
\label{sigma}
\big(B(A_{\sigma(n_1+\ldots+n_{l-1}+1)}),\ldots,B(A_{\sigma(l)})\big)
\end{equation}
has independent increments.
\end{prop}
\textbf{Proof.} It suffices to provide a such $\sigma$. We first transform $A_1,\ldots,A_n$ to their polar coordinates representations. For each $A_k$ where $k\in\{1,\ldots,n\}$, there exists $r_k\in[0,+\infty)$ and $\theta_k\in[0,2\pi)$ such that $A_k=r_ke^{i \theta_k}$. The following approach provides a permutation $\sigma$ satisfying (\ref{IndepIncre}): we choose $\sigma\in S_n$ such that
$$
\theta_{\sigma(1)}=\ldots=\theta_{\sigma(n_1)}<\theta_{\sigma_{n_1+1}}=\ldots=\theta_{\sigma_{n_1+n_2}}<\ldots<\theta_{\sigma_{n_1+\ldots+n_{q-1}+1}}=\ldots=\theta_{\sigma(n)}
$$
with $n_1+\ldots+n_q=n$ and for each group $\displaystyle \sigma\big(\sum_{m=1}^ln_m+1\big),\ldots,\displaystyle \sigma\big(\sum_{m=1}^{l+1}n_m\big)$,
$$
r_{ \sigma(\sum_{m=1}^ln_m+1)}\le r_{ \sigma(\sum_{m=1}^ln_m+2)}\le \ldots \le r_{ \sigma(\sum_{m=1}^{l+1}n_m)}.
$$
To show (\ref{IndepIncre}) and (\ref{sigma}), on one hand, by Theorem \ref{inverseEx3}, for each $l=1,\ldots,n$, the elements $\{A_k\}_{k=\sigma(\sum_{m=1}^ln_m+1),\ldots,\sigma(\sum_{m=1}^{l+1}n_m) }$ are on the same ray so they have independent increments. On the other hand, the random vectors
\begin{eqnarray*}
&&\left(B(A_{\sigma(1)}),\ldots,B(A_{\sigma(n_1)})\right),\left(B(A_{\sigma(n_1+1)}),\ldots,B(A_{\sigma(n_1+n_2)})\right),\\
&&\big(B(A_{\sigma(n_1+\ldots+n_{q-1}+1)}),\ldots,B(A_{\sigma(n)})\big)
\end{eqnarray*}
are independent, due to the fact that for $X$, $Y$ on different rays,
\begin{eqnarray*}
Cov(B(X),B(Y))&=&\frac{1}{2}\left(d_1(X,0)+d_1(Y,0)-d_1(X,Y)\right)\\
&=&\frac{1}{2}\left(|X|+|Y|-(|X|+|Y|)\right)=0.~\square
\end{eqnarray*}
Proposition \ref{simulation} leads to the following simulation algorithm.
\subsubsection{Algorithm of Simulating Brownian Motion Indexed by $(\mathbb R^2,d_1)$:}
If $A_1,\ldots,A_n$ verify the assumption given in Proposition \ref{simulation}, then
\begin{description}
\item Step $1$: Determine $\sigma\in S_n$ and $q\ge 1$ such that
\begin{eqnarray*}
&&\left(B(A_{\sigma(1)}),\ldots,B(A_{\sigma(n_1)})\right),\left(B(A_{\sigma(n_1+1)}),\ldots,B(A_{\sigma(n_1+n_2)})\right),\nonumber\\
&&\ldots,\big(B(A_{\sigma(n_1+\ldots+n_{q-1}+1)}),\ldots,B(A_{\sigma(n)})\big)
\end{eqnarray*}
are independent, and each vector has independent increments.
\item Step $2$: Generate $n$ independent zero mean Gaussian random variables $Z_1,\ldots,Z_n$, with
$$
Var(Z_k)=\left\{
\begin{array}{ll}
d_1(0,A_{\sigma(k)})&~\mbox{if $k=\sum\limits_{m=1}^{l}n_m+1$ for some $m$}\\
d_1(A_{\sigma(k-1)},A_{\sigma(k)})&~\mbox{otherwise}.
\end{array}\right.
$$
\item Step $3$: For $j=1,\ldots,n$, set
$$
B(A_{\sigma(j)})=\sum_{k=\sum_{m=1}^{l}n_m+1}^{j}Z_k,~\mbox{if $j\in \left\{\sum\limits_{m=1}^{l}n_m+1,\ldots,\sum\limits_{m=1}^{l+1}n_m\right\}$}.
$$
\end{description}
Now let us study the simulation of Brownian motion $B$ indexed by $(\mathbb R^2,d_2)$, an $\mathbb R$-tree with river metric. Similar to Proposition \ref{simulation}, we have the following proposition:
\begin{prop}
\label{simulation1}
Given $n$ points vertically and horizontally labelled, i.e., for $A_1,\ldots,A_n\in(\mathbb R^2,d_2)$ such that
$$
\{(0,0),(A_1^{(1)},0),\ldots,(A_n^{(1)},0)\}\subset \{A_1,\ldots,A_n\}
$$
and
\begin{eqnarray*}
&&A_{1}^{(1)}=\ldots=A_{n_1}^{(1)}<A_{n_1+1}^{(1)}=\ldots=A_{n_1+n_2}^{(1)}\\
&&<\ldots<A_{n_1+\ldots+n_{p-1}+1}^{(1)}=\ldots=A_{n_1+\ldots+n_{p}}^{(1)}< 0\\
&&\le \ldots<A_{n_1+\ldots+n_{q-1}+1}^{(1)}=\ldots=A_{n}^{(1)}
\end{eqnarray*}
with $n_1+\ldots+n_q=n$ and for each group $\sum_{m=1}^ln_m+1,\ldots,\sum_{m=1}^{l+1}n_m$,
$$
A_{\sum_{m=1}^ln_m+1}^{(2)}\le A_{\sum_{m=1}^ln_m+2}^{(2)}\le\ldots\le A_{\sum_{m=1}^{l+1}n_m}^{(2)}.
$$
Then there exists a sequence of independent Gaussian variables $(Z_1,\ldots,Z_{n-1})$ such that
\begin{equation}
\label{AZ}
\left(B(A_1),\ldots,B(A_n)\right)=\left(\sum_{k\in I_1}Z_k,\ldots,\sum_{k\in I_n}Z_k\right)~\mbox{in distribution}
\end{equation}
for some $I_k\subset\{1,\ldots,n\}$ for any $k=1,\ldots,n$.
\end{prop}
\textbf{Proof.} We define for $k=1,\ldots,n-1$,
\begin{equation}
\label{sigma1}
Z_k=\left\{\begin{array}{ll}
B(A_{k+1})-B(A_k)&\mbox{if $A_{k+1}^{(1)}=A_k^{(1)}$}\\
B((A_{k+1}^{(1)},0))-B((A_k^{(1)},0))&\mbox{if $A_{k+1}^{(1)}>A_k^{(1)}$}.
\end{array}\right.
\end{equation}
From Theorem \ref{inverseEx4}, we see $(Z_1,\ldots,Z_{n-1})$ is a sequence of independent random variables. Now we are going to find $I_1,\ldots,I_n$ such that (\ref{AZ}) holds true. Let's consider a directed graph $G=(V,E)$, with the set of vertices
$$
V=\{A_1,\ldots,A_n\}
$$
and the set of edges
$$
E=\{e_1,\ldots,e_{n-1}\},
$$
where
\begin{equation}
\label{e}
e_k=\left\{\begin{array}{ll}
\overrightarrow{A_kA_{k+1}}&\mbox{if $A_{k+1}^{(1)}=A_k^{(1)}\ge0$}\\
\overrightarrow{(A_{k}^{(1)},0)(A_{k+1}^{(1)},0)}&\mbox{if $A_{k+1}^{(1)}>A_k^{(1)}\ge0$}\\
\overrightarrow{A_kA_{k-1}}&\mbox{if $A_{k-1}^{(1)}=A_k^{(1)}<0$}\\
\overrightarrow{(A_{k}^{(1)},0)(A_{k-1}^{(1)},0)}&\mbox{if $A_{k-1}^{(1)}<A_{k}^{(1)}<0$}.
\end{array}\right.
\end{equation}
We denote by $A_{n_0}=(0,0)$. For $k=1,\ldots,n$, let $P_k$ be the shortest path from $A_{n_0}$ to $A_k$ in $G$. Namely, there exists a set $\{k_{1},\ldots,k_{\psi(k)}\}\subset\{1,\ldots,n\}$ such that
\begin{eqnarray*}
P_{k}&=&\left(\overrightarrow{A_{n_0}A_{k_{1}}},~\overrightarrow{A_{k_{1}}A_{k_{2}}},\ldots,\overrightarrow{A_{k_{\psi(k)-1}}A_{k_{\psi(k)}}}\right)\\
&=&(e_{j_1},\ldots,e_{j_{\psi(k)}}).
\end{eqnarray*}
Denote by $I_k=\{j_1,\ldots,j_{\psi(k)}\}$, then (\ref{AZ}) is satisfied for such a choice of $(I_k)_{k=1,\ldots,n}$. $\square$
Proposition \ref{simulation1} leads to the following simulation algorithm.
\subsubsection{Algorithm of Simulating Brownian Motion Indexed by $(\mathbb R^2,d_2)$:}
If $A_1,\ldots,A_n\in \mathbb R^2$, the following algorithm shows how to simulate $(B(A_1),\ldots,B(A_n))$:
\begin{description}
\item Step $1$:
Generate $n-1$ independent zero mean Gaussian random variables $Z_1,\ldots,Z_{n-1}$, with
$$
Var(Z_k)=\left\{\begin{array}{ll}
d_2(A_k,A_{k+1})&\mbox{if $A_{k+1}^{(1)}=A_k^{(1)}$}\\
d_2((A_{k+1}^{(1)},0),(A_k^{(1)},0))&\mbox{if $A_{k+1}^{(1)}>A_k^{(1)}$}.
\end{array}\right.
$$
\item Step $2$: For $k=1,\ldots,n$, determine $I_k$.
Finally,
$$
\left(B(A_k)\right)_{k=1,\ldots,n}=\left(\sum_{j\in I_k}Z_j\right)_{k=1,\ldots,n}~\mbox{in distribution}.
$$
\end{description}
|
2,877,628,090,193 | arxiv | \section{Introduction}
Complex networks are graphs representative of the intricate connections between elements in many natural and artificial systems \cite{strogatz,havlin,barabasi,physrep}, whose description in terms of statistical properties have been largely developed looking for a universal classification of them. However, when the networks are locally analyzed, some characteristics that become partially hidden in the global statistical description emerge. The most relevant is perhaps the discovery in many of them of {\em community structure}, meaning the existence of densely (or strongly) connected groups of nodes, with sparse (or weak) connections between these groups \cite{firstnewman}.
The study of the community structure helps to elucidate the organization of the networks and, eventually, could be related to the functionality of groups of nodes \cite{amaral}. The most successful solutions to the community detection problem, in terms of accuracy and computational cost required, are those based in the optimization of a quality function called {\em modularity} proposed by Newman and Girvan \cite{newgirvan} that allows the comparison of different partitioning of the network. The extension of modularity to weighted \cite{wnewman} and directed networks \cite{mesoscales,leicht} has been the first steps towards the analysis of the community structure in general networks.
Very often networks are defined from correlation data between elements. The common analysis of correlation matrices uses classical or advanced statistical techniques \cite{song}. Nevertheless an alternative analysis in terms of networks is possible. The network approach usually consists in filtering the correlation data matrix, by eliminating poorly correlated pairs according to a threshold, and by keeping unsigned the value of the correlation, producing a network of positive links and no self-loops (self-correlations). Recently, some authors pointed out the possibility to analyze these networks via spectral decomposition \cite{heimo1,heimo2}. We devise also the possibility to analyze them in terms of Newman's modularity to reveal the community structure (clusters) of the correlated data. However, any of these approaches can be misleading because of two facts: first, the sign of the correlation is important to avoid the mixing of correlated and anti-correlated data, and second, the existence of self-loops is critical for the determination of the community structure \cite{mesoscales}. Here we propose a method to extract the community structure in networks of correlated data, that accounts for the existence of signed correlations and self-correlations, preserving the original information. To this end, we extend the modularity to the most general case of directed, weighted and signed links. We will show the performance of our method in a real network of correlations between commercial activities, previously analyzed in \cite{jensen} using a Potts model.
\section{Generalization of modularity}
Given an undirected network partitioned into communities, the modularity of a given partition is, up to a multiplicative constant, the probability of having edges falling within groups in the network minus the expected probability in an equivalent (null case) network with the same number of nodes, and edges placed at random preserving the nodes' strength, where the strength of a node stands for the sum of the weights of its connections \cite{newanaly}. In mathematical form, modularity is expressed in terms of the weighted adjacency matrix $w_{ij}$, that represents the value of the weight in the link between $i$ and $j$ ($0$ if no link exists), as \cite{newanaly}
\begin{equation}
Q = \frac{1}{2w} \sum_i\sum_j \left(
w_{ij} - \frac{w_i w_j}{2w}
\right) \delta(C_i,C_j) \,,
\label{QW}
\end{equation}
where $C_i$ is the community to which node $i$ is assigned, the Kronecker delta function $\delta(C_i,C_j)$ takes the values, 1 if nodes $i$ and $j$ are into the same community, 0 otherwise, the strengths are $w_i=\sum_j w_{ij}$, and the total strength is $2w=\sum_i w_i =\sum_i \sum_j w_{ij}$.
The larger the modularity, the larger the deviation from the null case and the better the partitioning. Note that the optimization of the modularity cannot be performed by exhaustive search since the number of different partitions are equal to the Bell \cite{bell} or exponential numbers, which grow at least exponentially in the number of nodes $N$. Indeed, optimization of modularity is a NP-hard (Non-deterministic Polynomial-time hard) problem \cite{brandes}. Several authors have attacked the problem proposing different optimization heuristics \cite{newfast, clauset, rogernat, duch, pujol, newspect}.
To demonstrate the flaws of modularity when trying to extract the community structure of correlated data we show the following example. Suppose we have a network with a well defined community structure as the one presented in Fig.~\ref{toy}. Let us pretend that each community is indeed a functional community, in such a way that nodes in every group have different states. To simplify the mathematics we will consider that the nodes in community A are in a state $+1$, and nodes in community B are in a state $-1$. After, we define the correlation between these data as, for example, $R_{ij}=S_i S_j$, $S_i$ and $S_j$ being the corresponding states of nodes $i$ and $j$. The question is: can we infer communities A and B from the correlated data represented in matrix $R$? Applying modularity, the answer is negative. Let us sketch the proof. The matrix R is blockwise composed of submatrices $R_{AA}$, $R_{AB}$, $R_{BA}$, and $R_{BB}$. The blocks $R_{AA}$ and $R_{BB}$ are all valued $+1$, and $R_{AB}$ and $R_{BA}$ are valued $-1$. Any matrix of this form results in zero modularity for all partitions, since $R_{ij}=\frac{w_i w_j}{2w}$ for all pairs (see Eq.~\ref{QW}).
\begin{figure}[!t
\begin{center}
\begin{tabular}[b]{ccc}
\raisebox{-25pt}{\includegraphics[width=130pt]{squatri-net.eps}}
&
$\Rightarrow$\hspace{5pt}
&
\raisebox{-30pt}{\includegraphics[width=75pt]{squatri-cor.eps}}
\end{tabular}
\end{center}
\caption{Network with well-defined community structure and its correlation matrix.}
\label{toy}
\end{figure}
To reveal the community structure in the network presented in Fig.~\ref{toy} from
its correlation matrix, it is necessary to revise the formulation of modularity.
Let us suppose that we have a weighted undirected complex network with weights $w_{ij}$
as above. The relative strength $p_i$ of a node
\begin{equation}
p_i = \frac{w_i}{2 w}\,,
\end{equation}
may be interpreted as the probability that this node makes links to other
ones, if the network were random. This is precisely the approach taken
by Newman and Girvan to define the modularity null case term, which reads
\begin{equation}
p_i p_j = \frac{w_i w_j}{(2 w)^2}\,.
\end{equation}
The introduction of negative weights destroys this probabilistic interpretation
of $p_i$, since in this case the values of $p_i$ are not guaranteed to be between
zero and one. The problem is the implicit hypothesis that there is only one
unique probability to link nodes, which involves both positive and negative
weights. To solve this problem, we have to introduce two different probabilities to
form links, one for positive and the other for negative weights.
Let us formalize this approach. First, we separate the positive and negative
weights:
\begin{equation}
w_{ij} = w_{ij}^{+} - w_{ij}^{-}\,,
\end{equation}
where
\begin{eqnarray}
w_{ij}^{+} & = & \max\{0, w_{ij}\}\,, \\
w_{ij}^{-} & = & \max\{0, -w_{ij}\}\,.
\end{eqnarray}
The positive and negative strengths are given by
\begin{eqnarray}
w_i^{+} & = & \sum_j w_{ij}^{+}\,, \\
w_i^{-} & = & \sum_j w_{ij}^{-}\,,
\end{eqnarray}
and the positive and negative total strengths by
\begin{eqnarray}
2 w^{+} & = & \sum_i w_i^{+} = \sum_i \sum_j w_{ij}^{+}\,, \\
2 w^{-} & = & \sum_i w_i^{-} = \sum_i \sum_j w_{ij}^{-}\,.
\end{eqnarray}
Obviously,
\begin{equation}
w_i = w_i^{+} - w_i^{-}
\end{equation}
and
\begin{equation}
2 w = 2 w^{+} - 2 w^{-}\,.
\end{equation}
With these definitions at hand, the connection probabilities with
positive and negative weights are respectively
\begin{eqnarray}
p_i^{+} & = & \frac{w_i^{+}}{2 w^{+}}\,, \\
p_i^{-} & = & \frac{w_i^{-}}{2 w^{-}}\,.
\end{eqnarray}
Now, there are two terms which contribute to modularity: the first
one takes into account the deviation of actual positive weights
against a null case random network given by probabilities $p_i^{+}$,
and the other is its counterpart for negative weights. Thus, it
is useful to define
\begin{eqnarray}
Q^{+} & = & \frac{1}{2w^{+}} \sum_i \sum_j \left(
w_{ij}^{+} - \frac{w_i^{+} w_j^{+}}{2w^{+}}
\right) \delta(C_i,C_j)\,,
\\
Q^{-} & = & \frac{1}{2w^{-}} \sum_i \sum_j \left(
w_{ij}^{-} - \frac{w_i^{-} w_j^{-}}{2w^{-}}
\right) \delta(C_i,C_j)\,.
\end{eqnarray}
The total modularity must be a trade off between the tendency of
positive weights to form communities and that of negative weights
to destroy them. If we want that $Q^{+}$ and $Q^{-}$ contribute
to modularity proportionally to their respective positive and negative
strengths, the final expression for modularity $Q$ is
\begin{equation}
Q = \frac{2 w^{+}}{2 w^{+} + 2 w^{-}} Q^{+} -
\frac{2 w^{-}}{2 w^{+} + 2 w^{-}} Q^{-}\,.
\end{equation}
An alternative equivalent form for modularity $Q$ is
\begin{eqnarray}
Q = \frac{1}{2w^{+} + 2 w^{-}} & \displaystyle \sum_i \sum_j &
\left[ w_{ij} - \left(
\frac{w_i^{+} w_j^{+}}{2w^{+}}- \frac{w_i^{-} w_j^{-}}{2w^{-}}
\right) \right] \nonumber \\
&&\times \delta(C_i,C_j)\,.
\label{QWS}
\end{eqnarray}
The main properties of Eq.~(\ref{QWS}) are the following: without negative weights, the
standard modularity is recovered; modularity is zero when all nodes are together
in one community; and it is antisymmetric in the weights, i.e.
$Q(C,\{w_{ij}\}) = - Q(C,\{-w_{ij}\})$\,.
The extension to directed networks \cite{njp} is simply obtained by the substitutions
\begin{eqnarray}
w_{i}^{\pm} &\rightarrow & w_{i}^{\pm,\mbox{\scriptsize out}} = \sum_{k} w_{ik}\,,\\
w_{j}^{\pm} &\rightarrow & w_{j}^{\pm,\mbox{\scriptsize in}} = \sum_{k} w_{kj} \,.
\end{eqnarray}
\section{Comparison with other methods}
In Fig.~\ref{hexa} we show a simple example of a network for which the original Newman modularity Eq.~(\ref{QW}) and the Potts model in \cite{jensen} do not yield the expected partition in two communities, whereas our new modularity Eq.~(\ref{QWS}) succeeds. It consists in two cliques, formed by positive links, and connected by two edges, one positive and the other negative. All positive links have a weight $+1$, and the negative a weight $v<0$. Any size of the cliques greater than or equal to three does the job.
First, the Potts model in \cite{jensen} is based on a Hamiltonian which only takes into account the difference between positive and negative weights within the modules, and is equivalent to modularity but without the null case term. In the network Fig.~\ref{hexa}, if $|v|<1$, the strength between the two cliques is $1+v>0$, thus the Potts model is rewarded to join both cliques in the same module. Clearly, the absence of the null case is responsible of this incorrect result.
On the other hand, the original definition of modularity (Eq.~\ref{QW}), which does include a null case, was not designed to cope with negative weights. In this example, its optimal partition is again a single module containing all the nodes if the value of $|v|$ is greater than the number of positive links.
\begin{figure}[!t
\begin{center}
\includegraphics[width=200pt]{hexagons.eps}
\end{center}
\caption{Network with two well-defined communities. Solid lines correspond to positive links, and the dashed line to the only negative link, with weight $v<0$.}
\label{hexa}
\end{figure}
In \cite{traag} the authors propose an alternative definition of modularity for positive and negative links. Their work, also based on a Potts model representation of the network communities' assignment \cite{bornholdt}, is totally compatible with the definition found in the current work, and equivalent for the values of their parameters $\lambda = \gamma = 1$.
\section{Application to a real network}
We now turn to an example of community structure detection using our method in a specific social network. We deal with the spatial distribution of retail activities in the city of Lyon, thanks to data obtained at the Lyon's Commerce Chamber \footnote{The Commerce Chamber classifies retail activities according to commercial criteria derived from an experienced knowledge of the field. This classification is adopted here as reference.}. We have shown in \cite{jensen} how to transform data on locations into a matrix of correlated data, in this case of attractions/repulsions (i.e.\ positive and negative links) between retail activities. To compute the interaction between activities A and B, the idea is to compare the concentrations of B stores in the neighborhood of A stores to a reference concentration obtained by locating the B stores randomly. To compute the random reference, the idea \cite{duranton} is to locate the B stores on the array of {\em all existing} store sites. This is the best way to take into account automatically the geographical peculiarities of each town. The logarithm of the ratio of the actual concentration to the reference concentration gives the interaction coefficient, which is positive for attractions and negative for repulsions, as anticipated.
More precisely, the (self) interaction of $N_{A}$ A stores embedded in a larger set of $N_t$ locations is
\begin{equation}
\label{com-aaa}
a_{AA}(r) = \log_{10} \frac{N_t - 1}{N_A (N_A-1)} \sum_{i=1}^{N_A} \frac{ N_A
(A_i)}{N_t (A_i)} \,,
\end{equation}
where $N_A (A_i)$ and ${N_t (A_i)}$ represent the number of A stores and the
total number of stores in the neighborhood of store $A_i$, i.e. locations at a
distance smaller than {$r$}. Similarly, the coefficient characterizing the
spatial distribution of the $B_i$ around the $A_i$ is
\begin{equation}
\label{com-aab}
a_{AB}(r) = \log_{10} \frac{N_t - N_A}{N_A N_B} \sum_{i=1}^{N_A} \frac{ N_B
(A_i)}{N_t
(A_i) - N_A (A_i)} \,,
\end{equation}
where $N_A (A_i)$, $N_B (A_i)$ and $N_t (A_i)$ are respectively the $A$, $B$ and total number of locations in the neighborhood of point $A_i$ (not counting $A_i$). Both $a_{AA}$ and $a_{AB}$ are defined so that they take value 0 when there are no spatial correlations. In the case of the $a_{AB}$ coefficient, this means that the local $B$ spatial concentration is not perturbed, on average, by the presence of A stores, and is equal to the average concentration over the whole town, $\frac{N_B}{N_t-N_A}$. Only coefficients which deviate significantly from 0, using a Montecarlo sampling, are taken into account in the adjacency matrix. The final result of the analysis of the 11629 stores in Lyon is a directed network with 97 nodes (retail activities) and 1131 links, 715 positive and 416 negative.
We analyze the community structure of the resulting network using the modularity defined in Eq.~(\ref{QWS}). The optimization method used is Tabu search \cite{mesoscales} that for this case gave the highest modularity when compared to others \cite{jstat}. We perform a comparison between the different partitions obtained optimizing independently Eq.~(\ref{QW}) (resulting in 4~communities) and Eq.~(\ref{QWS}) (resulting in 6~communities), against the Lyon's Commerce Chamber retail activities classification (9~communities predefined). The similarity of the first two partitions to the third one is measured using three different indices, namely the Rand Index \cite{rand}, the Jaccard Index \cite{jaccard}, and the Normalized Mutual Information (NMI) \cite{strehl} (see Table~\ref{tab}). The larger their values, the more similar the partitions are. All indices show a better performance of Eq.~(\ref{QWS}) in recovering the actual communities provided by the Lyon's Commerce Chamber. Note that in both modularities we have used all the positive and negative links. Therefore, the increase in performance can only be attributed to a proper use of the information embedded in the links.
\begin{table}[t]
\caption{Comparison between the different partitions and the Lyon Chamber of Commerce classification.}
\begin{tabular}{lcc}
\hline
\hline
& optimal partition & optimal partition\\
& of Eq.~(\ref{QW}) & of Eq.~(\ref{QWS}) \\
\hline
Rand Index & 0.6168 & 0.6952\\
Jaccard Index & 0.1336 & 0.1426\\
NMI & 0.1458 & 0.2310\\
\hline
\end{tabular}
\label{tab}
\end{table}
Our method is also helpful to understand the spatial organisation of retail stores. To interpret the information conveyed by the network links, we use of the z-score \cite{rogernat}. The basic idea consists in computing the z-score (Z) of the internal strength of each node with respect to the average internal strength of the community to which is assigned. To be consistent with our approach along the paper both quantities should be evaluated consistently with the sign of the interactions and with the directionality of links, then
\begin{equation}
Z^{\pm,\mbox{\scriptsize in/out}}_{i} =
\frac{w^{\pm,\mbox{\scriptsize in/out}}_{i,\mbox{\scriptsize int}} -
\langle w^{\pm,\mbox{\scriptsize in/out}}_{\mbox{\scriptsize int}}\rangle}
{\sigma(w^{\pm,\mbox{\scriptsize in/out}}_{\mbox{\scriptsize int}})} \,,
\label{z}
\end{equation}
\noindent where subindices `int' express that links are restricted within the community to which node $i$ belongs to, `in/out' refer to the direction of links, and $\langle\cdots\rangle$ and $\sigma$ are the average and standard deviation of the corresponding variables, respectively.
Using the z-score we can answer some questions about the role of nodes in their communities. For example, one can study, for each community, which are the most attractive retailers (max $Z^{+,\mbox{\scriptsize out}}$), the most repulsive retailers (max $Z^{-,\mbox{\scriptsize out}}$), the most attracted retailers (max $Z^{+,\mbox{\scriptsize in}}$), and the most repelled retailers (max $Z^{-,\mbox{\scriptsize in}}$). In Table~\ref{tab2} we show the three highest results of these z-scores obtained for the largest community found (34 retail activities). This group gathers the proximity stores, which means mainly food stores. Here are some examples of the understanding of the spatial organisation of retail stores allowed by our method. Sports facilities and funeral services are peculiar because they strongly attract (and are attracted) by some specific activities that go along with them almost systematically, e.g.\ car repairs and small hardware stores. Gas stations enjoy a paradoxical situation in this group, since they represent the most attracted and the most repelled activity. There is an interesting commercial interpretation of this paradox: gas stations tend to have the most specific commercial environment, strongly attracting some of the group's activities (such as supermarkets) and being strongly repelled by others which however are in the proximity store group (for example, butchers or cake shops stores almost never have gas stations close to them). Dairy products and cake shops strongly repel some specific of the activities that belong to their same group, such as car repairs or firm's restaurants.
\begin{table}[t]
\caption{Roles of retailers within communities.}
\begin{tabular} {ccccc}
\hline
\hline
&+ attractive & + repulsive& + attracted & + repelled \\
\hline
& Funeral Services & Dairy products & Gas Station & Gas Station \\
& Sports facility & Cake shop & Sports facility & Flea market \\
& Car dealer & Drugstore & Funeral Services & Car dealer \\
\hline
\end{tabular}
\label{tab2}
\end{table}
\section{Conclusions}
Summarizing, we have proposed a new formulation of modularity that allows for the analysis of any complex network, in general with links directed, weighted, signed and with self-loops, preserving the original probabilistic semantics of modularity. With this definition one can analyze networks arising from correlated data without necessarily symmetrizing the network, skipping auto-correlation or considering only the unsigned value of the correlations. We devise that other methods are also likely to be appropriate for this task, after its pertinent adaptation, for example the analysis via clique percolation \cite{palla}, or specifically methods based on the minimization of the energy function of an equivalent spin glass system, were weighted signed links can be interpreted in terms of ferromagnetic and anti-ferromagnetic interactions between spins \cite{bornholdt}.
We have analyzed within the scope of the new modularity an interesting model of attraction-repulsion of retail stores in a large city, previously reported in \cite{jensen}. The results overcome those obtained using the original definition of modularity when compared to the Lyon Chamber of Commerce classification, and also point out the necessity of defining new roles of nodes based on directionality and sign of the weights of links, as we have proposed for the z-score.
\begin{acknowledgments}
We thank J. Borge for help with the simulations. This work has been partially supported by the Spanish DGICYT Project FIS2006-13321-C02-1. We gratefully acknowledge Christophe Baume and Bernard Gagnaire from Lyon's Commerce Chamber who have kindly provided the location data.
\end{acknowledgments}
|
2,877,628,090,194 | arxiv | \section{Introduction}\label{sec:intro}
Multi-robot collision avoidance in cluttered environments is a fundamental problem when deploying a team of autonomous robots for applications such as coverage \citep{Breitenmoser2016}, target tracking \citep{zhou2018resilient}, formation flying \citep{Zhu2019ICRA} and multi-view cinematography \citep{Nageli2017multiple}. Given the robot current states and goal locations, the objective is to plan a local motion for each robot to navigate towards its goal while avoiding collisions with other robots and obstacles in the environment. Most existing algorithms solve the problem in a deterministic manner, where the robot states and obstacle locations are perfectly known. Practically, however, robot states and obstacle locations are generally obtained by an estimator based on sensor measurements that have noise and uncertainty. Taking this uncertainty into consideration is of utmost importance for safe and robust multi-robot collision avoidance.
In this paper, we present a decentralized probabilistic approach for multi-robot collision avoidance under localization and sensing uncertainty that does not rely on communication. Our approach is built on the buffered Voronoi cell (BVC) method developed by \citet{Zhou2017}. The BVC method is designed for collision avoidance among multiple single-integrator robots, where each robot only needs to know the positions of neighboring robots. We extend the method into probabilistic scenarios considering robot localization and sensing uncertainties by mathematically formalizing a buffered uncertainty-aware Voronoi cell (B-UAVC). Furthermore, we consider static obstacles with uncertain locations in the environment. We apply our approach to double-integrator dynamics, differential-drive robots, and general high-order dynamical robots.
\subsection{Related Works}\label{sec:relatedWork}
\subsubsection{Multi-robot collision avoidance}
The problem of multi-robot collision avoidance has been well studied for deterministic scenarios, where the robots' states are precisely known. One of the state-of-the-art approaches is the reciprocal velocity obstacle (RVO) method \citep{VandenBerg2008}, which builds on the concept of velocity obstacles (VO) \citep{Fiorini1998}. The method models robot interaction pairwise in a distributed manner and estimates future collisions as a function of relative velocity. Based on the basic framework, RVO has been extended towards several revisions: the optimal reciprocal collision-avoidance (ORCA) method \citep{VanDenBerg2011} casting the problem into a linear programming formulation which can be solved efficiently, the generalized RVO method \citep{Bareiss2015} applying for heterogeneous teams of robots, and the $\varepsilon$-cooperative collision avoidance ($\varepsilon$CCA) method \citep{Alonso-Mora2018} accounting for the cooperation of nonholonomic robots. In addition to those RVO-based methods, the model predictive control (MPC) framework has also been widely used for multi-robot collision avoidance, which includes decentralized MPC \citep{Shim2003}, decoupled MPC \citep{Chen2015}, and sequential MPC \citep{Morgan2016, Luis2020}.
While those approaches typically require the robots position and velocity, or more detailed future trajectory information to be known among neighboring robots, the recent developed buffered Voronoi cell (BVC) method \citep{Zhou2017,pierson2020weighted} only requires the robots to know the
positions of other robots.
In this paper, we build upon the concept of BVC and extend it to probabilistic scenarios, where each robot only needs to estimate the positions of its neighboring robots.
\subsubsection{Collision avoidance under uncertainty}
Some of the above deterministic collision avoidance approaches have been extended to scenarios where robot localization or sensing uncertainty is considered.
Based on RVO, the COCALU method \citep{Claes2012} takes into account bounded localization uncertainty of the robots by constructing an error-bounded convex hull of the VO of each robot.
\citet{Gopalakrishnan2017} presents a probabilistic RVO method for single-integrator robots.
\citet{Kamel2017} presents a decentralized MPC where robot motion uncertainty is taken into account by enlarging the robots with their 3-$\sigma$ confidence ellipsoids.
A chance constrained MPC problem was formulated by \citet{Lyons2012} for planar robots, where rectangular regions were computed and inter-robot collision avoidance was transformed to avoid overlaps of those regions.
Using local linearization, \citet{Zhu2019RAL} proposed a chance constrained nonlinear MPC (CCNMPC) method to ensure that the probability of inter-robot collision is below a specified threshold.
Among these attempts to incorporate uncertainty into multi-robot collision avoidance, several limitations are observed.
Probabilistic VO-based methods are limited to systems with simple first-order dynamics, or limited to homogeneous teams of robots.
Probabilistic MPC-based methods typically demand communication of the planned trajectory of each robot to guarantee collision avoidance, which does not scale well with the number of robots in the system.
An alternative to communicating trajectories is to assume that all other robots move with constant velocity \citep{Kamel2017}, which has been shown to lead to collisions in cluttered environments \citep{Zhu2019RAL}.
Recently, \citet{luo2020multi} proposes probabilistic safety barrier certificates (PrSBC) to define the space of admissible control actions that are probabilistic safe, but it is only designed for single-integrator robots. In this paper, we define the probabilistic safe region for each robot directly based on the concept of buffered Voronoi cell (BVC).
\rebuttal{
The BVC method has also been extended to probabilistic scenarios by \cite{Wang2019}. Taking into account the robot measurement uncertainty of other robots, they present the probabilistic buffered Voronoi cell (PBVC) to assure a safety level given a collision probability threshold. However, since the PBVC of each robot does not have an analytic solution, they employ a sampling-based approach to approximate it. In contrast, our proposed B-UAVC has an explicit and analytical form, which is more efficient to be computed. Moreover, our B-UAVC can be incorporated with MPC to handle general nonlinear systems, while the PBVC method developed by \cite{Wang2019} cannot be directly applied within a MPC framework. }
\subsubsection{Spatial decomposition in motion planning}
Our method constructs a set of local safe regions for the robots, which decompose the workspace. Spatial decomposition is broadly used in robot motion planning. \citet{Deits2015} proposes the IRIS (iterative regional inflation by semi-definite programming) algorithm to compute safe convex regions among obstacles given a set of seed points. The algorithm is then used for UAV path planning \citep{deits2015efficient} and multi-robot formation control \citep{Zhu2019ICRA}. \citet{Liu2017} presents a simpler but more efficient iteratively inflation algorithm to compute a convex polytope around a line segment among obstacles and utilizes it to construct a safe flight corridor for UAV navigation \citep{tordesillas2019faster}. Similar safe flight corridors are constructed for trajectory planning of quadrotor swarms \rebuttal{\citep{Honig2018}}, by computing a set of max-margin separating hyperplanes between a line segment and convex polygonal obstacles. The max-margin separating hyperplanes are also used by \citet{Arslan2019} to construct a local robot-centric safe region in convex sphere worlds for sensor-based reactive navigation. While those spatial decomposition methods have shown successful application in robot motion planning, they all assume perfect knowledge on robots and obstacles positions. In this paper, we consider both the robot localization and obstacle position uncertainty and construct a local uncertainty-aware safe region for each robot.
\subsection{Contribution}
The main contribution of this paper is a decentralized and communication-free method for probabilistic multi-robot collision avoidance in cluttered environments. The method considers robot localization and sensing uncertainties and relies on the computation of buffered uncertainty-aware Voronoi cells (B-UAVC). At each time step, each robot computes its B-UAVC based on the estimated position and uncertainty covariance of itself, neighboring robots and obstacles, and plans its motion within the B-UAVC. Probabilistic collision avoidance is ensured by constraining each robot's motion to be within its corresponding B-UAVC, such that the inter-robot and robot-obstacle collision probability is below a user-specified threshold.
An earlier version of this paper was published by \citet{Zhu2019MRS}. In this version, three main additional extensions are developed: a) we further consider static obstacles with uncertain locations in the environment; b) we extend the approach to double-integrator dynamics and differential-drive robots and c) we provide thorough simulation and experimental results and analyses.
\subsection{Organization}
The remaining of this paper is organized as follows. In Section \ref{sec:preliminary} we present the problem statement and briefly summarize the concept of BVC. In Section \ref{sec:method_1} we formally introduce the buffered uncertainty-aware Voronoi cell (B-UAVC) and its construction method. We then describe how the B-UAVC is used for probabilistic multi-robot collision avoidance in Section \ref{sec:method_2}. Simulation and experimental results are presented in Section \ref{sec:sim_result} and Section \ref{sec:exp_result}, respectively. Finally, Section \ref{sec:conclsuion} concludes the paper.
\section{Preliminaries}\label{sec:preliminary}
Throughout this paper vectors are denoted in bold lowercase letters, $\mathbf{x}$, matrices in plain uppercase $M$, and sets in calligraphic uppercase, $\mathcal{S}$. \rebuttal{$I$ indicates the identity matrix.} A superscript $ \mathbf{x}^T $ denotes the transpose of $ \mathbf{x} $. $\norm{\mathbf{x}}$ denotes the Euclidean norm of $\mathbf{x}$ and $\norm{\mathbf{x}}_Q^{2} = \mathbf{x}^TQ\mathbf{x}$ denotes the weighted square norm. A hat $\hat{\mathbf{x}}$ denotes the mean of a random variable $\mathbf{x}$. $\textnormal{Pr}(\cdot)$ indicates the probability of an event and $p(\cdot)$ indicates the probability density function.
\subsection{Problem Statement}
Consider a group of $n$ robots operating in a $d$-dimensional space $\mathcal{W} \subseteq \mathbb{R}^d$, where $d~\rebuttal{\in}~\{2, 3\}$, populated with $m$ static polygonal obstacles.
For each robot $i \in \mathcal{I} = \{1,\dots,n\}$, $\mathbf{p}_i \in \mathbb{R}^d$ denotes its position, $\mathbf{v}_i = \dot{\mathbf{p}}_i$ its velocity and $\mathbf{a}_i = \dot{\mathbf{v}}_i$ its acceleration.
Let $\mathcal{G} = \{\mathbf{g}_1,\dots,\mathbf{g}_n\}$ denote their goal locations.
\rebuttal{A safety radius $r_s$ is given for all robots.}
We consider that the position of each robot is obtained by a state estimator and is described as a Gaussian distribution with covariance $\Sigma_i$, i.e. $\mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i, \Sigma_i)$.
\rebuttal{We also consider static polytope obstacles with known shapes but uncertain locations.
For each obstacle $o\in \mathcal{I}_o = \{1,\dots,m\}$, denote by $\hat{\mathcal{O}}_o \subset \mathbb{R}^d$ its occupied space when located at the expected (mean) position. $\hat{\mathcal{O}}_o$ is given by a set of vertices. Hence, the space actually occupied by the obstacle can be written as $\mathcal{O}_o = \{\mathbf{x}+\mathbf{d}_o~|~\mathbf{x}\in\hat{\mathcal{O}}_o,\mathbf{d}_o \sim \mathcal{N}(0, \Sigma_o) \} \subset \mathbb{R}^d$, where $\mathbf{d}_o $ is the uncertain translation of the obstacle's position, which has a zero mean and covariance $\Sigma_o$.
}
A robot $i$ in the group is collision free with another robot $j$ if their distance is greater than the sum of their radii, i.e. \rebuttal{$\tn{dis}(\mathbf{p}_i,\mathbf{p}_j) \geq 2r_s$}
and with the obstacle $o$ if the minimum distance between the robot and the obstacles is larger than its radius, i.e. \rebuttal{$\tn{dis}(\mathbf{p}_i, \mathcal{O}_o) \geq r_s$}.
\rebuttal{
The distance function $\tn{dis}(\cdot)$ between a robot with another robot or an obstacle are defined as $\tn{dis}(\mathbf{p}_i,\mathbf{p}_j) = \norm{\mathbf{p}_i - \mathbf{p}_j}$, and $\tn{dis}(\mathbf{p}_i, \mathcal{O}_o) = \min_{\mathbf{p} \in \mathcal{O}_o}\norm{\mathbf{p}_i - \mathbf{p}}$, respectively.
}
Note that the robots' and obstacles' positions are random variables following Gaussian distributions, which have an infinite support. Hence, the collision-free condition can only be satisfied in a probabilistic manner, which is defined as a chance constraint as follows.
\begin{definition}[Probabilistic Collision-Free]\label{def:pcollfree}
A robot $i$ at position $\mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i, \Sigma_i)$ is probabilistic collision-free with a robot $j$ at position $\mathbf{p}_j \sim \mathcal{N}(\hat{\mathbf{p}}_j, \Sigma_j)$ and an obstacle $o$ at position $\mathbf{p}_o \sim \mathcal{N}(\hat{\mathbf{p}}_o, \Sigma_o)$ if
\begin{align}
\textnormal{Pr}(\rebuttal{\tn{dis}(\mathbf{p}_i,\mathbf{p}_j) \geq 2r_s}) &\geq 1 - \delta, ~~\forall j\in\mathcal{I}, j\neq i, \label{eq:chanceConRobot} \\
\textnormal{Pr}(\rebuttal{\tn{dis}(\mathbf{p}_i, \mathcal{O}_o) \geq r_s}) &\geq 1 - \delta, ~~\forall o \in \mathcal{I}_o, \label{eq:chanceConObs}
\end{align}
where $\delta$ is the collision probability threshold for inter-robot and robot-obstacle collisions.
\end{definition}
The objective of probabilistic collision avoidance is to compute a local motion plan, $\mathbf{u}_i$, for each robot in the group, that respects its kinematic and dynamical constraints, makes progress towards its goal location, and is probabilistic collision free with other robots as well as obstacles in the environment.
In this paper, we first consider single-integrator dynamics for the robots,
\begin{equation}
\dot{\mathbf{p}}_i = \mathbf{u}_i,
\end{equation}
and then extend it to double-integrator systems, differential-drive robots and robots with general high-order dynamics.
\subsection{Buffered Voronoi Cell}\label{subsec:vc}
The key idea of our proposed method is to compute an uncertainty-aware collision-free region for each robot in the system, which is a major extension of the deterministic buffered Voronoi cell (BVC) method \citep{Zhou2017,pierson2020weighted}. In this section, we briefly describe the concept of BVC.
For a set of deterministic points $(\mathbf{p}_1, \dots, \mathbf{p}_n) \in \mathbb{R}^d$, the standard Voronoi cell (VC) of each point $i \in \mathcal{I}$ is defined as \citep{Okabe2009}
\begin{equation}\label{eq:vcDef}
\mathcal{V}_i = \{\mathbf{p}\in\mathbb{R}^d : \norm{\mathbf{p}- \mathbf{p}_i} \leq \norm{\mathbf{p} - \mathbf{p}_j}, \forall j\neq i \},
\end{equation}
which can also be written as
\begin{equation}\label{eq:vcPlane}
\mathcal{V}_i = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{p}_{ij}^T\mathbf{p} \leq \mathbf{p}_{ij}^T\frac{\mathbf{p}_i+\mathbf{p}_j}{2}, \forall j\neq i \},
\end{equation}
where $\mathbf{p}_{ij} = \mathbf{p}_j - \mathbf{p}_i$.
It can be observed that $\mathcal{V}_i$ is the intersection of a set of hyperplanes which separate point $i$ with any other point $j$ in the group, as shown in Fig. \ref{subfig:vc}. Hence, VC can be obtained by computing the separating hyperplanes between each pair of points.
To consider the footprints of robots, a buffered Voronoi cell for each robot $i$ is defined as follows:
\begin{equation}\label{eq:bvcPlane}
\mathcal{V}_i^b = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{p}_{ij}^T\mathbf{p} \leq \mathbf{p}_{ij}^T\frac{\mathbf{p}_i+\mathbf{p}_j}{2} - r_s\norm{\mathbf{p}_{ij}}, \forall j\neq i \},
\end{equation}
which is obtained by retracting the edges of the VC with a safety distance (buffer) \rebuttal{$r_s$}.
\rebuttal{In deterministic scenarios, if the robots are mutually collision-free, then the BVC of each robot is a non-empty set \citep{Zhou2017}.}
It is also trivial to prove that the BVCs are disjoint and if the robots are within their corresponding BVCs individually, they are collision free with each other. Using the concept of BVC, \citet{Zhou2017} proposed a control policy for a group of single-integrator robots whose control inputs are velocities. Each robot can safely and continuously navigate in its BVC, given that other robots in the system also follow the same rule. However, the guarantee does not hold for double-integrator dynamics or non-holonomic robots such as differential-drive robots.
\begin{figure*}[t]
\centering
\rebuttal{
\subfloat[]{\label{subfig:vc}
\includegraphics[width=.20\textwidth]{vc.pdf}}}
\quad
\rebuttal{
\subfloat[]{\label{subfig:uavc}
\includegraphics[width=.20\textwidth]{uvc.pdf}}}
\quad
\subfloat[]{\label{subfig:buavc1}
\includegraphics[width=.20\textwidth]{buvc1.pdf}}
\quad
\subfloat[]{\label{subfig:buavc2}
\includegraphics[width=.20\textwidth]{buvc2.pdf}}
\caption{Example of buffered uncertainty-aware Voronoi cells (B-UAVC). Blue dots are robots; blue dash-dot ellipses indicate the 3-$\sigma$ confidence ellipsoid of the position uncertainty. (a) Deterministic Voronoi cell (VC, the boundary in \rebuttal{gray solid} line). (b) Uncertainty-aware Voronoi cell based on the best linear separators (UAVC, the boundary in blue dashed line). (c) UAVC with robot raidus buffer (the boundary in green solid line). (d) Final B-UAVC with robot radius and collision probability buffer (the boundary in red solid line). }\label{fig:buavc}
\end{figure*}
\subsection{Shadows of Uncertain Obstacles}
To account for uncertain obstacles in the environment, we rely on the concept of obstacle shadows introduced by \citet{Axelrod2018}. The $\epsilon$-shadow is defined as follows:
\begin{definition}[$\epsilon$-Shadow]\label{def:shadow}
A set $\mathcal{S}_o \subseteq \mathbb{R}^d$ is an $\epsilon$-shadow of an uncertain obstacle $\mathcal{O}_o$ if the probability $\textnormal{Pr}(\mathcal{O}_o \subseteq \mathcal{S}_o) \geq 1 - \epsilon$.
\end{definition}
Geometrically, an $\epsilon$-shadow is a region that contains the uncertain obstacle with probability of at least $1 - \epsilon$, which can be non-unique. For example $\mathcal{S}_o = \mathbb{R}^d$ is an $\epsilon$-shadow of any uncertain obstacle. To preclude this trivial case, the maximal $\epsilon$-shadow is defined:
\begin{definition}[Maximal $\epsilon$-Shadow]\label{def:max_shadow}
A set $\mathcal{S}_o \subseteq \mathbb{R}^d$ is a maximal $\epsilon$-shadow of an uncertain obstacle $\mathcal{O}_o$ if the probability $\textnormal{Pr}(\mathcal{O}_o \subseteq \mathcal{S}_o) = 1 - \epsilon$.
\end{definition}
The above definition ensures that if there exists a maximal $\epsilon$-shadow $\mathcal{S}_o$ of the uncertain obstacle $\mathcal{O}_o$ that does not intersect the robot, i.e.
\rebuttal{$\tn{dis}(\mathbf{p}_i, \mathcal{S}_o) \geq r_s$},
then the collision probability between the robot and obstacle is below $\epsilon$, i.e. \rebuttal{$\textnormal{Pr}({\tn{dis}}(\mathbf{p}_i, \mathcal{O}_o) \geq r_s) \geq 1-\epsilon$}.
\rebuttal{Note that the maximal $\epsilon$-shadow may also be non-unique.
In this paper, we employ the method proposed by \cite{Dawson2020IROS} to construct such shadows.
Recall that the uncertain obstacle $\mathcal{O}_o$ is related to the nominal geometry $\hat{\mathcal{O}}_o$ by $\mathcal{O}_o = \{ \mathbf{x}+\mathbf{d}_o~|~ \mathbf{x}\in\hat{\mathcal{O}}_o, \mathbf{d}_o \sim \mathcal{N}(0, \Sigma_o) \}$.}
To construct the maximal $\epsilon$-shadow, we first define the following ellipsoidal set
\begin{equation}\label{eq:shadowset}
\mathcal{D}_o = \{ \mathbf{d}: \mathbf{d}^T\Sigma_o^{-1}\mathbf{d} \leq F^{-1}(1-\epsilon) \},
\end{equation}
where $F^{-1}(\cdot)$ is the inverse of the cumulative distribution function (CDF) of the chi-squared distribution with $d$ degrees of freedom.
Next, Let
\begin{equation}\label{eq:shadowobs}
\mathcal{S}_o = \hat{\mathcal{O}}_o + \mathcal{D}_o \rebuttal{=\{\mathbf{x} + \mathbf{d}~|~\mathbf{x}\in\hat{\mathcal{O}}_o,\mathbf{d}\in\mathcal{D}_o\}},
\end{equation}
be the Minkowski sum of \rebuttal{the nominal obstacle shape $\hat{\mathcal{O}}_o$ and the ellipsoidal set $\mathcal{D}_o$.
Then, we have the following lemma \citep{Axelrod2018} and theorem \citep{Dawson2020IROS}:
\begin{lemma}\label{lemma:probability_Do}
Let $\mathbf{d}_o \sim \mathcal{N}(0, \Sigma_o) \in \mathbb{R}^d$ and $\mathcal{D}_o = \{ \mathbf{d}:\mathbf{d}^T\Sigma_o^{-1}\mathbf{d} \leq F^{-1}(1-\epsilon) \} \subset \mathbb{R}^d$, then $\textnormal{Pr}(\mathbf{d}_o \in \mathcal{D}_o) = 1 - \epsilon$.
\end{lemma}
\begin{theorem}\label{theorem:maximal_shadow}
$\mathcal{S}_o$ is a maximal $\epsilon$-shadow of $\mathcal{O}_o$.
\end{theorem}
}
\rebuttal{Proofs of the above lemma and theorem are given in Appendix \ref{appendix:proof_probability_Do} and \ref{appendix:proof_maximal_shadow}.}
\section{Buffered Uncertainty-Aware Voronoi Cells} \label{sec:method_1}
In this section, we formally introduce the concept of buffered uncertainty-aware Voronoi cells (B-UAVC) and give its construction method.
\subsection{Definition of B-UAVC}\label{subsec:defBUAVC}
Our objective is to obtain a probabilistic safe region for each robot in the workspace given the robots and obstacles positions, and taking into account their uncertainties.
\begin{definition}[Buffered Uncertainty-Aware \\Voronoi Cell]\label{def:buavc}
Given a team of robots $i \in \{1,\dots,n\}$ with positions mean $\hat{\mathbf{p}}_i \in \mathbb{R}^d$ and covariance $\Sigma_i \in \mathbb{R}^{d\times d}$, and a set of convex polytope obstacles $o \in \{1,\dots,m\}$ with known shapes and locations mean $\hat{\mathbf{p}}_o \in \mathbb{R}^d$ and covariance $\Sigma_o \in \mathbb{R}^{d\times d}$, the buffered uncertainty-aware Voronoi cell (B-UAVC) of each robot is defined as a convex polytope region:
\begin{align}
\mathcal{V}_i^{u,b} = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{a}_{ij}^T\mathbf{p} &\leq b_{ij} - \beta_{ij}, \forall j\neq i, j\in\mathcal{I}, \\ \tn{and~~} \mathbf{a}_{io}^T\mathbf{p} &\leq b_{io} - \beta_{io}, \forall o\in\mathcal{I}_o \},
\end{align}
such that the probabilistic collision free constraints in Definition \ref{def:pcollfree} are satisfied.
\end{definition}
In the above B-UAVC definition, $\mathbf{a}_{ij}, \mathbf{a}_{io} \in \mathbb{R}^d$ and $b_{ij}, b_{io} \in \mathbb{R}$ are parameters of the hyperplanes that separate the robot from other robots and obstacles, which results in a decomposition of the workspace. $\beta_{ij}$ and $\beta_{io}$ are additional buffer terms added to retract the decomposed space for probabilistic collision avoidance. Accordingly, we further define
\begin{align}\label{eq:uavc}
\mathcal{V}_i^{u} = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{a}_{ij}^T\mathbf{p} &\leq b_{ij}, \forall j\neq i, j\in\mathcal{I}, \\ \tn{and~~} \mathbf{a}_{io}^T\mathbf{p} &\leq b_{io}, \forall o\in\mathcal{I}_o \},
\end{align}
that does not include buffer terms to be the uncertainty-aware Voronoi cell (UAVC) of robot $i$.
It can be observed the UAVC and B-UAVC of robot $i$ are the intersection of the following:
\begin{enumerate}
\item $n-1$ half-space hyperplanes separating robot $i$ from robot $j$ for all $j\neq i, j\in \mathcal{I}$;
\item $m$ half-space hyperplanes separating robot $i$ from obstacle $o$ for all $o \in \mathcal{I}_o$.
\end{enumerate}
In the following, we will describe how to calculate the separating hyperplanes with parameters $(\mathbf{a}_{ij}, b_{ij})$ and $(\mathbf{a}_{io}, b_{io})$ that construct the UAVC and then the corresponding buffer terms $\beta_{ij}, \beta_{io}$ constructing the B-UAVC for probabilistic collision avoidance.
\subsection{Inter-Robot Separating Hyperplane}\label{subsec:robotPlane}
In contrast to only separating two deterministic points in Voronoi cells, we separate two uncertain robots with known positions mean and covariance. To achieve that, we rely on the concept of the best linear separator between two Gaussian distributions \citep{Anderson1962}.
Given $\mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i,\Sigma_i)$ and $\mathbf{p}_j \sim \mathcal{N}(\hat{\mathbf{p}}_j,\Sigma_j)$, consider a linear separator $\mathbf{a}_{ij}^T\mathbf{p} = b_{ij}$ where $\mathbf{a}_{ij} \in \mathbb{R}^d$ and $b_{ij} \in \mathbb{R}$. The separator classifies the points $\mathbf{p}$ in the space into two clusters: $\mathbf{a}_{ij}^T\mathbf{p} \leq b_{ij}$ to the first one while $\mathbf{a}_{ij}^T\mathbf{p} > b_{ij}$ to the second. The separator parameters $\mathbf{a}_{ij}$ and $b_{ij}$ can be obtained by minimizing the maximal probability of misclassification.
The misclassification probability when $\mathbf{p}$ is from the first distribution is
\begin{equation*}
\begin{aligned}
\textnormal{Pr}_i(\mathbf{a}_{ij}^T\mathbf{p} > b_{ij}) &= \textnormal{Pr}_i\left(\frac{\mathbf{a}_{ij}^T\mathbf{p} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i}{\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}} > \frac{b_{ij} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i}{\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}}\right) \\
&= 1 - \Phi((b_{ij} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i)/\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}),
\end{aligned}
\end{equation*}
where $\Phi(\cdot)$ denotes the cumulative distribution function (CDF) of the standard normal distribution.
Similarly, the misclassification probability when $\mathbf{p}$ is from the second distribution is
\begin{equation*}
\begin{aligned}
\textnormal{Pr}_j(\mathbf{a}_{ij}^T\mathbf{p} \leq b_{ij}) &= \textnormal{Pr}_j\left(\frac{\mathbf{a}_{ij}^T\mathbf{p} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_j}{\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}} \leq \frac{b_{ij} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_j}{\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}}\right) \\
&= 1 - \Phi((\mathbf{a}_{ij}^T\hat{\mathbf{p}}_j - b_{ij})/\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}).
\end{aligned}
\end{equation*}
The objective is to minimize the maximal value of $\textnormal{Pr}_i$ and $\textnormal{Pr}_j$, i.e.
\begin{equation}\label{eq:best_linear_separator}
(\mathbf{a}_{ij}, b_{ij}) = \arg\underset{\mathbf{a}_{ij}\in\mathbb{R}^d,b_{ij}\in\mathbb{R}}{\min\max}(\textnormal{Pr}_i, \textnormal{Pr}_j),
\end{equation}
which can be solved using a fast minimax procedure.
\rebuttal{In this paper, we employ the procedure developed by \cite{Anderson1962} to compute the best linear separator parameters $\mathbf{a}_{ij}$ and $b_{ij}$. A brief summary of the procedure is presented in Appendix \ref{appendix:best_linear_separator}.}
\begin{remark}\label{rmk:sepPlane}
The best linear separator coincides with the separating hyperplane of Eq. (\ref{eq:vcPlane}) when $\Sigma_i = \Sigma_j = \sigma^2 I$. In this case, $\mathbf{a}_{ij} = \frac{2}{\sigma^2}(\hat{\mathbf{p}}_j - \hat{\mathbf{p}}_i)$ and $b_{ij} = \frac{1}{\sigma^2}(\hat{\mathbf{p}}_j-\hat{\mathbf{p}}_i)^T(\hat{\mathbf{p}}_i+\hat{\mathbf{p}}_j)$.
\end{remark}
\begin{remark}\label{rmk:sepMutual}
$\forall i\neq j \in \mathcal{I}, \mathbf{a}_{ji} = -\mathbf{a}_{ij}, b_{ji} = -b_{ij}$. This can be obtained according to the definition of the best linear separator.
\end{remark}
\begin{remark}\label{rmk:uavcNotFull}
In contrast to deterministic Voronoi cells, the UAVCs constructed from the best linear separators generally do not constitute a full tessellation of the workspace, i.e. $\bigcup_1^n\mathcal{V}_i^u \subseteq \mathcal{W}$, as shown in Fig. \ref{subfig:uavc}.
\end{remark}
\subsection{Robot-Obstacle Separating Hyperplane}\label{subsec:obsPlane}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{polygon_hyperplane.pdf}
\caption{Depiction of uncertainty-aware separating hyperplane calculation between a point and an arbitrary polytope obstacle with uncertain location. (Top left) A point and a polytope obstacle with uncertain location. (Top right) Effects of the transformation $W$ to normalize the error covariance. (Bottom left) $\epsilon$-shadow of the transformed obstacle and the max-margin separating hyperplane in the transformation space. (Bottom right) Inverse transformation to obtain the uncertainty-aware separating hyperplane. }%
\label{fig:poly_plane}%
\end{figure}
Our method to calculate the uncertainty-aware separating hyperplane between a robot and a convex polytope obstacle with uncertain location is illustrated in Fig. \ref{fig:poly_plane}. Given the mean position of the robot $\hat{\mathbf{p}}_i$ and the \rebuttal{uncertain obstacle $\mathcal{O}_o = \{\mathbf{x} + \mathbf{d}_o~|~\mathbf{x}\in\hat{\mathcal{O}}_o, \mathbf{d}_o \sim \mathcal{N}(0, \Sigma_o)$\}}, we first perform a linear coordinate transformation:
\begin{equation}\label{eq:coordinate_transformation}
W = (\sqrt{\Sigma_o})^{-1},
\end{equation}
Under the transformation, the robot mean position and obstacle information become
\begin{align}
{\hat{\mathbf{p}}}_i^W &= W\hat{\mathbf{p}}_i, \\
\hat{\mathcal{O}}_o^W &= W\hat{\mathcal{O}}_o, \\
{\mathbf{d}}_o^W &= W\mathbf{d}_o, \\
{\Sigma}_o^W &= W\Sigma_o W^T = I^{d\times d}.
\end{align}
\rebuttal{The transformed uncertain obstacle is then $\mathcal{O}_o^W = \{\mathbf{x}^W + \mathbf{d}_o^W~|~\mathbf{x}^W \in \hat{\mathcal{O}}_o^W, \mathbf{d}_o^W\sim\mathcal{N}(0,I)\}$.} Here we use the super-script ${\cdot}^W$ to indicate variables in the transformed space. Note that the obstacle position uncertainty covariance is normalized to an identity matrix under the transformation, as shown in Fig. \ref{fig:poly_plane} (Top right).
\rebuttal{This coordinate transformation technique to normalize the uncertainty covariance has also been applied to other motion planning under uncertainty works \citep{Hardy2013}.}
Then given the collision probability threshold $\delta$, we compute a $\epsilon$-shadow of the transformed uncertain obstacle $\mathcal{O}_o^W$ based on Eqs. (\ref{eq:shadowset})-(\ref{eq:shadowobs}):
\begin{align}
{\mathcal{D}}_o^W &= \{{\mathbf{d}}^W: {{\mathbf{d}}^{W}}^T{\mathbf{d}}^W \leq F^{-1}(1-\epsilon) \}, \label{eq:transD} \\
{\mathcal{S}}_o^W &= \hat{\mathcal{O}}_o^W + {\mathcal{D}}_o^W, \label{eq:transShadow}
\end{align}
where $\epsilon = 1-\sqrt{1-\delta}$, making that $\textnormal{Pr}({\mathcal{O}}_o^W \subseteq {\mathcal{S}}_o^W) = \sqrt{1-\delta}$.
\rebuttal{
Note that we assume $\hat{\mathcal{O}}_o$ is a convex polytope. Hence, the transformed $\hat{\mathcal{O}}_o^W$ is also a polytope. In addition, it can be observed the set ${\mathcal{D}}_o^W$ defined in Eq. (\ref{eq:transD}) is a circular (sphere in 3D) set with radius $\sqrt{F^{-1}(1-\epsilon)}$. Hence, we can compute the $\epsilon$-shadow in Eq. (\ref{eq:transShadow}) of the transformed uncertain obstacle by dilating its nominal shape by the diameter of the set ${\mathcal{D}}_o^W$, which results in an inflated convex polytope. Note that the resulted convex polytope is slightly larger than the exact Minkowski sum $\mathcal{S}_o^W$ which has smaller round corners. This introduces some conservativeness. For simplicity, we use the same notation $\mathcal{S}_o^W$ for the resulted inflated convex polytope and thus there is $\textnormal{Pr}({\mathcal{O}}_o^W \subseteq {\mathcal{S}}_o^W) > \sqrt{1-\delta}$.
}
Next, we separate ${\hat{\mathbf{p}}}_i^W$ from ${\mathcal{S}}_o^W$ by finding a max-margin separating hyperplane between them. Note that ${\mathcal{S}}_o^W$ is a bounded convex polytope that can be described by a list of vertices $({\psi}_1^W, \dots, {\psi}_{p_o}^W)$. Hence, finding a max-margin hyperplane between ${\hat{\mathbf{p}}}_i^W$ and ${\mathcal{S}}_o^W$ can be formulated as a support vector machine (SVM) problem \citep{Honig2018}, which can be efficiently solved using a quadratic program:
\begin{equation}\label{eq:quadprog}
\begin{aligned}
\min \quad & {\mathbf{a}_{io}^{W}}^T{\mathbf{a}}_{io}^W \\
\tn{s.t.} \quad & {{\mathbf{a}}_{io}^W}^T{\hat{\mathbf{p}}}_{io}^W - {b}_{io}^W \leq 1, \\
& {{\mathbf{a}}_{io}^W}^T{\psi}_k^W - {b}_{io}^W \geq 1, ~~\forall~k \in 1,\dots, p_o.
\end{aligned}
\end{equation}
The solution of the above quadratic program (\ref{eq:quadprog}) formulates a max-margin separating hyperplane with parameters $({\mathbf{a}}_{io}^W, {b}_{io}^W)$. We then shift it along its normal vector towards the obstacle shadow, resulting in a separating hyperplane exactly touching the shadow, as shown in Fig. \ref{fig:poly_plane} (Bottom left). Finally we perform an inverse coordinate transformation $W^{-1}$ and obtain the uncertainty-aware separating hyperplane between the robot and obstacle in the original workspace:
\begin{equation}\label{eq:an_obs}
\begin{aligned}
\mathbf{a}_{io} &= W^{T}{\mathbf{a}}_{io}^W, \\
b_{io} &= {b}_{io}^W,
\end{aligned}
\end{equation}
as shown in Fig. \ref{fig:poly_plane} (Bottom right), \rebuttal{in which the $\epsilon$-shadow in the transformed space $\mathcal{S}_o^W$ becomes $\mathcal{S}_o$ in the original space.}
\begin{remark}\label{rmk:shadow}
The linear coordinate transformation $W$ and its inverse $W^{-1}$ preserves relative geometries of $\mathcal{O}_o$. That is, $\textnormal{Pr}(\mathcal{O}_o \subseteq \mathcal{S}_o) = \textnormal{Pr}({\mathcal{O}}_o^W \subseteq {\mathcal{S}}_o^W) > \sqrt{1-\delta}$.
\end{remark}
\subsection{Collision Avoidance Buffer and B-UAVC}\label{subsec:buffer}
In Section \ref{subsec:robotPlane} and \ref{subsec:obsPlane} we have described the method to compute the hyperplanes that construct the UAVC. Now we introduce two buffer terms to the UAVC, to account for the robot physical safety radius and the collision probability threshold.
Recall Eq. (\ref{eq:uavc}) that the UAVC of robot $i$ can be written as the intersection of a set of separating hyperplanes
\begin{equation*}
\begin{aligned}
\mathcal{V}_i^{u} = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{a}_{ij}^T\mathbf{p} &\leq b_{ij}, \forall j\neq i, j\in\mathcal{I}, \\ \tn{and~~} \mathbf{a}_{io}^T\mathbf{p} &\leq b_{io}, \forall o\in\mathcal{I}_o \},
\end{aligned}
\end{equation*}
Let \rebuttal{$l\in\mathcal{I}_l = \{1,\cdots,n,n+1,\cdots,n+m\}, l \neq i$} denote any other robot or obstacle, we can write the UAVC in the following form
\begin{equation}
\mathcal{V}_i^u = \{ \mathbf{p}\in\mathbb{R}^d: \mathbf{a}_{il}^T\mathbf{p} \leq b_{il}, \forall l\in\mathcal{I}_l, l\neq i \}.
\end{equation}
which combines the notations for inter-robot and robot-obstacle separating hyperplanes. Next, we will describe the computation method of probabilistic collision avoidance buffer to extend the UAVC to B-UAVC.
\subsubsection{Robot safety radius buffer}
We compute the robot safety radius buffer by shifting the boundary of the UAVC towards the robot by a distance equal to the robot's radius. Hence the corresponding buffer for the hyperplane $(\mathbf{a}_{il}, b_{il})$ is
\begin{equation}\label{eq:radius_buffer}
\beta_i^r = \rebuttal{r_s}\norm{\mathbf{a}_{il}}.
\end{equation}
Figure \ref{subfig:buavc1} shows the buffered UAVC of each robot after taking into account their safety radius.
\subsubsection{Collision probability buffer}
\rebuttal{To achieve probabilistic collision avoidance,}
we further compute a buffer term $\beta_i^\delta$, which is defined as
\begin{equation}\label{eq:delta_buffer}
\beta_i^\delta = \sqrt{2\mathbf{a}_{il}^T\Sigma_i\mathbf{a}_{il}}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1),
\end{equation}
where $\tn{erf}(\cdot)$
is the Gauss error function \citep{Andrews1997} defined as $\tn{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$ and $\tn{erf}^{-1}(\cdot)$ is its inverse. In this paper, we assume the threshold satisfies $0 < \delta < 0.75 $, which is reasonable in practice. Hence, $\tn{erf}^{-1}(2\sqrt{1-\delta}-1) > 0, \beta_i^\delta>0$.
This buffer can be obtained by following the proof of forthcoming Theorem \ref{thm:buavc_robot} and Theorem \ref{thm:buavc_obs}.
Finally, the buffered uncertainty-aware Voronoi cell (B-UAVC) is obtained by combining the two buffers
\begin{equation}\label{eq:BUAVC}
\begin{aligned}
\mathcal{V}_{i}^{u, b}
= \{ \mathbf{p} \in \mathbb{R}^d:
\mathbf{a}_{il}^{T}\mathbf{p} \leq b_{il} - \beta_i^r - \beta_i^\delta,
\forall l\in\mathcal{I}_l, l\neq i \}.
\end{aligned}
\end{equation}
Figure \ref{subfig:buavc2} shows the final B-UAVC of each robot in the team.
\subsection{Properties of B-UAVC}\label{subsec:prop_buavc}
In this subsection, we justify the design of $\epsilon$ in Eq. (\ref{eq:transD}) when computing the shadow of uncertain obstacles, and computation of the collision probability buffer $\beta_i^\delta$ in Eq. (\ref{eq:delta_buffer}) by presenting the following two theorems.
\begin{theorem}[Inter-Robot Probabilistic Collision Free]\label{thm:buavc_robot}
$\forall \mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i, \Sigma_i)$ and $\mathbf{p}_j \sim \mathcal{N}(\hat{\mathbf{p}}_j, \Sigma_j)$, where $\hat{\mathbf{p}}_i \in \mathcal{V}_{i}^{u,b}$ and $\hat{\mathbf{p}}_j \in \mathcal{V}_{j}^{u,b}, i \neq j \in \mathcal{I}$, we have
\begin{equation*}
\textnormal{Pr}(\tn{dis}(\mathbf{p}_i,\mathbf{p}_j) \geq 2r_s) \geq 1-\delta,
\end{equation*}
i.e. the probability of collision between robots $i$ and $j$ is below the threshold $\delta$.
\end{theorem}
\begin{proof}
We first introduce the following lemma:
\begin{lemma}[Linear Chance Constraint \citep{Blackmore2011}]\label{lem:linChance}
A multivariate random variable $\mathbf{x} \sim \mathcal{N}(\hat{\mathbf{x}}, \Sigma)$ satisfies
\begin{equation}\label{eq:linChance}
\textnormal{Pr}(\mathbf{a}^T\mathbf{x} \leq b) = \frac{1}{2} + \frac{1}{2}\tn{erf}\left( \frac{b-\mathbf{a}^T\hat{\mathbf{x}}}{\sqrt{2\mathbf{a}^T\Sigma\mathbf{a}}} \right).
\end{equation}
\end{lemma}
According to Eq. (\ref{eq:BUAVC}), if $\hat{\mathbf{p}}_i\in\mathcal{V}_i^{u,b}$, there is
\begin{equation}\label{eq:iBUAVC}
\mathbf{a}_{ij}^T\hat{\mathbf{p}}_i \leq b_{ij} - r_s\norm{\mathbf{a}_{ij}} - \sqrt{2\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1).
\end{equation}
Applying Lemma \ref{lem:linChance} and substituting the above equation, we have
\begin{equation}\label{eq:iBUAVC_r}
\begin{aligned}
\textnormal{Pr}(\mathbf{a}_{ij}\mathbf{p}_i &\leq b_{ij} - r_s\norm{\mathbf{a}_{ij}}) \\
&= \frac{1}{2} + \frac{1}{2}\tn{erf}\left( \frac{b_{ij} - r_s\norm{\mathbf{a}_{ij}})-\mathbf{a}_{ij}^{T}\hat{\mathbf{p}}_{i}}{\sqrt{2\mathbf{a}_{ij}^{T}\mathbf{a}_{ij}}} \right) \\
&\geq \frac{1}{2} + \frac{1}{2}\tn{erf}\left( \tn{erf}^{-1}(2\sqrt{1-\delta}-1) \right) \\
&= \frac{1}{2} + \frac{1}{2}(2\sqrt{1-\delta}-1) \\
&= \sqrt{1-\delta}.
\end{aligned}
\end{equation}
Similarly for robot $j$, there is
\begin{equation}
\textnormal{Pr}(\mathbf{a}_{ji}\mathbf{p}_j \leq b_{ji} - r_s\norm{\mathbf{a}_{ji}}) \geq \sqrt{1-\delta}.
\end{equation}
Note that $\mathbf{a}_{ij} = -\mathbf{a}_{ji}$, $b_{ij} = -b_{ji}$ (Remark \ref{rmk:sepMutual}). It is trivial to prove that
\begin{equation}
\left.\begin{aligned}
\mathbf{a}_{ij}\mathbf{p}_i \leq b_{ij} - r_s\norm{\mathbf{a}_{ij}}\\
\mathbf{a}_{ji}\mathbf{p}_j \leq b_{ji} - r_s\norm{\mathbf{a}_{ji}}
\end{aligned}
\right\}
\implies \norm{\mathbf{p}_i-\mathbf{p}_j}\geq 2r_s.
\end{equation}
Hence, we have
\begin{equation}
\begin{aligned}
&\textnormal{Pr}(\tn{dis}(\mathbf{p}_i,\mathbf{p}_j) \geq 2r_s) = \textnormal{Pr}(\norm{\mathbf{p}_i-\mathbf{p}_j}\geq 2r_s) \\
&\geq \textnormal{Pr}(\mathbf{a}_{ij}\mathbf{p}_i \leq b_{ij} - r_s\norm{\mathbf{a}_{ij}})\cdot\textnormal{Pr}(\mathbf{a}_{ji}\mathbf{p}_j \leq b_{ji} - r_s\norm{\mathbf{a}_{ji}}) \\
&\geq \sqrt{1-\delta}\cdot\sqrt{1-\delta} \\
&= 1-\delta.
\end{aligned}
\end{equation}
This completes the proof.
\qed
\end{proof}
\begin{theorem}[Robot-Obstacle Probabilistic Collision Free]\label{thm:buavc_obs}
$\forall \mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i,\Sigma_i)$, where $\hat{\mathbf{p}}_i \in \mathcal{V}_{i}^{u,b}$, we have $\textnormal{Pr}(\tn{dis}(\mathbf{p}_i, \mathcal{O}_o)\geq r_s) \geq 1-\delta$, i.e. the probability of collision between robot $i$ and obstacle $o$ is below the threshold $\delta$.
\end{theorem}
\begin{proof}
Similar to Eq. (\ref{eq:iBUAVC_r}), we have
\begin{equation}
\textnormal{Pr}(\mathbf{a}_{io}\mathbf{p}_i \leq b_{io} - r_s\norm{\mathbf{a}_{io}}) \geq \sqrt{1-\delta}.
\end{equation}
Based on the computation of $\mathbf{a}_{io}$ and $b_{io}$ in Eq. (\ref{eq:quadprog})-(\ref{eq:an_obs}), it is straightforward to prove that
\begin{equation}
\mathbf{a}_{io}\mathbf{p}_i \leq b_{io} - r_s\norm{\mathbf{a}_{io}} \implies \tn{dis}(\mathbf{p}_i, \mathcal{S}_o) \geq r_s.
\end{equation}
Thus,
\begin{equation}
\textnormal{Pr}(\tn{dis}(\mathbf{p}_i, \mathcal{S}_o) \geq r_s) \geq \sqrt{1-\delta}.
\end{equation}
If $\mathcal{O}_o \subseteq \mathcal{S}_o$ and $\tn{dis}(\mathbf{p}_i, \mathcal{S}_o) \geq r_s$, there is $\tn{dis}(\mathbf{p}_i, \mathcal{O}_o) \geq r_s$. Hence, by combining with Remark \ref{rmk:shadow}, we have
\begin{equation}
\begin{aligned}
\textnormal{Pr}(\tn{dis}(\mathbf{p}_i, \mathcal{O}_o) \geq r_s) &\geq \textnormal{Pr}(\mathcal{O}_o \subseteq \mathcal{S}_o) \cdot \textnormal{Pr}(\tn{dis}(\mathbf{p}_i, \mathcal{S}_o) \geq r_s) \\
&> \sqrt{1-\delta}\cdot\sqrt{1-\delta} \\
&= 1-\delta,
\end{aligned}
\end{equation}
which completes the proof.
\qed
\end{proof}
\section{Collision Avoidance Using B-UAVC} \label{sec:method_2}
In this section, we present our decentralized collision avoidance method using the B-UAVC. We start by describing a reactive feedback controller for single-integrator robots, followed by its extensions to double-integrator and non-holonomic differential-drive robots. A receding horizon planning formulation is further presented for general high-order dynamical systems. We also provide a discussion on our proposed method.
\subsection{Reactive Feedback Control}
\subsubsection{Single integrator dynamics}
Consider robots with single-integrator dynamics $\dot{\mathbf{p}}_i = \mathbf{u}_i$, where $\mathbf{u}_i = \mathbf{v}_i$ is the control input. Similar to \citet{Zhou2017}, a fast reactive feedback one-step controller can be designed to make each robot move towards its goal location $\mathbf{g}_i$, as follows:
\begin{equation}\label{eq:reactive_controller}
\mathbf{u}_i = v_{i,\max}\cdot \frac{\mathbf{g}_i^* - \hat{\mathbf{p}}_i}{\norm{\mathbf{g}_i^* - \hat{\mathbf{p}}_i}},
\end{equation}
where $v_{i,\max}$ is the robot maximal speed and
\begin{equation}\label{eq:goal_projection}
\mathbf{g}_i^* := \argmin_{\mathbf{p}\in\mathcal{V}_i^{u,b}}\norm{\mathbf{p} - \mathbf{g}_i},
\end{equation}
is the closest point in the robot's B-UAVC to its goal location.
The strategy used in the controller, Eq. (\ref{eq:reactive_controller}), is also called the ``move-to-projected-goal'' strategy \citep{Arslan2019}. At each time step, each robot in the system first constructs its B-UAVC $\mathcal{V}_i^{u,b}$, then computes the closest point in $\mathcal{V}_i^{u,b}$ to its goal, i.e. the ``projected goal'', and generates a control input according to Eq. (\ref{eq:reactive_controller}). Note that the constructed B-UAVC is a convex polytope represented by the intersection of a set of half-spaces hyperplanes. Hence, finding the closest point, Eq. (\ref{eq:goal_projection}), can be recast as a linearly constrained least-square problem, which can be solved efficiently using quadratic programming in polynomial time \citep{kozlov1980polynomial}.
\subsubsection{Double integrator dynamics}
For single-integrator robots, the reactive controller Eq. (\ref{eq:reactive_controller}) guarantees the robot to be always within its corresponding B-UAVC and thus probabilistic collision free with other robots and obstacles. However, the controller may drive the robot towards to the boundary of its B-UAVC. Consider the double-integrator robot which has a limited acceleration, $\ddot{\mathbf{p}}_i = \mathbf{u}_i$, where $\mathbf{u}_i = \mathbf{a}\mathbf{c}\vc_i$ is the control input. It might not be able to continue to stay within its B-UAVC when moving close to the boundary of the B-UAVC. Hence, to \rebuttal{enhance} safety, as illustrated in Fig. \ref{fig:stopping} we introduce an additional safety stopping buffer, which is defined as
\begin{equation}\label{eq:double_int_extra_buffer}
\beta_i^s =
\begin{cases}
\frac{\norm{\mathbf{a}_{il}^T\mathbf{v}_i}^2}{2acc_{i,\max}}, &\tn{if~~} \mathbf{a}_{il}^T\mathbf{v}_i > 0; \\
0, &\tn{otherwise},
\end{cases}
\end{equation}
where $acc_{i,\max}$ is the maximal acceleration of the robot.
\rebuttal{This additional stopping buffer heuristically leaves more space for the robot to decelerate in advance before touching the boundaries of the original B-UAVC.}
Hence, the updated B-UAVC in Eq. (\ref{eq:BUAVC}) with an additional safety stopping buffer now becomes
\begin{equation}\label{eq:BUAVC_S}
\begin{aligned}
\mathcal{V}_{i}^{u, b}
= \{ \mathbf{p} \in \mathbb{R}^d:
\mathbf{a}_{il}^{T}\mathbf{p} \leq b_{il} - \beta_i^r - \beta_i^\delta - \beta_i^s, \\
\forall l\in\mathcal{I}_l, l\neq i \}.
\end{aligned}
\end{equation}
Accordingly, the reactive feedback one-step controller for double-integrator robots is as follows,
\begin{equation}\label{eq:double_int_controller}
\mathbf{u}_i = acc_{i,\max}\cdot \frac{\mathbf{g}_i^* - \hat{\mathbf{p}}_i}{\norm{\mathbf{g}_i^* - \hat{\mathbf{p}}_i}}.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{stopping.pdf}
\caption{Additional buffer is added to allow robots with double-integrator dynamics to have enough space to decelerate.}
\label{fig:stopping}
\end{figure}
\subsubsection{Differential-drive robots}\label{subsubsec:control_diff}
Consider differential-drive robots moving on a two dimensional space $\mathcal{W} \subseteq \mathbb{R}^2$, whose motions are described by
\begin{equation}
\begin{aligned}
&\dot{\hat{\mathbf{p}}}_i = v_i \mat \cos\theta_i \\ \sin\theta_i \mate, \\
&\dot{\theta}_i = \omega_i,
\end{aligned}
\end{equation}
where $\theta_i \in [-\pi, \pi)$ is the orientation of the robot, and $\mathbf{u}_i = (v_i, \omega_i)^T \in \mathbb{R}^2$ is the vector of robot control inputs in which $v_i$ and $\omega_i$ are the linear and angular velocity, respectively.
We adopt the control strategy developed by \citet{Arslan2019} and \citet{Astolfi1999} and briefly describe it in the following.
As shown in Fig. \ref{fig:diff_control}, firstly, two line segments
\begin{align}\label{eq:differ_drive_LUW}
L_v &= \mathcal{V}_i^{u,b} \cap H_N, \\
L_\omega &= \mathcal{V}_i^{u,b} \cap H_G,
\end{align}
are determined, in which $H_N$ is the straight line from the robot position towards its current orientation and $H_G$ is the straight line towards its goal location, respectively. Then the closest point in the robot's B-UAVC, $\mathbf{g}_i^*$, and in the two lines segments $\mathbf{g}_{i,v}^*$, $\mathbf{g}_{i,\omega}^*$ is computed. Finally the control inputs of the robot are given by
\begin{equation}\label{eq:diff_control}
\begin{aligned}
v_i &= -k\cdot[\cos(\theta)~\sin\theta](\hat{\mathbf{p}}_i - \mathbf{g}_{i,v}^*), \\
\omega_i &= k\cdot\tn{atan}\left(\frac{[-\sin(\theta)~\cos\theta](\hat{\mathbf{p}}_i - (\mathbf{g}_i^*+\mathbf{g}_{i,\omega}^*)/2)}{[\cos(\theta)~\sin\theta](\hat{\mathbf{p}}_i - (\mathbf{g}_i^*+\mathbf{g}_{i,\omega}^*)/2)}\right),
\end{aligned}
\end{equation}
where $k > 0$ is the fixed control gain. It is proved by \citet{Arslan2019} that if the local safe region is convex, then the robot will stay within the convex safe region under the control law of Eq. (\ref{eq:diff_control}).
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{differential_drive.pdf}
\caption{Reactive feedback control for differential-drive robots. }
\label{fig:diff_control}
\end{figure}
\subsection{Receding Horizon Planning}\label{subsec:control_mpc}
Consider general high-order dynamical systems with, potentially nonlinear, dynamics $\mathbf{x}_i^{k} = \mathbf{f}_i(\mathbf{x}_i^{k-1}, \mathbf{u}_i^{k-1})$, where $\mathbf{x}_i^k \in \mathbb{R}^{n_x}$ denotes the robot state at time step $k$ which typically includes the robot position $\mathbf{p}_i^k$ and velocity $\mathbf{v}_i^k$, and $\mathbf{u}_i^k \in \mathbb{R}^{n_u}$ the robot control input. To plan a local trajectory that respects the robot kinodynamic constraints, we formulate a constrained optimization problem with $N$ time steps and a planning horizon $\tau = N\Delta t$, where $\Delta t$ is the time step, as follows,
\begin{problem}[Receding Horizon Trajectory Planning]\label{prob:rhc}
\begin{subequations}
\begin{align}
\min\limits_{\hat{\mathbf{x}}_i^{1:N}, \mathbf{u}_i^{0:N-1}} ~~
& \sum_{k=0}^{N-1}\mathbf{u}_i^kR\mathbf{u}_i^k + (\hat{\mathbf{p}}_i^N - \mathbf{g}_{i}^N)^TQ_N(\hat{\mathbf{p}}_i^N - \mathbf{g}_{i}^N) \nonumber \\
\text{s.t.} ~~ & \mathbf{x}_i^0 = \hat{\mathbf{x}}_i, \\
& \hat{\mathbf{x}}_i^{k} = \mathbf{f}_i(\hat{\mathbf{x}}_i^{k-1}, \mathbf{u}_i^{k-1}), \\
& \hat{\mathbf{p}}_i^k \in \mathcal{V}_i^{u,b}, \label{eq:buavcCons} \\
& \mathbf{u}_i^{k-1} \in \mathcal{U}_i,\\
&\forall i\in\mathcal{I}, \,\forall k\in \{1,\dots,N\}.
\end{align}
\end{subequations}
\end{problem}
In Problem \ref{prob:rhc}, $\mathcal{U}_i \in \mathbb{R}^{n_u}$ is the admissible control space; $R \in \mathbb{R}^{n_u \times n_u}$, $Q_N \in \mathbb{R}^{d\times d}$ are positive semi-definite symmetric matrices. The constraint (\ref{eq:buavcCons}) restrains the planned trajectory to be within the robot's B-UAVC $\mathcal{V}_i^{u,b}$. According to the definition of $\mathcal{V}_i^{u,b}$ in Eq. (\ref{eq:BUAVC_S}), the constraint can be formulated as a set of linear inequality constraints:
\begin{equation}\label{eq:buavcLinCons}
\mathbf{a}_{il}^{T}\hat{\mathbf{p}}_i^k \leq b_{il} - \beta_i^r - \beta_i^\delta -\beta_i^s, ~\forall l\in\mathcal{I}_l, l\neq i.
\end{equation}
At each time step, the robot first constructs its corresponding B-UAVC $\mathcal{V}_i^{u,b}$ represented by a set of linear inequalities and then solves the above receding horizon planning problem.
\rebuttal{The problem is in general a nonlinear and non-convex optimization problem due to the robot's nonlinear dynamics formulated as equality constraints $\hat{\mathbf{x}}_i^{k} = \mathbf{f}_i(\hat{\mathbf{x}}_i^{k-1}, \mathbf{u}_i^{k-1})$.
}
While a solution of the problem including the planned trajectory and control inputs is obtained, the robot only executes the first control input $\mathbf{u}_i^0$. Then with time going on and at the next time step, the robot updates its B-UAVC and solves the optimization problem again. The process is performed until the robot reaches its goal location.
\rebuttal{
}
\rebuttal{
\begin{remark}[Probability of collision for the planned trajectory]
From Theorem {\ref{thm:buavc_robot}} and {\ref{thm:buavc_obs}}, constraint ({\ref{eq:buavcCons}}) guarantees that at each stage within the planning horizon, the collision probability of robot $i$ with any other robot or obstacle is below the specified threshold $\delta$. Hence, the probability of collision for the entire planning trajectory of robot $i$ with respect to each other robot and obstacle can be bounded by $\textnormal{Pr}(\cup_{k=1}^{N}\hat{\mathbf{p}}_i^k \notin \mathcal{V}_i^{u,b}) \leq \sum_{k=1}^{N}\textnormal{Pr}(\hat{\mathbf{p}}_i^k \notin \mathcal{V}_i^{u,b}) = N\delta$. Nevertheless, this bound is over conservative in practice. The real collision probability of the planned trajectory is much smaller than $N\delta$ \mbox{\citep{Schmerling2017}}. Hence, we impose the collision probability threshold $\delta$ for each individual stage in the context of receding horizon planning, thanks to the fast re-planning and relatively small displacement between stages \citep{luo2020multi}.
\end{remark}
}
\rebuttal{
Algorithm \ref{alg:ca_buavc} summarizes our proposed method for decentralized probabilistic multi-robot collision avoidance, in which each robot in the system first constructs its B-UAVC, and then compute control input accordingly to restrain its motion to be within the B-UAVC.
\begin{algorithm}[t]
\caption{Collision Avoidance Using B-UAVC for Each Robot $i\in\mathcal{I}$ in a Multi-robot Team}
\label{alg:ca_buavc}
\begin{algorithmic}[1]
\Statex ------------------ Construction of B-UAVC ------------------
\State Obtain $\mathbf{p}_i \sim \mathcal{N}(\hat{\mathbf{p}}_i, \Sigma_i)$ via state estimation
\For {Each other robot $j \in \mathcal{I}, j \neq i$}
\State Estimate $\mathbf{p}_j \sim \mathcal{N}(\hat{\mathbf{p}}_j, \Sigma_j)$
\State Compute the best linear separator parameters $(\mathbf{a}_{ij}, b_{ij})$ via Eq. (\ref{eq:best_linear_separator})
\EndFor
\For {Each static obstacle $o\in\mathcal{I}_o$}
\State Estimate $\mathbf{d}_o \sim \mathcal{N}(0, \Sigma_o)$ with known $\hat{\mathcal{O}}_o$
\State Compute the separating hyperplane parameters $(\mathbf{a}_{io}, b_{io})$ via Eqs. (\ref{eq:coordinate_transformation})-(\ref{eq:an_obs})
\EndFor
\For {Each separating hyperplane $l \in \mathcal{I}_l, l\neq i$}
\State Compute the safety radius buffer via Eq. (\ref{eq:radius_buffer}): $\beta_i^r = {r_s}\norm{\mathbf{a}_{il}}$
\State Compute the collision probability buffer via Eq. (\ref{eq:delta_buffer}): $\beta_i^\delta = \sqrt{2\mathbf{a}_{il}^T\Sigma_i\mathbf{a}_{il}}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1)$
\State Construct the B-UAVC via Eq. (\ref{eq:BUAVC})
\EndFor
\Statex ------------------ Collision Avoidance Action ------------------
\If {$i$ is single-integrator}
\State Compute control input via Eqs. (\ref{eq:reactive_controller})-(\ref{eq:goal_projection})
\ElsIf {$i$ is double-integrator}
\State Compute control input via Eqs. (\ref{eq:double_int_extra_buffer})-(\ref{eq:double_int_controller})
\ElsIf {$i$ is differential-drive}
\State Compute control input via Eqs. (\ref{eq:differ_drive_LUW})-(\ref{eq:diff_control})
\Else
\State Compute control input by solving Problem 1
\EndIf
\end{algorithmic}
\end{algorithm}
}
\rebuttal{
\subsection{Discussion}
\subsubsection{Uncertainty estimation}\label{subsubsec:estimation_discussion}
For each robot $i$ in the system, to construct its B-UAVC, the robot needs a) its own position estimation mean $\hat{\mathbf{p}}_i$ and uncertainty covariance $\Sigma_i$ from onboard measurements via a filter, e.g. a Kalman filter, and b) to know each other robot $j$'s position mean $\hat{\mathbf{p}}_i$ and uncertainty covariance $\Sigma_j$. In case communication is available, such position estimation information can be communicated among robots. However, in a fully decentralized system where there is no communication, each robot $i$ will need to estimate other robot $j$'s position mean and covariance, denoted by $\tilde{\mathbf{p}}_j$ and $\tilde{\Sigma}_j$, via its own onboard sensor measurements. In this case, we assume that robot $i$'s estimation of robot $j$'s position mean is the same as robot $j$'s own estimation, i.e. $\tilde{\mathbf{p}}_j = \hat{\mathbf{p}}_j$; while robot $i$'s estimation of the uncertainty covariance of robot $j$ is larger than its own localization uncertainty covariance, i.e. $|\tilde{\Sigma}_j| \geq |\Sigma_i|$. This assumption is reasonable in practice since the robot generally has more accurate measurements of its own position than other robots in the environment. Then robot $i$ computes its B-UAVC using $\hat{\mathbf{p}}_i, \Sigma_i, \tilde{\mathbf{p}}_j$, and $\tilde{\Sigma}_j$. According to the properties of the best linear separator, this assumption leads that each robot $i$ always partitions a smaller space when computing the separating hyperplane with another robot $j$, which results in a more conservative B-UAVC to ensure safety for robot $i$ itself.
\subsubsection{Empty B-UAVCs}
Taking into account uncertainty, the robots being probabilistic collision-free (Definition 1), i.e., $\textnormal{Pr}(\norm{\mathbf{p}_i-\mathbf{p}_j}\geq 2r_s) \geq$ $1 - \delta, \forall i,j \in \{1,\dots,n\}, i\neq j$, does not guarantee that the defined B-UAVC $\mathcal{V}_i^{u,b}$ is non-empty. Nevertheless, the case $\mathcal{V}_i^{u,b}$ being empty is rarely observed in our simulations and experiments. We handle this situation by decelerating the robot if its B-UAVC is empty.
}
\section{Simulation Results}\label{sec:sim_result}
We now present simulation results comparing our proposed B-UAVC method with state-of-the-art baselines as well as a performance analysis of the proposed method in a variety of scenarios.
\subsection{Comparison to the BVC Method}\label{subsec:com_bvc}
We first compare our proposed B-UAVC method with the BVC approach \citep{Zhou2017} that we extend in two-dimensional obstacle-free environments with single-integrator robots. Both the B-UAVC and BVC methods only need robot position information to achieve collision avoidance, in contrast to the well-known reciprocal velocity obstacle (RVO) method \citep{VanDenBerg2011} which also requires robot velocity information to be communicated or sensed. Comparison between BVC and RVO has been demonstrated by \citet{Zhou2017} in 2D scenarios, hence in this paper we focus on comparing the proposed B-UAVC with BVC.
We deploy the B-UAVC and BVC in a $10 \times 10$m environment with 2, 4, 8, 16 and 32 robots forming an \emph{antipodal circle swapping} scenario (\citet{VanDenBerg2011}). In this scenario, the robots are initially placed on a circle (equally spaced) and their goals are located at the antipodal points of the circle. We use a circle with a radius of 4.0 m in simulation.
Each robot has a radius of 0.2 m, a local sensing range of 2.0 m and a maximum allowed speed of 0.4 m/s.
The goal is assumed to be reached for each robot when the distance between its center and goal location is smaller than 0.1 m.
To simulate collision avoidance under uncertainty, two different levels of noise, $\Sigma_1 = \tn{diag}(0.04~\tn{m}, 0.04~\tn{m})^2$ and $\Sigma_2 = \tn{diag}(0.06~\tn{m}, 0.06~\tn{m})^2$, are added to the robot position measurements.
\rebuttal{Particularly, each robot's localization uncertainty covariance is $\Sigma_1$ and its estimation of other robots' position uncertainty covariance is $\Sigma_2$}.
The time step used in simulation is $\Delta t = 0.1$ s.
In the basic BVC implementation, an extra $10\%$ or $100\%$ radius buffer is added to the robot's real physical radius to account for measurement uncertainty for comparison \citep{Wang2019}. In the B-UAVC implementation, the collision probability threshold is set as $\delta = 0.05$. Any robot will stop moving when it arrives at its goal or is involved in a collision. Both the B-UAVC and BVC methods use the same deadlock resolution techniques proposed in this paper (\rebuttal{Appendix \ref{appendix:deadlock}}). We set a maximum simulation step $K = 800$ and the collision-free robots that do not reach their goals within $K$ steps are regarded to be in deadlocks/livelocks.
For each case (number of robots $n$) and each method, we run the simulation 10 times. In each single run, we evaluate the following performance metrics:
(a) collision rate,
(b) minimum distance among robots,
(c) average travelled distance of robots,
and (d) time to complete a single run.
The collision rate is defined to be the ratio of robots colliding over the total number of robots.
Time to complete a single run is defined to be the time when the last robot reaches its goal. Note that the metrics (2)(3)(4) are calculated for robots that successfully reach their goal locations.
Finally, statistics of 10 instances under each case are presented.
The simulation results are presented in Fig. \ref{fig:compare_bvc_si}. In all runs, no deadlocks are observed. In terms of collision avoidance, both the B-UAVC approach and BVC with additional 100\% robot radius achieve zero collision in all runs. The BVC with only 10\% robot radius leads to collisions when the total number of robots gets larger. In particular, when there are 32 robots an average of 28\% robots collide, as shown in Fig. \ref{subfig:com_bvc_si_col_rate}. While the BVC with 100\% additional robot radius can also achieve zero collision rate as our proposed B-UAVC, it is more conservative and less efficient. In average, the B-UAVC saves 10.1\% robot travelled distance (Fig. \ref{subfig:com_bvc_si_tra_dis}) and 14.4\% time for completing a single run (Fig. \ref{sub@subfig:com_bvc_si_tra_time}) comparing to the BVC with additional 100\% robot radius.
\rebuttal{
\begin{remark}
The ``BVC + $X\%$'' is a heuristic way to handle uncertainty. The above simulation results show that if $X$ is too small, then it cannot ensure safety; while if $X$ is too large, the results will be very conservative and less efficient. So generally reasoning about individual uncertainties using the proposed B-UAVC method will perform better than determining an extra $X\%$ buffer.
\end{remark}
}
\rebuttal{
\begin{remark}
In some cases we can design such an $X$ that it will have the same results as the B-UAVC method.
Consider the case where $\Sigma_i = \Sigma_j = \sigma^2I$. According to Remark \ref{rmk:sepPlane}, the best linear separator coincides with the separating hyperplane computed by the BVC method, whose parameters are denoted by $\mathbf{a}_{ij}$ and $b_{ij}$. The hyperplane parameters can be further normalized to make $\norm{\mathbf{a}_{ij}}=1$. In this case, our B-UAVC and the BVC have the same safety radius buffer $\beta_i^r = r_s$. Given a collision probability threshold $\delta$, our B-UAVC further introduces another buffer to handle uncertainty
\begin{equation*}
\begin{aligned}
\beta_i^\delta &= \sqrt{2\mathbf{a}_{il}^T\Sigma_i\mathbf{a}_{il}}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1) \\
&= \sigma\sqrt{2}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1).
\end{aligned}
\end{equation*}
If we choose an extra safety buffer $X\%$ such that
\begin{equation*}
X\%\cdot r_s = \sigma\sqrt{2}\cdot\tn{erf}^{-1}(2\sqrt{1-\delta}-1),
\end{equation*}
then the results of the ``BVC + $X\%$'' method are the same as our B-UAVC method.
However, our B-UAVC method can handle general cases where it is hard to design an $X\%$ to always achieve the same level of performance.
\end{remark}
}
\begin{figure}[t]
\centering
\subfloat[]{\label{subfig:com_bvc_si_col_rate}
\includegraphics[width=.23\textwidth]{sym_006_010_col_rate.pdf}}
\subfloat[]{\label{subfig:com_bvc_si_min_dis}
\includegraphics[width=.23\textwidth]{sym_006_010_min_dis.pdf}}
\\
\subfloat[]{\label{subfig:com_bvc_si_tra_dis}
\includegraphics[width=.23\textwidth]{sym_006_010_tra_dis.pdf}}
\subfloat[]{\label{subfig:com_bvc_si_tra_time}
\includegraphics[width=.23\textwidth]{sym_006_010_tra_time.pdf}}
\caption{Evaluation of the antipodal circle scenario with varying numbers of single-integrator robots. The (a) collision rate, (b) minimum distance, (c) travelled distance and (d) complete time are shown. Lines denote mean values and shaded areas around the lines denote standard deviations over 10 repetitions for each scenario.}
\label{fig:compare_bvc_si}
\end{figure}
\begin{figure*}[h]
\centering
\subfloat[$t = 1$ s.]{\label{}
\includegraphics[width=.185\textwidth]{r8o10_10-crop.pdf}}
\subfloat[$t = 6$ s.]{\label{}
\includegraphics[width=.185\textwidth]{r8o10_60-crop.pdf}}
\subfloat[$t = 12$ s.]{\label{}
\includegraphics[width=.185\textwidth]{r8o10_120-crop.pdf}}
\subfloat[$t = 18$ s.]{\label{}
\includegraphics[width=.185\textwidth]{r8o10_180-crop.pdf}}
\subfloat[$t = 27$ s.]{\label{}
\includegraphics[width=.185\textwidth]{r8o10_270-crop.pdf}}
\caption{A sample simulation run of the random moving scenario with 8 robots and 10\% obstacle density. The robot initial and goal locations are marked in circle disks and solid squares. Grey boxes are static obstacles. The B-UAVCs are shown in shaded patches with dashed boundaries. }
\label{fig:r8o10}
\end{figure*}
\subsection{Performance Analysis}
We then study the effect of collision probability threshold on the performance of the proposed B-UAVC method. Similarly, we deploy the B-UAVC in a $10 \times 10$m environment with 2, 4, 8, 16 and 32 robots in obstacle-free and cluttered environments with 10\% obstacle density. In the obstacle-free case for each number of robots $n$, 10 scenarios are randomly generated to form a challenging \emph{asymmetric swapping} scenario \citep{Serra2020}, indicating that the environment is split into $n$ sections around the center and each robot is initially randomly placed in one of them while required to navigate to its opposite section around the center. In the obstacle-cluttered case, 10 \emph{random moving} scenarios are simulated for each different number of robots in which robot initial positions and goal locations are randomly generated.
Fig. \ref{fig:r8o10} shows a sample run of the scenario with 8 robots and 10 obstacles.
We then run each generated scenario 5 times given a parameter setting (collision probability threshold). The robots have the same radius and maximal speed as in Section \ref{subsec:com_bvc}. Localization noise with zero mean and covariance $\Sigma = \tn{diag}(0.06~\tn{m}, 0.06~\tn{m})^2$ is added. For evaluation of performance, we focus on the robot collision rate, the robot deadlock rate,
and the minimum distance among successful robots.
We evaluate the performance of B-UAVC with different levels of collision probability threshold: $\delta =$ 0.05, 0.10, 0.20 and 0.30. The simulation results are presented in Fig. \ref{fig:compare_thresh}.
In the top row of the figure, we consider the collision rate among robots. The result shows that with a roughly small collision probability threshold $\delta = 0.05, 0.10 ,0.20$, no collisions are observed in both obstacle-free asymmetric swapping and obstacle-cluttered random moving scenarios, indicating that the B-UAVC method maintains a high level of safety. However, when $\delta$ is set to 0.3, the collision rate among robots increase dramatically, in particular when the number of robots is large. For example, in the asymmetric swapping scenario with 32 robots, there are 68.75\% robots involve in collisions in average.
In the bottom row of the figure, the minimum distance among robots are compared. The result shows that with smaller threshold, the minimum distance will be a little bit larger. The reason is that robots with a smaller threshold will have more conservative behavior and have smaller B-UAVCs during navigation.
\begin{figure}[t]
\centering
\subfloat[]{\label{subfig:thresh_col_rate_asy}
\includegraphics[width=.23\textwidth]{asy_006_010_thresh_col_rate.pdf}}
\subfloat[]{\label{subfig:thresh_col_rate_rand}
\includegraphics[width=.23\textwidth]{rand_006_010_thresh_col_rate.pdf}}
\\
\subfloat[]{\label{subfig:thresh_min_dis_asy}
\includegraphics[width=.23\textwidth]{asy_006_010_thresh_min_dis.pdf}}
\subfloat[]{\label{subfig:thresh_min_dis_rand}
\includegraphics[width=.23\textwidth]{rand_006_010_thresh_min_dis.pdf}}
\caption{Effect of the collision probability threshold on the method performance. The (a)-(b) collision rate, and (c)-(d) minimum distance among robots are shown. The evaluation has 2, 4, 8, 16, and 32 robot cases with 10 instances each. The left column shows results of the asymmetric swapping scenario and the right column shows results of the random moving scenario with 10\% obstacle density. Lines denote mean values and shaded areas around the lines denote standard deviations over 50 runs. }
\label{fig:compare_thresh}
\end{figure}
\subsection{Simulations with Quadrotors in 3D Space}
\begin{figure}[t]
\centering
\subfloat{\label{}
\includegraphics[width=.20\textwidth]{buavc_4_top.pdf}}
\subfloat{\label{}
\includegraphics[width=.20\textwidth]{ccmpc_4_top.pdf}}
\\
~
\setcounter{subfigure}{0}
\subfloat[]{\label{subfig:buavc_4}
\includegraphics[width=.20\textwidth]{buavc_4_side.pdf}}
\subfloat[]{\label{subfig:ccmpc_4}
\includegraphics[width=.20\textwidth]{ccmpc_4_side.pdf}}
~
\caption{Simulation with six quadrotors exchanging positions in 3D space. Solid lines represent executed trajectories of the robots. (a) Results of our B-UAVC method. (b) Results of the CCNMPC method \citep{Zhu2019RAL}.}%
\label{fig:simMPCResults}%
\end{figure}
We evaluate our receding horizon planning algorithm with quadrotors in 3D space and compare our method with one of the state-of-the-art quadrotor collision avoidance methods: the chance constrained nonlinear MPC (CCNMPC) with sequential planning \citep{Zhu2019RAL}, which requires communication of future planned trajectories among robots. For both methods, we adopt the same quadrotor dynamics model for planning. The quadrotor radius is set as $r = 0.3$ m and the collision probability threshold is set to $\delta = 0.03$. The time step is $\Delta t = 0.05$ s and the total number of steps is $N = 20$ resulting in a planing horizon of one second.
As shown in Fig. \ref{fig:simMPCResults}, we simulate with six quadrotors exchanging their initial positions in an obstacle-free 3D space. Each quadrotor is under localization uncertainty $\Sigma = \tn{diag}(0.04~\tn{m},~0.04~\tn{m},~0.04~\tn{m})^2$. For each method, we run the simulation 10 times and calculate the minimum distance among robots. Both our B-UAVC method and the CCNMPC method successfully navigates all robots without collision. An average minimum distance of 0.72 m is observed in our B-UAVC method, while the one of CCNMPC is 0.62 m, which indicates our method is more conservative than the CCNMPC. However, the CCNMPC is centralized and requires robots to communicate their future planned trajectories with each other, while the B-UAVC method only needs robot positions to be shared or sensed.
\section{Experimental Validation}\label{sec:exp_result}
In this section we describe the experimental results with a team of real robots. A video demonstrating the results accompanies this paper.
\begin{figure*}[t]
\centering
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{2j2o_240_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{2j2o_720_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{2j2o_960_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{2j2o_1800_crop.png}}
\\
\setcounter{subfigure}{0}
\subfloat[$t$ = 2 s.]{\label{}
\includegraphics[width=.24\textwidth]{2j2o_240.pdf}}
\subfloat[$t$ = 6 s.]{\label{}
\includegraphics[width=.24\textwidth]{2j2o_720.pdf}}
\subfloat[$t$ = 8 s.]{\label{}
\includegraphics[width=.24\textwidth]{2j2o_960.pdf}}
\subfloat[$t$ = 15 s.]{\label{}
\includegraphics[width=.24\textwidth]{2j2o_1800.pdf}}
\caption{Collision avoidance with two differential-drive robots and two static obstacles. The two robots are required to swap their positions. Top row: Snapshots of the experiment. Bottom row: Trajectories of the robots. Robot initial and goal positions are marked in circle disks and solid squares, respectively. Grey boxes are static obstacles. The B-UAVCs are shown in shaded patches with dashed boundaries. }%
\label{fig:exp_jackal}
\end{figure*}
\begin{figure}[h]
\centering
\subfloat[]{\label{subfig:jackal_dis}
\includegraphics[width=.23\textwidth]{2d2o_his.pdf}}
\subfloat[]{\label{subfig:jackal_dis_obs}
\includegraphics[width=.23\textwidth]{2d2o_his_obs.pdf}}
\caption{Experimental results with two differential-drive robots. (a) Histogram of inter-robot distance. (b) Histogram of distance between robots and obstacles. }%
\label{fig:exp_jackal_sta}
\end{figure}
\subsection{Experimental Setup}
We test our proposed approach on both ground vehicles and aerial vehicles in an indoor environment of 8m (L) $\times$ 3.4m (W) $\times$ 2.5m (H). Our ground vehicle platform is the Clearpath Jackal robot and our aerial vehicle platform is the Parrot Bebop 2 quadrotor. For ground vehicles, we apply the controller designed for differential-drive robots as shown in Section \ref{subsubsec:control_diff}. For quadrotors, the receding horizon trajectory planner presented in Section \ref{subsec:control_mpc} is employed.
\rebuttal{The quadrotor dynamics model $\mathbf{f}$ in Problem 1 is given in Appendix \ref{appendix:quad_model}. For solving Problem 1 which is a nonlinear programming problem, we rely on the solver Forces Pro \citep{Zanelli2020} to generate fast C code to solve it.}
Both types of robots allow executing control commands sent via ROS. The experiments are conducted in a standard laptop (Quadcore Intel i7 CPU@2.6 GHz) which connects with the robots via WiFi.
\rebuttal{An external motion capture system (OptiTrack) is used to track the pose (position and orientation) of each robot and obstacle in the environment running in real time at 120 Hz, which is regarded as the \emph{real} (ground-truth) pose. To validate collision avoidance under uncertainty, we then manually add Gaussian noise to the real pose data to generate noisy measurements. Taking the noisy measurements as inputs, a standard Kalman filter running at 120 Hz is employed to estimate the states of the robots and obstacles.} In all experiments, the added position measurements noise to the robots is zero mean with covariance $\Sigma_i^{\prime} = \tn{diag}(0.06~\tn{m}, 0.06~\tn{m}, 0.06~\tn{m})^2$, which results in an average estimated position uncertainty covariance $\Sigma_i = \tn{diag}(0.04~\tn{m}, 0.04~\tn{m}, 0.04~\tn{m})^2$. The added noise to the obstacles is zero mean with covariance $\Sigma_o^{\prime} = \tn{diag}(0.03~\tn{m}, 0.03~\tn{m}, 0.03~\tn{m})^2$ and the resulted estimated position uncertainty covariance is $\Sigma_o = \tn{diag}(0.02~\tn{m}, 0.02~\tn{m}, 0.02~\tn{m})^2$. The collision probability threshold is set as $\delta = 0.03$ as in previous works \citep{Zhu2019MRS,Zhu2019RAL}.
\begin{figure*}[h]
\centering
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{2d2o_swap_2_crop.png}}
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{2d2o_swap_100_crop.png}}
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{2d2o_swap_210_crop.png}}
\\
\setcounter{subfigure}{0}
\subfloat[$t$ = 0.1 s.]{\label{}
\includegraphics[width=.33\textwidth]{2d2o_swap_2.pdf}}
\subfloat[$t$ = 5 s.]{\label{}
\includegraphics[width=.33\textwidth]{2d2o_swap_100.pdf}}
\subfloat[$t$ = 10.5 s.]{\label{}
\includegraphics[width=.33\textwidth]{2d2o_swap_210.pdf}}
\caption{Collision avoidance with two quadrotors and two static obstacles. The two quadrotors are required to swap their positions. Top row: Snapshots of the experiment. Bottom row: Trajectories of the robots. Quadrotor initial and goal positions are marked in circles and diamonds. Solid lines represent travelled trajectories and dashed lines represent planned trajectories.}%
\label{fig:exp_quad_2}
\end{figure*}
\begin{figure*}[h]
\centering
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{3d0o_2_crop.png}}
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{3d0o_120_crop.png}}
\subfloat{\label{}
\includegraphics[width=.3\textwidth]{3d0o_200_crop.png}}
\\
\setcounter{subfigure}{0}
\subfloat[$t$ = 0.1 s.]{\label{}
\includegraphics[width=.33\textwidth]{3d0o_2.pdf}}
\subfloat[$t$ = 6 s.]{\label{}
\includegraphics[width=.33\textwidth]{3d0o_120.pdf}}
\subfloat[$t$ = 10 s.]{\label{}
\includegraphics[width=.33\textwidth]{3d0o_200.pdf}}
\caption{Collision avoidance with three quadrotors in a shared workspace. Top row: Snapshots of the experiment. Bottom row: Trajectories of the robots.}%
\label{fig:exp_quad_3}
\end{figure*}
\begin{figure}[t]
\centering
\subfloat[]{\label{subfig:quad_dis_his}
\includegraphics[width=.23\textwidth]{quad_3d0o_his.pdf}}
\subfloat[]{\label{subfig:quad_obs_dis_his}
\includegraphics[width=.23\textwidth]{quad_2d2o_his_obs.pdf}}
\caption{Experimental results with two/three quadrotors with/without obstacles. (a) Histogram of inter-robot distance. (b) Histogram of distance between robots and obstacles. }%
\label{fig:exp_quad_3_sta}
\end{figure}
\begin{figure*}[t]
\centering
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_2_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_480_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_720_crop.png}}
\subfloat{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_1440_crop.png}}
\\
\setcounter{subfigure}{0}
\subfloat[$t$ = 0.1 s.]{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_2.pdf}}
\subfloat[$t$ = 4 s.]{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_480.pdf}}
\subfloat[$t$ = 6 s.]{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_720.pdf}}
\subfloat[$t$ = 12 s.]{\label{}
\includegraphics[width=.24\textwidth]{1j1d2o_1440.pdf}}
\caption{Collision avoidance with a heterogeneous team of a differential-drive robot and a quadrotor. Top row: Snapshots of the experiment. Bottom row: Trajectories of the robots. }%
\label{fig:exp_het}
\end{figure*}
\subsection{Experimental Results}
\subsubsection{Experiments with differential-drive robots in 2D}
We first validated our proposed approach with two differential-drive robots. In the experiment, two robots are required to swap their positions while avoiding two static obstacles in the environment. The robot safety radius is set as 0.3 m. We run the experiment four times. The two robots successfully navigated to their goals while avoiding each other as well as the obstacles in all runs.
Fig. \ref{fig:exp_jackal} presents the results of one run. The top row of the figure shows a series of snapshots during the experiment, while the bottom row shows the robots' travelled trajectories and their corresponding B-UAVCs. It can be seen that each robot always keeps a very safe region (B-UAVC) taking into account its localization and sensing uncertainties. In Fig. \ref{fig:exp_jackal_sta} we cumulate the distance between the two robots (Fig. \ref{subfig:jackal_dis}) and distance between the robots and obstacles (Fig. \ref{subfig:jackal_dis_obs}) during the whole experiments. It can be seen that a minimum safe inter-robot distance of 0.6 m and a safe robot-obstacle distance of 0.3 m were maintained over all the runs.
\subsubsection{Experiments with quadrotors in 3D}
We then performed experiments with a team of quadrotors in two scenarios: with and without static obstacles. The quadrotor safety radius is set as 0.3 m.
\paragraph{Scenario 1}
Two quadrotors swap their positions while avoiding two static obstacles in the environment. We performed the swapping action four times and Fig. \ref{fig:exp_quad_2} presents one run of the results.
\paragraph{Scenario 2}
Three quadrotors fly in a confined space while navigating to different goal positions. The goal locations are randomly chosen such that the quadrotors' directions from initial positions towards goals are crossing. New goals are generated after all quadrotors reach their current goals. We run the experiment for a consecutive two minutes within which the goal of each quadrotor has been changed eight times.
Fig. \ref{fig:exp_quad_3} presents a series of snapshots during the experiment.
Fig. \ref{fig:exp_quad_3_sta} cumulates the inter-quadrotor distance in the experiments of both scenarios, and the distance between quadrotors and obstacles in Scenario 1. It can be seen that a minimum safety distance of 0.6 m among quadrotors and that of 0.3 m between quadrotors and obstacles were achieved during the whole experiments.
\subsubsection{Experiments with heterogeneous teams of robots}
We further tested our approach with one ground differential-drive robot and one quadrotor to show that it can be applied to heterogeneous robot teams. In the experiment, the ground robot only considers its motion and the obstacles in 2D (the ground plane) while ignoring the flying quadrotor. In contrast, the quadrotor considers both itself location and the ground robot's location as well as obstacles in 3D, in which it assumes the ground vehicle has a height of 0.6 m. To this end, the B-UAVC of the ground robot is a 2D convex region while that of the quadrotor is a 3D one.
Fig. \ref{fig:exp_het} shows the results of the experiment. It can be seen that the two robots successfully reached their goals while avoiding each other and the static obstacles. Particularly at $t$ = 4 s, the quadrotor actively flies upward to avoid the ground robot. In Fig. \ref{fig:exp_hete_sta} we cumulate the distance between the two robots and the distance between robots and obstacles, which show that a safe inter-robot clearance of 0.6 m and that of 0.3 m between robots and obstacles were maintained during the experiment.
\begin{figure}[t]
\centering
\subfloat[]{\label{subfig:hete_dis_his}
\includegraphics[width=.23\textwidth]{hete_2d2o_his-crop.pdf}}
\subfloat[]{\label{subfig:hete_obs_dis_his}
\includegraphics[width=.23\textwidth]{hete_2d2o_his_obs-crop.pdf}}
\caption{Experimental results with a ground differential-drive robot and a quadrotor. (a) Histogram of inter-robot distance. (b) Histogram of distance between robots and obstacles. }%
\label{fig:exp_hete_sta}
\end{figure}
\section{Conclusion}\label{sec:conclsuion}
In this paper we presented a decentralized and communication free multi-robot collision avoidance method that accounts for robot localization and sensing uncertainties. By assuming that the uncertainties are according to Gaussian distributions, we compute a chance-constrained buffered uncertainty-aware Voronoi cell (B-UAVC) for each robot among other robots and static obstacles. The probability of collision between robots and obstacles is guaranteed to be below a specified threshold by constraining each robot's motion to be within its corresponding B-UAVC. We apply the method to single-integrator, double-integrator, differential-drive, and general high-order dynamical multi-robot systems.
In comparison with the BVC method, we showed that our method achieves robust safe navigation among a larger number of robots with noisy position measurements where the BVC approach will fail.
In simulation with a team of quadrotors, we showed that our method achieves safer yet more conservative motions compared with the CCNMPC method, which is centralized and requires robots to communicate future trajectories.
We also validated our method in extensive experiments with a team of ground vehicles, quadrotors, and heterogeneous robot teams in both obstacle-free and obstacles-clutter environments.
Through simulations and experiments, two limitations of the proposed approach are also observed.
The approach can achieve a high level of safety under robot localization and sensing uncertainty, however, it also leads to conservative behaviours of the robots, particulary for agile vehicles (quadrotors) in confined space.
And, since the approach is local and efficient inter-robot coordination is not well investigated, deadlocks and livelocks may occure for large numbers of robots moving in complex environments.
For future work, we plan to employ the proposed approach as a low-level robust collision-avoidance controller, and incorporate it with other higher-level multi-robot trajectory planning and coordination methods to achieve more efficient multi-robot navigation.
\rebuttal{
\section*{Appendix}
\def\Alph{section}{\Alph{section}}
\setcounter{section}{0}
\section{Proofs of Lemmas and Theorems}
\subsection{Proof of Lemma \ref{lemma:probability_Do}}\label{appendix:proof_probability_Do}
\begin{proof}
First we can write the random variable $\mathbf{d}_o$ in an equivalent form $\mathbf{d}_o = \Sigma_o^\prime\mathbf{d}_o^\prime$, where $\mathbf{d}_o^\prime\sim\mathcal{N}(0,I) \in \mathbb{R}^d$ and $\Sigma_o^{\prime}\Sigma_o^{\prime T} = \Sigma_o$.
Note that $\mathbf{d}_o^{\prime T}\mathbf{d}_o^{\prime}$ is a chi-squared random variable with $d$ degrees of freedom. Hence, there is
\begin{equation*}
\textnormal{Pr}(\mathbf{d}_o^{\prime T}\mathbf{d}_o^{\prime} \leq F^{-1}(1-\epsilon)) = 1 - \epsilon.
\end{equation*}
Also note that $\Sigma_o^{-1} = (\Sigma_o^{\prime}\Sigma_o^{\prime T})^{-1} = \Sigma_o^{\prime T^{-1}}\Sigma_o^{\prime -1}$, thus $\mathbf{d}_o^T\Sigma_o^{-1}\mathbf{d}_o = \mathbf{d}_o^{\prime T}\Sigma_o^{\prime T}\Sigma_o^{\prime T^{-1}}\Sigma_o^{\prime -1}\Sigma_o^\prime\mathbf{d}_o^\prime = \mathbf{d}_o^{\prime T}\mathbf{d}_o^{\prime}$. Hence, it follows that $\textnormal{Pr}(\mathbf{d}_o^T\Sigma_o^{-1}\mathbf{d}_o \leq F^{-1}(1-\epsilon)) = 1-\epsilon$. Thus, let $\mathcal{D}_o = \{ \mathbf{d}: \mathbf{d}^T\Sigma_o^{-1}\mathbf{d} \leq F^{-1}(1-\epsilon) \}$, there is $\textnormal{Pr}(\mathbf{d}_o \in\mathcal{D}_o) = 1 - \epsilon$.
\end{proof}
\subsection{Proof of Theorem \ref{theorem:maximal_shadow}}\label{appendix:proof_maximal_shadow}
\begin{proof}
We need to prove that the set $\mathcal{S}_o$ contains the set $\mathcal{O}_o$ with probability $1-\epsilon$. It is equivalent to that for any point in $\mathcal{O}_o$, the set $\mathcal{S}_o$ contains this point with probability $1-\epsilon$. Recall the definition of $\mathcal{O}_o$, every $\mathbf{y} \in \mathcal{O}_o$ can be written as $\mathbf{x} + \mathbf{d}_o$ with some $\mathbf{x} \in \hat{\mathcal{O}}_o$. Also note the definition $\mathcal{S}_o = \{\mathbf{x} + \mathbf{d}~|~\mathbf{x}\in\hat{\mathcal{O}}_o,\mathbf{d}\in\mathcal{D}_o\}$. Hence the probability that $\mathcal{S}_o$ contains $\mathbf{y}$ is equal to the probability that $\mathcal{D}_o$ contains $\mathbf{d}_o$. That is, $\textnormal{Pr}(\mathbf{y} \in \mathcal{S}_o) = \textnormal{Pr}(\mathbf{d}_o \in \mathcal{D}_o)= 1 - \epsilon, \forall \mathbf{y} \in \mathcal{O}_o$. Thus, $\textnormal{Pr}(\mathcal{O}_o \subseteq \mathcal{S}_o) = 1-\epsilon$. $\mathcal{S}_o$ is a maximal $\epsilon$-shadow of $\mathcal{O}_o$.
\end{proof}
\section{Procedure to Compute the Best Linear Separator Between Two Gaussian Distributions}\label{appendix:best_linear_separator}
The objective is to solve the following minimax problem:
\begin{equation*}
(\mathbf{a}_{ij}, b_{ij}) = \arg\underset{\mathbf{a}_{ij}\in\mathbb{R}^d,b_{ij}\in\mathbb{R}}{\min\max}(\textnormal{Pr}_i, \textnormal{Pr}_j),
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\textnormal{Pr}_i(\mathbf{a}_{ij}^T\mathbf{p} > b_{ij}) &= 1 - \Phi((b_{ij} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i)/\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}), \\
\textnormal{Pr}_j(\mathbf{a}_{ij}^T\mathbf{p} \leq b_{ij}) &= 1 - \Phi((\mathbf{a}_{ij}^T\hat{\mathbf{p}}_j - b_{ij})/\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}).
\end{aligned}
\end{equation*}
Let $u_1 = \frac{b_{ij} - \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i}{\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}}$, $u_2 = \frac{\mathbf{a}_{ij}^T\hat{\mathbf{p}}_j - b_{ij}}{\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}}$. As the function $\Phi(\cdot)$ is monotonic, the original minimax problem is equivalent to
\begin{equation*}
(\mathbf{a}_{ij}, b_{ij}) = \arg\underset{\mathbf{a}_{ij}\in\mathbb{R}^d,b_{ij}\in\mathbb{R}}{\max\min}(u_1, u_2).
\end{equation*}
We can write $u_1$ in the following form for a given $u_2$,
\begin{equation*}
u_1 = \frac{\mathbf{a}_{ij}^T\hat{\mathbf{p}}_{ij} - u_2\sqrt{\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}}}{\sqrt{\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij}}},
\end{equation*}
where $\hat{\mathbf{p}}_{ij} = \hat{\mathbf{p}}_j - \hat{\mathbf{p}}_i$. For each given $u_2$, $u_1$ needs to be maximized. Hence, we can differentiate the above equation with respect to $\mathbf{a}_{ij}$ and set the derivative to equal to zero, which leads to
\begin{equation}\label{eq:procedure_a_ij}
\mathbf{a}_{ij} = [t\Sigma_i + (1-t)\Sigma_j]^{-1}\hat{\mathbf{p}}_{ij},
\end{equation}
where $t \in (0,1)$ is a scaler. Thus according to definition of $u_1$ and $u_2$, we have
\begin{equation}\label{eq:procedure_b_ij}
b_{ij} = \mathbf{a}_{ij}^T\hat{\mathbf{p}}_i + t\mathbf{a}_{ij}^T\Sigma_i\mathbf{a}_{ij} = \mathbf{a}_{ij}^T\hat{\mathbf{p}}_j - (1-t)\mathbf{a}_{ij}^T\Sigma_j\mathbf{a}_{ij}.
\end{equation}
It is proved that $u_1 = u_2$ must be hold for the solution of the minimax problem \citep{Anderson1962}, which leads to
\begin{equation}\label{eq:procedure_t}
\mathbf{a}_{ij}^T[t^2\Sigma_i - (1-t)^2\Sigma_j]\mathbf{a}_{ij} = 0.
\end{equation}
Thus, one can first solve for $t$ by combining Eqs. (\ref{eq:procedure_a_ij}) and (\ref{eq:procedure_t}) via numerical iteration efficiently. Then $\mathbf{a}_{ij}$ and $b_{ij}$ can be computed using Eqs. (\ref{eq:procedure_a_ij}) and (\ref{eq:procedure_b_ij}).
\section{Deadlock Resolution Heuristic}\label{appendix:deadlock}
We detect and resolve deadlocks in a heuristic way in this paper. Let $\norm{\Delta\mathbf{p}_i}$ be the position progress between two consecutive time steps of robot $i$, and $\Delta\mathbf{p}_{\min}$ a predefined minimum allowable progress distance for the robot in $n_{\tn{dead}}$ time steps. If the robot has not reached its goal and $\Sigma_{n_{\tn{dead}}}\norm{\Delta\mathbf{p}_i} \leq \Delta\mathbf{p}_{\min}$, we consider the robot as in a deadlock situation.
For the one-step controller, each robot must be at the ``projected goal'' $\mathbf{g}_i^*$ when the system is in a deadlock configuration \citep{Zhou2017}. In this case, each robot chooses one of the nearby edges within its B-UAVC to move along. For receding horizon planning of high-order dynamical systems, the robot may get stuck due to a local minima of the trajectory optimization problem. In this case, we temporarily change the goal location $\mathbf{g}_i$ of each robot by clockwise rotating it along the $z$ axis with $90\degree$, i.e.
\begin{equation}
\mathbf{g}_{i,\tn{temp}} = R_Z(-90\degree)(\mathbf{g}_i - \hat{\mathbf{p}}_i) + \hat{\mathbf{p}}_i,
\end{equation}
where $R_Z$ denotes the rotation matrix for rotations around $z$-axis. This temporary rotation will change the objective of the trajectory optimization problem, thus helping the robot to recover from a local minima. Once the robot recovers from stuck, its goal is changed back to $\mathbf{g}_i$.
Similar to most heuristic deadlock resolutions, the solutions presented here can not guarantee that all robots will eventually reach their goals since livelocks
(robots continuously repeat a sequence of behaviors that bring them from one deadlock situation to another one)
may still occur.
\section{Quadrotor Dynamics Model}\label{appendix:quad_model}
We use the Parrot Bebop 2 quadrotor in our experiments. The state of the quadrotor is
\begin{equation*}
\mathbf{x} = [\mathbf{p}^T, \mathbf{v}^T, \phi, \theta, \psi]^T \in \mathbb{R}^9,
\end{equation*}
where $\mathbf{p} = [p_x, p_y, p_z]^T \in \mathbb{R}^3$ is the position, $\mathbf{v} = [v_x, v_y, v_z]^T \in \mathbb{R}^3$ the velocity, and $\phi, \theta, \psi$ the roll, pitch and yaw angles of the quadrotor. The control inputs to the quadrotor are
\begin{equation*}
\mathbf{u} = [\phi_c, \theta_c, v_{z_c}, \dot{\psi}_c]^T \in \mathbb{R}^4,
\end{equation*}
where $\phi_c$ and $\theta_c$ are commanded roll and pitch angles, $v_{z_c}$ the commanded velocity in vertical $z$ direction, and $\dot{\psi}_c$ the commanded yaw rate.
The dynamics of the quadrotor position and velocity are
\begin{equation*}
\begin{cases}
\dot\mathbf{p} = \mathbf{v}, \\
\mat \dot{v}_x \\ \dot{v}_y \mate = R_Z(\psi)\mat \tan\theta \\ -\tan\phi \mate g - \mat k_{D_x}v_x \\ k_{D_y}v_y \mate, \\
\dot{v}_z = \frac{1}{\tau_{v_z}}(k_{v_z}v_{z_c} - v_z),
\end{cases}
\end{equation*}
where $g = 9.81~\tn{m}/\tn{s}^2$ is the Earth's gravity, $R_Z(\psi) = \mat \cos\psi &-\sin\psi \\ \sin\psi &\cos\psi \mate$ is the rotation matrix along the $z$-body axis, $k_{D_x}$ and $k_{D_y}$ the drag coefficient, $k_{v_z}$ and $\tau_{v_z}$ the gain and time constant of vertical velocity control.
The attitude dynamics of the quadrotor are
\begin{equation*}
\begin{cases}
\dot\phi = \frac{1}{\tau_{\phi}}(k_{\phi}\phi_c - \phi), \\
\dot\theta = \frac{1}{\tau_{\theta}}(k_{\theta}\theta_c - \theta), \\
\dot\psi = \dot{\psi}_c,
\end{cases}
\end{equation*}
where $k_{\phi}, k_{\theta}$ and $\tau_{\phi}, \tau_{\theta}$ are the gains and time constants of roll and pitch angles control respectively.
We obtained the dynamics model parameters $k_{D_x} = 0.25$, $k_{D_y} = 0.33$, $k_{v_z}=1.2270$, $\tau_{v_z}=0.3367$, $k_{\phi}=1.1260$, $\tau_{\phi}=0.2368$, $k_{\theta}=1.1075$ and $\tau_{\theta}=0.2318$ by collecting real flying data and performing system identification.
}
\bibliographystyle{spbasic}
\balance
|
2,877,628,090,195 | arxiv | \section{Introduction}
Quantum mechanics knows many examples of discrete states coupled to a
continuum. Much work has been devoted to these problems in the past. In
this paper we are going to review these works briefly, but reconsider them
as methods to prepare time dependent states. All tunnelling problems are of
this type, but in addition we have similar phenomena occurring when a
photon excitation is followed by leakage into continuous outgoing states.
A simple physical example is provided by autoionization, where an excited
state is coupled to atomic ionization according to
\begin{equation}
A+\hbar \omega \rightarrow A^{*}\rightarrow A^{+}+e^-. \label{e1}
\end{equation}
In molecules, the analogous process is the predissociation reaction
\begin{equation}
AB+\hbar \omega \rightarrow AB^{*}\rightarrow A+B, \label{e2}
\end{equation}
and in chemical reactions the transition state may give a similar behaviour
\begin{equation}
AB+C+\hbar \omega \rightarrow ABC^{*}\rightarrow A+BC. \label{e3}
\end{equation}
In a nucleus, $ABC^{*}$ would be a compound nucleus or a doorway state. In
particle physics, the particle production processes provide additional
examples.
When the final states form a continuous spectrum, the decay of the
initially prepared state is often described by an exponential. This was
first derived from quantum theory by Landau~\cite{r1}, but the procedure is
usually referred to by the names Weisskopf and Wigner~\cite{r2}. These
features were found to appear exactly in the treatment of the Lee
model~\cite{r3}, where, however, the primary interest was devoted to the
renormalizability of the theory. The appearance of decay as a consequence
of the analytic properties of the propagators became a central issue for
the theoretical discussion after Peierls had pointed out that the functions
needed to be continued to the second Riemann sheet~\cite{r4}. These
discussions can be found in Refs.~\cite{r5} and~\cite{r6}, and the results
are summarized in the context of scattering theory in the text book~\cite{r7}.
When several singularities are close to each other or multiple poles occur,
there appears non-exponential time dependence as discussed by Mower~\cite{r8};
the corresponding problem for a generalized Lee-model is
treated in Ref.~\cite{r9}. In molecules the high density of states gives
rise to a similar physical situation through nonradiative transfer of
excitation; for a review of this theory see Ref.~\cite{r10}. Many more
works from this early period could be cited, but these may suffice for the
present.
The works above are mainly formulated as scattering problems and, except
for the evaluation of probability decay, they tend not to look into the
detailed time evolution of the processes. The use of well controlled laser
pulses in the femtosecond time range has recently made it possible to
follow microscopic processes in space and time~\cite{ra1}. Thus one may
resolve the Schr\"odinger propagation following a fast preparation of an
initial state. It is then interesting to reconsider the theories above from
the point of view of state preparation. If we excite a resolved initial
state coupled to a continuum, which kind of wave packet can we prepare into
the continuum? How is this state emerging and how is it propagating?
This can be seen as a continuation of our earlier work on molecular
dynamics~\cite{r11} and on electrons in semiconductor structures~\cite{r12}.
Here we consider explicitly the time dependence following an
initial preparation. Our state functions can thus stay normalized at all
times in contrast to the situation in a scattering description. Thus we can
avoid paradoxes of exponentially growing amplitudes such as discussed by
Peierls in Ref.~\cite{r13}.
The preparation of well localized wave packets is an interesting and
challenging problem in atomic physics. Various excitation processes have
produced electronic wave packets on bound Rydberg manifolds; see e.g.
Refs.~\cite{rr0} and~\cite{rr1}, but it is not clear if these can be launched
into freely propagating states. Molecular dissociation, following coherent
excitation from the ground state~\cite{r11}, is expected to prepare
localized wave packets in the reaction coordinate. In simple cases, this
can be seen as propagation in the laboratory too. To prepare electronic
wave packets in semiconductors poses huge experimental difficulties~\cite{rr2};
to retain their coherence seems almost impossible. Excitonic
wave packets dephase more slowly, but their theoretical description is even
more difficult.
In this paper we adopt the presumption that an experimental procedure
exists to prepare a pure isolated state. The interaction with the continuum
is set in, and we can follow both the growth of the continuum states and
the decay of the initial one. This makes it possible to consider the
process as a state preparation. We can describe the disappearance of the
initial state and the growth and propagation of the state prepared in the
continuum.
In order to achieve our goal, we postulate the validity of a simple model
Hamiltonian. It is of the Lee model type and can hence be solved exactly to
give formal expressions for all quantities of interest. In order to treat
the initial value problem we utilize the Laplace transform, but in order to
retain agreement with the conventional Fourier transform treatments, we
denote the Laplace transform variable by $-i\omega$; the resulting
transform is named the Laplace-Fourier transform
(${\cal LF}$-transform).
We present the model in Sec.~2 and its formal solution in Sec.~3, where also
the general behaviour of the solution is discussed. In Sec.~4 we present
some simple models which illustrate the features of the general discussion.
In Sec.~4.1 we assume that the continuum spectral density can be described
by a simple complex pole, the generalization to several poles is
straightforward. This approximation has acquired renewed interest after
Garraway~\cite{r14} has showed that the multiple pole approximation can be
put into a Lindblad form amenable to a Monte Carlo simulation in terms of
hypothetical pseudomodes, which describe the non-Markovian effects.
However, he finds that consistency may require that the pseudomodes are
coupled in their Hamiltonian.
Section~4.2 presents a simple model where the role of analytic continuation
can be elucidated and the exact behaviour of the wave packet in the
continuum can be calculated. The role of the process as a preparation is
clearly seen in this model. In Sec.~4.3 we try to construct a physical
system having the properties of our earlier models.
This is selected from the field of molecular
physics~\cite{r11} and tests how weak coupling to a molecular continuum can be
utilized to prepare a wave packet. We find that this is, indeed, possible,
but the system has got many complementary features which we will discuss in
a forthcoming paper.
Section~5 presents a brief discussion of our results
and the conclusions.
\section{The model calculation}\setcounter{equation}{0}
We are considering a model with one single state embedded in a continuum.
This is described by the Hamiltonian
\begin{equation}
\begin{array}{lll}
H & = & H_0+V \\
& & \\
H_0 & = & \omega_0\mid 0\rangle \langle 0\mid +\int d\epsilon \mid
\epsilon \rangle \,\epsilon \,\langle \epsilon \mid \\ & & \\
V & = & \int d\epsilon \;\left( V_\epsilon \mid \epsilon \rangle \langle
0\mid +V_\epsilon ^{*}\mid 0\rangle \langle \epsilon \mid \right)
\end{array}
\label{a1}
\end{equation}
in an obvious notation. Observe that this is the form of a general
tunnelling Hamiltonian. The initial state is supposed to be the isolated
state
\begin{equation}
\mid \psi (t=0)\rangle =\mid 0\rangle . \label{a2} \end{equation}
Because we are going to consider time evolution, we introduce the
Laplace-Fourier transform in the form
\begin{equation}
\widetilde{\psi }(\omega )=-i\int\limits_0^\infty dt\, e^{i\omega t}\psi
(t)\,\equiv {\cal LF}(\psi ). \label{a3}
\end{equation}
This gives us analytic functions in the conventional half-plane ${\rm Im}(\omega
)\geq 0$. We have the usual properties
\begin{equation}
\begin{array}{lll}
{\cal LF}(\dot \psi ) & = & -i\left( \omega \widetilde{\psi }(\omega
)-\psi (t=0)\right) \\
& & \\
\lim\limits_{\omega \rightarrow \infty }\left( \omega \widetilde{\psi }%
(\omega )\right) & = & \psi (t=0)\equiv \psi_0 \\ & & \\
\lim\limits_{\omega \rightarrow 0}\left( \omega \widetilde{\psi }(\omega
)\right) & = & \psi (t=\infty ).
\end{array}\label{a4}
\end{equation}
The inversion is achieved by
\begin{equation}
\psi (t)=\frac i{2\pi }\int\limits_{-\infty +ia}^{+\infty +ia}e^{-i\omega
t}\;\widetilde{\psi }(\omega )\,dt, \label{a5}
\end{equation}
where $a>0$. With these definitions the solution of the Schr\"odinger
equations is given by
\begin{equation}
\widetilde{\psi }(\omega )=G(\omega )\psi_0, \label{a6}
\end{equation}
where the propagator is
\begin{equation}
G(\omega )=\frac 1{\omega -H}. \label{a7}
\end{equation}
It is easy to verify the consistency of these relations.
We are now going to partition the problem above in order to separate the
time evolution of the single state and that of the continuum. Thus we
introduce the projectors
\begin{equation}
\begin{array}{lll}
P & = & \mid 0\rangle \langle 0\mid \\
& & \\
Q & = & 1-P=\int d\epsilon \mid \epsilon \rangle \,\langle \epsilon \mid .
\end{array}\label{a8}
\end{equation}
Both projectors commute with the Hamiltonian $H_0.$
The results that we need were derived long ago in scattering theory, but
for easy reference we summarize them in the Appendix. We introduce the
definition
\begin{equation}
O^{XY}\equiv XOY, \label{a9}
\end{equation}
where $O$ is any operator and $X,Y=P$ or $Q;$ the operators $O^{XX}$ are
denoted by $O^X$ simply.
We introduce a connected operator $\Gamma $ which has the single state pole
removed from the intermediate states by writing \begin{equation}
\Gamma =V+V\,G_0^Q\,\Gamma , \label{a10} \end{equation}
where the unperturbed propagator is
\begin{equation}
G_0(\omega )=\frac 1{\omega -H_0}. \label{a11} \end{equation}
In terms of this operator, we can now write the exact solution for all the
propagators needed in the form
\begin{eqnarray}
G^P&=&\left( (G_0^P)^{-1}-\Gamma ^P\right) ^{-1} \label{a12a}\\
G^{QP}&=&G_0^QV^{QP}G^P \label{a12b}\\
G^Q&=&G_0^Q+G_0^Q\left( \Gamma ^Q+\Gamma ^{QP}G^P\Gamma ^{PQ}\right) G_0^Q.
\label{a12c}
\end{eqnarray}
Thus we can obtain all desired partitioned propagators if we can solve the
integral equation (\ref{a10}). In the present case, this can be solved
trivially if we notice that
\begin{equation}
V^P=V^Q=0. \label{a13}
\end{equation}
From this it follows that
\begin{eqnarray}
\begin{array}{lll}
\Gamma ^{QP} & = & V^{QP} \\
& & \\
\Gamma ^P & = & V^{PQ}G_0^QV^{QP}.
\end{array}\label{a14}
\end{eqnarray}
This gives the solutions
\begin{eqnarray}
\langle 0\mid G\mid 0\rangle &=&\frac 1{\omega -\omega_0-\Sigma (\omega )}
\label{a15a}\\
\langle \epsilon \mid G\mid 0\rangle &=& \left( \frac 1{\omega -\epsilon
}\right) V_\epsilon \left( \frac 1{\omega -\omega_0-\Sigma (\omega
)}\right) \label{a15b}\\
\langle \epsilon \mid G\mid \epsilon ^{^{\prime }}\rangle &=&\frac{\delta
(\epsilon -\epsilon ^{^{\prime }})}{\omega -\epsilon }+\left( \frac{%
V_\epsilon \,}{\omega -\epsilon }\right) \left( \frac{V_{\epsilon ^{\prime
}}^{*}}{\omega -\epsilon ^{^{\prime }}}\right) \left( \frac 1{\omega
-\omega_0-\Sigma (\omega )}\right) . \label{a15c}
\end{eqnarray}
The self-energy function is given by
\begin{equation}
\Sigma (\omega )=\langle 0\mid \Gamma \mid 0\rangle =\int d\epsilon \left(
\frac{D(\epsilon )}{\omega -\epsilon }\right) , \label{a16}
\end{equation}
where we have introduced the spectral density \begin{equation}
D(\epsilon )=\mid V_\epsilon \mid ^2\geq 0. \label{a19} \end{equation}
This notation is preferable because we can then include a density-of-states
function in the spectral density $D(\omega )$ when needed.
With the initial condition (\ref{a2}) we find the amplitude of the isolated
state to be
\begin{equation}
A(t)\equiv \langle 0\mid e^{-iHt}\mid 0\rangle =\frac i{2\pi
}\int\limits_{-\infty +ia}^{+\infty +ia}e^{-i\omega t}\;\left( \frac
1{\omega -\omega_0-\Sigma (\omega )}\right) \,dt. \label{a17}
\end{equation}
Because the integrated function is analytic in the upper half plane, the
time evolution is determined by the singularities of the integrand on the
real axis and below. We are thus looking for solutions of the equation
\begin{equation}
h(\omega )\equiv \omega -\omega_0-\Sigma (\omega )=0. \label{a18}
\end{equation}
As can be seen from Eq.~(\ref{a16}) the function $\Sigma (\omega )$ has got
a branch cut along the real axis. Its weight is given by $D(\omega ).$
\section{General properties of the solution} \setcounter{equation}{0}
From the definition (\ref{a16}) it follows directly that
\begin{equation}
{\rm Im}\,\Sigma (\omega \pm i\eta )=\mp \pi D(\omega ); \label{a20}
\end{equation} it is assumed that $\eta \rightarrow 0.$ Consequently, the
function has a branch cut at every value of $\omega $ such that $D(\omega )$
differs from zero.
All physical systems have energies bounded from below, thus all integrals
over the variable $\epsilon $ must start at a finite value $\mu .$ If the
energy of the isolated state lies below this spectral cut-off
\begin{equation}
\omega_0<\mu , \label{c1}
\end{equation}
the continuum only serves to renormalize the state. Then we may write
\begin{equation}
\Sigma (\omega)=\int d\epsilon \;\frac{D(\epsilon )}{\omega
-\omega_0-(\epsilon -\omega_0)}=\int d\epsilon \;\frac{D(\epsilon )}
{(\omega_0-\epsilon )}-\int d\epsilon \;\frac{D(\epsilon )}{(\omega_0-\epsilon
)^2} \;\left( \omega -\omega_0\right) + ... \label{c2}
\end{equation}
This gives the propagator (\ref{a15a}) the form
\begin{equation}
\langle 0\mid G\mid 0\rangle =\frac{Z}{\omega -\tilde{\omega}_0},
\label{c3}
\end{equation}
where the state renormalization constant is given by
\begin{equation}
Z=\left( 1+\int d\epsilon \;\frac{D(\epsilon )}{(\omega_0-\epsilon )^2}
\right) ^{-1} \label{c4}
\end{equation}
and the renormalized energy is
\begin{equation}
\tilde{\omega}_0=\omega_0+Z\int d\epsilon \;\frac{D(\epsilon )}{
(\omega_0-\epsilon)}<\omega_0. \label{c5}
\end{equation}
These results are valid to order $\left( \frac{\omega_0-\tilde{\omega}_0 }
{\mu -\omega_0}\right)$, and we still have $\tilde{\omega}_0<\mu$.
The only effect of the continuum is to push the isolated level
away and decrease the overlap between the initial bare state and the
renormalized state to $Z<1$. This state does not decay but continues
oscillating with the frequency $\tilde{\omega}_0$ forever. The
reminder of the initial state is lost into the continuum as the coupling is
switched on.
When the isolated state is embedded in the continuum, $\omega_0>\mu$, it
can decay. In the equation (\ref{a17}) we require the zeros of the function
$ h(\omega)$ to be in the lower half plane. Let us see if such
singularities exist by setting
\begin{equation}
\omega =\omega ^{\prime}-i\omega ^{\prime \prime } \label{a21}
\end{equation}
into $h(\omega)$. We find
\begin{equation}
\omega ^{\prime }-i\omega ^{\prime \prime }-\omega_0-\int d\epsilon
\;\frac{ D(\epsilon )}{\omega ^{\prime }-i\omega ^{\prime \prime
}-\epsilon }=0. \label{a22}
\end{equation}
Separating the real and imaginary parts of this equation should fix the
oscillational frequency $\omega ^{\prime }$ and the damping $\omega
^{\prime \prime }.$ We write down the imaginary part as \begin{equation}
\omega ^{\prime \prime }\left( 1+\int d\epsilon \;\frac{D(\epsilon
)}{\left( \omega ^{\prime }-\epsilon \right) ^2+\left( \omega ^{\prime
\prime }\right) ^2}\right) =0. \label{a23}
\end{equation}
This equation obviously lacks solutions $\omega ^{\prime \prime }\neq 0$
and does not correspond to a physical result. This is obtained when we
remember that the physical values of $\omega $ approach the real axis from
the upper half plane, and we should thus look for zeros when the function
$\Sigma (\omega )$ is continued analytically into the lower half plane.
This procedure will provide the singularities giving a contribution to the
integral (\ref{a17}). We consequently need to continue $\Sigma (\omega )$
across the real axis as discussed by Peierls~\cite{r14}.
The simplest manner to do this is to push the integration contour in
Eq.~(\ref {a17}) down below the real axis. Any isolated pole encountered in
this manner is included as a single contribution to the time evolution in
the manner shown in Fig.~1. For one single pole this works well, and the
result is simple exponential decay as in the Weisskopf-Wigner approach.
However, with several singularities only well isolated poles can be treated
this way. When the poles are close to each other, this method cannot be
justified, because its validity is based on a series expansion like that in
Eq.~(\ref{c2}) around each singularity; the radius of convergence can only be
extended to the nearest singularity, and hence close lying poles
interfere~\cite{r8}. Another limitation comes from the lower cut-off
in the spectral
weight $ D(\omega)$. Because $D(\omega)=0$ for $\omega <\mu $, this end
point must be a singularity of the function. Thus there is a branch cut
starting here, which can be moved around but must be pinned down at $\omega
=\mu$. Poles too close to this cut-off cannot have a large radius of
convergence and hence the branch cut cannot be neglected in their
treatment. In particle theory these cuts derive from particle creation
thresholds.
\begin{figure}
\vspace*{-0.5cm}
\centerline{\psfig{width=2.4in,file=JMOfig1.eps}}
\caption[f1]{In the complex $\omega $-plane, the Laplace-Fourier transform is
inverted by integration along the line $[-\infty +ia,+\infty +ia].$ The analytic
continuation can be effected by moving the contour into the lower
half-plane. When a pole is encountered, the contour must circle it, but can
continue down leaving the pole encircled by an isolated contour.}
\end{figure}
In the region where $D(\omega)$ is an analytic function of $\omega$, we
can achieve the analytic continuation of the self-energy by defining the
new function
\begin{equation}
\Sigma ^{+}(\omega )=\int d\epsilon \left( \frac{D(\epsilon )}{\omega
-\epsilon }\right) -2\pi iD(\omega ). \label{a24} \end{equation}
Using (\ref{a20}) one can easily verify that the function $\Sigma
^{+}(\omega )$ has no singularity when the real axis is crossed from above.
When this process works, it is a simple way to look for singularities on
the second sheet of the lower half plane.
When the result (\ref{a24}) is inserted into the equation (\ref{a18}) we
find the relation
\begin{equation}
\omega ^{^{\prime }}-\omega_0-\int d\epsilon \,D(\epsilon )\left( \frac{
\omega ^{^{\prime }}-\epsilon }{\left( \omega ^{^{\prime }}-\epsilon
\right) ^2+\omega ^{^{\prime \prime }2}}\right) +2\pi \frac{\partial
D(\omega ^{^{\prime }})}{\partial \omega }\omega ^{^{\prime \prime }}=0,
\label{a25} \end{equation}
where we have anticipated that the imaginary part is small. In addition we
have
\begin{equation}
\omega ^{^{\prime \prime }}=2\pi iD(\omega ^{^{\prime }})-\int d\epsilon
\,D(\epsilon )\left( \frac{\omega ^{^{\prime \prime }}}{\left( \omega
^{^{\prime }}-\epsilon \right) ^2+\omega ^{^{\prime \prime }2}}\right) .
\label{a26}
\end{equation}
The values of the oscillational frequency $\omega ^{^{\prime }}$ and the
damping $\omega ^{^{\prime \prime }}$ are determined by the coupled
equations (\ref{a25}) and (\ref{a26}). The oscillational part determined
from (\ref{a25}) is discussed in detail by Cohen-Tannoudji~\cite{rrr1}. In
the limit $\omega ^{^{\prime \prime }}\rightarrow 0$ the results simplify.
When the damping is small, we obtain
\begin{equation}
\omega ^{^{\prime \prime }}\Rightarrow \pi D(\omega ), \label{a26a}
\end{equation}
which directly provides a consistency check. The Weisskopf-Wigner result
thus emerges in the weak coupling limit from our prescription for analytic
continuation.
It is instructive to realize that in the weak coupling limit, the
prescription we have used for analytic continuation is to replace $\Sigma
(\omega )$ just below the real axis by its value just above; i.e. we are
performing an analytic continuation using only the zeroth order term in a
Taylor expansion across the real axis. Thus the analytically continued
equation (\ref{a22}) becomes
\begin{equation}
\omega ^{\prime }-i\omega ^{\prime \prime }-\omega_0-\int d\epsilon
\;\frac{ D(\epsilon )}{\omega ^{\prime }+i\omega ^{\prime \prime
}-\epsilon }=0. \label{a27}
\end{equation}
For small values of $\omega ^{\prime \prime }$ this gives back all results
derived previously. It is to be noted that, inside the integral, the
precise value of $\omega ^{\prime \prime }$ does not affect the result, and
we can obtain the correct damping using the conventional infinitesimal
$i\eta $ prescription. Our discussion is in fact a derivation of this rule.
An instructive way to consider the equation (\ref{a27}) is to regard the
imaginary part in the denominator as a small dissipative part of the
energies in the continuum
\begin{equation}
\epsilon -i\omega ^{\prime \prime }\Rightarrow \epsilon -i\epsilon
^{^{\prime }}. \label{a28}
\end{equation}
This provides an interesting physical interpretation of the results. Adding
any minute dissipative mechanism to the continuum we will find that the
isolated state will decay with the rate (\ref{a26a}) independently of the
details of the dissipation of the continuum. Thus any system coupled
perturbatively to a dissipative probability sink through a continuum will
decay with a rate determined by the coupling strength and not by the actual
dissipation of the continuum. In the perturbative limit, the probability
flow into the continuum will be the bottleneck. The dissipated continuum
acts as a proper reservoir unable to return the probability once it has
received it. This interpretation of the procedure directly provides the
right sign in the denominator of (\ref{a27}) to effect the proper analytic
continuation.
When we calculate the integral in Eq.~(\ref{a17}), we distort the contour
into the lower half plane until we encounter the pole defined by
(\ref{a25}) and separate a contour $c_0$ circling this. If no other
singularity were encountered, we could pull the contour to negative
imaginary infinity and find no corrections. In a physical system this is,
however, impossible because the spectral density $D(\omega )$ has
necessarily got the lower limit $\mu $ as explained above. The branch cut
starting here is pulled down to $-i\infty $ in the manner shown in Fig.~2.
This contributes an additional contour $c_1$ as shown in the figure. Thus
we find
\begin{eqnarray}
A(t) & = & \frac{i}{2\pi}\left[ \int_{c_1} dt\,e^{-i\omega t}\;\left(
\frac{1}{\omega -\omega_0-\Sigma (\omega )}\right)
+\int_{c_0} dt\,e^{-i\omega t}\;\left( \frac{1}{\omega -\omega
^{^{\prime }}+i\omega ^{^{\prime \prime }}}\right) \right] \nonumber\\
& = & I_1(t)+\exp \left( -i\omega ^{^{\prime }}t-\omega^{^{\prime \prime
}}t\right) .\label{a29}
\end{eqnarray}
The second term is as expected in a perturbative calculation of the
Weisskopf-Wigner type, and we have taken the residue $Z$ of (\ref{c4}) to
be unity. The decay rate of the probability is given by the well known
expression
\begin{equation}
\gamma =2\omega ^{^{\prime \prime }}=2\pi D(\omega ). \label{a30}
\end{equation}
However, the corrections from the cut are given as
\begin{equation}
I_1(t)=\frac{e^{-i\mu t}}{2\pi }\int\limits_0^\infty e^{-\xi t}\left(
\frac{ \Sigma_I(\mu -i\xi +\eta )-\Sigma_{II}(\mu -i\xi -\eta )}{\left(
\omega -\omega_0-\Sigma_I\right) \left( \omega -\omega_0
-\Sigma_{II}\right) } \right) d\xi , \label{a31}
\end{equation}
where the infinitesimal parameter $\eta $ ensures that the self energy is
evaluated on the two sides of the cut. From this form we can see that in
the long time limit $t\rightarrow \infty $, only the value $\xi \approx 0$
contributes. This would give no contribution without the singularity of $
\Sigma $ across the cut. Thus we can set $\xi =0$ everywhere except in the
integral
\begin{eqnarray}
\Sigma_I(\mu -i\xi +\eta )-\Sigma_{II}(\mu -i\xi -\eta ) & = &
\int_{\mu}^{\infty} d\epsilon \;D(\epsilon )\left[ \frac{1}{\mu -i\xi
+\eta -\epsilon }-\frac{1}{\mu -i\xi -\eta -\epsilon }\right] \nonumber\\
& = & \int\limits_0^\infty d\nu \;D(\mu +i\nu )\left[ \frac{1}{\xi +\nu
+i\eta }-\frac{1}{\xi +\nu -i\eta }\right] \label{a32}\\
& = & -2\pi iD(\mu +i\xi ).\nonumber
\end{eqnarray}
Because $\omega =\mu $ is a threshold, we expect that we have
\begin{equation}
D(\mu +i\xi )=\beta \;\left( i\xi \right) ^\alpha +... \label{a33}
\end{equation}
for small values of $\xi .$ Inserting this into the equation (\ref{a31}),
we find the leading term
\begin{equation}
I_1(t)=\frac{\beta \,e^{-i\mu t}\;i^{\alpha +3}}{\left( \mu -\omega_0
-\Sigma (\mu )\right) ^2}\Gamma (\alpha +1)\left( \frac 1{t^{\alpha
+1}}\right) . \label{a34}
\end{equation}
This type of asymptotic result was given for scattering theory already in
Ref.~\cite{r7}. When the exponential in (\ref{a29}) has decayed, the power
dependence (\ref{a34}) will dominate for long times.
\begin{figure}
\vspace*{-0.5cm}
\centerline{\psfig{width=3.4in,file=JMOfig2.eps}}
\caption[f2]{Because physical spectra have a lower limit $\mu $, there must be a
branch cut starting at this value. When isolated poles are encircled by closed
contours, here $c_0$, the branch cut can be drawn from $\mu $ to $-\infty$,
thus contributing another contour $c_1$.}
\end{figure}
The treatment given above does, of course, assume that there are no
additional poles encountered. If these are well separated, they contribute
simply additional exponential terms, but for poles situated close to each
other interferences occur; for a discussion of these cf. the discussion by
Mower~\cite{r8}.
\section{Simple models}
\subsection{A pole in the spectral density}
Many systems display a moderately sharp spectral feature in its density of
states. Hence one may often approximate this with a Lorentzian shape, which
simplifies the treatment considerably. Such assumptions are often utilized
to describe scattering from condensed matter~\cite{rrr2}, but it is useful
in theoretical discussions too. Recently Garraway has utilized it to
describe non-Markovian effects in terms of pseudomodes~\cite{r14}.
We assume that the spectral function has the simple shape \begin{equation}
D(\epsilon )=\frac{A^2}{\left( \epsilon -a\right) ^2+b^2}. \label{a35}
\end{equation}
This satisfies the positivity requirement, but it lacks the lower limit
necessary in physical systems. Thus no power law terms are expected.
With the expression (\ref{a35}), the analytic continuation to the second
sheet is trivial. Inserted into Eqs. (\ref{a16}) and (\ref{a15a}) it gives
the results
\begin{equation}
\begin{array}{lll}
\Sigma (\omega ) & = & \frac{\textstyle\pi A^2}{\textstyle b}\left(
\frac{\textstyle 1}{\textstyle\omega -a+ib}\right)
\\
& & \\
\langle 0\mid G(\omega )\mid 0\rangle & = & \frac{\textstyle\omega -a+ib}
{\textstyle (
\omega -a+ib)( \omega -\omega_0) -\frac{\textstyle\pi A^2}{\textstyle b}}.
\end{array}
\label{a36}
\end{equation}
This form is found to satisfy the correct initial condition
\begin{equation}
\lim_{\omega \rightarrow \infty }\omega \langle 0\mid G(\omega )\mid
0\rangle \,\psi (0)=\psi (0). \label{a37} \end{equation}
The general solution of the time evolution becomes \begin{equation}
A(t)=\left( R_1\,e^{i\Omega_{+}t}+R_2\,e^{i\Omega_{-}t}\right) ,
\label{a38}
\end{equation}
where $\Omega_{\pm }$ are the two roots of the denominator in
Eq.~(\ref{a36}); from the form of the equation it follows that the sum of
the residues is unity: $R_1+R_2=1$.
Inserting the ansatz (\ref{a21}) into the denominator of Eq.~(\ref{a36}) we
find that the imaginary part follows as the solution of the equation
\begin{equation}
\omega ^{^{\prime \prime }}=\left( \frac{\pi A^2}{b}\right) \frac{\left(
b-\omega ^{^{\prime \prime }}\right) }{\left( a-\omega ^{^{\prime }}\right)
^2+\left( b-\omega ^{^{\prime \prime }}\right) ^2}. \label{a39}
\end{equation}
Because the right hand side of this equation is positive for $\omega
^{^{\prime \prime }}<b$ there must be a solution for $0<\omega ^{^{\prime
\prime }}<b$, irrespective of the value of $\omega ^{^{\prime }}.$ This
proves that both roots $\Omega_{\pm }$ have negative imaginary parts and
the solution $A(t)$ decays to zero as we expect. Thus the pole
approximation is consistent and only fails to reproduce the long time
behaviour deriving from the threshold in the spectral density. In a similar
manner the appearance of several poles can be discussed. Of special
interest is the case when a pole with negative weight is encountered. This
case is discussed in detail by Garraway~\cite{r14a}.
\subsection{The branch cut model}
In order to illustrate the concepts introduced in Sec.~3 we introduce a
model where the Weisskopf-Wigner result emerges as an exact consequence. It
is then possible to follow how the analytic continuation to the physical
sheet works in detail, and the correction terms can be evaluated
explicitly.
The model is based on the following spectral density
\begin{equation}
D(\epsilon )=\left\{
\begin{array}{cc}
A^2; & \mid \epsilon \mid <L \\ & \\
0; & \mid \epsilon \mid >L.
\end{array}\right.
\label{a40}
\end{equation}
It is taken to be an essential feature of the model that $L$ is taken to be
large (infinite) in the final expressions. For finite $L,$ it is trivial to
compute the self-energy from (\ref{a16}) to be \begin{equation}
\Sigma (\omega )=A^2\log \left( \frac{-\left( L+\omega \right) }{\left(
L-\omega \right) }\right) . \label{a41}
\end{equation}
In this model we can explicitly see that the sign of the imaginary part
clearly depends on the choice of branch of the function involved. We use
the discussion in Sec.~3 to introduce the analytic continuation (\ref{a24})
\begin{equation}
\Sigma ^{+}(\omega )=A^2\left[ \log \left( \frac{-\left( 1+\frac \omega
L\right) }{\left( 1-\frac \omega L\right) }\right) -2\pi i\right] .
\label{a42}
\end{equation}
If we now introduce the ansatz
\begin{equation}
\omega =\omega ^{^{\prime }}-i\omega ^{^{\prime \prime }}. \label{a43}
\end{equation}
we find that
\begin{equation}
\log \left( \frac{-\left( 1+\frac \omega L\right) }{\left( 1-\frac \omega
L\right) }\right) =\log \left( \frac{R_{-}}{R_{+}}\right) +i\left(
\varphi_{-}-\varphi_{+}\right) , \label{a44}
\end{equation}
where we have
\begin{equation}
\frac{R_{-}}{R_{+}}=\sqrt{\frac{\left( 1+\frac{\omega ^{^{\prime
}}}L\right) ^2+\left( \frac{\omega ^{^{\prime \prime }}}L\right) ^2}{\left(
1-\frac{ \omega ^{^{\prime }}}L\right) ^2+\left( \frac{\omega ^{^{\prime
\prime }}} L\right) ^2}} \label{a45}
\end{equation}
and
\begin{equation}
\varphi_{\mp }=\arctan \left( \frac{\omega ^{^{\prime \prime }}}{\left(
L\pm \omega ^{^{\prime }}\right) }\right) . \label{a46} \end{equation}
These relations are illustrated in Fig.~3. It is now easy to see that
independently of whether we choose the cut of the logarithm function from
the origin to $+\infty $ or to $-\infty ,$ the limit becomes
\begin{equation}
\lim_{L\rightarrow \infty }\Sigma ^{+}(\omega )=-i\,\pi A^2, \label{a47}
\end{equation}
giving exactly the decay constant
\begin{equation}
\gamma =2\pi A^2; \label{a48}
\end{equation}
cf. Eq.~(\ref{a30}). The frequency shift vanishes in this limit. Our
treatment is not mathematically rigorous, and the contributions from the
branch cut are not evaluated. They disappear when the lower spectral limit
$ -L$ goes to infinity. It is, however, possible to evaluate the
correction terms to order $L^{-1}$ and verify the over-all validity of the
picture we have developed. The model can also be extended to treat an
asymmetric spectral weight function defined on the interval
$[L_{-},L_{+}].$ Here the limit appears slightly differently and
modifications of the results above can be found.
\begin{figure}[tbh]
\hspace*{-2cm}\psfig{width=6.4in,file=JMOfig3.ps}
\vspace*{-2.5cm}
\caption[f3]{The branch-cut model gives the self energy $\Sigma$ in terms of
two vectors $R_{\pm }\exp \left( i\varphi_{\pm }\right)$ in the complex plane
as shown.}
\end{figure}
With the result (\ref{a48}) we find an exact exponential disappearances of
the amplitude of the initial state in Eq.~(\ref{a17}). It is, however,
instructive to evaluate the state in the continuum part too. The
probability of emergence of this state cannot grow faster than the isolated
state disappears, and consequently we expect it to need a time of the order
$ \gamma ^{-1}$ to appear. From Eq.~(\ref{a15b}) we find for the continuum
wave function
\begin{eqnarray}
\Psi_{cont}(x,t) & = & \int \int d\epsilon \,\phi_\epsilon (x)\langle
\epsilon \mid G(\omega )\mid 0\rangle \;e^{-i\omega t}\frac{id\omega
}{2\pi } \nonumber\\
& = & \int d\epsilon \,\phi_\epsilon (x)\left( \frac{V_\epsilon
}{\epsilon -\omega_0+i\frac \gamma 2}\right) \left( e^{-i\epsilon
\,t}-e^{-i\omega_0t}e^{-\gamma t/2}\right) \label{a49} \\
& = & \Phi (x,t)-e^{-i\omega_0t}e^{-\gamma t/2}\Phi (x,0). \nonumber
\end{eqnarray}
Here $\phi_\epsilon (x)$ is the continuum eigenfunction corresponding to
the eigenvalue $\epsilon $, and $\Phi (x,t)$ is the state emerging from the
process on the corresponding Hilbert space. This has got the spectral
distribution
\begin{equation}
P(\epsilon )=\frac{\gamma /2\pi }{\left( \epsilon -\omega_0\right)
^2+\left( \frac \gamma 2\right) ^2} \label{a50} \end{equation}
as known from the Weisskopf-Wigner calculation. In addition, we can see how
the continuum state grows from an initial zero value to the emerging wave
state $\Phi (x,t),$ which will travel according to the dynamics in the
continuum as determined by the spectral distribution (\ref{a50}). This
feature has usually not been discussed in the ordinary decay problems
treated in the literature.
In the present model the analytic behaviour can be seen directly. This is
useful if we want to gain understanding of the features playing the central
role in the general treatment. We can also calculate the emergence of the
outgoing wave packet explicitly (\ref{a49}). In order to see the shaping of
this wave packet we consider one more model where the features discussed
can be followed numerically.
\subsection{A wave packet model}
\begin{figure}[tbh]
\hspace*{-2cm}\psfig{width=6.4in,file=JMOfig4.ps}
\vspace*{-2.5cm}
\caption[f4]{The molecular energy levels have one harmonic potential and one
straight slope. The parameters are chosen such that the energies of the two
levels coincide at $x=0$.}
\end{figure}
As a model we choose a coupled pair of energy levels with one harmonic
potential and one potential slope providing a continuum. The Schr\"odinger
equation is
\begin{equation}
i\frac \partial {\partial t}\left[
\begin{array}{c}
\psi_1 \\ \psi_2
\end{array} \right] =\left[
\begin{array}{cc}
\left( -\frac{\partial ^2}{\partial x^2}+\frac 12x^2\right) & V \\ V &
\left( -\frac{\partial ^2}{\partial x^2}-\beta x+\frac 1{\sqrt{2}}\right)
\end{array}\right] \left[
\begin{array}{c}
\psi_1 \\ \psi_2
\end{array}\right] . \label{a51}
\end{equation}
This corresponds to a scaled molecular problem with the parameters
\begin{equation}
m=\frac 12; \qquad \omega =\sqrt{2}. \label{a52}
\end{equation}
Because the zero point energy is given by \begin{equation}
\frac 12\omega =\frac 1{\sqrt{2}}, \label{a53} \end{equation}
the ground state energy of the harmonic oscillator coincides with the
continuum energy at the origin; see Fig.~4. To the extent we can neglect the
influence of the first excited harmonic oscillator state, we can consider
this model as a realization of a situation where the initial oscillator
ground state
\begin{equation}
\psi_1(x,t=0)=(\pi \sqrt{2})^{-1/4}\exp \left(
-\frac{x^2}{2\sqrt{2}}\right) \label{a54}
\end{equation}
is embedded in the continuous spectrum of the slope; the coupling is given
by $V.$ When this is in the perturbative regime \begin{equation}
\frac{V^2}\beta \ll \omega \label{a55}
\end{equation}
we expect the Weisskopf-Wigner treatment to hold and the initial state to
decay nearly exponentially. Setting
\begin{equation}
V=0.5; \qquad \beta =3.0 \label{a56}
\end{equation}
we satisfy (\ref{a55}) and obtain the decay of the initial state
\begin{equation}
P_1(t)=\int \mid \psi_1(x,t)\mid ^2dx \label{a57} \end{equation}
shown in Fig.~5. As seen, there is a clear range of times over which the
initial state decays exponentially as expected. By looking at the state
emerging on the second energy level $\psi_2(x,t)$ shown in Fig.~6 we can
see that the emerging state like (\ref{a49}) indeed produces a wave packet
which travels down the potential slope while it spreads according to the
requirements of quantum mechanics. In the present model, however, the
initial state is not depleted totally by the exponential decay. This
derives from the fact that the crossing potential surfaces form quasibound
states in the adiabatic representation, and the population trapped in these
states can then ooze out only over times much longer than those
characterizing the exponential decay.
The model calculation presented in this section shows that our system (\ref
{a51}) can serve as an illustration of Weisskopf-Wigner decay, and it
proves that such decay can serve as a wave packet preparation device.
However, we have found that even this simple system works only when the
parameters are chosen right, and it contains features not expected from the
simpler models discussed above. We consider these aspects in a forthcoming
publication, where we shall present the general behaviour of the harmonic
state resonantly coupled to a sloped potential. Here we have only presented
a single case as an illustration of our general theoretical considerations.
\section{Discussion}\setcounter{equation}{0}
\begin{figure}[tbh]
\vspace*{-0.5cm}
\centerline{\psfig{width=4.8in,file=JMOfig5.eps}}
\caption[f5]{The system is prepared on the ground state of the harmonic
potential,
and the probability that it remains there is plotted as a function of time. The
probability leaves the level in an exponential manner as this
semilogarithmic plot shows.}
\end{figure}
In this paper we have reviewed the scattering theory approach to
partitioning of the Hamiltonian time evolution in a quantum system. We are
especially interested in the situation where one (or more) discrete states
leak into a continuum, i.e., decay of quasistationary states. By choosing a
Laplace-Fourier transform in time, we concentrate on solving an initial
value problem and following the system as it irreversibly transfers its
probability into the continuum. This is regarded as a wave packet
preparation procedure.
The emphasis on time evolution has been motivated by the recent
experimental progress in pulsed laser technology. It is now possible to
excite a single state selectively, control its coupling to other states,
including continuum ones, and follow the ensuing time evolution of the
quantum states. Such work has experienced tremendous progress in molecular
investigations recently, but also time resolved spectroscopy of
semiconductor structures is possible.
Starting from a giving initial time, the future evolution of quantum
systems is determined by the analytic properties of the propagator
functions in the complex frequency plane. In order to obtain the correct
physical behaviour these have to be continued analytically to a second
Riemann sheet in the lower half plane. Any pole encountered will give rise
to a resonance but the physical behaviour must also receive contributions
from unavoidable branch cuts.
\begin{figure}[tbh]
\vspace*{-0.5cm}
\centerline{\psfig{width=4.5in,file=JMOfig6.ps}}
\caption[f6]{The probability $|\psi_2(x,t)|^2$ leaving the bound oscillator
state appears on the sloping continuum potential. Here we can see it appear
and fall down the slope in the form of a spreading wave packet. Part of the
probability gets captured near the position of the original bound state
because of trapping in the potential well formed by the adiabatic levels.
This contains a component on the continuous manifold, which explains the
trapped portion. Even this must eventually leave the trap but clearly over
a much longer time scale.}
\end{figure}
The necessary analytic continuation introduces simple relations between the
self-energy function in the upper half-plane and that in the lower. In the
weak coupling limit, the Weisskopf-Wigner case, the prescription for
continuation is found to be equivalent with adding a small dissipative
mechanism to the states in the continuum. The causality requirement on the
sign of this dissipation automatically affects the necessary analytic
continuation. Physically we can understand this as follows: When
probability leaks into the continuum through the weak coupling link, it is
dissipated by all modes and oozes into the unobserved degrees of freedom
providing the continuum damping. In scattering theory this mechanism is
simply the outgoing boundary condition at infinity; all scattered modes are
irreversibly lost at the edges of our universe. The bottle neck is provided
by the weak link, and the rate of loss of the initial state is fixed by
this. The dissipation in the continuum is only present to prevent any
return of probability. Hence its details are inessential to the decay, its
presence however is necessary.
We have discussed the analytic behaviour of the propagators in detail, and
illustrate the properties by simple model calculations. Most of the
treatment follows the tradition in this field and carries out the argument
in the energy representation. However, to prove the existence of the
phenomena discussed, we display the behaviour of a model describing laser
coupling of a bound molecular state into a dissociating continuum. In
addition to showing the expected exponential decay, the model system proves
that the state prepared on the continuum is in the form of an outgoing wave
packet. The model, however, also displays features not simply describable
in the model calculations performed in this paper; we will return to a
detailed exploration of the decay characteristics in a forthcoming
publication.
The purpose of this paper is to draw attention to the necessity to
investigate the time evolution in coupled quantum systems. This provides a
broad range of interesting quantum models, which may also shed light on the
phenomena occurring in dynamical processes initiated and explored by pulsed
lasers.
|
2,877,628,090,196 | arxiv | \section{INTRODUCTION}
Neutrino oscillograms of the Earth are contours of constant oscillation
or survival probabilities in the plane of neutrino energy and nadir
or zenith angle of neutrino trajectory. They encode information on both the
neutrino parameters and the Earth density profile and proved to be a very
useful and illuminating tool for analyzing neutrino oscillations in the
Earth.
The oscillograms exhibit a very rich structure with local and global
maxima and minima (including the MSW resonance maxima in the mantle and
core of the Earth), saddle points and the parametric enhancement ridges
(Fig.~\ref{fig:1}). It was shown in \cite{Akhmedov:2006hb} that all these
features can be readily understood in terms of various realizations of
just two conditions: the generalized amplitude (collinearity) and phase
conditions. The study in \cite{Akhmedov:2006hb} was performed in the
limit $\Delta m_{21}^2=0$; in the present talk, based on
\cite{Akhmedov:2008qt}, the results for $\Delta m_{21}^2\ne 0$ are
presented.
\section{OSCILLOGRAMS FOR 3-FLAVOUR NEUTRINO OSCILLATIONS}
In the full 3-flavour framework with $\Delta m_{21}^2\ne0$
and $\theta_{13}\ne 0$, the oscillograms have non-trivial structure
both at low and high energies. Oscillations at low energies are
dominated by the solar channel, whereas those at higher energies are
mainly driven by the atmospheric parameters $(\Delta
m_{31}^2,\theta_{13})$ (see Fig.\ref{fig:1}).
\begin{figure}[htb]
\hspace*{0.6cm}
{\includegraphics[width=5.4cm,angle=90]{smr1a.eps}}
\vspace*{-1.0cm}
\caption{\small Oscillograms for $1-P_{ee}$ for three different values
of $\theta_{13}$.}
\label{fig:1}
\end{figure}
The main qualitatively new feature of 3-flavour oscillations as compared
to 2-flavour ones is the possibility of CP violation. Its effect can be
conveniently studied in terms of CP-oscillograms, which are defined as
contour plots of equal probability difference
\begin{equation}
\Delta P_{\mu e}^{\rm CP} \equiv
P_{\mu e}(\delta) - P_{\mu e}(\delta_{\rm th})\,,
\label{eq:1}
\end{equation}
where $\delta$ is the true value of the CP violating phase and
$\delta_{\rm th}$ is the assumed, or theoretical, value which we want to test.
The CP-oscillograms have a rather complex domain-like structure (see
Fig.~\ref{fig:2}); however, this structure can be readily understood and
interpreted in terms of three grids of curves, as we discuss next.
\vspace*{-0.2cm}
\begin{figure}[htb]
\hspace*{-0.3cm}
\includegraphics[width=7.6cm]{fig-dcp-me}
\vspace*{-0.7cm}
\caption{
Oscillograms for the difference of probabilities $\Delta P_{\mu
e}^{\rm CP} = P_{\mu e}(\delta) - P_{\mu e}(\delta_{\rm th})$
with $\delta_{\rm th} = 0^\circ$. Shown are the solar (black),
atmospheric (white) and interference phase condition (cyan)
curves.}
\label{fig:2}
\end{figure}
The oscillation probability $P_{\mu e}$ can be conveniently written as
\begin{equation}
P_{\mu e}=|c_{23}\, A_{e2}\, e^{i \delta}+s_{23}\,A_{e3}|^2\,,
\label{eq:2}
\end{equation}
where the amplitudes $A_{e2}$ and $A_{e3}$ are independent of $\theta_{23}$
and CP-violating phase $\delta$. To leading order in small parameters $s_{13}$
and $\Delta m_{21}^2/\Delta m_{31}^2$, $A_{e2}$ depends only on the
solar oscillation parameters and $A_{e3}$, only on the atmospheric ones:
\begin{equation}
A_{e2}\simeq A_S(\theta_{12},\,\Delta m_{21}^2)\,,~~~
A_{e3}\simeq A_A(\theta_{13},\,\Delta m_{31}^2)\,,
\label{eq:3}
\end{equation}
i.e., these amplitudes approximately coincide with, respectively, the
solar
and atmospheric contributions to the amplitude of $\nu_\mu\leftrightarrow
\nu_e$ oscillations. {}From eq.~(\ref{eq:2}) we find
\begin{eqnarray}
P_{\mu e}\!\!\!\! &=&\!\!\!\!
c_{23}^2 |A_S|^2 + s_{23}^2 |A_A|^2 ~~~~~\nonumber \\
&&\!\!\!\!+ 2 s_{23} c_{23} |A_S| |A_A| \cos(\phi + \delta)\,,
\label{eq:4}
\end{eqnarray}
where $\phi = arg(A_S A_A^*)$. Only the last term in (\ref{eq:4}) depends
on $\delta$; the condition of vanishing $\Delta P_{\mu e}$ (i.e.
$P_{\mu e}(\delta)=P_{\mu e}(\delta_{\rm th})$) therefore corresponds to
\begin{equation}
|A_S| |A_A| \cos(\phi + \delta)=|A_S| |A_A| \cos(\phi + \delta_{th})\,.
\label{eq:5}
\end{equation}
There are three non-trivial realisations of this condition:
\begin{itemize}
\item
$A_S~=~0$ ~($\Rightarrow$ solar ``magic'' lines)
\item
$A_A~=~0$ ~($\Rightarrow$ atm. ``magic'' curves)
\item
$(\phi + \delta_{th}) = - (\phi + \delta) +
2\pi l$, ~~or~~\\
$
\phi = -(\delta + \delta_{th})/2 + \pi k.
$
\end{itemize}
There is also, of course, the trivial realization, $\delta_{\rm th}=\delta+
2\pi n$, when the true and assumed values of ~$\delta$ coincide.
The above three conditions give rise to three grids of curves, which delineate
the different domains in the CP oscillograms (Fig.~\ref{fig:2}); they have a
very simple physical interpretation. We shall now discuss these conditions
in the energy region between the 1-2 and 1-3 MSW resonances; the results
for other energy domains can be found in \cite{Akhmedov:2008qt}.
\subsection{Solar ``magic'' curves}
The condition of vanishing solar channel contribution to the oscillation
probability, $A_S=0$, in the constant matter density approximation can be
written as $\sin^2(\omega_{12} L)=0$ (i.e. $\omega_{12} L=\pi n$), where
$\omega_{12}$ is the solar oscillation frequency. At energies exceeding
the 1-2 resonance energies in the mantle and in the core of the Earth,
$E\gtrsim 0.5$ GeV, one has $\omega_{12}\simeq V/2$, where $V=\sqrt{2} G_F
N_e$ is the matter-induced potential of neutrinos.
The condition $A_S=0$ therefore takes the form
\begin{equation}
L(\Theta_\nu) \simeq \frac{2\pi n}{V} \,,
\label{eq:6}
\end{equation}
where $L(\Theta_\nu)$ is the nadir angle dependent length of the neutrino
trajectory in the Earth. Note that the condition (\ref{eq:6}) is energy
independent and determines the baselines for which the ``solar'' contribution
to the probability vanishes. In the plane $(\Theta_\nu, E_\nu)$ it represents
nearly vertical lines $\Theta_\nu \approx \text{const}$
(black lines in Fig.~\ref{fig:2}).
There are three solar magic lines which correspond to $n = 1$ (in the
mantle domain) and $n = 2, 3$ (in the core domain). The existence of a
baseline ($L\approx 7600$ km) for which the probability of
$\nu_e\leftrightarrow \nu_\mu$ oscillations in the Earth is
approximately independent of the ``solar'' parameters ($\Delta m_{21}^2$,
$\theta_{12}$) and of the CP-phase $\delta$ was first pointed out
in~\cite{Barger:2001yr} and later discussed in a number of publications
(see~\cite{Huber:2002uy,Smirnov:2006sm} and references therein).
This baseline
was dubbed ``magic'' in~\cite{Huber:2002uy}. The interpretation of this
baseline as corresponding to vanishing ``solar'' amplitude $A_{e2}$, according
to eq.~(\ref{eq:6}) with $n=1$, was given in~\cite{Smirnov:2006sm}.
It was also shown there that for neutrino trajectories crossing the core of
the Earth there exist two more solar ``magic'' baselines, corresponding to the
oscillation phase equal $\pi n $ with $n=2$ and 3, and the existence of the
atmospheric ``magic'' curves was pointed out.
\subsection{Atmospheric ``magic curves''}
The second realization of the condition (\ref{eq:5}) corresponds to
vanishing atmospheric amplitude, $A_A=0$. In this case, like on the
solar ``magic'' lines, the probabilities of $\nu_e\leftrightarrow \nu_\mu$
(as well as $\nu_e\leftrightarrow \nu_\tau$) oscillations are independent of
the CP-violating phase. In addition, they do not depend on the
atmospheric parameters $\Delta m_{31}^2$ and $\theta_{13}$.
The properties of atmospheric ``magic'' curves can be easily understood in
the constant density approximation, in which the condition $A_A=0$ is
satisfied when $\sin^2(\omega_{31} L(\Theta_\nu))=0$, i.e. $\omega_{31}
L(\Theta_\nu)=\pi k$, $k = 1, 2, \dots$. For energies which are not too close
to the atmospheric MSW resonance energy, this condition reduces to
\begin{equation}
E \simeq \frac{\Delta m_{31}^2 L(\Theta_\nu)}
{|4\pi k \pm 2 V L(\Theta_\nu) |} \,,
\label{eq:7}
\end{equation}
which corresponds to the bent curves in the $(\Theta_\nu, E_\nu)$
plane (shown in white in Fig.~\ref{fig:2}).
\subsection{The interference phase condition}
The third of the above mentioned realizations of the condition (\ref{eq:6})
depends on the interference phase between the amplitudes $A_S$ and $A_A$
and therefore we shall call it the interference phase condition.
In the energy region between the 1-2 and 1-3 resonances we have $\phi \approx
\Delta m_{31}^2 L/4E$, i.e. in the first approximation $\phi$ does not depend
on the matter density. The interference phase condition then takes the form
$\Delta m_{31}^2 L/4E=-(\delta + \delta_{\rm th})/2+\pi l$, or
\begin{equation}
E_\nu = \frac{\Delta m_{31}^2 L(\Theta_\nu)}
{4\pi l - 2(\delta + \delta_{\rm th})} \,.
\label{eq:intphase}
\end{equation}
{}From the comparison of eqs.~(\ref{eq:intphase}) and (\ref{eq:7}) it
follows that the interference phase curves (shown in cyan in the oscillogram
of Fig.~\ref{fig:2}) are similar to atmospheric curves, but are steeper than
the latter for $\nu$'s and less steep for $\bar{\nu}$'s.
\section{CONCLUSIONS}
It can be seen from Fig.~\ref{fig:2} that the three grids of curves
discussed above -- the solar and atmospheric ``magic'' ones and the
interference phase curves -- give a very accurate description of the borders
between the domains. On these borders the probability difference $\Delta
P_{\mu e}^{\rm CP}=P_{\mu e}(\delta)-P_{\mu e}(\delta_{\rm th})$
vanishes, and therefore there is no sensitivity to the
phase $\delta$. The regions of maximal sensitivity to the CP-violating
phase correspond to the central parts of the domains.
Our analysis has been performed in terms of the oscillation probabilities;
a more realistic study which considers the number of events in future
detectors is currently under way.
|
2,877,628,090,197 | arxiv | \section{Introduction}
\label{sec:intro}
Understanding the principles guiding neuronal organization has been a major goal in
neuroscience. The ability to reconstruct individual neuronal arbors is necessary, but not sufficient to achieve this goal: understanding how neurons of the same and different types co-locate themselves requires the reconstruction of the arbors of multiple neurons sharing similar molecular and/or physiological features from the same brain. Such denser reconstructions may allow the field to answer some of the fundamental questions of neuroanatomy: do cells of the same type tile across the lateral dimensions by avoiding each other? To what extent do the organizational principles within a brain region extend across the whole brain? While dense reconstruction of electron microscopy images provides a solution~\cite{denk2004serial, helmstaedter2013connectomic}, its field-of-view has been limited for studying region-wide and brain-wide organization.
Recent advances in tissue clearing~\cite{chung2013structural, susaki2014whole} and light microscopy enable a fast, and versatile approach to this problem. In particular, oblique light-sheet microscopy can image thousands of individual neurons at once from the entire mouse brain at a 0.406 $\times$ 0.406 $\times$ \SI{2.5}{\micro\meter^3} resolution~\cite{narasimhan17}. Moreover, by registering reconstructed neurons from multiple brains of different neuronal gene expressions to a common coordinate framework such as the Allen Mouse Brain Atlas~\cite{lein2007genome}, it is possible to study neuronal structure and organization across many brain regions and neuronal cell classes. Therefore, this method may soon produce hundreds of full brain images, each containing hundreds of sparsely labeled neurons. However, scaling neuronal reconstructions to such large sets is not trivial. The gold standard of manual reconstruction is a tedious and labor-intensive process with a single neuronal
reconstruction taking a few hours. This makes automated reconstruction the most viable alternative. Recently, many automated methods appeared for the reconstruction of neurons from light microscopy images. These include methods based on supervised learning with neuronal networks as well as other approaches~\cite{peng2017automatic, turetken2011automated, wang2011broadly, turetken2012automated, uygar14, gala2014active}. Some common problems include slow training and/or reconstruction speeds, tendency for topological mistakes despite high voxel-wise accuracy, and vulnerability to rare but important imaging artifacts such as stitching misalignments and microscope stage jumps. Here, we propose a supervised learning method based on a convolutional neural network architecture to address these shortcomings. In particular, we suggest (i) an objective function that penalizes topological errors more heavily, (ii) a data augmentation framework to increase robustness against multiple imaging artifacts, and (iii) a distributed scheme for scalability. Training data
augmentation for addressing microscopy image defects was initially demonstrated for automated tracing of neurons in electron microscopy images~\cite{lee2017superhuman}. Here, we adapt this approach to sparse light microscopy images.
The U-Net architecture~\cite{ronneberger15, cciccek20163d} has recently received significant interest, especially in the analysis of biomedical images. By segmenting all the voxels of an input patch rather than a central portion of it, the U-Net can learn robust segmentation rules faster, and decreases the memory and storage requirements. In this paper, we train a 3D U-Net convolutional network on a set of manually traced neuronal arbors. To overcome challenges caused by artifacts producing apparent discontinuities in the arbors, we propose a fast, connectivity-based regularization technique. While approaches that increase topological consistency exist~\cite{briggman2009maximin, jain2010boundary}, they are either too slow for peta-scale images, or are not part of an online training procedure. Our approach is a simple, differentiable modification of the cost function, and the computational overhead scales linearly with the voxel count of the input patch. On the other hand, while these regularization techniques can enforce proper connectivity, there are relatively few examples of the various imaging artifacts in the training set. In order to increase the examples of such artifacts, we simulate them through various data augmentations and present these simulations under a unified framework. Taken together, our approach produces a significant increase in the topological accuracy of neuronal reconstructions on a test set.
In addition to accuracy, an efficient, scalable implementation is necessary for reconstructing petavoxel-sized image datasets. We maintain scalability and increase the throughput by using a distributed framework for reconstructing neurons from brain images, in which the computation can be distributed across multiple GPU instances. Finally, we augment data at run-time to avoid
memory issues and computational bottlenecks. This significantly increases the throughput rate because data transfers are a substantial bottleneck. We report segmentation speeds exceeding 300 gigavoxels per hour and linear speedups in the presence of additional GPUs.
\section{Methods}
\label{sec:methods}
\subsection{Convolutional neural network regularization through
digital topology techniques}
\label{sec:regularization}
To create the training set, we obtain volumetric reconstructions of the manual arbor traces of neuronal images by a topology-preserving inflation of the traces~\cite{sumbul2014automated}. We use a 3D U-Net convolutional neural network architecture~\cite{ronneberger15, cciccek20163d, lee2017superhuman} to learn to segment the neurons from this volumetric training set. Since neuronal morphology is ultimately represented and analyzed as a tree structure, we consider the branching pattern of the segmented neuron more important than its voxelwise accuracy. Hence, to penalize topological changes between the ground-truth and the prediction at the time of training, we binarize the network output by thresholding and identify all non-simple points in this binarized patch based on $26$-connectivity~\cite{bertrand1994new} --- points when added or removed change an object's topology (e.g., splits and mergers) --- and assign larger weights to them in the binary cross-entropy cost function
\vspace{-.1cm}
\begin{equation}
\label{equ:loss}
J(\hat{y}, y) = -\frac{1}{N}\sum^{N}_{i=1} w_{i} \big[ y_{i} \log
(\hat{y}_{i}) + (1 - y_{i}) \log (1 - {\hat{y}_{i}})
\big]
\end{equation}
where $w_i=w>1$ if voxel $i$ is non-simple while $w_i=1$ otherwise, $N$ is the number of voxels, and $y_i$ and $\hat{y_i}$ are the label image and predicted segmentation, respectively.
Note that the simple-ness of a voxel depends only on its $26$-neighborhood, and therefore this operation scales linearly with the patch size.
\subsection{Simulation of image artifacts through data augmentations}
\label{sec:augmentations}
\begin{figure*}[htb]
\centering
\includegraphics[width=178.0mm]{Augmentations}
\caption{Data augmentations. (From left to right) No augmentation
provides no augmentation on the raw image or the
ground-truth. Occluded branches simulates a loss of a localized
signal due to lack of fluorescence. Stitching misalignment
simulates a stitching misalignment between two image
volumes. Light scattering simulates a blurred image due to light
scattering in the cleared tissue. Duplicate sections simulates a
halt of the stage and an imaging of duplicate sections. Dropped
sections simulates a jump of the stage and a missing image
section. Artifacts in the raw images are identified by arrows while the corresponding changes in the labels are identified in red. \vspace{-2ex}}
\label{fig:augmentations}
\end{figure*}
Data augmentation is a technique that augments the base training data with pre-defined transformations of it. By creating statistical invariances (e.g. against rotation) within the dataset or over-representing rarely occurring artifacts, augmentation can increase the robustness of the learned algorithm. Motivated by the fact that 3D microscopy is prone to several image artifacts, we followed a unified framework for data augmentation. In particular, our formalism requires explicit models of the underlying artifacts and the desired reconstruction in their presence to augment the original training set with simulations of these artifacts.
We define the class of ``artifact-generating'' transformations as $S$ such that if $\mathcal{T} \in S$, then $\mathcal{T} = \mathcal{T}_{R} \otimes \mathcal{T}_{L}$ for
$\mathcal{T}_{R}: \mathbb{R}^{n_1 \times n_2 \times n_3} \rightarrow \mathbb{R}^{n_1 \times n_2 \times n_3}$ and
$\mathcal{T}_{L}: {\{0, 1\}}^{n_1 \times n_2 \times n_3} \rightarrow
{\{0, 1\}}^{n_1 \times n_2 \times n_3}$, where $\mathcal{T}_R$ acts on an $n_1 \times n_2 \times n_3$ raw image and $\mathcal{T}_L$ acts on its corresponding label image. For example, the common augmentation step of rotation by $90^\circ$ can be realized by $\mathcal{T}_R$ and $\mathcal{T}_L$ both rotating their arguments by $90^\circ$. Data augmentation adds these rotated raw/label image pairs to the original training set (Fig.~\ref{fig:augmentations}).
{\bf Occluded branches:} Branch occlusions can be caused by
photobleaching or an absence of a fluorophore. We model the artifact-generating transformation for an absence of a fluorophore as $\mathcal{T} = \mathcal{T}_{R} \otimes \mathcal{I}$, where
\begin{equation}
\mathcal{T_{R}}(R; x, y, z) = R - \mathrm{PSF}(x, y, z)
\end{equation}
such that $\mathcal{I}$ denotes the identity transformation, $x$ denotes the position of
the absent fluorophore and
$\mathrm{PSF}$ is its corresponding point-spread function. Here, we approximated the $\mathrm{PSF}$ of a fluorophore with a multivariate Gaussian.
{\bf Duplicate sections:} The stage of a scanning 3D microscope can
intermittently stall, which can duplicate the imaging of a tissue section. The artifact-generating transformation for stage stalling is given by $\mathcal{T} =
\mathcal{T}_{R} \otimes \mathcal{I}$, where
\begin{equation}
\mathcal{T_R}(R; \mathbf{r}, \mathbf{r_0}, N)
=
\begin{cases}
R(\mathbf{r}), & \mathbf{r} \not\in N \\
R(\mathbf{r_{0}}), & \mathbf{r} \in N
\end{cases}
\end{equation}
for the region $\mathbf{r} = (x, y, z)$ and the plane $\mathbf{r_0} = (x_0, y, z)$
such that $\mathcal{T}_{R}$ duplicates the slice $\mathbf{r_{0}}$ in a rectangular neighborhood $N$.
{\bf Dropped sections:} Similar to the stalling of the stage, jumps that result in missed sections can occur intermittently. The corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{T}_{L}$, where
\begin{equation}
\mathcal{T}_{R}(R; \mathbf{r}, x_0, \Delta)
=
\begin{cases}
R(x, y, z), & x \leq x_0 \\
R(x + \Delta, y, z), & x > x_0
\end{cases}
\end{equation}
and
\begin{equation}
\mathcal{T}_{L}(L; \mathbf{r}, x_0, \Delta)
=
\begin{cases}
L(x, y, z), & x \leq x_0 - \Delta \\
L(D(x), y, z), & |x-x_0-\frac{\Delta}{2}| > \frac{3\Delta}{2} \\
L(x + 2\Delta, y, z), & x \geq x_0 + 2\Delta
\end{cases}
\end{equation}
such that $\mathbf{r}=(x, y, z)$, for $D(x, x_0, \Delta) = x_0 - \Delta + \frac{3}{2}\lceil x -
x_0 + \Delta \rceil$, which downsamples
the region to maintain partial connectivity in the label. Hence, $\mathcal{T}_{R}$ skips a small region given by $\Delta$ at $x_0$, and $\mathcal{T}_{L}$ is the corresponding desired transformation on the label image.
{\bf Stitching misalignment:} Misalignments can occur between 3D image
stacks, potentially causing topological breaks and mergers between
neuronal branches. The corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{T}_{L}$, where
\begin{equation}
\mathcal{T}_{R}(R;x,y,z,\Delta)
=
\begin{cases}
R(x, y, z), & x \leq x_{0} \\
R(x, y + \Delta, z), & x > x_{0}
\end{cases}
\end{equation}
and
\begin{equation}
\mathcal{T}_{L}(L;x,y,z,\Delta)
=
\begin{cases}
L, & x \leq x_{0} - \frac{1}{2} \Delta \\
\Sigma_{zy}(\Delta) L, & |x-x_{0}|<\frac{1}{2} \Delta \\
L(x, y + \Delta, z), & x > x_{0} + \frac{1}{2} \Delta
\end{cases}
\end{equation}
such that $\Sigma_{zy}(\Delta)$ is a shear transform on $L$. Hence,
$\mathcal{T}_{R}$ translates a region of $R$ to simulate a stitching
misalignment, and $\mathcal{T}_{L}$ shears a region around the
discontinuity to maintain 18-connectivity in the label.
{\bf Light scattering:} Light scattering by the cleared tissue can create an
inhomogeneous intensity profile and blur the image. To simulate this
image artifact, we assumed the scatter has a homogeneous profile and is anisotropic due to the oblique light-sheet. We approximate these characteristics with a Gaussian kernel:
$G(x,y,z)=G(\mathbf{r})=\mathcal{N}(\mathbf{r}; \mu, \Sigma)$. In addition, the global inhomogeneous intensity profile was simulated with an additive constant. Thus, the corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{I}$, where
\begin{equation}
\mathcal{T}_{R}(R) = R(x,y,z)*G(x,y,z)+\lambda
\end{equation}
\subsection{Fully automated, scalable tracing}
\label{sec:system}
To optimize the pipeline for scalability, we store images as parcellated HDF5 datasets.
For training, a file server software streams these images to the GPU server, which performs data augmentations on-the-fly, to minimize storage space requirements. For deploying the trained neural network, the file server similarly streams the datasets to a GPU server for segmentation. Once the segmentation is completed, the neuronal morphology is reconstructed automatically from the segmented image using the UltraTracer neuron tracing tool within the Vaa3D software package~\cite{peng2017automatic}
\section{Experimental Procedure}
\label{sec:experiment}
In our experiments, we used a dataset of 54 manually traced neurons
imaged using oblique light-sheet microscopy. These morphological
annotations were dilated while preserving topology for training the
neural network for segmentation. We partitioned the dataset into
training, validation, and test sets by randomly choosing 25, 8, and 21 neurons, respectively. The
software package PyTorch was used to implement the neural
network~\cite{paszke17}. The network was trained using an Adam
optimizer for gradient descent~\cite{kingma15}. Training and
reconstruction were conducted on two Intel Xeon Silver 4116 CPU, 256 GB
RAM, and 2 NVIDIA GeForce GTX 1080 Ti GPUs.
\vspace{-2ex}
\section{Results}
\label{sec:results}
\subsection{Topologically accurate reconstruction}
\label{sec:cortex}
\begin{figure}[htb]
\centering
\includegraphics[width=86.0mm]{Reconstruction-revised}
\caption{Neuronal
images reconstructed using the U-Net architecture. (Upper-left) Raw image. (Upper-right) Label. (Lower-left) Reconstruction performed without augmentations or regularization. (Lower-right) Reconstruction performed with augmentations and regularization.
\vspace{-2ex}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=80.0mm]{ISBI-results-revision}
\caption{Evaluation of segmentation results. The groups None,
Reg., Aug., and Both represent the trials with
no augmentations or regularization, with the connectivity-based
regularization, with augmentations, and with both augmentations and regularization,
respectively. The groups were quantified using the Jaccard index
and compared using a paired Student's t-test. ({\small$\star\star\star$} indicates $p < 0.001$ and {\small$\star\star\star\star$} indicates $p < 0.0001$)
\vspace{-2ex}}
\label{fig:results}
\end{figure}
To quantify the topological accuracy of the network on light-sheet microscopy data, we define the topological error as the number of non-simple points that must be added or removed from a prediction to obtain its corresponding label. Specifically, for binary images $\hat{L}$ and $L$, let $\mathcal{W}(\hat{L},L)$ denote a topology-preserving warping of $\hat{L}$ that minimizes the voxelwise disagreements between the warped image and $L$~\cite{jain2010boundary, uygar14}, $\hat{L}\cap L$ denote the binary image whose foreground is common to both $\hat{L}$ and $L$, and $c(L)$ denote the number of foreground voxels of $L$. We quantify the agreement between a reconstruction $\hat{L}$ and label $L$ using the Jaccard index as
\begin{equation}
J(\hat{L}, L) = \frac{c(\mathcal{W}(\hat{L}, L) \cap L)}{c(\mathcal{W}(\hat{L}, L) \cup L)}.
\end{equation}
We compared this score across different U-Net results: without any augmentations or regularization, with the augmentations, with the topological regularization, and with both the topological regularization and the augmentations. The U-Net results with augmentations and topological regularization performed significantly better compared to the results without augmentations or regularization (Figs 2, 3).
\subsection{Neuron reconstruction is efficient and scalable}
\label{sec:efficient}
To quantify the efficiency of the distributed framework, we measured
the framework’s throughput for augmenting data, training on the data,
and segmenting the data. Augmentations performed at 35.2 $\pm$ 9.2 gigavoxels per hour while training
performed at 16.8 $\pm$ 0.2 megavoxels per hour. Segmentation performed at 348.8 $\pm$ 1.9 gigavoxels per
hour. Both segmentation and training showed a linear speedup
with an additional GPU. For an entire mouse brain, neuronal reconstruction would take about 23 hours on a single GPU.
\section{Discussion}
\label{sec:discussion}
In this paper, we proposed an efficient, scalable, and accurate algorithm capable of reconstructing neuronal anatomy from light microscopy images of the whole brain. Our method employs topological regularization as well as simulates discontinuous image artifacts inherent to the imaging systems. These techniques help maintain topological correctness of the trace (skeleton) representations of neuronal arbors.
While we demonstrated the merit of our approach on neuronal images obtained by oblique light-sheet microscopy, our methods address some of the problems common to most 3D fluorescence microscopy techniques. Therefore, we hope that some of our methods will be useful for multiple applications. Combined with the speed and precision of oblique light-sheet microscopy, the distributed and fast nature of our approach enables the production of a comprehensive database of neuronal anatomy across many brain regions and cell classes. We believe that these aspects will be useful in discovering different cortical cell types as well as understanding the anatomical organization of the brain.
\newpage
\begin{small}
\bibliographystyle{IEEEbib}
\section{Introduction}
\label{sec:intro}
Understanding the principles guiding neuronal organization has been a major goal in
neuroscience. The ability to reconstruct individual neuronal arbors is necessary, but not sufficient to achieve this goal: understanding how neurons of the same and different types co-locate themselves requires the reconstruction of the arbors of multiple neurons sharing similar molecular and/or physiological features from the same brain. Such denser reconstructions may allow the field to answer some of the fundamental questions of neuroanatomy: do cells of the same type tile across the lateral dimensions by avoiding each other? To what extent do the organizational principles within a brain region extend across the whole brain? While dense reconstruction of electron microscopy images provides a solution~\cite{denk2004serial, helmstaedter2013connectomic}, its field-of-view has been limited for studying region-wide and brain-wide organization.
Recent advances in tissue clearing~\cite{chung2013structural, susaki2014whole} and light microscopy enable a fast, and versatile approach to this problem. In particular, oblique light-sheet microscopy can image thousands of individual neurons at once from the entire mouse brain at a 0.406 $\times$ 0.406 $\times$ \SI{2.5}{\micro\meter^3} resolution~\cite{narasimhan17}. Moreover, by registering reconstructed neurons from multiple brains of different neuronal gene expressions to a common coordinate framework such as the Allen Mouse Brain Atlas~\cite{lein2007genome}, it is possible to study neuronal structure and organization across many brain regions and neuronal cell classes. Therefore, this method may soon produce hundreds of full brain images, each containing hundreds of sparsely labeled neurons. However, scaling neuronal reconstructions to such large sets is not trivial. The gold standard of manual reconstruction is a tedious and labor-intensive process with a single neuronal
reconstruction taking a few hours. This makes automated reconstruction the most viable alternative. Recently, many automated methods appeared for the reconstruction of neurons from light microscopy images. These include methods based on supervised learning with neuronal networks as well as other approaches~\cite{peng2017automatic, turetken2011automated, wang2011broadly, turetken2012automated, uygar14, gala2014active}. Some common problems include slow training and/or reconstruction speeds, tendency for topological mistakes despite high voxel-wise accuracy, and vulnerability to rare but important imaging artifacts such as stitching misalignments and microscope stage jumps. Here, we propose a supervised learning method based on a convolutional neural network architecture to address these shortcomings. In particular, we suggest (i) an objective function that penalizes topological errors more heavily, (ii) a data augmentation framework to increase robustness against multiple imaging artifacts, and (iii) a distributed scheme for scalability. Training data
augmentation for addressing microscopy image defects was initially demonstrated for automated tracing of neurons in electron microscopy images~\cite{lee2017superhuman}. Here, we adapt this approach to sparse light microscopy images.
The U-Net architecture~\cite{ronneberger15, cciccek20163d} has recently received significant interest, especially in the analysis of biomedical images. By segmenting all the voxels of an input patch rather than a central portion of it, the U-Net can learn robust segmentation rules faster, and decreases the memory and storage requirements. In this paper, we train a 3D U-Net convolutional network on a set of manually traced neuronal arbors. To overcome challenges caused by artifacts producing apparent discontinuities in the arbors, we propose a fast, connectivity-based regularization technique. While approaches that increase topological consistency exist~\cite{briggman2009maximin, jain2010boundary}, they are either too slow for peta-scale images, or are not part of an online training procedure. Our approach is a simple, differentiable modification of the cost function, and the computational overhead scales linearly with the voxel count of the input patch. On the other hand, while these regularization techniques can enforce proper connectivity, there are relatively few examples of the various imaging artifacts in the training set. In order to increase the examples of such artifacts, we simulate them through various data augmentations and present these simulations under a unified framework. Taken together, our approach produces a significant increase in the topological accuracy of neuronal reconstructions on a test set.
In addition to accuracy, an efficient, scalable implementation is necessary for reconstructing petavoxel-sized image datasets. We maintain scalability and increase the throughput by using a distributed framework for reconstructing neurons from brain images, in which the computation can be distributed across multiple GPU instances. Finally, we augment data at run-time to avoid
memory issues and computational bottlenecks. This significantly increases the throughput rate because data transfers are a substantial bottleneck. We report segmentation speeds exceeding 300 gigavoxels per hour and linear speedups in the presence of additional GPUs.
\section{Methods}
\label{sec:methods}
\subsection{Convolutional neural network regularization through
digital topology techniques}
\label{sec:regularization}
To create the training set, we obtain volumetric reconstructions of the manual arbor traces of neuronal images by a topology-preserving inflation of the traces~\cite{sumbul2014automated}. We use a 3D U-Net convolutional neural network architecture~\cite{ronneberger15, cciccek20163d, lee2017superhuman} to learn to segment the neurons from this volumetric training set. Since neuronal morphology is ultimately represented and analyzed as a tree structure, we consider the branching pattern of the segmented neuron more important than its voxelwise accuracy. Hence, to penalize topological changes between the ground-truth and the prediction at the time of training, we binarize the network output by thresholding and identify all non-simple points in this binarized patch based on $26$-connectivity~\cite{bertrand1994new} --- points when added or removed change an object's topology (e.g., splits and mergers) --- and assign larger weights to them in the binary cross-entropy cost function
\vspace{-.1cm}
\begin{equation}
\label{equ:loss}
J(\hat{y}, y) = -\frac{1}{N}\sum^{N}_{i=1} w_{i} \big[ y_{i} \log
(\hat{y}_{i}) + (1 - y_{i}) \log (1 - {\hat{y}_{i}})
\big]
\end{equation}
where $w_i=w>1$ if voxel $i$ is non-simple while $w_i=1$ otherwise, $N$ is the number of voxels, and $y_i$ and $\hat{y_i}$ are the label image and predicted segmentation, respectively.
Note that the simple-ness of a voxel depends only on its $26$-neighborhood, and therefore this operation scales linearly with the patch size.
\subsection{Simulation of image artifacts through data augmentations}
\label{sec:augmentations}
\begin{figure*}[htb]
\centering
\includegraphics[width=178.0mm]{Augmentations}
\caption{Data augmentations. (From left to right) No augmentation
provides no augmentation on the raw image or the
ground-truth. Occluded branches simulates a loss of a localized
signal due to lack of fluorescence. Stitching misalignment
simulates a stitching misalignment between two image
volumes. Light scattering simulates a blurred image due to light
scattering in the cleared tissue. Duplicate sections simulates a
halt of the stage and an imaging of duplicate sections. Dropped
sections simulates a jump of the stage and a missing image
section. Artifacts in the raw images are identified by arrows while the corresponding changes in the labels are identified in red. \vspace{-2ex}}
\label{fig:augmentations}
\end{figure*}
Data augmentation is a technique that augments the base training data with pre-defined transformations of it. By creating statistical invariances (e.g. against rotation) within the dataset or over-representing rarely occurring artifacts, augmentation can increase the robustness of the learned algorithm. Motivated by the fact that 3D microscopy is prone to several image artifacts, we followed a unified framework for data augmentation. In particular, our formalism requires explicit models of the underlying artifacts and the desired reconstruction in their presence to augment the original training set with simulations of these artifacts.
We define the class of ``artifact-generating'' transformations as $S$ such that if $\mathcal{T} \in S$, then $\mathcal{T} = \mathcal{T}_{R} \otimes \mathcal{T}_{L}$ for
$\mathcal{T}_{R}: \mathbb{R}^{n_1 \times n_2 \times n_3} \rightarrow \mathbb{R}^{n_1 \times n_2 \times n_3}$ and
$\mathcal{T}_{L}: {\{0, 1\}}^{n_1 \times n_2 \times n_3} \rightarrow
{\{0, 1\}}^{n_1 \times n_2 \times n_3}$, where $\mathcal{T}_R$ acts on an $n_1 \times n_2 \times n_3$ raw image and $\mathcal{T}_L$ acts on its corresponding label image. For example, the common augmentation step of rotation by $90^\circ$ can be realized by $\mathcal{T}_R$ and $\mathcal{T}_L$ both rotating their arguments by $90^\circ$. Data augmentation adds these rotated raw/label image pairs to the original training set (Fig.~\ref{fig:augmentations}).
{\bf Occluded branches:} Branch occlusions can be caused by
photobleaching or an absence of a fluorophore. We model the artifact-generating transformation for an absence of a fluorophore as $\mathcal{T} = \mathcal{T}_{R} \otimes \mathcal{I}$, where
\begin{equation}
\mathcal{T_{R}}(R; x, y, z) = R - \mathrm{PSF}(x, y, z)
\end{equation}
such that $\mathcal{I}$ denotes the identity transformation, $x$ denotes the position of
the absent fluorophore and
$\mathrm{PSF}$ is its corresponding point-spread function. Here, we approximated the $\mathrm{PSF}$ of a fluorophore with a multivariate Gaussian.
{\bf Duplicate sections:} The stage of a scanning 3D microscope can
intermittently stall, which can duplicate the imaging of a tissue section. The artifact-generating transformation for stage stalling is given by $\mathcal{T} =
\mathcal{T}_{R} \otimes \mathcal{I}$, where
\begin{equation}
\mathcal{T_R}(R; \mathbf{r}, \mathbf{r_0}, N)
=
\begin{cases}
R(\mathbf{r}), & \mathbf{r} \not\in N \\
R(\mathbf{r_{0}}), & \mathbf{r} \in N
\end{cases}
\end{equation}
for the region $\mathbf{r} = (x, y, z)$ and the plane $\mathbf{r_0} = (x_0, y, z)$
such that $\mathcal{T}_{R}$ duplicates the slice $\mathbf{r_{0}}$ in a rectangular neighborhood $N$.
{\bf Dropped sections:} Similar to the stalling of the stage, jumps that result in missed sections can occur intermittently. The corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{T}_{L}$, where
\begin{equation}
\mathcal{T}_{R}(R; \mathbf{r}, x_0, \Delta)
=
\begin{cases}
R(x, y, z), & x \leq x_0 \\
R(x + \Delta, y, z), & x > x_0
\end{cases}
\end{equation}
and
\begin{equation}
\mathcal{T}_{L}(L; \mathbf{r}, x_0, \Delta)
=
\begin{cases}
L(x, y, z), & x \leq x_0 - \Delta \\
L(D(x), y, z), & |x-x_0-\frac{\Delta}{2}| > \frac{3\Delta}{2} \\
L(x + 2\Delta, y, z), & x \geq x_0 + 2\Delta
\end{cases}
\end{equation}
such that $\mathbf{r}=(x, y, z)$, for $D(x, x_0, \Delta) = x_0 - \Delta + \frac{3}{2}\lceil x -
x_0 + \Delta \rceil$, which downsamples
the region to maintain partial connectivity in the label. Hence, $\mathcal{T}_{R}$ skips a small region given by $\Delta$ at $x_0$, and $\mathcal{T}_{L}$ is the corresponding desired transformation on the label image.
{\bf Stitching misalignment:} Misalignments can occur between 3D image
stacks, potentially causing topological breaks and mergers between
neuronal branches. The corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{T}_{L}$, where
\begin{equation}
\mathcal{T}_{R}(R;x,y,z,\Delta)
=
\begin{cases}
R(x, y, z), & x \leq x_{0} \\
R(x, y + \Delta, z), & x > x_{0}
\end{cases}
\end{equation}
and
\begin{equation}
\mathcal{T}_{L}(L;x,y,z,\Delta)
=
\begin{cases}
L, & x \leq x_{0} - \frac{1}{2} \Delta \\
\Sigma_{zy}(\Delta) L, & |x-x_{0}|<\frac{1}{2} \Delta \\
L(x, y + \Delta, z), & x > x_{0} + \frac{1}{2} \Delta
\end{cases}
\end{equation}
such that $\Sigma_{zy}(\Delta)$ is a shear transform on $L$. Hence,
$\mathcal{T}_{R}$ translates a region of $R$ to simulate a stitching
misalignment, and $\mathcal{T}_{L}$ shears a region around the
discontinuity to maintain 18-connectivity in the label.
{\bf Light scattering:} Light scattering by the cleared tissue can create an
inhomogeneous intensity profile and blur the image. To simulate this
image artifact, we assumed the scatter has a homogeneous profile and is anisotropic due to the oblique light-sheet. We approximate these characteristics with a Gaussian kernel:
$G(x,y,z)=G(\mathbf{r})=\mathcal{N}(\mathbf{r}; \mu, \Sigma)$. In addition, the global inhomogeneous intensity profile was simulated with an additive constant. Thus, the corresponding artifact-generating transformation is given by $\mathcal{T} = \mathcal{T}_{R} \otimes
\mathcal{I}$, where
\begin{equation}
\mathcal{T}_{R}(R) = R(x,y,z)*G(x,y,z)+\lambda
\end{equation}
\subsection{Fully automated, scalable tracing}
\label{sec:system}
To optimize the pipeline for scalability, we store images as parcellated HDF5 datasets.
For training, a file server software streams these images to the GPU server, which performs data augmentations on-the-fly, to minimize storage space requirements. For deploying the trained neural network, the file server similarly streams the datasets to a GPU server for segmentation. Once the segmentation is completed, the neuronal morphology is reconstructed automatically from the segmented image using the UltraTracer neuron tracing tool within the Vaa3D software package~\cite{peng2017automatic}
\section{Experimental Procedure}
\label{sec:experiment}
In our experiments, we used a dataset of 54 manually traced neurons
imaged using oblique light-sheet microscopy. These morphological
annotations were dilated while preserving topology for training the
neural network for segmentation. We partitioned the dataset into
training, validation, and test sets by randomly choosing 25, 8, and 21 neurons, respectively. The
software package PyTorch was used to implement the neural
network~\cite{paszke17}. The network was trained using an Adam
optimizer for gradient descent~\cite{kingma15}. Training and
reconstruction were conducted on two Intel Xeon Silver 4116 CPU, 256 GB
RAM, and 2 NVIDIA GeForce GTX 1080 Ti GPUs.
\vspace{-2ex}
\section{Results}
\label{sec:results}
\subsection{Topologically accurate reconstruction}
\label{sec:cortex}
\begin{figure}[htb]
\centering
\includegraphics[width=86.0mm]{Reconstruction-revised}
\caption{Neuronal
images reconstructed using the U-Net architecture. (Upper-left) Raw image. (Upper-right) Label. (Lower-left) Reconstruction performed without augmentations or regularization. (Lower-right) Reconstruction performed with augmentations and regularization.
\vspace{-2ex}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=80.0mm]{ISBI-results-revision}
\caption{Evaluation of segmentation results. The groups None,
Reg., Aug., and Both represent the trials with
no augmentations or regularization, with the connectivity-based
regularization, with augmentations, and with both augmentations and regularization,
respectively. The groups were quantified using the Jaccard index
and compared using a paired Student's t-test. ({\small$\star\star\star$} indicates $p < 0.001$ and {\small$\star\star\star\star$} indicates $p < 0.0001$)
\vspace{-2ex}}
\label{fig:results}
\end{figure}
To quantify the topological accuracy of the network on light-sheet microscopy data, we define the topological error as the number of non-simple points that must be added or removed from a prediction to obtain its corresponding label. Specifically, for binary images $\hat{L}$ and $L$, let $\mathcal{W}(\hat{L},L)$ denote a topology-preserving warping of $\hat{L}$ that minimizes the voxelwise disagreements between the warped image and $L$~\cite{jain2010boundary, uygar14}, $\hat{L}\cap L$ denote the binary image whose foreground is common to both $\hat{L}$ and $L$, and $c(L)$ denote the number of foreground voxels of $L$. We quantify the agreement between a reconstruction $\hat{L}$ and label $L$ using the Jaccard index as
\begin{equation}
J(\hat{L}, L) = \frac{c(\mathcal{W}(\hat{L}, L) \cap L)}{c(\mathcal{W}(\hat{L}, L) \cup L)}.
\end{equation}
We compared this score across different U-Net results: without any augmentations or regularization, with the augmentations, with the topological regularization, and with both the topological regularization and the augmentations. The U-Net results with augmentations and topological regularization performed significantly better compared to the results without augmentations or regularization (Figs 2, 3).
\subsection{Neuron reconstruction is efficient and scalable}
\label{sec:efficient}
To quantify the efficiency of the distributed framework, we measured
the framework’s throughput for augmenting data, training on the data,
and segmenting the data. Augmentations performed at 35.2 $\pm$ 9.2 gigavoxels per hour while training
performed at 16.8 $\pm$ 0.2 megavoxels per hour. Segmentation performed at 348.8 $\pm$ 1.9 gigavoxels per
hour. Both segmentation and training showed a linear speedup
with an additional GPU. For an entire mouse brain, neuronal reconstruction would take about 23 hours on a single GPU.
\section{Discussion}
\label{sec:discussion}
In this paper, we proposed an efficient, scalable, and accurate algorithm capable of reconstructing neuronal anatomy from light microscopy images of the whole brain. Our method employs topological regularization as well as simulates discontinuous image artifacts inherent to the imaging systems. These techniques help maintain topological correctness of the trace (skeleton) representations of neuronal arbors.
While we demonstrated the merit of our approach on neuronal images obtained by oblique light-sheet microscopy, our methods address some of the problems common to most 3D fluorescence microscopy techniques. Therefore, we hope that some of our methods will be useful for multiple applications. Combined with the speed and precision of oblique light-sheet microscopy, the distributed and fast nature of our approach enables the production of a comprehensive database of neuronal anatomy across many brain regions and cell classes. We believe that these aspects will be useful in discovering different cortical cell types as well as understanding the anatomical organization of the brain.
\newpage
\begin{small}
\bibliographystyle{IEEEbib}
|
2,877,628,090,198 | arxiv |
\section{Introduction}
Recent years have seen a great improvement in text-independent speaker verification. The speaker verification system extracts speaker characteristic information from a given utterance and then verify the speaker ID. In the state-of-the-art methods of speaker verification, i-vector \cite{i-vector} is used to represent speaker characteristics, and probabilistic linear discriminant analysis (PLDA) \cite{plda0,plda1,plda2} is used as a verifier. While this system performs well on long utterances, the performance degrades drastically when only short utterances are available \cite{kanagasundaram2011vector}. The main cause of this problem is the biased phonetic distribution of short utterances, which makes the estimated speaker features become statistically unreliable. However, in many real world scenarios, users may be reluctant to provide several-minute-long utterances.
Significant efforts have been made to remedy the performance degradation in short utterance speaker verification. In \cite{kanagasundaram2014improving}\cite{7953206}\cite{duration}, the variance of i-vectors for short utterances are modeled and used for i-vector normalization. \cite{kenny2013plda} and \cite{lin2017fast} proposed to utilize duration information in PLDA model. \cite{yamamoto2015denoising} uses phonetic information to reconstruct reliable i-vectors.
\iffalse
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{ogan.jpg}
\caption{Generative Adversarial Networks.}
\label{fig:GAN}
\end{figure}
\fi
In the past years, deep learning has become very popular in the speaker verification field. Many approaches use deep neural networks to process i-vectors. For example, \cite{villalba2017tied} proposed a variational autoencoder as a back-end for i-vector based speaker recognition, \cite{mahto2017vector} used denoising autoencoders to compensate for noisy speech. However, a large amount of data is required for training deep neural networks \cite{e2e}, while the amount of data available for speaker verification are usually very small. This has been one of the biggest obstacles for building an end-to-end speaker verification system using deep learning. Hence, it may be better to improve the i-vector and PLDA framework by using deep learning.
Recently, a novel structure called generative adversarial network (GAN) \cite{goodfellow2014generative} has become extremely popular. GAN can learn a mapping from random noise to target domain, by playing a zero-sum game with two networks, a generator $G$ and a discriminator $D$: $G$ tries to generate ``real'' samples which can fool $D$, while $D$ tries to determine whether a given sample is from real data distribution or from $G$.
This paper describes an i-vector transformation method using conditional GAN for improving i-vector based short utterance speaker verification. The method uses GAN to estimate a generative model which can generate a reliable i-vector from an unreliable i-vector, in which we assume an i-vector from a long utterance is reliable, and an i-vector from a short utterance is unreliable. Specifically, we used the conditional version of GAN, where both the generator and the discriminator have an i-vector from a short utterance as the conditional input. The generator $G$ tries to generate a reliable i-vector from an unreliable one, and the discriminator $D$ tries to decide whether a given reliable i-vector is a real one extracted from a long utterance or a fake one generated by $G$. In order to stabilize GAN training, numerical difference (cosine distance) between generated i-vectors and target reliable i-vectors are used in the training stage. Moreover, inspired by \cite{mahto2017vector}, we tried to improve the speaker discriminative ability of generated i-vectors by adding an extra speaker label predicting task to $G$. This multi-task learning framework can better guide the training of GAN. In the testing stage, $G$ is used to generate reliable i-vectors from those extracted from short utterances, and then the generated i-vectors would be used in PLDA scoring.
This paper is organized as follows: Section 2 briefly introduces related works of our methods. Section 3 presents the proposed GAN-based structure for i-vector restoration. Section 4 describes experimental evaluations for speaker verification in two NIST SRE tasks. Section 5 summarizes this paper.
\section{Related Works}
\subsection{I-vector and PLDA}
I-vector and PLDA have been widely used in the state-of-the-art systems for text-independent speaker verification.
The i-vector approach aims to extract a fixed and low dimension representation from a given utterance based on a factor analysis model. As described in \cite{i-vector}, an utterance is projected onto a low-dimensional total variability space which contains both channel- and speaker-dependent information, as an i-vector. Given an utterance, the channel- and speaker-dependent GMM supervector $M$ can be written as:
\begin{equation}
M = m + Tw,
\label{eq-2}
\end{equation}
where $m$ is the speaker- and channel-independent supervector taken from the universal background model (UBM), $T$ is the total variability matrix (TVM) and $w$ is the i-vector.
Probability linear discriminant analysis (PLDA) \cite{plda2} is applied as a generative model for i-vectors, which can be written as follows,
\begin{equation}
w = \bar{w} + Ux + Vy + \epsilon
\label{eq-1}
\end{equation}
where $\bar{w}$ is the global mean of i-vectors, $U$ and $V$ are an eigenvoice and an eigenchannel matrix respectively, $x$ and $y$ are speaker- and channel-factors, and $\epsilon$ is residual noise.
Given two i-vectors, the log-likelihood ratio of the same-speaker and different-speaker hypotheses is computed by the PLDA model as the measure of their similarity.
\subsection{Generative Adversarial Networks Family}
Generative adversarial networks (GANs) were introduced in \cite{goodfellow2014generative} to estimate a generative model by an adversarial process, in which a generator G tries to generate a sample using a random noise vector $z$ and a discriminator D tries to compute the probability that a given sample is from real data $y$ rather than generated by G. Training of GAN is equivalent to optimizing the following min-max function,
\begin{equation}
\begin{aligned}
\min\limits_G \max\limits_D V_{\textnormal {GAN}}(D, G) =&\ E_{y}[\log D(y)] \\
&+ E_{z}[\log (1-D(G(z)))].
\label{eq1}
\end{aligned}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{gan.jpg}
\caption{Conditional Generative Adversarial Networks.}
\label{fig:conditional GAN}
\end{figure}
As our target is about transformation, we used GAN's conditional version (CGAN) \cite{mirza2014conditional} in our approach. The adversarial training procedure is almost the same as the original GAN, and the only difference is both the generator and the discriminator have a conditional input $x$, as in Figure 1. The min-max function is:
\begin{equation}
\begin{aligned}
\min\limits_G \max\limits_D V_{\textnormal {CGAN}}(D, G)=\ & E_{x,y}[\log D(y|x)]\\
&+ E_{x,z}[\log (1-D(G(z|x)))].
\label{eq2}
\end{aligned}
\end{equation}
There have already been several successful applications of CGAN in similar tasks. \cite{isola2017image} uses it to convert image styles. \cite{pascual2017segan} applies it to enhance speech. Inspired by their success, we apply CGAN in i-vector space to improve the performance of i-vector based short utterance speaker verification.
\section{Proposed Method}
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{general.jpg}
\caption{Framework of our method.}
\label{fig:general framework}
\end{figure}
Figure 2 shows the framework of our proposed method. At first, acoustic features (MFCC) are extracted from a short utterance, then an unreliable i-vector is extracted from them. Next, an i-vector transformation function is applied to the unreliable i-vector, and finally the transformed i-vector is fed into the PLDA model.
\subsection{GAN for i-vector Transformation}
Our target is to estimate a transformation function which can restore a reliable i-vector (extracted from a long utterance), from a short-utterance i-vector. We use a CGAN-based structure to estimate this function. Overall architecture of the proposed GAN is the one shown in Figure 1, where the conditional input $x$ is an i-vector extracted from a short utterance, the real sample $y$ is an i-vector from a long utterance. In the training stage, $G$ is optimized to generate a reliable i-vector using the one extracted from a short utterance, and $D$ is optimized to determine whether the given reliable i-vector is fake (generated by $G$) or real (extracted from a long utterance). In testing, $G$ is used as the transformation function for an i-vector extracted from a short utterance in the testing set.
In order to prevent several problems such as unstable gradient and model collapse in GAN training, we use a special GAN structure Wasserstein GAN (WGAN) \cite{arjovsky2017wasserstein}. Denoting $x$ as an unreliable i-vector, $y$ as a reliable i-vector and $z$ as random noise, the min-max function is represented as:
\begin{equation}
\begin{aligned}
\min\limits_G \max\limits_D V_{\textnormal {WCGAN}}(D, G) =\ &E_{x,y}D(y|x) \\
& - E_{x,z}D(G(z|x)),
\label{eq3}
\end{aligned}
\end{equation}
Then the objective function related to GAN for $G$ is
\begin{equation}
\min \textnormal {G}= - E_{x,z}D(G(z|x)),
\label{eq4}
\end{equation}
and for $D$, objective function is
\begin{equation}
\max \textnormal{D} = E_{x,y}D(y|x) - E_{x,z}D\left(G\left(z|x\right)\right).
\label{eq5}
\end{equation}
Regarding the training data for GAN, i-vectors extracted from short and long utterances are required. While only long utterances are present in the training dataset, we obtained short utterances by segmenting a long utterance into short utterances. I-vectors are extracted from both long and short utterances using the same extractor. Through this process we can obtain an i-vector pair consisting two i-vectors from the same speaker and session, but one is from a short utterance and the other is from a long utterance. The i-vector pairs are utilized in the next section.
\subsection{Speaker Verification-oriented Objective Functions}
To better guide the training of GAN for our task and make the best use of the training data, two additional learning tasks are added to the GAN framework.
\subsubsection{Numerical difference}
The most straight-forward approach to measure the performance of transformation is computing the numerical difference between the generated i-vector and the target. We compute this objective function using i-vector pairs mentioned above. In many other similar tasks, mean squared error (MSE) is used to measure such a numerical difference. However, for i-vectors, we believe cosine distance is more suitable. The objective function related to this task can be written as:
\begin{equation}
\min \textnormal {COS} = \frac{1}{m} \sum_{i=1}^{m} \left[\frac{1}{n_i} \sum_{j=1}^{n_i}\left(1 - \frac{G(z|x_{ij}) \cdot y_i}{\|G(z|x_{ij})\|\ \|y_i\|} \right)\right],
\label{eq6}
\end{equation}
where $m$ is the number of long utterances in the training set, $y_i$ refers to the i-vector extracted from the $i$-th long utterance in the training set, $n_i$ is the number of short utterances extracted from the $i$-th long utterance, $x_{ij}$ means the i-vector extracted from the $j$-th segment of the $i$-th long utterance and $z$ is random noise.
\begin{figure}[t]
\centering
\includegraphics[scale=0.45]{g1.jpg}
\caption{Training of the generator network $G$ and its application in the testing stage.}
\label{fig:modified G}
\end{figure}
\subsubsection{Speaker discrimination}
The training objectives explained above only compensate the variance brought by the biased phonetic distribution of short utterances, and the speaker labels provided by the training set are not used yet.
Clearly, improving the speaker discriminative ability of generated i-vectors can enlarge the inter-speaker differences among i-vectors, which would improve the verification performance in the PLDA scoring stage. As shown in Figure 3, in the training stage, a supplementary section, $G_{\textnormal{sup}}$, is concatenated after the generator $G$, which takes the generated i-vector as an input and predicts its speaker label. We minimize cross entropy between the prediction result and the ground truth:
\begin{equation}
\min \textnormal{CE} = \frac{1}{m}\sum_{i=1}^{m} \left[\frac{1}{n_i} \sum_{j=1}^{n_i} l_{ij}^k \left(\log o_{ij}^k\right)\right],
\label{eq7}
\end{equation}
where $l_{ij}^k$ is the empirical probability observed in the ground truth that the target i-vector belongs to the $k$-th class, and $o_{ij}^k$ is the predicted probability that the generated i-vector belongs to the $k$-th class. In summary, for training $G$, our goal is to minimize
\begin{equation}
a\ \textnormal{G} + b\ \textnormal{COS} + c\ \textnormal{CE},
\label{eq8}
\end{equation}
where $a$, $b$, $c$ are weight parameters for these three targets, respectively.
After training, as shown in Figure 3, only $G$ is used to generate a reliable i-vector, which is fed into a PLDA model for the next scoring step.
\section{Evaluation}
\subsection{Experimental setup}
We evaluated the performance of our method in the speaker verification tasks of the NIST SRE 2008 \cite{sre08}. We used the ``short2-10sec" and ``10sec-10sec" conditions as our trial sets, where each session is an excerpt of telephone speech, and ``short2" refers to five-minute-long speech while ``10sec" means that the active voice part in the sample is about 10 seconds. There are three sub-conditions in the trail sets: Condition 6 covers all the speech segments, Condition 7 involves only those spoken in English, and Condition 8 only has those spoken in English by native U.S. English speakers \cite{sre08}.
Performance measures for the evaluation were the equal error rate (EER) and the minimum detection cost function (minDCF) of NIST SRE 2008 \cite{sre08} on the trails calculated with DETware provided by NIST \cite{det}.
We compared our method, which is named as ``D-WCGAN'' (Discriminative WCGAN) in the experiments with a baseline i-vector and PLDA system that does not apply any short-utterance compensation techniques. To demonstrate the contribution of GAN to the performance improvement, we made an extra system, which shared almost the same structure with the proposed GAN but did not contain a discriminator and did not use GAN-related objective function. This system is named as ``Single G'' in the following part.
\subsubsection{Baseline system}
The baseline system is the i-vector and PLDA system shown in Section 2. In this system, the input speech segment was first converted to a time series of 60 dimensional feature vectors of Mel-frequency cepstral coefficients (20 dimensional features followed by their first and second derivatives) extracted from a frame of 20ms long and 10ms shift. An i-vector of 400 dimensions was then extracted from the acoustic features using a Gaussian mixture model with 2048 mixture components as a universal background model (UBM) and a total variability matrix (TVM). Length normalization was applied to i-vectors as a preprocessing step before being sent to the PLDA model. Kaldi speech recognition toolkit \cite{kaldi} was used to run these steps.
The UBM, the TVM, and PLDA models were all gender-dependent and trained with SRE08's development data, which contains the NIST SRE2004-2006 data, Switchboard, and Fisher corpus. This dataset as a whole consistes 34,925 utterances from 7,275 male speakers.
\subsubsection{Proposed GAN}
The training data of GAN is a subset of SRE08's development set mentioned above and SRE08's training set, which contains 1,986 male speakers in total. To make the short and long utterance pairs mentioned in Section 4, we used a sliding window of 20s long and 10s shift to cut one long utterance into short utterances. The UBM, TVM for extracting i-vectors are the same as the one used in the baseline system. Finally, we got 331,675 i-vector pairs for GAN training.
The activation function of hidden layers in the proposed GAN, if not specified, is a leaky ReLU \cite{lrelu} with an alpha value set to 0.3. As mentioned above, $G$ generates an i-vector and $G_{\textnormal{sup}}$ predicts its speaker label. The input layer of $G$ contains 450 nodes to accept the 400-dimension i-vectors and random noise vectors of 50 dimensions, followed by three hidden layers with 512 nodes. $G$'s output layer has 400 nodes, which holds the generated i-vector. The activation function for the output layer of $G$ is tanh. $G_{\textnormal{sup}}$ has one hidden layer, which contains 1,986 nodes. Output layer of $G_{\textnormal{sup}}$ also have 1,986 nodes and the activation function of each node is softmax.The random noise vectors were sampled from a Gaussian distribution with zero mean and standard deviation $0.5$. $D$ has four hidden layers and its input layer has 800 nodes, which accepts two concatenated i-vectors. Output layer of $D$ has only one node with a linear activation function. As we used the WGAN structure, weight clipping is done on $D$, where the clipping range is $-0.01$ to $0.01$.
We used the Tensorflow library \cite{tensorflow} for our neural networks implementation. The networks were optimized using RMSProp \cite{rmsprop} with a mini-batch of 64 samples. The learning rate was set to $0.0001$. For G training, we set the value of $a$, $b$, $c$ as 4, 7, 1, respectively.
In the testing phase, for the ``short2-10sec" condition, an i-vector extracted from an utterance in the testing set are transformed by $G$, then PLDA scoring is done on the i-vector extracted from the enrollment set and the transformed i-vectors. At last, score-wise fusion is done between the baseline system and the proposed method. For the ``10sec-10sec" case, almost all the steps are the same as the former one, but the i-vectors from both the enrollment and the testing set are transformed by $G$. The i-vector extractor and PLDA model are the same as those used in the baseline system.
\begin{table}[t]
\caption{The speaker verification results in terms of EER (\%) on all the three conditions of the SRE08 ``short2-10sec" male trail list.}
\label{tab:example}
\centering
\begin{tabular}{l@{} r r r r}
\toprule
\multicolumn{1}{c}{}&
\multicolumn{4}{c}{\textbf{EER (\%)}} \\
\cmidrule{2-5}
\multicolumn{1}{c}{\textbf{System}} &
\multicolumn{1}{c}{\textbf{Cond. 6}} &
\multicolumn{1}{c}{\textbf{Cond. 7}} &
\multicolumn{1}{c}{\textbf{Cond. 8}} &
\multicolumn{1}{c}{\textbf{Average}}
\\
\midrule
a) Baseline & 7.28 & 6.15 & 6.06 & 6.50 \\
b) Single G & 10.04 & 8.85 & 8.33 & 9.07 \\
c) a + b & 7.28 & 5.77 & 6.06 & 6.37 \\
d) D-WCGAN & 9.45 & 8.08 & 8.33 & 8.62 \\
e) a + d & \textbf{6.89} & \textbf{5.77} & \textbf{5.30} & \textbf{5.99} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\caption{The speaker verification results in terms of EER (\%) on all the three conditions of the SRE08 ``10sec-10sec" male trail list.}
\label{tab:example}
\centering
\begin{tabular}{l@{} r r r r}
\toprule
\multicolumn{1}{c}{}&
\multicolumn{4}{c}{\textbf{EER (\%)}} \\
\cmidrule{2-5}
\multicolumn{1}{c}{\textbf{System}} &
\multicolumn{1}{c}{\textbf{Cond. 6}} &
\multicolumn{1}{c}{\textbf{Cond. 7}} &
\multicolumn{1}{c}{\textbf{Cond. 8}} &
\multicolumn{1}{c}{\textbf{Average}}
\\
\midrule
a) Baseline & 11.97 & 10.32 & 9.60 & 10.63 \\
b) Single G & 15.32 & 13.89 & 12.00 & 13.77 \\
c) a + b & 11.16 & 10.71 & 9.60 & 10.49 \\
d) D-WCGAN & 15.42 & 13.89 & 13.60 & 14.30 \\
e) a + d & \textbf{10.75} & \textbf{8.73} & \textbf{8.80} & \textbf{9.43} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\caption{The speaker verification results in terms of minDCF on Condition 6 of the SRE08 ``short2-10sec" and ``10sec-10sec" male trail lists.}
\label{tab:example}
\centering
\begin{tabular}{l@{} r r}
\toprule
\multicolumn{1}{c}{}&
\multicolumn{2}{c}{\textbf{minDCF}} \\
\cmidrule{2-3}
\multicolumn{1}{c}{\textbf{System}} &
\multicolumn{1}{c}{\textbf{short2-10sec}} &
\multicolumn{1}{c}{\textbf{10sec-10sec}}
\\
\midrule
a) Baseline & \textbf{0.370} & 0.553 \\
b) Single G & 0.494 & 0.717 \\
c) a + b & 0.391 & 0.540 \\
d) D-WCGAN & 0.454 & 0.678 \\
e) a + d & 0.375 & \textbf{0.522} \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Results}
Table 1 shows the EERs of the ``short2-10sec" condition of NIST SRE 2008. The average EER of our proposed method was 5.99\%, and it outperformed that of the baseline i-vector PLDA system, 6.50\%. The reduction of average EER is 7.85\%. Table 2 shows the EERs of ``10sec-10sec" condition of NIST SRE 2008. The average EER of our proposed method was 9.43\%, and it outperformed the 10.63\% of baseline, and the reduction of average EER is 11.29\%. Although our method alone did not outperform the baseline system, it achieved better results when the score-wise fusion was done with the baseline method. We found that the best results was achieved when the score weight ratio of baseline system and our method is 7:3. Table 3 shows the minDCF of the Condition 6 of ``short2-10sec" and ``10sec-10sec" sets. The minDCF of our method is 1.33\% worse than the baseline's in ``short2-10sec", but 5.61\% better in ``10sec-10sec". These results showed that our proposed method can make i-vectors more reliable in most cases. However, in current stage, the amount of training data for the GAN is not enough, even smaller than the amount of PLDA's training data. If we have more training data for the GAN, the performance of the proposed methods may become much better.
Regarding the importance of GAN, our results (b, c in Table 1, 2 and 3) showed that performance became worse, but slightly better than the baseline system in EER, when $D$ was absent. This fact demonstrates the contribution of GAN.
\section{Conclusions}
This paper has proposed a GAN-based speaker feature restoration method for speaker verification using short utterances. The generator is trained to transform an unreliable i-vector extracted from a short utterance to a reliable i-vector which can be extracted from a long utterance. Speaker labels are also used in the training of GAN to improve the speaker discriminative ability of generated i-vectors. The evaluation results on NIST SRE 2008 task show that our proposed method improved the performance, especially when only short utterances are available for enrollment and testing.
Our future work includes collecting more data for GAN training, as well as applying the GAN-based framework to other cases when i-vectors become unreliable, for example, noise exists in utterances. In addition, we plan to make the discriminator network able to determine whether two given i-vectors are from one speaker or not, so that we can use the GAN model as a back-end for the text-independent speaker verification system.
\section{Acknowledgment}
This work was supported by JSPS KAKENHI 16H02845 and by JST CREST Grant Number JPMJCR1687, Japan.
\bibliographystyle{IEEEtran}
|
2,877,628,090,199 | arxiv |
\section{Introduction}
\indent
The link between Coronal Mass Ejection, the perturbations of the corona they induce and the production of Solar Energetic Particles (SEPs) is a topic of active research. During the launch of an energetic CME, moving fronts, or waves, are frequently observed in Extreme UltraViolet images (EUV) propagating away from the flaring source region (Thompson et al. 1999). There is a great event to event variability in the morphology and kinematic properties of these EUV fronts making their physical interpretation challenging. For a comprehensive discussion of all proposed theories concerning their origins, we here refer the reader to the extensive review by Warmuth (2015) on this topic. In addition, the formation of the White-Light (WL) signatures of CME-driven shocks was investigated observationally by Ontiveros and Vourlidas (2009) (see also review by Vourlidas and Ontiveros 2010) and numerically by Manchester al. (2008). The \emph{Sun-Earth Connection Coronal and Heliospheric Investigation} (\emph{SECCHI}; Howard et al. 2008) onboard the Solar-Terrestrial Relation Observatory (STEREO) mission (Kaiser et al. 2008) has provided since 2007, unprecedented imaging of solar storms from vantage points situated outside the Sun-Earth line. This capability combined with the images taken by the \emph{Atmospheric Imaging Assembly} (AIA) onboard the Solar Dynamics Observatory (SDO) (Lemen et al. 2012) has spurred a flurry of studies on Magnetic Flux Ropes (MFRs) and coronal pressure waves that form during CME events.\\
\indent The so-called 3-part structure of a CME often observed in WL images includes a filament, a dark core and a pile-up. The pile-up marks initially the outer contour of a CME (e.g. Hundhausen et al. 1972, see review by Thernisien et al. 2011); it corresponds to plasma lifted from the low corona and/or pushed aside by the dark core where the MFR acts as an expanding piston (Vourlidas et al. 2013). Remote-sensing observations combined with numerical simulations show that the subset of EUV fronts that form at the coronal base during CME onset is initially co-located with the 'pile-up' and corresponds to material compressed at low coronal heights by the lateral expansion of the flux rope (e.g. Patsourakos and Vourlidas 2009; Rouillard et al. 2012). When the lateral expansion ceases because the core has reached some pressure equilibrium with the surrounding coronal medium, it can no longer push material in the low corona along the surface and the EUV wave gradually becomes more freely propagating (Patsourakos and Vourlidas 2012; Warmuth, 2015). Its speed and direction are no longer dictated by the expanding core but gradually becomes altered by the local variations in the characteristic speed of the medium. This propagation phase was studied in a number of papers that, not only tracked the EUV signatures, but also the induced deflection of coronal material higher up in the corona (Rouillard et al. 2012; Kwoon et al. 2015). \\
\indent Fast CMEs form near active regions that are typically situated below helmet streamers where strong magnetic fields can prevail. In the direct vicinity of active regions, the characteristic speed of plasma can reach values greater than 1000 km/s but EUV front speeds are typically less than 1000 km/s (Nitta et al. 2013), hence many EUV fronts may not have enough time to steepen into shocks near active region. The EUV front could be initially a layer of compressed material separating the MFR with the ambient corona plasma. It is only when the ambient characteristic speed has sufficiently decreased away from the source region, that a fast pressure front driven by the expansion of the MFR may eventually steepen into a shock. It is impossible to tell from EUV or WL images alone if a shock has really formed at a particular height and a technique must be developed to infer where the propagating front moves faster that the local fast-mode speed . This is one of the challenging tasks undertaken in this paper.\\
\indent We also investigate here the relation between the evolving CME and the release of high-energy particles near the Sun on 17 May 2012. The physical mechanisms that produce solar particles with energies greater than several 100 MeV within a few minutes of the flare and/or CME occurrence are still highly debated. Different origins have been proposed, including magnetic reconnection in solar flares (e.g. Cane et al. 2003), betatron acceleration in the interaction region generated by the expanding CME (e.g. Kozarev et al. 2013), diffusive shock acceleration in the shock located around the rapidly expanding CME (e.g. Sandroos and Vainio 2009). Recent studies have exploited the unprecedented imaging capability offered by STEREO and SDO to track and compare the 3-D evolution of propagating fronts with the properties of SEPs near 1AU (Rouillard et al. 2012, Lario et al. 2014, Kozarev et al. 2015). To do that, the propagation time required for particles to reach the spacecraft making in-situ measurements must be accounted for by considering both their transit speed and the distance travelled. The latter is regulated by the length and variability of the interplanetary magnetic field. These very few studies show that the timing of SEP onsets can be understood in terms of the time taken by the fast coronal shock to reach the different magnetic field lines connected with particle detectors. The questions that remain unanswered are: (1) where along those fronts does the shock form and, (2) is the shock sufficiently strong in the corona to energise particles?
\indent The analysis of the 21 March 2011 event by Rouillard et al. (2012) showed that the 30 minute delay of the two onsets of SEP events measured at L1 relative to STEREO-A (STA) was the time for the propagating front to transit from the footpoint location of the magnetic field lines connected with STA to those connected with the L1 spacecraft. For the reasons discussed in the previous sections, testing the hypothesis that particles are accelerated at the CME shock cannot be limited to simply tracking propagating fronts in EUV images. Hence Rouillard et al. (2012), presented a combined analysis of the EUV and WL corona to derive an estimate of the 3-D speed of the pressure wave by tracking both the density variations ahead of the CME and deflected streamers higher up in the corona. No derivation was proposed in that study of the fast magnetosonic Mach number that would confirm the existence of a shock at any particular height. However as we shall see in this paper, the coronal heights considered in Rouillard et al. (2012) were likely high enough for the ambient fast mode speed to have dropped sufficiently for a shock to form. Since this study, we have developed a number of observationally-based techniques to derive quantitatively the 3-D properties of propagating fronts (including the $M_{FM}$) in order to test the hypothesis that high-energy SEP are produced at coronal shocks. \\
\indent
After presenting the properties of the 17 May 2012 event (sections 2, 3 and 4), we present a new method to extract shock wave parameters in 3-D (sections \ref{sec:PFSS} and \ref{sec:MHD} ) using a number of different techniques. We then compare those derived shock parameters with simultaneous radio measurements (section 7) and the properties of the SEP measured near Earth (section 8).
\section{The 17 May SEP event:}
\indent At 01:25UT on 17 May 2012, the Geostationary Operational Environmental Satellites (GOES) spacecraft detected a M5.1 X-ray flare after several days of relatively quiet solar activity marked by occasional C-class flares, weak CME events ($<$600 km/s) and relatively weak energetic particle fluxes measured in the inner heliosphere. This M-class flare was associated with the eruption of a fast ($>$1600 km/s) and impulsive CME and the detection of very energetic particles (GeV) near Earth. A previous study reported that this solar event was associated with the detection of a Ground Level Enhancement (GLE) by ground-based neutron monitors (Gopalswamy et al., 2013); evidence that proton exceeding several hundreds of MeV energies were released from the Sun. This is directly supported by space measurements of protons exceeding GeV energies (Adriani et al. 2015) by the The Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) instrument (Picozza et al. 2007). This event occured in isolation provides an excellent opportunity to study the link between a CME and the production of high-energy particles without the contamination from other events.\\
\indent There are several puzzling aspects of this event that were highlighted in previous articles. In particular the flare intensity (M5.1) was lower than flare intensities measured in previous GLE events. As noted by Gopalswamy et al. (2013) and discussed in detail later in this paper; despite the rather weak flare, the associated CME had a fast speed more typical of X-class flares. Based on the arrival time of the GeV particles detected in the GLE and an interplanetary magnetic field line of length 1.2AU, Gopalswamy et al. (2013) put the Solar Particle Release (SPR) time of particles near the Sun at 01:41UT or about 15 minutes after flare onset when the CME had already reached a height of 2.3 R$_\odot$ and roughly ten minutes after the onset time of the type II burst (01:30UT). Using simple geometric arguments, they put the height of a first shock formation roughly at 1.38 R$_\odot$ well below the height reached by the leading edge of the CME at their SPR time. In Appendix A, we use a velocity dispersion analysis to show that the SPR time derived by Gopalswamy is likely too late by some 4 minutes (01:37:20$\pm$00:00:02UT) because the pathlength followed by these particles is more likely to be about 1.89$\pm0.02$ AU. Provided that magnetic connectivity between the shock and the point of in-situ measurements is maintained from the time of shock formation onwards the shock would have about five minutes to accelerate particles to GeV energies. The hypothesis that diffusive-shock acceleration is the energisation mechanism of these particles assumes that magnetic connectivity is established between the Earth and a coronal shock. This paper presents a thorough analysis of the evolution of the shock and employs a new combination of observationally-based and numerical techniques to derive, not only the magnetic connectivity of the near-Earth environment with the shock, but also some of the shock properties before and during the GLE event.\\
\section{Observations:}
\begin{figure}
\epsscale{1.1}
\plotone{Figure1}
\caption{ A view of the ecliptic plane from solar north showing the positions of the Earth, STA and STB. The nominal Parker spiral connecting magnetically the Earth to the low corona is shown in black. The intersection of the COR2A (red), COR2-B (dark blue), SOHO C2 (light blue) fields of views with the ecliptic plane are shown as pairs of elongated triangles. The trajectory of the CME launched on the 17th of May results from the analyses of heliospheric imagery as given in Appendix A). The longitudinal extent of the CME (piston+shock) was chosen to fit with the observation of the shock by STA (as measured in situ: see Appendix A) and is here exactly 100 degrees. This figure and the analysis of the trajectory of the CME was made using the IRAP propagation tool and J-maps produced by the HELCATS project (see acknowledgements for details). {\it The Astrophysical Journal}}
\label{ORBITSEP}
\end{figure}
Figure~\ref{ORBITSEP} presents the positions of the STEREO spacecraft and the Earth on 17 May 2012, these three vantage points provided 360$^\circ$ views of the Sun. The longitudinal separation of STA and STB with respect to Earth were 114$^\circ$ and 117$^\circ$, respectively. The expansion of the CME could be tracked simultaneously from widely separated spacecraft allowing the 3-D volume of the expanding high-pressure fronts to be derived by using the comprehensive suite of optical instruments on STEREO, SDO and the \emph{Solar and Heliospheric Observatory} (\emph{SOHO}). The SECCHI package onboard \emph{STEREO} (Howard et al. 2008) consists of an Extreme Ultraviolet Imager (EUVI), two coronagraphs (COR-1 and COR-2), and the Heliospheric Imager (HI). At the time of the event studied here, the magnetic connectivity of the STEREO and the near-Earth orbiting spacecraft, also provides a circumsolar measurement of particles potentially released from widely separated source regions. \\
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.35]{Figure2}
\caption{This figure compares running-difference images (rows a,c,e) of the CME observed by STA (left hand-column and STB (right-hand column) with the results of applying the fitting technique (rows b,d,f) developed here. The images are all from the EUVI instruments except the left-hand image shown in row f obtained by COR1-A. Red crosses are superposed on the fitted ellipsoids, they show the contour of the propagating front observed in the running difference images and are used to constrain the extent and location of the ellipsoid at each time.}
\label{TRIANGWL}
\end{center}
\end{figure*}
\indent Rows (a), (c) and (e) of Figure~\ref{TRIANGWL} present images covering the first 20 minutes of the CME eruption as viewed along the Sun-STA, Sun-Earth and Sun-STB lines. With the exception of the image obtained by the \emph{Large Angle and Spectrometric Coronagraph Experiment} (\emph{LASCO}; Brueckner et al. 1995) C2 shown in the last row, the sequence of images shown in rows (a), (c) and (e) are at the closest times to 01:35, 01:45 and 01:55UT, respectively. These images show that the coronal region perturbed by the expanding CME increases with time in both EUV and in the WL images. \\
\indent The surface of the propagating front generated around the expanding CME is initially fairly regular and we found that an ellipsoid fits the outermost extent of this perturbed region very well. We manually extracted the location of the outermost extent of the CME off limb and on disk at all available times. These points are plotted as red crosses in the images given in rows (b), (d) and (f) and are used to outline the contour of the ellipsoids viewed from the three vantage points. When the CME is low in the corona, the high cadence of images taken by SDO and STEREO nearly guarantees that simultaneous images are obtained from the three vantage points at regular five minute intervals starting from the flare onset at 01:25UT. \\
\indent The dimensions of the ellipsoid are defined by a set of three parameters and its central position is defined in heliocentric coordinates (radius, latitude and longitude). An ellipsoid is considered a good visual fit when it intersects most of the red crosses. Off limb the ellipsoid must pass by the outermost extent of the CME. On disk the red crosses mark the location of the EUV front and must match the line of intersection of the ellipsoid with the solar surface. During the first 20 minutes of the event we used observations from STEREO and SDO. Beyond the SDO AIA field of view (1.3 R$_\odot$), coronal images from Earth's perspective are obtained at low cadence by the LASCO coronagraphs. To cross-check the inferred location of the CME extent in LASCO images, we interpolated the four parameters at the LASCO C2 recording times; the interpolated locations are shown in the middle panel of row (f), revealing very good agreement between the observations and the fitted geometrical surface. In addition to the different time cadence of the different optical instruments, we noted that the signal to noise ratio in the COR-1 images is reduced near the edge of the field of view. This has been noticed before (e.g. Rouillard et al. 2010) and it can hamper our ability to accurately track the outer edge of the pressure wave in COR-1 when the CME reaches these heights. For this reason, we rely on COR-2 towards the external part of COR-1, where the COR-1 and COR-2 fields of view overlap as shown in the first column of rows (e) and (f). \\
\section{Overal comparison between shock location and the in-situ measurements:}
\begin{figure*}
\begin{center}
\includegraphics[angle=00,scale=.4]{Figure3}
\caption{Center: a view of the near-Sun environment with the triangulated locations of the propagating fronts at four successive times. The relative locations of the magnetic field lines connecting the STA, STB and L1 points assuming a Parker spiral from the spacecraft to 2.5R$_\odot$ and the PFSS model from 2.5R$_\odot$ to the solar surface are shown as colored lines. The colored arrows mark the direction of hypothetical particles propagating outward towards the interplanetary medium.} Three panels show the time series of hourly-averaged 5-10 MeV/nuc Oxygen (blue lines) and Iron (red lines) fluxes measured over a 7-day interval by the LET instruments on the two STEREOs and the EPACT/LEMT instrument on the Wind spacecraft.
\label{SHOCKSEPs}
\end{center}
\end{figure*}
\indent Figure~\ref{SHOCKSEPs} presents, as superposed black ellipsoids, the location of the CME front at regular 5-minute intervals between 01:25 and 01:55 UT. We also show the Parker spiral connected with the ST-B (blue), ST-A (red) and near-Earth (green line) orbiting spacecraft. These spirals were defined by the speeds of the solar wind measured in situ at the three spacecraft (ST-B: 300 km/s , ST-A: 350 km/s, and near Earth: 400 km/s) near the times shown in Figure~\ref{SHOCKSEPs}. Below 2.5 R$_\odot$, we trace the magnetic field lines using a Potential Field Source Surface (PFSS) model made available on solarsoft by the Lockheed Martin Solar And Astrophysics Laboratory (LMSAL)\footnote {$http://www.lmsal.com/~derosa/pfsspack/$}. The extrapolation is based on evolving surface magnetic maps into which are assimilated data from the Helioseismic and Magnetic Imager \emph{Helioseismic and Magnetic Imaging} (HMI; Scherrer et al. 2012) onboard SDO. These maps account for the transport and dispersal of magnetic flux across the photospheric surface using a flux-transport model (Schrijver \& DeRosa 2003).The transport processes are differential rotation and supergranular diffusion, they modify continually the distribution of photospheric magnetic fields. The area of the corona of interest in the present study is situated near the West limb, hence the photospheric magnetic field measurements used in the present study were only a few days old at the time of the extrapolation.
These estimated field lines allow us to determine approximately how the three spacecraft connect to the corona. According to Figure~\ref{SHOCKSEPs}, the space environment situated near Earth is well connected with the emerging CME (green), whereas ST-A only connects with the CME much later and ST-B is not connected with the event. This is in qualitative agreement with particle measurements taken near Earth by EPACT/LEMT (ULEIS, Mason et al. 1998) on ACE and the Low Energy Telescope (LET, Mewaldt et al. 2008), one of four sensors that make up the Solar Energetic Particle (SEP) instrument of the IMPACT investigation on STEREO (Luhmann et al. 2008). The LET is designed to measure the elemental composition, energy spectra, angular distributions, and arrival times of H to Ni ions over the energy range from ~3 to ~30 MeV/nucleon.\\
The hourly-averaged flux of Oxygen ions in the 5-10 MeV/nuc energy range was very intense at the Wind spacecraft (10$^{-2}$ particles/cm$^2$-sr-s-MeV/nuc) , ST-A detected initially low Oxygen flux increasing steadily to peak at 2x10$^{-3}$ particles/cm$^2$-sr-s-MeV/nuc when the derived CME front intersects the spacecraft some 48 hours after the launch of the CME. In contrast ST-B measured no SEP event. Since this study is focused on the conditions that produced the GLE during the first few minutes of the CME launch, we do not discuss STA or STB particle measurements further, since the SEP either occurred much later for STA and not at all for STB.\\
\section{Properties of the emerging shock: the PFSS approach}
\label{sec:PFSS}
\subsection{Derivation of the 3-D shock speed}
\indent Once the parameters of the successive ellipsoids are obtained, we interpolate these parameters at steps of 150 seconds to generate a sequence of regularly time-spaced ellipsoids. To compute the 3-D expansion speed of the surface of the pressure wave, we find for a point $P$ on the ellipsoid at time $t$, the location of the closest point on the ellipsoid at previous time-step $t-\delta t$ by searching for the minimal distance between point P and all points on the ellipsoid at time $t-\delta t$. We then compute the distance travelled between these two points that we divide by the time interval $\delta t=150$ seconds to obtain an estimate of the speed $P$. This approach slightly underestimates the shock speed calculated at time $t$ during the acceleration phase of the CME. We considered a set of 70$\times$70 grid points distributed over the ellipsoid. This number of points is computationally tractable and is sufficiently high to compute accurately the speed of the expanding shock as well as the shock geometry and Mach number described later. We compared this approach with that of computing the normal to the point $P$ at time $t-\delta t$ and finding the intersection between this normal and the ellipsoid at time $t$. Speeds computed by dividing the distance between this intersection at time $t-\delta t$ and $P$ at time $t$ gave nearly identical speeds to those computed with the minimal distance providing the heliocentric latitudinal/longitudinal coordinates of the center of the ellipsoid varies very slowly ($<5^\circ$) between consecutive 150 second time intervals, thereby guaranteeing that the consecutive ellipsoids are quasi-concentric. This condition is fulfilled for this event since we found that its latitude shifted southward from a heliocentric latitude of 4$^\circ$ at 01:30UT to stabilise at -2$^\circ$ at 01:45UT, while its longitude shifted eastwards from a Carrington longitude of 190$^\circ$ at 01:30UT to stabilize at 180$^\circ$ at 01:55UT. \\
\indent Figure~\ref{MAthetaBnPFSS} presents the results of extracting the speed of the propagating front along the normal vector to the front surface and as a function of time. In addition to the location of the sphere, we trace open magnetic field lines using the PFSS model. Figure~\ref{MAthetaBnPFSS} shows that the CME front emerged from a region located inside a streamer.\\
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.45]{Figure4}
\caption{The results of the derivation of shock parameters based on the combined inversion of imagery data and the PFSS model at four successive times during the eruption of the CME. Each column shows a different parameter: the shock normal speed (left), $\theta_{Bn}$ (center) and Mach number (right). Coronal magnetic field lines are traced in black.}
\label{MAthetaBnPFSS}
\end{center}
\end{figure*}
\subsection{Derivation of the 3-D shock geometry and Mach number:}
We seek to derive the evolving properties of the CME-driven shock using the full set of available in-situ and imaging observations. In addition to the derivation of the shock speed, the parameters of interest are the shock geometry and the shock Mach number. The Mach number in an unmagnetised fluid is the ratio of the speed of the wave along the wave normal to the speed of sound of the ambient medium upstream of the wave. In a magnetised plasma, there are three modes: the fast and slow magnetosonic waves and the intermediate Alfven wave. In this paper, the characteristic speed to which the front speed will be compared is the fast-mode speed, defined as:
\begin{equation}
V_{FM}=\sqrt{\frac{1}{2}\big[ V_A^2+C_S^2+\sqrt{(V_A^2+C_S^2)^2-4V_A^2C_S^2cos^2(\theta_{Bn})}\big ]}
\label{eq:Mach}
\end{equation}
where $V_A$ is the Alfven speed, $C_{S}$ is the sound speed, $\theta_{Bn}$ is the angle between the wave vector and the magnetic field vector. The Mach number, $M_{FM}$, is here defined as:
\begin{equation}
M_{FM}=\frac{V_S}{V_{FM}}
\end{equation}
where $V_S$ is the shock speed. We assume that at the very low coronal heights imaged here, the wind speed is zero. To derive the fast-mode speed, we need to derive the shock geometry, the properties of the background coronal plasma including temperature, density and the magnetic field. Since direct measurements of the 3-D coronal magnetic field strength are not yet possible, we have to employ some magnetic field reconstruction or modelling of the corona to infer the 3-D magnetic field distribution.\\
\indent A shock is quasi-parallel when $\theta_{Bn}<45^\circ$ and quasi-perpendicular when $\theta_{Bn}>45^\circ$. Equation 1 leads to the property that for a parallel geometry, the fast mode speed becomes $V_{FM}=\sqrt{ \frac{1}{2} \big[ V_A^2+C_S^2+\left|V_A^2-C_S^2 \right| \big]}$ whereas for a perpendicular geometry, $V_{FM}=\sqrt{V_A^2+C_S^2}$. For a coronal temperature of $T = 1.4$ MK, the sound speed is roughly 180 km/s, generally lower than the ambient Alfven speed except near the tip of streamers where the magnetic field strength can decrease by an order of magnitude. At this location, the shock becomes simultaneously quasi-parallel, in that region the fast mode speed is controlled by the sound speed. To derive the sound speed, we use $T = 1.4$ MK for the present PFSS approach. \\
\indent Coronal shocks undergo different regimes that are related to the value of their Mach number. There is a critical Mach number (M$_c$) (Edminston and Kennel 1984; Mann et al. 1995) above which simple resistivity cannot provide the total shock dissipation. The microphysical structure of collisionless shocks is very different when the shock is sub or super-critical (e.g. Marcowith et al. 2016). In the super-critical case a significant part of upstream ions are reflected on the shock front gaining an amount of energy that enables them to be injected into the acceleration process. Sub-critical shock do not reflect ions, significantly diminishing the ion and electron acceleration efficiency. M$_c$ is a function of the various shock parameters, but it has been argued that it is at most 2.7 and usually much closer to unity (e.g. Mann et al. 1995, Schwartz, 1998). In the present study, a shock is said super-critical when $M_{FM}>3$. \\
\paragraph{Derivation of the ambient magnetic field properties:}
An extrapolation of the photospheric magnetic field to the corona using the PFSS technique can provide the magnetic field at all points on the surface of the triangulated shock surface. The PFSS model has a number of strong assumptions including a heliocentric spherical source surface, that no current is flowing in the corona and that the field is radial at the photospheric boundary (e.g. Wang and Sheeley, 1990). The line-of-sight component is measured by HMI and in the present study, the input is such that the line-of-sight component was converted to a radial component. It is common to also correct magnetograms for the poorly observed polar magnetic fields by applying a latitude dependent correction factor (Wang and Sheeley, 1991), here however no correction was applied since the magnetic maps used for the LMSAL PFSS model build up polar fields over time through transported processes. Finally, the measurements of surface magnetic fields are also prone to line profile saturation, including the HMI instrument, this saturation is not accounted for in the PFSS model used here. To derive the distribution of angles $\theta_{Bn}$ (Equation \ref{eq:Mach}) over the entire ellipsoid, we first derived numerically the vector normal to the ellipsoid at each point on its surface and then computed the angle between this normal vector and the coronal magnetic field vector obtained from PFSS at that point. \\
\indent The values of the magnetic field and plasma parameters modelled in the present section and in section \ref{sec:MHD} were interpolated between the numerical grid points at each location on the triangulated surface front. The modelled field is defined on a spherical grid $\{r_i,\theta_j,\phi_k\}$ with a constant step ${\rm d} \phi$ and variable steps in radial and co-latitude, i.e. ${\rm d} r$ and ${\rm d} \theta$ are non-constant. We adapted a linear interpolation method, described for a 2D axisymmetric grid by Cerutti et al. (2015), here generalized in 3D (volume weightening). The generalization is straightforward because the integration over $\phi$ is elemantary. \\
\indent Given an arbitrary position $(r,\theta, \phi)$ where we want to interpolate the field value between grid positions, we find the cell in which the point is located, indexed as $\{i,j,k\}$ cell. We then calculate the total volume of the cell using:
\begin{eqnarray}
V_{\rm cell} &=&\int_{\phi_k}^{\phi_{k+1}} \int_{\theta_j}^{\theta_{j+1}} \int_{r_i}^{r_{i+1}} r^2 \sin \theta {\rm d}r {\rm d} \theta {\rm d}\phi \nonumber \\
&=& {\phi_{k+1}-\phi_k \over 3} (\cos \theta_j-\cos \theta_{j+1})(r_{i+1}^3-r_i^3)
\end{eqnarray}
For the interpolation, 8 supplemantary sub-volumes are needed and are calculated in the same manner. The results are shown in the center column of figure \ref{MAthetaBnPFSS} and reveal that the geometry of the shock varies greatly in space and time. The shock is mostly quasi-perpendicular when situated below the streamers in closed field regions. It becomes quasi-parallel when the nose of the shock reaches the source surface near 01:37:30UT and enters open (and (radial) field regions. A quasi-perpendicular geometry occurs mostly near its flanks as shown previously for other events (Kozarev et al. 2015). The band of high $M_{FM}$ is co-located with the region of quasi-perpendicular geometry but evolves within 10 minutes into a quasi-parallel geometry, we discuss this transition later in the paper. \\
\indent The Alfven speed is proportional to the ambient magnetic field strength and inversely dependent on the square root of the plasma density. \\
\indent The open coronal fields computed with PFSS using uncorrected magnetograms can be at times much weaker than the open magnetic field measured near 1AU (Arden et al. 2014). Since we are interested in the process of shock formation along open magnetic field lines connected with near-Earth spacecraft, our approach has been to correct the magnetic fields derived from PFSS by using in-situ measurements. The total magnetic flux released into the interplanetary medium can be computed from PFSS extrapolations by simply averaging the unsigned radial field component at the source surface multiplied by its surface area. \\
\indent The expansion of the magnetic field leads to a more uniformly distributed radial field at the PFSS source surface than at the photosphere but the field has not yet spread out uniformly in latitudes and longitudes. The Ulysses spacecraft, that surveyed the radial component of the heliospheric magnetic field outside the ecliptic plane and as a function of heliospheric distance, revealed that beyond 1AU the absolute value of the radial field is independent of heliographic latitude (Smith and Balogh, 1995). This result implies that a re-distribution of the magnetic field continues beyond the source surface and for several tens of solar radii in the outer corona probably smoothing out differences in the tangential pressure and forcing the radial magnetic field to become uniform in latitude by 1AU. This redistribution occurs beyond the source surface and is more gradual than the strong re-distribution forced the source surface associated with the radial field boundary condition. As we shall see, MHD models with boundary conditions maintained at much higher coronal heights (30 R$_\odot$) suggest a more gradual redistribution of the radial field component with heliocentric distance than PFSS.\\
\indent The average of the radial field component extrapolated over the entire source surface is: 5.24 $10^{-6}$T at the time of extrapolation considered for this event. We also compared the full surface average (5.24 $10^{-6}$T) with an average of the radial field component taken over an area centered on the heliocentric coordinates of AR 11476 and extending 30 degrees around that region and we found an even lower value of 4.34 $10^{-6}$T or 82\% of the total surface average. We use this latter value to account for the possibility that the radial magnetic field may not have re-distributed uniformly over a sphere centered at the Sun and of radius 30 solar radii. \\
Since the Ulysses observations show that by 1AU the radial field measured in the ecliptic is representative of the radial field measured at any latitude, we can compare the average of the source surface field with the radial field values measured in situ in the ecliptic plane near 1AU. To derive the radial field value that is representative of the background magnetic field, we followed the procedure used in Rouillard et al. (2007) to derive the total open magnetic flux and averaged the absolute value of the hourly radial field values measured near 1AU over a full 27-day solar rotation period. The passage of large Interplanetary CMEs (ICMEs) will increase the background radial field values measured in situ near 1AU at a specific spacecraft. To obtain a more robust global estimate of the open magnetic field, we used not only the OMNI data but also the STA and STB magnetometer data. In all cases, the average radial field is close to 1.9$\pm$0.4 nT which we take as our reference radial field component representative of the 'background' solar wind with a 20$\%$ uncertainty in this estimate. \\
\indent To compare the PFSS data to this radial field, we simply account for the nearly spherical expansion of the field between the source surface and 1AU such that the estimated radial field at 1AU is $ 4.34 \times10^{-6}\times(2.5/215)^2=0.58$nT, a factor of about 3.3 less than the measured radial field of 1.9nT near 1AU. A similar value is obtained by comparing the average open field at the reference source surface location (4.34 $\times 10^{-6}T$) in a region limited to the streamer where the CME originated with the value of the radial field measured near 1AU around the onset time of the SEP event. We conclude that the PFSS extrapolation used in the present study underestimates significantly the total open flux released in the interplanetary medium and that a correction factor of 3.3 should be applied to the PFSS data in order to obtain more realistic field values in the corona. The correction factor was obtained by comparing the radial components of the magnetic field at the source surface and at 1AU. To preserve the global topology of the field, the correction factor was applied to all components of the magnetic field including closed field regions of the corona that cannot be related to in-situ measurements made near 1AU. We adopted this technique because the focus of the present paper is on the production of SEPs that travel along open magnetic field lines to 1AU. In a future study, we will exploit radio imaging of other events to show that this correction may be too severe in the closed field regions of the corona. The correction will have the effect to substantially decrease the computed Mach numbers of the shock thereby providing conservative estimates. \\
\paragraph{Derivation of the ambient density:}
Past derivations of coronal densities have considered 1D (Mann et al. 1999, LeBlanc et al. 1998) and 2D analytic models (Warmuth $\&$ Mann, 2005). These studies have shown that the use of a generic radial density model can lead to inaccurate derivations of local Alfven speeds due to the strong magnetic field gradients in the corona. In order to derive electron densities that are more representative of the (background) coronal conditions through which our pressure front is propagating we propose to invert remote-sensing observations. \\
\indent Estimates of the electron density distribution can be obtained by inverting EUV images using Differential Emission Measure (DEM) inverted from the SDO/AIA six coronal Fe filters (Aschwanden et al., 2001). For the density calculation using SOHO/LASCO, we use polarized brightness images. The brightness of the K-corona results from Thomson scattering of photospheric light by coronal electrons (Billings 1966). In the case of polarised brightness observations at small elongations (below 5 R$_\odot$ Mann 2003), the F-corona can be assumed unpolarized and thus does not contribute to the polarized signal; for this reason we restrict our derivation of electron densities to below 5 R$_\odot$. The technique employed to interpolate densities between the AIA and SOHO fields of view is detailed in the paper by Zucca et al. (2014). Beyond 5 solar radii, we assume that the plasma expands spherically to 1AU. We assume that electrons situated within only 3 degrees longitude of the plane of the sky contribute to the emission.\\
\indent In order to derive densities in the entire volume crossed by the triangulated front, we let the corona rotate in the plane of the sky for several days (spanning about 70 degrees of longitude) and repeat the aforementioned analysis every six hours (every 3.3 degrees of solar longitudes) between the 15 and 20 of May 2012. We then interpolate densities on a regular 1 degree longitudinal grid between each meridional plane to obtain a uniform 3-D grid of density values inside the entire volume crossed by the front. For this derivation, we checked that no large CME was present in the fields of view of AIA and C2 at the times used to derive the background densities. One of the assumptions made in this analysis is that the CME of interest here that passed through the LASCO field of view between 01 and 05UT did not alter the structure of the coronal streamer permanently and did not affect the density reconstruction between 17 and 20 May. This is supported by a smooth transition of the electron density variations derived before (00UT on 16 May) and after (06UT on the 17 of May) the CME passage. \\
\paragraph{The fast-magnetosonic Mach number:}
\indent The distribution of $M_{FM}$ is obtained by dividing the normal speed at each point on the triangulated front by the local fast mode speed of the medium. At these low coronal heights we can neglect the speeds of the solar wind plasma in this derivation of $M_{FM}$. The right-hand column of Figure~\ref{MAthetaBnPFSS} presents the distribution of $M_{FM}$ over the entire front. Before 01:32UT, the front is located well inside the streamer, the shock has not formed at these low heights ($M_{FM}<1$). Between 01:32 and 01:35UT, the front speed exceeds the local fast-mode speed in certain regions. This transition marks the formation of a shock and occurs near the onset time of the type II burst at 01:32UT (reference). A subset of the front reaches super-critical speeds ($M_{FM}>3$) when it enters the open magnetic field regions situated at the top of the streamer after 01:37UT. \\
\indent A band of high-$M_{FM}$ becomes very clear after that time along the front surface and located near the tip of the streamer and its associated neutral line. We found that the strongest rises in $M_{FM}$ mark drops in the background Alfven speeds. Such drops are mainly due to decreases in the strength of the coronal magnetic field and to a lesser extent to increases in the density since the Alfven speed is inversely proportional to the square root of the density.\\
\indent PFSS extrapolations show that the magnetic field lines that form the helmet streamers expand significantly between the photosphere and the source surface. We note that there is ample evidence that such a band of low magnetic field strength/high density (therefore enhanced plasma beta) exists from remote-sensing and in-situ measurements. The WL counterpart of this band is the plasma sheet typically observed as bright rays extending above helmet streamers (Bavassano et al. 1997; Wang 2009). The in-situ counterpart is thought to be the Heliospheric Plasma Sheet (HPS) typically measured during sector boundary crossings. This sheet is associated with very high plasma beta near 1AU due to an order of magnitude decrease in the magnetic field and signficant increases in the plasma density (Winterhalter et al. 1994, Crooker et al. 2004). The more the HPS/HCS is warped in latitude, usually in response to higher solar activity or weaker polar fields, the more the trajectory of a spacecraft making in situ measurements will be aligned to the normal of the local tangential plane of the HPS/HCS. During those times, the HPS measured near 1AU is well defined and of typically short duration, lasting at most 16 hours (Crooker et al. 2004). Assuming a typical rotation period of 25.38 days, we can convert that duration into a longitudinal width, this corresponds to about 10 degrees longitudinal width. The band of high $M_{FM}$ derived on the ellipsoid extends over a 10-15 degrees longitudinal width, near the upper limit of the size of the HPS typically measured near 1AU. \\
\indent The location of the source surface height at 2.5Rs is somewhat arbitrary. The justification for such a height stems from the coronagraphic observation that coronal electrons appear to flow rather radially beyond 2.5 R$_\odot$ (e.g. Wang and Sheeley 1990). Additionally, the position of coronal holes derived by PFSS agrees rather well with EUV observations and the sector boundary structure predicted by PFSS at 1AU agrees well with in-situ measurements during the different phases of the solar cycle (e.g. Wang et al. 2009). More recent studies have argued that a better agreement is obtained between the total open flux derived from the PFSS model and in-situ measurements by letting the source surface vary during the solar cycle (Arden et al. 2014). The large low-latitude coronal holes observed by STEREO during the solar minimum could be better interpreted by decreasing the height of the source surface to lower heights. Indeed decreasing the height of the source surface will allow more field lines to open to the interplanetary medium, this will increase the size of coronal holes and of the total open flux released to the interplanetary medium. \\
\indent We investigated the effect of changing the source surface height on the computed $M_{FM}$. The procedure described in section 4.1 to correct the total open flux values needs to be repeated for each new source surface height and we found correction factors ranging from 2.14 at 2Rs to 4.12 at 3Rs. We found that the band of high $M_{FM}$ retains its global shape for the three source surface radii, however lowering the radius to 2Rs induces a very broad (15-20$^\circ$) band of high $M_{FM}$. The broadness of this band of much lower magnetic field strength cannot be easily related with the heliospheric plasma sheet measured in the solar wind near 1AU. A heliospheric plama sheet measured for 16 hours and passing over a spacecraft at 300 km/s would correspond to a longitudinal extent of $10^\circ$. Increasing the source surface height to 3Rs delays the formation of the shock to after 01:32:30UT so that no shock has yet formed around the onset time of the type II bursts (01:32:00UT). A source surface height situated at 2.5 R$_\odot$ supports the existence of a shock already at 01:32:30UT and produces a broad but not too unrealistic band of high $M_{FM}$ perhaps akin to the heliospheric plasma sheet typically measured near 1AU. In addition we also checked that for a source surface at 2.5 R$_\odot$ the size of the coronal holes are similar to those observed by the EUVI instruments on STA. \\
\section{Properties of the emerging shock: the MHD approach}
\label{sec:MHD}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=0.85]{Figure5}
\caption{In the same format as figure \ref{MAthetaBnPFSS} for the derivation of shock parameters based on the MAST MHD model.}
\label{MAthetaBnMHD}
\end{center}
\end{figure*}
A strong assumption of the PFSS model is the spherically uniform source surface that forces magnetic field lines to diverge rapidly from the photosphere to become radial at the surface. While coronagraphic imaging and in situ measurements provide strong supporting evidence for the existence of a narrow region of combined dense plasma and much weaker magnetic fields near the coronal/heliospheric neutral line, we investigated whether MHD simulations, with no source surface assumed, provide additional evidence for the formation of this region. MHD simulations provide both derivations of the global magnetic field as well as plasma density and temperature.\\
\indent In this study, we used the two sets of 3-D MHD models developed by Predictive Sciences Inc. Like the PFSS model, these models use SDO HMI magnetograms as the inner boundary condition of the magnetic field. The outer boundary is set at 30 solar radii. The Magneto-Hydrodynamic Around a Sphere Polytropic (MASP) model is a polytropic MHD model and has a standard energy equation with a value of the polytropic index, gamma, close to 1 (typically 1.05) to crudely approximate the energy transport in the corona (Linker et al. 1999). The temperature at the lower boundary in this model is selected to be a coronal temperature (1.8 mega-Kelvin). For the times of interest to this study the densities derived by this model tend to be unrealistic. Indeed, applying an inverse square density fall off between 30 R$_\odot$ and 1AU to compare the model with in-situ measurements, we find that simulated densities are an order of magnitude too high compared with those measured in the solar wind. \\
\indent The Magneto-Hydrodynamic Around a Sphere Thermodynamic MAST model is a MHD model with improved thermodynamics including realistic energy equations with thermal conduction parallel to the magnetic field, radiative losses, and coronal heating. The effect of Alfven waves on the expanding coronal plasma is also included using the so-called Wentzel-Kramers-Brillouin approximation. The temperature at the lower boundary in this model is 20,000 K (approximately the upper chromosphere), and the transition region is captured in the model. Special techniques are used to broaden the transition region such that it is resolvable on 3D meshes and still gives accurate results for the coronal part of the solutions. The coronal heating description is empirical and the coronal densities arise entirely from the heating and its interaction with the other terms. A description of this model appears in Lionello et al. (2009). \\
\indent Extrapolating the simulated values from the outer boundary of the model (30Rs) to 1AU reveals that, like for PFSS, the simulated neutral line maps to the heliospheric current sheet. In addition, the density values are also well simulated and fall in the range of density values measured before the onset time of the SEP event. The average value of the coronal magnetic field threading a sphere centered at the Sun and located at 5 R$_\odot$ leads to values of $\sim$1.3nT at 1AU. We choose a height of 5 R$_\odot$ after tracing open and closed magnetic field lines in the MAST model; we found that beyond this height magnetic field lines are mostly open to the interplanetary medium. We remind the reader, that in contrast to MHD models, the height at which magnetic field lines are all open is set by that of the source surface in the PFSS model. The average radial field measured in MAST is lower than the measured radial field values near 1AU (1.9$\pm$0.4 nT). A correction was applied to magnetic field values of the MAST model by multiplying all field values by a factor of 1.5($=1.9/1.2$). Again the correction factor is applied to all components of the magnetic field to preserve the global topology. \\
\indent We show in Figure \ref{MAthetaBnMHD}, the front speed (left), $\theta_{Bn}$ (center) and $M_{FM}$ (right) on the surface. In the MHD model the neutral line forms at the same location as in the PFSS model, but is more oriented along the North-South direction than the neutral line derived with PFSS. Just like for the previous technique, the fast-mode speed drops to low values in the vicinity of the neutral line ($<$200 km/s), thereby boosting $M_{FM}$ because the magnetic field strength is low and the density high. We note that the field strength drops in this region due to a combination of the field expansion (like PFSS) and, since we are using a MHD model, some level of numerical diffusion which forces field lines to reconnect. Although here not physically resolved, such reconnection processes are thought to occur in the vicinity of the real neutral line and of the heliospheric plasma sheet since complex magnetic structures reminiscent of magnetic flux ropes and field line disconnections are frequently measured near 1AU in the heliospheric plasma sheet (Crooker et al. 1996, Rouillard et al. 2011c). The MHD model suggests additionally that the Alfven speed drops to 200-300 km/s along the southern flank of the CME structure. The $M_{FM}$ values typically range from below 1 to beyond 7 across the triangulated 3-D front after 01:37:30UT with the highest values occurring in the vicinity of the neutral line. This is in general agreement with the PFSS/DEM technique presented in the previous section. \\
\indent Finally, comparing the middle and right-hand columns of Figure \ref{MAthetaBnMHD} shows that the highest $M_{FM}$ tend to occur for a quasi-parallel geometry. We note however a region of oblique to quasi-perpendicular shock and high-Mach number along the southern flank of the structure. This contrasts with PFSS that predicted a similar band of super-critical quasi-perpendicular shock across the nothern flank of the CME. \\
\section{Comparison between the emerging shock and radio measurements:}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.55]{Figure6}
\caption{Top: a radio spectrogram from the Culgoora radioheliograph showing the type II burst starting at 01:32UT. Bottom: the distribution of M$_{FM}$ over the propagating front at the three first times (01:32:30UT, 01:35UT, 01:37:30UT) shown in the right-hand column of Figures \ref{MAthetaBnPFSS} and \ref{MAthetaBnMHD}. The color table was defined on a smaller range of values of M$_{FM}$ between 0 and 4 for this early phase of the eruption. Magnetic field lines are shown in black. }
\label{typeIIMA}
\end{center}
\end{figure*}
As mentioned in the introduction and visible in the spectrogram of Figure \ref{typeIIMA}, type II bursts were measured during the event by the Culgoora radioheliograph starting at 01:32UT and drifting from 140 to 18 MHZ. Both the fundamental and the harmonic are visible on the spectrogram. For comparison, the estimated distribution of $M_{FM}$ on the shock surface are repeated from Figure \ref{MAthetaBnPFSS} and \ref{MAthetaBnMHD} for the two models used, but we changed the range of the color table from $M_{FM}=0$ to $M_{FM}=4$. For the technique used here, the earliest time at which Mach numbers were derived was 01:32:30UT so 30 seconds after the onset of the type II burst and both models confirm that a part of the pressure wave has become a shock, with a maximum of $M_{FM}\sim1.5$ for PFSS and some more localised increases of $M_{FM}\sim3$ for MHD. The shock is sub-critical for the PFSS technique but is already becoming super-critical in a limited area near the nose of the pressure wave in the MHD approach. As the event evolves the shock becomes rapidly super-critical over large fractions of the surface in both approaches. Multiple portions of the pressure wave are becoming super-critical, this is perhaps providing an explanation for the complex nature of the type II at 01:32UT observed in this spectrogram. We also investigated how far our technique could explain the sudden onset of the type II burst by using additional SDO AIA images with 30 second time cadence at intermediate times between 01:30:00UT and 01:32:30UT but without SECCHI data available at such high cadence, the results of this analysis were not sufficiently conclusive as they were too limited by the single viewpoint. In essence the details of the initial expansion rate were not sufficiently resolved at these intermediate times.
\section{Deriving shock parameters along specific field lines:}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.40]{Figure7}
\caption{Properties of the interplanetary medium measured near Earth over a 4 day-interval centered on the onset of the GLE event. The parameters shown are the normalised spectrogram of suprathermal electrons measured by ACE SWEPAM at 272 eV (a), the magnetic field magnitude (black) and components (b) measured by Wind MFI, the plasma speed measured by Wind (c), the temperature measured by Wind (d) and the flux of particles with energies exceeding 60 MeV measured by GOES (d).}
\label{INSITUSEP}
\end{center}
\end{figure*}
\indent To compare the properties of energetic particles measured near 1AU with the properties of shocks inferred in the corona, we need to consider the path followed by energetic particles to propagate from the Sun to 1AU. Since these escaping particles gyrate along interplanetary magnetic field lines, a model for the interplanetary magnetic field is usually assumed; the simplest and most common approach is to model these field lines as an Archimedian spiral. The locus of these spirals is controlled by two parameters, the speed of the solar wind carrying the field line of interest and the rotation rate of the Sun. The speed is usually defined by the average solar wind speed measured in situ at the onset time of the SEP event. Typically the spiral connects at outer boundary of the coronal model used (2.5 R$_\odot$ for the PFSS model and 30 R$_\odot$ for the MHD model) and the spacecraft making the in-situ measurement.\\
\indent The assumption of an Archimedian spiral to connect near-Earth data with the shock requires that the SEP event occured during quiet solar wind conditions, both in the near-Earth environment and in the region situated between the Sun and Earth. However solar wind measurements made in situ near Earth reveal that a magnetic cloud was passing at the time of the GLE onset (Figure \ref{INSITUSEP}). The ACE spacecraft measured several common signatures of magnetic cloud including counter-streaming electrons (Figure \ref{INSITUSEP}a), a smooth rotation of the magnetic field (Figure \ref{INSITUSEP}b,c,d) and a low temperature (Figure \ref{INSITUSEP}e). The period preceding that magnetic cloud passage may also be another Interplanetary CME passage since complex magnetic fields and atypical suprathermal electron signatures were also measured at the time. \\
\indent We investigated the origin of these transients and whether they also erupted in AR 11476 that later produced the 17 May 2012 and its GLE event. The aim being to determine if magnetic connectivity between the vicinity of AR 11476 and the ACE spacecraft were likely to be established by the internal field of the magnetic cloud measured during the SEP event. The heliospheric imagers onboard STEREO were imaging this region continuously days before the GLE event and allow CMEs and CIRs to be located in 3D (e.g. Rouillard et al. 2008; 2011a,b). We considered the CME and CIR catalogues made available by the Heliospheric Cataloguing, Analysis and Techniques Service (HELCATS) FP7 project. This project produced the first systematic catalogues of CMEs (Harrison et al. 2016) and CIRs (Plotnikov et al. 2016) observed by the heliospheric imagers onboard STEREO. A detailed analysis of the state of the interplanetary medium days preceding and during the GLE event is presented in Appendix B for clarity purposes, since this paper focuses mostly on the 3-D expansion of the shock and its effect on energetic particles. The conclusion of the analysis is that, during the SEP event, the near-Earth environment is magnetically connected to the region where the shock forms by a magnetic cloud that erupted five days earlier (on 12 May 2012) from the vicinity of the same AR 11476 (see Appendix A for more details). \\
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=.536]{Figure8}
\caption{A smoothed HMI magnetograms projected on the solar disk with the position of the shock triangulated in this study at 01:45UT. Two magnetic field lines are derived from the PFSS (left) and the MAST MHD model (right). For each image, the black line passes through the band of high-Mach number while the dashed line passes through a weaker part of the shock. }
\label{FLshock}
\end{center}
\end{figure}
\indent The passage of a magnetic cloud at GLE onset makes a tracing of the field line linking the Earth to the low corona impossible with our current limited understanding of the internal structure of CMEs. Instead we decide to illustrate the variability of shock properties along different open magnetic field lines, by extracting shock parameters along two different lines for each model. These four lines are traced in Figure \ref{FLshock}a with a smoothed HMI magnetogram shown on the surface of the Sun, the strong bipole (black/white region) is AR 11476. Magnetic field lines (A,B) and (C,D) are open to the interplanetary medium and are from the PFSS and MHD models respectively. For both models, the solid lines pass through the region of high-Mach number while the dashed lines pass through low-Mach number. In addition to these magnetic field lines, Figures \ref{FLshock}b and \ref{FLshock}c show the reconstructed $M_{FM}$ values from, respectively, the PFSS (Figure \ref{MAthetaBnPFSS}) and the MHD (Figure \ref{MAthetaBnMHD}) models. \\
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.52]{Figure9}
\caption{Shock plasma parameters extracted using the PFSS/DEM technique at the intersection between the triangulated shock and a magnetic field line passing through the band of high Mach number shown in Figure\ref{MAthetaBnPFSS} (filled squares) and a magnetic field line intersecting the shock far from that band (open squares). These parameters are plotted as a function of time (in UT). Panels (a) and (b): the ambient coronal density (N, cm$^{-3}$) and magnetic field (B, G) upstream of the shock. Superposed on these plots are derivations of ambient plasma properties from other studies as detailed in the text. Panels (c), (d), (e) and (f) show the shock speed (V$_s$, kms$^{-1}$), Mach number ($M_{FM}$), $\theta_{Bn}$, heliocentric distance (R$_{\odot}$) at the intersection between the shock and different magnetic field lines of the helmet streamer. Panel (g): the flux of 2.64-10.4 MeV electrons as a function of time with superposed the times of the flare onset, and Type II burst. The SPR times derived by Gopalswamy et al. (2013) (GEA SPR) and derived by the velocity dispersion analysis in Appendix A (VD SPR) are shown as vertical blue and red lines, respectively. The uncertainty in these estimates are shown as the corresponding horizontal segments. }
\label{PFSSFLC}
\end{center}
\end{figure*}
\indent The open and filled squares in Figure \ref{PFSSFLC} and \ref{PSIFLC} correspond to shock parameters extracted along the dashed and continuous field lines in Figure \ref{FLshock}. Figure \ref{PFSSFLC}a shows the background coronal density at the shock-field line intersection derived using the differential emission measure. For comparison, the green dots show the densities that are obtained when assuming the Leblanc et al. (1998) profile at the height of field line-shock intersection (Figure \ref{PFSSFLC}f). The Leblanc et al. (1998) profile was derived from the drift of type III bursts and assumes a density at 1AU of about 7 cm$^{-3}$, very close to the density measured near 1AU. Our two curves of reconstructed densities differ initially by an order of magnitude but they rapidly converge with the Leblanc et al.'s densities above 3Rs.\\
\indent Figure \ref{PFSSFLC}b shows the background coronal magnetic field at the shock-field line intersection using the PFSS model. For comparison, the purple and blue diamonds show the magnetic fields derived from the relation of Poomvises et al. (2012) at the height of field line-shock intersection (Figure \ref{PFSSFLC}f). The Poomvises et al. (2012) profiles were derived from the stand-off distance between CME core and the driven shocks. The reader is referred to their paper for more information. We find that the field line threading the shock far from the neutral line is very similar to the Poomvises et al. (2012) profile while the magnetic field strength of the line passing near the neutral line is systematically an order of magnitude lower. Figure \ref{PFSSFLC}c shows the variation of the shock speed along the two field lines, due to the rather spherical expansion of the shock, the two speeds do not differ much between the field line locations. $M_{FM}$ is strongly dependent on the magnetic field strength. The very different magnetic field strengths are therefore reflected in the $M_{FM}$ values that are much higher for the field line passing near the neutral line. \\
\indent The flux of relativistic electrons (2.64-10.4 MeV) shown in Figure \ref{PFSSFLC}g were obtained by the Electron Proton Helium Instrument (EPHIN) part of the \emph{Suprathermal and Energetic Particle Analyzer} (COSTEP) (M\"{u}ller-Mellin et al. 1995 onboard SOHO ({PFSSFLC}d). In addition, the onset times of the flare and type II burst and the estimated Solar Release Time (SPR) of the GeV protons are is shown in Figure \ref{PSIFLC}d). According to PFSS the shock is initially confined to closed-field regions but at roughly 01:37 the shock enters the open field regions. At the time, the geometry is quasi-perpendicular (Figure \ref{PFSSFLC}c), but $M_{FM}$ is still small. It is not until 01:37:30UT, about the time of the SPR time, that the Mach number increases dramatically along field line connected to the vicinity of the neutral line. Also shown in Figure \ref{PFSSFLC}g) are the SPR times derived by Gopalswamy et al. 01:41 (+00:02/-00:05) UT and in Appendix A, 01:37:20 (+00:02/-00:02) UT, using the velocity dispersion analysis. The SPR times occur during the transition to super-criticality particularly along the field line passing near the neutral line. In addition the analysis suggests that the shock was quasi-perpendicular at the estimated SPR.\\
\begin{figure*}
\begin{center}
\includegraphics[angle=0,scale=.52]{Figure10}
\caption{In exactly the same format as Figure \ref{PFSSFLC} but for the parameters extracted using the MAST MHD model from Predictive Sciences Inc. Just as in Figure \ref{PFSSFLC}, the filled squares correspond to a field line passing inside the band of high-Mach number shown in Figure\ref{MAthetaBnPFSS} while the open squares correspond to a field line passing the shock far from the band.}
\label{PSIFLC}
\end{center}
\end{figure*}
\indent Figure \ref{PSIFLC} is in an identical format to Figure \ref{PFSSFLC} but it displays the results of extracting shock parameters using the MHD simulation instead of the PFSS model. The field line passing in the vicinity of the neutral line shows higher density values than the Leblanc et al. (1998) profile, however the density values are lower for the field line far from the line. Like PFSS the magnetic field magnitude is generally lower for the field line passing near the neutral but the difference is less pronounced between the two field lines than for PFSS. As already seen in Figure \ref{FLshock}, the connectivity between the triangulated front and open field lines is established much earlier than for PFSS at 01:30UT or about 5 minutes after the flare onset. At that time, the geometry is quasi-perpendicular (Figure \ref{PSIFLC}e) but the front is sub-critical (Figure \ref{PSIFLC}d). At the formation time of the shock, $\theta_{Bn}$ is close to 45$^\circ$ which points to the occurrence of an oblique shock. The variation of $M_{FM}$ is more gradual, the shock becomes super-critical for both field lines considered near the SPR release time of GeV particles. The SPR times occur for this model when the shock becomes super-critical just after the quasi-perpendicular phase when the shock has reached an oblique geometry.
\section{Discussion:}
The images taken by the STEREO and near-Earth orbiting spacecraft are sufficient to map the 3-D extent of propagating fronts that form during the eruption of CMEs. According to our geometrical fitting technique, the shape of the propagating front remained highly spherical during the eruption process. This is also seen in the event analysed by Kwon et al. (2015). In our analysis, the EUV front is considered to be the low-coronal signatures of the expanding WL front (see Figure~\ref{TRIANGWL}). During the 17 May 2012 event, the speed of the EUV front never exceeded 500 km/s (Figure 5) and the fastest lateral motions are not measured in EUV images but higher up in the corona (Figure \ref{MAthetaBnPFSS}). \\
\indent The novelty of the present technique, also exploited in Salas-Matamoros et al. (2016), resides in the derivation of the normal speed and the Mach number ($M_{FM}$) over the entire surface of the CME front (Figure \ref{MAthetaBnPFSS} and \ref{MAthetaBnMHD} ). This was obtained from a combination of different techniques incuding the inversion of coronal imaging, magnetic field reconstructions and MHD modelling. The CME drives a shock and even a super-critical shock with $M_{FM}$ values in excess of 3 (Figure \ref{MAthetaBnPFSS} and \ref{MAthetaBnMHD} ) with the highest values occurring near the nose of the CME. This is in agreement with the results of Bemporad et al. (2014) who used remote-sensing observations from SOHO to derive the coronal and shock properties of the 11 June 1999 CME in the plane of the sky. Our analysis shows additional structure in the distribution of the $M_{FM}$ induced by the complex topology of the background field. At the earliest time available (01:32:30UT), portions of the triangulated front have already steepened into a shock, this is in agreement with the detection at the time of a type II burst (Figure \ref{typeIIMA}). \\
\indent It is interesting to compare this result with the analysis of the 2011 March 21 CME event (Rouillard et al. 2012). In that analysis, the expanding front was not fitted using the technique presented in this paper, however the shape of the CME, as seen projected in the plane of the sky, appeared highly elliptical with its major axis orthogonal to the direction of propagation (see Figure 2 of Rouillard et al. 2012). The very strong lateral expansion of the front observed in WL tracks also the EUV front. This lateral expansion eventually pushed streamers located far from the source region, at the same time that the EUV wave reached the footpoints of those streamers. That lateral expansion speed was on average about 400 km/s, but the pushed streamers were launched with a speed of 900 km/s. That paper demonstrated that the speed parallel to the solar surface of the EUV and WL fronts was at the same. However contrary to the EUV wave that moves only along the solar surface, the pushed streamers have an additional high radial speed component. For the 17 May 2012 event, analysis of the Mach numbers (not shown here) at the very low height imaged by EUV instruments remains predominantly with $M_{FM}<1$ throughout the event with small patches of 1$<M_{FM}<$2, therefore while certain parts of the EUV wave may have steepened into a shock, it remains sub-critical throughout the event. The analysis presented in the present paper suggests that a super-critical shock is unlikely to develop at the heights we observe EUV waves but that a shock can rapidly become supercritical at the heights imaged by WL coronagraphs (2R$_\odot$) near the tip of streamers.\\
The formation of a super-critical shock means that early during the eruption process, instabilities develop along the shock front that could play a role in the acceleration of high-energy particles. Preliminary simulations that model the process of diffusive-shock acceleration (Sandroos and Vainio, 2009) using the magneto-plasma properties of the shock derived in the present paper suggest acceleration to 300 MeV in 80 seconds on some of the open magnetic field lines. \\
\indent For both models, the rapid rise of $M_{FM}$ values occurs when the propagating front reaches open magnetic field that diverge strongly near 2-3Rs where helmet streamers typically form. The highest values of $M_{FM}$ are associated with the coronal neutral line, the source location of the heliospheric current sheet and its surrounding heliospheric plasma sheet. The latter, frequently measured near 1AU, is typically associated with magnetic fields that are an order of magnitude smaller than those measured in the ambient solar wind. This is clearly predicted by the PFSS reconstruction. However the PFSS model may over-estimate the size of this region. Both PFSS and the MHD approach reveal that the $M_{FM}$ values extracted along open field lines crossing the shock far from the neutral line remain overall below super-critical values ($<$3) until high up in the corona ($>$2R$_\odot$).\\
\indent Past studies have used SOHO observations to infer the heights of WL CMEs at the onsets of GLEs measured since 1997 (Gopalswamy et al. 2012). These inferred heights were obtained from a single viewpoint and therefore less accurate than the technique used in the present paper. Additionally, the limited coverage of the corona obtained from SOHO required interpolation techniques. Nevertheless these approaches provide estimates of the time delay between the onset of the flare and the onset of the GLE. These heights are listed in Table 1, columns 2 and 3 list the height estimates made by Gopalswamy et al. (2012) and Reames et al. (2009), respectively, using different approaches. They found the heights of particle releases above 2-3Rs showing a long delay between the onset of the flare and the injection time of high-energy particles. The results of the present paper provides evidence for the delayed release times of GeV protons to be related with the time needed for the shock to become super-critical.\\
\begin{deluxetable*}{ccrrrrrrrrcrl}
\tabletypesize{\scriptsize}
\tablecaption{Characteristics of WL CMEs and in situ measurements during GLE events}
\tablewidth{0pt}
\tablehead{
\colhead{GLE no } & \colhead{ Date} & \colhead{Time} & \colhead{CME Ht at SPR} & \colhead{CME Ht at SPR} &
\colhead{State} }
\startdata
55 &1997 Nov 06 & 110000 - 120000 & NG & 2.34 & Solar Wind, before
ICME & 8 & No\\
(1) &(2) & (3) & (4) & (5) & (6) \\
\hline
56 &1998 May 02 & 110000 - 120000 & 2.9 $\pm$ 0.2 & 1.97 & Inside ICME \\
57 &1998 May 06 & 070000 - 080000 & 2.0 $\pm$ 0.2 & 2.21 & Inside ICME \\
58 &1998 Aug 24 & 190000 - 200000 & 5.7 $\pm$ 0.5 & 5.14 & Solar Wind \\
59 & 2000 Jul 14 & 090000 - 100000 & 2.6 $\pm$ 0.3 & 1.74 & Inside ICME\\
60 & 2001 Apr 15 & 110000 - 120000 & 2.4 $\pm$ 0.2 & 2.10 & Inside ICME\\
61 & 2001 Apr 18 & 010000 - 020000 & 4.8 $\pm$ 0.7 & 3.92 & Inside ICME\\
62 & 2001 Nov 04 & 150000 - 160000 & NG & 8.05 & Between two ICMEs \\
63 & 2001 Dec 26 & 040000 - 050000 & 3.6 $\pm$ 0.5 & 2.88 & 6 hours after MC \\
64 & 2002 Aug 24 & 010000 - 020000 & 2.4 $\pm$ 0.5 & 2.96 & Trail of ICME \\
65 & 2003 Oct 28 & 100000 - 110000 & 4.3 $\pm$ 0.4 & 2.39 & Trail of MC \\
66 & 2003 Oct 29 & 190000 - 200000 & 5.7 $\pm$ 1.0 & 4.15 & Trail of MC \\
67 & 2003 Nov 02 & 160000 - 170000 & 3.3 $\pm$ 0.5 & 2.85 & Trail of ICME \\
68 & 2005 Jan 17 & 090000 - 100000 & NG & 2.72 & Trail of MC \\
69 & 2005 Jan 20 & 053000 - 063000 & 2.6 $\pm$ 0.3 & 2.31 & Trail of MC \\
70 & 2006 Dec 13 & 010000 - 020000 & 3.8 $\pm$ 0.6 & 3.07 & Solar Wind \\
71 & 2012 May 17 & 013800 - 033000 & 3.8 $\pm$ 0.6 & 3.07 & Inside MC \\
\enddata
\tablecomments{Columns 1: The official GLE number, 2 and 3: the date, the start and end times of the GLE, 4: CME height (in solar radii, R$_\odot$) at the solar particle release time inferred derived by Reames (2009) using a velocity dispersion analyses based on near-Earth particle measurements, 5: CME height at GLE onset obtained by quadratic extrapolation (in solar radii, R$_\odot$) by Gopalswamy et al. (2012), 6: the state of the solar wind at GLE onset. Acronyms used: NG for not given, MC: Magnetic Cloud, ICME: Interplanetary Coronal Mass Ejection. \label{table1} }
\end{deluxetable*}
\indent Both the PFSS and MHD approach show that the shock progresses from a quasi-perpendicular shock to a quasi-parallel super-critical shock. The PFSS model suggests that a quasi-perpendicular shock forms around the SPR time derived by the velocity dispersion analysis (VD SPR). This result could support the idea that a quasi-perpendicular shock combined with a seed population of energetic particles may be more effective to accelerate particles to very high energies than a quasi-parallel shock (Tylka and Lee 2005). Sandroos and Vainio (2009) showed that the magnetic geometry of the ambient corona can have an effect of about one order of magnitude on the maximum energies reached by the process of diffusive-shock acceleration, and that for some field geometries 1 GeV energies are attainable, provided that seed particles with sufficiently high energies (100 keV) are available. The MHD model suggests that the quasi-perpendicular shock has already occured and changed to an oblique super-critical shock by the VD SPR time. It would be instructive to run particle acceleration models to see if the weak quasi-perpendicular shock pre-accelerated some particles that were eventually accelerated to very high energies by the super-critical quasi-parallel shock.
\indent The very significant rise in $M_{FM}$ at the tip of the streamer, where we infer that the HPS must generally form, occurs near the release time of very energetic particles inferred from Earth-based neutron monitors for both techniques. In-situ measurements of the HPS show that the ambient magnetic field drops by an order of magnitude and the density can increase by a factor of 4-5 (Winterhalter et al.1994), hence the Alfven speed decreases dramatically to favour the formation of super-critical shocks. Remote-sensing observations suggest this plasma sheet already exists in the corona forming above the tip of helmet streamers (Bavassano et al. 1997; Wang et al. 2009). GLEs are associated with CMEs that emerge within a few latitudinal degrees of the nominal footprint of the Parker spiral connecting the point of in-situ measurements (e.g. Gopalswamy et al. 2012), the present study would argue that a good connectivity is necessary to the shock regions crossing the vicinity of the tip of streamers and the associated neutral line. The curved field lines that form the streamer could also favor multiple field-line crossings of the shocks and efficient particle acceleration (Sandroos and Vainio 2006, 2009).\\
\indent The HPS has a number of other interesting properties that make it a favorable location for strong particle energisation. Beside potentially boosting the $M_{FM}$, the high densities typically observed in white-light and measured in situ in the HPS would provide a localised increase in seed particles to be accelerated by the shock. The last decades of research has revealed how variable the HPS is, both spatially and temporally. The HPS contains signatures of small-scale transients that are released continually from the tip of streamers including small-scale magnetic flux ropes. The HPS is often entirely missed at 1AU when ICMEs, magnetic clouds and smaller transients 'replace' the standard HCS crossing, such as seen in Figure\ref{MAthetaBnPFSS}. These transients are formed inside or at the top of helmet streamers in regions where magnetic reconnection between oppositely directed field lines is very likely happening continually to form bundles of twisted magnetic fields or flux ropes. This continual transient activity leaves systematic signatures in the outflowing solar wind along the heliospheric neutral line (e.g. Rouillard et al. 2010, Plotnikov et al. 2016). The formation of these complex field topologies involves the closed magnetic field lines situated inside helmet streamers that are in more direct proximity to the flaring active region than open field lines from coronal holes. We note that the tip of streamers may therefore provide an escape route for heavy ions and suprathermal particles that were previously confined to closed magnetic loops as either pre-energised particles by the quasi-perpendicular shock or by the concomittant flaring activity. Since the energetic particles move faster than the accelerator, they rapidly populate and scatter upstream of the forming shock in the open field region. These latter particles could also be an important population of seed particles for a prompt energisation by the mechanism of diffusive-shock acceleration (Tylka and Lee 2005). Even the MHD model used in this paper is not able to model such disruptions realistically, and therefore our derivation of the geometry of the shock, uncertain in these models, will be even more uncertain inside the HPS.
\indent Masson et al. (2012) analysed the near-Earth properties of the solar wind during 10 out of the 16 GLEs detected by neutron monitors since 1997. They showed that 7 of the GLE onsets occurred during disturbed solar wind conditions measured by ACE and Wind at 1AU including 2 very clear ICME passages. This frequent association is related to the finding made by Belov et al. (2009) that the accelerative and modulative efficiencies of solar storms are tightly correlated; CMEs followed by GLEs are associated with a high probability of a very large Forbusch decrease measured at Earth. We revisited the analysis by Masson et al. (2012) by (1) analysing the near-Earth solar wind conditions of the other official GLE events not listed in their paper, (2) considering in addition the suprathermal electrons measurements obtained by the ACE and Wind spacecraft, (3) the ICME list of Richardson and Cane (2009). The results are shown in column 6. As revealed by figure \ref{MAthetaBnPFSS}, the GLE event analysed in this paper occurred inside a clear magnetic cloud. Out of the 16 GLE events, we could confirm that the near-Earth environment was in the 'background' solar wind for two events only (GLEs 58 and 70). \\
Using the full set of SECCHI observations, we demonstrated in Appendix B that that the magnetic cloud measured in situ at the time of the GLE originated in a CME that erupted on 12 May 2012 from AR 11476. This agreement between the source longtiude of both the active region that produced the magnetic cloud measured in situ and the CME/GLE event of 17 May 2012 strengthens our argument that the near-Earth environment was magnetically connected, through a flux rope, to the coronal region that produced the shock, perhaps rooted in the direct vicinity of the active region. It remains to be demonstrated whether every ICME that occured during GLEs since 1997 erupted from the same region that produced the flare/CME responsible for the GLE event. For our event, the MHD simulation finds open field lines rooted near AR 11476, future simulations could investigate whether these open field lines were opened by the preceding CME activity. \\
\indent In light of the previous results, a limitation of our study resides in the rather static treatment of the background coronal magnetic field. The eruption of CMEs from the same active region days prior to the event of insterest in this paper may have induced time-dependent effects that are missed out by the PFSS and MHD model used in present study. Previous observational studies combined with numerical models of the coronal field have investigated the topological changes induced by CME eruption. They demonstrate that CMEs (1) open closed field lines that previously formed the streamer's helmet base (e.g. Fainshtein et al. 1998), (2) generate additional white-light rays in the trailing part of CMEs that appear for several hours (Kahler and Hundhausen 1992; Webb et al. 2003), (3) produce transient coronal holes in less than 1 hour that disappear in 1–2 days (e.g. de Toma et al. 2005). The event of 2012 May 12 occurred five days before the event studied here and these transient structures had faded by the 2012 May 17. The white-light rays in particular were clearly visible in the plane of the sky from STA and had largely faded away from the camera that same day. The MHD model suggested that some of the field lines connected to the plasma sheet were rooted in the AR11476 and could be remnants of this preceding CME activity. \\
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=.37]{Figure11}
\caption{The top two schematics are views onto helmet streamers with magnetic field lines shown in blue. The location where oppositely directed field lines meet is the heliospheric plasma and current sheet. The fall off of the coronal fast-mode speed with distance is shown as a fading black color illustrating the abrupt drop usually seen inside the plasma sheet. In green, we show the relative locations of the flare and the pressure wave that forms around the CME acting as a piston. Two scenarii are sketched during 'quiet' (top image) and more 'disrupted' (middle image) coronal conditions during the reformation of the plasma sheet over the several days that follow the eruption of a CME. Bottom panel: a schematic of the interplanetary conditions during the GLE event. A magnetic cloud with closed field topology is connecting the coronal shock with the near-Earth environment, channeling particles to 1AU. }
\label{HPSFR}
\end{center}
\end{figure}
\indent The presence of a magnetic cloud linking the shock to the near-Earth environment could change the magnetic connectivity of the near-Earth environment with the coronal plasma sheet. As stated above the helmet streamers retrieve an equilibrium configuration at most a few tens of hours after the eruption of a CME, during this process it is unclear where the magnetic footpoints of CMEs end up connected to in the low corona but presumably they will be part of the new equilibrium configuration found by the streamer and its reformed plasma sheet. The top two schematics of Figure \ref{HPSFR} present an illustration of the relation between the emerging shock and the plasma sheet with no CME activity prior to the event and for a disturbed plasma sheet reaching a new equilibrium but threaded by magnetic field remnants of prior CME activity rooted near the active region. \\
\indent The nearly systematic association between the occurrence of GLEs and the passage near Earth of ICMEs will also change the likelihood of being connected with the accelerator near the Sun. We note that contrary to a single Parker spiral, a magnetic flux rope occupies a very large volume of the interplanetary medium; this will increase the probability that the near-Earth environment (or any point inside the flux rope) becomes magnetically connected with the coronal region producing very high-energy particles. If the neutral line is indeed a favorable but spatially limited region of particle acceleration, the presence of a large-scale magnetic flux rope (or another complex magnetic field structure) will increase the chances of being magnetically connected with that narrow region. This is illustrated in the bottom schematic of Figure \ref{HPSFR}. The magnetic cloud passing over the Earth on 2012 May 17 transports counter-streaming suprathermal electrons just before the onset of the GLE event. In addition, the magnetic connectivity to the solar corona was therefore occurring at both ends of the flux rope and SEPs. Strong beams of counter-streaming electrons are often measured after the onset of GLE events but these are likely associated with back-propagating electrons due to the ICME acting as a magnetic mirror and an associated higher flux of electron due to the GLE and a higher level of scattering due to the ICME magnetic fields. \\
\section{Conclusion:}
Assuming that the particle accelerator is situated near the shock-sheath system, the delay typically seen between the flare and solar particle release times should depend initially on the time necessary:
\begin{itemize}
\item for the pressure wave to steepen into a shock: the formation processes of the shock will depend on the 3-D expansion speed of the driver gas and the spatial variations of the characteristic speed of the ambient medium in which it is propagating,
\item for the shock to propagate longitudinally: this is particularly true during the progression through predominantly radial magnetic fields since cross-field diffusion is much weaker than field-aligned diffusion.
\end{itemize}
The use of different magnetic models points to the considerable uncertainties that are faced when attempting to derive the topology of the background magnetic field through which coronal shocks propagate. However our approach has revealed that, regardless of the model used, a shock has formed at the time of the onset of the type II burst and a super-critical shock has formed at the release time of high-energy particles.
An alternative hypothesis not investigated here is of course that particles are accelerated in the solar flare. The delayed GLE onset would then be interpreted as the time required for the closed loops that drive the expansion of the piston and also channel the flare particles, to reconnect with the open magnetic field lines that are connected with the spacecraft measuring the GLE. This mechanism was investigated numerically by Masson et al. (2013). For the event analysed here, the reconnection process would occur between the erupting piston and the magnetic field lines of the magnetic cloud measured in situ. A delayed onset could only occur here if the magnetic field lines of the magnetic cloud are initially topologically distinct to the flaring loops or the erupting piston.\\
\indent We are currently repeating the analysis presented in the present paper on other events measured by near 1AU orbiting spacecraft with the hope to decipher the nature of the particle accelerator. Clearly the presence of additional spacecraft situated closer to the Sun (Solar Probe+) and outside of the ecliptic plane (Solar Orbiter) should provide (1) radically better timing of particle onsets than inferred by in-situ measurements made near 1AU and (2) unprecedented views from outside the ecliptic plane to disentangle more easily the different delays in particle onsets.\\
\acknowledgments
We acknowledge usage of the tools made available by the plasma physics data center (Centre de Données de la Physique des Plasmas; CDPP; http://cdpp.eu/), the Virtual Solar Observatory (VSO; $http://sdac.virtualsolar.org$), the Multi Experiment Data $\&$ Operation Center (MEDOC; $https://idoc.ias.u-psud.fr/MEDOC$), the French space agency (Centre National des Etudes Spatiales; CNES; https://cnes.fr/fr) and the space weather team in Toulouse (Solar-Terrestrial Observations and Modelling Service; STORMS; https://stormsweb.irap.omp.eu/). This includes the data mining tools AMDA (http://amda.cdpp.eu/) and CLWEB (clweb.cesr.fr/) and the propagation tool (http://propagationtool.cdpp.eu). R.F.P. and I.P. acknowledge financial support from the HELCATS project under the FP7 EU contract number 606692. R.V. acknowledges the financial support from the HESPERIA project under the EU/H2020 contract number 637324. A. W. acknowledges the support by DLR under grant No. 50 QL 0001. A.P.R. acknowledges funding from CNES and the Leibniz Institute f\"{u}r Astrophysik Potsdam (AIP) to visit AIP and to collaborate with A.W. and G.M. on the present project. The \emph{STEREO} \emph{SECCHI} data are produced by a consortium of \emph{RAL} (UK), \emph{NRL} (USA), \emph{LMSAL} (USA), \emph{GSFC} (USA), \emph{MPS} (Germany), \emph{CSL} (Belgium), \emph{IOTA} (France) and \emph{IAS} (France). The \emph{ACE} data were obtained from the \emph{ACE} science center. The WIND data were obtained from the \emph{Space Physics Data Facility}.
|
2,877,628,090,200 | arxiv |
\section{Introduction}
Multivariate regression analysis is an important tool for empirical research in strategic management and economics. These methods account for confounding influence factors between a treatment and an outcome by including a set of control variables in order to obtain unbiased causal effect estimates. Notwithstanding their importance for causal inference, in practice scholars often overstate the role of control variables in regressions. In this note we argue that, while essential for the identification of treatment effects, control variables generally have no structural interpretation themselves. This is because even valid controls are often correlated with other unobserved factors, which renders their marginal effects uninterpretable from a causal inference perspective \citep{Westreich2013, Keele2020}. Consequently, researchers need to be careful with attaching too much meaning to control variables and should consider to ignore them entirely when interpreting the results of their analysis.
Drawing substantive conclusions from control variables is common however in applied research. Authors frequently make use of formulations such as: \emph{``control variables have expected signs"} or \emph{``it is worth noting the results of our control variables"}. Based on the volume of papers published in the last five years in \emph{Strategic Management Journal}, we found that 47 percent of papers that made use of parametric regression models also explicitly discussed the estimated effects of controls.\footnote{
We analyzed all research articles published in Strategic Management Journal between January 2015 and May 2020 and found that, out of a total number of 458 papers which included parametric regression models, 213 proceeded to explicitly interpret and draw substantive conclusions based on the marginal effects of control variables.}
Moreover, in our own experience as authors of empirical research papers, we encountered instances in which reviewers specifically asked us to provide an economic interpretation of control variable coefficients. The argument was that, although they were not the main focus of the analysis, the controls could still provide valuable information to other researchers in the field who are investigating related research questions. In the following, we will explain why this approach is potentially misleading and should therefore better be avoided.
\section{The structural interpretation of control variables}
\begin{figure}[t]
\centering
\subcaptionbox{\label{fig_large_graph}}[.48\linewidth]{
\begin{tikzpicture}[>=triangle 45]
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={below:{$X$}}] (X) at (0,0) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={below:{$Y$}}] (Y) at (5,0) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={left:{$Z_1$}}] (Z1) at (0.8,1.7) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={right:{$Z_2$}}] (Z2) at (4.2,1.7) {};
\draw[->,shorten >= 1pt] (X)--(Y);
\draw[->,shorten >= 1pt] (Z1)--(X);
\draw[->,shorten >= 1pt] (Z2)--(Y);
\draw[<->,dashed,shorten >= 1pt] (Z1) to[bend left=45] (Z2);
\end{tikzpicture}
}
\subcaptionbox{\label{frontdoor}}[.5\linewidth]{
\begin{tikzpicture}[>=triangle 45]
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={below:{$X$}}] (X) at (0,0) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={below:{$Y$}}] (Y) at (6,0) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={left:{$Z_1$}}] (Z1) at (0.8,1.7) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={[yshift=0.1cm]above:{$Z_2$}}] (Z2) at (3,3) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={left:{$Z_3$}}] (Z3) at (3,1) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={left:{$Z_4$}}] (Z4) at (4.5,1.5) {};
\node[fill,circle,inner sep=0pt,minimum size=5pt,label={right:{$Z_5$}}] (Z5) at (5.7,1.9) {};
\draw[->,shorten >= 1pt] (X)--(Y);
\draw[->,shorten >= 1pt] (Z1)--(X);
\draw[->,shorten >= 1pt] (Z1)--(Z2);
\draw[->,shorten >= 1pt] (Z2)--(Z3);
\draw[->,shorten >= 1pt] (Z2)--(Z4);
\draw[->,shorten >= 1pt] (Z2)--(Z5);
\draw[->,shorten >= 1pt] (Z3)--(Y);
\draw[->,shorten >= 1pt] (Z4)--(Y);
\draw[->,shorten >= 1pt] (Z5)--(Y);
\draw[<->,dashed,shorten >= 1pt] (Z1) to[bend left=60] (Z2);
\draw[<->,dashed,shorten >= 1pt] (Z2) to[bend left=60] (Z5);
\end{tikzpicture}
}
\caption{}
\label{fig1}
\end{figure}
The relationships between the main explanatory variables and the controls in a regression can be complex, therefore it is useful to explicitly depict them in a causal diagram \citep{Pearl2000,Huenermund2019b}. \citet{Durand2009} were the first to introduce causal graphs in the strategic management literature, by arguing their usefulness as a tool for empirical research. Figure \ref{fig1}a presents a simple economic model with a treatment variable $X$ and an outcome variable $Y$. Both variables are connected by an arrow, denoting the direction of causal influence factor between them. In addition, there are two confounding factors, $Z_1$ and $Z_2$, affecting the treatment and the outcome. $Z_1$ and $Z_2$ are correlated, as a result of a common influence they share, which is denoted by the dashed bidirected arc in the graph. The fact that $Z_1$ and $Z_2$ are correlated creates what is known as a backdoor path between the treatment and the outcome \citep{Pearl2000}. $X$ and $Y$ are not only connected by the genuinely causal path $X \rightarrow Y$, but also by a second path, $X \leftarrow Z_1 \dashleftarrow\dashrightarrow Z_2 \rightarrow Y$, which creates a spurious, non-causal correlation between them.
The role of control variables in regression analysis is exactly to block such backdoor paths, in order to get at the uncontaminated effect of $X$ on $Y$. For this purpose, it is sufficient to control for any variable that lies on the open path.\footnote{
Technical note: Requiring the path to be previously unblocked rules out that the variable which is adjusted for is a collider \citep{Huenermund2019b}. A discussion of collider variables in causal graphs goes beyond the scope of this note.}
Thus, in the example of Figure \ref{fig1}a, the researcher has the choice between either controlling for $Z_1$ or $Z_2$, since both would allow to identify the causal effect of interest. The choice between different admissible sets of control variables is thereby of high practical relevance. Researchers often have a fairly detailed knowledge about the treatment assignment mechanism $Z_1 \rightarrow X$, for example, because there are specific organizational or administrative rules that can be exploited for identification purposes \citep{Angrist1990,Flammer2017,Huenermund2019}. At the same time, the set of variables $Z_2$ that are direct influence factors of $Y$ will likely be quite large. Thus, in practical applications it might be much easier to control the treatment assignment mechanism than trying to include all the variables that have an effect on the outcome measure in a regression.
Nevertheless, although controlling for $Z_1$ is sufficient to obtain an unbiased estimate for $X$, its marginal effect will itself not correspond to any causal effect of $Z_1$ on $Y$. This is because $Z_1$ is correlated with $Z_2$ and will thus partially pick up an effect of $Z_2$ on $Y$ too \citep{Cinelli2020}.\footnote{
To illustrate this phenomenon quantitatively, we parametrize the causal graph in Figure \ref{fig1}a in the following way:
\begin{align*}
z_1 &\leftarrow u + \varepsilon_1, \\
z_2 &\leftarrow u + \varepsilon_2, \\
x &\leftarrow z_1 + \varepsilon_3, \\
y &\leftarrow x + z_2 + \varepsilon_4,
\end{align*}
with $N=1000$, and $u$ and $\varepsilon_m$ being standard normal. We then run a regression of $Y$ on $X$ and $Z_1$, which gives us a consistent coefficient estimate for $X$ (= 0.970, std.\ err.\ = 0.052; bootstrapped with 1000 replications), while the effect of $Z_1$ (= 0.541, std.\ err.\ = 0.064) turns out to be biased. By contrast, if we also include $Z_2$ in the regression, the coefficient of $Z_1$ drops to zero (= -0.016, std.\ err.\ = 0.042), which corresponds to its true causal effect on $Y$ in this example.
}
The danger of interpreting estimated effects of potentially endogenous control variables, such as $Z_1$, is referred to as \emph{table 2 fallacy} in epidemiology \citep{Westreich2013}.\footnote{
Epidemiologists usually present the results of multivariate regression analyses right after a table with descriptive statistics of the data, therefore the name \emph{table 2 fallacy}.}
A similar point has been made recently by \citet{Keele2020} in the field of political science.
Figure \ref{fig1}b depicts a more complex example, with several admissible sets of controls, each sufficient to identify the causal effect of $X$ on $Y$ \citep{Textor2011}. One possibility in this situation would be to simply control for $Z_1$, which is the only direct influence factor of $X$, and thus blocks all paths entering $X$ through the backdoor. Similarly, controlling for the direct influence factors of $Y$ ($Z_3$, $Z_4$, and $Z_5$) would also block all backdoor paths. A third alternative is to control for the entire set of confounders ($Z_1$, $Z_2$, $Z_3$, $Z_4$, and $Z_5$), although this would be the most data-intensive identification strategy and lead to less precise estimates, due to lower degrees of freedom. This example illustrates that the minimally sufficient set of controls (here: $Z_1$) for identifying the causal effect of $X$ is often much smaller than the total number of confounding variables in a model. At the same time, the estimated marginal effects for the control variables only have a structural interpretation themselves if all the direct influence factors of $Y$ (here: $Z_3$, $Z_4$, and $Z_5$) are accounted for in the regression. As we argued above, this is unlikely to be the case though, since in many real-world settings the number of causal factors determining $Y$ can be prohibitively large.
\section{Implications for research practice}
Attaching economic meaning to the marginal effects of biased control variables is problematic, as researchers could develop false intuitions or come to erroneous policy conclusions based on them. Therefore it is advisable to not discuss the results obtained for control variables in empirical papers, unless the researchers can be sure that they have accounted for all relevant influence factors of the outcome in a regression (\emph{all-causes regression}). Since in many practical settings this is unlikely, however, we recommend to treat controls as \emph{nuisance parameters}, which are included in the analysis for identification purposes but are not reported themselves in the output tables \citep{Liang1995}. Our suggestion thereby corresponds to the way control variables are treated by non-parametric matching estimators \citep{Heckman1998} and modern machine learning techniques for high-dimensional settings \citep{Belloni2014}. These methods similarly do not report estimation results related to controls, either because there are simply too many covariates in the analysis, which is the primary use-case for machine learning, or marginal effects of control variables are not even returned by the estimation protocol, as in the matching case.
In short, there is no reason to be worried if the estimated coefficients of control variables do not have expected signs, since they are likely to be biased anyways in practical applications. Instead, researchers should rather focus on interpreting the marginal effects of the main variables of interest in their manuscripts. The estimation results obtained for controls, by contrast, have little substantive meaning and can therefore safely be omitted---or relegated to an appendix. This approach will not only prevent researchers from drawing wrong causal conclusions based on endogenous controls, but will also allow to streamline the discussion sections of empirical research papers and save on valuable journal space.
\newpage\singlespacing
\bibliographystyle{smj}
|
2,877,628,090,201 | arxiv | \section{Introduction}
\label{sec:introduction}
\textit{Multi-agent reinforcement learning} (MARL) is a machine learning paradigm which enables multiple agents to concurrently learn behaviour from interactions with the environment as well as interactions with each other. MARL methods have become increasingly capable of learning complex behaviour~\citep{berner2019dota,vinyals2019grandmaster}, but their learned behaviours are usually highly task-specific. This can be desirable to maximise effectiveness in specific tasks, but limits the applicability in the real-world, which often requires the learned behaviour to be robust to small perturbations and changes in the environment~\citep{dulac2021challenges,akkaya2019solving}.
Our work addresses the challenge of \textbf{teamwork adaptation} in which a team of agents is trained in a set of training tasks, and then has to adapt to novel, previously unseen testing tasks.
As an example, consider a warehouse environment in which a team of agents must navigate in the warehouse to collect and deliver shelves with requested items (\Cref{fig:rware_generalisation}). Tasks can vary in their warehouse layout, and agents need to adjust their team strategy depending on the layout of the warehouse to optimise deliveries. Whereas the warehouses with shelves on one side (\Cref{fig:rware_generalisation}, left) require all agents to move to the same side of the warehouse, the warehouse with shelves on both ends (\Cref{fig:rware_generalisation}, right) requires agents to effectively split between both ends to minimise waiting and travel time.
\begin{figure}
\centering
\includegraphics[width=.8\textwidth]{media/generalisation/rware_generalisation_wide_label.pdf}
\caption{Teamwork adaptation in multi-robot warehouse environment: In the training tasks with different layouts, both agents (orange) need to deliver shelves (blue) with requested items (green) to the delivery zone (black). After training, the agents are fine-tuned and evaluated in the unseen testing task which requires novel coordination to effectively distribute the workload across both agents.}
\label{fig:rware_generalisation}
\end{figure}
The above example illustrates that a team of agents may need to adapt their individual behaviour and coordination to novel tasks in a non-trivial way. However, such adaptation requires agents to identify the task they are in. Current MARL approaches are not designed for such adaptation, and in our experiments we show their limitations at adapting to novel tasks. %
Motivated by the challenge of teamwork adaptation, we propose to equip agents with the ability to infer the task they are in by learning \textbf{multi-agent task embeddings} (MATE)\footnote{Open-source implementation available at \url{https://github.com/uoe-agents/MATE}}. Through interacting with the environment, each agent builds up a task embedding using an encoder conditioned on the trajectories of agents in the task.
Motivated by the observation that each task is uniquely identifiable by its transition and reward functions, the encoder is jointly trained with a separate decoder which conditioned on task embeddings is optimised to reconstruct transitions and rewards. %
Learned embeddings allow a team of agents to adapt to previously unseen but related tasks by conditioning their policies on task embeddings. We hypothesise that access to task embeddings simplifies teamwork adaptation. %
We consider and compare three paradigms of learning such embeddings: independent MATE, centralised MATE, and mixed MATE. For independent MATE, each agent independently encodes local information to obtain an embedding of the task, whereas centralised MATE trains a single encoder conditioned on the joint information for all agents. Mixed MATE trains independent encoders for agents but optimises them using a mixture of task embeddings with weights for each task embedding being conditioned on the state of the environment.
We empirically evaluate all MATE variants in four multi-agent environments which require increasing degrees of adaptation. We find that MATE improves teamwork adaptation in several cases compared to fine-tuning MARL agents without task embeddings or training agents in testing tasks with no prior training. In particular, in tasks where all agents need to coordinate, MATE provides useful information for agent adaptation, with mixed MATE performing the best among all three paradigms across a wide range of adaptation scenarios. We further demonstrate that learned task embeddings produce clusters which clearly identify tasks, and the learned mixture of mixed MATE focuses on task embeddings of agents which currently observe the most information about the task.
\section{Related Work}
\label{sec:related_work}
\textbf{Meta Reinforcement Learning:} Meta reinforcement learning (Meta RL) aims to leverage concepts from meta learning~\citep{hospedales2020meta} to learn policies which can adapt to novel testing tasks using limited interactions. Such adaptation can be achieved by computing meta gradients to find a model initialisation from which an effective policy in testing tasks can be obtained using few optimisation steps~\citep{finn2017model}. %
Another common approach for meta RL implicitly builds up a latent context using recurrent neural networks which enable adaptation to new tasks~\citep{duan2016rl,wang2016learning,fakoor2020meta}. %
However, these approaches rely on the recurrent context to learn useful information for adaptation without any explicit optimisation enforcing such usefulness. \citet{rakelly2019efficient} disentangle the objectives of task inference and learning control policies and leverage variational inference techniques to learn a context inference model. \citet{zintgraf2020varibad} also propose to learn a variational inference model to explicitly learn task beliefs and incorporate task uncertainty into the policies for adaptation. Their approach is similar in architecture and optimisation to MATE but is limited to the adaptation of policies of individual agents. In contrast, we address the challenge of adapting the policies of multiple agents cooperating in a team.
\textbf{Transfer Learning:} Established techniques from transfer learning~\citep{taylor2009transfer} address a similar challenge to meta RL by extracting representations, action selection or other components from already learned models to improve the capabilities of agents in novel testing tasks. Unlike meta RL which typically adapts to novel tasks using fine-tuning over few episodes, transfer learning requires a dedicated transferring procedure for each new task. \citet{da2019survey} provide an overview specifically for methods aiming to transfer policies of multiple agents. They distinguish between inter-agent transfer, aiming to leverage information from one agent with possibly more expertise to transfer another agent~\citep{da2017simultaneously,da2018autonomously}, and intra-agent transfer which aims to transfer knowledge across tasks for all involved agents. Related to transfer learning are approaches focusing on multi-task learning in which agents are trained to generalise over a set of multiple tasks~\citep{vithayathilvargheseSurveyMultiTaskDeep2020,omidshafiei2017deep,caruana1997multitask}.%
\looseness=-1
\textbf{Multi-Agent Reinforcement Learning:} There has been significant progress in MARL to solve challenging coordination problems~\citep{papoudakis2021benchmarking} with many approaches focusing on the paradigm of centralised training and decentralised execution (CTDE)~\citep{christianos2020shared,rashid2018qmix,foerster2018counterfactual,sunehag2017value,lowe2017multi}. Under this paradigm, information is shared across agents at training time without conditioning agent policies on such joint information. In this way, training can leverage privileged information without preventing agent policies from being deployed in a decentralised manner.
We train MATE using the reconstruction of transition and reward functions of tasks defined over all agents, so MATE also falls under the paradigm of CTDE. In multi-agent systems, it is appealing to explicitly learn models of the behaviour of other agents in the environment to infer their possible behaviour~\citep{papoudakis2021agent,zintgraf2021deep,albrecht2018autonomous}. Such approaches demonstrate the ability for a single agent within a multi-agent system to adapt to different agents to cooperate or compete with. Similarly, the challenges of ad hoc teamwork~\citep{stone2010ad} and zero-shot coordination~\citep{hu2020other} address the problem of training agents to be able to coordinate with new partners without prior training in a team or established team strategies. Approaches in these settings prominently model other agents~\citep{rahman2021towards,barrett2015cooperating} or learn best response strategies to a diverse population of agents maintained during training~\citep{lupu2021trajectory}. However, few work exist which address the challenge of adapting the policies of multiple agents to novel tasks. \citet{hu2021updet} propose a novel architecture leveraging transformer models and self-attention~\citep{vaswani2017attention} to be able to reuse policies for varying team sizes, and \citet{zhang2021learning} train a latent model to encode coordination information to generalise to varying team sizes. \citet{vezhnevetsOptionsResponsesGrounding2020} train agents using a hierarchical approach with a high-level policy choosing the low-level behaviour which should be deployed given currently available information about the task and other agents in the environment. \citet{mahajan2022generalization} formalise the problem of combinatorial generalisation in which a team of agents needs to generalise over varying capabilities of agents. In contrast to all these approaches, we focus on the problem of adapting a fixed team of agents to varying tasks.
\section{Problem Definition}
\label{sec:problem_setting}
\paragraph{Partially Observable Stochastic Games} We consider multi-agent tasks modelled as partially observable stochastic games (POSGs) for $N$ agents~\citep{hansen2004dynamic}. %
A POSG is given by the tuple $(\mathcal{I},\mathcal{S}, \{\mathcal{O}^i\}_{i\in \mathcal{I}}, \{A^i\}_{i\in \mathcal{I}}, \Omega, \mathcal{P}, \{R^i\}_{i\in \mathcal{I}})$.
Agents are indexed by $i\in\mathcal{I} = \{1,\ldots,N\}$. $\mathcal{S}$ denotes the state space of the environment and $\mathbf{\mathcal{A}} = A^1\times\ldots\times A^N$ denotes its joint action space. $\mathcal{P}: \mathcal{S} \times \mathbf{\mathcal{A}} \times \mathcal{S} \mapsto [0, 1]$ denotes the transition function of the environment, defining a distribution of successor states given the current state and the applied joint action. At each timestep $t$, each agent $i$ receives an observation $o_t^i \in \mathcal{O}^i$ defined by the observation function $\Omega: \mathcal{S} \times \mathbf{\mathcal{A}} \mapsto \Delta(\mathbf{\mathcal{O}})$ conditioned on the current state and applied joint action for joint observation space $\mathbf{\mathcal{O}} = \mathcal{O}^1\times\ldots\times \mathcal{O}^N$. Each agent learns a policy $\pi_i(a^i_t | o^i_{1:t})$ conditioned on its history of observations, denoted by $o^i_{1:t} = (o^i_1, \ldots, o^i_t)$. After timestep $t$, agent $i$ receives a reward $r^i_t$ given by its reward function $R^i: \mathcal{S} \times \mathcal{A} \mapsto \mathbb{R}$. The objective is to learn a joint policy $\boldsymbol{\pi} = (\pi_1, \ldots, \pi_N)$ such that the discounted returns of each agent $G^i = \sum_{t=1}^\infty \gamma^{t-1} r^i_t$ are maximised with respect to the policies of all other agents, formally $\forall_{i}: \pi_{i} \in \arg \max _{\pi_{i}^{\prime}} \mathbb{E}\left[R^i | \pi_{i}^{\prime}, \pi_{-i}\right]$, with discount factor $\gamma \in [0; 1)$ and $\pi_{-i} = \boldsymbol{\pi} \setminus \{\pi_i\}$.
\paragraph{Teamwork Adaptation}
\looseness=-1
We consider the challenge of transferring the joint policies $\boldsymbol{\pi}$ of a fixed team of agents from a set of training tasks $\mathcal{T}_{\text{train}}$ to novel testing tasks $\mathcal{T}_{\text{test}}$ with $\mathcal{T}_{\text{train}} \cap \mathcal{T}_{\text{test}} = \emptyset$. Each task is represented as a POSG. %
Without any further assumptions, training and testing tasks could be arbitrarily different and hence no generalisation could be feasibly expected. %
We assume that training and testing tasks have identical number of agents $N$, action space $\mathcal{A}$ and identical dimensionality of observations.
In our problem setting, agents are first trained in $\mathcal{T}_{\text{train}}$ for $N_\text{train}$ timesteps and can be trained in $\mathcal{T}_{\text{test}}$ for a limited number of timesteps $N_{\text{test}}$ to fine-tune the policy. We refer to this setting as teamwork adaptation which is similar to the problem of domain adaptation for RL~\citep{eysenbach2020off}.
\section{Learning Multi-Agent Task Embeddings}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{media/MATE/All-MATE.pdf}
\caption{MATE recurrently encodes (blue) trajectories of states, actions and rewards into variational task embeddings. We consider three paradigms of MATE: independent MATE, centralised MATE and mixed MATE. Policies (orange) are conditioned on task embeddings and the decoder (green) receives sampled (purple) task embeddings, current observations and actions to predict the transition and reward function of all agents.}
\label{fig:mate}
\end{figure}
For agents to be able to adapt their behaviour to the current task, they need to infer the task they are in as they interact with their environment. Therefore, we propose to learn multi-agent task embeddings (MATE) $\mathcal{N}(\mathbf{z})$ from agent trajectories, consisting of observations, actions and rewards, which provide agent policies $\pi_i(a^i_t | o^i_{1:t}, \mathcal{N}(\mathbf{z}))$ with information about the task as well as a measure of uncertainty about the task. We hypothesise that explicitly modelling and learning task embeddings is advantageous for teamwork adaptation. %
In order to learn task embeddings, we train a variational encoder-decoder architecture~\citep{kingma2013auto}. Agents use an encoder $q$ to compute a variational distribution $q(\mathbf{z} | \boldsymbol{\tau}_{1:t}) = \mathcal{N}(\mathbf{z}; \boldsymbol{\mu}, \text{diag}(\boldsymbol{\sigma}))$ representing the task embeddings. These task embeddings are given by a diagonal Gaussian distribution with mean $\boldsymbol{\mu} \in \mathbb{R}^d$ and variance $\boldsymbol{\sigma} \in \mathbb{R}_+^d$. We concisely denote this distribution over task embeddings as $\mathcal{N}(\mathbf{z})$. The encoder is represented as a recurrent neural network which outputs $\boldsymbol{\mu}$ and $\log\left(\text{diag}(\boldsymbol{\sigma})\right)$. At each timestep, information about the trajectory consisting of observations, actions and rewards is fed into the encoder together with the previous hidden state to update the task embeddings. The hidden state is reset after each episode to enable training on multiple tasks sampled for each episode. In order to train the encoder, we define a decoder $p$ to reconstruct the transition and reward functions across all tasks conditioned on task embedding samples $\mathbf{z} \sim \mathcal{N}(\mathbf{z})$.
The encoder and decoder, parameterised by $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ respectively, are jointly optimised to maximise the following evidence lower bound (ELBO) given trajectory $\boldsymbol{\tau}_{1:t}$
\begin{equation}
\text{ELBO}(\boldsymbol{\phi}, \boldsymbol{\psi} | \boldsymbol{\tau}_{1:t}) = \mathbb{E}_{q\left({\mathbf{z}_t | \boldsymbol{\tau}_{1:t}; \boldsymbol{\phi}}\right)}\left[\log p({\mathbf{o}_{t+1}, \mathbf{r}_t | \mathbf{o}_t, \mathbf{a}_t, \mathbf{z}; \boldsymbol{\psi}})\right] - \beta \text{KL}\left(q\left({\mathbf{z} | \boldsymbol{\tau}_{1:t}; \boldsymbol{\phi}}\right)|| p(\mathbf{z})\right)
\label{eq:vae_elbo}
\end{equation}
with additional hyperparameter $\beta$ to control the regularisation of the KL prior~\citep{higgins2017beta}. This objective is motivated by the observation that each task can be identified by its unique transition and reward functions. Modelling the decoder as a multivariate Gaussian model over observations and rewards with constant diagonal covariance matrix and assuming a standard Gaussian prior allows us to minimise the following loss, equivalent to maximising the ELBO (see \Cref{app:mate_loss_derivation} for derivation):
\begin{multline}
\mathbb{L}(\boldsymbol{\phi}, \boldsymbol{\psi} | \boldsymbol{\tau}_{1:t}) = \mathbb{E}_{q\left({\mathbf{z}_t | \boldsymbol{\tau}_{1:t}; \boldsymbol{\phi}}\right)}\Big[\left(p({\mathbf{o}_{t+1} | \mathbf{o}_t, \mathbf{a}_t, \mathbf{z}; \boldsymbol{\psi}}) - \mathbf{o}_{t+1}\right)^2 \\+ \left(p({\mathbf{r}_t | \mathbf{o}_t, \mathbf{a}_t, \mathbf{z}; \boldsymbol{\psi}}) - \mathbf{r}_t\right)^2\Big] - \beta \frac{1}{2} \sum_{j=1}^{d}\left(1+\log \left(\sigma_{j}^{2}\right)-\mu_{j}^{2}-\sigma_{j}^{2}\right)
\label{eq:mate_loss}
\end{multline}
The intuition of this loss is that the decoder $p$ is optimised to model the mean of the generative multivariate Gaussian distribution over reconstructed observations and rewards. %
We consider three paradigms of learning MATE: (1) independent MATE, (2) centralised MATE and (3) mixed MATE which vary in the information used to encode task embeddings.
\textbf{Independent Multi-Agent Task Embeddings (Ind-MATE)} independently trains separate encoders $q^i(z^i | \tau^i_{1:t}; \phi^i)$ with $\tau^i_{1:t} = \{(o^i_u, a^i_u, r^i_u)\}_{u=1}^t$ for each agent $i$ conditioned only on its individual trajectory. The centralised decoder is shared across all agents and used to decode individual task embeddings of all agents, and the policy $\pi_i(a_t^i | o_t^i, \mathcal{N}(z^i))$ of agent $i$ is conditioned only on its individual task embedding. We hypothesise that such task embeddings are limited in the encoded information which cannot fully represent the task under partial observability.
\textbf{Centralised Multi-Agent Task Embeddings (Cen-MATE)} instead shares a single encoder $q(\mathbf{z} | \boldsymbol{\tau}_{1:t}; \boldsymbol{\phi})$ with $\boldsymbol{\tau}_{1:t} = \{(\mathbf{o}_u, \mathbf{a}_u, \mathbf{r}_u)\}_{u=1}^t$ and decoder conditioned on the joint information across all agents. The policy of each agent $i$ is conditioned on the joint, shared task embedding $\mathcal{N}(\mathbf{z})$. Such a shared, centralised task embedding has access to information from all agents and can therefore encode more information about the task than Ind-MATE. However, policies depend on the computation of the task embedding which requires access to the joint information across all agents. The access to this privileged information prevents decentralised execution of agents.
\looseness=-1
\textbf{Mixed Multi-Agent Task Embeddings (Mix-MATE)} is similar to Ind-MATE in that each agent trains an individual encoder conditioned only on its local trajectory to ensure decentralised execution. However, instead of decoding each variational task embedding independently, a mixture of task embeddings is computed $q(\mathbf{z} | \boldsymbol{\tau}_{1:t}; \boldsymbol{\phi}) = \sum_{i=1}^N w_i q^i(z^i | \tau^i_{1:t}; \phi^i)$ and sampled from. Mixture weights $\mathbf{w} = f_m(\mathbf{o}_t)$ are computed using a single-layer network with softmax output conditioned on the joint observations. %
We hypothesise that such mixing allows individual task embeddings to be more representative of the full task while preserving decentralised execution. We also note that the mixture distribution provides insight into which agent's task embedding is considered most important in a given state.
\section{Experimental Evaluation}
\label{sec:evaluation}
In this section, we evaluate our proposed approach of learning multi-agent task embeddings. In particular, we will investigate (1) whether MATE improves teamwork adaptation given by returns in testing tasks after limited fine-tuning, (2) how the three paradigms of MATE compare to each other, and (3) what information is encoded by MATE. To answer these questions, we conduct an evaluation in four multi-agent environments, visualised in \Cref{fig:envs}. In all experiments, we train agents using the multi-agent synchronous advantage actor-critic (MAA2C)~\citep{papoudakis2021benchmarking,mnih2016asynchronous} algorithm. During fine-tuning, we freeze MATE encoders and only fine-tune MARL policies and critics. For implementation details and hyperparameters, see \Cref{app:imp_details}, and details for each task can be found in \Cref{app:marl_envs}.
\subsection{Multi-Agent Environments}
\label{sec:evaluation_envs}
\textbf{Multi-Robot Warehouse:} In our motivational example of the multi-robot warehouse (RWARE)~\citep{papoudakis2021benchmarking,christianos2020shared} agents need to collect and deliver requested items from shelves within warehouses. Each agent observes a $5\times5$ grid centred around the agent containing information about nearby shelves, agents and delivery zones. Agents can move forward, pick-up shelves, and rotate which also rotates their observation. Tasks include warehouses with two or four agents, and varying layouts which lead to static, but significant variation requiring adaptation in coordination behaviour outlined in \Cref{fig:rware_generalisation}. Training and testing sets include multiple warehouses of similar layout but a range of sizes. Each episode consists of $500$ timesteps.
\looseness=-1
\textbf{Multi-Agent Particle Environment:} In the cooperative navigation task of the multi-agent particle environment (MPE)~\citep{mordatch2018emergence,lowe2017multi}, three agents need to navigate a continuous two-dimensional, fully-observable world to cover all three landmarks which are initialised in random locations while avoiding collisions with each other. In the training task, agents only receive a small punishment of $-1$ for collisions with each other whereas testing tasks punish agents with significantly higher negative rewards of $-5$ and $-50$, respectively. Episodes terminate after $25$ timesteps.
\looseness=-1
\textbf{Boulder-Push:} In this new environment (BPUSH), agents navigate a gridworld and need to push a box towards a goal location. Agents observe the box and agents in a $9\times9$ grid and the direction the box is required to be pushed in. These tasks require significant coordination with agents only receiving rewards for successfully pushing the box forward which requires cooperation of all agents. Episodes terminate after the box has been pushed to its goal location or after at most $50$ timesteps. In the training task, two agents need to push a box in a small $8\times8$ gridworld. Testing tasks include tasks with gridworld sizes of $12\times12$ (medium) and $20\times20$ (large) as well as tasks with a small negative penalty of $-0.01$ for unsuccessful pushing attempts of agents. Tasks with penalty terms make the exploration of the optimal behaviour of only pushing the box together with the other agent significantly harder.
\textbf{Level-Based Foraging:} In the level-based foraging (LBF) environment~\citep{Albrecht2013ASystems,albrecht2018autonomous}, multiple agents need to coordinate in a gridworld to pick-up food. Food and agents are assigned levels with agents only being able to pick-up adjacent food if the sum of levels of all agents cooperating to pick-up is greater or equal to the level of the food. Agents observe a $5\times5$ grid centred on themselves and are rewarded for picking-up food depending on its level and their contribution. Episodes terminate after all food has been collected or after at most $50$ timesteps. We train in two comparably simple LBF tasks with gridworld sizes of $8\times8$ and two and four agents, respectively. Testing tasks for two and four agents contain a task with gridworld size $10\times10$ in which each food will require at least two agents to coordinate to pick it up, tasks with larger gridworld size of $15\times15$ and tasks with a small penalty of $-0.1$ for unsuccessful picking attempts of food.
\begin{figure}
\centering
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[height=10em]{media/tasks/rware-tiny-2ag}
\caption{\centering Multi-robot warehouse (RWARE)}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\fbox{\includegraphics[height=8.9em]{media/tasks/mpe-simple-spread}}
\caption{\centering Multi-agent particle (MPE)}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[height=10em]{media/tasks/bpush-small-2ag}
\caption{\centering Boulder-push (BPUSH)}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[height=10em]{media/tasks/lbf-grid-8x8-2p-2f}
\caption{\centering Level-based foraging (LBF)}
\end{subfigure}
\caption{Visualisations of all four multi-agent environments}
\label{fig:envs}
\end{figure}
\subsection{Baselines}
\label{sec:evaluation_baselines}
\textbf{No MATE (fine-tune)}: For the fine-tune baseline, we train MARL agents without any task embeddings in the training task and fine-tune in testing tasks for further $N_{\text{test}}$ timesteps. This baseline allows us to evaluate the benefits of task embeddings for teamwork adaptation.
\textbf{No MATE (scratch)}: For the scratch baseline, we train MARL agents without any task embeddings only in the testing tasks for $N_{\text{test}}$ timesteps. This baseline allows us to distinguish adaptation settings where prior training in potentially simpler tasks can improve returns in testing tasks from cases of negative transfer where prior training hurts the performance of agents in the testing tasks.
\subsection{Evaluation Results}
\label{sec:evaluation_results}
\begin{table}[t]
\caption{Fine-tuning performance given by IQM and standard deviation across final returns over five random seeds. Highest IQM per testing task (within one standard deviation) are shown in bold.}
\label{tab:finetuning_results}
\robustify\bf
\resizebox{\textwidth}{!}{
\begin{tabular}{@{}p{3em} l l S S S S S @{}}
\toprule
& $\mathcal{T}_{\text{train}}$ ($N_\text{train}$) & $\mathcal{T}_{\text{test}}$ ($N_{\text{test}}$) & {Scratch} & {Fine-tune} & {Ind-MATE} & {Cen-MATE} & {Mix-MATE}\\ \midrule
\multirow{6}{*}{RWARE} & \multirow{2}{*}{tiny-2ag (25M)} & small-2ag (25M) & 0.02(0)& \bf 8.54(229)& 5.97(341)& \bf 10.90(297)& \bf 7.96(411)\\
& & corridor-2ag (25M) & \bf 23.00(291)& \bf 22.26(558)& \bf 26.98(406)& \bf 28.59(710)& \bf 25.39(1017)\\
& \multirow{2}{*}{tiny-4ag (25M)} & small-4ag (25M) & 0.12(87)& \bf 28.48(92)& 26.61(114)& 26.58(132)& \bf 27.69(121)\\
& & corridor-4ag (25M) & \bf 50.13(184)& 42.75(370)& 43.12(231)& 45.53(242)& 43.96(330)\\
& wide-both (50M) & wide-one-sided (50M) & \bf 12.39(675)& 0.03(349)& 0.02(0)& 3.08(450)& 0.02(0)\\
& wide-one-sided (50M) & wide-both (50M) & 0.04(62)& \bf 7.06(395)& \bf 5.26(369)& 3.45(311)& \bf 7.34(339)\\\midrule
\multirow{2}{*}{MPE} & \multirow{2}{*}{cooperative navigation (10M)} & penalty navigation, pen=5 (10M) & \bf -259.65(1109)& -258.81(113)& \bf -257.50(31)& -258.47(75)& -258.45(68)\\
& & penalty navigation, pen=50 (10M) & -2019.35(395)& \bf -2016.83(354)& -2019.20(379)& -2020.60(439)& \bf -2015.15(201)\\\midrule
\multirow{4}{*}{BPUSH} & \multirow{4}{*}{small-2ag (5M)} & small-pen-2ag (20M) & \bf -0.01(1)& \bf -0.00(105)& \bf 0.88(130)& \bf -0.00(103)& \bf 0.89(130)\\
& & medium-pen-2ag (20M) & \bf -0.01(1)& \bf -0.00(0)& \bf -0.00(0)& \bf 0.00(0)& \bf 0.00(0)\\
& & medium-2ag (20M) & 0.69(36)& 1.24(47)& \bf 2.04(29)& \bf 2.07(46)& \bf 2.32(52)\\
& & large-2ag (20M) & 0.00(0)& 0.39(10)& \bf 0.60(16)& 0.43(14)& \bf 0.97(40)\\\midrule
\multirow{6}{*}{LBF} & \multirow{3}{*}{8x8-2p-2f (5M)} & 10x10-2p-2f-coop (5M) & \bf 0.88(2)& 0.80(3)& \bf 0.84(4)& 0.81(3)& \bf 0.84(4)\\
& & 15x15-2p-4f (5M) & \bf 0.71(3)& \bf 0.67(4)& \bf 0.73(8)& \bf 0.71(1)& \bf 0.71(2)\\
& & 15x15-2p-2f-pen (5M) & \bf 0.70(0)& \bf 0.64(8)& 0.55(6)& 0.65(4)& 0.56(12)\\
& \multirow{3}{*}{8x8-4p-4f (5M)} & 10x10-4p-2f-coop (5M) & \bf 0.78(1)& \bf 0.73(5)& 0.74(3)& 0.72(2)& 0.74(2)\\
& & 15x15-4p-6f (5M) & \bf 0.73(0)& 0.66(3)& \bf 0.71(2)& 0.69(2)& \bf 0.71(3)\\
& & 15x15-4p-4f-pen (5M) & \bf 0.44(0)& 0.30(8)& \bf 0.24(24)& \bf 0.35(10)& \bf 0.39(7)\\\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[t]
\begin{subfigure}{\textwidth}
\centering
\includegraphics[trim={0 0.5em 0 0.5em},clip,width=.8\textwidth]{media/results_nolabels/finetuning/legend.pdf}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/rware/tiny-2ag_corridor-2ag.pdf}
\caption{RWARE corridor-2ag}
\label{fig:rware_corridor_2ag}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/rware/wide-sides_wide.pdf}
\caption{RWARE wide-both}
\label{fig:rware_wide_both}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/mpe/zone-spread-v4_penalty-spread-v1.pdf}
\caption{MPE penalty navigation (pen=5)}
\label{fig:mpe_penalty_5}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/bpush/small-2ag_small-pen-2ag.pdf}
\caption{BPUSH small-pen-2ag}
\label{fig:bpush_small_pen}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/bpush/small-2ag_medium-2ag.pdf}
\caption{BPUSH medium-2ag}
\label{fig:bpush_medium}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\textwidth]{media/results_nolabels/finetuning/lbf/8x8-4p-4f_15x15-4p-4f-pen.pdf}
\caption{LBF 15x15-4p-4f-pen}
\label{fig:lbf_4p_pen}
\end{subfigure}
\caption{IQM and 95\% confidence intervals for fine-tuning returns in selected testing tasks.}
\label{fig:finetuning_results}
\end{figure}
We report episodic returns as the sum of episodic returns of all agents. We train and fine-tune each algorithm for five random seeds and report the interquartile mean (IQM)~\citep{agarwal2021deep} and standard deviation across final fine-tuning returns over all seeds in \Cref{tab:finetuning_results}. Learning curves with IQM and stratified bootstrap 95\% confidence intervals computed over 100,000 samples are shown in \Cref{fig:finetuning_results} for some testing tasks. Learning curves for all training and testing scenarios can be found in \Cref{app:training_plots,app:finetuning_plots}.
In RWARE (\Cref{fig:finetuning_rware_all}), agents need to heavily explore in order to receive positive rewards as they need to execute long sequences of actions to deliver requested items. Therefore, agents significantly benefit from the initial training phase during fine-tuning for testing tasks with larger or more complicated environments. In particular, in small warehouse instances and wide warehouses with shelves on both sides (\Cref{fig:rware_wide_both}) we find the fine-tuning of agents with and without MATE improves returns significantly compared to training from scratch. In contrast, in simpler environments such as the corridor warehouses (\Cref{fig:rware_corridor_2ag}) agents appear to suffer from the initial training phase starting with policies which exhibit less exploration than randomly initialised policies with entropy regularisation. For MATE, we find that Cen-MATE with access to observations of all agents performs the best in most tasks due to significant partial observability in these tasks.
Fine-tuning agents in MPE (\Cref{fig:finetuning_mpe_all}) from simple spread to tasks with larger collision penalties appears to lead to no significant benefits over training agents from scratch. Agent policies remain almost identical throughout fine-tuning for both testing tasks, exhibiting almost identical returns at the beginning and end of fine-tuning.%
For BPUSH (\Cref{fig:finetuning_bpush_all}), we find MATE significantly outperforms all baselines in three out of four fine-tuning scenarios. In particular Mix-MATE outperforms all other approaches. Adding a penalty to unsuccessful pushing attempts of agents makes the exploration of the optimal behaviour significantly more challenging, so training agents from scratch is unsuccessful. Agents previously trained on the simpler task without such a penalty perform well early during fine-tuning, but throughout fine-tuning agents exhibit more defensive behaviour and refrain from pushing the box to avoid penalties leading to decreasing overall returns. We verified that this behaviour was consistent even for lower entropy regularisation coefficient $10^{-4}$ for reduced penalty-inducing exploration. We find Ind-MATE and Mix-MATE are significantly more robust at informing agents about the task and allow them to adapt their behaviour by limiting unsuccessful pushing attempts. We hypothesise that Cen-MATE is less successful in these tasks due to its worse sample efficiency exhibiting lower returns at the end of the initial training phase. Agents fine-tuned with MATE are also better at adapting to tasks with larger gridworlds, reaching higher returns in particular for Mix-MATE and Ind-MATE.
\looseness=-1
In LBF (\Cref{fig:finetuning_lbf_all}), we find that the majority of tasks require comparably little adaptation with agents reaching similar returns at the beginning and end of fine-tuning. We hypothesise that the high degree of variability within individual LBF tasks, through agent and food placement as well as level allocation, allows agents to already learn most capabilities required for testing tasks during the initial training phase. In tasks with penalty for unsuccessful picking-up of food, significant adaptation of behaviour is required but policies obtained after the initial training phase tend to frequently attempt to pick-up food. This behaviour results in lower returns and consequentially negative initial transfer performance.
Overall, we find MATE to improve teamwork adaptation in several cases compared to both baselines. In particular in the BPUSH environment where all agents need to coordinate, we find MATE consistently improves returns during fine-tuning. Across all three paradigms of MATE, we find Cen-MATE to perform the best in environments with significant partial observability such as most RWARE tasks. However, training the shared encoder on the larger joint observations of all agents makes training with Cen-MATE less sample efficient leading to worse returns at the end of fine-tuning in several BPUSH tasks and the corridor task with four agents (\Cref{fig:finetuning_rware_corridor_4ag}). Lastly, we find the mixture of task embeddings appears to improve adaptation with Mix-MATE outperforming Ind-MATE in almost all fine-tuning tasks. Due to these benefits, we recommend to learn task embeddings using Mix-MATE whenever decentralised execution is of importance. For testing tasks with significant partial observability, Cen-MATE should be considered for its privileged encoded information but prevents decentralised deployment of agents.
\begin{figure}[t]
\begin{subfigure}{.02\textwidth}
\centering
\includegraphics[angle=90, height=15em]{media/mate_tsne/legend.pdf}
\end{subfigure}
\begin{subfigure}{.40\textwidth}
\centering
\includegraphics[width=.8\textwidth]{media/tasks/rware-wide-19w-2ag}
\includegraphics[width=.8\textwidth]{media/tasks/rware-wide-19w-left-2ag}
\includegraphics[width=.8\textwidth]{media/tasks/rware-wide-19w-right-2ag}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[trim={0 0 0 3em},clip,width=\textwidth]{media/mate_tsne/cen_mate_wide_tsne_cropped.pdf}
\end{subfigure}
\caption{t-SNE projection of task embeddings in wide warehouses with colours representing warehouse layouts across a single episode. Task embeddings were obtained using Cen-MATE trained in one-sided wide warehouses. For visual clarity task embedding are visualised every 10 timesteps with increasing opacity throughout the episode to highlight embeddings at latter timesteps.}
\label{fig:mate_tsne}
\end{figure}
\subsection{Learned Multi-Agent Task Embeddings}
\label{sec:evaluation_analysis_embeddings}
After evaluating the benefits of MATE for teamwork adaptation, we analyse the task embeddings learned by MATE to gain a deeper understanding for the proposed method. First, we address the questing whether task embeddings encode meaningful information to identify tasks. \Cref{fig:mate_tsne} visualises the t-SNE~\citep{van2008visualizing} projection of task embeddings obtained using Cen-MATE for both one-sided wide warehouses as well as the wide warehouse with shelves on both sides. More specifically, we execute a policy trained without MATE in one-sided warehouses and sequentially encode the trajectory using Cen-MATE trained in the same one-sided warehouses. \Cref{fig:mate_tsne} clearly illustrates separate clusters of embeddings for different tasks. Whereas the warehouse with shelves on both sides (blue) is clearly separated from one-sided warehouses, embeddings for both symmetric one-sided warehouses are close together. However, embeddings of one-sided warehouses are also separated with embeddings for the wide-left (orange) warehouse concentrated within a ring of embeddings for the wide-right (green) warehouse. Note that these embeddings were not trained in the wide-both warehouse task, demonstrating that MATE is able to generalise to novel tasks and produce task embeddings which identify tasks not encountered during training.
In \Cref{sec:evaluation}, we find Mix-MATE to consistently outperform Ind-MATE for teamwork adaptation. It appears the mixture model significantly improves the impact of learned task embeddings on teamwork adaptation. \Cref{fig:mate_mixing} visualises mixture weights for Mix-MATE in the wide-left warehouse environment. The mixing has no significant emphasis on the embedding of either agent until the first agent reaches the block of shelves on the left. The discovery and encoding of this task-relevant information leads to a rise in weighting on task embedding of the first agent. This importance ends after the second agent also reaches the block of shelves. After both agents discovered the shelves identifying the task, the mixing appears to again focus on the first agent which has loaded a shelf with requested items.
\begin{figure}[t]
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{media/mixing/rware_agents_mixing_obs.pdf}
\end{subfigure}
\begin{subfigure}{.50\textwidth}
\centering
\includegraphics[width=.95\textwidth]{media/mixing/rware_wide_left_mix_weights_line.pdf}
\end{subfigure}
\caption{Mix-MATE mixing weights for 25 timesteps in the wide-left warehouse environment. At timestep 10, only agent 1 is observing the shelves in the warehouse (agent 2 does not observe any shelves) so higher weighting is assigned to its respective embedding. At timestep 17, agent 2 now observes a larger part of the shelves which results in higher weights on the embedding of agent 2.}
\label{fig:mate_mixing}
\end{figure}
\section{Conclusion}
In this work, we discussed the problem of teamwork adaptation in which a team of agents needs to adapt their policies to solve novel tasks with limited fine-tuning. We presented three paradigms to learn multi-agent task embeddings (MATE), which recurrently encode information about the current task identified by its unique transition and reward functions. Our experiments in four multi-agent environments demonstrated that MATE provides useful information for teamwork adaptation, with most significant improvements in tasks where all agents need to coordinate. Our analysis of learned task embeddings shows that they form clusters which identify tasks. Among the three paradigms proposed to learn MATE, we find mixed MATE to perform the best across a wide range of testing tasks. Learned mixtures of task embeddings provide interpretable insight into the training of these embeddings, with higher weights being put on agents which currently observe more information about the task.
Future work could aim to relax the assumption that training and testing tasks have identical observation dimensionality and number of agents. This assumption limits the settings in which MATE can be applied, and could be addressed using network architectures leveraging attention~\citep{vaswani2017attention}. Lastly, future work should evaluate the feasibility of learning a mixture of task embeddings and centrally encoding joint information for centralised embeddings in tasks with many agents to investigate the scalability of our proposed approach.
\bibliographystyle{plainnat}
|
2,877,628,090,202 | arxiv | \section{Introduction}
Investigating molecular dynamics of glass-forming liquids is one of the most intriguing topics. It has been experimentally established that the structural relaxation time ($\tau_\alpha$) reflecting the time scale for liquid structure reorganization systematically deviates from the simple Arrhenius behaviour during cooling process on approaching to the glass transition temperature, $T_g$, defined by $\tau_\alpha(T_g)=100$ s \cite{61,62}. The non-Arrhenius dependence of the structural relaxation time, $\tau_\alpha$ has universal character because it has been observed for different groups of glass-forming liquids (van der Waals and associated liquids, polymers, ionic liquids, molten metals, etc). However, degree of deviation of $\tau_\alpha$ form the Arrhenius law at $T_g$ is material dependent and is characterized by means of fragility or steepness index, $m=\left[\partial\log_{10}\tau_\alpha/\partial (T_g/T)\right]_{T=T_g}$. Consequently, the parameter $m$ was used to introduce the strong versus fragile liquid classification scheme. According to this classification strong liquids reveal temperature evolution of structural relaxation time less deviating from the Arrhenius behaviour than fragile ones.
Much efforts have been spent in the last decades to formulate satisfactory models being able to capture and explain all experimentally observed features of structural dynamics of glass forming liquids. One of such successful approaches is the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory of bulk relaxation \cite{2,6,7,10,8,42,9}. In this theory, a single molecular motion is considered as a consequence of its interactions with the nearest neighbours and molecular cooperativity outside the cage of neighbouring molecules. The treatment leads to two strongly-related but distinct barriers corresponding to local and elastically collective dynamics. Plugging these two barriers into the Kramer's theory gives the structural alpha relaxation times. To determine the temperature dependence of the structural relaxation times, Mirigian and Schweizer have used a thermal mapping, which is based on an equality between hard-sphere-fluid and experimental isothermal compressibility. From this, the ENCLE theory has successfully described the alpha relaxation event of polymers \cite{2,8}, and thermal liquids \cite{6,7,10} over 14 decades in time. However, amorphous drugs and many materials have no experimental data for the thermal mapping. It is impossible to compare ECNLE calculations with experiments. Recently, Phan and his coworkers \cite{42,9,50} proposed another density-to-temperature conversion based on the thermal expansion process to handle this issue.
The rapid cooling of liquid to obtain the glass is not the only way. Alternative method to vitrify it is squeezing (compression) \cite{63,64}. Therefore, by changing the hydrostatic pressure of liquid one can also control its molecular dynamics \cite{63,64}. Compression brings about increase in the molecular packing, in consequence, leading to increase of the structural relaxation time. Numerous experimental results \cite{63,64} show that the pressure counterpart of the Arrhenius law:
\begin{eqnarray}
\tau_\alpha=\tau_0\exp\left(\frac{P\Delta V}{k_BT}\right),
\label{eq:1}
\end{eqnarray}
derived based on transition state theory fails to grasp pressure dependence of $\tau_\alpha$, where $\Delta V$ is activation volume, $P$ is the pressure, and $k_B$ is Boltzmann constant. The experimentally measured relaxation times are found to change with pressure much faster than predicted by Eq. (\ref{eq:1}) \cite{64,65,66,67,68}. It indicates that the activation volume is not constant but in general increases with increasing pressure on approaching to glassy state. An extension of ECNLE theory \cite{10} was introduced in 2014 to understand compression effects on the glass transition. Authors used Schweizer's the thermal mapping associated with the compressibility data measured at different pressures. However, theoretical predictions are more sensitive to pressure than experiments. Thus, it is crucial to propose a better model to determine quantitatively the pressure-dependent structural dynamics.
The main goal of this paper is to develop the ECNLE theory in a new approach to describe the pressure dependence of $\tau_\alpha$. To validate our development, we implement new dielectric spectroscopy measurements on three different rigid and non-polymeric supramolecules at a wide range of pressures and temperatures. Then, theoretical calculations are quantitatively compared to experimental results. Theoretical limitations are clearly discussed.
\section{Theoretical Methods}
\subsection{Formulation}
To theoretically investigate the structural relaxation time of amorphous materials, these materials are described as a fluid of disconnected spheres (a hard-sphere fluid) and we formulate calculations for activation events of a single particle. The hard-sphere fluid is characterized by a particle diameter, $d$, and the number of particles per volume, $\rho$. According to the ECNLE theory \cite{2,6,7,10,8,42,9,3,4}, the dynamic free energy quantifying interactions of an arbitrary tagged particle with its nearest neighbors at temperature $T$ is
\begin{eqnarray}
\frac{F_{dyn}(r)}{k_BT} &=& \int_0^{\infty} dq\frac{ q^2d^3 \left[S(q)-1\right]^2}{12\pi\Phi\left[1+S(q)\right]}\exp\left[-\frac{q^2r^2(S(q)+1)}{6S(q)}\right]
\nonumber\\ &-&3\ln\frac{r}{d},
\label{eq:2}
\end{eqnarray}
where $\Phi = \rho\pi d^3/6$ is the volume fraction, $S(q)$ is the static structure factor, $q$ is the wavevector, $r$ is the displacement of the particle. The dynamic free energy is constructed without considering effects of rotational motions. We use the Percus-Yevick (PY) integral equation theory \cite{1} for a hard-sphere fluid to calculate $S(q)$. The PY theory defines $S(q)$ via the direct correlation function $C(q)=\left[S(q)-1 \right]/\rho S(q)$. The Fourier transform of $C(q)$ is \cite{1}
\begin{eqnarray}
C(r) &=& -\frac{(1+2\Phi)^2}{(1-\Phi)^4} + \frac{6\Phi(1+\Phi/2)^2}{(1-\Phi)^4}\frac{r}{d}\nonumber\\
&-&\frac{\Phi(1+2\Phi)^2}{2(1-\Phi)^4}\left(\frac{r}{d}\right)^3 \quad \mbox{for} \quad r \leq d \\
C(r) &=& 0 \quad \mbox{for} \quad r > d.
\end{eqnarray}
The free energy profile gives us important information for local dynamics. For $\Phi \leq 0.43$, $F_{dyn}(r)$ monotonically decreases with increasing $r$ and particles are not localized \cite{3,4,1}. In denser systems ($\Phi > 0.43$), one observes the dynamical arrest of particles within a particle cage formed by its neighbors occurs and a free-energy barrier emerges as shown in Fig. \ref{fig:3}. We determine the particle cage radius, $r_{cage}$, as a position of the first minimum in the radial distribution function, $g(r)$. The localization length ($r_L$) and the barrier position ($r_B$) are the local minimum and maximum of the dynamic free energy. The separation distance between these two positions, $\Delta r =r_B-r_L$, is a jump distance. The local energy-barrier height is calculated by $F_B=F_{dyn}(r_B)-F_{dyn}(r_L)$.
Compression effects modify motion of a single particle. Motion of a particle is governed by both nearest-neighbor interparticle interactions and applied pressure. Under a high pressure condition, when a particle is displaced by a small distance ($r \ll d$), the applied pressure acts on a volume $\Delta V(r) \approx d^2r$ and causes the mechanical work. In addition, the free volume and the molecular volume are reduced with compression. For simplification purposes, we suppose that the volume fraction is insensitive to pressure. Thus, we propose a new and simple expression for the dynamic free energy
\begin{eqnarray}
\frac{F_{dyn}(r)}{k_BT}
&\approx& \int_0^{\infty} dq\frac{ q^2d^3 \left[S(q)-1\right]^2}{12\pi\Phi\left[1+S(q)\right]}\exp\left[-\frac{q^2r^2(S(q)+1)}{6S(q)}\right]
\nonumber\\ &-&3\ln\frac{r}{d} + \frac{P}{k_BT/d^3}\frac{r}{d}.
\label{eq:pressure}
\end{eqnarray}
The diffusion of a particle through its cage is decided by rearrangement of particles in the first shell. The reorganization process slightly expands the particle cage and excites collective motions of other particles in surrounding medium by propagating outward radially a harmonic displacement field $u(r)$. By using Lifshitz's continuum mechanics analysis \cite{5}, the distortion field in a bulk system is analytically found to be
\begin{eqnarray}
u(r)=\frac{\Delta r_{eff}r_{cage}^2}{r^2}, \quad {r\geq r_{cage}},
\label{eq:3}
\end{eqnarray}
where $\Delta r_{eff}$ is the cage expansion amplitude \cite{6,7}, which is
\begin{eqnarray}
\Delta r_{eff} = \frac{3}{r_{cage}^3}\left[\frac{r_{cage}^2\Delta r^2}{32} - \frac{r_{cage}\Delta r^3}{192} + \frac{\Delta r^4}{3072} \right].
\end{eqnarray}
Since $\Delta r_{eff}$ is relatively small, particles beyond the first coordination is supposed to be harmonically oscillated with a spring constant at $K_0 = \left|\partial^2 F_{dyn}(r)/\partial r^2\right|_{r=r_L}$. Thus, the oscillation energy of the oscillator at a distance $r$ is $K_0u^2(r)/2$. By associating with the fact that the number of particles at a distance between $r$ and $r + dr$ is $\rho g(r)4\pi r^2dr$, we can calculate the elastic energies of cooperative particles outside the cage to determine effects of their collective motions. The elastic barrier, $F_e$, is
\begin{eqnarray}
F_{e} = 4\pi\rho\int_{r_{cage}}^{\infty}dr r^2 g(r)K_0\frac{u^2(r)}{2}.
\label{eq:5}
\end{eqnarray}
For $r \geq r_{cage}$, $g(r)\approx 1$. The calculations allow us to determine contributions of nearest-neighbor interactions and collective rearrangement to the activated relaxation of a particle.
Due to chemical and biological complexities, conformational configuration, and chain connectivity, local and non-local dynamics is non-universally coupled. In our recent work \cite{9}, an adjustable parameter $a_c$ is introduced to scale the collective elastic barrier as $F_e \rightarrow a_cF_e$. The treatment has simultaneously provided quantitatively good agreements between theory and experiment in both the dynamic fragility and temperature dependence of structural relaxation time for 22 amorphous drugs and polymers \cite{9}. According to Kramer's theory, the structural (alpha) relaxation time defined by the mean time for a particle to diffuse from its particle cage is
\begin{eqnarray}
\frac{\tau_\alpha}{\tau_s} = 1+ \frac{2\pi}{\sqrt{K_0K_B}}\frac{k_BT}{d^2}\exp\left(\frac{F_B+a_cF_e}{k_BT} \right),
\label{eq:6}
\end{eqnarray}
where $K_B$=$\left|\partial^2 F_{dyn}(r)/\partial r^2\right|_{r=r_B}$ is absolute curvatures at the barrier position and $\tau_s$ is a short time scale of relaxation. The explicit expression of $\tau_s$ is \cite{6,7}
\begin{eqnarray}
\tau_s=g^2(d)\tau_E\left[1+\frac{1}{36\pi\Phi}\int_0^{\infty}dq\frac{q^2(S(q)-1)^2}{S(q)+b(q)} \right],
\label{eq:6-1}
\end{eqnarray}
where $\tau_E$ is the Enskog time scale, $b(q)=1/\left[1-j_0(q)+2j_2(q)\right]$, and $j_n(x)$ is the spherical Bessel function of order $n$. In various works \cite{2,6,7,42,9} of thermal liquids, polymers and amorphous drugs, $\tau_E \approx 10^{-13}$ s.
To compare our hard-sphere calculations with experiment, a density-to-temperature conversion (thermal mapping) is required. The initial thermal mapping proposed by Schweizer \cite{10} is
\begin{eqnarray}
S_0^{HS}(\Phi)&=&\frac{(1-\Phi)^4}{(1+2\Phi)^2}\equiv S_0^{exp}(T,P),
\label{eq:Cmapping}
\end{eqnarray}
where $S_0$ is the isothermal compressibility. Clearly, this mapping requires experimental equation-of-state (EOS) data. The superscripts $HS$ and $exp$ correspond to hard sphere and experiment, respectively. Although this mapping has successfully provided both qualitative and quantitative descriptions for $\tau_\alpha(T)$ for 17 polymers and thermal liquids \cite{2,6,7,8,10}, the EOS data is unknown for our three polymers presented in next sections.
Thus, we employ another thermal mapping \cite{42,9,50} constructed from the thermal expansion process of materials. During a heating process, the number of molecules remains unchanged while the volume of material increases linearly. This analysis leads to $\rho \approx \rho_0\left[1-\beta\left(T-T_{0}\right)\right]$ \cite{42,9,50}. Here $\beta$ is the volume thermal expansion coefficient, and $\rho_0$ and $T_{0}$ are the initial number density and temperature, respectively. From this, we can convert from a volume fraction to temperature of experimental material via
\begin{eqnarray}
T \approx T_0 - \frac{\Phi - \Phi_0}{\beta\Phi_0}.
\label{eq:7}
\end{eqnarray}
For most organic materials and amorphous drugs (22 materials) \cite{42,9,50}, $\beta \approx 12\times 10^{-4}$ $K^{-1}$. This value is consistent with Schweizer's the original mapping \cite{2}. $\Phi_0 \approx 0.5$ is the characteristic volume fraction estimated in our prior works \cite{42,9,50}. The parameter $T_0$ captures material-specific details such as molar mass and particle size. This density-to-temperature conversion has been used in the cooperative-string model for supercooled dynamics \cite{51}. In our calculations, the parameters $T_0$ and $a_c$ are tuned to obtain the best quantitative agreement between theoretical and experimental temperature dependence of structural relaxation times.
\subsection{Ultra-local limit}
Figure \ref{fig:3} shows an example dynamic free energy for $\Phi = 0.57$ at different pressures in unit of $k_BT/d^3$ and defines key length and energy scales. The localization length is nearly insensitive to compression. Meanwhile, the barrier position increases and the local barrier height is raised with increasing the applied pressure. The result implies that the compression induces more constraint to the local dynamics of the tagged particle.
When the local barrier is beyond a few $k_BT$, much insight for key length scales of the dynamic free energy has been gained using the approximate "ultra-local" analytic analysis. In the ultra-local limit, since $r_L/d \ll 1$, high wavevectors are dominant in calculations of $F_{dyn}(r)$. We can ignore the wavevector integral below a cutoff $q_c$, and exploit $C(q)=-4\pi d^3 g(d)\cfrac{\cos(qd)}{(qd)^2}$ in the exact PY theory for $q \geq q_c$ \cite{1,11,12} and $S(q)\approx 1$.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Graph163.pdf}
\caption{\label{fig:3}(Color online) Dynamic free energy as a function of reduced particle displacement for a hard sphere fluid of packing fraction $\Phi = 0.57$ at several pressures in unit of $k_BT/d^3$. The inset shows a growth of the barrier height with $\Phi$ at $p = 0, 1,$ and 2 $k_BT/d^3$.}
\end{figure}
Combining the analytical expression of $C(q)$ and $S(q)\approx 1$ with $\left[\partial F_{dyn}(r)/\partial r\right]_{r=r_L}=0$ gives a self-consistent equation for the localization length and barrier position
\begin{eqnarray}
\frac{9d^2}{r_{L,B}^2}-\frac{3P}{k_BT/d^3}\frac{d}{r_{L,B}}&\approx& \frac{24\Phi g^2(d)}{\pi}\int_{q_c}^{\infty}dq e^{-q^2r_{L,B}^2/3}\nonumber\\
&\approx& \frac{12\Phi g^2(d)}{\pi}\frac{\sqrt{3\pi}d}{r_{L,B}}erfc\left(\frac{q_cr_{L,B}}{\sqrt{3}}\right).\nonumber\\
\label{eq:12}
\end{eqnarray}
Now, since $q_cr_L/\sqrt{3} \ll 1$, one obtains
\begin{eqnarray}
r_L\equiv r_L(P)=\frac{r_L(P=0)}{1+\sqrt{\cfrac{\pi}{3}}\cfrac{P}{k_BT/d^3}\cfrac{1}{4g^2(d)\Phi}},
\label{eq:13}
\end{eqnarray}
where $r_L(P=0)=\cfrac{\sqrt{3\pi} d}{4g^2(d)\Phi}$ is the localization length at $P = 0$ or ambient pressure \cite{11,12}. Equation (\ref{eq:13}) quantitatively reveals how the external pressure restricts molecular motions. The localization length is reduced with increasing the compression. In addition, the Percus-Yevick (PY) theory for the contact number \cite{1} gives $g(d)=\cfrac{(1+\Phi/2)}{(1-\Phi)^2}$. Thus, $4\pi g^2(d)\Phi \approx 110$ for $\Phi=0.57$ is much larger than the considered values of $P/(k_BT/d^3)$. This finding explains why $r_L(P)$ is nearly unchanged as seen in Fig. \ref{fig:3}.
When $q_cr_B/\sqrt{3}$ is sufficiently large, one can use $\ce{erfc}(x)\approx e^{-x^2}/(\sqrt{\pi}x)$ to approximate $r_B$ in Eq. (\ref{eq:12}) and then obtain
\begin{eqnarray}
\frac{P}{k_BT/d^3}\frac{r_B}{d}&\approx& 3 - \frac{12\Phi g^2(d)}{\pi q_cd}\exp\left(-\frac{q_c^2r_B^2}{3} \right).
\label{eq:14}
\end{eqnarray}
The analytic form in Eq. (\ref{eq:14}) qualitatively indicates an increase of $r_B$ with increasing pressure as observed in Fig. \ref{fig:3}. Since prior works \cite{11,12} shows very poor quantitative accuracy of Eq. (\ref{eq:14}) compared to the numerical predictions at ambient pressure ($P\approx 0$), the deviation is expected to be large at elevated pressures. Thus, we do not show the corresponding curves.
The local barrier height $F_B$ in the ultra-local limit \cite{11,12} can be analytically calculated as
\begin{widetext}
\begin{eqnarray}
\frac{F_B}{k_BT} &=& -3\ln\frac{r_B}{r_L}-\frac{12\Phi g^2(d)}{\pi d} \int_{q_c}^\infty\frac{dq}{q^2}\left[e^{-q^2r_B^2/3}-e^{-q^2r_L^2/3} \right]\nonumber\\
&=& -3\ln\frac{r_B}{r_L}+\frac{12\Phi g^2(d)}{\sqrt{\pi}q_cd}\left[ \frac{q_cr_B}{\sqrt{3}}erfc\left(\frac{q_cr_B}{\sqrt{3}}\right)+\frac{e^{-q^2r_L^2/3}-e^{-q^2r_B^2/3}}{\sqrt{\pi}}\right].
\label{eq:16}
\end{eqnarray}
\end{widetext}
Clearly, the growth of $r_B$ with pressure is faster than that of $\ln(r_B)$ and it leads to the pressure-induced rise of $F_B$. At a given compression condition, we find that $F_B$ increases linearly with $\Phi g^2(d)= \cfrac{\Phi(1+\Phi/2)^2}{(1-\Phi)^4}$. Thus, $F_B$ grows with $\Phi$. The findings are consistent with numerical results shown in the inset of Fig. \ref{fig:3}. This analysis also reveals that adding the pressure term to the dynamic free energy as written in Eq. (\ref{eq:pressure}) exhibits the same manner as using Eq. (\ref{eq:2}) for hard-sphere fluids at higher effective volume fractions.
In addition, based on analysis in prior works \cite{11,12}, one can also perform an the dynamic shear modulus in the ultra-local limit as
\begin{eqnarray}
G(P) &=& \frac{9\Phi k_BT}{5\pi d r_L^2(P)}\nonumber\\
&=&\frac{9\Phi k_BT}{5\pi d r_L^2(P=0)}\left(1+\frac{P\sqrt{\pi/3}}{k_BT/d^3}\cfrac{1}{4g^2(d)\Phi}\right)^2\nonumber\\
&=& G(P=0)\left(1+\frac{P\sqrt{\pi/3}}{k_BT/d^3}\cfrac{1}{4g^2(d)\Phi}\right)^2.
\label{eq:15}
\end{eqnarray}
Equation (\ref{eq:15}) shows that $G(P)$ hardly changes with the applied pressure.
\section{Experimental Section}
\subsection{Materials}
The experiments were performed on three rigid and non-polymeric supramolecules. Two of the tested samples are planar, linear, and their chemical structure (shown in Figure \ref{fig:1}) differs only in the end of the group (the diphenylamine-fluorene moiety is the same). In the material referred to M67 has the metoxy -OCH3 group is the end, while in sample named M68 the end of the group is the -CF3 moiety. The third material, entitled M71, enclose other motif (i.e. carbazole-carbazole group) than M67 and M68, which lead to deflection of the chemical structure. All of tested samples were synthesized by Sonogashira coupling reaction between 4-iodoanisole (M67) or 4-iodobenzotrifluoride (M68 and M71) and ethynyl derivative of diphenylamine-fluorene motif (M67 and M68) or ethynyl derivative of carbazole-carbazole moiety. The obtained compounds were purified by column chromatography, giving 99 $\%$ purity of the samples.
\begin{figure}[htp]
\center
\includegraphics[width=8.5cm]{Figure1_compression.pdf}
\caption{\label{fig:1}(Color online) Chemical structures of tested compounds.}
\end{figure}
\subsection{Dielectric spectroscopy at ambient pressure}
The isobaric dielectric measurements at ambient pressure were carried out using Novo-Control GmbH alpha impedance analyzer in the frequency range from $10^{-2}$ to $10^6$ Hz at various temperature conditions (329-353 K for M67, 326-371 K for M68, and for M71 320-386 K). The temperature was controlled by Quatro temperature controller using a nitrogen gas cryostat with temperature stability better than 0.1 K. The tested sample was placed between two stainless steel electrodes of a capacitor (20 mm diameter) with a fixed gap between electrodes (0.1 mm) provided by fused silica spacer fibers. The dielectric measurements of M67 and M68 were performed after the vitrification by fast cooling from melting point (430, and 425 K, respectively), while M71 was measured during slow cooling from 386 K.
\subsection{Dielectric spectroscopy at elevated pressure}
The isothermal dielectric measurements at elevated pressure were performed utilizing a high-pressure system with an MP5 micropump (Unipress) and an alpha impedance analyzer (Novocontrol GmbH). The pressure was controlled with an accuracy better than 1 MPa by an automatic pressure pump, the silicone oil was used as a pressure-transmitting fluid. The sample cell was the same as used during the measurements at ambient pressure (15 mm diameter of the capacitor, 0.1 distance between electrodes provided by Teflon spacer). To avoid contact between sample and pressure-transmitting fluid, the capacitor was placed in a Teflon ring and additionally wrapped by the Teflon tape. The temperature was controlled by Weiss Umwelttechnik GmbH fridge with the precision better than 0.1 K. The measurements were performed at 347 K (5-45 MPa) for M67, 338 K (0.1 to 28 MPa) for M68 and 338 K (0.1 to 28 MPa) for M71.
\section{Results and Discussion}
Representative dielectric spectra measured for M71 above the glass transition temperature are presented in Fig. \ref{fig:2}. As can be seen, the structural relaxation process and dc-conductivity (on the low-frequency flank of the $\alpha$-process) move towards lower frequencies with decreasing temperature (or with squeezing at isothermal condition). From analysis of the dielectric loss peak we obtained the relaxation time, $\tau_\alpha$, using the following definition: $\tau_\alpha=1/2\pi f_{max}$ , where $f_{max}$ is the maximum frequency of the structural relaxation peak. The $\log\tau_\alpha$ as a function of (i) inverse of temperature is presented in Fig. \ref{fig:4}, while (ii) $\log\tau_\alpha$ as a function of $P/P_g$ is depicted in Fig \ref{fig:5}.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Figure2_compression.pdf}
\caption{\label{fig:2}(Color online) The dielectric loss spectra of M71 measured above glass transition temperature at ambient pressure.}
\end{figure}
Figure \ref{fig:4} shows theoretical and experimental $\log_{10}\tau_\alpha$ of M67, M68, and M71 under atmospheric pressure ($P \approx 0$) as a function of $1000/T$. We use Eqs. (\ref{eq:6}), (\ref{eq:6-1}), and (\ref{eq:7}) to calculate the temperature dependence of $\tau_\alpha$. To obtain the quantitatively good accordance, we use $T_0=465$ $K$ and $a_c=4$ for M67, $T_0=499$ $K$ and $a_c=1$ for M68, and $T_0=524$ $K$ and $a_c=0.36$ for M71. Different chemical end groups cause the different relative importance of the collective elastic distortion and give various values of $a_c$. Overall, the ENCLE calculations agree quantitatively well with experimental data.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Graph134.pdf}
\caption{\label{fig:4}(Color online) Temperature dependence of structural relaxation time of M67, M68, and M71 under ambient pressure ($P \approx 0$). Open points are experimental data and solid curves correspond to our ECNLE calculations.}
\end{figure}
Under high compression effects, motion of particles has more constraint and the relaxation process is significantly slowed down. From the previous section, we know that the barrier height $F_B$ and jump distance $\Delta r=r_B-r_L$ are increased with a pressure rise. Thus, the collective barrier $F_e \sim K_0\Delta r^4$ also grows. For simplification, we assume that the correlation between local and collective molecular dynamics in substances does not change when applying pressure. In addition, the thermal expansion coefficient $\beta$ and the characteristic temperature $T_0$ are supposed to remain unchanged with pressure. The assumption allows us to calculate the pressure dependence of structural relaxation time. Since pressure entering to the dynamic free energy in Eq. (\ref{eq:pressure}) is in unit of $k_BT/d^3$, our numerical results can be compared to experimental data without introducing additional parameters by the pressure normalization.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Graph133.pdf}
\caption{\label{fig:5}(Color online) Logarithm of structural relaxation time of M67 at 347 $K$, M68 at 338 $K$, and M71 at 338 $K$ versus pressure normalized by $P_g$, which is defined by $\tau_\alpha(P_g)=1$s. Open points are experimental data and solid curves correspond to our ECNLE calculations.}
\end{figure}
Theoretical calculations and experimental data for $\log_{10}\tau_\alpha$ versus normalized pressure of our three materials in an isothermal condition are contrasted in Fig. \ref{fig:5}. At a fixed temperature, we use Eq. (\ref{eq:7}) to map from temperature to a packing fraction of the effective hard-sphere fluid in ECNLE calculations. Then, the pressure dependence of physical quantities for local dynamics and the alpha relaxation time are calculated using Eq. (\ref{eq:pressure}) when varying pressure. We define the glass transition pressure $P_g$ at $\tau_\alpha(P_g)=1s$ to normalize pressure. One observes a quantitatively good accordance between theory and experiment shown in Fig. \ref{fig:5}. This agreement suggests that our simple assumption of ignoring effects of chemical and biological structures seems plausible. We do not need to consider steric repulsion between molecules since the hard-sphere models are still applicable during compression. However, this simplicity may cause deviation between theory and experiment. Numerical results in Fig.\ref{fig:5} also reveal that our extended ECNLE theory is a predictive approach to investigate effects of pressure when only knowing parameters $T_0$ and $a_c$ from molecular mobility at ambient conditions.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Graph135.pdf}
\caption{\label{fig:6}(Color online) Logarithm of structural relaxation time of M67 at 347 $K$, M68 at 338 $K$, and M71 at 338 $K$ versus pressure in unit of MPa. Open points are experimental data and solid curves correspond to our ECNLE calculations.}
\end{figure}
To compare with experiment in real unit of pressure (MPa), we establish an equality between the theoretical and experiment $P_g$ to calculate the particle diameter. Results are $d = 0.434$ nm for M67, $d = 0.567$ nm for M68, and $d = 0.575$ nm for M71, respectively. Experimental data and theoretical calculations for the pressure dependence of $\tau_\alpha$ of our three pure amorphous materials in isothermal processes are shown in Figure \ref{fig:6}. One can see better quantitative consistency between theory and experiment than in Fig. \ref{fig:5} since $d$ is fixed and calculated at $P = P_g$. At high-pressure regime, molecules are incompressible while at low pressures (and/or ambient condition), molecules are internally relaxed and their volume becomes relatively larger. The curves of ECNLE calculations are slightly above those of experimental data. The theory-experiment deviation becomes more important at low compression.
Obviously, there is no universal way to determine $d$. If the diameter $d$ is calculated at a low pressure regime, the behavior is reversed and theoretical predictions deviate from experiment at high pressures. These results clearly indicate that the external pressure not only reduces the free volume, but also change the molecular size. All factors change the packing fraction $\Phi$. In Fig. \ref{fig:7}a, we show the temperature or density dependence of $\tau_\alpha$ for a representative material (M71) under various pressure conditions. Increasing the packing fraction $\Phi$ and compression slows down the molecular dynamics in the same manner. The shrinking-down process of molecules under large compression can be quantified by tuning the value of $d$ to obtain the best quantitative fit between theoretical and experimental $\log_{10}\tau_\alpha(P)$.
\begin{figure}[htp]
\center
\includegraphics[width=9cm]{Graph188.pdf}
\includegraphics[width=9cm]{Graph189.pdf}
\caption{\label{fig:7}(Color online) (a) Logarithm of structural relaxation time of M71 at different external pressures. A horizontal blue dashed line indicates a vitrification time scale
criterion of 1 $s$. (b) The pressure dependence of the glass transition temperature of M71. The inset shows the theoretical fragility plotted versus external pressures in unit of $k_BT/d^3$.}
\end{figure}
Based on theoretical calculations in Fig. \ref{fig:7}a, one can determine $T_g(P)$ defined as $\tau_\alpha(T_g)=1$ s and the dynamic fragility of M71
\begin{eqnarray}
m = \left. \frac{\partial\log_{10}(\tau_\alpha)}{\partial(T_g/T)}\right |_{T=T_g}.
\label{eq:fragility}
\end{eqnarray}
Numerical results are shown in Fig. \ref{fig:7}b. Generically, both $T_g$ and $m$ increase with compression. It means this glass former becomes more fragile at elevated pressure. In the ECNLE theory, the higher fragility corresponds to more collective elasticity or greater effects of collective motions on the glass transition \cite{8,9}. This finding is consistent with prior simulations \cite{71,73} and experiments \cite{72,74}. We can explain this behavior using a nontrivial correlation among the cooling rate ($h$), glass transition temperature, and dynamic fragility \cite{9}
\begin{eqnarray}
h\tau_\alpha(T_g) = \frac{T_g}{m\ln(10)}.
\label{eq:18}
\end{eqnarray}
Since $h\tau_\alpha(T_g)$ is a constant, $m$ monotonically vary with $T_g$. Consequently, at a fixed temperature, the pressure-induced slowing down of the relaxation time shifts $T_g$ towards a larger value and causes an increase of $m$. We emphasize that this analysis can be changed if glass-forming liquids have strong electrostatic interactions and chemical/biological complexities.
\section{Conclusions}
We have developed the ECNLE theory of bulk relaxation to capture the pressure effects on the glass transition of glass-forming liquids. Amorphous materials are described as a hard sphere fluid. Under compression condition, a mechanical work done by the pressure acting on a tagged particle modifies its the dynamic free energy. The free energy profile provides the pressure dependence of key physical quantities of the local dynamics by only considering nearest-neighbor interactions. The localization length is slightly reduced with increasing pressure, while the barrier position and local-barrier height grows. These variations in the ultra-local limit (high densities or low temperatures) have been analytically analyzed. Our calculations indicate that further restrictions apply to the local dynamics. It leads to a significantly slowing-down of molecular mobility when applying pressure. The validity of our theoretical approach has been supported by dielectric spectroscopy experiments. We measured the dielectric loss spectra of three different materials to determine the alpha structural relaxation time at ambient and elevated pressures over a wide range of temperature. Our theoretical temperature and pressure dependence of the structural relaxation time quantitatively agree with experimental data.
\begin{acknowledgments}
This work was supported by JSPS KAKENHI Grant Numbers JP19F18322 and JP18H01154. M. Paluch is deeply grateful for the financial support by the National Science Centre within the framework of the Maestro10 project (grant no UMO- 2018/30/A/ST3/00323). This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2019.318.
\end{acknowledgments}
\section*{Conflicts of interest}
There are no conflicts to declare.
|
2,877,628,090,203 | arxiv | \section{Introduction}
\subsection{Background}
In many scientific problems, the parameter of interest is a component-wise monotone function. In practice, an estimator of this function may have several desirable statistical properties, yet fail to be monotone. This often occurs when the estimator is obtained through the pointwise application of a statistical procedure over the domain of the function. For instance, we may be interested in estimating a conditional cumulative distribution function $\theta_0$, defined pointwise as $\theta_0(a,y)=P_0(Y \leq y \mid A = a)$, over its domain $\mathscr{D}\subset \mathbb{R}^2$. Here, $Y$ may represent an outcome and $A$ an exposure. The map $y \mapsto \theta_0(a,y)$ is necessarily monotone for each fixed $a$. In some scientific contexts, it may be known that $a \mapsto \theta_0(a,y)$ is also monotone for each $y$, in which case $\theta_0$ is a bivariate component-wise monotone function. An estimator of $\theta_0$ can be constructed by estimating the regression function $(a,y) \mapsto E_{P_0} \left[ I(Y \leq y) \mid A = a\right]$ for each $(a,y)$ on a finite grid using kernel smoothing, and performing suitable interpolation elsewhere. For some types of kernel smoothing, including the Nadaraya-Watson estimator, the resulting estimator is necessarily monotone as a function of $y$ for each value of $a$, but not necessarily monotone as a function of $a$ for each value of $y$. For other types of kernel smoothing, including the local linear estimator, which often has smaller asymptotic bias than the Nadaraya-Watson estimator, the resulting estimator need not be monotone in either component.
Whenever the function of interest is component-wise monotone, failure of an estimator to itself be monotone can be problematic. This is most apparent if the monotonicity constraint is probabilistic in nature -- that is, the parameter mapping is monotone under all possible probability distributions. This is the case, for instance, if $\theta_0$ is a distribution function. In such settings, returning a function estimate that fails to be monotone is nonsensical, like reporting a probability estimate outside the interval $[0,1]$. However, even if the monotonicity constraint is based on scientific knowledge rather than probabilistic constraints, failure of an estimator to be monotone can be an issue. For example, if the parameter of interest represents average height or weight among children as a function of age, scientific collaborators would likely be unsatisfied if presented with an estimated curve that were not monotone. Finally, as we will see, there are often finite-sample performance benefits to ensuring that the monotonicity constraint is respected.
Whenever this phenomenon occurs, it is natural to seek an estimator that respects the monotonicity constraint but nevertheless remains close to the initial estimator, which may otherwise have good statistical properties. A monotone estimator can be naturally constructed by projecting the initial estimator onto the space of monotone functions with respect to some norm. A common choice is the $L_2$-norm, which amounts to using multivariate isotonic regression to correct the initial estimator.
\subsection{Contribution and organization of the article}
In this article, we discuss correcting an initial estimator of a multivariate monotone function by computing the isotonic regression of the estimator over a finite grid in the domain, and interpolating between grid points. We also consider correcting an initial confidence band by using the same procedure applied to the upper and lower limits of the band. We provide three general results regarding this simple procedure.
\begin{enumerate}
\item Building on the results of \cite{robertson1988order} and \cite{chernozhukov2009rearrangement}, we demonstrate that the corrected estimator is at least as good as the initial estimator, meaning:
\begin{enumerate}[(a)]
\item its uniform error over the grid used in defining the projection is less than or equal to that of the initial estimator for every sample;
\item its uniform error over the entire domain is less than or equal to that of the initial estimator asymptotically;
\item the corrected confidence band contains the true function on the projection grid whenever the initial band does, at no cost in terms of average or uniform band width.
\end{enumerate}
\item We provide high-level sufficient conditions under which the uniform difference between the initial and corrected estimators is $o_P(r_n^{-1})$ for a generic sequence $r_n \longrightarrow \infty$.
\item We provide simpler lower-level sufficient conditions in two special cases:
\begin{enumerate}[(a)]
\item when the initial estimator is uniformly asymptotically linear, in which case the appropriate rate is $r_n = n^{1/2}$;
\item when the initial estimator is kernel-smoothed with bandwidth $h_n$, in which case the appropriate rate is $r_n = (n h_n)^{1/2}$ for univariate kernel smoothing.
\end{enumerate}
\end{enumerate}
We apply our theoretical results to two sets of examples: nonparametric efficient estimation of a G-computed distribution function for a binary exposure, and local linear estimation of a conditional distribution function with a continuous exposure.
Other authors have considered the correction of an initial estimator using isotonic regression. To name a few, \cite{mukarjee1994} used a projection-like procedure applied to a kernel smoothing estimator of a regression function, whereas \cite{patra2016} used the projection procedure applied to a univariate cumulative distribution function in the context of a mixture model. These articles addressed the properties of the projection procedure in their specific applications. In contrast, we provide general results that are applicable broadly.
\subsection{Alternative projection procedures}
The projection approach is not the only possible correction procedure. \cite{dette2006}, \cite{chernozhukov2009rearrangement}, and \cite{chernozhukov2010quantile} studied a correction based on monotone rearrangements. However, monotone rearrangements do not generalize to the multivariate setting as naturally as projections -- for example, \cite{chernozhukov2009rearrangement} proposed averaging a variety of possible multivariate monotone rearrangements to obtain a final monotone estimator. In contrast, the $L_2$ projection of an initial estimator onto the space of monotone functions is uniquely defined, even in the context of multivariate functions.
\cite{daouia2012isotonic} proposed an alternative correction procedure that consists of taking a convex combination of upper and lower monotone envelope functions, and they demonstrated conditions under which their estimator is asymptotically equivalent in supremum norm to the initial estimator. There are several differences between our contributions and those of \cite{daouia2012isotonic}. For instance, \cite{daouia2012isotonic} did not study correction of confidence bands, which we consider in Section~\ref{confidence}, or the important special case of asymptotically linear estimators, which we consider in Section~\ref{linear}. Our results in these two sections apply equally well to our correction procedure and to the correction procedure considered by \cite{daouia2012isotonic}.
Perhaps the most important theoretical contribution of our work beyond that of existing research is the weaker form of stochastic equicontinuity that we require for establishing asymptotic equivalence of the initial and projected estimators. In contrast, \cite{daouia2012isotonic} explicitly required the usual uniform asymptotic equicontinuity, while application of the Hadamard differentiability results of \cite{chernozhukov2010quantile} requires weak convergence to a tight limit, which is stronger than uniform asymptotic equicontinuity. Our weaker condition allows us to use our general results to tackle a broader range of initial estimators, including kernel smoothed estimators, which are typically not uniformly asymptotically equicontinuous at useful rates, but nevertheless can frequently be shown to satisfy our condition. We discuss this in detail in Section~\ref{sec:kern}. We illustrate this general contribution in Section~\ref{cond_dist} by studying the bivariate correction of a conditional distribution function estimated using local linear regression, which would not be possible using the stronger asymptotic equicontinuity condition. In numerical studies, we find that the projected estimator and confidence bands can offer substantial finite-sample improvements over the initial estimator and bands in this example.
\section{Main results}\label{monotone}
\subsection{Definitions and statistical setup}
Let $\s{M}$ be a statistical model of probability measures on a probability space $(\s{X}, \s{B})$. Let $\theta : \s{M} \to \ell^{\infty}(\s{T})$ be a parameter of interest on $\mathscr{M}$, where $\s{T}:= [0,1]^d$ and $\ell^{\infty}(\s{T})$ is the Banach space of bounded functions from $\s{T}$ to $\d{R}$ equipped with supremum norm $\|\cdot\|_{\s{T}}$. We have specified this particular $\s{T}$ for simplicity, but the results established here apply to any bounded rectangular domain $\s{T}\subset \d{R}^d$. For each $P \in \s{M}$, denote by $\theta_P$ the evaluation of $\theta$ at $P$ and note that $\theta_P$ is a bounded real-valued function on $\s{T}$. For any $t \in \s{T}$, denote by $\theta_P(t) \in \d{R}$ the evaluation of $\theta_P$ at $t$.
For any vector $t\in\d{R}^d$ and $1\leq j\leq d$, denote by $t_j$ the $j^{th}$ component of $t$. Define the partial order $\leq$ on $\d{R}^d$ by setting $t \leq t'$ if and only if $t_j \leq t_j'$ for each $1 \leq j \leq d$. A function $f: \d{R}^d \to \d{R}$ is called (component-wise) monotone non-decreasing if $t \leq t'$ implies that $f(t) \leq f(t')$. Denote $\|t\| = \max_{1 \leq j \leq d} |t_j|$ for any vector $t \in \d{R}^d$. Additionally, denote by $\bs\Theta \subset \ell^{\infty}(\s{T})$ the convex set of bounded monotone non-decreasing functions from $\s{T}$ to $\d{R}$. For concreteness, we focus on non-decreasing functions, but all results established here apply equally to non-increasing functions
Let $\mathscr{M}_0:=\{P\in\mathscr{M}:\theta_P\in\bs{\Theta}\}\subseteq \s{M}$ and suppose that $\mathscr{M}_0$ is nonempty. Generally, this inclusion is strict only if, rather than being implied by the rules of probability, the monotonicity constraint stems at least in part from prior scientific knowledge. Also, define $\bs\Theta_0 := \{ \theta \in \bs\Theta : \theta = \theta_P\mbox{ for some }P \in \mathscr{M}\} \subseteq \bs\Theta$. We are primarily interested in settings where $\bs\Theta_0 = \bs\Theta$, since in this case there is no additional knowledge about $\theta$ encoded by $\s{M}$, and in particular there is no danger of yielding a corrected estimator that is compatible with no $P \in \s{M}$.
Suppose that observations $X_1,X_2,\ldots,X_n$ are sampled independently from an unknown distribution $P_0\in\s{M}_0$, and that we wish to estimate $\theta_0:=\theta_{P_0}$ based on these observations. Suppose that, for each $t \in \s{T}$, we have access to an estimator $\theta_n(t)$ of $\theta_0(t)$ based on $X_1,X_2,\ldots,X_n$. We note that the assumption that the data are independent and identically distributed is not necessary for Theorems~\ref{thm:barlow} and~\ref{monotone_supnorm} below. For any suitable $f:\s{X}\rightarrow \mathbb{R}$, we define $Pf := \int f(x) \, P(dx)$ and $\mathbb{G}_nf := n^{1/2}\int f(x)(\d{P}_n-P_0)(dx)$, where $\d{P}_n$ is the empirical distribution based on $X_1, X_2,\dotsc, X_n$.
The central premise of this article is that $\theta_n(t)$ may have desirable statistical properties for each $t$ or even uniformly in $t$, but that $\theta_n$ as an element of $\ell^{\infty}(\s{T})$ may not fall in $\bs\Theta$ for any finite $n$ or even with probability tending to one. Our goal is to provide a corrected estimator $\theta_n^*$ that necessarily falls in $\bs\Theta$, and yet retains the statistical properties of $\theta_n$. A natural way to accomplish this is to define $\theta_n^*$ as the closest element of $\bs\Theta$ to $\theta_n$ in some norm on $\s{T}$. Ideally, we would prefer to take $\theta_n^*$ to minimize $\| \theta-\theta_n\|_{\s{T}}$ over $\theta \in \bs\Theta$. However, this is not tractable for two reasons. First, optimization over the entirety of $\s{T}$ is an infinite-dimensional optimization problem, and is hence frequently computationally intractable. To resolve this issue, for each $n$, we let $\s{T}_n = \{t_1,t_2, \dotsc, t_{m_n}\} \subseteq \s{T}$ be a finite rectangular lattice in $\s{T}$ over which we will perform the optimization, and define and consider $\|\cdot\|_{\s{T}_n}$ as the supremum norm over $\s{T}_n$. While it is now computationally feasible to define $\theta_{n,\infty}^*$ as a minimizer over $\theta \in \bs\Theta$ of the finite-dimensional objective function $\| \theta - \theta_n\|_{\s{T}_n}$, this objective function is challenging due to its non-differentiability. Instead, we define
\begin{equation} \theta_n^* \in \argmin_{\theta \in \bs\Theta} \sum_{t \in \s{T}_n} \left[\theta(t) -\theta_n(t)\right]^2\ .\label{projection}\end{equation}
The squared-error objective function is smooth in its arguments. In dimension $d=1$, $\theta_n^*$ thus defined is simply the isotonic regression of $\theta_n$ on the grid $\s{T}_n$, which has a closed-form representation as the greatest convex minorant of the so-called cumulative sum diagram. Furthermore, since $\| \theta_n^* - \theta_n\|_{\s{T}_n} \geq \| \theta_{n,\infty}^* - \theta_n\|_{\s{T}_n}$, many of our results also apply to $\theta_{n,\infty}^*$.
We note that $\theta_n^*$ is only uniquely defined on $\s{T}_n$. To completely characterize $\theta_n^*$, we must monotonically interpolate function values between elements of $\s{T}_n$. We will permit any monotonic interpolation that satisfies a weak condition. By the definition of a rectangular lattice, every $t \in \s{T}$ can be assigned a hyper-rectangle whose vertices $\{s_1, s_2\dotsc, s_{2^d}\}$ are elements of $\s{T}_n$ and whose interior has empty intersection with $\s{T}_n$. If multiple such hyper-rectangles exist for $t$, such as when $t$ lies on the boundary of two or more such hyper-rectangles, one can be assigned arbitrarily. We will assume that, for $t \notin \s{T}_n$, $\theta_n^*(t) = \sum_k \lambda_{k,n}(t) \theta_n^*(s_k)$ for weights $ \lambda_{1,n}(t),\lambda_{2,n}(t),\ldots,\lambda_{2^d,n}(t)\in(0,1)$ such that $\sum_k \lambda_{k,n}(t) = 1$. In words, we assume that $\theta_n^*(t)$ is a convex combination of the values of $\theta_n^*$ on the vertices of the hyper-rectangle containing $t$. A simple interpolation approach consists of setting $\theta_n^*(t) = \theta_n^*(t')$ with $t'$ the element of $\s{T}_n$ closest to $t$, and choosing any such element if there are multiple elements of $\s{T}_n$ equally close to $t$. This particular scheme satisfies our requirement.
Finally, for each $n$, we let $\ell_n(t) \leq u_n(t)$ denote lower and upper endpoints of a confidence band for $\theta_0(t)$. We then define $\ell_n^*$ and $u_n^*$ as the corrected versions of $\ell_n$ and $u_n$ using the same projection and interpolation procedure defined above for obtaining $\theta_n^*$ from $\theta_n$.
In dimension $d = 1$, $\theta_n^*(t)$, $\ell_n^*(t)$, and $u_n^*(t)$ can be obtained for $t \in \s{T}_n$ via the Pool Adjacent Violators Algorithm, as implemented in the \texttt{R} command \texttt{isoreg} \citep{Rlang}. In dimension $d = 2$, the corrections can be obtained using the algorithm described in \cite{bril1984bivariate}, which is implemented in the \texttt{R} command \texttt{biviso} in the package \texttt{Iso} \citep{Isopackage}. In dimensions $d \geq 3$, no tailored algorithm for computation of the isotonic regression estimate yet exists to our knowledge. However, general-purpose algorithms for minimization of quadratic criteria over convex cones have been developed an implemented in the \texttt{R} package \texttt{coneproj} and may be used in this case \citep{meyer1999cone, coneproj}.
\subsection{Properties of the projected estimator}
The projected estimator $\theta_n^*$ is the isotonic regression of $\theta_n$ over the grid $\s{T}_n$. Hence, many existing finite-sample results on isotonic regression can be used to deduce properties of $\theta_n^*$. Theorem~\ref{thm:barlow} below collects a few of these properties, building upon the results of \cite{barlow1972order} and \cite{chernozhukov2009rearrangement}. We denote $\omega_n := \sup_{t \in \s{T}} \min_{s\in\s{T}_n} \| t - s\|$ as the mesh of $\s{T}_n$ in $\s{T}$.
\begin{theorem}\label{thm:barlow}
\begin{itemize}
\item[(i)] It holds that $\|\theta_n^* - \theta_0\|_{\s{T}_n} \leq \|\theta_n - \theta_0\|_{\s{T}_n}$.
\item[(ii)] If $\omega_n = o_P(1)$ and $\theta_0$ is continuous on $\s{T}$, then $\| \theta_n^* - \theta_0\|_{\s{T}} \leq \| \theta_n - \theta_0\|_{\s{T}} + o_P(1)$.
\item[(iii)] If there exists some $\alpha>0$ for which $\sup_{s,t\in\s{T}:\|t - s\| \leq \delta} |\theta_0(t) - \theta_0(s)| = o(\delta^\alpha)$ as $\delta \to 0$, then $\| \theta_n^* - \theta_0\|_{\s{T}} \leq \|\theta_n - \theta_0\|_{\s{T}} + o_P(\omega_n^{\alpha})$.
\item[(iv)] Whenever $\theta_0(t)\in[\ell_n(t),u_n(t)]$ for all $t\in\s{T}_n$, $\theta_0(t) \in [\ell_n^*(t), u_n^*(t)]$ for all $t\in\s{T}_n$.
\item[(v)] It holds that $\sum_{t \in \s{T}_n} [ u_n^*(t) - \ell_n^*(t)] = \sum_{t \in \s{T}_n} [ u_n(t) - \ell_n(t)]$ and $\| u_n^* - \ell_n^* \|_{\s{T}_n} \leq \|u_n - \ell_n \|_{\s{T}_n}$.
\end{itemize}
\end{theorem}
Before presenting the proof of Theorem~\ref{thm:barlow}, we remark briefly on its implications. Part (i) says that the estimation error of $\theta_n^*$ over the grid $\s{T}_n$ is never worse than that of $\theta_n$, whereas parts (ii) and (iii) say that the estimation error of $\theta_n^*$ on all of $\s{T}$ is asymptotically no worse than the estimation error of $\theta_n$ in supremum norm. Similarly, part (iv) says that the isotonized band $[\ell_n^*, u_n^*]$ never has worse coverage than the original band over $\s{T}_n$. Finally, part (v) says that the potential increase in coverage comes at no cost to the average or supremum width of the bands over $\s{T}_n$. We note that parts (i), (iv) and (v) hold true for each $n$.
While comprehensive in scope, Theorem~\ref{thm:barlow} does not rule out the possibility that $\theta_n^*$ performs strictly better, even asymptotically, than $\theta_n$, or that the band $[\ell_n^*, u_n^*]$ is asymptotically strictly more conservative than $[\ell_n, u_n]$. In order to construct confidence intervals or bands with correct asymptotic coverage, a stronger result is needed: it must be that $\|\theta_n^* - \theta_n\|_{\s{T}} = o_P(r_n^{-1})$, where $r_n$ is a diverging sequence such that $r_n \|\theta_n - \theta_0\|_{\s{T}}$ converges in distribution to a non-degenerate limit distribution. Then, we would have that $r_n\|\theta_n^* - \theta_0\|_{\s{T}}$ converges in distribution to this same limit, and hence confidence bands constructed using approximations of this limit distribution would have correct coverage when centered around $\theta_n^*$, as we discuss more below.
We consider the following conditions on $\theta_0$ and the initial estimator $\theta_n$:
\begin{description}
\item[(A)] there exists a deterministic sequence $r_n$ tending to infinity such that, for all $\delta>0$,
\[\sup_{\|t -s\| < \delta/r_n} \left|r_n\left[\theta_n(t) - \theta_0(t)\right] - r_n\left[\theta_n(s) - \theta_0(s)\right] \right| \inproblow 0;\]
\item[(B)] there exists $K_1 < \infty$ such that $|\theta_0(t) - \theta_0(s)| \leq K_1\|t-s\|$ for all $t,s\in \s{T}$;
\item[(C)] there exists $K_0 > 0$ such that $K_0\|t-s\| \leq |\theta_0(t) - \theta_0(s)|$ for all $t,s\in \s{T}$.
\end{description}
Based on these conditions, we have the following result.
\begin{theorem}\label{monotone_supnorm}
If conditions (A)--(C) hold and $\omega_n = o_P(r_n^{-1})$, then $\|\theta_n^* - \theta_n\|_{\s{T}} = o_P(r_n^{-1})$.
\end{theorem}
This result indicates that the projected estimator is uniformly asymptotically equivalent to the original estimator in supremum norm at the rate $r_n$.
Condition (A) is related to, but notably weaker than, uniform stochastic equicontinuity \citep[p.\ 37]{van1996weak}. (A) follows if, in particular, the process $\{r_n [\theta_n(t) - \theta_0(t)] : t\in \s{T}\}$ converges weakly to a tight limit in the space $\ell^{\infty}(\s{T})$. However, the latter condition is sufficient but not necessary for (A) to hold. This is important for application of our results to kernel smoothing estimators, which typically do not converge weakly to a tight limit, but for which condition (A) nevertheless often holds. We discuss this at length in Section~\ref{cond_dist}. The results of \cite{daouia2012isotonic} (see in particular condition (C3) therein) and \cite{chernozhukov2010quantile} rely on uniform stochastic equicontinuity in demonstrating asymptotic equivalence of their correction procedures, which essentially limits the applicability of their procedures to estimators that converge weakly to a tight limit in $\ell^{\infty}(\s{T})$.
Condition (B) constrains $\theta_0$ to be Lipschitz. Condition (C) constrains the variation of $\theta_0$ from below, and is slightly more restrictive than a requirement for strict monotonicity. If, for instance, $\theta_0$ is differentiable, then (C) is satisfied if all first-order partial derivatives of $\theta_0$ are bounded away from zero. Condition (C) excludes, for instance, situations in which $\theta_0$ is differentiable with null derivative over an interval. In such cases, $\theta_n^*$ may have strictly smaller variance on these intervals than $\theta_n$ because $\theta_n^*$ will pool estimates across the flat region while $\theta_n$ may not. Hence, in such cases, $\theta_n^*$ may potentially asymptotically improve on $\theta_n$, so that $\theta_n^*$ and $\theta_n$ are not asymptotically equivalent at the rate $r_n$. Theoretical results in these cases would be of interest, but are beyond the scope of this article. In addition to conditions (A)--(C), Theorem~\ref{monotone_supnorm} requires that the mesh $\omega_n$ of $\s{T}_n$ tend to zero in probability faster than $r_n^{-1}$. Since $\s{T}_n$ is chosen by the user, this is not a problem in practice.
We prove Theorem~\ref{monotone_supnorm} via three lemmas, which may be of interest in their own right. The first lemma controls the size of deviations in $\theta_n$ over small neighborhoods, and does not hinge on condition (C) holding.
\begin{lemma}
If (A)--(B) hold and $b_n = o_P(r_n^{-1})$, then $\displaystyle \sup_{\|t-s\| \leq b_n} \left| \theta_n(t) - \theta_n(s)\right| = o_P(r_n^{-1})$.
\label{lemma:moduli_of_continuity}
\end{lemma}
The second lemma controls the size of neighborhoods over which violations in monotonicity can occur. Henceforth, we define $\kappa_n := \sup\left\{\|t-s\| : s, t \in \s{T}, s \leq t, \theta_n(t) \leq \theta_n(s)\right\}.$ In this lemma we again require condition (A), but now require (C) rather than (B).
\begin{lemma}\label{neighborhoods}
If conditions (A) and (C) hold, then $\kappa_n = o_P(r_n^{-1})$.
\end{lemma}
Our final lemma bounds the maximal absolute deviation between $\theta_n^*$ and $\theta_n$ over the grid $\s{T}_n$ in terms of the supremal deviations of $\theta_n$ over neighborhoods smaller than $\kappa_n$. This lemma does not depend on any of the conditions (A)--(C).
\begin{lemma}
The inequality $\max_{t \in \s{T}_n} |\theta_n^*(t) - \theta_n(t)| \leq \sup_{\|s- t\| \leq \kappa_n} |\theta_n(s) - \theta_n(t)|$ holds.
\label{pava}
\end{lemma}
The proof of Theorem~\ref{monotone_supnorm} follows easily from Lemmas~\ref{lemma:moduli_of_continuity},~\ref{neighborhoods}, and~\ref{pava}. The proof of these Lemmas dn Theorem~\ref{monotone_supnorm} are presented in Appendix~B.
\subsection{Construction of confidence bands}\label{confidence}
Suppose there exists a fixed function $\gamma_{\alpha} : \s{T} \to \d{R}$ such that $\ell_n$ and $u_n$ satisfy:
\begin{description}
\item[(a)] $\| r_n (\theta_n - \ell_n) - \gamma_\alpha \|_{\s{T}} \inproblow 0$,
\item[(b)] $\| r_n (u_n - \theta_n) - \gamma_\alpha \|_{\s{T}} \inproblow 0$,
\item[(c)] $P_0\left( r_n | \theta_n(t) - \theta_0(t)| \geq \gamma_\alpha(t)\mbox{ for all }t \in \s{T}\right)\longrightarrow 1 - \alpha$.
\end{description} As an example of a confidence band that satisfies conditions (a)--(c), suppose that $\sigma_0:\s{T}\rightarrow (0,+\infty)$ is a scaling function and $c_{\alpha}$ is a fixed constant such that, as $n$ tends to infinity,
\[P_0\left(r_n \left\| \frac{\theta_n - \theta_0}{\sigma_0} \right\|_{\s{T}} \geq c_{\alpha}\right)\longrightarrow 1 - \alpha\ .\]
If $\sigma_n$ is an estimator of $\sigma_0$ satisfying $\| \sigma_n -\sigma_0\|_{\s{T}} \inproblow 0$ and $c_{\alpha,n}$ is an estimator of $c_{\alpha}$ such that $c_{\alpha,n} \inproblow c_{\alpha}$, then the Wald-type band defined by lower and upper endpoints $\ell_n(t) := \theta_n(t) - c_{\alpha,n}r^{-1}_n\sigma_n(t)$ and $u_n(t) := \theta_n(t) + c_{\alpha}r^{-1}_n\sigma_n(t)$ satisfies (a)--(c) with $\gamma_\alpha = c_{\alpha} \sigma_0$. However, the latter conditions can also be satisfied by other types of bands, such as those constructed with a consistent bootstrap procedure.
Under conditions (a)--(c), the confidence band $[\ell_n,u_n]$ has asymptotic coverage $1-\alpha$. When conditions (A) and (B) also hold, the corrected band $[\ell_n^*, u_n^*]$ has the same asymptotic coverage as the original band $[\ell_n, u_n]$, as stated in the following result.
\begin{corollary}\label{cor:band}
If conditions (A)--(B) and (a)--(c) hold, $\gamma_\alpha$ is uniformly continuous on $\s{T}$, and $\omega_n = o_P(r_n^{-1})$, then the confidence band $[\ell_n^*, u_n^*]$ has asymptotic coverage $1-\alpha$.
\end{corollary}
The proof of Corollary~\ref{cor:band} is presented in Appendix~\ref{app:cor1}. We also note that Theorem~\ref{monotone_supnorm} immediately implies that Wald-type confidence bands constructed around $\theta_n$ have the same asymptotic coverage if they are constructed around $\theta_n^*$ instead.
\section{Refined results under additional structure}
In this section, we provide more detailed conditions that imply condition (A) in two special cases: when $\theta_n$ is asymptotically linear, and when $\theta_n$ is a kernel smoothing-type estimator.
\subsection{Special case I: asymptotically linear estimators}\label{linear}
Suppose that the initial estimator $\theta_n$ is uniformly asymptotically linear: for each $t\in \s{T}$, there exists $\phi_{0,t}:\s{X}\mapsto \mathbb{R}$ depending on $P_0$ such that $\int \phi_{0,t}(x)dP_0(x)=0$, $\int \phi^2_{0,t}(x)dP_0(x)<\infty$, and
\begin{equation}\label{asy_linear}
\theta_n(t) = \theta_0(t)+ \frac{1}{n}\sum_{i=1}^{n} \phi_{0,t}(X_i) + R_{n, t}
\end{equation}
for a remainder term $R_{n,t}$ with $n^{1/2}\sup_{t\in \s{T}} |R_{n,t}| = o_P(1)$. The function $\phi_{0,t}$ is the influence function of $\theta_n(t)$ under sampling from $P_0$. It is desirable for $\theta_n$ to have representation \eqref{asy_linear} because this immediately implies its uniform weak consistency as well as the pointwise asymptotic normality of $n^{1/2}\left[\theta_n(t) - \theta_0(t)\right]$ for each $t \in \s{T}$. If in addition the collection $\{\phi_{0,t} : t \in \s{T}\}$ of influence functions forms a $P_0$-Donsker class, $\{n^{1/2}\left[\theta_n(t) - \theta_0(t)\right] : t \in \s{T}\}$ converges weakly in $\ell^{\infty}(\s{T})$ to a Gaussian process with covariance function $\Sigma_0:(t,s)\mapsto\int \phi_{0,t}(x)\phi_{0,s}(x)dP_0(x)$. Uniform asymptotic confidence bands based on $\theta_n$ can then be formed by using appropriate quantiles from any suitable approximation of the distribution of the supremum of the limiting Gaussian process.
We introduce two additional conditions:
\begin{description}
\item[(A1)] the collection $\{\phi_{0,t} : t \in \s{T}\}$ of influence curves is a $P_0$-Donsker class;
\item[(A2)] $\Sigma_0$ is uniformly continuous in the sense that $\limsup_{\|t - s\| \to 0} |\Sigma_0(s, t) - \Sigma_0(t,t)| = 0.$
\end{description} Whenever $\theta_n$ is uniformly asymptotically linear, Theorem~\ref{monotone_supnorm} can be shown to hold under (A1), (A2) and (B), as implied by the theorem below. The validity of (A1) and (A2) can be assessed by scrutinizing the influence function $\phi_{0,t}$ of $\theta_n(t)$ for each $t \in \s{T}$. This fact renders the verification of these conditions very simple once uniform asymptotic linearity has been established.
\begin{theorem} For any estimator $\theta_n$ satisfying \eqref{asy_linear}, (A1) and (A2) together imply (A).
\label{thm:emp_process}
\end{theorem}
The proof of Theorem~\ref{thm:emp_process} is provided in Appendix~\ref{app:thm3}. In Section~\ref{survival}, we illustrate the use of Theorem~\ref{thm:emp_process} for the estimation of a G-computed distribution function.
We note that conditions (A1) and (A2) are actually sufficient to establish uniform asymptotic equicontinuity, which as discussed above is stronger than (A). Therefore, Theorem~\ref{thm:emp_process} can also be used to prove asymptotic equivalence of the majorization/minorization correction procedure studied in \cite{daouia2012isotonic}.
\subsection{Special case II: kernel smoothed estimators}\label{sec:kern}
For certain parameters, asymptotically linear estimators are not available. In particular, this is the case when the parameter of interest is not sufficiently smooth as a mapping of $P_0$. For example, density functions, regression functions, and conditional quantile functions do not permit asymptotically linear estimators in a nonparametric model when the exposure is continuous. In these settings, a common approach to nonparametric estimation is kernel smoothing.
Recent results suggest that, as a process, the only possible weak limit of $\{ r_n[\theta_n(t) - \theta_0(t)] : t \in \s{T}\}$ in $\ell^{\infty}(\s{T})$ may be zero when $\theta_n$ is a kernel smoothed estimator. For example, in the case of the Parzen-Rosenblatt estimator of a density function with bandwidth $h_n$, Theorem 3 of \cite{stupfler2016} implies that if $c_n := r_n \left(n h_n / |\log h_n|\right)^{-1/2} \to 0$, then $\{ r_n[\theta_n(t) - \theta_0(t)] : t \in \s{T}\}$ converges weakly to zero in $\ell^{\infty}(\s{T})$, whereas if $c_n \to c \in (0, \infty]$, then it does not converge weakly to a tight limit in $\ell^{\infty}(\s{T})$. As a result, $\{ r_n[\theta_n(t) - \theta_0(t)] : t \in \s{T}\}$ only satisfies uniform stochastic equicontinuity for $r_n$ such that $r_n \left(n h_n / |\log h_n|\right)^{-1/2} \to 0$. However, for any such rate $r_n$, $r_n^{-1}$ is slower than the pointwise and uniform rates of convergence of $\theta_n - \theta_0$. As a result, $\theta_n$ and $\theta_n^*$ may not be asymptotically equivalent at the uniform rate of convergence of $\theta_n - \theta_0$, so that confidence intervals and regions based on the limit distribution of $\theta_n - \theta_0$, but centered around $\theta_n^*$, may not have correct coverage. We note that, while \cite{stupfler2016} establishes formal results for the Parzen-Rosenblatt estimator, we expect that the results therein extend to a variety of kernel smoothed estimators.
As a result of the lack of uniform stochastic equicontinuity of $r_n(\theta_n - \theta_0)$ for useful rates $r_n$, establishing (A) is much more difficult for kernel smoothed estimators than for asymptotically linear estimators. However, since (A) is weaker than uniform stochastic equicontinuity, it may still be possible. Here, we provide alternative sufficient conditions that imply condition (A) and that we have found useful for studying a kernel smoothed estimator $\theta_n$.
When the initial estimator $\theta_n$ is kernel smoothed, we can frequently show that
\begin{equation}
\sup_{t \in \s{T}} \left| r_n \left[ \theta_n(t) - \theta_0(t)\right] - a_n b_0(t) - R_n(t) \right| \inprob 0\ , \label{eq:smooth}
\end{equation}
where $b_0 : \s{T} \to \d{R}$ is a deterministic bias, $a_n$ is sequences of positive constants, and $R_n :\s{T} \to \d{R}$ is a random remainder term. We then have
\begin{align*}
& \sup_{\| t - s\| < \delta / r_n} \left| r_n \left[ \theta_n(t) - \theta_0(t) \right] - r_n \left[ \theta_n(s) - \theta_0(s) \right] \right| \\
&\qquad= \sup_{\| t - s\| < \delta / r_n} a_n \left| b_0(t) - b_0(s) \right| +\sup_{\| t - s\| < \delta / r_n}\left| R_n(t) - R_n(s) \right| +o_P(1) \ .
\end{align*}
If $b_0$ is uniformly continuous on $\s{T}$ and $a_n = O(1)$, or $b_0$ is uniformly $\alpha$-H{\"o}lder on $\s{T}$ and $a_n = O\left( r_n^\alpha\right)$, then the first term on the right hand side tends to zero in probability. Attention may then be turned to demonstrating that the second term vanishes in probability. It appears difficult to provide a general characterization of the form of $R_n$ that encompasses kernel smoothed estimators. However, in our experience, it is frequently the case that $R_n(t)$ involves terms of the form $\d{G}_n \nu_{n,t}$, where $\nu_{n,t} : \s{X} \to \d{R}$ is a deterministic function for each $n \in \{1, 2, \dots\}$ and $t \in \s{T}$. In the course of demonstrating that $\sup_{\| t - s\| < \delta / r_n}\left| R_n(t) - R_n(s) \right| \inprob 0$, a rate of convergence for $\sup_{\| t - s\| < \delta / r_n}\left| \d{G}_n\left( \nu_{n,t} - \nu_{n,s}\right) \right|$ is then required. Defining $\s{F}_{n,\eta} := \{ \nu_{n,t} - \nu_{n,s} : \|t -s \| < \eta\}$ for each $\eta > 0$, this is equivalent to establishing a rate of convergence for the local empirical process $\| \d{G}_n \|_{\s{F}_{n,\delta / r_n}} := \sup_{\xi \in \s{F}_{n,\delta / r_n}} | \d{G}_n \xi|$. Such rates can be established using tail bounds for empirical processes. We briefly comment on two approaches to obtaining such tail bounds.
We first define bracketing and covering numbers of a class of functions $\s{F}$ -- see \cite{van1996weak} for a comprehensive treatment. We denote by $\| F\|_{P,2} = [ P( F^2)]^{1/2}$ the $L_2(P)$ norm of a given $P$-square-integrable function $F:\mathscr{X}\rightarrow\mathbb{R}$. The bracketing number $N_{[]}(\varepsilon, \s{F}, L_2(P))$ of a class of functions $\s{F}$ with respect to the $L_2(P)$ norm is the smallest number of $\varepsilon$-brackets needed to cover $\s{G}$, where an $\varepsilon$-bracket is any set of functions $\{ f: \ell \leq f \leq u\}$ with $\ell$ and $u$ such that $\|\ell-u\|_{P,2} < \varepsilon$. The covering number $N(\varepsilon, \s{F}, L_2(Q))$ of $\s{F}$ with respect to the $L_2(Q)$ norm is the smallest number of $\varepsilon$-balls in $L_2(Q)$ required to cover $\s{F}$. The uniform covering number is the supremum of $N(\varepsilon\|F\|_{2,Q}, \s{F}, L_2(Q))$ over all discrete probability measures $Q$ such that $\|F\|_{Q,2} > 0$, where $F$ is an envelope function for $\s{F}$. The bracketing and uniform entropy integrals for $\s{F}$ with respect to $F$ are then defined as
\begin{align*}
J_{[]}(\delta, \s{F}) &:= \int_0^{\delta} \left[ 1 + \log N_{[]}\left(\varepsilon \| F\|_{P_0,2}, \s{F}, L_2(P_0)\right) \right]^{1/2}\, d\varepsilon \\
J( \delta, \s{F}) &:= \sup_Q \int_0^{\delta} \left[ 1 + \log N\left(\varepsilon \| F\|_{Q,2}, \s{F}, L_2(Q)\right) \right]^{1/2} \, d\varepsilon \ .
\end{align*}
We discuss two approaches to controlling $\| \d{G}_n \|_{\s{F}_{n,\delta / r_n}}$ using these integrals. Suppose that $\s{F}_{n,\eta}$ has envelope function $F_{n,\eta}$ in the sense that $|\xi(x)| \leq F_{n,\eta}$ for all $\xi \in \s{F}_{n,\eta}$ and $x \in \s{X}$. The first approach is useful when $\| F_{n,\delta / r_n} \|_{P_0, 2}$ can be adequately controlled. Specifically, if either $J(1, \s{F}_{n, \delta/r_n})$ or $J_{[]}(1, \s{F}_{n, \delta/r_n})$ is $O(1)$, then $\| \d{G}_n \|_{\s{F}_{n,\delta / r_n}} \leq M_\delta \| F_{n,\delta / r_n} \|_{P_0, 2}$ for all $n$ and some constant $M_\delta \in (0, \infty)$ not depending on $n$ by Theorems 2.14.1 and 2.14.2 of \cite{van1996weak}.
The second approach we consider is useful when the envelope functions do not shrink in expectation, but the functions in $\s{F}_{n,\eta}$ still get smaller in the sense that $\gamma_{n, \delta} := \sup_{\xi \in \s{F}_{n,\delta / r_n}} \|\xi\|_{P_0,2}$ tends to zero. For example, if $\nu_{n,t}$ is defined as $\nu_{n,t}(x) := I(0 \leq x \leq t)$ for each $x \in \s{X} \subseteq \d{R}$, $t \in [0,1]$, and $n$, then $F_{n,\eta} : x \mapsto I(0 \leq x \leq 1)$ is the natural envelope function for $\s{F}_{n,\eta}$ for all $n$ and $\eta$, so that $\| F_{n,\delta / r_n} \|_{P_0, 2}$ does not tend to zero. However, $\gamma_{n,\delta} \leq \left(\bar{p}_0 \delta / r_n\right)^{1/2}$ if the density $p_0$ corresponding to $P_0$ is bounded above by $\bar{p}_0$, which does tend to zero. In these cases, the basic tail bounds in Theorem 2.14.1 and 2.14.2 of \cite{van1996weak} are too weak. Sharper, but slightly more complicated, bounds may be used instead. Specifically, if $F_{n, \delta / r_n} \leq C < \infty$ for all $n$ large enough and either
\begin{align*}
J\left( \gamma_{n, \delta}, \s{F}_{n,\delta / r_n}\right) + \frac{J\left( \gamma_{n, \delta}, \s{F}_{n,\delta / r_n}\right)^2}{\gamma_{n, \delta}^2 n^{1/2}} \, \text{ or } \, J_{[]}\left( \gamma_{n, \delta}, \s{F}_{n,\delta / r_n}\right) + \frac{J_{[]}\left( \gamma_{n, \delta}, \s{F}_{n,\delta / r_n}\right)^2}{\gamma_{n, \delta}^2 n^{1/2}}
\end{align*}
are $o\left(z_n^{-1} \right)$, then $\| \d{G}_n \|_{\s{F}_{n,\delta / r_n}} = o_P\left(z_n^{-1}\right)$ by Lemma 3.4.2 of \cite{van1996weak} and Theorem 2.1 of \cite{vandervaart2011}. Analogous statements hold if these expressions are $O\left(z_n^{-1} \right)$.
In some cases, both of these approaches must be used to control different terms arising within $R_n(t)$, as for the conditional distribution function discussed in Section~\ref{cond_dist}.
\section{Illustrative examples}
\subsection{Example 1: Estimation of a G-computed distribution function}\label{survival}
We first demonstrate the use of Theorem~\ref{thm:emp_process} in the particular problem in which we wish to draw inference on a G-computed distribution function. Suppose that the data unit is the vector $X=(Y, A, W)$, where $Y$ is an outcome, $A \in \{0,1\}$ is an exposure, and $W$ is a vector of baseline covariates. The observed data consist of independent draws $X_1,X_2,\ldots,X_n$ from $P_0\in\mathscr{M}$, where $\mathscr{M}$ is a nonparametric model.
For $P\in\mathscr{M}$ and $a_0 \in \{0,1\}$, we define the parameter value $\theta_{P,a_0}$ pointwise as $\theta_{P,a_0}(t) := E_P\left\{ P\left( Y \leq t \mid A = a_0, W\right)\right\}$, the G-computed distribution function of $Y$ evaluated at $t$, where the outer expectation is over the marginal distribution of $W$ under $P$. We are interested in estimating $\theta_{0,a_0} :=\theta_{P_0, a_0}$. This parameter is often of interest as an interpretable marginal summary of the relationship between $Y$ and $A$ accounting for the potential confounding induced by $W$. Under certain causal identification conditions, $\theta_{0, a_0}$ is the distribution function of the counterfactual outcome $Y(a_0)$ defined by the intervention that deterministically sets exposure to $A=a_0$ \citep{robins1986,gill2001}.
For each $t$, the parameter $P\mapsto \theta_{P,a_0}(t)$ is pathwise differentiable in a nonparametric model, and its nonparametric efficient influence function at $P\in\mathscr{M}$ is given by \[\varphi_{P,a_0,t}(y, a, w):=\frac{I(a = a_0)}{g_P(a_0 \mid w)}\left[I(y\leq t)-\bar{Q}_{P}(t \mid a_0, w)\right]+\bar{Q}_P(t \mid a_0, w)-\theta_{P,a_0}(t)\ ,\] where $g_P( a_0 \mid w):=P(A=a_0\mid W=w)$ is the propensity score and $\bar{Q}_P(t \mid a_0, w):=P\left(Y\leq t \mid A=a_0,W=w\right)$ is the conditional exposure-specific distribution function, as implied by $P$ \citep{van2003unified}. Given estimators $g_n$ and $\bar{Q}_n$ of $g_0:=g_{P_0}$ and $\bar{Q}_0:=\bar{Q}_{P_0}$, respectively, several approaches can be used to construct, for each $t$, an asymptotically linear estimator of $\theta_0(t)$ with influence function $\phi_{0,a_0,t}=\varphi_{P_0,a_0,t}$. For example, the use of either optimal estimating equations or the one-step correction procedure leads to the doubly-robust augmented inverse-probability-of-weighting estimator \[\theta_{n,a_0}(t) := \frac{1}{n}\sum_{i=1}^{n}\frac{I(A_i = a_0)}{g_n(a_0 \mid W_i)}\left[I(Y_i\leq t)-\bar{Q}_n(t \mid a_0, W_i)\right]+\frac{1}{n}\sum_{i=1}^{n}\bar{Q}_n(t \mid a_0, W_i)\] as discussed in detail in \cite{van2003unified}. Under conditions on $g_n$ and $\bar{Q}_n$, including consistency at fast enough rates, $\theta_{n,a_0}(t)$ is asymptotically efficient relative to $\mathscr{M}$. In this case, $\theta_{n,a_0}(t)$ satisfies \eqref{asy_linear} with influence function $\phi_{0,a_0,t}$. However, there is no guarantee that $\theta_{n,a_0}$ is monotone.
In the context of this example, we can identify simple sufficient conditions under which conditions (A)--(B), and hence the asymptotic equivalence of the initial and isotonized estimators of the G-computed distribution function, are guaranteed. Specifically, we find this to be the case when: \begin{enumerate}[\ \ \ \ (i)]
\item there exists some $\eta>0$ such that $g_0(a_0 \mid W) \geq \eta$ almost surely under $P_0$, and;
\item there exist non-negative real-valued functions $K_1,K_2$ such that \[K_1(w)|t-s|\ \leq\ |\bar{Q}_0(t \mid a_0, w) - \bar{Q}_0(s \mid a_0, w)|\ \leq\ K_2(w)|t - s|\] for all $t, s \in \s{T}$, and such that, under $P_0$, $K_1(W)$ is strictly positive with non-zero probability and $K_2(W)$ has finite second moment.
\end{enumerate}
We conducted a simulation study to validate our theoretical results in the context of this particular example. For samples sizes $n\in\{100, 250, 500, 750, 1000\}$, we generated $1000$ random datasets as follows. We first simulated a bivariate covariate $W$ with independent components $W_1$ and $W_2$, respectively distributed as a Bernoulli variate with success probability $0.5$ and a uniform variate on $(-1,1)$. Given $W=(w_1,w_2)$, exposure $A$ was simulated from a logistic regression model with $P_0(A = 1 \mid W_1=w_1,W_2=w_2) = \text{expit}(0.5 + w_1- 2 w_2)$. Given $W=(w_1,w_2)$ and $A=a$, $Y$ was simulated as the inverse-logistic transformation of a normal variate with mean $0.2 -0.3 a-4w_2$ and variance $0.3$.
For each simulated dataset, we estimated $\theta_{0,0}(t)$ and $\theta_{0,1}(t)$ for $t$ equal to each outcome value observed between $0.1$ and $0.9$. To do so, we used the estimator described above, with propensity score and conditional exposure-specific distribution function estimated using correctly-specified parametric models. We employed two correction procedures for the estimators $\theta_{n,0}$ and $\theta_{n,1}$. First, we projected $\theta_{n,0}$ and $\theta_{n,1}$ onto the space of monotone functions separately. Second, noting that $\theta_{0,0}(t) \leq \theta_{0,1}(t)$ for all $t$, so that $(a,t)\mapsto\theta_{0,a}(t)$ is component-wise monotone for this particular data-generating distribution, we considered the projection of $(a, t) \mapsto \theta_{n,a}(t)$ onto the space of bivariate monotone functions on $\{0,1\} \times \s{T}$. For each simulation and each projection procedure, we recorded the maximal absolute differences between (i) the initial and and projected estimates, (ii) the initial estimate and the truth, and (iii) the projected estimate and the truth. We also recorded the maximal widths of the initial and projected confidence bands
Figure~\ref{fig:sim_results} displays the results of this simulation study, with output from the univariate and bivariate projection approaches summarized in the top and bottom rows, respectively. The left column displays the empirical distribution of the scaled maximum absolute discrepancy between $\theta_n$ and $\theta_n^*$ for all sample sizes studied. This plot confirms that the discrepancy between these two estimators indeed decreases faster than $n^{-1/2}$, as our theory suggests. Furthermore, for each $n$, the discrepancy is larger for the two-dimensional projection.
The middle column of Figure~\ref{fig:sim_results} displays the empirical distribution function of the ratio between the maximum discrepancy between $\theta_n$ and $\theta_0$ and that of $\theta_n^*$ and $\theta_0$. This plot confirms that $\theta_n^*$ is always at least as close to $\theta_0$ than is $\theta_n$ over $\s{T}_n$. The maximum discrepancy between $\theta_n$ and $\theta_0$ can be more than 25\% larger than that between $\theta_n^*$ and $\theta_0$ in the univariate case, and up to 50 \% larger in the bivariate case.
The right column of Figure~\ref{fig:sim_results} displays the empirical distribution function of the ratio between the maximum size of the initial uniform 95\% influence function-based confidence band and that of the isotonic band. For large samples, the maximal widths are often close, but for smaller samples, the initial confidence bands can be up to 50\% larger than the isotonic bands, especially for the bivariate case. The empirical coverage of both bands is provided in Table~\ref{tab:coverages}. The coverage of the isotonic band is essentially the same as the initial band for the univariate case, whereas it is slightly larger than that of the initial band in the bivariate case.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{simulation_plots.pdf}
\caption{Summary of simulation results for G-computed distribution function. Each plot shows cumulative distributions of a particular discrepancy over 1000 simulated datasets for different values of $n$. Left panel: maximal absolute difference between the initial and isotonic estimators over the grid used for projecting, scaled up by root-$n$. Middle panel: ratio of the maximal absolute difference between the initial estimator and the truth and the maximal absolute difference between the isotonic estimator and the truth. Right panel: ratio of the maximal width of the initial confidence band and the maximal width of the isotonic confidence band. The top row shows the results for the univariate projection, and the bottom row shows the results for the bivariate projection.}
\label{fig:sim_results}
\end{figure}
\begin{table}
\caption{Coverage of 95\% confidence bands for the true counterfactual distribution function.}
\vspace*{1em}
\label{tab:coverages}
\centering
\begin{tabular}{crrrrrr}
&$n$ & 100 & 250 & 500 & 750 & 1000 \\
\hline
\multirow{2}{*}{d=1}&Initial band & 92.5 & 94.1& 96.0 & 94.5 & 95.5 \\
&Monotone band & 92.5 & 94.1& 96.0 & 94.5 & 95.5 \\
\hline
\multirow{2}{*}{d=2}&Initial band & 93.9 &94.0 & 95.0 & 94.6 & 94.9 \\
&Monotone band & 95.7& 95.9& 95.5 &95.3& 95.1 \\
\hline
\end{tabular}
\end{table}
\subsection{Example 2: Estimation of a conditional distribution function}\label{cond_dist}
We next demonstrate the use of Theorem~\ref{monotone_supnorm} with dimension $d=2$ for drawing inference on a conditional distribution function. Suppose that the data unit is the vector $X = (A, Y)$, where $Y$ is an outcome and $A$ is now a continuous exposure. The observed data consist of independent draws $(A_1, Y_1),(A_2,Y_2),\ldots,(A_n,Y_n)$ from $P_0\in\mathscr{M}$, where $\mathscr{M}$ is a nonparametric model. We define the parameter value $\theta_P$ pointwise as $\theta_{P}(t_1, t_2) := P\left( Y \leq t_1 \mid A = t_2 \right)$. Thus, $\theta_P$ is the conditional distribution function of $Y$ at $t_1$ given $A = t_2$. The map $(t_1, t_2) \mapsto \theta_P(t_1, t_2)$ is necessarily monotone in $t_1$ for each fixed $t_2$, and in some settings, it may be known that it is also monotone in $t_2$ for each fixed $t_1$. This parameter completely describes the conditional distribution of $Y$ given $A$, and can be used to obtain the conditional mean, conditional quantiles, or any other conditional parameter of interest.
For each $t_1$, the true function $\theta_0(t_1, t_2) = \theta_{P_0}(t_1, t_2)$ may be written as the conditional mean of $I(Y \leq t_1)$ given $A = t_2$. Hence, any method of nonparametric regression can be used to estimate $t_2\mapsto \theta_0(t_1, t_2)$ for fixed $t_1$, and repeating such a method over a grid of values of $t_1$ yields an estimator of the entire function. We expect that our results would apply to many of these methods. Here, we consider the local linear estimator \citep{fan1996local}, which may be expressed as
\[ \theta_n(t_1, t_2) := \frac{1}{nh_n} \sum_{i=1}^n I(Y_i \leq t_1)\left[ \frac{s_{2,n}(t_2) - s_{1,n}(t_2) \left(A_i - t_2\right)}{s_{0,n}(t_2)s_{2,n}(t_2) - s_{1,n}(t_2)^2}\right] K\left( \frac{A_i - t_2}{h_n}\right) \ ,\
where $K:\d{R}\to\d{R}$ is a symmetric and bounded kernel function, $h_n \to 0$ is a sequence of bandwidths, and $s_{j,n}(t_2) := \frac{1}{nh_n} \sum_{i=1}^n \left(A_i - t_2\right)^j K\left( \frac{A_i - t_2}{h_n}\right)$ for $j \in \{0,1,2\}$. Under regularity conditions on the true distribution function $\theta_0$, the marginal density $f_0$ of $A$, the bandwidth sequence $h_n$, and the kernel function $K$, for any fixed $(t_1, t_2)$, $\theta_n$ satisfies
\[ (nh_n)^{1/2} \left[ \theta_n(t_1, t_2) - \theta_0(t_1, t_2) - h_n^2 V_K b_0(t_1, t_2)\right] \indist N\left(0, S_K v_0(t_1, t_2)\right), \]
where $V_K := \int x^2K(x) dx$ is the variance of $K$, $S_K := \int K(x)^2 dx$, and $b_0(t_1, t_2)$ and $v_0(t_1, t_2)$ depend on the derivatives of $\theta_0$ and on $f_0$. If $h_n$ is chosen to be of order $n^{-1/5}$, the rate that minimizes the asymptotic mean integrated squared error of $\theta_n$ relative to $\theta_0$, then $n^{2/5} \left[ \theta_n(t_1, t_2) - \theta_0(t_1, t_2) \right]$ converges in law to a normal random variate with mean $V_K b_0(t_1, t_2)$ and variance $S_K v_0(t_1, t_2)$. Under stronger regularity conditions, the rate of convergence of the uniform norm $\|\theta_n - \theta_0\|_\s{T}$ can be shown to be $(n h_n / \log n)^{1/2}$ \citep{hardle1988}.
Theorem~\ref{thm:emp_process} cannot be used to establish (A) in this problem, since $\theta_n$ is not an asymptotically linear estimator. Furthermore, as discussed above, recent results suggest that $\{ r_n[\theta_n(t) - \theta_0(t)] : t \in \s{T}\}$ does not converge weakly to a tight limit in $\ell^{\infty}(\s{T})$ for any useful rate $r_n$. Despite this lack of weak convergence, condition (A) can be verified directly in the context of this example under smoothness conditions on $\theta_0$ and $f_0$ using the tail bounds for empirical processes outlined in Section~\ref{sec:kern}. Denoting by $\theta_{0,t_2}'$ and $\theta_{0,t_2}''$ the first and second derivatives of $\theta_0$ with respect to its second argument, we define
\begin{align*}
R_{\theta}^{(2)}(t, \delta) &:= \theta_0(t_1, t_2 + \delta) - \theta_0(t_1, t_2) - \delta\theta_{0,t_2}'(t_1, t_2) - \tfrac{1}{2} \delta^2\theta_{0,t_2}''(t_1, t_2)\ ,
\end{align*}
and $R_{f}^{(1)}(t, \delta) := f_0(t_2 +\delta ) - f_0(t_2) - \delta f_0'(t_2)$, where $f_0'$ is the derivative of $f_0$. We then introduce the following conditions on $\theta_0$, $f_0$, and $K$:
\begin{description}
\item[(d)] $\theta_{0,t_2}''$ exists and is continuous on $\s{T}$, and as $\delta \to 0$, $\sup_{t \in \s{T}} | R_{\theta}^{(2)}(t, \delta)| = o(\delta^2)$;
\item[(e)] $\inf_{t\in \s{T}} f_0(t_2) > 0$, $f_0'$ exists and is continuous on $\s{T}$, and $\sup_{t \in \s{T}}| R_{f}^{(1)}(t, \delta)|= o(\delta)$;
\item[(f)] $K$ is a Lipschitz function supported on $[-1,1]$ and satisfies condition (M) of \cite{stupfler2016}.
\end{description}
We also define $\nu_{n,t}(y,a) := \left[ I(y \leq t_1) - \theta_0(t_1, a)\right] K\left( \frac{a - t_2}{h_n}\right)$, $g_n(t_2) := s_{0,n}(t_2)s_{2,n}(t_2) - s_{1,n}(t_2)^2$, and $R_n(t) := h_n^{-1/2} \left[ \frac{ s_{2,n}(t_2)}{g_n(t_2)} \d{G}_n \nu_{n,t} - \frac{ s_{1,n}(t_2)}{g_n(t_2)} \d{G}_n \left( \ell_t \nu_{n,t}\right)\right]$. We then have the following result.
\begin{proposition}\label{prop:kern_unif}
Suppose conditions (d)-(f) hold, $nh_n^5 =O(1)$, and $nh_n^4/ \log h_n^{-1} \longrightarrow \infty$. Then
\[ \sup_{t \in \s{T}} \left| \left(nh_n \right)^{1/2} \left[ \theta_n(t_1, t_2) - \theta_0(t_1, t_2) \right] - \left(n h_n^5\right)^{1/2} \tfrac{1}{2}\theta_{0, t_2}''(t_1, t_2) K_2 - R_n(t) \right| \inprob 0 \ .\]
\end{proposition}
Proposition~\ref{prop:kern_unif} aids in establishing the following result, which formally establishes asymptotic equivalence of the local linear estimator of a conditional distribution function and its correction obtained via isotonic regression at the rate $r_n = (n h_n)^{1/2}$.
\begin{proposition}\label{prop:cond_dist}
Suppose conditions (d)-(f) hold and $nh_n^5 \longrightarrow c \in (0, \infty)$. Then condition (A) holds for the local linear estimator with $r_n = (nh_n)^{1/2}$.
\end{proposition}
The proof of Propositions~\ref{prop:kern_unif} and~\ref{prop:cond_dist} are provided in Supplementary Material. These results may be of interest in their own right for establishing other properties of the local linear estimator.
As with the first example, we conducted a simulation study to validate our theoretical results. For samples sizes $n\in\{100, 250, 500, 750, 1000\}$, we generated $1000$ random datasets as follows. We first simulated $A$ as a Beta$(2,3)$ variate. Given $A=a$, $Y$ was simulated as the inverse-logistic transformation of a normal variate with mean $0.5 \times [1 + (a - 1.2)^2]$ and variance one.
For each simulated dataset, we estimated $\theta_0(y,a)$ for each $(y,a)$ in an equally spaced square grid of mesh $\omega_n = n^{-4/5}$. For each unique $y$ in this grid, we estimated the function $a\mapsto \theta_0(y, a)$ using the local linear estimator, as implemented in the \texttt{R} package \texttt{KernSmooth} \citep{KernSmooth, wand1995kernsmooth}. For each value of $y$ in the grid, we computed the optimal bandwidth based on the direct plug-in methodology of \cite{ruppert1995effective} as implemented by the \texttt{dpill} function, and we then set our bandwidth as the average of these $y$-specific bandwidths. We constructed initial confidence bands using a variable-width nonparametric bootstrap \citep{hall2001bootstrap}.
We first note that, for all sample sizes considered, over 99\% of simulations had monotonicity violations in both the $y$- and $a$-directions. Figure~\ref{fig:sim_results_cond_dist} displays the results of this simulation study. The left exhibit of Figure~\ref{fig:sim_results_cond_dist} confirms that the discrepancy between $\theta_n$ and $\theta_n^*$ decreases faster than $r_n^{-1}=n^{-2/5}$, as our theory suggests. The middle exhibit indicates that in roughly 50\% of simulations, there is less than 5\% difference between $\| \theta_n^* - \theta_0\|_{\s{T}_n}$ and $\|\theta_n - \theta_0\|_{\s{T}_n}$, but even for $n=1000$, in roughly 25\% of simulations, $\theta_n^*$ offers at least a 25\% improvement in estimation error. In smaller samples, the estimation error of $\theta_n^*$ is less than half that of $\theta_n$ in 5-10\% of simulations. The rightmost exhibit indicates that the projected confidence bands regularly reduce the uniform size of the initial bands by 10-20\%. Finally, the empirical coverage of uniform 95\% bootstrap-based bands and their projected versions is provided in Table~\ref{tab:coverages2}. As before, the projected band is always more conservative than the initial band, and the difference in coverage diminishes as $n$ grows. However, the initial bands in this example are anti-conservative, even at $n=1000$, likely due to the slower rate of convergence, and the corrected bands offer a much more substantial improvement in this example than in the first.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{simulation_plots_cond_dist_loclin.pdf}
\caption{Summary of simulation results for conditional distribution function. The three columns display the same results as those in Figure~\ref{fig:sim_results}.}
\label{fig:sim_results_cond_dist}
\end{figure}
\begin{table}[h]
\caption{Coverage of 95\% confidence bands for the true conditional distribution function.}
\vspace*{1em}
\label{tab:coverages2}
\centering
\begin{tabular}{rrrrrr}
\hline
$n$ & 100 & 250 & 500 & 750 & 1000 \\
\hline
Initial band & 37.6 & 64.9 & 83.2 & 86.3 & 89.7\\
Monotone band & 60.8 & 80.4 & 90.3 & 92.3 & 93.9 \\
\hline
\end{tabular}
\end{table}
\section{Discussion}\label{conclusion}
Many estimators of function-valued parameters in nonparametric and semiparametric models are not guaranteed to respect shape constraints on the true function. A simple and general solution to this problem is to project the initial estimator onto the constrained parameter space over a grid whose mesh goes to zero fast enough with sample size. However, this introduces the possibility that the projected estimator has different properties than the original estimator. In this paper, we studied the important shape constraint of multivariate component-wise monotonicity. We provided results indicating that the projected estimator is generically no worse than the initial estimator, and that if the true function is strictly increasing and the initial estimator possesses a relatively weak type of stochastic equicontinuity, the projected estimator is uniformly asymptotically equivalent to the initial estimator. We provided especially simple sufficient conditions for this latter result when the initial estimator is uniformly asymptotically linear, and provided guidance on establishing the key condition for kernel smoothed estimators.
We studied the application of our results in two examples: estimation of a G-computed distribution function, for use in understanding the effect of a binary exposure on an outcome when the exposure-outcome relationship is confounded by recorded covariates, and of a conditional distribution function, for use in characterizing the marginal dependence of an outcome on a continuous exposure. In numerical studies, we found that the projected estimator yielded improvements over the initial estimator. The improvements were especially strong in the latter example.
In our examples, we only studied corrections in dimensions $d=1$ and $d=2$. In future work, it would be interesting to consider corrections in dimensions higher than 2. For example, for the conditional distribution function, it would be of interest to study multivariate local linear estimators for a continuous exposure $A$ taking values in $\d{R}^{d-1}$ for $d > 2$. Since tailored algorithms for computing the isotonic regression do not yet exist for $d > 2$, it would also be of interest to determine whether a version of Theorem~\ref{monotone_supnorm} could be established for the relaxed isotonic estimator proposed by \cite{fokianos2017integrated}. Alternatively, it is possible that the uniform stochastic equicontinuity currently required by \cite{chernozhukov2010quantile} and \cite{daouia2012isotonic} for asymptotic equivalence of the rearrangement- and envelope-based corrections, respectively, could be relaxed along the lines of our condition (A). Finally, our theoretical results do not give the exact asymptotic behavior of the projected estimator or projected confidence band when the true function possesses flat regions. This is also an interesting topic for future research.
\vspace{.2in}
\singlespacing
{\footnotesize
\section*{Acknowledgements}
The authors gratefully acknowledge support from the Career Development Fund of the Department of Biostatistics at the University of Washington (MC) and from NIAID grants 5UM1AI058635 (TW, MC) and 5R01AI074345 (MJvdL).
}
\doublespacing
|
2,877,628,090,204 | arxiv | \section{Introduction}
What happens to orbits subject to linear frictional drag?
In typical physical settings, such as Rydberg
atoms or stellar binaries, the effective frictional
forces are nonlinear and, typically, lead to
the circularization of the orbit. Orbital evolution under
linear friction is special in that, as we show below,
the eccentricity and the apsides do not change to leading order
in the damping. The purpose of this note
is to understand this elementary result from the underlying
dynamical symmetry of the Kepler problem,
thus demonstrating the utility of a
Hamiltonian notion in
its non-Hamiltonian generalization.
In many astrophysical
situations, the secular evolution due to friction of orbits in a two body
system is towards circular orbits.
In an orbit in a central field the
angular momentum scales with the momentum while the energy generally scales
with the
momentum-squared. Friction, assumed to be spatially isotropic and
homogeneous but time
odd, typically scales the momentum. This means generally that the resultant
secular evolution in central force systems is that in which the energy is
minimized
at fixed angular momentum.
This is clearly the circular orbit. The velocity dependence of the
frictional force is quite relevant, in particular as referenced against the
velocity dispersion of the (undamped) motion in that central potential.
Clearly, under the action of such dissipative forces, a consequence
of symmetry is that the flow in orbital shape (not size!)
has two fixed points, circular orbits and
strictly radial (infall) orbits.
Few physical problems have received
more scrutiny than
bounded orbits in the two- and few- body system.
Among these, the two-body Kepler problem
is arguably the most experimentally relevant and best
studied example, having
been illuminated by intense theoretical inquiry spanning
hundreds of years leading to
important insights even in relatively recent times
\cite{Ermano, Hermann, Bernoulli, Laplace, Runge, Lenz, Pauli, Hulthen, Fock, Bargmann, Moser, Belbruno, Ligon, Cushman, Stehle, Iosifescu, Gyorgyi, Guillemin, Cordani, Keane1, Fradkin}.
We do not present a systematic review or histiography of this
celebrated problem (though we thankfully acknowledge also
\cite{Keane2, Goldstein, Jose, Quesne, Kustaanheimo, Chen,
Stahlhofen, Landau1}
which we have found quite useful for our study).
We do not aim to contribute
to the vast literature on astrophysically and microphysically
relevant models of friction in orbital problems (though the interested
reader may find references
\cite{Colosimo, Casertano, Mayor, Mardling, Kozai1, Kozai2, Apostolatos, Dotti, Glampedakis, Apostalakis, Lightman, Schutz, Bellomo}
a useful launching point for such review).
Instead, the our purpose here is to accomodate from the
dynamical symmetry group point-of-view the result that
linear frictional damping (to leading order) preserves
the orbit's shape.
Although Hamiltonian systems may lose dynamical symmetry
completely when dissipative forces are included, it can be shown
that some structure may remain under a modified symplectic form.
After a brief introduction to the method by which Tarasov
extends symplectic structure of Hamiltonian mechanics to
dissipative systems, we apply it to the determination of the
time averages of dynamical quantities. Damping
invariably introduces new dynamical timescales and the
time averaging we implement is over times short compared
with these timescales (but still long compared with the
orbital timescales in the undamped problem).
Tarasov's construction
reveals the relevance of the dynamical symmetry algebra
to the damped Kepler problem.
We then compare this aproach to
the classic ``variations of constants'' method of
orbit parameter evolution
by describing an
improvement
that follows from our study. The elementary method
can be generalized to non-Kepler homogeneous potentials and also
determines orbital shape evolution for linearly
damped Kepler orbits beyond leading order.
\section{Dynamical Symmetry and Tarasov's Construction}
In the undamped Kepler problem the lack of precession is generally
understood as a consequence of a dynamical symmetry, the
celebrated $so(4)$ symmetry formed from the two commuting $so(3)$,
one from the angular momentum ${\vec L} = {\vec r}\times{\vec p}$
the other from the Runge-Lenz vector,
${\vec S} = {\vec L}\times{\vec p} + k{{\vec r}\over{|{\vec r}|}}$
(\cite{Runge, Lenz, Pauli})
being the maximal set of local, algebraically independent
operators that commute with the Hamiltonian, $H={{ {\vec p}^2}\over{2}} + V(r) $, with $V(r) = kr^\alpha$ for $k < 0$ and $\alpha = -1$.
(though see \cite{Fradkin} for a more precise and general statement of the
connection between algebra and orbits in a central field).
\begin{equation}
\fl
\{L_i, L_j\} = 2\epsilon_{ijk} L_k \qquad \{L_i, S_j\} = 2\epsilon_{ijk} S_k \qquad
\{S_i, S_j\} = -2H\epsilon_{ijk} S_k
\label{14}
\end{equation}
The length of ${\vec S}$ is
proportional to the eccentricity (and points along the
semi-major axis of the orbit, in the
direction to the periastron from the focus).
Defining ${\vec L}$ and ${\vec S}$ has utility beyond their being
constants in the 2-body
Kepler problem, for example, parameterizing the secular evolution of orbits
under various
Hamiltonian perturbations \cite{Stahlhofen, Morehead}.
This $so(4)$ is one of the maximal compact factor groups of the
$so(4,2)$ (the conformal group)
extended symmetry formed by ${\vec L}, {\vec A}, H$,
the generalization of the scaling operator
${\cal R} = {\vec r}\cdot{\vec p}$ and the
Virial operator
${\cal V} ={{ {\vec p}^2}\over{2}} - {{r}\over{2}}\partial_r V(r) $
(\cite{Iosifescu, Cordani})
The other central potential posessing an easily recognizable
dynamical symmetry is the
multi-dimensional harmonic oscillator ($V$ as given with $k > 0$,
$\alpha$ = 2). As is well known, the
isotropic $D$-dimensional
harmonic oscillator's naive $O(D)$ symmetry is part of a larger $U(D)$
dynamical symmetry. For $D=2$ harmonic oscillator, note that the
$U(2)$ symmetry does enlarge further to a $so(3,2)$ when including
${\cal R}, {\cal V}$ and their generalization (the virial subalgebra \Eref{4} through
\Eref{6} of each oscillator alone and closes
to a $sl(2,R)$ subgroup of the $so(3,2)$). Note
further that it is this later algebra that is isomorphic to the
dimensionally reduced
$so(4,2)$ of the 3-d Kepler problem, by which we mean the reduction of that algebra
to generators associated with the orbital plane only. These considerations
can also be understood from the KS construction\cite{Kustaanheimo, Chen}
of the Kepler problem, in which a four-dimensional
isotropic harmonic oscillator is the starting point.
In that construction the
$u(4) = su(4) ~{\times}~ u(1)$ is, of itself, not preserved by the KS construction.
Instead, it is the $u(2,2)$ subgroup of the four
identical, independent
oscillator's
$sp(8,{\bf {\rm R}})$ symmetry in which the overall $u(1)$ can be isolated as
the angular momentum constraint of the KS construction\cite{Quesne}.
The residual symmetry $su(2,2) \sim so(4,2)$ is
that of the 3-d Kepler problem.
The analytical connection between the Kepler problem and the
isotropic harmonic oscillator has deep historical roots, going back to
Newton and Hooke (see \cite{Grant} and references therein).
Finally, the geometric
construction of the undamped Kepler problem as geodesic flow on (spatial)
a 3-manifolds of constant curvature relates the $so(4)$ dynamical
symmetry to the isometry group generated by Killing
vector fields on the spatial slice\cite{Guillemin, Keane1, Keane2}.
These various connections between the Kepler problem and
the isotropic harmonic oscillator do not lead to a
simple structural connection between the associated damped problems.
To leading order in the damping, Kepler
orbits subject to linear frictional force
do not change shape or precess as they decay.
It would be satisfying to understand this elementary
result as a consequence of the preservation of
the dynamical algebra under linear friction. Although this is
reminiscent of the damped N-dimensional harmonic oscillator,
there is no simple way to relate the damped problems.
Since the subgroup
associated with the shape and precession (through the ${\vec S}$) is rank one
it is suggestive that the entire group structure is preserved
to leading order in the linear friction.
A recent paper by Tarasov\cite{Tarasov}
suggests a straightforward generalization
of the Poisson structure to systems with dissipative forces.
There are many other approaches to addressing
structural questions of dissipative systems
(for one example, see \cite{Sergi1, Sergi2}). We find the approach of
\cite{Tarasov} to be most useful for addressing questions of the
dynamical symmetries that survive including dissipation.
For completeness we now briefly review Tarasov's construction, and
apply it to dissipation in the central field problem
in the following section.
To preserve as much of the algebraic structure as
possible, Tarasov constructs a one-parameter
family of two forms (that define a generalized Poisson structure)
that -in a sense-
interpolate between different dampings.
In the zero damping limit it smoothly matches onto the
canonical symplectic form.
Dimensionally, any damping parameter introduces a new time
scale into the problem, thus this new interpolating two form
must also be explicitely time-dependent.
Tarasov requires this family of two forms to
have the following useful properties
(1) Non-degeneracy: The two form
${\bf \omega} = \omega_{ij}(t) {\bf {\rm d}}x^i\wedge{\bf {\rm d}}x^j$
is antisymmetric and non-degenerate along the entire flow. The $x^i$ are the
$2N$ (local) phase space co-ordinates. In positive terms, the
inverse $\omega^{ij}\omega_{jk} = \delta^i_j$ exists almost
globally \footnote{Since we do not formulate this entirely in the
exterior calculus, we must allow for higher codimension singularites
that may not be resolvable in the dissipative system.}.
(2) Jacobi Identity: the two form is used to define
a new Poisson bracket
$\{A, B\}_T = \omega^{ij}\partial_i A \partial_j B$ that
forms an associative algebra. Explicitely it satisfies.
\begin{equation}
\{A,\{B,C\}_T\}_T + \{B,\{C,A\}_T\}_T +\{C,\{A,B\}_T\}_T =0
\label{jacobi}
\end{equation}
Here we use the subscript '$T$' to distinguish this bracket
from the Poisson bracket of the undamped problem.
(3) Derivation property of time translation:
with respect to this new
bracket the time derivative of the new
Poisson bracket satisfies the derivation
property (also called the Liebnitz rule)
\begin{equation}
{{\rm d}\over{\rm d}t} \{A, B\}_T = \{ { {{\rm d}A}\over{\rm d}t}, B\}_T +
\{A, {{{\rm d}B}\over{\rm d}t} \}_T
\label{derivation}
\end{equation}
These requirements are remarkable for several reasons.
First, property (1) indicates that (2) and (3) are possible.
The deeper relevance of property (1) is that we
can regard the two-form as (essentially a) global metric
on the phase space. Property (2) indicates local mechanical
observables in
this 'dissipation deformed' algebra form a lie algebra.
Property (3) is key to the utility of Tarasov's construction for
understanding constants of motion in dissipative systems.
It stipulates that time development
in the dissipative system, while no longer just $\{~, H\}$ (or even
$\{~, H\}_T$), must
be compatible with the
structure of the symplectic algebra in the new bracket
and thus the (new) bracket of
time independent quantities in the dissipative system are themselves
time independent. Thus, just as in the Hamiltonian case,
time independent quantities form a closed subalgebra.
Note that for a Hamiltonian system property
(3) is automatic since in that case time translation is an
inner automorphism of the symplectic algebra.
In a dissipative system by contrast
the Hamiltonian is no longer the operator of time translation, but,
if Tarasov's construction can be implemented, time translation is still
an automorphism of the algebra, and as such may be regarded
as an outer automorphism.
Finally, from property (3) it follows after a brief calculation
that the two-form ${\bf \omega}$ must be time idependent in the full
dissipative system, ${ {{\rm d}{\bf \omega}}\over{\rm d}t} =0$. In
terms of symplectic geometry, this is metric compatibility of the
dissipative flow.
To proceed with the construction, consider the general
flow ${\dot x}^i = \chi^i({\vec x}, t)$. Again, these are not assumed to
be Hamiltonian flows.
Assuming property (1) and using $\omega^{ij}$ to form a bracket
$\{A,B\}_T = \omega^{ij} \partial_i A \partial_j B$ ,
property (2) leads to the condition
\begin{equation}
\omega^{im}\partial_{m}\omega^{jk} + \omega^{jm}\partial_{m}\omega^{ki}
+ \omega^{km}\partial_{m}\omega^{ij} = 0 .
\label{jacobi2}
\end{equation}
Total time derivatives and derivatives along phase space
directions do not commute in the flow,
\begin{equation}
[{{ {\rm d}}\over{{\rm d} t}}, \partial_i ] A = -\partial_j A \partial_i \chi^j .
\label{time}
\end{equation}
Using this and the jacobi identity \eref{jacobi}, one sees that
property (3) implies a condition relating the form $\bf \omega$
and the flow $\chi^i$,
\begin{equation}
{{\partial \omega_{ij}(t)}\over{\partial t}} = \partial_i \chi_j - \partial_j \chi_i
\qquad {\rm where} \qquad \chi_j = \omega_{jk}(t) \chi^k
\label{derivation_explicit}
\end{equation}
Given $\chi^i$, we proceed by solving
\eref{jacobi2} for an $\omega^{ij}$
that satisfies \eref{derivation_explicit}. This
completes Tarasov's construction.
We breifly offer a few further remarks
helpful to orient the reader.
First, in the more familiar context of Hamiltonian flows, there
${\dot x}^i = \chi^i = \{x^i, H\}$ for a local function $H$ on the
phase space. For this case we can compute in the Darboux
frame and learn that the usual symplectic form
(automatically satisfying \eref{jacobi2})
is a solution also to \eref{derivation_explicit} since the RHS in that
case is zero. We recognize the RHS of \eref{derivation_explicit} as
exactly the obstruction to the flow, $\chi^i$ being Hamiltonian.
Conformal transformation
of the two-form, $\tilde {\bf \omega} = \Omega {\bf \omega}$,
where $\Omega$ is a a scalar function, can only relate two solutions
of \eref{jacobi2} and \eref{derivation_explicit} IFF the $\Omega$
is a constant of the motion ${ {{\rm d}\Omega}\over{{\rm d}t}} = 0$.
For in that case \eref{derivation_explicit} indicates that
\begin{equation}
{{\partial \Omega}\over{ \partial t}} \omega_{ij} =
\chi_j\partial_i\Omega - \chi_i\partial_j\Omega
\label{timeprime}
\end{equation}
whereas \eref{jacobi2} yields
\begin{equation}
\omega_{jk}\partial_j\Omega + \omega_{kl}\partial_j\Omega +
\omega_{lj}\partial_k\Omega = 0
\label{derivation_explicitprime}
\end{equation}
so, contracting by $\chi^k$ and comparing with \eref{timeprime}, we
learn that $\Omega$ must be a constant of the motion. Thus, each
solution is conformally unique.
We do not know what conditions on $\chi^i$
lead to the existence of even one non-singular
{\it simultaneous} solution ${\bf \omega}$
of \eref{jacobi2}
and
\eref{derivation_explicit}.
Tarasov\cite{Tarasov} provides an explicit solution
for a general Hamiltonian system ammended by a general
linear frictional force.
The general question of the
existence of ${\bf \omega}(t)$ for a more general $\chi^i$ is at this
point unclear, but beyond the scope of this present effort.
\section{Dynamical Symmetry in a Damped System}
Consider damped orbital motion in a central field;
\begin{equation}
{\dot {\vec x}} = {\vec p}
\label{7}
\end{equation}
\begin{equation}
{\dot {\vec p}} = -\partial_r V {{\vec x}\over{r}}
-\beta(p) {\vec p}
\label{8}
\end{equation}
with $r=|{\vec x}|$ and
$V(r)$ the interparticle potential (throughout we take the reduced
mass to be normalized to 1). The function $\beta(p)$ is some general
function parameterizing the speed dependence of the damping, and
this form of the damping function is the most general consistent
with isotropy and homogeniety of the damping forces.
Note that we can understand this set as descending from a limit
in which the central mass is very much larger than the orbital mass though,
as in general,
damping does inextricably mix the center of mass motion and the
relative motion.
We call linear damping the choice of
$\beta$ constant.
The
\Eref{derivation_explicit} takes the form,
\begin{equation} {{\partial \omega_{xp}(t)}\over{\partial t}} =
\partial_x (\omega_{px} \chi^x) - \partial_p (\omega_{xp} \chi^p)
= \partial_p \omega_{xp'} (\beta(p) p')
\label{20}
\end{equation}
Again, we do not know
if solutions to
\Eref{20} exist and satisfy Jacobi for every choice of $\beta(p)$.
However, for
$\beta(p)=const.$ there is a
simple solution to \Eref{20} that satisfies Jacobi\cite{Tarasov},
\begin{equation} \omega_{ij}(t) = e^{\beta t}{\hat \omega}_{ij}
\label{21}
\end{equation}
where $\hat \omega$ is the usual symplectic form of the undamped
Kepler problem. Physically this
corresponds to the uniform shrinkage of phase space volumes
under linear damping.
Clearly, in going from
$\{,\}$ (Poisson bracket) to the new bracket $\{,\}_T$ the relations in
\Eref{14} gain a factor of $e^{-\beta t}$. The algebra
in the new bracket resulting from
this simple rescaling is still $so(4)$. The utility of this simple change
to the algebra of \Eref{14} (which was for the undamped system) is that
it is now compatible
with the evolution under \Eref{7} and \Eref{8} of the damped system. To see this in
an example, take the first relation in \Eref{14} and take the (total) time derivative
of both sides. Then note
${{ {\rm d} \{L_i,L_j\} }\over{{\rm d}t}} = -2\beta (2\epsilon_{ijk} L_k) \ne
2\epsilon_{ijk} {\dot L}_k$; {\it i.e.} the usual Poisson bracket is no longer
compatible with time evolution. Duplicating the previous line for
$\{L_i, L_j\}_T = 2e^{-\beta t}\epsilon_{ijk} L_k$ one learns that this is compatible
with the flow \Eref{7} and \Eref{8}. Similarly, one may check that all the brackets in
\Eref{14} (after replacing $\{, \}$ with $\{, \}_T$) are as well.
Also note that $\{L_i, H\}_T = 0 = \{S_i, H\}_T$, though since
brackets with $H$
no longer delineate time evolution, these equations do not
imply that
$\vec L$ and $\vec S$ are constants of the motion in the dissipative system
(also clear from \Eref{15} below).
The critique here is familiar to any attempt to reconcile
symplectic structure and dissipation; fundamentally, \Eref{7} and \Eref{8} still treat
$x$ and $p$ differently so that time evolution is no longer an element in the dynamical
algebra of $\{,\}$ or $\{, \}_T$.
To relax the category of 'constants of the motion'
sufficiently for dissipative systems,
consider to what extent
dynamical quantities
averaged over some number of orbits change
on a longer time scale, {\it i.e.} on
a timescale relevant to the dissipation (note $1/\beta(p)$
is essentially that timescale).
Let $<>$ denote time averages over many orbits, ${\cal O}$ a
classical observable,
and suppose that $\omega$
is a
solution to
\Eref{derivation_explicit}
and the Jacobi identity for the system as in
\Eref{7} and \Eref{8}.
In general,
\begin{equation}
<\{ {\cal O}, H\}_T> = < \omega^{xp}(t) \bigl({\dot x}\partial_x {\cal O}
-(-{\dot p} - \beta(p)p) \partial_p {\cal O} \bigr) >
\end{equation}
\begin{equation}
\qquad \qquad \qquad = < \omega(t) \biggl[ { {{\rm d}{\cal O}}\over{ {\rm d} t}} -
{{\partial {\cal O}}\over{\partial t}}\biggr] + \omega^{xp}(t)\beta(p) p \partial_p {\cal O} >
\label{23}
\end{equation}
Note that sums are implied in the $x, p$ indices of the $\omega^{xp}(t)$,
the new symplectic form.
Above we have used isotropy to rewrite the sum in the first term in terms of
the (normalized) symplectic trace of $\omega^{xp}(t)$ which we denote simply as $\omega(t)$.
To show one intermediate step, integrating by parts and using \Eref{20} we arrive at
$$
<\{ {\cal O}, H\}_T> = {{1}\over{T}} \Delta(\omega_T {\cal O}) +< \omega^{x\alpha}\omega^{p\beta}
(\partial_\alpha \chi_\beta - \partial_\beta \chi_\alpha + \chi^l\partial_l \omega_{\alpha\beta}){\cal O}
$$
\begin{equation}
\qquad \qquad \qquad
+ \omega^{xp}\beta(p)p\partial_p {\cal O} - \omega {{\partial {\cal O}}\over{\partial t}}>
\label{24}
\end{equation}
Where $\Delta(G)$ refers simply to the overall change of the
quantity $G$ over time $T$.
Finally, using the \Eref{time} and the fact that $\omega^{xp}$ satisfies the Jacobi
identity we reduce the above to
\begin{equation}
\fl <\{ {\cal O}, H\}_T> = {{1}\over{T}} \Delta(\omega {\cal O})
+ < (\omega^{xp'}\partial_{p'}\chi^p
- \omega^{px'}\partial_{x'}\chi^x){\cal O} + \omega^{xp}\beta(p)p\partial_p{\cal O} -
\omega{{\partial {\cal O}}\over{\partial t}}>
\label{25}
\end{equation}
We now specialize to vector fields
of the general form \Eref{7} and \Eref{8} to find,
\begin{equation}
\fl {{1}\over{T}} \Delta(\omega {\cal O}) = <\{ {\cal O}, H\}_T + \omega { {\partial {\cal O}}\over
{\partial t}}
- \omega^{xp} \bigl(\partial_p(\beta(p)) p {\cal O}-\beta(p) p \partial_p
{\cal O}\bigr) >
\label{26}
\end{equation}
and so making the RHS zero indicates conserved quantities in the non-Hamiltonian
system. Again, this last result
was derived for general $\beta(p)$,
which assumes only that the friction is
isotropic and homogeneous.
In the linear friction case $\beta(p) = \beta = const$.
For that case, using ${\cal O } = L/\omega^2$ in the above equation
implies that $L/\omega $
are constants of the motion in this system.
Similarly, taking ${\cal O} = S/\omega$
indicates that $\Delta S$ is proportional to $(2\beta {\vec L}/\omega)\times <\omega {\vec p}> $
which, again, is zero to first order in $\beta$.
This result then applied to the case of bounded Kepler orbits with linear damping
indicates that the (orbit-averaged) Runge-Lenz vector, and thus the dynamical algebra of the
Kepler problem, is conserved to leading order in the linear friction coefficient.
In elementary terms, although
angular momentum ${\vec L}$ and
${\vec S}$ are constants in the Hamiltonian system for
$V(r) \sim {{1}\over{r}}$
they evolve under linear damping of \eref{7}, \eref{8} as,
\begin{equation}
{\dot {\vec L}} = -\beta {\vec L} \qquad \qquad \qquad {\dot {\vec S}} = -2\beta {\vec L}\times{\vec p}.
\label{15}
\end{equation}
Note that in the weak damping limit, since ${\vec L}$ is conserved to
${\cal O}(\beta^0)$, the
second equation time averages to
$-2<\beta {\vec p}>\times<{\vec L}>$.
Thus, again we learn that if the damping were
strictly linear ($\beta$ constant)
then since $<{\vec p}> = 0$, the time average
of $\dot {\vec S}$ is $0$, again indicating that the eccentricity
vector would be conserved to leading order.
Note also that it is straightforward to
integrate the ${\vec L}$ equation explicitely, finding
${\vec L} = {\vec L}_0e^{-\beta t}$ the initial condition ${\vec L}_0$
being identified now a conserved quantity of the dissipative
system. We use these results in the next section
of this paper to ammend the 'textbook' orbital secular evolution
equations.
\section{The Damped Kepler Problem}
The previous section suggests that (linear-) damped bounded Kepler orbits
shrink but retain their aspect ratio and do not precess to leading order
in the damping. It is well known that
superlinear damping does lead to circularization whereas
sublinear damping leads to infall orbits in the Kepler case.
So far this begs the questions of whether this generalizes to other
central field problems, and, if so, then at what order in the linear
damping coefficient do orbits undergo shape and precessional change.
In this section we
address both questions,
first describing a problem that arises
using a time-honored
pertubative method for treating general perturbing forces
in the Kepler problem, and second,
generalize the result of the preceeding section
to a broad class of central field potentials.
We then
establish in precise terms the fate of Kepler orbits under
linear damping.
Consider the usual
secular orbital evolution method (called ``the variations of
constants'') most common in literature on cellestial mechanics,
for example, in \cite{Danby} (Chapter 11 Section 5, pg. 323, though
see also the treatments of non-linear friction
in \cite{Watson, MurrayDermott, BC}).
In the ``variations of constants' method, orbital response
to an applied force
${\vec F} = R{\vec x} + N{\vec L} + B {\vec L}\times{\vec x}$, in
the orbit's tilt $\Omega$, the orbital plane's axis, $i$, the
eccentricity $\epsilon$, the angle of the ascending node $\omega$ the
semi-major axis $a$ and the period $T = 2\pi/n$ (in their notation)
evolve following\cite{Danby},
\begin{equation}
{{{\rm d}\Omega}\over{{\rm d}t}} = {{nar}\over{\sqrt{1-\epsilon^2}}} N {{\sin{u}}\over{\sin{i}}}
\label{d1}
\end{equation}
\begin{equation}
{{{\rm d} i}\over{{\rm d}t}} = {{nar}\over{\sqrt{1-\epsilon^2}}} N \cos{u}
\label{d2}
\end{equation}
\begin{equation}
{{{\rm d} \omega}\over{{\rm d}t}} =
{{n a^2 \sqrt{1-\epsilon^2}}\over{\epsilon}}
\bigl[-R\cos \theta + B (1+{{r}\over{P}})\sin\theta \bigr] - \cos{i} {{ {\rm d}\Omega}\over{{\rm d} t}}
\label{d3}
\end{equation}
\begin{equation}
{{{\rm d} \epsilon}\over{{\rm d}t}} = n a^2 \sqrt{1-\epsilon^2}
\bigl[R\sin \theta + B (\cos\theta + \cos E)\bigr]
\label{d4}
\end{equation}
\begin{equation}
{{\rm d}a\over{{\rm d}t}} = {2n a^2} \bigl[ R {{a\epsilon}\over{\sqrt{1-\epsilon^2}}}\sin \theta + B {{a^2}\over{r}} \sqrt{1-\epsilon^2}\bigr]
\label{d5}
\end{equation}
and where
\begin{equation}
{{\rm d}n\over{{\rm d}t}} = -{{3n}\over{2a}} {{{\rm d}a}\over{ {\rm d}t}}
\label{d6}
\end{equation}
with $u=\theta+\omega$ and for the unperturbed Kepler orbit,
$ {{P}\over{r}} = 1 + \epsilon \cos\theta$, $P$ is the latus rectum, and $E$ is the anomaly,
{\it i.e.} $r = P(1-\epsilon\cos E)$.
The central angle $\theta$ is found via the usual definition of angular
momentum.
When we specialize these
Kepler orbit evolution equations to the
case of isotropic and homogeneous friction we learn that
(see \cite{Danby}, Chapter 11, section 7 but using $\beta(p) p$
for $T$ in that reference)
\begin{equation}
{{{\rm d} a}\over{ {\rm d}t}} = 2pa^2 \beta(p) p
\label{d7}
\end{equation}
\begin{equation}
{{{\rm d}\omega}\over{ {\rm d}t}} = {{2\sin\theta}\over{\epsilon}} \beta(p)
\label{d8}
\end{equation}
and
\begin{equation}
{{{\rm d} \epsilon}\over{ {\rm d}t}} = 2(\cos\theta + \epsilon)\beta(p)
\label{d9}
\end{equation}
We can now specialize further to the marginal case, linear friction
$\beta(p) = \beta = const$. To integrate
these equations, note $r^2 {{{\rm d}\theta}\over{{\rm d}t}} = L =
L_0 e^{-\beta t}$ and, in terms of the force components, $N=0$, and
$R = \beta(p) p \cos{\upsilon}$ and $B = \beta(p) p \sin{\upsilon}$
where $\upsilon$ is the angle between the radius vector and the
tangent to the orbit. That angle can be written
using the parameteric form of $r$ in terms of the
constants of the orbit and the angle $\theta$,
($\sin\upsilon = L/rp$ and $\cos\upsilon = \epsilon\sin\theta/Lp$)
resulting in a self-contained
pair of ODE's in $\epsilon$, $\theta$ and $t$,
\begin{equation}
{{{\rm d}\theta}\over{{\rm d}t}} = {{e^{+3\beta t}}\over{L_0^3}}
(1+\epsilon\cos\theta)^2
\label{d10}
\end{equation}
\begin{equation}
{{{\rm d}\epsilon}\over{{\rm d}t}} = -2\beta(\cos\theta + \epsilon)
\label{d11}
\end{equation}
If we integrate these to leading order in $\beta$ only (by, for example,
using the first equation to eliminate the time derivative to leading order
in $\beta$) we do indeed find that the eccentricity is an orbit-averaged
constant of the
motion.
But difficulty arises when we try to understand these equations
beyond leading order in the damping, as a direct numerical integration
of the equation set reveals (Figure 1). For a broad set of initial
angles and small initial eccentricities,
the $\epsilon$ passes through zero and
goes negative. For comparison, the eccentricity (i.e. the
square root of the length of the ${\vec S}$ vector) computed by numerical
integration of the original equations of motion for precisely
the same mechanical parameters and initial conditions is included on that
figure.
\begin{figure}
\includegraphics{eccentricity_vs_time.eps}
\caption{The eccentricity in the actual damped Kepler problem (solid curve) compared with the eccentricity from \eref{d10} and \eref{d11} (dashed curve) versus time. Note
that for the later the eccentricity can oscillates through zero and can even, as in this case, asymptote to a negative value.}
\label{fig1}
\end{figure}
Even if one only wanted to assign importance to the
asymptotic change in the eccentricity, that asymptotic change
from integrating the equation pair \eref{d10} , \eref{d11}
does not scale correctly with the
damping coefficient, as may be checked numerically
(see \cite{Danby} for further admonisions against using the
``variations of constants'' method over long timescales).
Clearly the ``variations of constants'' method at higher orders in the
evolution leads to unphysical results at short and long timescales.
The fault is traceable to
the fact that in higher order there are $\beta$-
(the damping coefficient) dependent terms in the orbit shape
whose
contributions are ignored substituting for $r$
using the undamped Kepler shape of the ellipse.
This substitution is however
inectricably part of the
``variation of constants'' method.
To further clarify this problem with the
``variations of constants'' method,
it is not
due to some ambiguity in the
eccentricity of a non-closed orbit,
since eccentricity itself, rendered as the length of
the ${\vec S}$ vector, has a local definition. Algebraically,
with this definition of the eccentricity,
note that $\epsilon^2 - 2L^2 U = k^2$ in the $1/r$ potential
{\it even under arbitrary damping}. A more useful
algebraically identical form is
$\epsilon^2 = 4{\cal V}^2r^2 -2 {\cal R}^2 H$,
from which, since $H$ is negative for any damping
function on a bounded orbit, we see immediately
that $\epsilon^2$ is bounded away from zero.
We now, in two parts, describe an approach emphasising the
secular evolution of the dynamical symmetry, that addresses
this mismatch with the usual ``variation of constants'' method.
For simplicity we focus in the main on
potentials with fixed scaling wieght $\alpha$, deined through
$ V(r) = kr^\alpha$. Orbits in any central potential are
characterized by a fixed orbital plane and a single dimensionless
parameter, the ratio $d/c$ of the perihelion distance $d$ to the
aphelion distance $c$. Let $L$ denote the angular momentum so that
$ V_{eff}(r) = {{L^2}\over{2r^2}} + V(r)$ is the effective potential.
Then from $V_{eff}(c) = U = V_{eff}(d)$ where $U$ is the total energy
for a $V(r)$ of a fixed $\alpha$, we have,
\begin{equation}
{{U}\over{k}} = {{c^{\alpha+2}-d^{\alpha+2}}\over{c^2-d^2}} \qquad \qquad
{{L^2}\over{2k}} = {{c^{\alpha}-d^{\alpha}}\over{c^2-d^2}} c^2 d^2
\label{9}
\end{equation}
that then can be reduced to a 'dispersion relation' between $U$ and $L$,
\begin{equation}
{{U^{\alpha+2}}\over{k^2 L^{2\alpha}}} = f(d/c)
\label{10}
\end{equation}
where $f$ in this case is a monotonic function on $[0,1]$. Note also
that $f(x) = f(1/x)$. We call $d/c$ the aspect ratio of the orbit
(related to the eccentricity in the $\alpha = -1$ case).
Thus in leading order (only) in the damping we think of the
RHS as a function of the orbital eccentricity only.
In applications, the differential form of \eref{10} is particularly useful,
\begin{equation}
\biggl[(\alpha+2) {{\delta U}\over{\delta L}} -2\alpha {{U}\over{L}} \biggr] {{\delta L}\over{U}}=
{{f'}\over{f}} \delta (d/c)
\label{11}
\end{equation}
Equations of this sort are often written down when referring to the secular evolution
of orbital system (see for example \cite{PJ} and references therein).
As before consider further only damping forces that are isotropic
and homogeneous; they can be written in the form from the
previous section, ${\vec F} _{drag} = -\beta(p) {\vec p}$.
(to simplify notation we henceforth drop the vector symbol over the $p$
denoting by $p$ both $|p|$ and ${\vec p}$, unambigious by context).
In the limit of weak damping we expect $L$ to
be approximately constant so that, time averaging, we arrive at
$<\delta L> = -<\beta(p)> L$
to leading order in $\beta(p)$.
Note also that
$\delta U = -\beta(p) p^2$ to leading order in $\beta$.
Note that the time derivative of ${\cal R}$ is the sum of a Poisson
bracket with H plus a term probprtional to $\beta$ (see \Eref{4}).
This is, ${{{\rm d}{\cal R}}\over{{\rm d t}}} = 2{\cal V} + \cal{O}(\beta)$ which,
averaged over bounded orbits, indicates
(the virial theorem) that $<{\cal V}> =\cal{O}(\beta)$.
Thus for $V(r) = kr^\alpha$ this implies that
$<p^2> = \alpha <V>+{\cal O}(\beta)$ so that
$<U> = {{\alpha+2}\over{2}} <V> + {\cal O}(\beta)$,
which to leading order in $\beta$ in \Eref{11} indicates
\begin{equation}
-{{2\alpha}\over{<p^2>}} \biggl( <\beta(p)p^2> - <\beta(p)><p^2> \biggr) =
{{f'}\over{f}} \delta(d/c)
\label{12}
\end{equation}
Thus restricted to
linear damping
($\beta$ constant), but for any
$\alpha$, the averages in \eref{12} factorize trivially
and the aspect ratio
is unchanged to leading order under linear damping.
The \Eref{12} also indicates that
this will, in general, not be the case for
a velocity dependent damping coefficient.
Although for potentials with a
fixed scaling exponent
$\alpha$ there is but one dimensionless parameter
(See LHS \Eref{10}), the introduction
of the damping coefficient $\beta$ introduces new length and time scales,
indicating that the orbital aspect ratio $d/c$ may be a
function of $\beta$ and time.
The fact that $\beta$ is time odd does apparently not preclude
its inclusion to linear order in the orbital aspect ratio in general.
Thus, we repeat, the conclusion that for any monomial
potentials linear damping preserves the orbital shape is not a
consequence of dimensional analysis and discrete symmetries.
As a final check, note that
relation \Eref{12} and \Eref{26} are both
consistent with the attractors of the secular flow in the orbital shape.
For circular orbits $p^2$ is a constant of the motion (again to
leading order in $\beta$) and thus the
LHS of \Eref{12} is zero, as expected by symmetry.
Note that in contrast to \eref{12} in \Eref{26} the change in the
eccentricity is proportional to the eccentricity for any
$\beta(p)$, and since the eccentricity vanishes in this limit its
orbit-averaged change by \eref{26} does as well.
Also the strictly radial infall orbit limit is one in which
the inner radius, $d \rightarrow 0$, and so the LHS of \Eref{12}
being non-zero in this limit
looks inconclusive. But, by the definition of $f$ via \Eref{10} we
see that in this limit
$f \rightarrow 0$ or $f \rightarrow \infty$ depending on the
sign of $\alpha$. Thus, by \eref{9}, $L=0$ and remain zero for
any $\beta(p)$.
In Tarasov's formulation, since \eref{26} is fully vector covariant
for isotropic and homogeneous (but otherwise arbitrary $\beta(p)$)
the change in the ${\vec S}$ must be along the vector itself
for the radial infall case.
Furthermore, as indicated in the discussion following \eref{26}, the
change $\Delta {\vec S}$ is linear in ${\vec L}$ (for any $\beta(p)$)
which vanishes in the radial infall case.
In summary, both prescriptions indicate
that circular orbits and radial infall
must satisfy $<\dot {\vec S}> = 0$
{\it for any damping function} as expected on the grounds by
symmetry.
Furthermore, it is straightforward to go further and perturbatively
show using \eref{12} and the equations of motion
that $\beta = const.$ is the {\it only}
shape-preserving damping function for the Kepler potential ($\alpha = -1$).
The result is clearly common to all monomial central potentials only, as it
is straightforward to demonstrate a counterexample in a more complicated
potential.
This is due to the fact that there are no additional length
scales in the potential and is not the case with other
potentials, such as the effective potential in General Relativity
(where the Schwarzschild radius arises as a second length scale in
the potential).
Returning to the rather general statement \eref{26}, in the Tarasov
formulation, the explicit time dependence of a candidate constant of motion
${\cal O}$ gives a second term which
cancels the last two terms. If the operator has a
fixed momentum scaling weight (for example, $L$ is weight 1 and $S$ is
essentially wieght 2), the last two terms will be of that same scaling weight
only for the case of linear friction, $\beta(p) = const.$
Note that this argument does not rule out the existence of
additional constants of the motion in the dissipative system
that scale to zero as one goes to the Hamiltonian limit.
The argument does, however, certify that in the case of linear
friction the original Hamiltonian symmetries do survive to leading order
in that friction.
Having shown that linear friction preserves the eccentricity to
leading order begs the question of what happens in higher order in the damping. In the
spirit of the discussion after \eref{10} where the Virial played a
key role, consider the time evolution of that part of the dynamical algebra
\begin{equation}
{\dot {\cal R}} = \{ {\cal R} , H\} -\beta(p) {\cal R}= 2{\cal V} - \beta(p)
{\cal R}
\label{4}
\end{equation}
\begin{equation}
\fl {\dot {\cal V}} =
\{{\cal V} , H\} -2\beta(p)(2{\cal V} -H)
= -{{1}\over{r}} (\partial_r V +
{{1}\over{2}} \partial_r(r\partial_r V)) {\cal R}
-2\beta(p)(2{\cal V} -H)
\label{5}
\end{equation}
\begin{equation}
{\dot H} = -2\beta(p) (2{\cal V} - H)
\label{5prime}
\end{equation}
and for completeness, we have
\begin{equation}
\{{\cal R} , {\cal V} \} = H +{\cal V} -V+{{r}\over{2}} \partial_r V
+ {{r}\over{2}}\partial_r(r\partial_r V)
\label{6}
\end{equation}
To orient the reader to the content of these, first note the
Hamiltonian limit ({\it i.e.} $\beta(p) \rightarrow 0$ limit)
for the Kepler case ($\alpha = -1$), both
$<{\cal V}> \rightarrow 0$ and $<{\cal R}/r^3> \rightarrow 0$
as expected.
We thus expect both of these time averages to be atleast proportional
to some positive power of $\beta$. Now, in the abscence of damping
${\cal V}$ is time even and ${\cal R}$ is time odd. Formally,
taking $\beta$ to be time odd preserves this discrete symmetry of the
above evolution equations. Since we expect the $<{\cal V}>$ and $<{\cal R}>$
to be analytic functions of $\beta$, it must thus
be that $<{\cal V}>$ vanishes
quadratically as $\beta \rightarrow 0$.
An elementary argument now certifies that the $<{\cal V}>$ must
be nonpositive in the damped system. Take $\beta(p) = \beta$ a constant.
Consider the radial component of the velocity, ${\cal R}/r$.
It must average to zero in the $\beta \rightarrow 0$ limit. Since the
damped orbit must shrink, we thus expect $<{\cal R}/r> \sim < -C\beta>$ for
some positive quantity $C$ (a function
of the other orbital parameters, etc.). But now take the evolution equation
\eref{4} divide by $r$ and time average. Clearly, integrating
by parts, $<{\dot {\cal R}}/r> = -<{\cal R}/r^2 ~ {\dot r}>
= -<{\cal R}^2/r^3>$ implying that the time average
$<2{\cal V}/r - \beta {\cal R}/r>$
must also be strictly negative. But since $<{\cal R}/r>$ must already
be negative, the $<{\cal V}>$ must also be strictly
negative in the damped system.
Specializing to Kepler ($\alpha = -1$), differentiating
$\epsilon^2 = |{\vec S}|^2$ in time and applying the
equations of motion of the system with friction, we learn
that $\frac{d}{dt}\epsilon^2 = -8\beta L^2 {\cal V}$. Using the fact that
$<\cal V>$ is negative and order $\beta^2$ and integrating both sides,
we learn that the asymptotic change in the eccentricity to leading order
is positive and also of order $\beta^2$ (note the integral
itself scales as $1/\beta$). Furthermore, since these are exact evolution
equations, we have shown that the integration is
well behaved throughout. Thus linear friction causes Kepler
orbits to become more eccentric by a fixed
amount that scales with the square of the linear damping coefficient.
\section{Conclusion}
Typically Hamiltonian symmetries lose thier
relevance to the geometry of the trajectories
when damping forces are added to the Hamiltonian system.
If the damping is weak, homogeneous and isotropic, then
for linear damping in monomial potentials, we have shown
that orbit-averaged shape is stationary.
This can understood most easily through
Tarasov's generalization of conserved
quantities from the Hamiltonian context to the non-Hamiltonian setting.
This approach also quantifies in precise analytic
terms the fate and subsequent utility of the dynamical symmetry
algebra in the associated non-Hamiltonian system.
There are three main frameworks for understanding orbital motion in a
perturbed central
field. The first
is directly from the equations of motion; this admits straightforward
generalization to the
non-Hamiltonian case but
somewhat obscures the structure and fate of the dynamical
symmetry group.
The second, namely the KS construction, embeds the
Kepler orbit
problem in the higher dimensional set of
harmonic oscillators with constraints; this illuminates the dynamical symmetry
group but does not
seem to readily admit a generalization to the non-Hamiltonian system. Lastly,
the geometrical
approach, namely that which associates the Kepler Hamilton equations to
geodesic flow on manifolds
of constant curvature, also illuminates the dynamical symmetry
group while making
the generalization to the non-Hamiltonian case somewhat unclear.
In light of these difficulties,
we used Tarasov's framework (and applied to the damped central
field problem here) for extending Poisson symmetries to dissipative
systems, emphasising its utility
in making crisp connections between
dynamics, algebra and the geometric character of the solutions.
Finally,
the dynamical algebra remains whole in first order in the linear
dissipative system, but flow at higher order
is not trivial.
The secular perturbative
method ``variation of constants'' is not
adequate to explain this, however an elementary method based on
the Virial subalgebra explains the change in the shape of kepler
orbits in higher order in linear damping.
\ack This work was supported in part by
the National Science Foundation through a grant to the Institute of
Theoretical Atomic and Molecular Physics at Harvard University and the
Smithsonian Astrophysical Observatory, where this work was begun and
by a fellowship from the Radcliffe Institute for Advanced Studies where
this work was concluded. It is a pleasure to acknowledge useful discussions
with Harvard-Smithsonian Center for Astrophysics
personell Mike Lecar, Hosein Sadgehpour, Thomas Pohl and Matt Holman.
\vfill
\par
\eject
\ \
\vskip .4in\
\centerline{\bf Bibliography}
|
2,877,628,090,205 | arxiv | \section{Introduction}\label{sec:intro}
In neutron star low-mass X-ray binaries (NS-LMXBs), a neutron star accretes matter from a low-mass companion star. The vast majority of NS-LMXBs contains weakly magnetized neutron stars, and these systems are traditionally divided into two main subclasses based on their correlated spectral and timing properties: the Z sources and atoll sources \citep{hasinger1989}. The names derive from the shapes of the tracks the sources trace out in the X-ray color--color and hardness--intensity diagrams (CDs and HIDs). The Z sources are the more luminous, with near-Eddington X-ray luminositities, whereas the atoll sources are thought to in general have $L_\mathrm{X}\lesssim0.5L_\mathrm{Edd}$. Further divisions of these two main NS-LMXB types have also been made; in particular, the Z sources have been divided into the Cyg-like and Sco-like subtypes (\citealt{kuulkers1997} and references therein).
In addition to differences in X-ray luminosity, rapid-variability characteristics, and spectral properties (see, e.g., \citealt{vanderklis2006} and references therein), the NS-LMXB subclasses also differ with respect to behavior in the radio band \citep{migliari2006} and the rates and properties of type I X-ray bursts exhibited \citep{galloway2008}. Understanding what physical factors underlie this variety in characteristics has been a long-standing problem in the study of X-ray binaries. It has long been clear that the mass accretion rate must play an important role (e.g., it seems evident that Z sources must in general accrete at a higher rate than atoll sources, given the former's significantly higher luminosity), but exactly what that role is, and where other physical parameters enter the picture, has been debated.
Another long-standing question concerns the physical nature of the motion of Z sources in the CD/HID. The Z sources trace out characteristic tracks in the CD and HID (consisting of the so-called horizontal, normal, and flaring branches) on typical timescales of hours to a few days. Initially, it was assumed that motion along the Z track was driven by changes in the mass accretion rate, with $\dot{M}$ increasing monotonically from the tip of the horizontal branch to the tip of the flaring branch (e.g., \citealt{hasinger1990}), but later observations cast doubts on this, and other scenarios have been proposed (see, e.g., \citealt{vanderklis2001,homan2002,homan2007,church2012}). In addition to motion along the tracks these sources show another type of motion, usually on longer timescales (days to weeks), where the tracks can shift, primarily in the HID. This has been referred to as secular motion/shifts/changes/variations. In most cases these shifts are small and do not lead to significant changes in the shapes of the tracks. However, a few sources have been known to exhibit much stronger secular changes where the tracks shift and change shape radically in both the CD and HID.
A breakthrough in our understanding of the Z/atoll phenomenology and the nature of secular evolution came with the transient NS-LMXB \mbox{XTE J1701--462}, which over the course of its 2006--2007 outburst evolved through all subclasses of low-magnetic-field NS-LMXBs---from a Cyg-like Z source at the highest luminosities to a Sco-like one, followed by a phase in the atoll source soft state (during which type~I X-ray bursts were seen), and ending with a transition to the atoll source hard state before returning to quiescence \citep{lin2009a,homan2010}. \citet{lin2009a} performed a detailed spectral analysis of the entire outburst, using data from the \textit{Rossi X-ray Timing Explorer} (\textit{RXTE}), and their results indicate that changes in the mass accretion rate were responsible for the evolution of the source. \citet{homan2010}---hereafter referred to as \citetalias{homan2010}---further argue that the observed behavior of the source implies that differences in the mass accretion rate can alone explain the existence of the various NS-LMXB subclasses, and that it is not necessary to invoke differences in other parameters, such as the magnetic field of the neutron star \citep{hasinger1989,psaltis1995} or the inclination angle of the system \citep{kuulkers1995}.
The main goal of this paper is to study to what extent (if any) evolution similar to that of \mbox{XTE J1701--462}\ is observed in other sources. Comparisons of \mbox{XTE J1701--462}'s evolution at the low-luminosity end with atoll sources have been made in \citet{lin2009a}, \citetalias{homan2010}, and \citet{munoz2014}, showing that there the source behaved similar to other low-luminosity (atoll) NS-LMXBs; in this paper we focus mainly on the high-luminosity (Z source) portion of \mbox{XTE J1701--462}'s evolution. To this end we performed a detailed analysis of the CDs and HIDs of three luminous neutron star X-ray binaries---\mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}---using all available \textit{RXTE}\ Proportional Counter Array (PCA) data. Similar to \mbox{XTE J1701--462}, these sources are known to have shown strong secular evolution \citep[e.g.,][]{oosterbroek1995,kuulkers1996a,wijnands1997,shirey1999a,schnerr2003}. Since we are mainly interested in sources that cross subclass boundaries, we do not include in our study persistent Z sources that show only mild secular shifts and no significant changes in the shapes of their tracks. Although small subsets of the \textit{RXTE}\ data sets for \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}\ have been studied in several papers---usually with an emphasis on spectroscopy and/or timing analysis---we are not aware of any papers presenting a comprehensive study of the secular evolution of any of these sources using a large amount of \textit{RXTE}\ data, with the exception of \citet{schnerr2003}, who studied a large set of observations of \mbox{GX 13+1}\ made in 1998 (see discussion in Section~\ref{sec:gx_13+1}). However, since then the amount of \mbox{GX 13+1}\ data in the \textit{RXTE}\ archive has grown considerably. We also note that Shirey and collaborators studied CDs and HIDs of \mbox{Cir X-1}\ using a number of observations made in 1996 and 1997 in a series of papers (\citealt{shirey1996,shirey1998,shirey1999a}; see discussion in Section~\ref{sec:cir_x-1}), but these observations represent only a small fraction of the currently available \textit{RXTE}\ data for this source. In addition to our analysis of \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}\ we also perform some reanalysis of \mbox{XTE J1701--462}\ in order to ensure complete consistency with the analysis of the other three sources and to facilitate comparisons between them and \mbox{XTE J1701--462}. We note that although \mbox{Cir X-1}\ may be a rare low-magnetic-field neutron star high-mass X-ray binary (as further discussed in Section~\ref{sec:cir_x-1}) we will for simplicity in general refer to all four sources studied in this paper as NS-LMXBs.
The structure of the paper is as follows. In Section~\ref{sec:analysis} we describe the general data analysis steps common to all four sources, but leave the description of analysis specific to each source to Section~\ref{sec:results}. In Section~\ref{sec:1701} we briefly describe our analysis and present our results for \mbox{XTE J1701--462}. This analysis largely follows that previously performed by \citetalias{homan2010} (and includes creating a sequence of CD/HID tracks that shows the secular evolution of the source); we therefore mostly refer to that paper for details. In this section we also tie this source's behavior to the various source states/branches generally seen in the CDs and HIDs of NS-LMXBs. In Sections \ref{sec:cyg_x-2}--\ref{sec:gx_13+1} we describe our analysis and present our results for \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}, respectively. For each source we construct a sequence of CD/HID tracks (analogous to the one for \mbox{XTE J1701--462}) that illustrates its secular evolution; we also give a brief background on each of these three sources. In Section~\ref{sec:discussion} we discuss our results, and in Section~\ref{sec:summary} we give a summary of our results and conclusions.
\section{Data Analysis}\label{sec:analysis}
\subsection{Data Extraction}\label{sec:extraction}
We used the PCA data from all pointed \textit{RXTE}\ observations of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}\ obtained during the lifetime of the mission. In Table~\ref{tab:source_sample} we list the total amount of useful data (i.e., data remaining after the removal of X-ray bursts and bad data of any kind) for each of these sources. In the case of \mbox{Cir X-1}, however, the quoted exposure time includes data affected by local absorption; see Section~\ref{sec:cir_x-1}. Because of the large number of observations (over 2300 individual ObsIDs), most of the data analysis steps described below were automated.
\begin{deluxetable}{lc}[t]
\tablewidth{8.6cm}
\tablecaption{Total Exposure Times for Analyzed Sources \label{tab:source_sample}}
\tablehead{\colhead{\hspace{-1.24cm}Source} & \colhead{Exp.\ Time (Ms)}}
\startdata
\mbox{XTE J1701--462}\ & 2.71 \bigstrut[t] \\[0.6ex]
\mbox{Cyg X-2}\ & 2.28 \\[0.6ex]
\mbox{Cir X-1}\ & 2.57 \\[0.6ex]
\mbox{GX 13+1}\ & 0.58 \bigstrut[b]
\enddata
\end{deluxetable}
The data were analyzed using HEASOFT, versions 6.9--6.12, as well as locally developed software. We used the \textit{Standard-2} data, extracting counts in \mbox{16 s} time bins, combined from all three xenon layers, from all active Proportional Counter Units (PCUs) at any given time. The data were filtered following the recommendations of the \textit{RXTE}\ Guest Observer Facility (GOF);\footnote[4]{See \url{http://heasarc.gsfc.nasa.gov/docs/xte/xhp_proc_analysis.html}.} this included the removal of data around PCU voltage breakdown events (but only from the relevant PCU in each case). We also corrected for dead time using the standard procedure recommended by the \textit{RXTE}\ GOF and subtracted background using the faint or bright background model as appropriate in each observation, based on the average count rate during the observation after exclusion of any type I X-ray bursts.
\subsection{Burst and Particle Flare Removal}\label{sec:removal}
We removed all data obtained during type I X-ray bursts; to identify bursts in an automated fashion we used a method similar to the one used by \citet{remillard2006b}. We note that although some bursts observed from \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ do not show cooling along their tails---as is usually observed in type~I X-ray bursts---the origin of these faint bursts is nevertheless very likely thermonuclear \citep{linares2011} and they were removed.
PCUs~0 and 1 lost their propane layers in 2000 and 2006, respectively, diminishing their ability to reject events due to particle background. Based on a comparison with data from PCU~2---which did not suffer from such particle flares and was nearly always on---these events were identified and subsequently removed from the data. We note that since PCU~2 data exist for all times during which PCU~0/1 data were removed, we did not exclude data from any given point in time entirely, but simply reduced the number of PCUs contributing to a particular time bin in these cases.
\subsection{Response Correction}\label{sec:correction}
From the \textit{Standard-2} data we extracted count rates in several different energy bands and used those to calculate colors for CDs and HIDs. The response of the \textit{RXTE}\ PCUs evolved over the duration of the mission due to several factors, and these changes in response must be corrected for before data across the lifetime of the mission can be combined into a single CD or HID for a given source. For this correction we used observations of the Crab Nebula, which was observed on average $\sim$3 times per month throughout the mission. For each Crab observation we extracted the average count rate in each of the energy bands of interest for each active PCU, using the \textit{Standard-2} data. We then fitted the mission-long light curve (where each data point represented the average count rate during a single observation) for each PCU in each energy band with a piecewise-linear function. We (somewhat arbitrarily) chose MJD 50800 (1997 December 18) as our reference epoch and normalized all data points to that date, first for each PCU individually, and then in each case the count rates from PCUs 0, 1, 3, and 4 to that of PCU~2.
Recently it has become apparent that X-ray emission from the Crab Nebula shows significant intrinsic variability. \citet{wilson-hodge2011} show that from 2001 to 2010 the \textit{RXTE}\ PCA count rate from the Crab (after correcting for changes in response) varied rather irregularly by several percent. The (response-corrected) PCU~2 variability is $\sim$5\% in the \mbox{2--15 keV} band and $\sim$8\% in the \mbox{15--50 keV} band. This can compromise our correction for the variation in the PCA response. Comparing our mission-long Crab light curves to those of \citet{wilson-hodge2011} we can estimate the magnitude of the effect on our derived rates and colors. The relative amplitude of the Crab count rate fluctuations gradually increases with energy, but otherwise the variability behaves in more or less the same way in the different energy bands. Due to this energy dependence of the variability it should have some effect on the colors. However, given the relatively small difference in the strength of the variability between adjacent energy bands, we expect that this variability will largely cancel out in the colors. We estimate that shifts in our colors due to the Crab variability are at most $\sim$1\%--2\% and in most cases significantly less; we therefore expect the influence of this on our results to be negligible (cf.\ discussion about uncertainty due to counting statistics in Section~\ref{sec:rebinning}). The intensity is affected more strongly; we expect that shifts there can possibly be as high as $\sim$6\% in the worst case. However, they are generally much smaller---probably less than $\sim$4\% in almost all cases and typically in the 0\%--3\% range---and we expect the effect of this on our results to in general be negligible as well.
\subsection{Construction of CDs and HIDs}\label{sec:rebinning}
For the creation of CDs (i.e., hard color versus soft color) and HIDs (hard color versus intensity) we used color definitions similar to those used in \citetalias{homan2010}: we defined our soft color as the net counts in the \mbox{4.0--7.3 keV} band divided by those in the \mbox{2.4--4.0 keV} band, and our hard color as net counts in the \mbox{9.8--18.2 keV} band divided by those in the \mbox{7.3--9.8 keV} band. The intensity we used for the HID was the net count rate per PCU in the \mbox{2--60 keV} band. Before creating our CDs and HIDs we combined the counts from all active PCUs for each \mbox{16 s} time bin (after performing the corrections and filtering described above) and then rebinned the data in a given observation in order to maintain a more uniform size in the error bars across different values in count rate. We set a minimum of 16,000~counts in the \mbox{2--60 keV} band (after background subtraction) for each rebinned data point. The rebinning was done in an adaptive/dynamic fashion---i.e., we in general did not use a single value for the time binning factor over an entire observation, but instead allowed the factor to vary (while imposing our counts limit) to adapt to a possibly varying count rate. In a few cases we applied a larger counts limit; these will be mentioned explicitly. In some instances entire observations did not contain enough counts to reach the 16,000~counts minimum; in those cases we created a single data point from the whole observation if the total number of counts was larger than 10,000, but excluded it from further consideration if it had fewer counts. (Any exceptions to this will be explicitly mentioned.) We note that for most observations of the four bright sources studied here no rebinning was necessary, since each \mbox{16 s} time bin usually contained more than the minimum 16,000~counts. With this minimum counts limit the uncertainty due to counting statistics is at most $\sim$2\% in the soft color and at most $\sim$3\% in the hard color (often much less); the uncertainty in the intensity is always less than 1\%.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm,trim=0 0 0 -15]{f1_300dpi.png}}
\caption{CD and HID representing the entire \textit{RXTE}\ PCA data set of \mbox{XTE J1701--462}.}\label{fig:1701_all_data}
\end{figure}
\section{Results}\label{sec:results}
\subsection{\mbox{XTE J1701--462}}\label{sec:1701}
\mbox{XTE J1701--462}\ was discovered with the \textit{RXTE}\ All-Sky Monitor (ASM) on 2006 January 18 as the source was entering outburst \citep{remillard2006a}. The first \textit{RXTE}\ PCA observation took place on 2006 January 19, and the source was subsequently observed with the PCA on average \mbox{$\sim$1.5--2} times per day for the remainder of the 19 month outburst, apart from a $\sim$50~day period in late 2006 and early 2007 during which the source could not be observed due to proximity to the Sun. The source returned to quiescence in early 2007 August and has since remained inactive apart from occasional low-level flaring (up to \mbox{$\sim$$10^{35}\textrm{ erg s}^{-1}$}), which has been observed by \textit{Swift}\ and \textit{XMM-Newton}\ \citep{fridriksson2010,fridriksson2011}. As in \citetalias{homan2010}, dates during the outburst will in this paper be referred to as days since the start of 2006 January 19 (MJD 53754.0).
\begin{figure*}[t]
\centering
\includegraphics[height=11.9cm]{f2a_300dpi.png}
\hspace{-0.04cm}
\includegraphics[height=11.9cm]{f2b_300dpi.png}
\hspace{-0.04cm}
\includegraphics[height=11.9cm]{f2c_300dpi.png}
\caption{CDs and HIDs for \mbox{XTE J1701--462}, showing the secular evolution of the source. Each of the 12 panels (consisting of a CD on the left and an HID on the right) corresponds to a particular selection of data from the entire \textit{RXTE}\ PCA data set for the source; red data points correspond to a particular subset (exhibiting minimal secular shifts) within a given selection (see Table~\ref{tab:selections}). Particular source states/branches are indicated in a few of the panels (see text for definitions). The dashed and solid lines show the approximate paths followed by the NB/FB and HB/NB vertices, respectively, as the tracks evolve. Data are binned to a minimum of 16,000~counts per data point, except for the IS and EIS data in selection L, which are binned to a minimum of 64,000~counts. Note the change in the intensity scale between the different HID columns. We also note that essentially the same tracks (with only small modifications here) were presented in slightly different form in Figures 3 and 4 in \citetalias{homan2010}.}\label{fig:1701_sequence}
\end{figure*}
The combined CD and HID for the entire 2006--2007 outburst of \mbox{XTE J1701--462}\ are shown in Figure~\ref{fig:1701_all_data}; the diagrams clearly illustrate that the source tracks exhibited strong secular motion during the outburst. As in \citetalias{homan2010} we divided the observations of the source into 12 subsets---hereafter referred to as selections---which we label A--L. In Figure~\ref{fig:1701_sequence} we show the CDs and HIDs for all 12 selections, which illustrate the secular evolution of the source during the outburst. We label the source states/branches of a few representative tracks: a Cyg-like Z track (selection B), a close-to Sco-like Z track (E), and two atoll-like tracks (K and L). For the Z-like tracks these are the horizontal branch (HB), the HB upturn, the normal branch (NB), the flaring branch (FB), and the dipping FB. For the atoll-like tracks these are the lower and upper banana branches (LB and UB), the island state (IS), and the extreme island state (EIS). The banana branch, IS, and EIS are also referred to as the soft, intermediate/transitional, and hard state, respectively. (We refer to \citetalias{homan2010} for examples of CDs/HIDs of several Cyg/Sco-like Z sources and atoll sources and a comparison of those with the CD/HID tracks of \mbox{XTE J1701--462}.)
The 12 data selections mostly correspond to particular ranges in the low-energy (\mbox{2.0--2.9 keV}) count rate, which \citetalias{homan2010} found to closely trace the secular changes during most of the outburst. This tracing of the secular evolution with the low-energy count rate breaks down when the source enters the HB or a dipping FB. For days~0--28 time-based selections (A and B) were therefore used, rather than ones based on the low-energy count rate. In addition, data from several HB excursions taking place after day~28 had to be moved to the same selection as neighboring NB data in the low-energy light curve. The time and count rate intervals we used are given in Table~\ref{tab:selections}. Figure~2 in \citetalias{homan2010} shows a low-energy light curve of the entire outburst, indicating the count rate intervals used there (which are for the most part equivalent to ours) and the HB data moved between selections.
\begin{deluxetable}{lcc}[]
\tablewidth{8.6cm}
\tablecaption{Time or Count Rate Intervals Used for Data Selections \label{tab:selections}}
\tablehead{\colhead{\hspace{-0.17cm}Selection} & \colhead{Full Interval} & \colhead{Subinterval} \\[0.4ex]
& \colhead{(counts s$^{-1}$ PCU$^{-1}$)\tablenotemark{a}} & \colhead{(counts s$^{-1}$ PCU$^{-1}$)\tablenotemark{a}}}
\startdata
A & Days 13.5--20.5 & Days 13.5--19.5 \bigstrut[t] \\[0.6ex]
B & Days 0--13.5 and 20.5--28 & Days 2--13.5 \\[0.6ex]
C & 106--114/Days 28--32.5 & \nodata \\[0.6ex]
D & 83--106 & 89--97 \\[0.6ex]
E & 66.5--83 & 72.5--78 \\[0.6ex]
F & 56--66.5 & 56--58.36 \\[0.6ex]
G & 46--56 & 50--52 \\[0.6ex]
H & 38.5--46 & 39.6--43 \\[0.6ex]
I & 31--38.5 & 33.5--35.5 \\[0.6ex]
J & 23.5--31 & 26--28.5 \\[0.6ex]
K & 15--23.5 & 20--22.5 \\[0.6ex]
L & 0.4--15/Days 550--564 & \nodata \bigstrut[b]
\enddata
\tablenotetext{a}{Count rates are in the \mbox{2.0--2.9 keV} band. Time intervals refer to days since MJD 53754.0.}
\tablecomments{}
\end{deluxetable}
Within most of the selections some secular motion is still evident, mostly in the HID. We therefore in 10 of the 12 selections color red the data points from a subset of the observations used in each case, to show tracks with minimal secular shifts; the subintervals in time or count rate defining these subsets are given in Table~\ref{tab:selections}. We also show, in both the CDs and HIDs, two straight lines that the NB/FB (lower) and HB/NB (upper) vertices of the tracks are seen to follow closely (as pointed out by \citealt{lin2009a}). We note that selections A--L do not represent a strict monotonic progression in time (although overall the evolution in time was from A to L); the source moved back and forth between selections during the outburst, mostly within the range F--H (see Figure~2 in \citetalias{homan2010}). It is clear from the sequence in Figure~\ref{fig:1701_sequence} that going from panel A to L the tracks smoothly evolve in shape from Cyg-like Z to Sco-like Z to atoll tracks as the upper and lower vertices both move monotonically along the vertex lines to higher color values and the overall intensity of the tracks decreases.
After day~566 (as the outburst was ending) the count rate from the source reached a roughly constant level of \mbox{$\sim$2 counts s$^{-1}$ PCU$^{-1}$} in the \mbox{2--60 keV} band (\mbox{$\sim$0.2 counts s$^{-1}$ PCU$^{-1}$} in the low-energy band). This residual flux can be attributed to diffuse Galactic background emission (see \citealt{fridriksson2010}). Figure~\ref{fig:1701_atoll} shows an alternative version of panel L in Figure~\ref{fig:1701_sequence} where we have binned the data more heavily and have subtracted this diffuse background emission based on data from observations made during the three weeks following the end of the outburst. In addition, we include in this version data from two observations made during days~564--566 (constituting the leftmost data point), which had a low-energy count rate of \mbox{$\sim$0.36 counts s$^{-1}$ PCU$^{-1}$}. We furthermore use a logarithmic scale for the horizontal axis in the HID so as to better illustrate the behavior of the source at the lowest count rates.
\subsection{\mbox{Cyg X-2}}\label{sec:cyg_x-2}
Cygnus X-2 (\mbox{Cyg X-2}) is one of the longest-known and most extensively studied X-ray binaries. \mbox{Cyg X-2}\ was classified as a Z source by \citet{hasinger1989} based on \mbox{\textit{EXOSAT}} data and is the prototype of the Cyg-like subgroup of the persistent Z sources. However, as was clear already from pre-\textit{RXTE}\ data \mbox{Cyg X-2}\ is unique among the six ``classic'' (persistent Galactic) Z sources in that it shows by far the strongest secular evolution \citep[e.g.,][]{kuulkers1996a,wijnands1997}.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm, trim = 0 0 0 -15]{f3_300dpi.png}}
\caption{Alternative version of panel L in Figure~\ref{fig:1701_sequence} with the data binned into fewer groups and Galactic background emission subtracted. Data in the LB (i.e, intensity \mbox{$\gtrsim$65 counts s$^{-1}$ PCU$^{-1}$} and hard color $\lesssim$0.85) are binned to a minimum of 128,000~counts per data point; data in the IS and EIS are binned to approximately one data point per day. The data represented by the leftmost point were not included in Figure~\ref{fig:1701_sequence}.}\label{fig:1701_atoll}
\end{figure}
\begin{figure}[b]
\centerline{\includegraphics[width=8.5cm]{f4_300dpi.png}}
\caption{CD/HID representing the entire \textit{RXTE}\ PCA data set of \mbox{Cyg X-2}.}\label{fig:cyg_x-2_all_data}
\end{figure}
\subsubsection{Analysis}\label{sec:cyg_x-2_analysis}
In Figure~\ref{fig:cyg_x-2_all_data} we show a CD/HID based on the entire \textit{RXTE}\ PCA data set for the source; strong variations in the shape and location of tracks are readily apparent. This combined CD/HID for \mbox{Cyg X-2}\ has strong similarities to the one for \mbox{XTE J1701--462}\ (Figure~\ref{fig:1701_all_data}). However, the fragmented nature of the data set (obtained over a period of 15~years) forces us to analyze the data in a manner different from the \mbox{XTE J1701--462}\ analysis.
\begin{figure*}[t]
\centering
\includegraphics[height=14.75cm]{f5a_300dpi.png}
\hspace{-0.055cm}
\includegraphics[height=14.75cm]{f5b_300dpi.png}
\hspace{-0.055cm}
\includegraphics[height=14.75cm]{f5c_300dpi.png}
\caption{A sequence of CDs and HIDs for \mbox{Cyg X-2}\ illustrating the secular evolution of the source. Table~\ref{tab:cyg_x-2_obs} lists the data used in each panel. In general, a panel combines data from observations widely separated in time (by months or years); exceptions to this are panels A, B, K, and N. In each of the other panels, a particular subset of the data---obtained over a period of at most a few days---is shown in red. The dashed and solid lines show the approximate paths followed by the lower and upper vertices, respectively. Note the change in the intensity scale between the different HID columns.}\label{fig:cyg_x-2_sequence}
\end{figure*}
For this analysis we considered all 591 individual ObsIDs for pointed observations of \mbox{Cyg X-2}\ made during the \textit{RXTE}\ mission (13 of which contained no useful data). Often multiple observations were fairly densely clustered together in time. Going chronologically through the data set the source jumps erratically back and forth around the CD/HID, in any given observation usually tracing out only short partial track segments. As a first step we went through the entire data set in time order, at any point combining into a single CD/HID track as many consecutive ObsIDs as possible without introducing clearly noticable secular shifts. We refer to such clusters of data (containing all data from a certain time interval) as \textit{subsets}. In a few cases significant secular motion took place during a single ObsID, requiring the observation to be split up between two different subsets. To identify secular motion we examined the tracks not only in the CD and HID, but also in a diagram of soft color versus intensity (which we refer to as an SID). The SID frequently yielded useful extra information/constraints. The result of this process was $\sim$300 subsets, which span periods as long as several days, although the vast majority is less than a day in length.
Considering only individual subsets results in a few tracks that seem to a large extent complete, but most of the subsets consist of shorter segments (i.e., incomplete tracks). By an ``incomplete'' track we mean that had the source stayed long enough at a particular stage in its secular progression (and had our observational coverage of the source been sufficiently comprehensive during that time) we expect that the source would have traced out a fuller (i.e., continuous and possibly more extended) track. It is important to note that this ``completeness'' is a function of location in the secular progression, as not all branches of the Z track are present at every stage of the secular evolution, and that there is always some uncertainty regarding what constitutes a ``complete'' track.
Fortunately, it was in general possible to combine subsets from various times throughout the \textit{RXTE}\ mission to form more complete tracks than are available from individual subsets. Usually, the most complete individual-subset tracks served as a foundation on which these combined tracks were built. To guide us in this process we took advantage of overlapping track segments and made sure that they lined up in all three diagrams---the CD, HID, and SID---which together provided a fairly stringent criterion for the appropriateness of combining particular segments.
\setlength\bigstrutjot{3pt}
\tabletypesize{\scriptsize}
\begin{deluxetable*}{clclc}[]
\tablewidth{18.0cm}
\tablecolumns{5}
\tablecaption{Time Intervals and Observations Used to Create \mbox{Cyg X-2}\ Tracks \label{tab:cyg_x-2_obs}}
\tablehead{ & & \colhead{Interval Length} & & \colhead{Exp.\ Time} \\[0.3ex]
\colhead{\hspace{-0.2cm}Panel} & \colhead{\hspace{-0.55cm}Time Interval (MJD)} & \colhead{days (hr)\tablenotemark{a}} & \colhead{\hspace{-4.03cm}ObsIDs} & \colhead{ks (hr)\tablenotemark{b}}}
\startdata
A & 50316.592--50316.764 & 0.17 (4.1)\phn\phn & 10063-[09:10]-01-00 & 9.0 (2.5) \bigstrut[b] \\
\hline
B & 51697.550--51698.705 & 1.15 (27.7)\phn & 40019-04-05-[00:10] & 33.9 (9.4)\phn \bigstrut \\
\hline
C & 50996.450--51000.593 (R) & 4.14 (99.4)\phn & 30418-01-[(01:05)-00,02-01] & 37.4 (10.4) \bigstrut[t] \\[0.4ex]
& 51561.344--51561.517 & 0.17 (4.1)\phn\phn & 40017-02-19-00 & 10.1 (2.8)\phn \\[0.4ex]
& 53138.767--53138.855 & 0.09 (2.1)\phn\phn & 90030-01-16-00 & 5.2 (1.4) \\[0.4ex]
& 53788.496--53788.577 & 0.08 (2.0)\phn\phn & 91009-01-42-[00:01] & 2.7 (0.7) \\ [0.4ex]
& 54649.890--54650.626 & 0.74 (17.6)\phn & 93443-01-01-[02,14:21] & 23.6 (6.6)\phn \bigstrut[b] \\
\hline
D & 51009.488--51009.725 & 0.24 (5.7)\phn\phn & 30046-01-01-00 & 12.9 (3.6)\phn \bigstrut[t] \\[0.4ex]
& 51349.631--51349.669 & 0.04 (0.9)\phn\phn & 40017-02-09-01 & 3.3 (0.9) \\[0.4ex]
& 53079.687--53079.785 & 0.10 (2.3)\phn\phn & 90030-01-04-00 & 5.2 (1.4) \\[0.4ex]
& 53222.440--53222.532 & 0.09 (2.2)\phn\phn & 90030-01-33-[00:01] & 4.8 (1.3) \\[0.4ex]
& 53286.420--53286.512 & 0.09 (2.2)\phn\phn & 90030-01-46-00 & 5.3 (1.5) \\[0.4ex]
& 53291.340--53291.433 & 0.09 (2.2)\phn\phn & 90030-01-47-00 & 5.4 (1.5) \\[0.4ex]
& 54007.232--54007.331 & 0.10 (2.4)\phn\phn & 92039-01-15-00 & 5.7 (1.6) \\[0.4ex]
& 54009.195--54010.341 & 1.15 (27.5)\phn & 92039-01-[17,18]-00 & 8.2 (2.3) \\[0.4ex]
& 54452.324--54452.554 & 0.23 (5.5)\phn\phn & 90022-08-04-[00:01] & 9.5 (2.6) \\[0.4ex]
& 54648.066--54649.329 (R) & 1.26 (30.3)\phn & 93443-01-01-[00,000,03:13] & 37.2 (10.3) \bigstrut[b] \\
\hline
E & 50168.900--50169.731 (R) & 0.83 (20.0)\phn & 10066-01-01-[00,000,001] & 37.3 (10.4) \bigstrut[t] \\[0.4ex]
& 51048.360--51048.599 & 0.24 (5.7)\phn\phn & 30046-01-07-00 & 13.4 (3.7)\phn \bigstrut[b] \\
\hline
F & 51055.495--51055.782 & 0.29 (6.9)\phn\phn & 30046-01-08-00 & 13.4 (3.7)\phn \bigstrut[t] \\[0.4ex]
& 51266.719--51267.126 & 0.41 (9.8)\phn\phn & 40017-02-05-[00:01] & 15.9 (4.4)\phn \\[0.4ex]
& 51413.523--51413.746 & 0.22 (5.4)\phn\phn & 40017-02-12-00 & 13.0 (3.6)\phn \\[0.4ex]
& 52429.372--52429.908 & 0.54 (12.9)\phn & 70016-01-01-[02,04:06] & 12.4 (3.4)\phn \\[0.4ex]
& 53511.868--53512.546 & 0.68 (16.3)\phn & 91010-01-01-[01,06:09] & 29.9 (8.3)\phn \\[0.4ex]
& 53685.118--53685.193 & 0.08 (1.8)\phn\phn & 90030-01-82-00 & 4.3 (1.2) \\[0.4ex]
& 54395.861--54398.037 (R) & 2.18 (52.2)\phn & 92038-01-[08-(00:02),09-(00:02),10-(00:03)] & 12.3 (3.4)\phn \bigstrut[b] \\
\hline
G & 51081.447--51081.725 & 0.28 (6.7)\phn\phn & 30046-01-12-00 & 14.3 (4.0)\phn \bigstrut[t] \\[0.4ex]
& 51444.401--51445.215 & 0.81 (19.5)\phn & 40019-04-[01-01G,01-02G,01-03,02-00,02-000] & 17.6 (4.9)\phn \\[0.4ex]
& 52533.929--52535.087 & 1.16 (27.8)\phn & 70015-02-[01-01G,02-00] & 19.1 (5.3)\phn \\[0.4ex]
& 53311.026--53311.105 & 0.08 (1.9)\phn\phn & 90030-01-51-[00:01] & 2.6 (0.7) \\[0.4ex]
& 53335.431--53335.724 & 0.29 (7.0)\phn\phn & 90022-08-01-00, 90030-01-56-[00:01] & 8.2 (2.3) \\[0.4ex]
& 53375.032--53375.053 & 0.02 (0.5)\phn\phn & 90030-01-64-00 & 1.8 (0.5) \\[0.4ex]
& 53394.710--53394.812 & 0.10 (2.5)\phn\phn & 90030-01-68-00 & 6.3 (1.8) \\[0.4ex]
& 53488.223--53488.310 & 0.09 (2.1)\phn\phn & 91009-01-12-[00:01] & 3.4 (0.9) \\[0.4ex]
& 53493.152--53493.227 (R) & 0.07 (1.8)\phn\phn & 91009-01-13-[00:01] & 2.8 (0.8) \\[0.4ex]
& 53498.069--53498.142 & 0.07 (1.8)\phn\phn & 91009-01-14-00 & 3.1 (0.9) \bigstrut[b] \\
\hline
H & 51539.192--51539.562 (R) & 0.37 (8.9)\phn\phn & 40017-02-18-[00:02] & 14.4 (4.0)\phn \bigstrut[t] \\[0.4ex]
& 53119.069--53119.174 & 0.10 (2.5)\phn\phn & 90030-01-12-00 & 6.8 (1.9) \\[0.4ex]
& 53330.721--53330.805 & 0.08 (2.0)\phn\phn & 90030-01-55-[00:01] & 2.9 (0.8) \\[0.4ex]
& 53347.497--53347.527 & 0.03 (0.7)\phn\phn & 90022-08-02-00 & 2.6 (0.7) \\[0.4ex]
& 53744.187--53744.278 & 0.09 (2.2)\phn\phn & 90030-01-94-00 & 5.2 (1.4) \\[0.4ex]
& 54388.862--54389.040 & 0.18 (4.3)\phn\phn & 92038-01-01-[00:02] & 3.8 (1.1) \bigstrut[b] \\
\hline
I & 50629.832--50631.933 (R) & 2.10 (50.4)\phn & 20053-04-01-[00:04,06,010,020,030]\tablenotemark{c} & 77.5 (21.5) \bigstrut[t] \\[0.4ex]
& 51029.751--51030.032 & 0.28 (6.7)\phn\phn & 30046-01-04-[00:01] & 13.8 (3.8)\phn \\[0.4ex]
& 53266.795--53266.821 & 0.03 (0.6)\phn\phn & 90030-01-42-00 & 2.3 (0.6) \\[0.4ex]
& 53758.990--53759.021 & 0.03 (0.7)\phn\phn & 90030-01-97-00 & 2.7 (0.7) \\[0.4ex]
& 54427.903--54428.970 & 1.07 (25.6)\phn & 93082-02-04-[00:01] & 2.4 (0.7) \bigstrut[b] \\
\hline
J & 53123.996--53129.015 (R) & 5.02 (120.5) & 90030-01-[13:14]-00 & 8.8 (2.4) \bigstrut[t] \\[0.4ex]
& 53320.893--53320.934 & 0.04 (1.0)\phn\phn & 90030-01-53-[00:01] & 1.4 (0.4) \bigstrut[b] \\
\hline
K & 51528.887--51529.451 & 0.56 (13.5)\phn & 40019-04-[03-01,04-(00:01),04-000] & 22.8 (6.3)\phn \bigstrut \\
\hline
L & 50719.393--50719.759 & 0.37 (8.8)\phn\phn & 20057-01-01-[00,000] & 20.2 (5.6)\phn \bigstrut[t] \\[0.4ex]
& 51536.473--51536.968 (R) & 0.50 (11.9)\phn & 40021-01-02-[00:04]\tablenotemark{d} & 19.2 (5.3)\phn \bigstrut[b] \\
\hline
M & 51535.886--51536.436 (R) & 0.55 (13.2)\phn & 40021-01-[01-01,02-00,02-000]\tablenotemark{d} & 26.7 (7.4)\phn \bigstrut[t] \\[0.4ex]
& 53202.754--53202.834 & 0.08 (1.9)\phn\phn & 90030-01-29-00 & 4.3 (1.2) \bigstrut[b] \\
\hline
N & 51535.010--51535.571 & 0.56 (13.5)\phn & 40021-01-01-[00,000,02] & 25.8 (7.2)\phn \bigstrut
\enddata
\tablenotetext{a}{The interval length is shown in units of both days and hours.}
\tablenotetext{b}{The total exposure time is shown in units of both ks and hours.}
\tablenotetext{c}{As the source showed significant secular motion during observation 20053-04-01-04, only an early part of it was used.}
\tablenotetext{d}{The source showed clear secular motion during observation 40021-01-02-00. Data from the first two orbits were used in panel M and data from the third (and final) orbit was used in panel L.}
\tablecomments{Subsets colored in red in Figure~\ref{fig:cyg_x-2_sequence} are denoted by (R) in the Time Interval column. In the ObsIDs column a colon denotes a range.}
\end{deluxetable*}
In Figure~\ref{fig:cyg_x-2_sequence} we show a sequence of 14 tracks, chosen (from a larger set of tracks) to illustrate as clearly as possible the overall secular evolution exhibited by the source---both of the individual branches and the locations of the tracks---as they move through the CD/HID. As is the case for \mbox{XTE J1701--462}\ the lower and upper vertices approximately follow straight line paths in both the CD and HID as the tracks evolve and shift in the diagrams. We show illustrative lines in Figure~\ref{fig:cyg_x-2_sequence} and order the panels based on the vertex locations of the tracks, starting (panel A) at the highest intensities and lowest color values.
The number of individual subsets in the tracks in the \mbox{Cyg X-2}\ sequence ranges from 1 to 10. The subsets (i.e., time intervals) and ObsIDs used for each panel in Figure~\ref{fig:cyg_x-2_sequence} are listed in Table~\ref{tab:cyg_x-2_obs}. For the panels that consist of more than one data subset (i.e., all except A, B, K, and N), a single representative subset is plotted in red. The combined exposure time of the data shown in Figure~\ref{fig:cyg_x-2_sequence} is $\sim$31\% of the total exposure time of the \mbox{Cyg X-2}\ data set (see Table~\ref{tab:source_sample}). As far as we can tell practically all the remaining data seem to belong to tracks similar to (and usually intermediate between) the ones shown in Figure~\ref{fig:cyg_x-2_sequence} (i.e., none of the other subsets is inconsistent with belonging to such tracks).
Although the tracks in panels K and L do not line up well enough in the HID (contrary to the CD) to be combined into a single track, together they should give a reasonably good idea of the rough shape of a (near-)complete track at this point in the secular progression. It is clear that such a track is similar in shape to those of the Sco-like Z sources.
We note that small secular shifts do occur in some of these tracks; some of the individual segments used have a broad appearance (e.g., the red-colored segment in panel D), which is in some cases probably due to mild secular motion, and in some instances we matched up segments despite their not lining up perfectly in the HID or SID if the overall appearance of the track is only minimally affected by this. There was inevitably sometimes some ambiguity regarding whether certain segments used in a given track rather belonged to a track slightly shifted from the one in question. In particular, the HB upturn was usually observed in short isolated segments in the CD/HID, and it was often hard to judge exactly with which tracks those should be combined. However, the overall conclusions about the secular evolution of the source that we infer from the data are not sensitive to these ambiguities in the combining process. Finally, we note that many of the tracks in Figure~\ref{fig:cyg_x-2_sequence} likely still suffer from some incompleteness, either because data segments that would serve to complete them are simply not available in the \mbox{Cyg X-2}\ data set, or because we felt that there was too much ambiguity in whether candidate segments were appropriate. In particular, we conclude that the track in panel A is likely missing the HB, the tracks in panels B and F are presumably missing (most of) the HB upturn, the one in panel H is missing most of the HB, the track in panel J is missing the NB and HB, and the one in panel K is missing the FB.
\subsubsection{Comparison with \mbox{XTE J1701--462}}\label{sec:comparison_cyg}
\mbox{Cyg X-2}\ exhibits secular evolution that is for the most part very similar to that of \mbox{XTE J1701--462}. As the overall intensity decreases the tracks smoothly evolve in shape from Cyg-like to Sco-like Z tracks. The tracks in \mbox{Cyg X-2}\ panels A--E (Figure~\ref{fig:cyg_x-2_sequence}) are very similar to the Cyg-like Z tracks of \mbox{XTE J1701--462}\ in panels A and B. Taken together, the tracks in \mbox{Cyg X-2}\ panels K and L show that at this point in the secular progression of the source its tracks look similar to (and perhaps intermediate between) those in \mbox{XTE J1701--462}\ panels E and F, the latter of which is very similar to the tracks of the persistent Sco-like Z sources, such as \mbox{GX 17+2} (see, e.g., \citetalias{homan2010}; \citealt{lin2012}). Finally, the tracks in \mbox{Cyg X-2}\ panels M and N look very similar to the tracks in \mbox{XTE J1701--462}\ panels G and H. As the \mbox{Cyg X-2}\ tracks shift and evolve in shape, both the upper and lower vertices quite closely follow straight lines in the CD and HID; as the overall intensity of the tracks decreases, the NB is squeezed between the converging vertex lines and gradually shortens, as observed in \mbox{XTE J1701--462}.
The main differences between \mbox{Cyg X-2}\ and \mbox{XTE J1701--462}\ lie in the fact that \mbox{Cyg X-2}\ is persistently luminous and has never been seen to enter the atoll regime (although the track in panel N does not seem to have an NB, short NB segments seem to exist down to the very lowest intensities observed in the \mbox{Cyg X-2}\ data set) and in the somewhat different FB behavior exhibited by \mbox{Cyg X-2}. In both sources the FB gradually evolves from being a purely dipping FB in the earliest (highest-intensity, Cyg-like) panels to a ``proper'' flaring (Sco-like) FB in later panels. However, in \mbox{Cyg X-2}\ the rotation of the FB is for the most part in the opposite direction to that seen in \mbox{XTE J1701--462}\ in both the CD and HID. In \mbox{Cyg X-2}\ the FB also develops a more complicated morphology in the HID, making a counterclockwise twist of between $180^\circ$ and $270^\circ$ around the lower vertex in panels G and H (such a twist was also observed in \textit{EXOSAT} data by \citealt{kuulkers1996a}) and then assuming a jagged S-like shape in panels I and J that gradually straightens out going from panel I to N. We note that a similar, although less pronounced, S-shaped FB was observed in the HID of panel E in \mbox{XTE J1701--462}.
\subsection{\mbox{Cir X-1}}\label{sec:cir_x-1}
Circinus X-1 (\mbox{Cir X-1}) features some of the richest and most complex phenomenology seen among the known neutron star X-ray binaries, and although it has been extensively studied for over four decades many of its properties remain poorly understood. The binary has an orbital period of $\sim$16.6~days, identified from periodic flaring first observed in the X-ray band \citep{kaluzienski1976}, and later in the radio, infrared, and optical \citep{whelan1977,glass1978,moneti1992}. These flares are thought to be due to enhanced accretion near periastron passage in a highly eccentric orbit \citep[e.g.,][]{murdin1980,tauris1999,jonker2004}. \citet{heinz2013} identify the radio nebula surrounding the binary as a relatively young ($\lesssim$5000~yr old) supernova remnant, which makes \mbox{Cir X-1}\ the youngest known X-ray binary and provides an explanation for the eccentricity of the orbit. This implies that the donor is likely an early-type star \cite[see also][]{jonker2007}. However, type~I X-ray bursts detected from the source \citep{tennant1986a,tennant1986b,linares2010} show that the neutron star is weakly magnetized, which is very unusual for a high-mass companion.
The X-ray emission from \mbox{Cir X-1}\ is highly variable on a wide range of timescales. In Figure~\ref{fig:cir_x-1_lc} we show an \textit{RXTE}\ ASM light curve of the source. The source was in a historically high state during the first few years of the \textit{RXTE}\ mission \citep{saz2003}, with an average flux of \mbox{$\sim$1.3 Crab} (and a maximum observed flux with the ASM of \mbox{$\sim$3.5--4 Crab}). The flux started gradually decreasing in mid-to-late 1999, and kept doing so until the source became undetectable with the ASM and showed no measurable activity over a two-year period in 2008--2010. In 2009 the source was observed with \textit{Chandra}\ at a flux of only a few tenths of a milliCrab \citep{sell2010}. Since 2010 May, however, the source has shown sporadic activity. When active the source usually exhibits complex variability over the course of an orbital period, featuring both absorption dips and flaring in the X-ray band; the effects of this on the ASM light curve can easily be seen in Figure~\ref{fig:cir_x-1_lc}. In the inset we show five orbital cycles from the bright phase of the source early in the \textit{RXTE}\ mission. X-ray flaring is strongest during the first few days after it commences; it then gradually decreases in strength as the orbit progresses. The source usually exhibits strong X-ray dipping during the last $\sim$0.5$-$1~days before the onset of flaring and then intermittent dips for up to two days afterwards \citep{shirey1998}; we discuss this dipping in more detail below.
\citet{oosterbroek1995} reported indications of both Z and atoll behavior in \textit{EXOSAT} data of \mbox{Cir X-1}. \citet{shirey1999a} analyzed data from 10~days of \textit{RXTE}\, PCA observations in 1997. They detected all three Z branches---although with some differences in shape compared to the classic Z sources---and found that the tracks moved around the CD/HID and evolved in shape. The identifications of the Z branches were supported by timing data.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm]{f6_300dpi.png}}
\caption{\textit{RXTE}\ ASM light curve of \mbox{Cir X-1}\ covering the entire lifetime of the \textit{RXTE}\ mission. Data points in the main plot are one day averages. The inset zooms in on five orbital cycles in early 1997; each data point there corresponds to a single dwell of the ASM. The long-dashed lines in the main plot and inset show the typical ASM count rate level for the Crab Nebula.}\label{fig:cir_x-1_lc}
\end{figure}
\subsubsection{Analysis}\label{sec:cir_x-1_analysis}
In our analysis we considered all 811 individual ObsIDs for pointed observations of \mbox{Cir X-1}\ made during the \textit{RXTE}\ mission (18 of which contained no useful data). A CD/HID using the entire data set (which spans $\sim$15 years) is shown in the upper plot in Figure~\ref{fig:cir_x-1_all_data}. The diagrams are heavily affected by both absorption and secular shifts and shape changes in the source tracks. As mentioned above, most of the dipping occurs close to the time of presumed periastron passage, which is often associated with flaring. \citet{clarkson2004} fitted a quadratic ephemeris to the times of dips observed in \textit{RXTE}\ ASM data from 1996 to 2003 and found that it provides a good predictor of the X-ray light curve. The strongest dipping often produces a characteristic track in the CD and HID; this track goes up to high color values and has two sharp bends in the CD. This can be seen in Figures~\ref{fig:cir_x-1_all_data} and \ref{fig:cir_x-1_dipping_removal}; in the latter we show various diagrams for 7~days of observations in 1997 June, which we discuss further below. (The observations used in Figure~\ref{fig:cir_x-1_dipping_removal} constitute the bulk of the data analyzed by \citealt{shirey1999a}). The shape of these dipping tracks can be understood if the X-ray emission is composed of two components: a bright component that is subject to heavy absorption and a faint component that is unaffected by the absorption, the latter perhaps due to X-rays from the central source scattered into our line of sight by surrounding material \citep{shirey1999b}. We note that the source also often exhibits shallower dips (also seen in Figure~\ref{fig:cir_x-1_dipping_removal}) which do not result in a (full) track of the sort described above.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm]{f7a_300dpi.png}}
\vspace{0.2cm}
\centerline{\includegraphics[width=8.6cm]{f7b_300dpi.png}}
\caption{CD and HID representing the entire \textit{RXTE}\ PCA data set of \mbox{Cir X-1}\ before (upper plot) and after (lower plot) the removal of data affected by absorption. A small portion of the data at very high soft and hard color values falls outside the diagrams in the upper panel. Note the differences in scale in soft and hard color between the upper and lower panels.}
\label{fig:cir_x-1_all_data}
\end{figure}
\begin{figure*}[t]
\centerline{
\includegraphics[width=5.71cm]{f8a_300dpi.png}
\hspace{0.5cm}
\includegraphics[width=10.5cm,trim=0 -280 0 0]{f8b_300dpi.png}
}
\caption{Diagrams demonstrating the removal of data affected by absorption from 7~days of observations of \mbox{Cir X-1}\ in 1997 June; the left plot shows intensity and soft color as a function of time, and the right plot shows a CD/HID. We identified the red data points with absorption and removed them from the data set before proceeding with further analysis. In the left plot the dashed vertical line shows the time of zero phase according to the dipping ephemeris of \citet{clarkson2004}.}\label{fig:cir_x-1_dipping_removal}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[height=11.9cm]{f9a_300dpi.png}
\hspace{-0.04cm}
\includegraphics[height=11.9cm]{f9b_300dpi.png}
\hspace{-0.078cm}
\includegraphics[height=11.9cm]{f9c_300dpi.png}
\caption{A sequence of CDs/HIDs for \mbox{Cir X-1}\ illustrating the secular evolution of the source. The data points in each panel were obtained within a relatively short period of time, ranging from $\sim$4~hr to $\sim$7~days, except for panels A and I, which combine three and two segments, respectively, widely separated in time (with each segment shown in a particular color). The dashed and solid lines are lower and upper vertex lines similar to those shown for \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}. Data are binned to a minimum of 16,000~counts per data point, except for IS and EIS points in panel L, which are binned to a minimum of 32,000~counts. Note the change in the intensity scale between the different HID columns.}\label{fig:cir_x-1_sequence}
\end{figure*}
Before proceeding with our analysis of the \mbox{Cir X-1}\ data set we removed by eye (to the extent possible) data points affected by absorption dips. An example of this is shown in Figure~\ref{fig:cir_x-1_dipping_removal}, where removed data points are colored red. As can be seen in the figure, during these observations the source showed intense dipping shortly before the start of flaring (at day $\sim$611.5), producing the characteristic dipping track in the CD/HID. Some shallower dips were then observed during the first $\sim$2~days of flaring. We note that the periodic ``flaring'' during an orbital cycle is in general associated with motion along all three (Cyg-like) Z branches, rather than being exclusively associated with motion along the FB (which is mostly a dipping FB in these particular observations). As is apparent from the figure, it is almost impossible to identify absorption dips on the basis of the light curve alone during periods of flaring. However, tracking the behavior of the soft color (and, to a lesser extent, the hard color) as a function of time, as well as inspecting the CD and HID, greatly aids in identifying dipping. In addition, we took into account the dipping ephemeris of \citet{clarkson2004} when performing the removal, since the vast majority of dipping events take place in the $\sim$1--1.5~days immediately before or after phase~0. However, we note that (shallow) dips are sometimes seen later in an orbital cycle; one example can be seen in Figure~\ref{fig:cir_x-1_dipping_removal}. In general, when unsure whether a given data segment was afflicted by dipping or not, we in general opted to rather err on the side of caution and remove the segment in question. The ``cleaned'' CD/HID resulting from our manual removal of absorption-affected data is shown in the lower plot in Figure~\ref{fig:cir_x-1_all_data}. It was of course unavoidable that some instances of minor dipping remain and that a small amount of unaffected data be removed. However, we expect that the effects of this on the conclusions we draw from the data are negligible.
After removing data points affected by absorption dips, as described above, we organized the data into subsets, similar to our \mbox{Cyg X-2}\ analysis described in the previous section. This resulted in $\sim$300 subsets spanning periods as long as several days, although most are less than a day in length. From these we created a sequence of 12 CD/HID (partial) tracks, shown in Figure~\ref{fig:cir_x-1_sequence}. These tracks were chosen to illustrate as best possible the overall secular behavior of the source. In all cases except two the data in a given panel are from a single subset, whose time intervals range from $\sim$2~hr to $\sim$7~days. The exceptions are panels A and I, where 2--3 independent segments (shown in different colors) were combined. It was not viable to form a whole sequence of combined and more complete tracks, as we did for \mbox{Cyg X-2}, since the data for \mbox{Cir X-1}\ show a much greater dynamic range and do not sample the HID densely enough for this. Overall, the upper and lower vertices of the \mbox{Cir X-1}\ tracks move systematically up and to the right in the CD, and up and to the left in the HID, as the tracks evolve in shape---similar to the behavior seen in \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}. The paths of the vertices can be approximated by the straight lines shown in Figure~\ref{fig:cir_x-1_sequence}. We order the tracks based on the position of the lower and/or upper vertex in the CD. (When both are present, the two vertices give consistent results.) In cases where some ambiguity remains due to very similar vertex locations (in particular, panels F and G), we use the shape of the track in the CD to decide the ordering, in particular the FB (i.e., we choose the ordering that produces a gradual evolution in the shape of the track in the CD). The combined exposure time of the data shown in Figure~\ref{fig:cir_x-1_sequence} is $\sim$10\% of the total exposure time of the \mbox{Cir X-1}\ data set. The data subsets (time intervals) and ObsIDs used in each panel are listed in Table~\ref{tab:cir_x-1_obs}.
We note that a few of the tracks in Figure~\ref{fig:cir_x-1_sequence} show some signs of secular motion in the HID (but much less in the CD); however, in most cases this has a negligible effect on the overall appearance of the track. The instance where this is most noticable is in the HID of panel E, where the upper parts of the track (NB and especially HB) show shifting toward lower intensities relative to the FB. A similar, but smaller, shift affects the HID in panel F. We also note that the behavior of \mbox{Cir X-1}\ is not as regular as that of \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}, and the vertices in the HID (and to a lesser extent in the CD) often deviate significantly from the lines shown. As will be discussed in Section~\ref{sec:discussion}, the relationship between a track's shape and its location in the HID is also not nearly as tight for \mbox{Cir X-1}\ as for \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}, where the motion of both vertices in both the CD and HID, as well as the decrease in overall intensity, is monotonic along the entire sequence. The behavior of \mbox{Cir X-1}\ is more regular in the CD than the HID, which is why we chose to use the CD as the basis for our ordering of the tracks. However, apart from some irregularities in panels F--I, the \mbox{Cir X-1}\ sequence does show a gradual decrease in overall intensity.
Panel L shows what seems to be an atoll transition from the banana branch through the IS and to an EIS. These data were obtained in 2010 May/June as the intensity steadily decreased over a $\sim$7~day period (part of a $\sim$50~day minioutburst following two years of nondetection by the \textit{RXTE}\ ASM). Observations preceding this 7~day period show what looks like a sequence of segments at successively lower count rates from tracks similar to the one in panel K. Figure~\ref{fig:cir_x-1_atoll} shows the same data as in panel L (in black) but with more binning. \citet{dai2012} studied the spectral evolution of the source during the entire 2010 May--June minioutburst with spectral fitting and also concluded that the source transitioned from the atoll soft to hard state. In 2010 August the source showed a similar transition (again with an intensity decline over $\sim$7~days), which we show with red data points in Figure~\ref{fig:cir_x-1_atoll}. We note that the apparent decrease in hard color with decreasing intensity in the EIS in both cases may well be due to soft diffuse background emission (which could not be subtracted) affecting the data points at the lowest intensities.
\subsubsection{Comparison with \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}}\label{sec:comparison_cir}
Like \mbox{XTE J1701--462}, \mbox{Cir X-1}\ has been observed in all NS-LMXB subclasses (Cyg-like Z, Sco-like Z, atoll). The overall secular evolution has many similarities to that observed for \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}, and many of the individual tracks have shapes similar to those seen for those two sources. Progressing along the \mbox{Cir X-1}\ sequence the NB grows shorter and rotates counterclockwise in the HID, while the HB rotates clockwise as it shortens, similar to what was seen for \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}. As in those two sources the FB shows the most complex behavior of the three branches in \mbox{Cir X-1}. In the CD of \mbox{Cir X-1}\ the FB gradually rotates counterclockwise and evolves in shape in a similar fashion to \mbox{XTE J1701--462}. In the HID the behavior of the FB is more irregular in \mbox{Cir X-1}\ than \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}, but overall it seems to rotate counterclockwise as it evolves from a dipping FB in the higher-intensity tracks to a Sco-like FB at lower intensities.
In panel G, the \mbox{Cir X-1}\ FB in both the CD and HID has a shape similar to those of the persistent Cyg-like Z sources \mbox{GX 340+0} and \mbox{GX 5--1} (see, e.g., \citealt{jonker1998}; \citealt{jonker2002}; \citetalias{homan2010}). We note that in the CD of \mbox{XTE J1701--462}\ the shape and orientation of the FB in panels A and B seems intermediate between \mbox{Cir X-1}\ panels E and F, whereas the \mbox{XTE J1701--462}\ FB in panels C (which has a very incomplete track) and D seems intermediate between \mbox{Cir X-1}\ panels G and H. \mbox{XTE J1701--462}\ may therefore have traced out a CD (and HID) track similar to those of \mbox{GX 340+0} and \mbox{GX 5--1} in between panels B and C; this portion of the secular progression of the source was missed due to a gap in \textit{RXTE}\ coverage and rapid secular evolution. However, we note that in none of the four sources compared in this paper do we see a full track in both the CD and HID where the shapes of all three spectral branches closely match those of \mbox{GX 5--1} and \mbox{GX 340+0}.
The atoll transitions of \mbox{Cir X-1}---especially the one in 2010 May/June---resemble the one of \mbox{XTE J1701--462}\ (see Figures~\ref{fig:1701_atoll}~and~\ref{fig:cir_x-1_atoll}); one notable similarity is that the soft color decreases significantly throughout the transitions, in contrast to the increase observed in most atoll sources \citep[e.g.,][]{mythesis2011}. However, there is also a striking difference between the atoll transitions of the two sources: the data from the banana branch preceding the ascent to the hard state extend to very high soft color for \mbox{Cir X-1}, occupying parameter space in the CD never explored before by the source in the \textit{RXTE}\ archive except during absorption dips or (partly) during traversals to the tip of the (nondipping) FB. The data points in question are unlikely to be associated with the FB, given the small changes in intensity and hard color observed, and the long timescale involved (more than a day). This pre-atoll-transition behavior of \mbox{Cir X-1}\ is in stark contrast to \mbox{XTE J1701--462}, where no such excursion to high soft color values was seen, and the region occupied by the source in the CD in selection L before moving to the IS was a logical extension of the movement of the source in the preceding selections. We also note that the observations in 2010 May/June were the first time in almost 15~years of \textit{RXTE}\ observations that \mbox{Cir X-1}\ was seen to transition to the atoll hard state.
\setlength\bigstrutjot{4pt}
\tabletypesize{\footnotesize}
\begin{deluxetable*}{clclc}[]
\tablewidth{18.0cm}
\tablecolumns{5}
\tablecaption{Time Intervals and Observations Used to Create \mbox{Cir X-1}\ Tracks \label{tab:cir_x-1_obs}}
\tablehead{ & & \colhead{Interval Length} & & \colhead{Exp.\ Time} \\[0.3ex]
\colhead{\hspace{-0.13cm}Panel} & \colhead{\hspace{-0.65cm}Time Interval (MJD)} & \colhead{days (hr)\tablenotemark{a}} & \colhead{\hspace{-4.4cm}ObsIDs} & \colhead{ks (hr)\tablenotemark{b}}}
\startdata
A & 50365.216--50365.307 & 0.09 (2.2)\phn\phn & 10068-08-02-00 & 5.5 (1.5) \\[0.6ex]
& 50497.364--50497.437 (G) & 0.07 (1.8)\phn\phn & 20095-01-01-00 & 4.8 (1.3) \\[0.6ex]
& 50711.692--50711.763 (R) & 0.07 (1.7)\phn\phn & 20095-01-18-00 & 5.9 (1.6) \bigstrut[b] \\
\hline
B & 50613.800--50614.155 & 0.36 (8.5)\phn\phn & 20094-01-02-[04,040]\tablenotemark{c} & 21.6 (6.0)\phn \bigstrut \\
\hline
C & 51603.985--51604.316 & 0.33 (8.0)\phn\phn & 40059-01-01-[00,02]\tablenotemark{c} & 13.1 (3.6)\phn \bigstrut \\
\hline
D & 52615.803--52618.276 & 2.47 (59.4)\phn & 70020-01-[02-(01:02),03-01,03-04,04-(00:04)] & 37.9 (10.5) \bigstrut \\
\hline
E & 53013.822--53017.845 & 4.02 (96.6)\phn & 70020-03-04-[00:20] & 34.6 (9.6)\phn \bigstrut \\
\hline
F & 53163.821--53165.483 & 1.66 (39.9)\phn & 80027-02-[02-02,02-06,03-(00:02),03-06] & 31.0 (8.6)\phn \bigstrut \\
\hline
G & 51831.981--51832.459 & 0.48 (11.5)\phn & 50136-01-04-[00:06] & 18.7 (5.2)\phn \bigstrut \\
\hline
H & 53168.550--53168.789 & 0.24 (5.7)\phn\phn & 80027-02-04-01 & 14.1 (3.9)\phn \bigstrut \\
\hline
I & 52787.744--52787.894 (R) & 0.15 (3.6)\phn\phn & 80114-04-01-[02:04] & 4.4 (1.2) \bigstrut[t] \\[0.6ex]
& 52951.992--52952.229 & 0.24 (5.7)\phn\phn & 80027-01-01-[02:03] & 9.4 (2.6) \bigstrut[b] \\
\hline
J & 53003.507--53003.590 & 0.08 (2.0)\phn\phn & 70020-03-01-01 & 5.6 (1.6) \bigstrut \\
\hline
K & 53271.381--53271.787 & 0.41 (9.8)\phn\phn & 90025-01-02-[02,24,25,27] & 7.9 (2.2) \bigstrut \\
\hline
L & 55343.361--55350.716 & 7.36 (176.5) & 95422-01-[03-(03:04),04-(00:13)] & 49.4 (13.7) \bigstrut
\enddata
\tablenotetext{a}{The interval length is shown in units of both days and hours.}
\tablenotetext{b}{The total exposure time is shown in units of both ks and hours.}
\tablenotetext{c}{The early parts of observations 20094-01-02-040 and 40059-01-01-00 were omitted due to secular motion.}
\tablecomments{Subsets colored in red/green in Figure~\ref{fig:cyg_x-2_sequence} are denoted by (R)/(G) in the Time Interval column. In the ObsIDs column a colon denotes a range.}
\end{deluxetable*}
\begin{figure}[]
\centerline{\includegraphics[width=8.6cm]{f10_300dpi.png}}
\caption{CD/HID showing two instances of \mbox{Cir X-1}\ undergoing an atoll transition from the soft to hard state. Black data points are from a $\sim$7~day period in 2010 May/June (also shown in panel L in Figure~\ref{fig:cir_x-1_sequence}) and red data points are from a $\sim$7~day period in 2010 August. The data were binned with a minimum of 128,000~counts per data point. In some cases entire observations did not have that many counts and were combined with other ones close in time.}\label{fig:cir_x-1_atoll}
\end{figure}
The \mbox{Cir X-1}\ track in panel D bears a strong resemblance in shape to the highest-intensity tracks (panels A and B) of \mbox{XTE J1701--462}\ (in particular when the HID is plotted on the same scale) and occupies a similar color range (especially in hard color). But \mbox{Cir X-1}\ also shows tracks that reach much higher intensities (panels A--C). Going from panel D to A in the \mbox{Cir X-1}\ sequence, the overall color values of tracks decrease and the color range spanned drops sharply as the intensity swings become larger, resulting in very stretched and flat tracks in the HID. In these ``extreme'' Cyg-like Z tracks in panels A and B \mbox{Cir X-1}\ exhibits much lower color values than ever observed for any of the other three sources analyzed in this paper, and it is striking how small the color variations along the tracks are despite the large changes in intensity.
\vspace{0.7cm}
\subsection{\mbox{GX 13+1}}\label{sec:gx_13+1}
\mbox{GX 13+1}\ is a bright X-ray binary whose classification as a Z or atoll source is ambiguous, although it has usually been labeled an atoll source. \citet{hasinger1989} classified \mbox{GX 13+1}\ as a bright atoll source based on CDs and power spectra from \textit{EXOSAT} observations. In \textit{RXTE}\ observations from 1996, \citet{homan1998} discovered a \mbox{57--69 Hz} QPO, which showed similar behavior to the horizontal-branch oscillation (HBO) seen in Z sources. \citet{schnerr2003} performed a combined CD, HID, and power-spectral analysis of a large number of \textit{RXTE}\ observations made in 1998. They found that the source traced out, on a timescale of hours, a curved two-branched track in the CD, which showed strong secular motion on a timescale of $\sim$1~week. The shape of the track was similar to the lower part (IS, LB, UB) of an atoll track or the NB/FB part of a Z track; the location of the vertex between the two branches was seen to approximately follow a straight line in the CD. They also found that the source showed peculiar CD/HID and rapid-variability behavior compared to most other Z or atoll sources, but overall they favored an atoll classification. \citet{homan2004} analyzed two simultaneous \textit{RXTE}/radio observations of \mbox{GX 13+1}\ performed in 1999. Based on the results of spectral fits, rapid-variability properties, behavior in the radio band, and the scarcity of type~I X-ray bursts observed from the source since its discovery, they concluded that the properties of \mbox{GX 13+1}\ were more similar to Z sources than atolls.
\subsubsection{Analysis}\label{sec:gx_13+1_analysis}
In our analysis we considered all 92 individual ObsIDs for pointed observations of \mbox{GX 13+1}. These span a period of $\sim$14~years. In Figure~\ref{fig:gx_13+1_all_data} we show a CD/HID based on all the \mbox{GX 13+1}\ data; strong secular motion is apparent. However, we note that \mbox{GX 13+1}\ overall shows the smallest range in secular evolution among the four sources studied here (e.g., as quantified by the range in soft or hard color over which the lower vertex is observed), and the other three sources show secular motion on shorter timescales than does \mbox{GX 13+1}. As for \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ we organized the data into subsets; this resulted in $\sim$50 such sets, which could in a few cases span intervals as long as several days with little or no visible secular motion. We constructed a sequence of six CD/HID tracks (shown in Figure~\ref{fig:gx_13+1_sequence}), which illustrates the secular evolution of the source. Similar to our analysis of \mbox{Cyg X-2}\ we combined subsets widely separated in time in three of these tracks to create more complete tracks than otherwise available and thereby give a fuller depiction of the overall secular behavior of the source. For these three tracks we also indicate in red a representative subset obtained in a short time interval. The number of individual subsets used for each track ranges from 1 to 4; these are listed in Table~\ref{tab:gx_13+1_obs} along with the corresponding ObsIDs. Similar to our \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ analysis we order the tracks based on the location of the lower vertex in the CD, starting at the lowest soft and hard color. (See discussion of this vertex in the following paragraph.) The data used in the six tracks together constitute $\sim$42\% of the total exposure time of the \mbox{GX 13+1}\ data set. We note that in a few of the subsets used we do see indications of small secular shifts---especially in panels B and F in Figure~\ref{fig:gx_13+1_sequence}. These shifts are apparent mostly in the HID (and SID), rather than the CD. However, the overall conclusions we draw from these tracks and the sequence as a whole are not affected by this.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm]{f11_300dpi.png}}
\caption{CD and HID representing the entire \textit{RXTE}\ PCA data set of \mbox{GX 13+1}.}\label{fig:gx_13+1_all_data}
\end{figure}
As can be seen from Figure~\ref{fig:gx_13+1_sequence} the tracks in the CD mostly have a two-branched form. The vertex between these branches follows rather closely a straight line (as observed by \citealt{schnerr2003}), which we show in the figure. We classify \mbox{GX 13+1}\ as a Z source based on its overall secular evolution in the CD/HID, the timescales on which it traces out its CD/HID tracks, and its rapid-variability properties. (We discuss this further in Section~\ref{sec:gx_13+1_discussion}.) We identify the vertex in the CD as the lower Z track vertex and the branches to the left/right of the vertex line in the CD as the normal/flaring branch. In addition, the tracks in panels D and E include subsets that we identify as the HB (plus upturn). In the CD these segments look very similar to excursions onto the HB and HB upturn in the Cyg-like Z tracks of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}. In the HID these segments also stand out and are located above and to the left of the rest of the track. We note that there is considerable ambiguity regarding where within panels C--E to place these presumed HB segments. What we show in Figure~\ref{fig:gx_13+1_sequence} is therefore our best guess but only one of a few possible ways of incorporating them into the sequence. What does seem clear from inspection of the entire \mbox{GX 13+1}\ data set is that these segments cannot plausibly be combined with other track segments---or accommodated in our sequence---as NB segments similar to the other ones we see. (We discuss these tracks further in Section~\ref{sec:discussion}).
The tracks in the HID look quite different from those of the other three sources. As we discuss below, different track branches can be identified in the HIDs; however, the fact that these branches are in general very broad and rather irregularly shaped makes them less useful for judging in what cases observations can be appropriately combined to form CD/HID tracks without significant secular shifts. The SID---although also displaying broad and somewhat irregular track branches---turned out to be more useful in this respect. To better illustrate the behavior of the source in the HID we show in Figure~\ref{fig:gx_13+1_colcode_example} a color-coded version of the track in panel C. The locations of the NB/FB vertex in the HID are in general less well defined than in the CD and they do not seem to follow a straight line.
\subsubsection{Secular Evolution}\label{sec:gx_13+1_evolution}
\begin{figure*}[t]
\centering
\includegraphics[height=9.3cm]{f12a_300dpi.png}
\hspace{-0.035cm}
\includegraphics[height=9.3cm]{f12b_300dpi.png}
\caption{A sequence of CDs/HIDs for \mbox{GX 13+1}\ illustrating the secular evolution of the source. Table~\ref{tab:gx_13+1_obs} indicates the data used in each panel. In panels C, D, and E data from observations widely separated in time were combined; in each of these three panels a subset of the data---obtained over a period of at most $\sim$1.8~days---\,is shown in red. The dashed line in the CDs shows the approximate path followed by the NB/FB vertex.}\label{fig:gx_13+1_sequence}
\end{figure*}
\begin{deluxetable*}{clclc}[]
\tablewidth{18.0cm}
\tablecolumns{5}
\tablecaption{Time Intervals and Observations Used to Create \mbox{GX 13+1}\ Tracks \label{tab:gx_13+1_obs}}
\tablehead{ & & \colhead{Interval Length} & & \colhead{Exp.\ Time} \\[0.3ex]
\colhead{\hspace{-0.13cm}Panel} & \colhead{\hspace{-0.65cm}Time Interval (MJD)} & \colhead{days (hr)\tablenotemark{a}} & \colhead{\hspace{-3.9cm}ObsIDs} & \colhead{ks (hr)\tablenotemark{b}}}
\startdata
A & 51278.745--51278.971 & 0.23 (5.4)\phn\phn & 40023-03-02-03 & 10.3 (2.9)\phn \bigstrut[b] \\
\hline
B & 51276.746--51278.117 & 1.37 (32.9)\phn & 40023-03-[01-00,01-02,02-(00:02),02-04] & 26.0 (7.2)\phn \bigstrut \\
\hline
C & 50990.407--50990.711 & 0.30 (7.3)\phn\phn & 30051-01-09-[00:01] & 9.8 (2.7) \bigstrut[t] \\[0.6ex]
& 53767.006--53767.601 (R) & 0.60 (14.3)\phn & 91007-08-02-[00,000] & 29.5 (8.2)\phn \bigstrut[b] \\
\hline
D & 50981.074--50982.844 (R) & 1.77 (42.5)\phn & 30050-01-01-[04:08,050,080] & 54.4 (15.1) \bigstrut[t] \\[0.6ex]
& 50984.407--50984.576 & 0.17 (4.1)\phn\phn & 30050-01-02-03 & 8.1 (2.3) \\[0.6ex]
& 51390.995--51394.488 & 3.49 (83.8)\phn & 40022-01-[01-(00:01),02-00,02-000] & 39.0 (10.8) \\[0.6ex]
& 51447.322--51447.428 & 0.11 (2.5)\phn\phn & 40023-03-04-00 & 6.8 (1.9) \bigstrut[b] \\
\hline
E & 51007.409--51007.560 & 0.15 (3.6)\phn\phn & 30051-01-12-01 & 8.0 (2.2) \bigstrut[t] \\[0.6ex]
& 53409.846--53409.949 (R) & 0.10 (2.5)\phn\phn & 90173-01-01-00 & 6.1 (1.7) \\[0.6ex]
& 54740.627--54743.339 & 2.71 (65.1)\phn & 93046-08-[02-00,03-(00:01)] & 7.8 (2.2) \\[0.6ex]
& 55409.062--55413.805 & 4.74 (113.8) & 95338-01-[01-(00:07),03-00,03-05] & 24.3 (6.8)\phn \bigstrut[b] \\
\hline
F & 50997.607--51003.646 & 6.04 (144.9) & 30051-01-[11-(00:03),12-00] & 17.5 (4.8)\phn \bigstrut
\enddata
\tablenotetext{a}{The interval length is shown in units of both days and hours.}
\tablenotetext{b}{The total exposure time is shown in units of both ks and hours.}
\tablecomments{Subsets colored in red in Figure~\ref{fig:gx_13+1_sequence} are denoted by (R) in the Time Interval column. In the ObsIDs column a colon denotes a range.}
\end{deluxetable*}
\mbox{GX 13+1}\ shows behavior that is in some ways quite different from that of the other three sources, especially in the HID. A more detailed description of the secular evolution of this source is therefore warranted. We note at the outset that, as Figure~\ref{fig:gx_13+1_all_data} indicates, the largest portion of the total exposure time for the source was spent on tracks near the middle of the sequence in Figure~\ref{fig:gx_13+1_sequence} (especially panel D), and it is therefore natural that the tracks in these panels would be the most complete ones. Conversely, the amount of data available at the lowest (panels A and B) and highest (panel F) vertex locations (all of which is shown in Figure~\ref{fig:gx_13+1_sequence}) is very small. These tracks therefore seem most likely to suffer from incompleteness.
We first focus on the evolution of the tracks in the CD. As the tracks move up the vertex line the FB gradually rotates counterclockwise and becomes longer and straighter (more Sco-like). Starting at the other end of the sequence, the NB gradually becomes longer as the tracks move down the vertex line from panel F to D. The NB seems to shorten again going from panel C to A, although we suspect that this may be due to incompleteness in panels A, B, and possibly C. These three tracks may also be missing HB and HB upturn portions.
The evolution in the HID is less obvious. At the lower vertex locations the NB is strongly tilted to the right and for a given hard-color value spans a large intensity range, especially in its upper part. The NB gradually becomes more compact going to higher vertex locations (although it is always broad compared to the NBs observed in the other three sources). The behavior of the FB in the HID is even harder to discern than that of the NB. The morphology of the FB in panel C can more easily be seen in the color-coded version in Figure~\ref{fig:gx_13+1_colcode_example}. The upward-pointing part of the FB is essentially vertical (but very short) in panel C, and is at the lowest intensities. Going counterclockwise along this track in the CD, the highest intensities are near the top of the NB and then the intensity becomes gradually lower moving along the track until the FB reaches its lowest point (in hard color), after which the intensity becomes approximately constant. (The same behavior is seen in panel B.) In panels D and E, the flat part of the FB in the CD forms a broad patch at the bottom of the track in the HID. The rising part of the FB extends from the left side of this patch, tilted slightly to the right from vertical, and overlaps with the broad NB (see, e.g., the red-colored data in panel E).
\subsubsection{Comparison with \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}}\label{sec:comparison_gx}
The tracks in the CD of \mbox{GX 13+1}\ show many similarities to those of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}. The evolution of the FB in the CD is similar to that seen for \mbox{XTE J1701--462}\ in selections A--H---with the FB rotating counterclockwise and growing longer (and more Sco-like)---and likewise similar to that of \mbox{Cir X-1}. There are strong similarities between the shape of the FB of \mbox{GX 13+1}\ in panels D--F and, e.g., the FB of \mbox{XTE J1701--462}\ in selections E--H, \mbox{Cyg X-2}\ in panel J, and \mbox{Cir X-1}\ in panels H and I. As in the other three sources, a lower vertex in the CD of \mbox{GX 13+1}\ moves along a straight line with positive slope. The NB is also similar to the ones of the other three sources (being mostly straight and oriented up and to the right), and the possible HB and HB upturn segments in \mbox{GX 13+1}\ have a similar appearance to those of the other sources as well.
Although less obvious at first sight, there are also similarities in the HID between GX~13+1 and the other three sources. The NB is oriented up and to the right, becoming gradually shorter and closer to vertical as the tracks move up in hard color---i.e., the intensity swings along the NB become smaller---and at the same time the maximum intensity observed on the NB gradually decreases (although for this to extend to panels A and B we need to assume they have incomplete NBs). This is also the case for the other three sources. What we interpret as traversals onto the HB and HB upturn in \mbox{GX 13+1}\ are manifested as movement toward lower intensities and higher hard color; this is in general the case for Z sources, including the other three discussed in this paper. In the earlier panels in the \mbox{GX 13+1}\ sequence the initial part of the FB is toward lower intensities (i.e., dipping FB behavior), as is seen in the other three sources. The \mbox{GX 13+1}\ FB rotates slightly clockwise in the HID (similar to \mbox{Cyg X-2}) as the source moves up the vertex line in the CD, evolving into an FB that shows an intensity increase rather than decrease. A notable difference between \mbox{GX 13+1}\ and the other three sources is the movement of the lower vertex in the HID. The location of the lower vertex in the HID of \mbox{GX 13+1}\ is in general rather poorly defined, and does not seem to follow a straight line. However, it seems clear that the lower vertices in panels A and B in Figure~\ref{fig:gx_13+1_sequence} are at a significantly lower intensity (\mbox{$\sim$550--600 counts s$^{-1}$ PCU$^{-1}$}) than the ones in panels C--F (\mbox{$\sim$800--900 counts s$^{-1}$ PCU$^{-1}$}), in contrast to the other three sources, where in general the intensity at the lower vertex decreases as it moves up the vertex line in the CD.
As is clear from the above \mbox{GX 13+1}\ shows similarities to both Cyg-like and Sco-like Z sources. Overall, the earlier panels in the track sequence are more Cyg-like and the later ones more Sco-like, although the distinction between the two is less clear than for the other three sources.
\begin{figure}[t]
\centerline{\includegraphics[width=8.6cm]{f13_300dpi.png}}
\caption{Color-coded version of the track in panel C in Figure~\ref{fig:gx_13+1_sequence}, illustrating the portions of the HID track corresponding to several segments along the track in the CD.}\label{fig:gx_13+1_colcode_example}
\end{figure}
\section{Discussion}\label{sec:discussion}
\subsection{Secular Evolution}\label{sec:secular_evolution}
The main goal of this paper is to study secular evolution in the CDs and HIDs of NS-LMXBs, using three sources historically known to show substantial changes in the shape and location of their CD/HID tracks: \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}. In particular, we want to determine to what extent the secular evolution of these three sources is similar to that seen in \mbox{XTE J1701--462}, the first source found to evolve through all NS-LMXB subclasses.
In Figure~\ref{fig:all_tracks_single_plot} we provide an overview of the secular evolution of the CD/HID tracks of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}. In each of the four panels we show all tracks from the sequences in Figures~\ref{fig:1701_sequence}, \ref{fig:cyg_x-2_sequence}, \ref{fig:cir_x-1_sequence}, and \ref{fig:gx_13+1_sequence} in a single CD/HID. (For \mbox{XTE J1701--462}\ we show only the red-colored subsets from Figure~\ref{fig:1701_sequence}, where applicable.) Strong secular evolution is found in all four sources, consistent with reports in the literature. While for \mbox{XTE J1701--462}\ all data come from the dense monitoring of a single 19~month outburst, the data for the other three sources were collected sporadically over a time span of $\sim$14--15~years. As a result, the secular evolution in these sources could not be followed in ``real time," as was possible for \mbox{XTE J1701--462}, but had to be reconstructed from multiple isolated (partial) tracks. To some extent this limited our ability to compare the secular evolution between sources, in particular for \mbox{GX 13+1}. Despite these issues, it is obvious from Figure~\ref{fig:all_tracks_single_plot} that the secular evolution of the four sources has many common characteristics.
\begin{figure*}[]
\vspace{0.1cm}
\centerline{
\hspace{0.915cm}
\includegraphics[width=13.8cm]{f14a_300dpi.png}\\
}
\vspace{0.2cm}
\centerline{
\hspace{0.915cm}
\includegraphics[width=13.8cm]{f14b_300dpi.png}\\
}
\vspace{0.2cm}
\centerline{
\hspace{0.915cm}
\includegraphics[width=13.8cm]{f14c_300dpi.png}\\
}
\vspace{0.2cm}
\centerline{
\hspace{0.915cm}
\includegraphics[width=13.8cm]{f14d_300dpi.png}\\
}
\caption{CDs/HIDs illustrating the overall secular evolution of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}. In the panels we show (from top to bottom) all the tracks from Figures~\ref{fig:1701_sequence}, \ref{fig:cyg_x-2_sequence}, \ref{fig:cir_x-1_sequence}, and \ref{fig:gx_13+1_sequence}, respectively. Different colors are used to distinguish between the individual tracks in each panel.}\label{fig:all_tracks_single_plot}
\end{figure*}
In the following we summarize our key findings, focusing on the similarities between the systems.
\begin{enumerate}
\item As part of their secular evolution, we see clear and continuous transitions between different NS-LMXB subclasses in all four sources. While some of the behavior we report has been described in previous works, our work for the first time unambigously links strong secular evolution to transitions between various NS-LMXB subclasses in \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}. Of the four sources \mbox{XTE J1701--462}\ and \mbox{Cir X-1}\ have shown the largest range in behavior; they have displayed Cyg-like and Sco-like Z tracks, atoll soft and hard states, and have both at some point entered quiescence or near-quiescence (\mbox{$\sim$$10^{35}$ erg s$^{-1}$} in the case of \mbox{Cir X-1}; \citealt{sell2010}). \mbox{Cyg X-2}\ and \mbox{GX 13+1}\ have only shown Cyg-like and Sco-like Z tracks, with the secular changes in \mbox{GX 13+1}\ being the most moderate of the four sources.
\item{Cyg-like Z source behavior (with large variations in intensity along the NB and HB, and a ``dipping'' FB that is directed toward lower intensities) is observed at the highest overall intensities, Sco-like Z source behavior (with small intensity variations along the NB and, if present, the HB, and large increases in intensity along the FB) at lower intensities, and (in \mbox{XTE J1701--462}\ and \mbox{Cir X-1}) atoll behavior at the lowest overall intensities.}
\item{Although possible incompleteness in some of the CD/HID tracks clouds the picture somewhat, the order in which the Z source branches evolve and disappear in \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ seems to be consistent with what is seen in \mbox{XTE J1701--462}. As the overall intensity decreases, the HB shortens and disappears first, followed by the NB, and finally the FB. The situation is less clear for \mbox{GX 13+1}; the earlier tracks in Figure~\ref{fig:gx_13+1_sequence} would need to be missing an HB (and possibly part of the NB) for the behavior of \mbox{GX 13+1}\ to be consistent with \mbox{XTE J1701--462}\ as well.}
\item{As the shapes of the Z tracks and branches gradually change, the vertices of the tracks shift along well-defined lines in the CD, moving toward higher color values as the overall intensity decreases. (We note, however, that the overall intensity evolution is less clear-cut for \mbox{GX 13+1}, and that \mbox{Cir X-1}\ shows some deviations from the general intensity trend.) For \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}\ this also applies to the HID, and to a large extent for \mbox{Cir X-1}\ as well.}
\item{As the overall intensity of the Z tracks increases and they become spectrally softer, their dynamic range (i.e., ratio of maximum to minimum intensity) increases as well. Whether this trend also extends to the tracks at the lowest color values in \mbox{GX 13+1}\ is unclear due to possible incompleteness issues.}
\end{enumerate}
Despite the fact that differences exist in their detailed behavior, based on the above findings we conclude that the secular evolution in the four NS-LMXBs that we studied largely follows similar patterns. This strongly suggests that the sequences of NS-LMXB subclasses reported in this work are representative of the class of NS-LMXB as a whole.
\subsection{Evolution of the Flaring Branch}\label{sec:flaring_branch_evolution}
Of the three main Z source branches, the FB undergoes the most dramatic changes in its shape and orientation. In all four sources it rotates in both the CD and HID and evolves from a branch that shows a strong decrease in intensity with respect to the NB/FB vertex (in the Cyg-like Z tracks), to one that shows a strong increase in intensity with respect to that vertex (in the Sco-like Z tracks), although the amplitude of these intensity changes is significantly smaller in \mbox{GX 13+1}\ than in the other three sources. From the dense ``coverage'' of the secular evolution of \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}\ it is clear that this FB evolution from dipping to flaring is a very gradual process, with the morphology changing smoothly throughout the track sequences. This suggests that the dipping Cyg-like FB and the Sco-like FB are related phenomena, and it seems unlikely that the overall observed FB behavior can be explained by two presumably unconnected mechanisms: absorption by the outer disk, which was proposed as an explanation for the \mbox{Cyg X-2}\ dipping behavior by \citet{balucinska-church2011}, and nuclear burning on the neutron star (combined with increases in the mass accretion rate in the Sco-like Z sources), proposed as an explanation for the flaring FB by \citet{church2012}. (See also \citealt{balucinska-church2012}.)
\subsection{Other Z Source Transients}\label{sec:other_z_transients}
We note that two additional Z source transients discovered after we began work on this paper, \mbox{IGR J17480--2446} (also known as \mbox{Terzan 5 X-2}) and \mbox{MAXI J0556--332}, have shown similar behavior to the sources studied here. \textit{RXTE}\ observations of \mbox{IGR J17480--2446} revealed clear Cyg-like Z source behavior at the peak of its $\sim$2.5~month outburst and atoll behavior at lower intensities; due to the rapid source evolution and sparsity of the data Sco-like Z behavior could not be identified unambiguously (\citealt{altamirano2010}; \citealt{chakraborty2011}; \citealt{altamirano2012}; D.~Altamirano et al.\ 2015, in preparation).
\mbox{MAXI J0556--332}\ showed both Cyg-like and Sco-like Z behavior during its 16~month outburst (\citealt{homan2011}; \citealt{sugizaki2013}; \citealt{homan2014}; J.~Homan et al.\ 2015, in preparation). Figure~2 in \citet{homan2014} shows a combined HID using all the \textit{RXTE}\ data for the source (panel a) as well as four time-selected HID tracks illustrating the secular evolution of the source (panels b--e). The total HID looks similar to that of \mbox{Cyg X-2}\ (Figure~\ref{fig:cyg_x-2_all_data}). The higher-intensity \mbox{MAXI J0556--332}\ tracks (panels b and c) are Cyg-like Z tracks; the track in panel b shows an extended dipping FB and resembles the highest-intensity tracks of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}. The lower-intensity tracks of \mbox{MAXI J0556--332}\ (panels d and e) are Sco-like Z tracks, resembling the tracks in panels F and G--H, respectively, in the \mbox{XTE J1701--462}\ sequence, and \mbox{K+L} and M in the \mbox{Cyg X-2}\ sequence. As the overall intensity changes and the \mbox{MAXI J0556--332}\ tracks gradually evolve in shape, both the upper and lower vertices follow straight lines in the HID, as observed for \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}. The \textit{RXTE}\ coverage of \mbox{MAXI J0556--332}\ ended before the outburst came to an end, while the source was still showing Sco-like Z behavior. It may subsequently have passed through the atoll soft and hard states on its way back to quiescence.
Finally, we note that the atoll transient \mbox{XTE J1806--246} likely entered the Sco-like Z source regime briefly at the peak of its outburst in 1998 \citep{wijnands1999}. Observations during the outburst peak show the source tracing out a curved (partial) track resembling the lower NB+FB part of a Sco-like Z track. As one would expect for a transition from the atoll to Z regime, this track is at lower color values (and higher intensities) with respect to the observations in the atoll soft state taking place before and after the outburst peak. While the source traced out this track, a \mbox{7--14 Hz} QPO was detected, whose frequency increases as a function of position along the track (going from the NB to FB), consistent with the behavior of the normal/flaring-branch oscillation in Z sources.
The three sources discussed in this section, along with the four sources analyzed in this paper, constitute the entire sample of NS-LMXBs known or believed to have shown transitions between at least two of the three main NS-LMXB subclasses (Cyg-like Z, Sco-like Z, atoll). The fact that these three additional transients show behavior seemingly largely consistent with the other four sources (e.g., the relative luminosity dependence of the subclasses) strengthens our conclusion at the end of Section~\ref{sec:secular_evolution}.
\subsection{Role of the Mass Accretion Rate}\label{sec:mass_accretion_rate}
The spectral fitting results of \citet{lin2009a} indicate that the secular evolution of \mbox{XTE J1701--462}\ was driven by changes in the mass accretion rate---they find that the $\dot{M}$ inferred from the accretion disk component in the spectra at the upper and lower vertices decreases monotonically when moving left along the two vertex lines in the HID. (See also discussion in \citetalias{homan2010}.) Based on the similarities in secular behavior between \mbox{XTE J1701--462}\ and the three sources studied in this paper we conjecture that the secular evolution of \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}\ is also driven by changes in the mass accretion rate. The case is clearest for \mbox{Cyg X-2}, whose behavior is most regular of the three sources and closest to that of \mbox{XTE J1701--462}. The fact that the overall intensity of the \mbox{Cyg X-2}\ tracks decreases monotonically as they gradually evolve in shape and move along the vertex lines to higher color values makes it plausible that the mass accretion rate in \mbox{Cyg X-2}\ decreases monotonically along the sequence, as seems to be the case for \mbox{XTE J1701--462}.
Although there are some important differences between the evolution of \mbox{Cir X-1}\ and \mbox{XTE J1701--462}, the global properties are still very similar. The overall large intensity decrease from the Cyg-like tracks in the earlier panels to the Sco-like (and finally atoll) tracks in the later panels suggests a decreasing mass accretion rate going from panel A to L. However, the weaker correlation between track shape and location in the HID compared to the other sources---in particular manifested in the jumping back and forth in the HID seen during part of the sequence (panels F--I)---suggests that the mass accretion rate progression may not be strictly monotonic along the sequence. In Section~\ref{sec:cir_x-1_discussion} we suggest that these ``discrepancies'' between the track shape and location in the HID may be the result of rapid changes in the mass accretion rate.
Despite the differences in the behavior of \mbox{GX 13+1}\ compared to the other three sources we find it most likely that changes in the mass accretion rate are also responsible for the secular evolution of \mbox{GX 13+1}\ and that, as in the other sources, $\dot{M}$ decreases as the source moves up the vertex line in the CD---i.e., going from panel A to F in Figure~\ref{fig:gx_13+1_sequence}. As the tracks become overall spectrally harder the maximum intensity of a given track decreases, and the intensity swings along the track become smaller, as observed for the other three sources, although incompleteness in the tracks in panels A and B needs to be invoked to explain their deviations from this trend. Such incompleteness can, however, not explain the fact that the lower vertex in the HID is at lower intensities in those two tracks than in tracks that we suggest exhibit lower mass accretion rates, opposite to the behavior of \mbox{XTE J1701--462}.
Our results support the idea that the mass accretion rate is responsible for changes between subclasses in individual sources, with Sco-like Z behavior appearing at lower accretion rates and overall luminosities than Cyg-like Z behavior (and atoll behavior appearing at accretion rates that are lower still). However, in this picture other parameters, such as the neutron star spin, mass, and magnetic field, may still affect the luminosity scale of the subclass sequence---i.e., the range in luminosity (and mass accretion rate) over which a given subclass appears may differ between sources.
\citet{lin2009b,lin2009a} showed that for its best-estimate distance of \mbox{8.8 kpc}, \mbox{XTE J1701--462}\ reached super-Eddington luminosities during its Cyg-like stage. Even for a distance that is 30\% smaller this would have been true. The super-Eddington nature of the Cyg-like tracks is further supported by observations of two Sco-like Z sources with accurately known distances, \mbox{Sco X-1} (\mbox{2.8$\pm$0.3 kpc}; \citealt{bradshaw1999}) and \mbox{LMC X-2} (\mbox{50$\pm$2 kpc}; \citealt{pietrzynski2013}). Spectral fits indicate that those sources are either around Eddington or mildly super-Eddington \citep{bradshaw2003,agrawal2009}. Given that Cyg-like behavior is observed at significantly higher overall intensities than Sco-like behavior in \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}, this suggests that Cyg-like Z tracks may in general exhibit luminosities well above the Eddington limit.
As the sources analyzed in this paper evolve from Sco-like to Cyg-like Z behavior, it is interesting to see a large increase in the dynamic range (i.e., ratio of maximum to minimum intensity) of the Z tracks. This is most clearly seen in the tracks of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}, although this also holds for \mbox{GX 13+1}, apart from the possibly very incomplete track in panel A (Figure~\ref{fig:gx_13+1_sequence}). The most extreme intensity swings are seen in \mbox{Cir X-1}, where we observe changes in the intensity (going between the tip of the dipping FB and the upper vertex) by factors up to 8 in periods as short as 20~minutes in the highest-intensity tracks. In these extreme Cyg-like Z tracks, \mbox{Cir X-1}\ seems likely to have reached significantly higher luminosities (perhaps by a factor of $\sim$3) and exhibited significantly higher mass accretion rates than observed in \mbox{XTE J1701--462}\ at the peak of its outburst. Given that the large Cyg-like intensity swings observed in \mbox{Cir X-1}\ (as well as \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}) occur in inferred luminosity ranges that are near- or super-Eddington, it seems likely that radiation pressure effects play an important role. While it is unlikely that the mass accretion rate toward the inner parts of the accretion flow varies this strongly on such a short timescale, strong radiation pressure may result in substantial changes in the geometry of the inner accretion flow, with a (large) fraction of the mass inflow possibly being redirected into wind or jet outflows, resulting in large intensity swings.
Given the near-/super-Eddington nature of the Z sources, it is also interesting to compare them with the transient ultraluminous X-ray source \mbox{M82 X-2}. This source was recently identified as an accreting neutron star \citep{bachetti2014} and has been seen to reach a maximum luminosity of \mbox{$\sim$1.8$\times10^{40}$~erg~s$^{-1}$} \citep{feng2010}, which is a factor of $\sim$30 higher than the peak luminosity of \mbox{XTE J1701--462}\ \citep{lin2009a}. However, the short-term light curves of \mbox{M82 X-2} presented in \citet{kong2007} do not show any of the strong luminosity swings that characterize the Cyg-like Z sources. A possible explanation for this is that due to the much higher neutron star magnetic field in \mbox{M82 X-2} (as inferred from the strong pulsations in the light curve; \citealt{bachetti2014}) the accretion flow geometry in that source is very different from that in the Cyg-like Z sources, perhaps preventing the strong feedback mechanisms that we think may cause the large luminosity swings.
Three of the sources studied in this paper (\mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{GX 13+1}) were also part of a large-sample study of NS-LMXB behavior by \citet{munoz2014}. These authors find that the Cyg-like and Sco-like Z sources occupy a region of the rms--luminosity diagram that partly overlaps with that of the luminous black hole X-ray binary \mbox{GRS 1915+105}. We note, however, that overall the Z sources and \mbox{GRS 1915+105} behave very differently \citep[e.g.,][]{belloni2000}, suggesting that at near/super-Eddington luminosities the observed properties of NS-LMXBs and black hole X-ray binaries are quite different, perhaps (in part) due to the increased importance of radiation feedback from the neutron star surface.
\subsection{Nonstationary Accretion in \mbox{Cir X-1}}\label{sec:cir_x-1_discussion}
As discussed in Sections~\ref{sec:cir_x-1} and \ref{sec:mass_accretion_rate}, the secular behavior of \mbox{Cir X-1}\ is not as regular as that of \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}, as for example manifested in deviations of the track vertices from a straight line path (especially in the HID). We suspect that this may be the result of the highly nonstationary accretion in \mbox{Cir X-1}, thought to arise from the binary's eccentric orbit. From transient LMXBs it is known that highly nonstationary accretion can lead to substantial hysteresis (e.g., \citealt{maccarone2003,homan2005}), with certain source states being observed over a much larger luminosity range than would be the case for slowly changing accretion rates. \citet{yu2009} showed that these effects become stronger as the changes in mass accretion rate become faster. We speculate that, analogously, highly nonstationary accretion could be a possible explanation for certain Z track shapes being observed over a range of luminosities, rather than at a single location along the vertex lines. While all four sources in our sample undergo nonstationary accretion, which gives rise to their secular evolution, the rate at which the mass accretion rate varies with time is likely considerably higher in \mbox{Cir X-1}\ than in the other three sources. This is suggested by the fastest timescales on which we observe secular evolution in \mbox{Cir X-1}, which are shorter than in the other sources. The fastest secular evolution seen for \mbox{XTE J1701--462}\ took place during the earliest stages of the outburst (panels A--C). Inspection of the SID indicates that the lower vertex of the HID track in panel A is at \mbox{$\sim$1900 counts s$^{-1}$ PCU$^{-1}$} rather than the \mbox{$\sim$1600 counts s$^{-1}$ PCU$^{-1}$} implied by the vertex line shown; we see indications for a similar, but smaller, shift in panel B. We also note that \mbox{MAXI J0556--332}\ showed strong deviations from a straight vertex line (which it otherwise followed) during the rapid rise of its outburst (J.~Homan et al.\ 2015, in preparation). This lends support to our hypothesis that rapid changes in the mass accretion rate underlie the observed deviations in \mbox{Cir X-1}.
An additional factor that may play a role in the somewhat irregular behavior of \mbox{Cir X-1}\ is the fact that the neutron star spin, the companion spin, and the orbital axis are likely mutually misaligned in this system \citep{brpo1995,heinz2013}. This can lead to precession of the neutron star spin axis, which may in turn affect the rotation of the inner accretion flow. Changes in the projected area of the inner accretion flow might then result in similar track shapes being observed at different X-ray fluxes.
\subsection{The Z Source Classification of \mbox{GX 13+1}}\label{sec:gx_13+1_discussion}
Overall, we find that the shape and evolution of the CD/HID tracks of \mbox{GX 13+1}\ show much stronger similarities to the other three sources analyzed in this paper than to atoll sources. Most generally, just the fact that we see gradual evolution in the shape of large-scale tracks traced out by the source as the locations of the tracks shift over a considerable range in the CD and HID points to a Z source, since such secular evolution in the CD/HID has never been observed in atoll sources---only in Z sources. Moreover, the specifics of this evolution show many similarities to those of \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}, as discussed in Section~\ref{sec:gx_13+1}. Furthermore, our identification of track segments in the \mbox{GX 13+1}\ sequence (panels D and E in Figure~\ref{fig:gx_13+1_sequence}) as the HB and HB upturn (see discussion in Section~\ref{sec:gx_13+1_analysis}) is supported by the detection of a highly significant \mbox{$\sim$22--29 Hz} QPO in the corresponding timing data of these segments (J.~K.~Fridriksson et al.\ 2015, in preparation). These frequencies are considerably lower than the \mbox{57--69 Hz} QPO found by \citet{homan1998} in parts of the tracks that we now classify as the upper part of the NB (see also \citealt{schnerr2003}). If we interpret the QPO as an HBO, the decrease in its frequency as the source moves onto the HB from the upper NB is consistent with what has been seen in other Z sources (see, e.g., \citealt{jonker2000} and \citealt{homan2002}).
\citet{munoz2014} find that in their source sample there are systematic differences between the ranges in fast-variability level (broadband fractional rms) shown by the different NS-LMXB subclasses. The Cyg-like Z sources reach the highest fractional rms values (up to $\sim$10\%--15\%), followed by the Sco-like Z sources (with peak values in the $\sim$5\%--10\% range), while bright persistent atoll sources and transient atolls in a bright soft state are typically observed at $\lesssim$5\%. \citet{munoz2014} find that the rms values oberved for \mbox{XTE J1701--462}\ in the different subclasses are consistent with this. The highest fractional rms in the Z sources is observed on the HB, and we find that on the presumed HB segments \mbox{GX 13+1}\ reaches fractional rms values as high as $\sim$9\%--10\% (J.~K.~Fridriksson et al.\ 2015, in preparation). This seems more consistent with Z than bright atoll behavior. We also note that the hard color range over which these \mbox{GX 13+1}\ HB segments are observed ($\sim$0.6--0.8) is consistent with HBs observed in both persistent and transient Z sources, but clearly too low to be consistent with the atoll hard state \citep{mythesis2011}.
Considering, in addition to the above, the timescale on which the \mbox{GX 13+1}\ tracks are traced out (hours to $\sim$1~day instead of the days/weeks timescales usually observed in atoll sources), the extreme scarcity of observed type~I X-ray bursts from the source \citep{fleischman1985,matsuba1995,galloway2008}, and the reported radio brightness/behavior \citep{fender2000,homan2004} we confidently classify \mbox{GX 13+1}\ as a Z source, one that switches between Sco- and Cyg-like Z behavior. However, the behavior of the source in the HID remains somewhat puzzling when compared with \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}---especially the broad appearance of the tracks and the fact that in the spectrally harder tracks the NB/FB vertex is at higher, not lower, intensities than in the softer tracks.
This behavior may be connected to the fact that \mbox{GX 13+1}\ is probably viewed at a high inclination angle (perhaps \mbox{$60^\circ$--\,$80^\circ$}) as indicated by observations of absorption dips in this system on several occasions \citep{diaz2012,dai2014,iaria2014}. It was recently shown for black hole X-ray transients that the shape of their HID tracks depends on the inclination angle \citep{munoz2013}. While it is not fully understood how inclination affects the shape of the HID, it is likely that an inclination dependence of the HID is present in NS-LMXBs as well, with different physical components, such as the accretion disk and boundary layer, being affected differently by changes in the viewing angle.
\section{Summary and Conclusions}\label{sec:summary}
It was recently shown that during its 2006--2007 outburst, the super-Eddington neutron star transient XTE J1701$-$462 evolved through all subclasses of low-magnetic-field NS-LMXBs as a result of changes in a single physical parameter: the mass accretion rate (\citealt{lin2009a}; \citetalias{homan2010}). The main goal of this paper is to study secular evolution in the CDs and HIDs of NS-LMXBs and, more specifically, to investigate whether evolution similar to that of \mbox{XTE J1701--462}\ occurs in other sources. To this end we studied the evolution of the CD/HID tracks of three sources that show complicated secular behavior---\mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}---using all \textit{RXTE}\ PCA data obtained during the lifetime of the mission. No comprehensive study of the CD/HID behavior of these sources---using most or all of the \textit{RXTE}\ data now available---has previously been performed. We created sequences of CD/HID tracks---in many cases carefully piecing together data obtained throughout the \textit{RXTE}\ mission to construct more complete individual tracks than otherwise available---that demonstrate the secular progression in the track morphologies and locations. In the case of \mbox{Cir X-1}, this analysis had to be preceded by the filtering out of data affected by intrinsic absorption, which otherwise strongly influences the appearance of the CD/HID.
We find that \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ show strong similarities in their secular behavior to \mbox{XTE J1701--462}; in particular, both sources display gradual evolution between Cyg- and Sco-like Z tracks, with the latter occuring at lower intensities. \mbox{Cir X-1}\ shows especially extreme versions of Cyg-like tracks at the highest observed intensities, with very large and rapid intensity swings but only small changes in spectral color. At the lowest observed intensities we also see clear transitions in the CD/HID from the atoll soft to hard state for \mbox{Cir X-1}. The very gradual evolution of the FB morphology that is observed (especially in \mbox{XTE J1701--462}\ and \mbox{Cyg X-2}), from a dipping (Cyg-like) FB to a flaring Sco-like one, suggests that these two different forms of the FB are related phenomena.
Although \mbox{GX 13+1}\ shows behavior that is in some ways peculiar---especially in the HID---we find that the source also displays similarities to \mbox{XTE J1701--462}, \mbox{Cyg X-2}, and \mbox{Cir X-1}, and overall we conclude that the properties of \mbox{GX 13+1}\ are strongly indicative of a Z source.
The fact that \mbox{Cyg X-2}\ and \mbox{Cir X-1}\ show evolution between different NS-LMXB subclasses that is similar to the evolution seen in \mbox{XTE J1701--462}\ lends support to the suggestion of \citetalias{homan2010} that the behavior of \mbox{XTE J1701--462}\ is representative of the entire class of NS-LMXBs. It also strengthens their conclusion that (at least within individual sources) Cyg-like Z behavior takes place at higher luminosities and mass accretion rates than Sco-like Z behavior. Based on the similarities to \mbox{XTE J1701--462}\ we conjecture that the secular evolution of \mbox{Cyg X-2}, \mbox{Cir X-1}, and \mbox{GX 13+1}\ is due to changes in the mass accretion rate in the systems, and overall our results support the notion that differences in mass accretion rate are the primary factor underlying the existence of the various NS-LMXB subclasses. We conclude that \mbox{Cyg X-2}\ and \mbox{Cir X-1}---like \mbox{XTE J1701--462}---probably reach super-Eddington luminosities, with \mbox{Cir X-1}\ likely reaching significantly higher luminosities and mass accretion rates than the other sources.
\acknowledgements
We thank the referee for constructive comments. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.
|
2,877,628,090,206 | arxiv | \section{Introduction}
Supersymmetric Grand Unified Theories
(SGUT)~\cite{GeorgiGlashow74,Raby,Mohapatra,Raby2} have achieved some
degree of success: unification of gauge couplings, charge
quantization, prediction of the weak mixing angle, the mass-scale
of neutrinos. Detection of weak scale superpartners or proton
decay, as well as some patterns of FCNC/LFV and CP violation
phenomena would indicate that some form of SGUT lays beyond the
SM~\cite{JLDC}. Although this degree of success is already present
in the minimal models (SO(10) or some variant of
SU(5))~\cite{Murayama,HMY93,Barr}, there are open problems that suggest
the need to incorporate more elaborate constructions~\cite{HMTY},
specifically the use of higher-dimensional representations in the
Higgs sector (i.e. SU(5) representations with dimension $> 24$).
For instance, a \textbf{45} representation is often
included to obtain correct mass relations for the first and second
families of d-type quarks and leptons~\cite{GeorgiJarlskog}, while
a \textbf{75} representation has been employed to address the
doublet-triplet problem~\cite{MuOkYa}.
When one adds these higher-dimensional Higgs representation within
the context of ${\cal N}=1$ SUSY GUTs, one must verify the
cancellation of anomalies associated to their fermionic partners,
i.e. the Higgsinos. The most straightforward solution to
anomaly-cancellation is obtained by including vector-like
representations i.e. including both $\psi$ and $\bar{\psi}$ chiral
supermultiplets; up to our knowledge this seems to be the option
chosen by most models builders. It is one of the purposes of this
paper to find alternatives to this option, namely to create an
anomaly free Higgs sector, including some representation $\psi$
and a set of other representations of lower-dimension
$\left\lbrace \phi_{1},\phi_{2},...\right\rbrace $. It turns out
that different anomaly-free combinations of representations are not
equivalent in terms of their $\beta$~functions.
It is also known that the unification condition imposes some
restrictions on the GUT-scale masses of the gauge bosons,
gauginos, Higgses, and Higgsinos~\cite{Terning2}. However the
addition of complete GUT multiplets does not change the unified
gauge couplings, and neither modifies the unification scale.
On the other hand, the
evolution of the gauge couplings above the GUT scale, up to the
Planck scale, depends on the matter and Higgs content, thus the
perturbative validity of the model is affected by
the inclusion of additional multiplets. This is important in order
to determine whether gravitational effects should be invoked for
the viability of the model~\cite{Calmet}. In this paper we also
study the effect of the higher-dimensional Higgs multiplets on
the evolution of the gauge coupling up to the Planck scale,
focusing on models that invoke different sets of
representations in order to satisfy the anomaly-free
conditions.
Our paper is organized as follows. In section II we review the different
mechanisms proposed in the literature to get anomaly free gauge
theories for general gauge groups. Focusing on SU(N)-type models,
we look for new alternatives to anomaly cancellation. The implications of our results
for specific SUSY GUT SU(5) models
are presented in section III. The issue of gauge coupling unification,
and the effect of higher-dimensional representations is discussed
in section IV, where we include the 2-loop effect on the gauge unification
that is brought by the Yukawa couplings associated with those representations
at the 1-loop level.
Finally our conclusions are presented in section V.
\section{Anomalies in gauge theories}
Whether a symmetry that holds at the classical level is respected
or not at the quantum level is signaled by the presence of
anomalies. The importance of anomalies was recognized almost
immediately after the proof that Yang-Mills theories with SSB are
renormalizable was presented in~\cite{Hooft Veltman}. Anomalies can be associated
with both global or local symmetries, the latter being most
dangerous for the consistency of the theory. The so-called
perturbative anomalies arise in abelian gauge symmetries while
non-abelian symmetries can have anomalies of a non-perturbative
origin that turn out to be of topological
nature~\cite{AdlerBellJackiw}. The need to require anomaly
cancellation in any gauge theory stems from the fact that their
presence destroys the quantum consistency of the
theory~\cite{Bardeen}. It turns out that all one needs in order
to identify the anomaly is to calculate the triangle diagrams
of the form AVV, with A=Axial, and V=Vector currents.
For a given fermionic representation $D$ of a gauge group $G$, the
anomaly can then be written as~\cite{Terning2}:
\begin{equation}
A(D)d^{abc}\equiv Tr\left[ \left\lbrace T_a^{D_i},T_b^{D_i}\right\rbrace T_c^{D_i} \right] ,
\end{equation}
where $T_a^{D_i}$ denotes the generators of the gauge group $G$ in
the representation $D_i$, and $d^{abc}$ denotes the anomaly
associated with the fundamental representation. The anomaly
coefficients $a_D\equiv A(D)$ for the most common representations
are shown in Table~\ref{table:SUNreps1} for SU(N) groups; a result
that is known in the literature~\cite{Terning2}. In order to
obtain these results one makes use of the following relations:
\begin{enumerate}
\item For a representation $R$ that is a direct sum of two
representations, $R=R^{1}\oplus R^{2}$, the anomaly is given
by
\begin{equation}\label{directsum}
A_{R}=A(R_{1}\oplus R_{2})=A(R_{1})+A(R_{2}).
\end{equation}
\item For a representation $R$ that is the tensor product of two
representations, the anomaly is given by:
\begin{equation}\label{tensorproduct}
A_{R}=A(R)=A(R_{1} \otimes R_{2})=D(R_{1})A(R_{2})+
D(R_{2})A(R_1),
\end{equation}
with $D(R_{i})$ denoting the dimensions of representations
$R_{i}$.
\end{enumerate}
Then, starting from the fundamental representations $F$, we
have taken the tensor products and evaluated the
unknown coefficients that appear in the products in terms of
$A(F)$. The dimension of the representations has been verified
using the chain notation $(\alpha,\beta,\gamma,\ldots)$.
Results for some SU(N) representations can be read off from tables
in~\cite{Slansky}. We have extended these results to include
additional higher-dimensional representations, with the corresponding
expressions shown in Table~\ref{table:SUNreps2}.
\begin{table}[ht]
\[\begin{array}{|c|c|c|c|}\hline
Irrep & dim(r)& 2T(r) & A(r)\\ \hline
\includegraphics[scale=.15]{1.eps}&N&1&1\\
Ad &N^{2}-1&2N&0\\
\includegraphics[scale=.15]{2.eps}&\frac{N(N-1)}{2}&N-2&N-4\\
\includegraphics[scale=.15]{11.eps}&\frac{N(N+1)}{2}&N+2&N+4\\
\includegraphics[scale=.15]{3.eps}&\frac{N(N-1)(N-2)}{6}&\frac{(N-3)(N-2)}{2}&\frac{(N-3)(N-6)}{2}\\
\includegraphics[scale=.15]{111.eps}&\frac{N(N+1)(N+2)}{6}&\frac{(N+2)(N+3)}{2}&\frac{(N+3)(N+6)}{2}\\
\includegraphics[scale=.15]{21.eps}&\frac{N(N-1)(N+1)}{3}&N^2-3&N^{2}-9\\
\includegraphics[scale=.15]{22.eps}&\frac{N^{2}(N+1)(N-1)}{12}&\frac{N(N-2)(N+2)}{3}&\frac{N(N-4)(N+4)}{3}\\
\includegraphics[scale=.15]{1111.eps}&\frac{N(N+1)(N+2)(N+3)}{24}&\frac{(N+2)(N+3)(N+4)}{6}&\frac{(N+3)(N+4)(N+8)}{6}\\
\includegraphics[scale=.15]{31.eps}&\frac{N(N+1)(N-1)(N-2)}{8}&\frac{(N-2)(N^2-N-4)}{2}&\frac{(N-4)(N^{2}-N-8)}{2}\\
\hline
\end{array}\]
\caption{Dimensions, Dynkin indexes, and anomaly coefficients for some
representations of SU(N).}
\label{table:SUNreps1}
\end{table}
\begin{table}[ht]
\[\begin{array}{|c|c|c|c|}\hline
Irrep & dim(r) & A(r)\\ \hline
\includegraphics[scale=.15]{4.eps}&\frac{N(N-1)(N-2)(N-3)}{24}&\frac{(N-4)(N-3)(N-8)}{6}\\
\includegraphics[scale=.15]{11111.eps}&\frac{N(N+1)(N+2)(N+3)(N+4)}{120}&\frac{(N+3)(N+4)(N+5)(N+10)}{24}\\
\includegraphics[scale=.15]{2111.eps}&\frac{N(N+1)(N+2)(N+3)(N-1)}{30}&\frac{(N-2)(N+3)(N+5)^{2}}{6}\\
\includegraphics[scale=.15]{221.eps}&\frac{N^{2}(N+1)(N+2)(N-1)}{24}&\frac{N(N+5)(5N^{2}-3N-50)}{24}\\
\includegraphics[scale=.15]{32.eps}&\frac{N^{2}(N+1)(N-1)(N-2)}{24}&\frac{N(N-5)(5N^{2}+3N-50)}{24}\\
\includegraphics[scale=.15]{311.eps}&\frac{N(N+1)(N+2)(N-1)(N-2)}{20}&\frac{(N^{4}-17N^{2}+100)}{4}\\
\includegraphics[scale=.15]{41.eps}&\frac{(N-3)(N-2)(N-1)N(N+1)}{30}&\frac{(N-5)^{2}(N-3)(N+2)}{6}\\
\includegraphics[scale=.15]{5.eps}&\frac{N(N-1)(N-2)(N-3)(N-4)}{120}&\frac{(N-5)(N-4)(N-3)(N-10)}{24}\\
\includegraphics[scale=.15]{111111.eps}&\frac{N(N+1)(N+2)(N+3)(N+4)(N+5)}{720}&\frac{(N+3)(N+4)(N+5)(N+6)(N+12)}{120}\\
\includegraphics[scale=.15]{21111.eps}&\frac{N(N+1)(N+2)(N+3)(N+4)(N-1)}{144}&\frac{(N+3)(N+4)(N+6)(N^{2}+5N-12)}{24}\\
\includegraphics[scale=.15]{2211.eps}&\frac{(N-1)N^{2}(N+1)(N+2)(N+3)}{80}&\frac{3}{40}(N-3)N(N+3)(N+4)(N+6)\\
\includegraphics[scale=.15]{3111.eps}&\frac{(N-2)(N-1)N(N+1)(N+2)(N+3)}{72}&\frac{(N+3)(N^{4}+3N^{3}-16N^{2}-36N+144)}{12}\\
\includegraphics[scale=.15]{2222.eps}&\frac{(N-1)N^{2}(N+1)^{2}(N+2)}{144}&\frac{(N-4)N(N+1)(N+3)(N+6)}{24}\\
\includegraphics[scale=.15]{321.eps}&\frac{(N-2)(N-1)N^{2}(N+1)(N+2)}{45}&\frac{2}{15}(N-4)(N-3)N(N+3)(N+4)\\
\includegraphics[scale=.15]{411.eps}&\frac{(N-3)(N-2)(N-1)N(N+1)(N+2)}{72}&\frac{(N-3)(N^{4}-3N^{3}-16N^{2}+36N+144)}{12}\\
\includegraphics[scale=.15]{33.eps} &\frac{(N-2)(N-1)^{2}N^{2}(N+1)}{144}&\frac{(N-6)(N-3)(N-1)N(N+4)}{24}\\
\includegraphics[scale=.15]{42.eps}&\frac{(N-3)(N-2)(N-1)N^{2}(N+1)}{80}&\frac{3}{40}(N-6)(N-4)(N-3)N(N+3)\\
\includegraphics[scale=.15]{51.eps}&\frac{(N-4)(N-3)(N-2)(N-1)N(N+1)}{144}&\frac{(N-6)(N-4)(N-3)(N^{2}-5N-12)}{24}\\
\includegraphics[scale=.15]{6.eps}&\frac{(N-5)(N-4)(N-3)(N-2)(N-1)N}{720}&\frac{(N-6)(N-5)(N-4)(N-3)(N-12)}{120}\\
\hline
\end{array}\]
\caption{Dimensions and anomaly coefficients for higher-dimensional
representations of SU(N).}
\label{table:SUNreps2}
\end{table}
Then, given the previous results, one can try to identify
possible ways that will enable us to construct anomaly free
models. As it has been considered in the literature \cite{GeorgiGlashow72,BanksGeorgi},
there are several ways to obtain anomaly free theories, namely:
\begin{description}
\item[i)] The gauge group itself is safe, i.e. it is always free
of anomalies. This happens, for instance, for SO(10) but not for
SU(5).
\item[ii)] The gauge group is a subgroup of an anomaly free
group, and the representations form a complete representation
of the anomaly free group. For instance, this happens in the SU(5) case for the
$\mathbf{5}+\mathbf{\overline{10}}$ representations, which together are anomaly
free, and this can be understood because they belong to the $\mathbf{16}$ representation of
SO(10), i.e. under SU(5) the \textbf{16} decomposes as: $\mathbf{16=5+\overline{10}+1}$.
\item[iii)] The fermionic representations appear in conjugate
pairs, i.e., they are vector-like. This is the most common choice
when the Higgs sector of SUSY GUT is extended \footnote{Although
Higgs scalars do not contribute to the anomaly, in SUSY models
they come with the Higgsinos, their fermionic partners, which can
contribute to the anomaly.}.
For instance, a
$\mathbf{45}+\mathbf{\overline{45}}$ pair is considered to
solve the problem associated with the wrong Yukawa unification for
first and second families within SU(5) models.
\end{description}
Here, we shall show that there are also other accidental
possibilities that result when several lower-dimensional Higgs
multiplets contribute to the anomaly associated with a larger-dimensional
Higgs representation. This will be illustrated with the
SU(5) case in the following section.
\section{Anomaly Cancellation in SUSY SU(5)}
Let us consider an SU(5) SUSY GUT model. There are three copies of
$\mathbf{\overline{5}}$ and $\textbf{10}$ representations to
accommodate the three families of quarks and leptons. Breaking of
the GUT group to the SM: $SU(5) \rightarrow SU(3)_C\times SU(2)_L
\times U(1)$, is achieved by including a (chiral) Higgs
supermultiplet in the adjoint representation (\textbf{24}).
Regarding anomalies, the $\mathbf{\overline{5}}$ and $\textbf{10}$
contributions cancel each other. This situation
corresponds to case ii in the previous section, that results
from the fact that the SU(5) gauge symmetry is a subgroup of SO(10).
On the other hand, the \textbf{24} representation is itself
anomaly free. The minimal Higgs sector
needed to break the SM gauge group can be formed with a pair
of \textbf{5} and $\mathbf{\overline{5}}$ representations, which is
indeed vectorial and therefore anomaly free (this corresponds
to case iii discussed above).
Now, within this minimal model with a Higgs sector consisting of
\textbf{5}+$\mathbf{\overline{5}}$, one obtains the mass relations
$m_{d_i}=m_{e_i}$, which works well for the third family,
but not for the second family,
while it may or may not work for the first family, depending on
whether or not one includes weak scale threshold
effects\cite{DMP}. One way to solve this problem is to add a
\textbf{45} representation, which couples to the d-type quarks
but not to the up-type, and one then obtains the Georgi-Jarlskog
factor~\cite{GeorgiJarlskog} needed for the correct mass
relations. Most models that obtain these relations with an
extended Higgs sector, include the conjugate representation
in order to cancel the anomalies, i.e.
$\mathbf{45}+\mathbf{\overline{45}}$ ~\cite{Pavel Fileviez}.
This is however not the only possibility, and this is
one of the main results of our paper.
\begin{table}[ht]
\[\begin{array}{|c|c|c|c|c|}\hline
Irrep &Multiplet &dim(r) & A(r)&2T(r)\\
\hline
\left[5\right]&(0,0,0,0)&1&0&0\\
\left[1\right]&(1,0,0,0)&5&1&1\\
\left[2\right]&(0,1,0,0)&10&1&3\\
\left[1,1\right]&(2,0,0,0)&15&9&7\\
\left[4,1\right]&(1,0,0,1)&24&0&10\\
\left[1,1,1\right]&(3,0,0,0)&35&44&28\\
\left[2,1\right]&(1,1,0,0)&40&16&22\\
\left[3,1\right]&(1,0,1,0)&45&6&24\\
\left[2,2\right]&(0,2,0,0)&50&15&35\\
\hline
\end{array}\]
\caption{Dimension, anomaly coefficients, and Dynkin indexes for different representations of SU(5).}
\label{table:SU(5)}
\end{table}
The results for the anomaly coefficients for some representations
of SU(5) (and their conjugates) are shown
in table~\ref{table:SU(5)}; we can see that the \textbf{45}
anomaly coefficient is 6. Then taking into consideration that the
\textbf{5} and the \textbf{10} have the anomaly coefficient $A=1$, we can
write down the following anomaly-free combinations
\begin{eqnarray}
A(\mathbf{45})+A(\mathbf{\overline{45}})=0& \ , \\
A(\mathbf{45})+6A(\mathbf{\overline{5}})=0& \ , \\
A(\mathbf{45})+6A(\mathbf{\overline{10}})=0& \ .
\end{eqnarray}
Alternatively we can write a general anomaly-free condition with these fields,
\begin{equation}
A(\mathbf{45})+fA(\mathbf{\overline{5}})+f'A(\mathbf{\overline{10}})=0, \ \text{with }f+f'=6 \ .
\end{equation}
One could also invoke a $\mathbf{\overline{15}}$ representation, which has
$A=-9$, through the following anomaly-free combination:
\begin{equation}\label{}
A(\mathbf{45})+A(\mathbf{\overline{15}})+3A(\mathbf{5})=0 \ .
\end{equation}
These are non-equivalent models with different physical
consequences. This is explicitly shown in the next section
where we discuss the issue of gauge coupling unification.
\section{Gauge coupling unification and perturbative validity.}
The $\beta$ functions for a general SUSY theory with
gauge group $G$ and matter fields appearing in chiral supermultiplets,
at the 1-loop level, are given by:
\begin{equation}\label{betafunctions}
\beta_1=\sum_R T_R - 3C_A,
\end{equation}
where $ T_R $ denotes the Dynkin index for the representation $R$,
and $C_A$ is the quadratic Casimir invariant for the adjoint
representation. For SU(N) type gauge groups $C_A=N$, while the
$T_R$ index for most common SU(5) representations are also shown
in Table~\ref{table:SU(5)}.
The RGE's with 1-loop $\beta$ functions for the gauge couplings of
the MSSM are
\begin{equation}\label{oneloopRGE}
\frac{d\alpha_i}{dt}=\beta_i \alpha_i^2,
\end{equation}
where
\begin{equation}\label{oneloopBetas}
\beta_i=\left(%
\begin{array}{c}
33/5 \\
1 \\
-3 \\
\end{array}%
\right)+\beta^X
\end{equation}
and $t=(2\pi)^{-1}\ln M$, with $M =$ mass scale. The index $i=1,2,3$ refers to
the U(1), SU(2) and SU(3) gauge groups respectively. The term $ \beta^X=\sum_{\Phi}
T(\Phi)$ denotes the contributions of the additional representations
beyond those included in the MSSM (the
sum is over all SU(5) additional multiplets $\Phi$). Assuming
$M_{SUSY}\approx M_{t}$, one obtains that the unified gauge coupling is
approximately $g(M_{GUT}) = 0.0416$, and unification
occurs at $M_{GUT}=2\times 10^{16}$~GeV~\cite{Hempfling}.
These simple 1-loop results can be improved by
using the 2-loop RGEs~\cite{Jones:1981we,Yamada:1993ga,Barger:1992ac}.
In such case we solve numerically
the corresponding RGE and we find that at the GUT scale
$M_{GUT}=1.28\times 10^{16}$~GeV, the unified gauge coupling is
$g_5(M_{GUT}) = 0.040$, and $h^t(M_{GUT})=0.6572$.
Now we are interested in evaluating the effect of the different
representations in the running from $M_{GUT}$ up to the Planck
scale. Besides evaluating the effect of the different anomaly free
combinations, we are also interested in finding which
representations are perturbatively valid up to the Planck scale.
The unified gauge coupling obeys the 1-loop RGE
\begin{equation}
\mu \frac{d \alpha_5^{-1}}{d\mu}=\frac{-\beta_1}{2\pi}=\frac{3-\beta^X}{2\pi} \ ,
\end{equation}
where $-\beta_1=\beta_{MIN}-\beta^X$, with $\beta_{MIN}=3 $
denoting the contribution to the SU(5) $\beta$~function
from the MSSM multiplets, including the one from the gauge sector.
The 1-loop $\beta$~functions for some interesting anomaly-free
combinations are found to be:
\begin{eqnarray}\label{betafuntions} \nonumber
\beta^X(\mathbf{45}+\mathbf{\overline{45}} )& = & 24 \ , \\ \nonumber
\beta^X(\mathbf{45}+ 6 (\mathbf{\overline{5}})) & = & 15 \ , \\ \nonumber
\beta^X(\mathbf{45}+ 6 (\mathbf{\overline{10}})) & = & 21 \ , \\ \nonumber
\beta^X(\mathbf{45}+ \mathbf{\overline{15}}+2 (\mathbf{10})+\mathbf{5}) & = & 19 \ , \\
\beta^X(\mathbf{50}+\mathbf{\overline{40}}+\mathbf{5}) & = & 29 \ .
\end{eqnarray}
As shown in Figure~\ref{fig:grunning1}, the
model with $\beta^X=29$ induces a running of the unified gauge coupling
that blows at the scale $M=6.61\times 10^{18}$~GeV, while for
$\beta^X=24$ this happens at $M=2.63\times 10^{19}$~GeV. The models
with $\beta^X=15,19,21$ are found to evolve safely all the way up to the
Planck scale.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{GrapBetas}
\caption{Evolution of the unified gauge coupling for the free
anomaly combinations listed in the text. The evolution
is shown all the way up to the Planck scale. }
\label{fig:grunning1}
\end{figure}
It is also interesting to consider the RGE effect associated with
the Yukawa coupling that involve the additional Higgs representations.
In order to do this we shall consider the 2-loop $\beta$~functions
for the gauge coupling~\cite{Martin:1993zk}, but will keep only the 1-loop RGE for the new
Yukawa couplings. Thus, we shall consider the following superpotential for the
SUSY SU(5) GUT model:
\begin{eqnarray}\label{} \nonumber
W &=& \frac{f}{3}Tr\Sigma^3+\frac{1}{2}f VTr\Sigma^2
+\lambda\bar{H}_{\alpha}(\Sigma^{\alpha}_{\beta}+3V\delta^{\alpha}_{\beta})H^\beta \\
&+&\frac{h^{ij}}{4} \varepsilon_{\alpha\beta\gamma\delta\epsilon}\psi^{\alpha\beta}_{i}\psi^{\gamma\delta}_{j}H^{\epsilon}
+\sqrt{2}f^{ij}\psi^{\alpha\beta}_{i}\phi_{j\alpha}\bar{H}_{\beta} \ .
\end{eqnarray}
Note that this superpotential involves the Higgs representations
$\mathbf{5}$, $\bar{\mathbf{5}}$ y $\mathbf{24}$.
The 1-loop RGEs for the Yukawa parameters are given by~\cite{HMY93}:
\begin{eqnarray}\label{RGE SUSY SU(5)}
\mu \frac{d\lambda}{d\mu} & = &
\frac{1}{(4\pi)^2}\left(-\frac{98}{5}g_5^2+\frac{53}{10}\lambda^2
+\frac{21}{40}f^2+3(h^t)^2\right)\lambda, \\
\mu \frac{d f}{d\mu} & = &
\frac{1}{(4\pi)^2}\left(-30g_5^2+\frac{3}{2}\lambda^2
+\frac{63}{40}f^2\right)f, \\
\mu \frac{d h^t}{d\mu} & = &
\frac{1}{(4\pi)^2}\left(-\frac{96}{5}g_5^2+\frac{12}{5}\lambda^2
+6(h^t)^2\right)h^t \ ,
\end{eqnarray}
while the 2-loop RGE for the unified gauge coupling is given by:
\begin{equation}\label{g5}
\mu\frac{d
g_5}{d\mu}=\frac{1}{(4\pi)^2}(-3g_5^{3})+\frac{1}{(4\pi)^4}\frac{794}{5}g_5^{5}-
\frac{1}{(4\pi)^4}\left\{\frac{49}{5}\lambda^2
+\frac{21}{4}f^2+12(h^t)^2\right\}g^3 \ .
\end{equation}
We use values of the coefficients $\lambda$, $h^t$ and $f$ that
are safe at the Planck scale, and look for their effects on the
unified gauge coupling. The resulting evolution is shown in one of the lines in
Figure~\ref{fig:grunning2}, where we show the 1-loop results, as well as the the
2-loop results with and without the Yukawas 1-loop contributions. The parameters used in
the plots are $ M_{GUT}=1.28 \times 10^{16}$~GeV, $\alpha(M_{GUT})=0.040$,
$h^t(M_{GUT})=0.6572$, $\lambda(M_{GUT})=0.6024$, and $f(M_{GUT})=1.7210$.
We notice that there are appreciable differences
for the evolution of the gauge coupling when going from the one to the 2-loop
cases, but this difference is reduced when one includes Yukawa couplings at
the 1-loop level.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{L7fs}
\caption{Evolution of the unified gauge coupling for three different cases:
i) the 1-loop result, ii) the 2-loop result without including Yukawas, and iii)
the 2-loop result including the (1-loop) running of the Yukawas.}
\label{fig:grunning2}
\end{figure}
\section{Conclusions}
We have studied the problem of anomalies in SUSY gauge theories
in order to search for alternatives to the usual vector-like
representations used in extended Higgs sectors.
The known results have been extended to include higher-dimensional
Higgs representations, which in turn have been applied to discuss
anomaly cancellation within the context of realistic GUT models of
SU(5) type. We have succeeded in
identifying ways to replace the $\mathbf{45}+\bar{\mathbf{45}}$
models within SU(5) SUSY GUTs. Then, we have studied the $\beta$~functions
for all the alternatives, and we find that they are not
equivalent in terms of their values.
We have also considered the RGE effect associated with
the Yukawa coupling that involve the additional Higgs representations.
We found that there are appreciable differences
for the evolution of the gauge coupling when going from the 1 to the 2-loop
RGE, but this difference is reduced when one includes the 1-loop Yukawa couplings at
the 2-loop level.
These results have important implications for the perturbative
validity of the GUT models at scales higher than the unification
scale.
\begin{acknowledgments}
This work was supported in part by CONACYT and SNI. A.A. acknowledges the
Benem\'erita Universidad Aut\'onoma de Puebla for its warm hospitality while
part of this work was being done.
\end{acknowledgments}
|
2,877,628,090,207 | arxiv | \section{Introduction}
Let $G$ be a compact connected Lie group and let $H\subset G$ be the
centralizer of a one--parameter subgroup in $G$. The homogeneous space $G/H$
is known as a \textsl{flag variety}. We fix a maximal torus $T\subseteq H $
and write $W$ (resp. $W^{^{\prime }}$) for the Weyl group of $G$ (resp. $H$).
In the founding article [AH] of topological $K$--theory as a ``generalized
cohomology theory'', Atiyah and Hirzebruch raised also the problem to
determine the ring $K(G/H)$, with the expectation that ``\textsl{the new
cohomology theory can be applied to various topological questions and may
give better results than the ordinary cohomology theory}''. As an initial
step they showed that the $K(G/H)$ is a free $\mathbb{Z}$-module with rank
equal to the quotient of the order of $W$ by the order of $W^{^{\prime }}$
[AH, Theorem 3.6].
In the subsequent years two distinguished additive bases of $K(G/H)$ have
emerged from algebraic geometry\footnote{%
The Grothendieck cohomology of coherent sheaves on $G/H$ is canonically
isomorphic to the $K$-theory of complex vector bundles on $G/H$ [Br$_{2}$,PR$%
_{2}$].}. The first of these, valid for the case $H=T$ and indexed by
elements from the Weyl group: $\{a_{w}\in K(G/T)\mid w\in W\}$, is due to
Demazure [D, Proposition 7]. The basis element $a_{w}$ will be called the
\textsl{Demazure class} relative to $w\in W$.
The second basis goes back to the classical works of Grothendieck and
Chevalley. Let $l:W\rightarrow \mathbb{Z}$ is the length function relative
to a fixed maximal torus $T\subset G$ and identify the set $W/W^{\prime }$
of left cosets of $W^{\prime }$ in $W$ with the subset $\overline{W}=\{w\in
W\mid l(w)\leq l(w_{1})$ for all $w_{1}\in wW^{\prime }\}$ of $W$. According
to Chevalley [Ch] the flag variety $G/H$ admits a canonical partition into
Schubert varieties, indexed by elements of $\overline{W}$,
\begin{center}
$G/H=\underset{w\in \overline{W}}{\cup }X_{w}(H)$, $\quad \dim
X_{w}(H)=2l(w) $.
\end{center}
\noindent The coherent sheaves $\Omega _{w}(H)\in K(G/H)$ of the Schubert
variety Poincare dual to the $X_{w}(H)$, $w\in \overline{W}$, form a basis
for the $\mathbb{Z}$-module $K(G/H)$ by Grothendieck (cf. [C, Lecture 4]).
The $\Omega _{w}(H)$ is called the \textsl{Grothendieck class} relative to $%
w\in \overline{W}$.
\bigskip
A complete description of the ring $K(G/T)$ (resp. $K(G/H)$) now amounts to
specify the structure constants $C_{u,v}^{w}\in \mathbb{Z}$ (resp. $%
K_{u,v}^{w}(H)\in \mathbb{Z}$) required to express the products of basis
elements
\begin{enumerate}
\item[(1.1)] $\qquad a_{u}\cdot a_{v}=\sum C_{u,v}^{w}a_{w}$ (resp. $\Omega
_{u}(H)\cdot \Omega _{v}(H)=\sum K_{u,v}^{w}(H)\Omega _{w}(H)$),
\end{enumerate}
\noindent where $u,v,w\in W$ (resp. $\in \overline{W}$).
Based on combinatorial methods, partial information concerning the ring $%
K(G/H)$ has been achieved during the past decade. Generalizing the classical
Pieri-Chevalley formula in the ordinary cohomology [Ch], various
combinatorial formulae expressing the product $L\cdot \Omega _{u}$ in terms
of the $\Omega _{w}$ were obtained, where $L$ is a line bundle on $G/T$.
This was originated by Fulton and Lascoux [FL] for the unitary group $U(n)$
of rank $n$ (see also Lenart [L]), extended to general $G$ by Pittie and Ram
[PR$_{1,2}$], Mathieu [M], Littelmann and Seshadri [LS] by using very
different methods. Another progress is when $G=U(n)$ and $H=U(k)\times
U(n-k) $, the flag variety $G/H$ is the Grassmannian $G_{n,k}$ of $k$-planes
through the origin in $\mathbb{C}^{n}$ and a combinatorial description for
the $K_{u,v}^{w}(H)$ was obtained by Buch [Bu].
In general, using purely geometric approach, Brion [Br$_{1}$,Br$_{2}$]
proved that the $K_{u,v}^{w}$ have alternating signs, confirming a
conjecture of Buch in [Bu].
\bigskip
In this paper we present a formula that expresses the constants $C_{u,v}^{w}$
(resp. the $K_{u,v}^{w}(H)$ ) by the Cartan numbers of $G$, see Theorem 1 in
\S 2 (resp. Theorem 2 in \S 5). These results are natural generalization of
the formula in [Du$_{2}$] for multiplying Schubert classes in the cohomology
of $G/H$ for, in the special cases $l(w)=l(u)+l(v)$, the number $C_{u,v}^{w}$
(resp. the $K_{u,v}^{w}(H)$ ) agrees with the coefficient of the Schubert
class $P_{w}$ in the product $P_{u}\cdot P_{v}$ (see in the notes after
Theorem 1 in \S 2).
An important problem in algebraic combinatorics is to find a combinatorial
description of the $C_{u,v}^{w}$ (resp. the $K_{u,v}^{w}(H)$ ) that has the
advantage to reveal their signs [L,Bu]. Apart from the combinatorial concern,%
\textsl{\ }effective computation in the $K$--theory of such classical
manifolds as the $G/H$\ is decidedly required by many problems from geometry
and topology. We emphasis that the formulae as given in this paper are
computable, although it is not readily to reveal the signs of the constants.
With some additional works the existing program [DZ$_{2}$] for multiplying
Schubert classes can be extended to implement the $C_{u,v}^{w}$ (resp. $%
K_{u,v}^{w}(H)$). This program uses the Cartan matrix of $G$ as the only
input and, therefore, is functional uniformly for all $G/H$.
\bigskip
The author feels very grateful to S. Kumar for valuable communication.
Indeed, our exposition benefits a lot from certain results (cf. Lemma 4.2;
Lemma 5.1) developed in the classical treatise [KK] on this subject.
Thanks are also due to my referee for many improvements on the earlier
version of the paper, and for his kindness in informing me the work [GR] by
S. Griffeth and A. Ram, where a combinatorial method to multiply two
elements of the Grothendieck basis of the equivariant K--ring of a flag
variety is given, and the article [W$_{2}$] by M. Willems, where similar
methods are used in the setting of equivariant K--theory.
\section{The formula for $C_{u,v}^{w}$}
We introduce notations (from Definition 1 to 4) in terms of which the
formula for $C_{u,v}^{w}$ will be presented in Theorem 1.
\bigskip
Fix once and for all a maximal torus $T$ in $G$. Set $n=\dim T$. Equip the
Lie algebra $L(G)$ with an inner product $(,)$ so that the adjoint
representation acts as isometries of $L(G)$.
The restriction of the exponential map $\exp :L(G)\rightarrow G$ to $L(T)$
defines a set $D(G)$ of $\frac{1}{2}(\dim G-n)$ hyperplanes in $L(T)$, i.e.
the set of\textsl{\ singular hyperplanes\ }through the origin in $L(T)$. The
reflections of $L(T)$ in these planes generate the Weyl group $W$ of $G$
([Hu, p.49]).
Take a regular point $\alpha \in L(T)$ and let $\Delta $ be the set of
simple roots relative to $\alpha $ [Hu, p.47]. For a $\beta \in \Delta $ the
reflection $r_{\beta }$ in the hyperplane $L_{\beta }\in D(G)$ relative to $%
\beta $ is called a \textsl{simple reflection}. If $\beta ,\beta ^{\prime
}\in \Delta $, \textsl{the Cartan number\ }
\begin{center}
$\beta \circ \beta ^{\prime }=2(\beta ,\beta ^{\prime })/(\beta ^{\prime
},\beta ^{\prime })$
\end{center}
\noindent is always an integer (only $0,\pm 1,\pm 2,\pm 3$ can occur). It is
also customary to use $(\beta ,\beta ^{\prime \vee })$ instead of $\beta
\circ \beta ^{\prime }$.
\bigskip
It is well known that the set of simple reflections $\{r_{\beta }\mid \beta
\in \Delta \}$ generates $W$. That is, any $w\in W$ admits a factorization
of the form
\begin{enumerate}
\item[(2.1)] \ \ \ \ \ \ \noindent $w=r_{\beta _{1}}\cdot \cdots \cdot
r_{\beta _{m}}$,$\quad \beta _{i}\in \Delta $.
\end{enumerate}
\begin{quote}
\textbf{Definition 1.}\textit{\ }The \textsl{length} $l(w)$\ of a $w\in W$\
is the least number of factors in all decompositions of $w$\ in the form
(2.1). The decomposition (2.1) is said \textsl{reduced} if $m=l(w)$.
If (2.1) is a reduced decomposition, the $m\times m$\ (strictly upper
triangular) matrix $A_{w}=(a_{i,j})$\ with
\end{quote}
\begin{center}
$a_{i,j}=\{%
\begin{array}{c}
0\text{ \ if }i\geq j\text{;\qquad } \\
\beta _{j}\circ \beta _{i}\text{\ if }i<j%
\end{array}%
$
\end{center}
\begin{quote}
\noindent will be called \textsl{the Cartan matrix of }$w$\ relative to the
decomposition (2.1).
\textbf{Definition 2.} Given a sequence $(\beta _{1},\cdots ,\beta _{m})$\
of simple roots and a $w\in W$, let $[i_{1},\cdots ,i_{k}]\subseteq \lbrack
1,\cdots ,m]$\ be the subsequence maximal in the inverse-lexicographic order
so that
\end{quote}
\begin{center}
$l(w)>l(wr_{\beta _{i_{k}}})>$\ $l(wr_{\beta _{i_{k}}}r_{\beta
_{i_{k-1}}})>\cdots >l(wr_{\beta _{i_{k}}}r_{\beta _{i_{k-1}}}\cdots
r_{\beta _{i_{1}}})$.
\end{center}
\begin{quote}
\noindent We call $(\beta _{1},\cdots ,\beta _{m})$\ \textsl{a derived
(simple root) sequence of }$w$, written $(\beta _{1},\cdots ,$\ $\beta
_{m})\thicksim w$, if $k=l(w)$\ (i.e. $r_{\beta _{i_{1}}}\cdots r_{\beta
_{i_{k}}}$\ is a reduced decomposition of $w$).
\end{quote}
\textbf{Remark 1.} It is clear that $(\beta _{1},\cdots ,\beta
_{m})\thicksim w$ implies $m\geq l(w)$, while the equality holds if and only
if $w=r_{\beta _{1}}\cdots r_{\beta _{m}}$.
The Definition 2 implies also that if $e\in W$ is the group unit, then $%
(\beta _{1},\cdots ,\beta _{m})\sim e$ for any sequence $(\beta _{1},\cdots
,\beta _{m})$\ of simple roots.
\bigskip
Let $\mathbb{Z}[y_{1},\cdots ,y_{m}]$ be the ring of integral polynomials in
$y_{1},\cdots ,y_{m}$, graded by $\mid y_{i}\mid =1$, and let $\mathbb{Z}%
[y_{1},\cdots ,y_{m}]_{(n)}$ be the submodule of all polynomials of degree $%
\leq n$. We introduce the additive maps
\begin{center}
$_{(n)}:\mathbb{Z}[y_{1},\cdots ,y_{m}]\rightarrow \mathbb{Z}[y_{1},\cdots
,y_{m}]_{(n)}$, $n\geq 0$,
\end{center}
\noindent by the following rule. If $f\in \mathbb{Z}[y_{1},\cdots ,y_{m}]$,
then
\begin{center}
$f=f_{(n)}+$ a sum of monomials of degree $>n$.
\end{center}
\begin{quote}
\textbf{Definition 3.} Let $A=(a_{i,j})_{m\times m}$\ be a strictly upper
triangular matrix (with integer entries). In terms of the entries of $A$\
define two sequences $\{q_{1},\cdots ,q_{m}\},$\ $\{\overline{q}_{1},\cdots ,%
\overline{q}_{m}\}\subset \mathbb{Z}[y_{1},\cdots ,y_{m}]$\ of polynomials
inductively as follows. Put $q_{1}=\overline{q}_{1}=1$,\ and for $k>1$\ let
\end{quote}
\begin{center}
$\qquad
q_{k}=\prod\limits_{a_{i,k}>0}(y_{i}+1)^{a_{i,k}}\prod%
\limits_{a_{i,k}<0}(-q_{i}y_{i}+1)^{-a_{i,k}}$;
$\qquad \overline{q}_{k}=\underset{a_{i,k}>0}{\prod }%
(-q_{i}y_{i}+1)^{a_{i,k}}\underset{a_{i,k}<0}{\prod }(y_{i}+1)^{-a_{i,k}}$.
\end{center}
\textbf{Remark 2.} As an example if $A=\left(
\begin{array}{ccc}
0 & 1 & 2 \\
0 & 0 & -1 \\
0 & 0 & 0%
\end{array}%
\right) $, then
\begin{center}
$\qquad \{%
\begin{array}{c}
q_{2}=y_{1}+1,\ \\
\overline{q_{2}}=1-y_{1}\text{;}%
\end{array}%
$ and $\{%
\begin{array}{c}
q_{3}=(y_{1}+1)^{2}[-(y_{1}+1)y_{2}+1], \\
\overline{q_{3}}=[-(y_{1}+1)y_{1}+1]^{2}(y_{2}+1)\text{.}%
\end{array}%
$
\end{center}
\noindent Note also that, since $A$ is strictly upper triangular, we always
have
\begin{center}
$q_{k},\overline{q}_{k}\in \mathbb{Z}[y_{1},\cdots ,y_{k-1},q_{1},\cdots
,q_{k-1}]=\mathbb{Z}[y_{1},\cdots ,y_{k-1}]$.
\end{center}
\begin{quote}
\textbf{Definition 4.} Given a strictly upper triangular matrix $%
A=(a_{i,j})_{m\times m}$\ of rank $m$\ define the operator $\Delta _{A}:%
\mathbb{Z}[y_{1},\cdots ,y_{m}]_{(m)}\rightarrow \mathbb{Z}$\ as the
composition
\end{quote}
\begin{center}
$\mathbb{Z}[y_{1},\cdots ,y_{m}]_{(m)}\overset{D_{m-1}}{\rightarrow }\mathbb{%
Z}[y_{1},\cdots ,y_{m-1}]_{(m-1)}\overset{D_{m-2}}{\rightarrow }\cdots
\overset{D_{1}}{\rightarrow }\mathbb{Z}[y_{1}]_{(1)}\overset{D_{0}}{%
\rightarrow }\mathbb{Z}$,
\end{center}
\begin{quote}
\noindent where the operator $D_{k-1}:\mathbb{Z}[y_{1},\cdots
,y_{k}]_{(k)}\rightarrow \mathbb{Z}[y_{1},\cdots ,y_{k-1}]_{(k-1)}$\ is
given by the following elimination rule.
\textsl{Expand each }$f\in \mathbb{Z}[y_{1},\cdots ,y_{k}]_{(k)}$\textsl{\
in terms of the powers of }$y_{k}$
\end{quote}
\begin{center}
$f=h_{0}+h_{1}y_{k}+h_{2}y_{k}^{2}+\cdots +h_{k}y_{k}^{k},$\textsl{\ \ }$%
h_{i}\in \mathbb{Z}[y_{1},\cdots ,y_{k-1}],$
\end{center}
\begin{quote}
\textsl{\noindent then put}
\end{quote}
\begin{center}
$D_{k-1}(f)=[h_{1}+h_{2}(\overline{q_{k}}-1)+\cdots +h_{k}(\overline{q_{k}}%
-1)^{k-1}]_{(k-1)}$\textsl{,}
\end{center}
\begin{quote}
\textsl{\noindent where the }$\overline{q_{k}}$\textsl{\ is given by }$A$%
\textsl{\ as in Definition 3 (see also Remark 2).}
\end{quote}
\textbf{Remark 3. }The $\Delta _{A}$ can be easily evaluated, as the formula
shows.
\begin{center}
$D_{k-1}(f)=[\sum\limits_{n\geq 1}(\frac{1}{n!}\frac{\partial ^{n}f}{%
(\partial y_{k})^{n}}\mid _{y_{k}=0})(\overline{q_{k}}-1)^{n-1}]_{(k-1)}$.
\end{center}
\textbf{Remark 4.} The operator\textbf{\ }$\Delta _{A}$ extends the idea of
\textsl{triangular operator} $T_{A}$ in [Du$_{1}$,Du$_{2}$,DZ$_{1}$,DZ$_{2}$%
] in the following sense. If $f\in \mathbb{Z}[y_{1},\cdots ,y_{m}]$\ is of
homogeneous degree $m$, then $\Delta _{A}(f)=T_{-A}(f)$\textit{.}
The operator $T_{A}$ appears to be a useful tool in computing with the
cohomology of $G/H$. It was applied to evaluate the degrees of Schubert
varieties in [Du$_{1}$], to present a formula for multiplying Schubert
classes in [Du$_{2}$,DZ$_{2}$], and to compute the Steenrod operations on
the $\mathbb{Z}_{p}$--cohomologies of $G/H$ \ in [DZ$_{1}$]. Apart from the $%
\Delta _{A}$, another generalization of the $T_{A}$ was given by Willems [W,
Definition 5.2.1], which was useful for multiplying Schubert classes in the $%
T$-equivariant cohomology of $G/T$ [W$_{1}$, Theorem 5.3.1].
\bigskip
Assume that $w=r_{\beta _{1}}\cdots r_{\beta _{m}}$, $\beta _{i}\in \Delta $%
, is a reduced decomposition of $w\in W$, and let $A_{w}=(a_{i,j})_{m\times
m}$ be the associated Cartan matrix. For a subsequence $L=[i_{1},\cdots
,i_{k}]\subseteq \lbrack 1,\cdots ,m]$ we set
\begin{center}
$\beta (L)=(\beta _{i_{1}},\cdots ,\beta _{i_{k}})$,\qquad\ $%
y_{L}=y_{i_{1}}\cdots y_{i_{k}}\in \mathbb{Z}[y_{1},\cdots ,y_{m}]$.
\end{center}
\begin{quote}
\textbf{Theorem 1.} \textsl{For }$u,v\in W$\textsl{\ we have}
\end{quote}
\begin{enumerate}
\item[(2.2)] $\qquad (-1)^{l(w)}C_{u,v}^{w}=(-1)^{l(u)+l(v)}\Delta
_{A_{w}}[(\sum\limits_{\beta (L)\thicksim u}y_{L})(\sum\limits_{\beta
(K)\thicksim v}y_{K})]_{(m)}$
$\qquad \qquad \qquad \qquad \quad -\sum\limits_{\substack{ l(u)+l(v)\leq
l(x)\leq l(w)-1 \\ \beta (1,\cdots ,m)\thicksim x\in W}}%
(-1)^{l(x)}C_{u,v}^{x}$\textsl{,}
\end{enumerate}
\begin{quote}
\noindent \textsl{where }$L,K\subseteq \lbrack 1,\cdots ,m]$.
\end{quote}
As suggested by Theorem 1, the job to compute all $C_{u,v}^{x}$ for given $%
u,v\in W$ may be organized as follows.
\begin{enumerate}
\item[(1)] If $l(w)(=m)<l(u)+l(v)$, then
$\qquad \qquad \lbrack (\sum\limits_{\beta (L)\thicksim
u}y_{L})(\sum\limits_{\beta (K)\thicksim v}y_{K})]_{(m)}=0$
implies that $C_{u,v}^{w}=0$.
\item[(2)] If $l(w)(=m)=l(u)+l(v)$ the formula becomes
\end{enumerate}
\ \ \ \ $\qquad C_{u,v}^{w}=\Delta _{A_{w}}[(\sum\limits_{\beta (L)\thicksim
u}y_{L})(\sum\limits_{\beta (K)\thicksim v}y_{K})]_{(m)}$
$\qquad =\Delta _{A_{w}}[(\sum\limits_{r_{L}=u,\mid L\mid
=l(u)}y_{L})(\sum\limits_{r_{K}=v,\mid K\mid =l(v)}y_{K})]$ (cf. Remark 1)
$\qquad =T_{-A_{w}}[(\sum\limits_{r_{L}=u,\mid L\mid
=l(u)}y_{L})(\sum\limits_{r_{K}=v,\mid K\mid =l(v)}y_{K})]$ (cf. Remark 4),
\begin{enumerate}
\item[ ] \noindent where $r_{L}=r_{\beta _{i_{1}}}\cdots r_{\beta _{i_{k}}}$%
, $\mid L\mid =k$ if $L=[i_{1},\cdots ,i_{k}]$. This recovers the formula
for multiplying Schubert classes in the ordinary cohomology of $G/T$ [Du$%
_{2} $,DZ$_{1}$].
\item[(3)] In general, assuming that all the constants $C_{u,v}^{x}$ with $%
l(x)<m$ have been obtained, the theorem gives $C_{u,v}^{w}$ with $l(w)=m$ in
terms of the operator $\Delta _{A_{w}}$ as well as those $C_{u,v}^{x}$ ($%
l(x)<m$, $(\beta _{1},\cdots ,\beta _{m})\thicksim x$) calculated before.
\end{enumerate}
\noindent It is clear from the discussion that Theorem 1 reduces the $%
C_{u,v}^{w}$ to the operators $\Delta _{A_{x}}$, hence to the matrices $%
A_{x} $ formed by Cartan numbers.
\bigskip
Theorem 1 is originated from the celebrated Bott-Samelson cycles on flag
manifolds [BS]. This may be seen from the geometric consideration that
underlies the algebraic formation from Definition 1 to 4. Indeed, the Cartan
matrix of a $w$ (Definition 1) with respect to the decomposition (2.1) gives
the structural data characterizing the Bott-Samelson cycle $S(\alpha ;\beta
_{1},\cdots ,\beta _{m})$ as a twisted products of $2$--spheres (Lemma 4.3);
the polynomials $\overline{q}_{k}$'s ( Definition 3) provide the relations
in describing the $K$--ring of $S(\alpha ;\beta _{1},\cdots ,\beta _{m})$ as
the quotient of a polynomial ring (Lemma 4.4); the operator $\Delta _{A}$
(Definition 4) handles the integration along the top cell of $S(\alpha
;\beta _{1},\cdots ,\beta _{m})$ in the $K$--theory (Lemma 4.4); and the
idea of derived sequence of a Weyl group element(Definition 2) is required
to specify the induced map of a Bott-Samelson cycle in $K$--theory (Lemma
4.5).
The remaining sections are so arranged. Before involving the specialities of
flag manifolds, Section 3 studies the $K$-theory of \textsl{twisted products
of }$2$\textsl{-spheres}, a family of manifolds that generalizes the
classical Bott-Samelson cycles on $G/T$ [BS] (Lemma 3.4). In addition,
\textsl{divided difference} in $K$-theory is introduced for \textsl{%
spherical represented involutions }(cf. \textbf{3.2}). In Section 4, by
resorting to the geometry of the adjoint representation, we interpret the
Bott-Samelson cycles on $G/T$ as certain twisted products of $2$-spheres,
and describe their $K$--rings in terms of the Cartan numbers of $G$ (Lemma
4.4). After determining the image of Demazure classes in the $K$--ring of a
Bott-Samelson cycle (Lemma 4.5), the theorems are established in Section 5.
\section{Preliminaries in topological $K$-theory}
All homologies (resp. cohomologies) will have integer coefficients unless
otherwise stated. If $f:M\rightarrow N$ is a continuous map between two
topological spaces $M$ and $N$, $f_{\ast }$ (resp. $f^{\ast }$) is the
homology (resp. cohomology) map induced by $f$, and $f^{!}:K(N)\rightarrow
K(M)$ is the induced map on the Grothendieck groups of topological complex
bundles. The involution on $K(M)$ by the complex conjugation is denoted by $%
\xi \rightarrow \overline{\xi }$, $\xi \in K(M)$.
Write $S^{2}$ for the $2$--dimensional sphere. If $M$ is an oriented closed
manifold $[M]\in H_{\dim M}(M)$ stands for the orientation class. The
Kronecker pairing, between cohomology and homology of a space $M$, will be
denoted by $<,>:H^{\ast }(M)\times H_{\ast }(M)\rightarrow \mathbb{Z}$.
Let $L_{\mathbb{C}}(M)$ be the set of isomorphism classes of complex line
bundles (i.e. of oriented real $2$--plane bundles) over $M$. The trivial
complex line bundle on $M$ is denoted by $1$. It is well known that
\begin{enumerate}
\item[(3.1)] sending a complex line bundle $\xi $ to its first Chern class $%
c_{1}(\xi )$ yields a one-to-one correspondence $c_{1}:L_{\mathbb{C}%
}(M)\rightarrow H^{2}(M)$ which is natural with respect to maps $%
M\rightarrow N$.
\end{enumerate}
\noindent Recall also from [AH] that
\begin{enumerate}
\item[(3.2)] if $M$ is a cell complex with even dimensional cells only, the
\textsl{Chern character} $Ch:K(M)\rightarrow H^{\ast }(M;\mathbb{Q})$ is
injective.
\end{enumerate}
\textbf{3.1. }$S^{2}$\textbf{--bundle with a section. }Let $p:E\rightarrow M$
be a smooth, oriented $S^{2}$--bundle over an oriented manifold $M$ with a
section $s:M\rightarrow E$. As the normal bundle $\eta $ of the embedding $s$
is oriented by $p$ and has real dimension $2$, we may regard $\eta \in L_{%
\mathbb{C}}(M)$. We put $\xi =p^{!}(\eta )\in L_{\mathbb{C}}(E)$, $%
c=c_{1}(\xi )\in H^{2}(E)$.
The integral cohomology $H^{\ast }(E)$ can be described as follows. Denote
by $i:S^{2}\rightarrow E$ the fiber inclusion over a point $z\in M$, and
write $J:$ $E\rightarrow E$ for the involution given by the antipodal map in
each fiber sphere.
\begin{quote}
\textbf{Lemma 3.1 }(cf. [Du$_{2}$, Lemma 3.1])\textbf{. }\textsl{There
exists a unique class }$x\in H^{2}(E)$\textsl{\ such that }$s^{\ast
}(x)=0\in H^{\ast }(M)$\textsl{\ and }$<i^{\ast }(x),[S^{2}]>=1$\textsl{.
Furthermore }
\textsl{(1) }$H^{\ast }(E)$\textsl{, as a module over }$H^{\ast }(M)$\textsl{%
, has the basis }$\{1,x\}$\textsl{\ subject to the relation }$x^{2}+cx=0$%
\textsl{;}
\textsl{(2) the }$J^{\ast }$\textsl{\ acts} \textsl{identically on }$H^{\ast
}(M)\subset H^{\ast }(E)$ \textsl{and }$J^{\ast }(x)=-x-c$\textsl{.}
\end{quote}
Let $X\in L_{\mathbb{C}}(E)$ be with $c_{1}(X)=x\in H^{2}(E)$, where $x$ as
that in Lemma 3.1. Put $y=X-1\in K(E)$. The next result is seen as the $K$%
--theoretic analogous of Lemma 3.1, in which (1) is classical (cf. [A,
Proposition 2.5.3)).
\begin{quote}
\textbf{Lemma 3.2.} \textsl{Assume that }$M$\textsl{\ has a cell
decomposition with even dimensional cells only. Then}
\textsl{(1) }$K(E)$, \textsl{as a module over }$K(M)$\textsl{, has the basis
}$\{1,y\}$\textsl{\ subject to the relation }$y^{2}=(\overline{\xi }-1)y$%
\textsl{. Furthermore}
\textsl{(2) }$\overline{y}=-\xi y$\textsl{;}
\textsl{(3) the }$J^{!}:K(E)\rightarrow K(E)$\textsl{\ acts identically\ on }%
$K(M)$\textsl{\ and satisfies }$J^{!}(y)=-y+(\overline{\xi }-1)$\textsl{.}
\end{quote}
\textbf{Proof.} Since $X$ restricts to the Hopf-line bundle on the fiber
sphere, $K(E)=K(M)[1,y]$. To verify the relation in (1) we compute
\begin{quote}
$Ch(y)=e^{x}-1=x+\frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}+\cdots $
$\qquad \quad =-\frac{x}{c}(e^{-c}-1)$ (for $x^{n}=(-c)^{n-1}x$ by (1) of
Lemma 3.1).
\end{quote}
\noindent It follows that
\begin{quote}
$Ch(y^{2})=[\frac{x}{c}(e^{-c}-1)]^{2}=-\frac{x}{c}(e^{-c}-1)^{2}$ ($%
x^{2}=-cx$ by Lemma 3.1)
$\qquad \quad =Ch[(\overline{\xi }-1)y]$.
\end{quote}
\noindent This implies that $y^{2}=(\overline{\xi }-1)y$ by (3.2). This
completes (1).
In view of (1) we may assume that $\overline{y}(=\overline{X}-1)=a+by$, $%
a,b\in K(M)$. Multiplying both sides by $X=y+1$ yields
\begin{quote}
$\qquad -y=(a+by)(y+1)=a+(a+b\overline{\xi })y$ (by (1)).
\end{quote}
\noindent Coefficients comparison gives $a=0$; $a+b\overline{\xi }=-1$. This
shows (2).
Finally we show (3). From $J^{\ast }(x)=-x-c$ (by Lemma 3.1) and $%
c_{1}J^{!}=J^{\ast }c_{1}$ (by the naturality of (3.1)) we get $J^{!}(X)=%
\overline{X}\cdot \overline{\xi }$. From this one obtains
\begin{quote}
$J^{!}(y)=\overline{X}\cdot \overline{\xi }-1=(\overline{y}+1)\overline{\xi }%
-1=(-\xi y+1)\overline{\xi }-1$ (by (2))
$\qquad =-y+(\overline{\xi }-1)$. $\square $
\bigskip
\end{quote}
\textbf{3.2. Divided difference in }$K$\textbf{--theory. }A self-map $r$ of
a manifold $M$ is called \textsl{an involution} if $r^{2}=id:M\rightarrow M$%
. A $2$\textsl{--spherical representation} of the involution $(M;r)$ is a
system $f:(E;J)\rightarrow (M;r)$ in which $p:E\rightarrow M$ is an oriented
$S^{2}$--bundle with a section $s$; and $f$ is a continuous map $%
E\rightarrow M$ that satisfies the following two constrains
\begin{enumerate}
\item[(3.3)] \noindent $\qquad f\circ s=id:M\rightarrow M$;$\qquad f\circ
J=r\circ f:$ $E\rightarrow M$,
\end{enumerate}
\noindent where $J$ is the involution on $E$ given by the antipodal map on
the fibers.
In view of the $K(M)$--module structure on $K(E)$ (cf. (1) of Lemma 3.2), a $%
2$--spherical representation (3.3) of the involution $(M,r)$ gives rise to
an additive operator $\Lambda _{f}:K(M)\rightarrow K(M)$ characterized
uniquely by (3.4) below, where the constraint $f\circ s=id$ implies that the
coefficient of $1$ is $p^{!}(z)$.
\textsl{The induced homomorphism }$f^{!}:K(M)\rightarrow K(E)$\textsl{\
satisfies}
\begin{enumerate}
\item[(3.4)] $\qquad \qquad \qquad f^{!}(z)=p^{!}(z)\cdot 1+p^{!}(\Lambda
_{f}(z))\cdot y$
\end{enumerate}
\noindent \textsl{for all }$z\in K(M)$\textsl{.}
The operator $\Lambda _{f}$ will be called \textsl{the divided difference}
associated to the $2$--spherical representation $f$ of the involution $(M;r)$%
.
\begin{quote}
\textbf{Lemma 3.3.} \textsl{Let }$\eta \in L_{\mathbb{C}}(M)$\textsl{\ the
normal bundle of the section }$s$\textsl{.} \textsl{Then}
\textsl{(1) }$r^{!}=Id+(\overline{\eta }-1)\Lambda _{f}$\textsl{\ }$:$%
\textsl{\ }$K(M)\rightarrow K(M)$\textsl{;}
\textsl{(2) }$\Lambda _{f}\circ r^{!}=-\Lambda _{f}$\textsl{.}
\end{quote}
\textbf{Proof.} Applying $J^{!}$ to (3.4) and using (3) of Lemma 3.2 to
write the resulting equality yield (note that $\xi =p^{!}(\eta )$ in Lemma
3.2)
\begin{center}
$\qquad J^{!}f^{!}(z)=p^{!}[z+(\overline{\eta }-1)\Lambda
_{f}(z)]-p^{!}[\Lambda _{f}(z)]y$.
\end{center}
\noindent On the other hand one gets from (3.4) that
\begin{center}
$\qquad f^{!}(r^{!}(z))=p^{!}(r^{!}(z))+p^{!}[\Lambda _{f}(r^{!}(z))]y$.
\end{center}
\noindent Since $J^{!}f^{!}=f^{!}r^{!}$ by (3.3), coefficients comparison
shows (1) and (2). $\square $
\bigskip
\textbf{3.3. The }$K$\textbf{-theory of a twisted product of }$S^{2}$. We
determine the $K$-rings for a class of manifolds specified below.
\begin{quote}
\textbf{Definition 5.} A smooth manifold $M$\ is called an oriented \textsl{%
twisted product of }$2$\textsl{--spheres\ of rank} $m$, written $M=\underset{%
1\leq i\leq m}{\propto }S^{2}$, if there is a tower of smooth maps
\end{quote}
\begin{center}
$M=M_{m}\overset{p_{m}}{\rightarrow }M_{m-1}\overset{p_{m-1}}{\rightarrow }%
\cdots \overset{p_{3}}{\rightarrow }M_{2}\overset{p_{2}}{\rightarrow }M_{1}%
\overset{p_{1}}{\rightarrow }M_{0}=\{z_{0}\}$
\end{center}
\begin{quote}
\noindent in which
1) $M_{0}$\ consists of a single point (as indicated);
2) each $p_{k}$\ is an oriented $S^{2}$--bundle with a fixed section $%
s_{k}:M_{k-1}\rightarrow M_{k}$.\
\end{quote}
Let $M=\underset{1\leq i\leq m}{\propto }S^{2}$ be a twisted product of $2$%
-spheres. Assign each $M_{k}$ with the base point $z_{k}=s_{k}\circ \cdots
\circ s_{1}(z_{0})\in M_{k}$ and denote by $h_{k}:$ $S^{2}\rightarrow M_{k}$
the inclusion of the fiber sphere of $p_{k}$ over the point $z_{k}$, $1\leq
k\leq m$. Consider the embedding $\iota _{k}:S^{2}\rightarrow M$ given by
the composition
\begin{center}
$s_{m}\circ \cdots \circ s_{k+1}\circ h_{k}:S^{2}\rightarrow
M_{k}\rightarrow M$.
\end{center}
\noindent Then the set $\{\iota _{1\ast }[S^{2}],\cdots ,\iota _{m\ast
}[S^{2}]\in H_{2}(M)\}$ of $2$-cycles forms a basis of $H_{2}(M)$ [Du$_{2}$,
Lemma 3.3]. Consequently, if we let $x_{i}\in H^{2}(M)$, $1\leq i\leq m$, be
the classes Kronecker dual to $\iota _{k\ast }[S^{2}]$ as $<x_{i},\iota
_{k\ast }[S^{2}]>=\delta _{ik}$, then
\begin{enumerate}
\item[(3.5)] the set $\{x_{1},\cdots ,x_{m}\}$ is a basis of $H^{2}(M)$ that
satisfies
$\qquad \qquad (s_{m}\circ \cdots \circ s_{k})^{\ast }(x_{k})=0$, $1\leq
k\leq m$.
\end{enumerate}
A set of numerical invariants for $M$ can now be extracted as follows. Let $%
\eta _{k}\in L_{\mathbb{C}}(M_{k-1})$ be the normal bundle of the embedding $%
s_{k}:M_{k-1}\rightarrow M_{k}$ with the induced orientation. Put $\xi
_{k}=(p_{k}\circ \cdots \circ p_{m})^{!}\eta _{k}\in L_{\mathbb{C}}(M)$. In
view of (3.5) we must have the expression in $H^{2}(M)$
$\qquad c_{1}(\xi _{1})=0$;
$\qquad c_{1}(\xi _{2})=a_{1,2}x_{1}$;
$\qquad c_{1}(\xi _{3})=a_{1,3}x_{1}+a_{2,3}x_{2};$
$\qquad \vdots $
$\qquad c_{1}(\xi _{m})=a_{1,m}x_{1}+\cdots +a_{m-1,m}x_{m-1}$
\noindent with $a_{i,j}\in \mathbb{Z}$.
\begin{quote}
\textbf{Definition 6.} With $a_{i,j}=0$\ for all $i\geq j$\ being
understood, the strictly upper triangular matrix $A=(a_{i,j})_{m\times m}$\
is called \textsl{the structure matrix of }$M=\underset{1\leq i\leq m}{%
\propto }S^{2}$\ relative to the basis $\{x_{1},\cdots ,x_{m}\}$ of $%
H^{2}(M) $.
\end{quote}
The ring $K(M)$ is determined by $A$ as follows. Let $X_{i}\in L_{\mathbb{C}%
}(M)$ be defined by $c_{1}(X_{i})=x_{i}$ (cf. (3.1), (3.5)) and set $%
y_{i}=X_{i}-1\in K(M)$. For a subset $I=[i_{1},\cdots ,i_{k}]\subseteq
\lbrack 1,\cdots ,m]$ we put
\begin{center}
$y_{I}=\{%
\begin{array}{c}
1\text{ if }k=0\text{ (i.e. }I=\{\emptyset \}\text{)}, \\
y_{I}=y_{i_{1}}\cdots y_{i_{k}}\text{ if }k\geq 1\text{.\ }%
\end{array}%
$
\end{center}
\noindent Let $q_{k},\overline{q_{k}}\in \mathbb{Z}[y_{1},\cdots ,y_{m}]$ be
defined in terms of $A$ as in Definition 3.
\begin{quote}
\textbf{Lemma 3.4. }\textsl{If }$M$\textsl{\ has the structure matrix }$%
A=(a_{i,j})_{m\times m}$\textsl{, then}
\textsl{(1) the set }$\{y_{I}\mid I\subseteq \lbrack 1,\cdots ,m]\}$\textsl{%
\ is a basis for }$K(M)$\textsl{;}
\textsl{(2) }$K(M)=\mathbb{Z}[y_{1},\cdots ,y_{m}]/<y_{k}^{2}=(\overline{%
q_{k}}-1)y_{k}$\textsl{, }$1\leq k\leq m>$\textsl{.}
\end{quote}
\textbf{Proof.} If $m=1$ then $M=S^{2}$ and it is clearly done. Assume next
that Lemma 3.4 holds for $m=n-1$, and consider now the case $m=n$.
Applying Lemma 3.2 to the $S^{2}$--bundle $M=M_{m}\overset{p_{m}}{%
\rightarrow }M_{m-1}$ we get
\begin{center}
$K(M)=K(M_{m-1})[1,y_{m}]$ ($y_{m}^{2}=(\overline{\xi _{m}}-1)y_{m}$).
\end{center}
\noindent This already shows (1).
For (2) it remains only to show $\overline{\xi _{m}}=\overline{q_{m}}\in
\mathbb{Z}[y_{1},\cdots ,y_{m-1}]$. In view of (1) of Lemma 3.2 and by the
inductive hypothesis, we can assume that
\begin{enumerate}
\item[(3.6)] $\qquad \xi _{k}=q_{k}$,\quad\ $\overline{\xi _{k}}=\overline{%
q_{k}}$ for all $k\leq m-1$.
\end{enumerate}
\noindent From $c_{1}(\xi _{m})=a_{1,m}x_{1}+\cdots +a_{m-1,m}x_{m-1}$ we
find that
\begin{quote}
$\overline{\xi _{m}}=\prod_{a_{i,m}>0}\overline{X_{i}}^{a_{i,m}}%
\prod_{a_{i,m}<0}X_{i}^{-a_{i,m}}$
$\qquad =\prod_{a_{i,m}>0}(\overline{y_{i}}+1)^{a_{i,m}}%
\prod_{a_{i,m}<0}(y_{i}+1)^{-a_{i,m}}$ (since $y_{i}=X_{i}-1$)
$\qquad =\prod_{a_{i,m}>0}(-\xi
_{i}y_{i}+1)^{a_{i,m}}\prod_{a_{i,m}<0}(y_{i}+1)^{-a_{i,m}}$ (by Lemma 3.2)
$\qquad
=\prod_{a_{i,m}>0}(-q_{i}y_{i}+1)^{a_{i,m}}%
\prod_{a_{i,m}<0}(y_{i}+1)^{-a_{i,m}}$ (by (3.6))
$\qquad =\overline{q_{m}}$ (cf. Definition 3).
\end{quote}
\noindent Similarly, we have $\xi
_{m}=\prod_{a_{i,m}>0}X^{a_{i,m}}\prod_{a_{i,m}<0}\overline{X_{i}}%
^{-a_{i,m}}=q_{m}$.$\square $
\bigskip
According to (1) of Lemma 3.4, any polynomial $f\in \mathbb{Z}[y_{1},\cdots
,y_{m}]$ can be expanded (uniquely) as a linear combination of the $y_{I}$
\begin{center}
$f=\sum a_{I}(f)y_{I}$, $I\subseteq \lbrack 1,\cdots ,m]$,
\end{center}
\noindent where the correspondences $a_{I}:\mathbb{Z}[y_{1},\cdots
,y_{m}]\rightarrow \mathbb{Z}$ by $f\rightarrow a_{I}(f)$ are clearly
additive. Indeed, problems concerning computing in the $K(M)$ ask an
effective algorithm to evaluate the $a_{I}$. The case $I=[1,\cdots ,m]$ will
be relevant to us and whose solution brings us the operator $\Delta _{A}$
given by Definition 4.
\begin{quote}
\textbf{Lemma 3.5. }\textsl{If }$M$\textsl{\ has the structure matrix }$%
A=(a_{i,j})_{m\times m}$\textsl{, then}
\end{quote}
\begin{center}
$a_{[1,\cdots ,m]}=\Delta _{A}\circ \quad _{(m)}:\mathbb{Z}[y_{1},\cdots
,y_{m}]\overset{(m)}{\rightarrow }\mathbb{Z}[y_{1},\cdots
,y_{m}]_{(m)}\rightarrow \mathbb{Z}$\textsl{. }
\end{center}
\begin{quote}
\noindent \textsl{In particular, }$a_{[1,\cdots ,m]}(f)=0$\textsl{\ if }$%
f_{(m)}=0$\textsl{.}
\end{quote}
\textbf{Proof.} This is parallel to the proof of Proposition 2 in [Du$_{1}$].%
$\square $
\section{Bott-Samelson cycles and Demazure classes}
With respect to the fixed regular point $\alpha \in L(T)$ the adjoint
representation $Ad:G\rightarrow L(G)$ gives rise to a smooth embedding
\begin{center}
$\varphi :G/T\rightarrow L(G)\quad \quad $by $\varphi (gT)=Ad_{g}(\alpha )$.
\end{center}
\noindent In this way $G/T$ becomes a submanifold of the Euclidean space $%
L(G)$. By resorting to the geometry of this embedding we recover the
Demazure operators on $K(G/T)$ in \textbf{4.3}, and\textbf{\ }the classical
Bott-Samelson cycle $\varphi _{0,\beta _{1},\cdots ,\beta _{k}}:S(\alpha
;\beta _{1},\cdots ,\beta _{k})\rightarrow G/T$ (associated to a sequence of
simple roots $\beta _{1},\cdots ,\beta _{k}$) on $G/T$ in \textbf{4.4}. As
application of Lemma 3.4, the ring $K(S(\alpha ;\beta _{1},\cdots ,\beta
_{k}))$ is described in terms of the Cartan numbers of $G$ (cf. Lemma 4.4).
The main result in this section is Lemma 4.5, which specifies the $\varphi
_{0,\beta _{1},\cdots ,\beta _{k}}^{!}$--image of a Demazure class in $%
K(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$.\ \
\textbf{4.1.} \textbf{Geometry from the adjoint representation.} Let $\Phi
^{+}\subset L(T)$ (resp. $\Delta \subset L(T)$) be the set of positive
(resp. simple) roots relative to $\alpha $ ([Hu, p.35]). Assume that the
Cartan decomposition of the Lie algebra $L(G)$ relative to $T\subset G$ is
\begin{center}
$L(G)=L(T)\oplus _{\beta \in \Phi ^{+}}F_{\beta }$,
\end{center}
\noindent where $F_{\beta }$ is the root space, viewed as a real $2$-plane,
belonging to the root $\beta \in \Phi ^{+}$ ([Hu, p.35]). It is known (cf.
[HPT,p.426-427; or Du$_{2}$, Sect.4]) that
\begin{enumerate}
\item[(4.1)] The subspaces $\oplus _{\beta \in \Phi ^{+}}F_{\beta }$ and $%
L(T)$ of $L(G)$ are tangent and normal to $G/T$ at $\alpha $ respectively;
\item[(4.2)] The tangent bundle to $G/T$ has a canonical orthogonal
decomposition into the sum of integrable $2$-plane bundles $\oplus _{\beta
\in \Phi ^{+}}E_{\beta }$ with $E_{\beta }(\alpha )=F_{\beta }$.
\item[(4.3)] The leaf of the integrable subbundle $E_{\beta }$ through a
point $z\in G/T$, denoted by $S(z;\beta )$, is a $2$-sphere that carries a
preferred orientation.
\item[(4.4)] Via the embedding $\varphi $, the canonical action of $W$ on $%
G/T$ can be given in terms of the $W$ action on $L(T)$ as $%
w(z)=Ad_{g}(w(\alpha ))$ if $z=Ad_{g}(\alpha )\in G/T$, $w\in W$ (cf. [BS]).
\end{enumerate}
\textbf{4.2. Demazure basis of }$K(G/T)$. For a complete subvariety $%
Y\subset G/T$ the \textsl{Euler--Poincar\'{e} characteristic} relative to $Y$
is the homomorphism $\chi (Y,-):K(G/T)\rightarrow \mathbb{Z}$ defined by
\begin{center}
$[\digamma ]\rightarrow \chi (Y,\digamma
)=\sum\limits_{j}(-1)^{j}h^{j}(\digamma \mid Y)$,
\end{center}
\noindent where $\digamma \mid Y$ means the restriction of $\digamma $ on $Y$%
, and where $h^{j}(\digamma \mid Y)$ denotes the dimension of the $j^{th}$
cohomology group of $h^{j}(\digamma \mid Y)$. The following characterization
of Demazure basis is due to B. Kostant and S. Kumar (compare [KK, (3.39)
Proposition] with [D, Proposition 7]).
\begin{quote}
\textbf{Definition 7. }The \textsl{Demazure basis} $\{a_{w}\in K(G/T)\mid
w\in W\}$ of the ring $K(G/T)$ is defined by
\end{quote}
\begin{center}
$\chi (X_{w},a_{u})=\delta _{w,u}$, $w,u\in W$,
\end{center}
\begin{quote}
where $X_{w}$ is the Schubert class on $G/T$ associated to $w$.
\end{quote}
\bigskip
\textbf{4.3. Divided difference on }$K(G/T)$\textbf{\ associated to a root. }%
Each root\textbf{\ }$\beta \in \Phi ^{+}$ gives rise to an involution $%
r_{\beta }:G/T\rightarrow G/T$ in the fashion of (4.4), and defines also the
subspace
\begin{center}
$S(\beta )=\{(z,z_{1})\in G/T\times G/T\mid z_{1}\in S(z;\beta )\}$
\end{center}
\noindent in view of (4.3). Projection $p_{\beta }:S(\beta )\rightarrow G/T$
onto the first factor is easily seen to be a $S^{2}$--bundle (with $%
S(z;\beta )$ as the fiber over $z\in G/T$). The map $s_{\beta
}:G/T\rightarrow S(\beta )$ by $s_{\beta }(z)=(z,z)$ furnishes $p_{\beta }$
with a ready-made section.
Let $x\in H^{2}(S(\beta ))$ be such that $s_{\beta }^{\ast }(x)=0$ and $%
<i^{\ast }(x),[S(z;\beta )]>=1$\ (cf. Lemma 3.1) and set $y=X-1\in K(S(\beta
))$, where $X\in L_{\mathbb{C}}(S(\beta ))$ is defined by $c_{1}(X)=x$ (cf.
(3.1)). Since the normal bundle of the embedding $s_{\beta }$ is easily seen
to be $E_{\beta }\in L_{\mathbb{C}}(G/T)$, one infers from Lemma 3.2 that
\begin{quote}
\textbf{Lemma 4.1.}\textsl{\ }$K(S(\beta ))=K(G/T)[1,y]/<y^{2}=(\overline{%
E_{\beta }}-1)y>$.
\end{quote}
Let $J_{\beta }$ be the involution on $S(\beta )$ given by the antipodal
maps in the fiber spheres, and let $f_{\beta }:S(\beta )\rightarrow G/T$ be
the projection onto the second factor. Then, as is clear,
\begin{center}
$f_{\beta }\circ s_{\beta }=id:G/T\rightarrow G/T$;$\quad f_{\beta }\circ
J_{\beta }=r_{\beta }\circ f_{\beta }:S(\beta )\rightarrow G/T$.
\end{center}
\noindent That is, \textsl{the map }$f_{\beta }:(S(\beta ),J_{\beta
})\rightarrow (G/T,r_{\beta })$\textsl{\ is a }$2$\textsl{-spherical
representation of the involution }$(G/T,r_{\beta })$\textsl{\ }(cf. \textbf{%
3.2}).
Abbreviate the divided difference $\Lambda _{f_{\beta }}:K(G/T)\rightarrow
K(G/T)$ associated to $f_{\beta }$ by $\Lambda _{\beta }$. The next result,
essentially due to Kostant and Kumar [KK], is the key in the proof of Lemma
4.5.
\begin{quote}
\textbf{Lemma 4.2.} Let $\{a_{w}\in K(G/T)\mid w\in W\}$ be the \textsl{%
Demazure basis} of $K(G/T)$, and let $\beta \in \Delta $ be a simple root.
Then
\end{quote}
\begin{center}
$\Lambda _{\beta }(\overline{a_{w}})=\{%
\begin{array}{c}
E_{\beta }\cdot \overline{a_{w\cdot r_{\beta }}}\text{\quad if }%
l(w)>l(wr_{\beta })\text{;} \\
-E_{\beta }\cdot \overline{a_{w}}\text{\quad if }l(w)<l(wr_{\beta })\text{.}%
\end{array}%
$
\end{center}
\textbf{Proof.} Recall from [KK,PR$_{1}$,D] that the classical \textsl{%
Demazure operator} $T_{\beta }:K(G/T)\rightarrow K(G/T)$ is given by
\begin{center}
$T_{\beta }(u)=$ $\frac{E_{\beta }\cdot u-r_{\beta }^{!}(u)}{E_{\beta }-1}$.
\end{center}
\noindent Substituting in $r_{\beta }^{!}=Id+(\overline{E_{\beta }}%
-1)\Lambda _{\beta }$ (lemma 3.3) yields
\begin{center}
$T_{\beta }(u)=u+\overline{E_{\beta }}\Lambda _{\beta }(u)$.
\end{center}
\noindent That is $\Lambda _{\beta }=E_{\beta }(T_{\beta
}-Id):K(G/T)\rightarrow K(G/T)$.
On the other hand combining [KK, Proposition (2.22),(d)] with [KK,
Proposition (3.39)] one gets
\begin{center}
$T_{\beta }(\overline{a_{w}})=\{%
\begin{array}{c}
\overline{a_{w}}+\overline{a_{wr_{\beta }}}\text{\quad if }l(w)>l(wr_{\beta
})\text{;} \\
0\text{, otherwise.\ \qquad \quad \qquad \qquad }%
\end{array}%
$
\end{center}
\noindent This completes the proof.$\square $
\bigskip
\textbf{4.4.} \textbf{Bott-Samelson cycles and their }$K$\textbf{-rings. }%
Given an ordered sequence $(\beta _{1},\cdots ,\beta _{k})$ of simple roots
(repetitions like $\beta _{i}=\beta _{j}$ for some $1\leq i<j\leq k$ may
occur), we set
\begin{center}
$S(\alpha ;\beta _{1},\cdots ,\beta _{k})=\{(z_{0},z_{1},\cdots ,z_{k})\in
G/T\times \cdots \times G/T\mid z_{0}=\alpha $; $z_{i}\in S(z_{i-1};\beta
_{i})\}$.
\end{center}
\noindent It is furnished with the structure of oriented twisted product of $%
2$-spheres of rank $k$ by the maps
\begin{center}
$S(\alpha ;\beta _{1},\cdots ,\beta _{i})\overset{p_{i}}{\underset{s_{i}}{%
\rightleftarrows }}S(\alpha ;\beta _{1},\cdots ,\beta _{i-1})$,
$p_{i}(z_{0},\cdots ,z_{i})=(z_{0},\cdots ,z_{i-1})$; $s_{i}(z_{0},\cdots
,z_{i-1})=(z_{0},\cdots ,z_{i-1},z_{i-1})$.
\end{center}
\noindent One has also the ready-made maps
\begin{center}
$\varphi _{0,\beta _{1},\cdots ,\beta _{k}}:S(\alpha ;\beta _{1},\cdots
,\beta _{k})\rightarrow G/T$ by $(z_{0},\cdots ,z_{k})\rightarrow z_{k}$
$\widehat{\varphi }_{0,\beta _{1},\cdots ,\beta _{k}}:S(\alpha ;\beta
_{1},\cdots ,\beta _{k})\rightarrow S(\beta _{k})$ by $(z_{0},\cdots
,z_{k})\rightarrow (z_{k-1},z_{k})$
\end{center}
\noindent that clearly satisfy
\begin{enumerate}
\item[(4.5)] \noindent\ $\varphi _{0,\beta _{1},\cdots ,\beta _{k}}=f_{\beta
_{k}}\circ \widehat{\varphi }_{0,\beta _{1},\cdots ,\beta _{k-1}}:S(\alpha
;\beta _{1},\cdots ,\beta _{k})\rightarrow G/T$;
\item[(4.6)] the commutative diagrams
\end{enumerate}
\begin{center}
$%
\begin{array}{ccccc}
S(\alpha ;\beta _{1},\cdots ,\beta _{k}) & \overset{\widehat{\varphi }%
_{0,\beta _{1},\cdots ,\beta _{k-1}}}{\rightarrow } & S(\beta _{k}) & & \\
p_{k-1}\downarrow \uparrow s_{k-1} & & p_{\beta _{k}}\downarrow \uparrow
s_{\beta _{k}} & \overset{f_{\beta _{k}}}{\searrow } & \\
S(\alpha ;\beta _{1},\cdots ,\beta _{k-1}) & \overset{\varphi _{0,\beta
_{1},\cdots ,\beta _{k-1}}}{\rightarrow } & G/T & \overset{f_{\beta
_{k}}\circ s_{\beta _{k}}=id}{\rightarrow } & G/T%
\end{array}%
$
\end{center}
\noindent in which $\widehat{\varphi }_{0,\beta _{1},\cdots ,\beta _{k-1}}$
is a bundle map over $\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}$.
\begin{quote}
\textbf{Definition 8 }([Du$_{2}$, \textbf{7.2]}). \textsl{The map (4.5) is
called the Bott-Samelson cycle associated to the sequence }$\beta
_{1},\cdots ,\beta _{k}$\textsl{\ of simple roots.}
\end{quote}
Let $\iota _{i}:S(\alpha ,\beta _{i})\rightarrow S(\alpha ;\beta _{1},\cdots
,\beta _{k})$ be the embedding specified by
\begin{center}
$\iota _{i}(\alpha ,z^{\prime })=(z_{0},\cdots ,z_{k})$,
\end{center}
\noindent where $z_{0}=\cdots =z_{i-1}=\alpha $, $z_{i}=\cdots
=z_{k}=z^{\prime }$. Then the cycles $\iota _{i\ast }[S(\alpha ,\beta
_{i})]\in H_{2}(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$, $1\leq i\leq k$,
form a basis of $H_{2}(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$ (cf.
\textbf{3.3}). Let $x_{i}\in H^{2}(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$
be the basis Kronecker dual to the $\iota _{j\ast }[S(\alpha ,\beta _{j})]$
as $<x_{i},\iota _{j\ast }[S(\alpha ,\beta _{j})]>=\delta _{ij}$. The next
result was shown in [Du$_{2}$, Lemma 4.5].
\begin{quote}
\textbf{Lemma 4.3.} \textsl{The structure matrix }$A=(a_{i,j})_{k\times k}$
\textsl{of }$S(\alpha ;\beta _{1},\cdots ,\beta _{k})$ \textsl{relative to} $%
\{x_{1},\cdots ,x_{k}\}$ \textsl{is given by the Cartan numbers of }$G$%
\textsl{\ as}
\end{quote}
\begin{center}
$a_{i,j}=\{%
\begin{array}{c}
\beta _{j}\circ \beta _{i}\ \quad \text{\textsl{if} }i<j\text{;} \\
0\ \quad \text{\textsl{if} }i\geq j\text{\textsl{.}\quad \quad }%
\end{array}%
$
\end{center}
Let $X_{i}\in L_{\mathbb{C}}(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$ be
defined by $c_{1}(X_{i})=x_{i}$. Set $y_{i}=X_{i}-1\in K(S(\alpha ;\beta
_{1},\cdots ,\beta _{k}))$. Let $\overline{q_{k}}\in \mathbb{Z}[y_{1},\cdots
,y_{m}]$ be defined in terms of $A$ as in Definition 3. Combining Lemma 3.4,
Lemma 3.5 with Lemma 4.3 yields the next result.
\begin{quote}
\textbf{Lemma 4.4. }\textsl{Let }$M=S(\alpha ;\beta _{1},\cdots ,\beta _{k})$%
\textsl{. Then}
\textsl{(1) the set }$\{y_{I}\mid I\subseteq \lbrack 1,\cdots ,m]\}$\textsl{%
\ is a basis of }$K(M)$\textsl{;}
\textsl{(2) }$K(M)=\mathbb{Z}[y_{1},\cdots ,y_{k}]/<y_{i}^{2}=(\overline{%
q_{i}}-1)y_{i}$\textsl{, }$1\leq i\leq k>$\textsl{;}
\textsl{(3) if }$f\in \mathbb{Z}[y_{1},\cdots ,y_{k}]$\textsl{\ with }$%
f=\sum a_{I}(f)y_{I}$\textsl{, then }
\end{quote}
\begin{center}
$a_{[1,\cdots ,k]}(f)=\Delta _{A}(f_{(k)})$\textsl{. }
\end{center}
\begin{quote}
\textsl{In particular, }$a_{[1,\cdots ,k]}(f)=0$\textsl{\ if} \textsl{\ }$%
f_{(k)}=0$\textsl{.}
\end{quote}
\textbf{4.5. The induced map of a Bott-Samelson cycle. }Given a sequence $%
\beta _{1},\cdots ,\beta _{k}$ of simple roots consider the induced ring map
\begin{center}
$\varphi _{0,\beta _{1},\cdots ,\beta _{k}}^{!}:K(G/T)\rightarrow K(S(\alpha
;\beta _{1},\cdots ,\beta _{k}))$.
\end{center}
\noindent The ring $K(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$ has the
additive basis $\{y_{I}\mid $\textsl{\ }$I\subseteq $\textsl{\ }$[1,\cdots
,k]\}$ by Lemma 4.4. \textsl{\ }Let $\{a_{w}\mid w\in W\}$\ be the Demazure
basis of $K(G/T)$.
\begin{quote}
\textbf{Lemma 4.5.}\textsl{\ The induced map} $\varphi _{0,\beta _{1},\cdots
,\beta _{k}}^{!}$ \textsl{is given by}
\end{quote}
\begin{center}
$[\varphi _{0,\beta _{1},\cdots ,\beta
_{k}}]^{!}(a_{w})=(-1)^{l(w)}\sum\limits_{I\subseteq \lbrack 1,\cdots
,k],\beta (I)\thicksim w}y_{I}$\textsl{.}
\end{center}
\textbf{Proof.} It suffices to show that
\begin{enumerate}
\item[(4.7)] $\qquad \lbrack \varphi _{0,\beta _{1},\cdots ,\beta _{k}}]^{!}(%
\overline{a_{w}})=(-1)^{l(w)}\sum\limits_{I\subseteq \lbrack 1,\cdots
,k],\beta (I)\thicksim w}\overline{y}_{I}$\textsl{,}
\end{enumerate}
\noindent for, the complex conjugation of (4.7) yields the Lemma. To this
end we compute
\begin{quote}
$\varphi _{0,\beta _{1},\cdots ,\beta _{k}}^{!}(\overline{a_{w}})=\widehat{%
\varphi }_{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(f_{\beta _{k}}^{!}((%
\overline{a_{w}})))$ (by (4.5))
\noindent $\quad =\widehat{\varphi }_{o,\beta _{1},\cdots ,\beta
_{k-1}}^{!}[p_{\beta _{k}}^{!}(\overline{a_{w}})+p_{\beta _{k}}^{!}(\Lambda
_{\beta _{k}}(\overline{a_{w}}))y_{k}]$ (by (3.4))
\noindent $\quad =\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{%
a_{w}})+\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\Lambda _{\beta
_{k}}(\overline{a_{w}}))y_{k}$ (by (4.6))
\noindent $\quad =\{%
\begin{array}{c}
\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})+\varphi
_{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{wr_{\beta _{k}}}}%
)E_{\beta _{k}}\cdot y_{k}\text{\quad if }l(w)>l(wr_{\beta })\text{;} \\
\varphi _{0.\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})-\varphi
_{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})E_{\beta
_{k}}\cdot y_{k}\text{,\quad otherwise,\qquad \quad \quad }%
\end{array}%
$
\end{quote}
\noindent where the last equality is by Lemma 4.2. From $\overline{y_{k}}%
=-E_{\beta _{k}}\cdot y_{k}$ (Lemma 3.2) we obtain
\begin{center}
\noindent $\varphi _{0,\beta _{1},\cdots ,\beta _{k}}^{!}(\overline{a_{w}}%
)=\{%
\begin{array}{c}
\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})-\varphi
_{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{wr_{\beta _{k}}}})%
\overline{y_{k}}\text{ if }l(w)>l(wr_{\beta })\text{;} \\
\varphi _{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})+\varphi
_{0,\beta _{1},\cdots ,\beta _{k-1}}^{!}(\overline{a_{w}})\overline{y_{k}}%
\text{, otherwise.\quad \qquad \quad }%
\end{array}%
$
\end{center}
\noindent This, together with an easy induction on $k$, reduces the proof of
(4.7) to the general properties of the Demazure classes. Let $f:X\rightarrow
G/T$ be a continuous map from a connected $2m$--dimensional $CW$-complex $X$%
. Then
(1) $f^{!}(\overline{a_{u}})=0$ whenever $l(u)>m$;
(2) $f^{!}(\overline{a_{e}})=1$ if $X$ is a single point, where $e\in W$ is
the group unit,
\noindent Indeed, (1) follows from Assertion III in [KK, (3.26)] and [KK,
(3.39)]; (2) can be deduced from (1) and
(i) $\Omega _{e}=1$ (cf. [PR$_{2}$, Corollary 2.5] and the footnote in
\textbf{5.2});
(ii) $\Omega _{e}=\sum_{w\in W}a_{w}$ (cf. Lemma 5.1 below).$\square $
\begin{quote}
\textbf{Corollary 1.} \textsl{Let }$e\in W$\textsl{\ be the group unit. Then}
\textsl{(1)} $[\varphi _{0,\beta _{1},\cdots ,\beta
_{k}}]^{!}(a_{e})=\prod\limits_{1\leq i\leq k}$ $(1+y_{i})$\
\noindent \textsl{and for} $w\neq e$
\textsl{(2)} $[\varphi _{0,\beta _{1},\cdots ,\beta _{k}}]^{!}(a_{w})=\{%
\begin{array}{c}
0\text{\quad \textsl{if} }l(w)>k\text{\textsl{;}\qquad \quad \qquad \qquad
\qquad } \\
(-1)^{k}\delta _{w,r_{\beta _{1}}\cdots r_{\beta _{k}}}y_{[1,\cdots ,m]}%
\text{\quad \textsl{if} }l(w)=k\text{\textsl{.}}%
\end{array}%
$
\end{quote}
\textbf{Example 1. }Let $\Delta =\{\alpha _{1},\alpha _{2}\}$ be a set of
simple roots of $G_{2}$ in which $\alpha _{1}$ is the short root [Hu, p.57].
The Weyl group $W$ of $G_{2}$ is generated by $\sigma _{i}$, $i=1,2$, the
reflection in the hyperplane $L_{i}\subset L(T)$ perpendicular to $\alpha
_{i}$. If we take $u=\sigma _{1}\sigma _{2}\sigma _{1}\sigma _{2}\sigma _{1}$%
, then Lemma 4.5 yields that
\begin{quote}
(a) $[\varphi _{0,(\alpha _{1},\alpha _{2},\alpha _{1},\alpha _{2},\alpha
_{1})}]^{!}(a_{u})=x_{1}x_{2}x_{3}x_{4}x_{5}$;
(b) $[\varphi _{0,(\alpha _{2},\alpha _{1},\alpha _{2},\alpha _{1},\alpha
_{2})}]^{!}(a_{u})=0$;
(c) $[\varphi _{0,(\alpha _{1},\alpha _{2},\alpha _{1},\alpha _{2},\alpha
_{1},\alpha _{2})}]^{!}(a_{u})=x_{1}x_{2}x_{3}x_{4}x_{5}(1+x_{6})$.
\end{quote}
\section{Multiplication in the ring $K(G/H)$}
Lemma 4.5 enables us to establish Theorem 1 (resp. Theorem 2) by computation
in the simpler ring $K(S(\alpha ;\beta _{1},\cdots ,\beta _{k}))$ (cf. Lemma
4.4).
\bigskip
\textbf{5.1. Proof of Theorem 1} (cf. \S 2)\textbf{. }Let $w=r_{\beta
_{1}}\cdot \cdots \cdot r_{\beta _{m}}$, $\beta _{i}\in \Delta $ be a
reduced decomposition of a $w\in W$, and let $A_{w}=(a_{i,j})_{m\times m}$
be the associated Cartan matrix (cf. Definition 1). Consider the
Bott-Samelson cycle $\varphi _{0,\beta _{1},\cdots ,\beta _{m}}:S(\alpha
;\beta _{1},\cdots ,\beta _{m})\rightarrow G/T$ associated to the sequence $%
\beta _{1},\cdots ,\beta _{m}$ of simple roots. Applying the induced ring
map $\varphi _{0,\beta _{1},\cdots ,\beta _{m}}^{!}$ to (1.1) yields in $%
K(S(\alpha ;\beta _{1},\cdots ,\beta _{m}))$ that
\begin{quote}
$\qquad \qquad \varphi _{0,\beta _{1},\cdots ,\beta _{m}}^{!}[a_{u}\cdot
a_{v}]=\sum\limits_{x\in W}C_{u,v}^{x}\varphi _{0,\beta _{1},\cdots ,\beta
_{m}}^{!}[a_{x}]$
$\qquad =(-1)^{l(w)}C_{u,v}^{w}y_{[1,\cdots ,m]}+\sum_{l(x)\leq
l(w)-1}C_{u,v}^{x}\varphi _{0,\beta _{1},\cdots ,\beta _{m}}^{!}[a_{x}]$,
\end{quote}
\noindent where the second equality follows from (2) of Corollary 1. Using
Lemma 4.5 to rewrite this equation yields
\begin{quote}
\bigskip $\qquad (-1)^{l(u)+l(v)}[\sum\limits_{\beta (L)\thicksim
u}y_{L}][\sum\limits_{\beta (K)\thicksim v}y_{K})]$
$=(-1)^{l(w)}C_{u,v}^{w}y_{[1,\cdots ,m]}+\sum_{l(x)\leq
l(w)-1}(-1)^{l(x)}C_{u,v}^{x}[\sum\limits_{\beta (J)\thicksim x}y_{J}]$,
\end{quote}
\noindent where $L,K,J\subseteq \lbrack 1,\cdots ,m]$. Finally, comparing
the coefficients of the monomial $y_{[1,\cdots ,m]}$ on both sides by using
(3) of Lemma 4.4, we obtain
\begin{quote}
\bigskip $\qquad \qquad (-1)^{l(u)+l(v)}\Delta _{A_{w}}[(\sum\limits_{\beta
(L)\thicksim u}y_{L})(\sum\limits_{\beta (K)\thicksim v}y_{K})]_{(m)}$
$\qquad =(-1)^{l(w)}C_{u,v}^{w}+\sum\limits_{\substack{ l(u)+l(v)\leq
l(x)\leq l(w)-1 \\ (\beta _{1},\cdots ,\beta _{m})\thicksim x}}%
(-1)^{l(x)}C_{u,v}^{x}$.
\end{quote}
\noindent This finishes the proof.$\square $
\bigskip
\textbf{Example 2. }Continuing from Example 1 we take
\begin{center}
$u=\sigma _{1}\sigma _{2}\sigma _{1}\sigma _{2}\sigma _{1}$;$\quad v=\sigma
_{2}\sigma _{1}\sigma _{2}\sigma _{1}\sigma _{2}$;$\quad w=\sigma _{1}\sigma
_{2}\sigma _{1}\sigma _{2}\sigma _{1}\sigma _{2}$.
\end{center}
\noindent Then the $u,v\in W$ are the only elements of length $5$, and $w$
is the element of highest length. Applying Theorem 1 we compute the
structure constants appearing in the expansion
\begin{center}
$a_{e}a_{u}=C_{e,u}^{u}a_{u}+C_{e,u}^{v}a_{v}+C_{e,u}^{w}a_{w}$.
\end{center}
\noindent In views of (a) in Example 1 and (1) in Corollary 1 we have
\begin{quote}
$C_{e,u}^{u}=\Delta _{A_{u}}[x_{1}x_{2}x_{3}x_{4}x_{5}\cdot \prod_{1\leq
i\leq 5}(1+x_{i})]_{(5)}$
$\qquad =\Delta _{A_{u}}(x_{1}x_{2}x_{3}x_{4}x_{5})=1$.
\end{quote}
\noindent Similarly, we get
\begin{quote}
$C_{e,u}^{v}=\Delta _{A_{v}}[0\cdot \prod_{1\leq i\leq 5}(1+x_{i})]_{(5)}=0$;
$C_{e,u}^{w}=(-1)^{5}\Delta _{A_{w}}[x_{1}x_{2}x_{3}x_{4}x_{5}(1+x_{6})\cdot
\prod_{1\leq i\leq 5}(1+x_{i})]_{(6)}$
$\qquad \qquad -[(-1)^{5}(C_{e,u}^{u}+C_{e,u}^{v})]$
$\qquad =-\Delta _{A_{w}}[x_{1}x_{2}x_{3}x_{4}x_{5}(x_{1}+\cdots
+x_{5}+2x_{6})]+1$
$\qquad =-2+1=-1$,
\end{quote}
\noindent where the three $\Delta _{A}(f)$'s concerned in the above
computation are directly evaluated from Definition 4 without resorting to
the specialities of $A$, thanks to the simplicities of the polynomials $f$
involved. Summarizing we get, in the ring $K(G_{2}/T)$, that
\begin{center}
$a_{e}a_{u}=a_{u}-a_{w}$.
\end{center}
\bigskip
\textbf{5.2.} The method establishing Theorem 1 is directly applicable to
find a formula for the structure constants $K_{u,v}^{w}(H)$ for multiplying
Grothendieck classes in the $K(G/H)$.
We begin with the simpler case $H=T$ (a maximal torus in $G$). Abbreviate $%
X_{w}(H)$ by $X_{w}$, $\Omega _{w}(H)$ by $\Omega _{w}$\footnote{%
The $\Omega _{w}$ corresponds to $\mathcal{O}_{w_{0}w}$ \ in [Br$_{2}$,KK,PR$%
_{2}$].} and $K_{u,v}^{w}(H)$ by $K_{u,v}^{w}$. The transition between the
two bases $\{a_{w}\mid w\in W\}$ and $\{\Omega _{w}\mid w\in W\}$ of $K(G/T)$
has been determined by Kostant and Kumar in [KK, Proposition 4.13; 3.39].
\begin{quote}
\textbf{Lemma 5.1.} \textsl{In the ring }$K(G/T)$\textsl{\ one has }
\end{quote}
\begin{center}
$\Omega _{w}=\sum\limits_{w\leq u}a_{u}$\textsl{,\qquad }$%
a_{w}=\sum\limits_{w\leq u}(-1)^{l(u)-l(w)}\Omega _{u}$
\end{center}
\begin{quote}
\noindent \textsl{where }$w\leq u$\textsl{\ means }$X_{w}\subseteq X_{u}$%
\textsl{.}
\end{quote}
Combining Lemma 5.1 with Lemma 4.5 gives the next result.
\begin{quote}
\textbf{Lemma 5.2.} \textsl{Let }$\beta _{1},\cdots ,\beta _{k}$ \textsl{be
a sequence of simple roots. With respect to the Grothendieck basis the
induced map} $\varphi _{0,\beta _{1},\cdots ,\beta _{k}}^{!}$\textsl{\ is
given by}
\end{quote}
\begin{center}
$[\varphi _{0,\beta _{1},\cdots ,\beta _{k}}]^{!}(\Omega
_{w})=\sum\limits_{I\subseteq \lbrack 1,\cdots ,k]}b_{I}(w)y_{I}$\textsl{,}
\end{center}
\begin{quote}
\noindent \textsl{where }$b_{I}(w)=\sum\limits_{\beta (I)\sim u,u\geq
w}(-1)^{l(u)}$\textsl{.}
\end{quote}
\bigskip
Based on Lemma 5.2, an argument parallel to the proof of Theorem 1 yields
\begin{quote}
\textbf{Theorem 2.} \textsl{Assume that }$w=r_{\beta _{1}}\cdot \cdots \cdot
r_{\beta _{m}}$\textsl{, }$\beta _{i}\in \Delta $\textsl{, is a reduced
decomposition of a }$w\in W$\textsl{, and let }$A_{w}=(a_{i,j})_{m\times m}$%
\textsl{\ be the associated Cartan matrix.} \textsl{For }$u,v\in W$\textsl{\
we have}
\end{quote}
\begin{center}
$(-1)^{l(w)}K_{u,v}^{w}=\Delta _{A_{w}}[(\sum b_{I}(u)y_{I})(\sum
b_{L}(v)y_{L})]_{(m)}$
$-\sum\limits_{\substack{ l(u)+l(v)\leq l(x) \\ x<w}}b_{[1,\cdots
,m]}(x)K_{u,v}^{x}$\textsl{,}
\end{center}
\begin{quote}
\noindent \textsl{where }$I,L\subseteq \lbrack 1,\cdots ,m]$\textsl{, and
where the numbers }$b_{K}(x)$\textsl{, }$K\subseteq \lbrack 1,\cdots ,m]$%
\textsl{, }$x\in W$\textsl{, are given as that in Lemma 5.2.}
\end{quote}
\bigskip
Proceeding to the general case let $H\subset G$ be the centralizer of a
one--parameter subgroup in $G$. Take a maximal torus $T\subset H$ and
consider the standard fibration $p:G/T\rightarrow G/H$. It is well known
that (cf. [PR$_{2}$, Proposition 1.6])
\begin{quote}
\textsl{The induced ring map }$p^{!}:K(G/H)\rightarrow K(G/T)$\textsl{\ is
injective and satisfies }$p^{!}[\Omega _{w}(H)]=\Omega _{w}$\textsl{, }$w\in
\overline{W}$\textsl{.}
\end{quote}
\noindent Consequently one gets
\begin{quote}
\textbf{Corollary 2.} $K_{u,v}^{w}(H)=K_{u,v}^{w}$\textsl{\ for }$u,v,w\in
\overline{W}$\textsl{.}
\end{quote}
\begin{center}
\textbf{References}
\end{center}
\begin{enumerate}
\item[{[A]}] M. Atiyah, $K$-theory, Benjamin, Inc., New York-Amsterdam, 1967
\item[{[AH]}] M. Atiyah and F. Hirzebruch, Vector bundles and homogeneous
spaces. 1961 Proc. Sympos. Pure Math., Vol. III pp. 7--38 American
Mathematical Society, Providence, R.I.
\item[{[Br$_{1}$]}] M. Brion, Positivity in the Grothendieck group of complex
flag varieties. Special issue in celebration of Claudio Procesi's 60th
birthday. J. Algebra 258 (2002), no. 1, 137--159.
\item[{[Br$_{2}$]}] M. Brion, Lectures on the geometry of flag varieties,
\textsl{preprint available on} arXiv: math.AG/0410240
\item[{[BS]}] R. Bott and H. Samelson, Application of the theory of Morse to
symmetric spaces, Amer. J. Math., Vol. LXXX, no. 4 (1958), 964-1029.
\item[{[Bu]}] A.S. Buch, A Littlewood-Richardson rule for the $K$-theory of
Grassmannians, Acta Math. 189 (2002), no. 1, 37--78.
\item[{[C]}] C. Chevalley, Les classes d'equivalence rationelle, I,II,
Seminaire C. Chevalley, Anneaux de Chow et applications (mimeographed
notes), Paris, 1958.
\item[{[Ch]}] C. Chevalley, Sur les D\'{e}compositions Cellulaires des
Espaces $G/B$, in Algebraic groups and their generalizations: Classical
methods, W. Haboush ed. Proc. Symp. in Pure Math. 56 (part 1) (1994), 1-26.
\item[{[D]}] M. Demazure, D\'{e}singularisation des vari\'{e}t\'{e}s de
Schubert g\'{e}n\'{e}ralis\'{e}es, Ann. Sci. \'{E}cole. Norm. Sup. (4)
7(1974), 53-88.
\item[{[Du$_{1}$]}] H. Duan, The degree of a Schubert variety, Adv. Math.,
180(2003), 112-133.
\item[{[Du$_{2}$]}] H. Duan, Multiplicative rule of Schubert classes, Invent.
Math.159(2005), 407-436.
\item[{[DZ$_{1}$]}] H. Duan and Xuezhi Zhao, A unified formula for Steenrod
operations in flag manifolds, \textsl{preprint available on} arXiv:
math.AT/0306250
\item[{[DZ$_{2}$]}] H. Duan and Xuezhi Zhao, Algorithm for multiplying
Schubert classes, \textsl{preprint available on} arXiv: math.AG/0309158.
\item[{[FL]}] W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck
ring of a flag bundle, Duke Math. J. 76 (1994) 711--729.
\item[{[GR]}] S. Griffeth, and A. Ram, Affine Hecke algebras and the Schubert
calculus, European J. Combin. 25 (2004), no. 8, 1263--1283.
\item[{[HPT]}] W. Y. Hsiang, R. Palais and C. L. Terng, The topology of
isoparametric submanifolds, J. Diff. Geom., Vol. 27 (1988), 423-460.
\item[{[Hu]}] J. E. Humphreys, Introduction to Lie algebras and
representation theory, Graduated Texts in Math. 9, Springer-Verlag New York,
1972.
\item[{[KK]}] B. Kostant and S. Kumar, $T$-equivariant $K$-theory of
generalized flag varieties. J. Differential Geom. 32 (1990), no. 2, 549--603.
\item[{[L]}] C. Lenart, A $K$-theory version of Monk's formula and some
related multiplication formulas. J. Pure Appl. Algebra 179 (2003), no. 1-2,
137--158.
\item[{[LS]}] P. Littelmann and C.S. Seshadri, A Pieri-Chevalley type formula
for $K(G/B)$ and standard monomial theory. Studies in memory of Issai Schur
(Chevaleret/Rehovot, 2000), 155--176, Progr. Math., 210, Birkhauser Boston,
Boston, MA, 2003.
\item[{[M]}] O. Mathieu, Positivity of some intersections in $K_{0}(G/B)$, J.
Pure Appl. Algebra 152 (2000) 231--243.
\item[{[PR$_{1}$]}] H. Pittie and A. Ram, A Pieri-Chevalley formula in the $K$%
-theory of a $G/B$-bundle, Electron. Res. Announc. Amer. Math. Soc. 5 (1999)
102--107.
\item[{[PR$_{2}$]}] H. Pittie and A. Ram, A Pieri-Chevalley formula for $%
K(G/B)$, \textsl{preprint available on} arXiv: math.RT/0401332.
\item[{[W$_{1}$]}] M. Willems, Cohomologie et $K$--th\'{e}orie \'{e}%
quivariantes des tours de Bott et des vari\'{e}t\'{e}s de drapeaux.
Application au calcul de Schubert, \textsl{preprint available on} arXiv:
math.AG/0311079.
\item[{[W$_{2}$]}] M. Willems, K-th\'{e}orie \'{e}quivariante des vari\'{e}t%
\'{e}s de Bott-Samelson. Application \`{a} la structure multiplicative de la
K-th\'{e}orie \'{e}quivariante des vari\'{e}t\'{e}s de drapeaux, \textsl{%
preprint available on} arXiv: math.AG/0412152.
\end{enumerate}
\end{document}
|
2,877,628,090,208 | arxiv | \section{Introduction}\label{sec1}
A two-dimensional normal density or Student $t$-density is constant on
boundaries of certain ellipses. Outside such an ellipse the density is
lower than inside. It is straightforward to find such an outer region
and its contour (line), for a given small probability. We can consider
such contour as a natural multidimensional extension of a
(one-dimensional) quantile. Even for extreme sets, that is,\ very low
density levels, the calculations are straightforward.
In this paper we consider, much more general, \textit{multi}variate
regularly varying distributions [for a review, see \citet{JesMik06}]. We consider the latter distributions,
since we want to explore in particular extreme sets, that is,\ sets far
removed from the origin.
A random vector $\mathbf{X}$ is multivariate regularly varying if
there exist
a constant $\alpha>0$, the index and an arbitrary probability measure
$\Psi$ on $\Theta=\{\mathbf{z}\in\mathbb{R}^d\dvtx \| \mathbf{z}\|
=1\}$, the unit
hypersphere, such that
\begin{equation}\label{mrv}
\lim_{t\rightarrow\infty }\frac{\mathbb{P} ( \|\mathbf{X}\|\geq
tx , \mathbf{X}/\|\mathbf{X}
\|\in A)}{\mathbb{P} (\|\mathbf{X}\| \geq t)}=x^{-\alpha}\Psi(A)
\end{equation}
for every $x>0$ and Borel set $A$ in $\Theta$ with $\Psi(\partial
A)=0$, with $ \|\mathbf{X}\|$ the $L_2$-norm of~$\mathbf{X}$; see
Rva\v{c}eva
(\citeyear{Rva62}).
An equivalent statement is
\begin{equation}\label{rad}
\lim_{t\rightarrow\infty }\frac{\mathbb{P} ( \|\mathbf{X}\|\geq
tx )}{\mathbb
{P} (\|\mathbf{X}\| \geq t)}=x^{-\alpha}\qquad \mbox{for } x>0,
\end{equation}
and there exists a measure $\nu$ such that
\begin{eqnarray}\label{(1.1)}
\lim_{t\rightarrow\infty }\frac{\mathbb{P} ( \mathbf{X}\in
tB)}{\mathbb{P}
(\|\mathbf{X}\| \geq t)}=\nu(B)<\infty
\end{eqnarray}
for every Borel set $B$ on $\mathbb{R}^d$ that is bounded away from
the origin and satisfies $\nu(\partial B)=0$; here $tB=\{t\mathbf
{z}\dvtx\mathbf{z}\in
B\}$.
Note that $\nu$ is homogeneous,
that is,\ for all
$a>0$,
\begin{eqnarray}
\nu(aB)=a^{-\alpha }\nu(B).\label{(1.2)}
\end{eqnarray}
Clearly, on $\{\mathbf{z}\in\mathbb{R}^d\dvtx \|\mathbf{z}\|\geq1\}$, $\nu$ is a \textit
{probability} measure.
The limit relation in (\ref{(1.1)}) is a multivariate analogue of the
``peaks-over-threshold'' or ``generalized Pareto limit'' method in
one-dimensional extreme value theory. Particular cases of (\ref{mrv})
are distributions in the sum-domain of attraction of $\alpha$-stable
distributions and heavy tailed elliptical distributions such as
multivariate $t$-distributions [see \citet{Has06}].
We require the convergence in (\ref{rad}) and (\ref{(1.1)}) at the density
level:
\begin{longlist}[(a)]
\item[(a)] Suppose that the distribution of $\mathbf{X}$ has a continuous
and positive density $f$ and that for some positive function $q$ and
some positive function $V$ regularly varying at infinity with negative
index $-\alpha $, we have
\begin{equation}\label{pw}
\lim_{t\rightarrow\infty }\frac{f(t\mathbf
{z})}{t^{-d}V(t)}=q(\mathbf{z})\qquad \mbox{for
all } \mathbf{z}\neq\mathbf{0}
\end{equation}
and
\begin{equation}\label{(1.3)}
\lim_{t\rightarrow\infty }\sup_{\mathbf{z}\in\Theta} \biggl|\frac
{f(t\mathbf{z}
)}{t^{-d}V(t)}-q(\mathbf{z}) \biggr|=0.
\end{equation}
Then $q$ is continuous on $\mathbb{R}^d\setminus\{\mathbf{0}\}$ and
$q(a\mathbf{z})=a^{-d-\alpha }q(\mathbf{z})$ for all $a>0$ and
$\mathbf{z}\neq\mathbf{0}$.
Throughout, \textit{we can and will take} $V(t)=\mathbb{P}(\|\mathbf{X}
\|>t)$ (see Lemma~\ref{lem1}, Section~\ref{sec5}).
From Lemma~\ref{lem1}, it follows that doing so (\ref{(1.1)}) holds with $
\nu(B)=\int_B q(\mathbf{z})\,d\mathbf{z}
$.
\end{longlist}
The extreme region will be of the form
\[
Q=\{\mathbf{z}\in\mathbb{R}^d\dvtx f(\mathbf{z})\leq\beta \},
\]
where $f$ is the probability density of the random vector $\mathbf
{X}$; $\beta
$ is determined in such a way that the probability of $Q$ is equal to a
given very small number $p$, like $1/10$,$000$.
It is the purpose of this paper to estimate $Q$ based on $n$ i.i.d.\
copies of $\mathbf{X}$. Note that the shape of $Q$ is
not predetermined, it depends on the density~$f$. For the estimation of
$Q$, we will use an approximation of $f$ based on the density of $\Psi
$. The values of $p$ we consider are typically of order $1/n$. This
means that the number of data points that fall in $Q$ is small and can
even be zero, that is,\ we are extrapolating outside the sample. This
lack of relevant data points makes estimation difficult. The estimation
of $Q$ is a multivariate analogue of the estimation of extreme
quantiles in the univariate setting; see, for example,\ de Haan and
Ferreira (\citeyear{deHFer06}), Chapter~4. The multivariate case
is much more complicated, however, since we have to estimate a whole
set instead of only one value.
Having an estimate of $Q$ can be important in various settings. It can
be used as an alarm system in risk management: if a new observation
falls in the estimated $Q$ it is a signal of extreme risk. See
\citet{EinLiLiu09} for an application to aviation safety along
these lines.
In a financial or insurance setting, points on the boundary of the
estimate of $Q$ can be used for stress testing. The estimate of $Q$ can
also be used to rank extreme observations (see Remark~\ref{rem3},
Section~\ref{sec2}).
For the ``central'' part of the distribution, that is,\ $\beta$ is
fixed (and ``not too small''), nonparametric estimation of density
level sets has been studied in depth in the literature. Two approaches
are used, the plug-in approach using density estimation [see
\citet{BalCueCue01} and \citet{RigVer09}], and the excess
mass approach [see M\"{u}ller and Sawitzki (\citeyear{MulSaw91}),
\citet{Pol95} and \citet{Tsy97}]. Our estimation problem
and (hence) our approach are quite different from these.
This paper is organized as follows. In Section~\ref{sec2}, we derive our
estimator and show a refined form of consistency. A simulation and
comparison study is presented in Section~\ref{sec3} and a financial application
is given in Section~\ref{sec4}. Section~\ref{sec5} contains the proof of the main result.
\section{Main result}\label{sec2}
Consider a random sample $\mathbf{X}_1, \mathbf{X}_2,\ldots, \mathbf
{X}_n $ with
$\mathbf{X}_i\stackrel{d}{=}\mathbf{X}$, for $i=1,\ldots, n$; their common
probability measure on $\mathbb{R}^d$ is denoted with $P$. Write $R_i$
for the
radius $\|\mathbf{X}_i\|$ and $\mathbf{W}_i$ for the direction
$\mathbf{X}_i/\|\mathbf{X}_i\|$ of
$\mathbf{X}_i$. We wish to
estimate an extreme risk region of the form
\[
Q=\{\mathbf{z}\in\mathbb{R}^d\dvtx f(\mathbf{z})\leq\beta \},
\]
where $\beta $ is such that $PQ=p>0$, where $p=p_n\to0$, as $n \to
\infty$.
This means that $Q$ and $\beta $ depend on $n$, that is, $Q=Q_n$ and
$\beta =\beta _n$.
We shall
connect $Q_n$ to a fixed set $S$ not depending on $n$, defined by
\[
S=\{\mathbf{z}\dvtx q(\mathbf{z})\leq1\}.
\]
It will
turn out that $Q_n$ can be approximated by a properly inflated version
of $S$. In fact, it follows from (\ref{(1.3)}) that the risk regions
are asymptotically homothetic as a function of $p$, for small values of $p$.
Define $H(s)=1-V(s)=\mathbb{P}(R\leq s)$
and
$U(t)=H^{-1}(1-\frac{1}{t})$. Note that $U$ is regularly varying at
infinity with index $1/\alpha $.
We will approximate $Q_n$ in two steps by a (deterministic) region
$\widetilde{Q}_n$. This approximation satisfies
\begin{equation}\label{appro}
\frac{P(Q_n \triangle\widetilde{Q}_n)}{p}\to0
\end{equation}
($\triangle$ denotes ``symmetric difference'') and is based on the
above limit relations. The region $\widetilde{Q}_n$ can therefore be
estimated using extreme value theory.
The first step is to establish an approximation of $\beta =\beta
(p)$. Let
\begin{longlist}[(b)]
\item[(b)] $k=k_n(<n)$ be a sequence of positive
integers such that $k\rightarrow\infty $ and $k/ n\rightarrow
0$.
\end{longlist}
The region $Q_n$ is approximated by
\[
\bar{Q}_n= \biggl\{\mathbf{z}\dvtx f(\mathbf{z})\leq\biggl(\frac{np}{k\nu(S)}
\biggr)^{1+{d}/{\alpha
}}\frac{1}{({n}/{k})(U({n}/{k}))^d} \biggr\}.
\]
Next, we approximate $\bar{Q}_n$ by a further region $\widetilde
{Q}_n$ defined in
terms of the limit density $q$ rather than $f$:
\begin{equation}\label{til}
\widetilde{Q}_n=U\biggl(\frac{n}{k}\biggr)\biggl(\frac{k\nu(S)}{np}\biggr)^{1/\alpha }S.
\end{equation}
Indeed, $S$ and this approximation of $Q_n$ are homothetic.
Write
\[
B_{r,A}=\{\mathbf{z}\dvtx \|\mathbf{z}\|\geq r, {\mathbf{z}}/{\|\mathbf
{z}\|} \in A\}
\]
for a Borel set $A$ on $\Theta$. Clearly,
$B_{r,A}=rB_{1,A}$ and hence
$
\nu(B_{r,A})=r^{-\alpha }\nu(B_{1,A}).
$
The relation between the spectral measure $\Psi$ and $\nu$ is [cf.\
(\ref{mrv}) and (\ref{(1.1)})]
\[
\Psi(A)=\nu(B_{1,A})
\]
for
a Borel set $A\subset\Theta$. Recall that the spectral measure is a
probability measure.
The existence of a density $q$ of $\nu$ implies the existence of a density
$\psi$ of $\Psi$, that is,
\[
\Psi(A)=\int_A \psi(\mathbf{w})\,d\lambda(\mathbf{w}),
\]
where $\lambda$ is the Hausdorff measure (surface area) on
$\Theta$ and
\[
q(r\mathbf{w})=\alpha r^{-\alpha -d}\psi(\mathbf{w}).
\]
Next, we write $S$ and $\nu(S)$ in terms of the spectral
density:
\[
S= \bigl\{\mathbf{z}=r\mathbf{w}\dvtx r\geq(\alpha \psi(\mathbf{w}))^{{1}/{(\alpha
+d)}}, \mathbf{w}\in
\Theta\bigr\}
\]
and hence
\[
\nu(S)=\alpha ^{-{\alpha }/{(\alpha +d)}}\int_{\Theta}
(\psi(\mathbf{w}))^{{d}/{(\alpha +d)}}\,d\lambda(\mathbf{w}).
\]
To estimate $\widetilde Q_n$, we need estimators for $U(n/{k}), \alpha
, S$
and $\nu(S)$. From the above expressions
for $S$ and $\nu(S)$, we see that this means that we have to estimate
$U(n/{k})$,
$\alpha $ and $\psi$. First, we define
\[
\widehat{U}\biggl(\frac{n}{k}\biggr)=R_{n-k: n}
\]
[the $(n-k)$th order statistic of the $R_i$, $i=1, \ldots, n$].
Since the tail of the distribution function of
$R$ is regularly varying with index $-\alpha $, we can use one of
the well-known estimators of the extreme value index $1/\alpha $,
based on
the $R_i, i=1, \ldots,
n$; see, for example, \citet{Hil75}, \citet{Smi87} and
\citet{DekEindeH89}. It remains to estimate $\psi$.
Let $K\dvtx [0,1]\rightarrow[0,1]$ be a continuous and
nonincreasing (kernel) function with $K(0)=1$ and $K(1)=0$.
For
$\mathbf{w}\in\Theta$, define an estimator of $\psi(\mathbf{w})$ by
\[
\widehat{\psi}_n(\mathbf{w})=\frac{c(h, K)}{k}\sum_{i=1}^n K\biggl(\frac
{1-\mathbf{w}^T\mathbf{W}_i}{h}\biggr)1_{[R_i>R_{n-k: n}]}
\]
with $0<h<1$ and
\[
c(h, K)=\biggl(\int_{C_\mathbf{w}(h)}K\biggl(\frac{1-\mathbf{v}^T\mathbf{w}}{h}\biggr)\,d\lambda
(\mathbf{v})\biggr)^{-1},\quad
C_\mathbf{w}(h)=\{\mathbf{v}\in\Theta\dvtx \mathbf{w}^T\mathbf{v}\geq
1-h\};
\]
cf.\
\citet{HalWatCab87}.
For estimating $Q_n$ it suffices to estimate $\widetilde{Q}_n$, see
(\ref
{appro}). Hence, in view of~(\ref{til}), we define
\begin{eqnarray}\label{defqhat}
\widehat{Q}_n=\widehat{U}\biggl(\frac{n}{k}\biggr)\biggl(\frac{k\widehat{\nu
(S)}}{np}\biggr)^{{1}/{\widehat{\alpha
}}}\widehat{S}
\end{eqnarray}
with
\[
\widehat{S}= \bigl\{\mathbf{z}=r\mathbf{w}\dvtx r\geq(\widehat{\alpha
}\widehat{\psi
}_n(\mathbf{w}))^{{1}/({\widehat{\alpha }+d})}, \mathbf{w}\in
\Theta\bigr\}
\]
and
\[
\widehat{\nu(S)}=\widehat{\alpha }^{-{\widehat{\alpha
}}/{(\widehat{\alpha }+d})}\int_{\Theta}
(\widehat{\psi}_n(\mathbf{w}))^{{d}/{(\widehat{\alpha
}+d)}}\,d\lambda(\mathbf{w}).
\]
In the definition of the set $S$, the choice of the value 1 was not
motivated. We could have taken any number $c>0$ instead.
Such an alternative definition of $S$ would lead to exactly the same
estimator $\widehat{Q}_n$, which shows that the value 1 plays no role.
Assume
{\renewcommand{\theequation}{c}
\begin{equation}\label{eqc}
\lim_{t\rightarrow\infty }\frac{U(t)}{t^{1/\alpha}}=c\qquad \mbox{for
some } c\in(0,\infty).
\end{equation}}
\vspace*{-\baselineskip}\vspace*{10pt}
\setcounter{equation}{9}
\noindent Note that this simple condition is weaker than the usual second order
condition with negative second order parameter $\rho$ [see, e.g.,
Theorem 4.3.8 in de Haan and Ferreira (\citeyear{deHFer06})]; indeed,
there exist functions $U$ with $\rho=0$ that satisfy condition~(\ref{eqc}).
\begin{theorem}\label{thm1}
Let $p\to0$ as $n\rightarrow\infty $.
Assume conditions \textup{(a), (b), (\ref{eqc})} hold and that $\widehat{\alpha }$ is
such that
$\sqrt{k}(\widehat{\alpha }-\alpha
)=O_{\mathbb{P}}(1)$.
Also assume that $({\log np})/{\sqrt{k}}\rightarrow0$, \mbox{$h\to0$} and
$k/(c(h, K)\log k)\to\infty$, as $n\to\infty$.
Then we have
\begin{equation} \label{(theorem)}
\frac{P(\widehat{Q}_n\triangle Q_n)}{p} \stackrel{\mathbb{P}}{\rightarrow}0\qquad
\mbox{as }
n\rightarrow\infty ,
\end{equation}
and hence
\[
\frac{P(\widehat Q_n)}{ p}\stackrel{\mathbb{P}}{\to} 1.
\]
\end{theorem}
\begin{remark}\label{rem1}
The tuning parameter $k$ is used in the estimators of $\alpha,
U(n/{k})$ and $\psi$. It is important to be able to choose three
different values for $k$, denoted with $k_\alpha , k_U$ and $k_\psi$,
respectively. (Note that ``good'' values of $k_\alpha $ and $k_U$ are
determined by the tail of $H$---the distribution function of $R_1$---whereas a good $k_\psi$ is determined by the conditional distribution
of $\mathbf{W}_1$, given that $R_1>r$, for large $r$.) If we adapt the
conditions of the theorem, in particular if (b) holds for $k_\alpha , k_U,
k_\psi$ and if $(\log np) / \sqrt{k_\alpha } \to0$, $k_\psi/(c(h,
K)\log
k_\psi)\to\infty$ and $(\log k_U)/\sqrt{k_\alpha }\to0$, then~(\ref{(theorem)}) remains true for the generalized estimator that
allows for the aforementioned different $k$-values. We will use this
generalized estimator in the simulation study and the real data
application.
The actual choice of these $k$-values is a notorious problem in extreme
value theory. A solution of this problem is far beyond the scope of the
present paper. We will only give heuristic guidelines here. First,
consider the estimation of $\alpha $. Plot $\widehat{\alpha }$ as a
function of $k$. Now
find the first stable, that is,\ approximately constant, region in the
graph of this function. This vertical level is the final estimate of
$\alpha $. It is also possible to use (complicated) \textit{asymptotically}
optimal procedures; see, for example, Danielsson et al.\ (\citeyear
{Danetal01}). Once the
estimate $\widehat{\alpha }$ is fixed, we plot $\widehat{U}(\frac
{n}{k})(\frac{k}{n})^{1/\widehat{\alpha }}$
against $k$ and we search for the first stable part in this graph. The
vertical level is now the estimate of the constant $c$ in condition
(c). Observe that $\widehat{U}(\frac{n}{k})(\frac{k}{n})^{1/\widehat
{\alpha }}$ is a building block of
$\widehat Q_n$, so we do not need to estimate $U(\frac{n}{k})$
separately. Also
observe that we do not (need to) determine $k_\alpha $ and $k_U$, but
only a
region of good values. Finally, using again the already fixed $\widehat
{\alpha }$, we
plot $\widehat{\nu(S)}$ as a function of $k$ and again we search for
the first stable region; we take $k_\psi$ to be the midpoint of this
region of $k$-values.
\end{remark}
\begin{remark}\label{rem2}
The class of multivariate regularly varying distributions is quite
large. It contains, for example, all elliptical distributions with a heavy
tailed radial distribution and all distributions in the domain of a
sum-attraction of a multivariate (nonnormal) stable distribution. It
seems natural, however, to try to extend the assumption of multivariate
regular variation to the case of nonequal tail indices~$\alpha$. It is
an important feature of the present model that all directions are
equally important: the marginal distributions do not play a special
role. An extension to nonequal tail indices would be possible in
principle, but it will be of limited value since it only works if
\textit{marginal} transformations lead to the present model. Also note
that basically all linear combinations of the components inherit the
lowest of the marginal tail indices: the tail index is not a smooth
function of the direction (if it is not constant). Moreover, the
statistical theory that will be needed will be challenging and will
lead to a new and different project.
\end{remark}
\begin{remark}\label{rem3}
Note that the estimated extreme risk region $\widehat Q_n=\widehat{Q}_n(p)$
depends on $p$ in a continuous way and has the property that $p_1<p_2$
implies $\widehat{Q}_n(p_1)\subset\widehat{Q}_n(p_2)$. Hence, we can
find the smallest $p$
such that an observation is on the boundary of $\widehat Q_n(p)$. The
corresponding observation can be considered the largest one and we
know its ``$p$-value.'' This is helpful in deciding whether some
observation is the most extreme or if it is an outlier. Also, by
continuing this procedure we can rank the larger observations.
\end{remark}
\section{Simulation study}\label{sec3}
In this section, a detailed simulation study is performed in order to
investigate the finite sample performance of our estimator [with
$1/\alpha$ estimated using the moment estimator of \citet
{DekEindeH89} and with $K(u)=1-u$].
We consider five multivariate distributions.
\begin{itemize}
\item The bivariate Cauchy distribution with
density
\begin{equation}\label{cauchy2}f(x,y)=\frac{1}{2\pi
(1+x^2+y^2)^{3/2}} ,\qquad (x,y) \in\mathbb{R}^2.
\end{equation}
This is a very heavy tailed density, with $\alpha=1$ and
$\psi(\mathbf{w})=1/(2\pi)$, for $\mathbf{w}\in\Theta$.
\item The trivariate Cauchy distribution with
density
\begin{equation}\label{cauchy3}f(x,y,z)=\frac{1}{\pi
^2(1+x^2+y^2+z^2)^2} ,\qquad (x,y, z) \in\mathbb{R}^3 .\vadjust{\goodbreak}
\end{equation}
This is also a very heavy tailed density, with $\alpha=1$
and $\psi(\mathbf{w})=1/(4\pi)$, for $\mathbf{w}\in\Theta$.
\item A bivariate elliptical distribution with density ($r_0\approx1.2481$)
\begin{equation}\label{ellip}
f(x,y)= \cases{
\displaystyle \frac{3}{4\pi}r_0^4(1+r_0^6)^{-3/2}, &\quad $x^2/4+y^2<
r_0^2,$\vspace*{2pt}\cr
\displaystyle \frac{3(x^2/4+y^2)^2}{4\pi(1+(x^2/4+y^2)^3)^{3/2}} ,&\quad $ x^2/4+y^2\geq
r_0^2. $
}
\end{equation}
It is less heavy tailed. We have $\alpha=3$ and
$\psi(w_1,w_2)=c(1+3w_2^2)^{-5/2}, \mathbf{w}=(w_1,w_2)\in\Theta$, with
$c\approx0.6028$.
\item A bivariate ``clover'' distribution with density ($r_0\approx1.2481$)
\begin{equation}\label{clov}
f(x,y)= \cases{
\displaystyle \frac{3}{10\pi}r_0^4(1+r_0^6)^{-3/2} \biggl(5+\frac
{4(x^2+y^2)^2-
32x^2y^2}{r_0(x^2+y^2)^{3/2}} \biggr),\vspace*{3pt}\cr
\qquad x^2+y^2< r_0^2,\vspace*{4pt}\cr
\displaystyle \frac{3 (9(x^2+y^2)^2-32x^2y^2 )}{10\pi(1+({x^2}+y^2)^3)^{3/2}}
,\vspace*{3pt}\cr
\qquad x^2+y^2\geq r_0^2.
}\hspace*{-8pt}
\end{equation}
We have $\alpha=3$, again, and
$\psi(w_1,w_2)=(9-32w_1^2w_2^2)/(10\pi), \mathbf{w}=(w_1,\break w_2)\in
\Theta$.
\item A bivariate asymmetric shifted distribution with density
[$r_0\approx1.2331$, $\widetilde{r}(x,y):=r_0\vee((x+5)^2+y^2)^{1/2}$]
\begin{equation}\label{asymm}
f(x,y) = \cases{
\displaystyle \frac{\widetilde{r}^2(x,y)}{6\pi(1+\widetilde
{r}^4(x,y))^{5/4}} \biggl(3+\frac{x+5}{\widetilde{r}(x,y)} \biggr),
\hspace*{69pt}\qquad y\geq0, \vspace*{5pt}\cr
\displaystyle \frac{\widetilde{r}^2(x,y)}{6\pi(1+\widetilde{r}^4(x,y))^{5/4}}
\biggl(3+\frac{(x+5)^3-3(x+5)y^2}{\widetilde{r}^3(x,y)} \biggr),
\qquad y<0.
}\hspace*{-15pt}
\end{equation}
This distribution is not symmetric and the ``center'' is not the origin,
but $(-5,0)$; $\alpha=1$ and
$\psi(w_1,w_2)= \frac{1}{6\pi}(3+w_1)$, if $ w_2\geq0$, and
$ \psi(w_1,w_2)= \frac{1}{6\pi}(3+4w_1^3-3w_1)$, if $w_2<0$,
$\mathbf{w}=(w_1,w_2)\in\Theta$.
\end{itemize}
First, we simulated single data sets of size 5,000 of the bivariate
Cauchy distribution, the elliptical distribution in (\ref{ellip}), the
clover distribution in (\ref{clov}) and the asymmetric shifted
distribution in (\ref{asymm}). We computed the true and estimated risk regions
for $p=1/2\mbox{,}000$, 1$/$5,000 or $1/10\mbox{,}000$. This is depicted in Figure \ref
{clouds}. We see that the estimated regions are relatively close to the
true risk regions. It is interesting to note that the $p$-value (see
Remark~\ref{rem3}) of the largest observation for the Cauchy sample is
0.000209, which is about $1/n$. This shows that this observation is a
typical one. (Looking at the data only, one might want to conclude that
this observation is an outlier.) Also note that for the bivariate
Cauchy distribution, for, for example, $p=1/10\mbox{,}000$, the density $f$ at the
boundary of the true risk region is less than $10^{-12}$. This
emphasizes that we are estimating in an ``almost empty'' part of the
plane and that a fully nonparametric procedure could not work here.
\begin{figure}
\includegraphics{891f01.eps}
\caption{True and estimated risk regions based on
one sample of size 5,000 from the bivariate Cauchy distribution, the
elliptical distribution in (\protect\ref{ellip}), the clover distribution in
(\protect\ref{clov}) and the asymmetric shifted distribution in (\protect\ref{asymm}).}\label{clouds}
\end{figure}
In addition, we simulated one sample of the bivariate distribution with
independent $t_3$-components. This distribution does \textit{not}
satisfy condition (a), since the spectral measure is discrete and
concentrated on the intersection of the coordinate axes with the unit
circle. We also simulated one sample of a bivariate ``logarithmic''
distribution with $\alpha=1$ and uniform spectral measure, but where
the radial distribution satisfies $U(t)/(t\log t)$ tends to a constant
and hence $U(t)/t \to\infty$ as $t\to\infty$, that is,\ this
distribution does \textit{not} satisfy condition (c). Although both
distributions do not satisfy our conditions, we see nevertheless
satisfactory behavior of the estimator in Figure~\ref{clouds2}. In the
left panel, the estimated region has about the right size and the
difficult shape is approximated reasonably well; in the right panel, we
see that both the shape and the size are approximated quite well.
\begin{figure}
\includegraphics{891f02.eps}
\caption{True and estimated risk regions based on
one sample of size $5\mbox{,}000$ from the bivariate distribution with
independent $t_3$-components and the ``logarithmic'' distribution.}\label{clouds2}
\end{figure}
After this visual assessment of our estimator based on one sample at a
time, we now investigate its performance based on 100 simulated samples
of size 5,000. We will compare our estimator (denoted EVT) to a
nonparametric and to a more parametric estimator. The nonparametric
estimator is only defined in case $p=1/n$ and tries to mimic the
largest order statistic as an estimator of the $(1-1/n)$th quantile in
the univariate case. It aims at elliptical level sets. It is defined as
follows. First, calculate the smallest ellipsoid containing half of the
data, the so-called MVE. Then inflate this ellipsoid, such that the
``largest'' observation lies on its boundary. Now the region
outside this ellipsoid is the estimator.
For $d=2$, the more parametric estimator is defined similarly to
$\widehat{Q}_n$
in (\ref{defqhat}), but (only) the estimation of $(\nu(S))^{1/\alpha
}S$ is done parametrically. Therefore, this estimator has the same size
as $\widehat{Q}_n$, but a different shape. (Note that the fully parametric
estimator based on multivariate normality would have a very bad
performance.) Take the $k$ observations with radius $R_i> R_{n-k: n}$
and consider the transformed data
$(R_i/R_{n-k: n}, \mathbf{W}_i)$.
In line with the limit result in (\ref{mrv}), assume that these
data have a ``distribution'' $(\cdot)^{-\alpha}\Psi$, where $\Psi$
depends on a parameter $\rho$. To be precise, we assume for the density
\[
\psi_\rho(\theta)=(4\pi)^{-1}\bigl(2+\sin\bigl(2(\theta-\rho)\bigr)\bigr),\qquad 0\leq
\theta< 2\pi, 0\leq\rho<\pi.
\]
(Here a point on the unit circle is represented by its angle $\theta
\in[0,2\pi)$.) Now $\alpha$ and $\rho$ are estimated by maximum
likelihood; observe that this yields the Hill estimator for $1/\alpha$.
\begin{table}
\tablewidth=265pt
\caption{Median of the relative errors $P(\widehat
{Q}_n\triangle Q_n)/p$
of the three estimators, for $p=1/5\mbox{,}000$ (p1) and $1/10\mbox{,}000$ (p2)}
\label{tab}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccccc@{}}
\hline
\textbf{Density}& \textbf{EVT p1} & \textbf{Par p1} & \textbf{NP p1} & \textbf{EVT p2} & \textbf{Par p2} \\
\hline
Biv.\ Cauchy & $0.28$ & $0.29$ & $0.72$ & $0.31$ & $0.32$ \\
Triv.\ Cauchy & $0.22$ & -- & $0.54$ & $0.24$ & -- \\
Elliptical & $0.36$ & $0.51$ & $0.80$ & $0.39$ & $0.54$ \\
Clover & $0.44$ & $0.57$ & $0.58$ & $0.49$ & $0.61$\\
Asymm.\ shifted & $0.26$ & $0.27$ & $0.61$ & $0.30$ & $0.32$\\
\hline
\end{tabular*} \vspace*{-3pt}
\end{table}
\begin{figure}
\includegraphics{891f03.eps}
\caption{Boxplots of ${P(\widehat{Q}_n\triangle Q_n )}/{p}$
for the here
proposed estimator and for the parametric and the nonparametric
estimator, based on
$100$ simulated data sets of size 5,000 from the five presented
densities for $p=1/5\mbox{,}000$ (p1) and $1/10\mbox{,}000$ (p2).}\label{f3}
\end{figure}
Table~\ref{tab} shows for the three different estimators the median
of the 100 relative errors $P(\widehat{Q}_n\triangle Q_n)/p$ for
$p=1/5\mbox{,}000$ (p1)
and $1/10\mbox{,}000$ (p2).
In Figure~\ref{f3}, boxplots are shown of the relative error
$P(\widehat{Q}_n\triangle Q_n)/p$ for $p=1/5\mbox{,}000$ (p1) and $1/10\mbox{,}000$
(p2). From
this table and figure, we see a good performance of our estimator. Its
behavior does not change much if $p$ changes from $1/5\mbox{,}000$ to $1/10\mbox{,}000$.
The parametric estimator performs reasonably well, but it is
outperformed by our estimator, in particular for the elliptical and
clover densities. Recall that this estimator can be seen as a
modification of our estimator, since it uses the same estimated
inflation factor, but the shape is estimated differently. We see a
moderate performance of the nonparametric estimator; also, it cannot be
adapted to $p=1/10\mbox{,}000$. Given that the estimation of these extreme
risk regions is a statistically difficult problem, we see decent
behavior of the three estimation methods. Obviously the parametric and
the nonparametric estimator do not perform well if the parametric part
of the model is not adequate or if the shape of the region is not
elliptical, respectively. The EVT estimator, presented in this paper,
does not suffer from these shortcomings and performs well for many
multivariate distributions.
\begin{figure}
\includegraphics{891f04.eps}
\caption{Estimator of $\psi$ of bivariate exchange rate
returns.}\label{psi}\vspace*{-3pt}
\end{figure}
\begin{figure}[b]
\includegraphics{891f05.eps}
\caption{Estimated extreme risk regions of exchange rate
returns.}\label{66}
\end{figure}
\section{Application}\label{sec4}
In this section, an application of our method to foreign exchange rate
data is presented. The data
are the daily exchange rates of Yen-Dollar and Pound-Dollar from
January 4, 1999 to July 31, 2009. Consider the daily log returns given
by
$
X_{i,j}=\log{({Y_{i+1,j}}/{Y_{i,j}})}$,
with $i=1,\ldots,2\mbox{,}664$, $j=1,2$, and $Y_{i,1}$ is the daily
exchange rate of the Yen to the Dollar and $Y_{i,2}$ of the Pound to
the Dollar.
First, we check the equality of the extreme value indices (the
reciprocals of the tail indices) of the right and left tails of both
marginal distributions and that of the radius. This yields 5 extreme
value indices; the 5 estimates in increasing order are: 0.141, 0.191,
0.223, 0.242, 0.256. Hence, the maximal difference is 0.115. Based on
the asymptotic normality of\vadjust{\goodbreak} the moment estimator of the extreme value
index, we compute an approximate upper bound for the maximal difference
of the 5 estimators under the null hypothesis of equality: 0.264.
Hence, there is no evidence that the 5 extreme value indices are
different. Other exchange rate data sets share this property. There are
also economic arguments supporting this claim. Therefore, we estimate
$\alpha$ based on the radius and find $\widehat\alpha=3.90$.
As a next step, we estimate the density $\psi$ of the spectral
measure. The estimate $\widehat\psi_n$
is depicted in Figure 4;
it is almost periodic with period $\pi$. This yields that the boundary
of the
estimated extreme risk region is not like a circle, but more like an
ellipse. The location of the maxima of $\widehat\psi_n$ correspond to
the major axis of the region. We estimate the extreme risk regions for
$p=1/2\mbox{,}000,1/5\mbox{,}000$ and $1/10$,$000$; see Figure\vadjust{\goodbreak} 5.
For risk management of financial institutions in the U.S.,\ it is
important to know which extreme exchange rate returns w.r.t.\ the Pound
and the Yen can occur and which returns essentially never occur. Our
estimate answers this question.
More specifically,
points on the boundary of the estimated extreme risk region can be used
as multivariate stress test scenarios.
A scenario on the intersection of the major axis of the
ellipse-like boundary of the extreme risk region and the boundary
itself corresponds to a larger shock than a scenario on the
intersection of the minor axis of the ellipse-like boundary and the
boundary itself, but our method shows
that their ``extremeness'' is about the same.
\section{Proofs}\label{sec5}
For the proof of the theorem, we need several lemmas and propositions.
We assume throughout that the conditions of the theorem are in force.
We start with a lemma on regular
variation in $\mathbb{R}^d$.
\begin{lemma}\label{lem1}
Write $l=1/\int_{\{\|\mathbf{z}\|\geq1\}
}q(\mathbf{z}
)\,d\mathbf{z}$. For any $\varepsilon >0$,
\begin{equation}\label{(4.1)}
\lim_{t\rightarrow\infty }\sup_{\|\mathbf{z}\|\geq\varepsilon }
\biggl|\frac{f(t\mathbf{z}
)}{t^{-d}V(t)}-q(\mathbf{z}) \biggr|=0.
\end{equation}
Moreover
\begin{equation}\label{ne}
\lim_{t\rightarrow\infty }\frac{\mathbb{P}(\mathbf{X}\in
tB)}{V(t)}=\int
_{B}q(\mathbf{z})\,d\mathbf{z}
\end{equation}
for any Borel set $B$ bounded away from the origin.
Define $q_t(\mathbf{z})=t(U(t))^d\times f(U(t)\mathbf{z})$. Then
\begin{eqnarray}\label{(4.4)}
\lim_{t\rightarrow\infty }\sup_{\|\mathbf{z}\|\geq\varepsilon }
|q_t(\mathbf{z})-lq(\mathbf{z})|=0.
\end{eqnarray}
Let $\widetilde h$ be the density of $H$, then
\begin{equation}\label{(4.5)}
\lim_{t\rightarrow\infty }\frac{\widetilde h(t)}{t^{-1}V(t)}=\frac
{\alpha}{l} .
\end{equation}
\end{lemma}
\begin{pf}
For any $\|\mathbf{z}\|\geq\varepsilon >0$
[cf. Theorem 2.1 in de Haan and Resnick (\citeyear{deHRes87})],
\begin{eqnarray*}
&& \biggl|\frac{f(t\mathbf{z})}{t^{-d}V(t)}-q(\mathbf{z}) \biggr|\\[-2pt]
&&\qquad = \biggl|\frac{f(t\|\mathbf{z}\|(\mathbf{z}/{\|\mathbf{z}
\|}))}{(t\|z\|)^{-d}V(t\|z\|)}\cdot\frac{(t\|z\|)^{-d}V(t\|z\|)}{t^{-d}V(t)}-
q(\mathbf{z}) \biggr|\\[-2pt]
&&\qquad \leq\|\mathbf{z}\|^{-d-\alpha } \biggl|\frac{f(t\|\mathbf{z}\|
({\mathbf{z}}/{\|\mathbf{z}
\|}))}{(t\|z\|)^{-d}V(t\|z\|)}
-q \biggl(\frac{\mathbf{z}}{\|\mathbf{z}\|} \biggr) \biggr|\\[-2pt]
&&\qquad \quad {}
+\frac{f(t\|\mathbf{z}\|({\mathbf{z}}/{\|\mathbf
{z}\|}))}{(t\|z\|)^{-d}V(t\|z\|)}
\biggl|\frac{(t\|z\|)^{-d}V(t\|z\|)}{t^{-d}V(t)}-\|\mathbf{z}\|^{-d-\alpha
} \biggr|
\end{eqnarray*}
Then ({\ref{(4.1)}}) follows from condition (a).\vadjust{\goodbreak}
Let a Borel set $B$ be such that
$B\subset\{\|\mathbf{z}\|\geq\gamma\}$, for some $\gamma>0$. Then
for $\mathbf{z}\in B$ and sufficiently large $t$,
${f(t\mathbf{z})}/{t^{-d}V(t)}$ is bounded by $q(\|\mathbf
{z}\|^{-1}\mathbf{z})\|\mathbf{z}\|^{-a/2-d}$.
Hence, (\ref{ne}) holds by Lebesgue's dominated convergence
theorem.
We have from (\ref{ne}), as $t\rightarrow\infty $,
\[
tV(U(t))=\frac{V(U(t))}{\mathbb{P}(R\geq U(t))}\rightarrow l.
\]
Hence (\ref{(4.1)}) implies, uniformly for $\| \mathbf{z}\|
\geq\varepsilon$,
\[
q_t(\mathbf{z})=tV(U(t))\frac{f(U(t)\mathbf
{z})}{(U(t))^{-d}V(U(t))}\rightarrow l
q(\mathbf{z}).
\]
Note that
\[
1-H(t)=\mathbb{P}(R> t)=\int_t^\infty \int_{\Theta}f(r\mathbf{w})\,d\lambda
(\mathbf{w})r^{d-1}\,dr.
\]
By taking derivatives, (\ref{(4.1)}) and the homogeneity of $q$, we obtain
\[
\lim_{t\rightarrow\infty }\frac{\widetilde h(t)}{t^{-1}V(t)}
=\int_{\Theta}q(\mathbf{w})\,d\lambda (\mathbf{w})
=\alpha \int_{\{\|\mathbf{z}\|\geq1\}}q(\mathbf{z})\,d\mathbf
{z}=\alpha /l.
\]
\upqed
\end{pf}
We now see that (\ref{pw}) and (\ref{(1.3)}) hold with $V=1-H$. From
now on, we will make the choice $V=1-H$ and hence $l=1$.
Note that with this choice the relations (\ref{(1.1)}) [with $\nu
(B)=\int_B q(\mathbf{z})\,d\mathbf{z}$] and (\ref{(1.2)}) readily
follow from ({\ref{ne}}).
\begin{coro}\label{coro1}
For all Borel sets $B$ with
positive distance from the origin,
\begin{equation}\label{(4.3)}
\lim_{t\rightarrow\infty }tP(U(t)B)=\nu(B)
\end{equation}
and
\begin{equation}\label{(4.32)}
\lim_{n\rightarrow\infty } \frac{\nu(S)}{p}P\biggl(U \biggl( \frac{n}{k}
\biggr)\biggl(\frac
{k\nu(S)}{np}\biggr)^{1/{\alpha }}B\biggr)=\nu(B).
\end{equation}
\end{coro}
\begin{pf}
From $\mathbb{P}(R\geq U(t))=1/t$ and (\ref{(1.1)}), we
obtain (\ref{(4.3)}).
It
follows from (c)
that
\begin{equation}\label{(uvsp)}
\frac{U({\nu(S)}/{p})}{U({n}/{k})({k\nu
(S)}/{(np)})^{{1}/{\alpha }}}
\rightarrow1.
\end{equation}
This yields (\ref{(4.32)}).
\end{pf}
\begin{lemma}\label{lem2}
For each $\varepsilon >0$, there exists a
$\delta >0$ and $t_0>0$ such that for $t>t_0$
\[
\biggl\{\mathbf{z}\dvtx \frac{f(t\mathbf{z})}{t^{-d}V(t)}\leq\varepsilon \biggr\}
\subset
\{\mathbf{z}\dvtx\|\mathbf{z}\|>\delta \}.
\]
\end{lemma}
\begin{pf}
It is sufficient to prove
$\{\mathbf{z}\dvtx \|\mathbf{z}\|\leq\delta \}\subset\{\mathbf
{z}\dvtx f(t\mathbf{z})/(t^{-d}V(t))>\varepsilon \}$.
First, by (\ref{(1.3)}) and the continuity of $q$, for some $c_1>0$,
there exists $s_0>0$ such that
for $s>s_0$
\[
\inf_{\mathbf{w}\in\Theta}\frac{f(s\mathbf{w})}{s^{-d}V(s)}\geq c_1
\]
and also for $s_1, s_2>s_0$ [cf.\ Proposition B.1.9.5 in de
Haan and Ferreira (\citeyear{deHFer06})]
\[
\frac{V(s_1)}{V(s_2)}>\frac{1}{2}\biggl(\frac{s_1}{s_2}\biggr)^{-\alpha /2}.
\]
Now for $t>s_0$ and any $\mathbf{z}\in\{\mathbf{z}\dvtx \|\mathbf
{z}\|\leq\delta \}$, there
are two possibilities.
\begin{longlist}[(ii)]
\item[(i)] $t\|\mathbf{z}\| > s_0$, then
\[
\frac{f(t\mathbf{z})}{t^{-d}V(t)}
=\frac{f(t\|\mathbf{z}\|(\mathbf{z}/{\|\mathbf
{z}\|}))}{(t\|\mathbf{z}\|)^{-d}V(t\|\mathbf{z}
\|)}\cdot\frac{(t\|\mathbf{z}\|)^{-d}V(t\|\mathbf{z}\|)}{t^{-d}V(t)}
> \frac{1}{2}c_1 \delta ^{-\alpha /2-d}>\varepsilon ;
\]
\item[(ii)] $t\|\mathbf{z}\| \leq s_0$, then by continuity of
$f$ and $f>0$, we have for some $c_2>0$, $f(t\mathbf{z})\geq
c_2$, and hence, since $\lim_{t\to\infty }t^{-d}V(t)=0$, we
obtain for $t>t_0(\geq s_0)$
\[
\frac{f(t\mathbf{z})}{t^{-d}V(t)}>\varepsilon .
\]
\end{longlist}
\upqed
\end{pf}
\begin{lemma}\label{lem3}
For $\varepsilon >0$ and large $n$,
\[
\bar{Q}_n\subset U \biggl(\frac{\nu(S)}{p} \biggr)\{\mathbf{z}\dvtx q(\mathbf
{z})\leq1+\varepsilon \}
\]
and
\[
\bar{Q}_n\supset U \biggl(\frac{\nu(S)}{p} \biggr)\{\mathbf{z}\dvtx q(\mathbf
{z})\leq1-\varepsilon \}.
\]
\end{lemma}
\begin{pf}
Recall that
$\bar{Q}_n= \{\mathbf{z}\dvtx f(\mathbf{z})\leq(\frac{np}{k\nu
(S)})^{1+({d}/{\alpha})}\frac{1}{({n}/{k})(U({n}/{k}))^d} \}$.
It follows from (\ref{(uvsp)}) that for $n$ large enough and
$\varepsilon _1>0$
\begin{eqnarray*}
\bar{Q}_n&=&U\biggl(\frac{\nu(S)}{p}\biggr)
\biggl\{\mathbf{z}\dvtx f\biggl(U\biggl(\frac{\nu(S)}{p}\biggr)\mathbf{z}\biggr)\leq\biggl(\frac{np}{k\nu
(S)}\biggr)^{1+
({d}/{\alpha })}\frac{1}{({n}/{k})(U({n}/{k}))^d} \biggr\}\\
&=&U\biggl(\frac{\nu(S)}{p}\biggr)
\biggl\{\mathbf{z}\dvtx q_{\nu(S)/p}(\mathbf{z})\leq\biggl(\frac{np}{k\nu
(S)}\biggr)^{{d}/{\alpha }}
\biggl(U\biggl(\frac{n}{k}\biggr)\biggr)^{-d}\biggl(U\biggl(\frac{\nu(S)}{p}\biggr)\biggr)^{d} \biggr\}\\
&\subset& U\biggl(\frac{\nu(S)}{p}\biggr) \bigl\{\mathbf{z}\dvtx q_{\nu(S)/p}(\mathbf
{z})\leq1+\varepsilon _1\bigr\}.
\end{eqnarray*}
Now Lemma~\ref{lem2} implies
$ \{\mathbf{z}\dvtx q_{\nu(S)/p}(\mathbf{z})\leq1+\varepsilon_1 \}
\subset\{\mathbf{z}\dvtx \|\mathbf{z}
\|>\delta \}$,
hence we have by (\ref{(4.4)})
\[
\bar{Q}_n\subset U \biggl(\frac{\nu(S)}{p} \biggr)\{\mathbf{z}\dvtx q(\mathbf
{z})\leq1+\varepsilon \}.
\]
The other inclusion follows in the same way (but Lemma~\ref{lem2} is not
needed).
\end{pf}
\begin{lemma}\label{lem4}
For $\varepsilon >0$ and large $n$,
\[
\widetilde{Q}_n\subset U \biggl(\frac{\nu(S)}{p} \biggr)\{\mathbf{z}\dvtx q(\mathbf
{z})\leq1+\varepsilon \}
\]
and
\[
\widetilde{Q}_n\supset U \biggl(\frac{\nu(S)}{p} \biggr)\{\mathbf{z}\dvtx q(\mathbf
{z})\leq1-\varepsilon \}.
\]
\end{lemma}
\begin{pf}
Recall that
$\widetilde{Q}_n=U ( \frac{n}{k} )(\frac{k\nu(S)}{np})^{
{1}/{\alpha }}\{\mathbf{z}\dvtx q(z)\leq1\}$.
Put $T_n=(U(\frac{\nu(S)}{p}))^{-1}U (\frac{n}{k} )(\frac{k\nu
(S)}{np})^{{1}/{\alpha }}$,
then
\begin{eqnarray*}
\widetilde{Q}_n&=&U\biggl(\frac{\nu(S)}{p}\biggr) \{T_n\mathbf{z}\dvtx q(\mathbf
{z})\leq1 \}
=U\biggl(\frac{\nu(S)}{p}\biggr) \{T_n\mathbf{z}\dvtx q(T_n\mathbf{z})\leq
T_n^{-d-\alpha } \}\\
&=&U\biggl(\frac{\nu(S)}{p}\biggr) \{\mathbf{z}\dvtx q(\mathbf{z})\leq
T_n^{-d-\alpha } \}.
\end{eqnarray*}
Since $T_n\rightarrow1$ as $n\rightarrow\infty $ by
(\ref{(uvsp)}), the result follows.
\end{pf}
\begin{prop}\label{prop1}
We have
\[
\lim_{n\rightarrow\infty }\frac{P(Q_n\bigtriangleup\widetilde{Q}_n)}{p}=0.
\]
\end{prop}
\begin{pf}
Note that $P(Q_n\bigtriangleup\widetilde
{Q}_n)\leq
P(Q_n\bigtriangleup\bar{Q}_n)+P(\bar{Q}_n\bigtriangleup\widetilde
{Q}_n)$. Observe that
$Q_n\subset\bar{Q}_n$ or $\bar{Q}_n\subset Q_n$, hence
$P(Q_n\bigtriangleup\bar{Q}_n
)\leq|p-P(\bar{Q}_n)|$. By Lemma~\ref{lem3} and Corollary~\ref{coro1},
for any $\varepsilon >0$ and large $n$
\begin{eqnarray*}
\frac{\nu(S)}{p}P(\bar{Q}_n)&\leq& \frac{\nu(S)}{p}P\biggl(U\biggl(\frac{\nu
(S)}{p}\biggr)\{\mathbf{z}\dvtx q(\mathbf{z})\leq1+\varepsilon \}\biggr)\\
&\rightarrow& \nu\bigl(\{\mathbf{z}\dvtx q(\mathbf{z})\leq1+\varepsilon \}
\bigr)\\
&=&\nu\bigl(\bigl\{\mathbf{z}\dvtx q\bigl(\mathbf{z}(1+\varepsilon )^{1/(d+\alpha
)}\bigr)\leq1\bigr\}\bigr)\\
&=&\nu\bigl(\bigl\{(1+\varepsilon )^{-1/(d+\alpha )}\mathbf{z}\dvtx q(\mathbf
{z})\leq1\bigr\}\bigr)\\
&=&(1+\varepsilon )^{\alpha /(d+\alpha )}\nu(S).
\end{eqnarray*}
Thus, $\limsup_{n\rightarrow\infty }\frac{P(\bar{Q}_n)}{p}\leq
(1+\varepsilon )^{\alpha
/(2+\alpha )}$.\vadjust{\goodbreak}
Similarly, we have $\liminf_{n\rightarrow\infty }\frac{P(\bar
{Q}_n)}{p}\geq
(1-\varepsilon )^{\alpha /(2+\alpha )}$.
Hence,\break $\lim_{n\rightarrow\infty }\frac{P(\bar{Q}_n)}{p}=1$, that
is,\
$\lim_{n\rightarrow\infty }\frac{P(Q_n\bigtriangleup\bar{Q}_n)}{p}=0$.
In the same way, it follows from Lemmas~\ref{lem3} and~\ref{lem4} that
\begin{eqnarray*}
\frac{\nu(S)}{p}P(\bar{Q}_n\bigtriangleup\widetilde{Q}_n)
&\leq& \frac{\nu(S)}{p}P\biggl(U\biggl(\frac{\nu(S)}{p}\biggr)\{\mathbf
{z}\dvtx 1-\varepsilon \leq q(\mathbf{z}
)\leq1+\varepsilon \}\biggr)\\
&\rightarrow& \nu\bigl(\{\mathbf{z}\dvtx 1-\varepsilon \leq q(\mathbf
{z})\leq1+\varepsilon \}\bigr)\\
&=&\nu(S)\bigl((1+\varepsilon )^{\alpha /(d+\alpha )}-(1-\varepsilon
)^{\alpha /(d+\alpha )}\bigr).
\end{eqnarray*}
Hence,
$\lim_{n\rightarrow\infty }\frac{P(\bar{Q}_n\bigtriangleup
\widetilde{Q}_n)}{p}=0$.
\end{pf}
The following proposition shows uniform consistency of $\widehat{\psi
}_n$ and
might be of independent interest. There is an abundant literature on
density estimation for directional data. In particular, uniform
consistency of density estimators for directional data has been
established in \citet{BaiRaoZha88}. Here, however, the data do
not have a \textit{fixed} probability density on $\Theta$: the
density $\psi$ is defined via a limit relation. Hence, $\psi$ is only
an approximate model for the directional data. As a consequence, a more
general result is required.
\begin{prop}\label{prop2}
As $n\rightarrow\infty $,
\[
\sup_{\mathbf{w}\in\Theta} |\widehat{\psi}_n(\mathbf{w})-\psi
(\mathbf{w})
|\stackrel{\mathbb{P}}{\rightarrow}0.
\]
\end{prop}
\begin{pf}
It is easy to see that, for any $\eta>0$, there exists a function
\[
K^*=\sum_{j=1}^m \alpha _j 1_{[r_{j-1}, r_j)}
\]
with $1 \geq\alpha _1\geq\alpha _2\geq\cdots\geq\alpha _m\geq
0$ and
$0=r_0<r_1<\cdots<r_m=1$, such that
\[
\sup_{u\in[0,1]}|K(u)-K^*(u)|\leq\eta.
\]
Write $U_i=1-H(R_i)$, $i=1,\ldots,n$, and denote the corresponding order
statistics with $U_{i: n}$. Let $\widetilde{P}$ be the probability
measure on
$\Theta
\times(0,1)$ corresponding to $(\mathbf{W}_1, U_1)$ and let
$\widetilde{P}_n$ be the
empirical
measure of the $(\mathbf{W}_i, U_i)$ $i=1,\ldots, n$.
Define
\[
\psi_n^*(\mathbf{w})=\frac{c(h, K)}{k}\sum_{i=1}^n K^*\biggl(\frac
{1-\mathbf{w}^T\mathbf{W}_i}{h}\biggr)1_{[R_i>R_{n-k: n}]}
\]
and
\[
\psi_{n,j}^*(\mathbf{w})=\frac{nc(h, K)}{k}\widetilde
{P}_n\bigl(D_{\mathbf{w},j}\times
(0,U_{k: n}]\bigr)\vadjust{\goodbreak}
\]
with $D_{\mathbf{w},j}= \{\mathbf{v}\in\Theta\dvtx 1-hr_{j}<\mathbf{w}^T\mathbf{v}
\leq1-hr_{j-1} \}$.
Observe that $\psi_n^*(\mathbf{w})=\sum_{j=1}^m\alpha _j\psi
_{n,j}^*(\mathbf{w})$. Also write
\[
\psi_{n,j}(\mathbf{w})=\frac{nc(h, K)}{k}\widetilde{P}\bigl(D_{\mathbf{w},j}\times(0,U_{k: n}]\bigr).
\]
Let $\varepsilon >0$. It is sufficient to show that for large $n$
\begin{eqnarray}
\mathbb{P}\Biggl(\sup_{\mathbf{w}\in\Theta} \Biggl|\widehat{\psi
}_n(\mathbf{w})-\sum
_{j=1}^m\alpha _j\psi_{n,j}(\mathbf{w}) \Biggr|\geq2\varepsilon \Biggr)&\leq&
2\varepsilon \label{(p2.1)},\\
\mathbb{P}\Biggl(\sup_{\mathbf{w}\in\Theta} \Biggl|\sum_{j=1}^m\alpha_j\bigl(\psi
_{n,j}(\mathbf{w})-c(h, K)\Psi(D_{\mathbf{w},j})\bigr) \Biggr|\geq
2\varepsilon \Biggr)&\leq&\varepsilon, \label{(p2.2)}\\
\sup_{\mathbf{w}\in\Theta} \Biggl|c(h, K)\sum_{j=1}^m\alpha _j\Psi
(D_{\mathbf{w},j})-\psi(\mathbf{w}) \Biggr|&\leq&\varepsilon . \label{(p2.3)}
\end{eqnarray}
For $\mathbf{w}\in\Theta$ and $\delta \in(0,1)$, write
$\mathcal{C}_{\delta }=\{C_\mathbf{w}(a)\dvtx \mathbf{w}\in\Theta
,a\leq\delta
\}$. Note that, as $n\rightarrow\infty $,
\begin{equation}\label{(p2.4)}
\sup_{C\in\mathcal{C}_{1}, 0<s\leq2}\frac{1}{\lambda (C)}
\biggl|\frac{n}{k}\widetilde{P}\bigl(C\times(0, sk/n ]\bigr)-s\Psi(C) \biggr|\rightarrow
0.
\end{equation}
This readily follows from
\begin{eqnarray*}
\frac{n}{k}\widetilde{P}\bigl(C\times(0, sk/n ]\bigr)
&=&\frac{n}{k}\mathbb{P}\biggl(\mathbf{W}\in C, R\geq U\biggl(\frac{n}{sk}\biggr)\biggr)\\
&=&\frac{n}{k}\int_{U({n}/{(sk)})}^{\infty }\int_C\frac
{f(r\mathbf{w})}{r^{-d}V(r)}\,d\lambda (\mathbf{w})\,r^{-1}V(r)\,dr
\end{eqnarray*}
and (\ref{(4.1)}) and (\ref{(4.5)}).
Now we prove (\ref{(p2.1)}).
It is easy to show that
\[
c(h,K)=\biggl(\frac{2\pi^{{(d-1)}/{2}}}{\Gamma({(d-1)}/{2})}
\int_{1-h}^1K\biggl(\frac{1-t}{h}\biggr)(1-t^2)^{{(d-3)}/{2}}\,dt\biggr)^{-1}
\]
and hence
\begin{eqnarray}
\limsup_{h\downarrow0}c(h, K)\lambda (C_\mathbf{w}(h))<\infty .
\label{(p2.0)}
\end{eqnarray}
We have
\begin{eqnarray} \label{(p2.1.2)}
&&|\widehat{\psi}_n(\mathbf{w})-\psi_n^*(\mathbf{w}) |\nonumber \\
&&\qquad =\frac{c(h, K)}{k} \Biggl|\sum_{i=1}^n\biggl(K\biggl(\frac{1-\mathbf{w}^T\mathbf{W}
_i}{h}\biggr)-K^*\biggl(\frac{1-\mathbf{w}^T\mathbf
{W}_i}{h}\biggr)\biggr)1_{[R_i>R_{n-k: n}]} \Biggr|
\nonumber\\
&&\qquad \leq \frac{c(h, K)}{k}\sum_{i=1}^n\eta1_{[\mathbf{W}_i\in
C_\mathbf{w}(h), R_i>R_{n-k: n}]} \\
&&\qquad \leq \eta\frac{n c(h, K)}{k}\widetilde{P}\bigl(C_\mathbf{w}(h)\times
(0,U_{k: n}]\bigr) \nonumber\\
&&\qquad \quad {} +\eta\frac{n c(h, K)}{k} \bigl|(\widetilde{P}_n-\widetilde
{P})\bigl(C_\mathbf{w}(h)\times
(0,U_{k: n}]\bigr) \bigr|.\nonumber
\end{eqnarray}
By (\ref{(p2.4)}), for $\eta$ small enough the first term is
less than $\varepsilon $, with probability tending to one, uniformly in
$\mathbf{w}\in\Theta$. Also,
\begin{eqnarray} \label{(p2.1.1)}
&&\Biggl|\psi_n^*(\mathbf{w})-\sum_{j=1}^m\alpha _j\psi_{n,j}(\mathbf{w})
\Biggr|\nonumber\\
&&\qquad \leq \sum_{j=1}^m\alpha _j |\psi_{n,j}^*(\mathbf{w})-\psi
_{n,j}(\mathbf{w}) | \\
&&\qquad \leq \sum_{j=1}^m\alpha _j \frac{nc(h,K)}{k} \bigl|(\widetilde
{P}_n-\widetilde{P})\bigl(D_{\mathbf{w},j}\times(0,U_{k: n}]\bigr) \bigr|.\nonumber
\end{eqnarray}
From (\ref{(p2.1.1)}), (\ref{(p2.1.2)}) and (\ref{(p2.0)}), we see
that for a proof of
(\ref{(p2.1)}) it remains to show that
\[
\frac{n}{k\lambda (C_\mathbf{w}(h))}\sup_{\mathbf{w}\in\Theta
}\sup
_{0<a\leq1}
\bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C_\mathbf{w}(ah)\times(0,U_{k: n}]\bigr)
\bigr|\stackrel{\mathbb{P}}{\rightarrow}0.
\]
It can be shown that there exists a constant $c=c(d)$ and
finitely many $\mathbf{w}_l$, $l=1,\ldots,l_h$ such that $l_h=O(c(h, K))$
as $h\downarrow0$, and for every $\mathbf{w}\in\Theta$ and $0<a\leq1$
\[
C_\mathbf{w}(ah)\in C_{\mathbf{w}_l}(ch)\qquad \mbox{for some } l.
\]
Hence for $\varepsilon _1>0$,
\begin{eqnarray*}
&&\mathbb{P}\biggl(\frac{n}{k\lambda (C_\mathbf{w}(h))}\sup_{\mathbf{w}\in\Theta
}\sup_{0<a\leq1}
\bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C_\mathbf{w}(ah)\times(0,U_{k: n}]\bigr)
\bigr|\geq\varepsilon _1\biggr)\\
&&\qquad \leq\mathbb{P}\Bigl(\max_{1\leq l\leq l_h}\mathop{\sup_{C\subset
C_{\mathbf{w}_l}(ch)}}_{ C\in\mathcal{C}_h}\sup_{0<s\leq2}
\bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C\times(0,sk/n]\bigr) \bigr|\geq\varepsilon
_1k/n\lambda (C_\mathbf{w}(h))\Bigr)\\
&&\qquad \quad {}+\mathbb{P}(U_{k: n}>2k/n)\\
&&\qquad \leq\sum_{l=1}^{l_h}\mathbb{P}\Bigl(\mathop{\sup_{C\subset
C_{\mathbf{w}_l}(ch)}}_{ C\in\mathcal{C}_h}\sup_{0<s\leq2}
\bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C\times(0,sk/n]\bigr) \bigr|\geq\varepsilon
_1k/n\lambda (C_\mathbf{w}(h))\Bigr)\\
&&\qquad \quad {}+\mathbb{P}(U_{k: n}>2k/n).
\end{eqnarray*}
The latter probability tends to $0$, so it suffices to consider
the sum of the $l_h$ probabilities. Write $b=\varepsilon _1k\lambda
(C_\mathbf{w}(h))$. Fix $l$ and define
$N=n\widetilde{P}_n(C_{\mathbf{w}_l}(ch)\times(0,2k/n])$,
$\mu=n\widetilde{P}(C_{\mathbf{w}_l}(ch)\times(0,2k/n])$. Define the
conditional probability measure
$\widetilde{P}_c=\frac{n \widetilde{P}}{\mu}$ on $C_{\mathbf{w}_l}(ch)\times(0,2k/n]$
and let $\widetilde{P}_{c,r}$ be the corresponding empirical measure,
based on
$r$ observations. We have
\begin{eqnarray} \label{(p2.1.3)}
&&\mathbb{P}\Bigl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal{C}_h}\sup_{0<s\leq2}
n \bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C\times(0,sk/n]\bigr) \bigr|\geq b\Bigr)
\nonumber\\[-1pt]
&&\qquad \leq\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\Bigl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal{C}_h}\sup_{0<s\leq2}
n \bigl|(\widetilde{P}_n-\widetilde{P})\bigl(C\times(0,sk/n]\bigr) \bigr|\geq b
\big|N=r\Bigr)\nonumber\\[-1pt]
&&\qquad \quad\hphantom{\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}} {}\times\mathbb{P}(N=r) +\mathbb{P}(|N-\mu|\geq b/3) \nonumber\\[-1pt]
&&\qquad \leq\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\biggl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal{C}_h}\sup_{0<s\leq2}
n \biggl|\biggl(\widetilde{P}_n-\frac{N}{\mu}\widetilde{P}\biggr)\bigl(C\times(0,sk/n]\bigr)
\biggr|\nonumber \\[-1pt]
&&\hspace*{252pt}\geq\frac{b}{2}
\Big|N=r\biggr)\mathbb{P}(N=r)\nonumber \\[-1pt]
&&\qquad \quad {} +\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\biggl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal{C}_h}\sup_{0<s\leq2}
n \biggl|\frac{(N-\mu)}{\mu}\widetilde{P}\bigl(C\times(0,sk/n]\bigr) \biggr|\nonumber \\[-1pt]
&&\hspace*{255pt}\geq\frac{b}{2}
\Big|N=r\biggr)\mathbb{P}(N=r)\nonumber \\[-1pt]
&&\qquad \quad {}+\mathbb{P}(|N-\mu|\geq b/3) \\[-1pt]
&&\qquad \leq \sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\biggl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal{C}_h} \sup_{0<s\leq2}
r \bigl|(\widetilde{P}_{c,r}-\widetilde{P}_c)\bigl(C\times(0,sk/n]\bigr) \bigr|\geq
\frac{b}{2}\biggr)\nonumber\\[-1pt]
&&\qquad \quad\hphantom{\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}} {}\times\mathbb{P}
(N=r)\nonumber\\[-1pt]
&&\qquad \quad {} + \sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\biggl(|r-\mu|\geq\frac{b}{2}\biggr)\mathbb{P}(N=r)+\mathbb
{P}(|N-\mu|\geq b/3).\nonumber
\end{eqnarray}
Note that the first probability of the second sum in the
right
side of (\ref{(p2.1.3)}) is equal to~0.
From Bennett's inequality [cf.\ \citet{ShoWel86}, page~851],
it follows that for some constant $c_1$
\[
\mathbb{P}(|N-\mu|\geq b/3)\leq2\exp\biggl(-\varepsilon _1^2c_1\frac
{k}{c(h, K)}\biggr).\vadjust{\goodbreak}
\]
Hence, since $l_h=O(c(h, K))$,
\[
\sum_{l=1}^{l_h}\mathbb{P}(|N-\mu|\geq b/3)
=O\biggl(c(h, K)\exp\biggl(-\varepsilon _1^2c_1\frac{k}{c(h, K)}\biggr)\biggr)=o(1).
\]
To complete the proof of (\ref{(p2.1)}), we need to consider
the first sum in the right side of (\ref{(p2.1.3)}). For the
first probability in there, we use Corollary 2.9 in \citet
{Ale84}, a
good probability bound for empirical processes on VC classes.
We obtain as an upper bound
\[
16\exp\biggl(-\frac{b^2}{4r}\biggr).
\]
Using $r\leq\mu+b/3$, we find for some constant $c_2$
\begin{eqnarray*}
&&\sum_{l=1}^{l_h}\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}
\mathbb{P}\biggl(\mathop{\sup_{C\subset C_{\mathbf{w}_l}(ch)}}_{ C\in
\mathcal
{C}_h}\sup_{0<s\leq2}
r \bigl|(\widetilde{P}_{c,r}-\widetilde{P}_c)\bigl(C\times(0,sk/n]\bigr) \bigr|\geq
\frac{b}{2}\biggr)\\
&&\hphantom{\sum_{l=1}^{l_h}\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor\mu+b/3\rfloor}}
{}\times\mathbb{P}(N=r)\\
&&\qquad \leq16 \sum_{l=1}^{l_h}\sum_{r=\lceil\mu-b/3\rceil}^{r=\lfloor
\mu+b/3\rfloor}
\exp\biggl(-\varepsilon _1^2c_2\frac{k}{c(h, K)}\biggr)\mathbb{P}(N=r)\\
&&\qquad \leq16 \sum_{l=1}^{l_h} \exp\biggl(-\varepsilon _1^2c_2\frac{k}{c(h, K)}\biggr)\\
&&\qquad =o(1).
\end{eqnarray*}
Next, we show (\ref{(p2.2)}). From (\ref{(p2.0)}) and
(\ref{(p2.4)}), we obtain for $\varepsilon _2>0$ small enough,
\begin{eqnarray*}
&&\sup_{\mathbf{w}\in\Theta} \Biggl|\sum_{j=1}^m\alpha _j\bigl(\psi
_{n,j}(\mathbf{w})-c(h, K)\Psi(D_{\mathbf{w},j})\bigr) \Biggr|\\
&&\qquad =\sup_{\mathbf{w}\in\Theta} \Biggl|\sum_{j=1}^m\alpha _j c(h,
K)\bigl(n/k\widetilde{P}
\bigl(D_{\mathbf{w},j}\times(0,U_{k: n}]\bigr)-\Psi(D_{\mathbf{w},j})\bigr) \Biggr|\\
&&\qquad \leq \varepsilon _2\sum_{j=1}^m\alpha _j c(h, K)\lambda
(C_\mathbf{w}(h))+
\sup_{\mathbf{w}\in\Theta} \Biggl|\sum_{j=1}^m\alpha _jc(h,
K)(nU_{k: n}/k-1)\Psi(D_{\mathbf{w},j}) \Biggr|\\
&&\qquad \leq\varepsilon + \biggl|\frac{n}{k}U_{k: n}-1 \biggr|\sum_{j=1}^m\alpha
_jc(h, K)\lambda
(C_\mathbf{w}(h))\sup_{\mathbf{w}\in\Theta}\psi(\mathbf{w})<2\varepsilon
\end{eqnarray*}
with probability tending to one.
It remains to prove (\ref{(p2.3)}). It is readily seen that
$\int_{C_\mathbf{w}(h)}K^*(\frac{1-\mathbf{w}^T\mathbf
{v}}{h})\,d\lambda (\mathbf{v}
)=\sum_{j=1}^m\alpha _j\lambda (D_{\mathbf{w},j})$.
Hence, for $\varepsilon _3>0$ small enough
\begin{eqnarray*}
&&\sup_{\mathbf{w}\in\Theta} \Biggl|c(h, K)\sum_{j=1}^m\alpha _j\Psi
(D_{\mathbf{w},j})-\psi(\mathbf{w}) \Biggr|\\[-1pt]
&&\qquad \leq \sup_{\mathbf{w}\in\Theta}\psi(\mathbf{w}) \Biggl|c(h, K)\sum
_{j=1}^m\alpha _j\lambda (D_{\mathbf{w},j})-1 \Biggr|+\varepsilon _3 c(h,
K)\sum_{j=1}^m\alpha
_j\lambda (D_{\mathbf{w},j})\\[-1pt]
&&\qquad \leq \sup_{\mathbf{w}\in\Theta}\psi(\mathbf{w}) \biggl|\frac{\int
_{C_\mathbf{w}(h)}K^*({(1-\mathbf{w}^T\mathbf{v})}/{h})\,d\lambda
(\mathbf{v})}
{\int_{C_\mathbf{w}(h)}K({(1-\mathbf{w}^T\mathbf
{v})}/{h})\,d\lambda (\mathbf{v})}-1
\biggr|\\[-1pt]
&&\qquad \quad {}+\varepsilon _3 c(h, K)\lambda (C_{\mathbf{w}}(h))\sum
_{j=1}^m\alpha _j\\[-1pt]
&&\qquad \leq \eta c(h, K)\lambda (C_{\mathbf{w}}(h))\sup_{\mathbf{w}\in
\Theta
}\psi(\mathbf{w})+\varepsilon _3 c(h, K)\lambda (C_{\mathbf{w}}(h))\sum_{j=1}^m\alpha
_j\\[-1pt]
&&\qquad \leq \varepsilon .
\end{eqnarray*}
\upqed
\end{pf}
From Proposition~\ref{prop2} and the consistency of $\widehat\alpha$, we obtain
immediately, as $n\rightarrow\infty $,
\[
\widehat{\nu(S)}\stackrel{\mathbb{P}}{\rightarrow}\nu(S)
\]
and, for $\varepsilon>0$,
\begin{equation}\label{inclu}
\mathbb{P}\bigl( (1+\varepsilon )S\subset\widehat{S}\subset
(1-\varepsilon )S\bigr)\to1.
\end{equation}
\begin{prop}\label{prop3}
As $n\rightarrow\infty $,
\[
\frac{P(\widetilde{Q}_n\triangle\widehat{Q}_n)}{p} \stackrel{\mathbb{P}}{\rightarrow} 0.
\]
\end{prop}
\begin{pf}
Note that as $n\rightarrow\infty $, we have
\begin{eqnarray*}
\widehat{U}\biggl(\frac{n}{k}\biggr)\Big/U\biggl(\frac{n}{k}\biggr)&\stackrel{\mathbb{P}}{\rightarrow}&1,\\[-1pt]
(\widehat{\nu(S)})^{{1}/{\widehat{\alpha
}}}&\stackrel{\mathbb{P}}{\rightarrow}&
(\nu(S))^{1/\alpha },
\\[-1pt]
\biggl(\frac{k}{np}\biggr)^{1/\widehat{\alpha }-1/\alpha }
&=&\exp\biggl(\frac{\sqrt{k}(\alpha -\widehat{\alpha })}{\widehat
{\alpha }\alpha }
\biggl(\frac{\log k}{\sqrt{k}}-\frac{\log(np)}{\sqrt{k}}\biggr)\biggr)
\stackrel{\mathbb{P}}{\rightarrow}1.
\end{eqnarray*}
Combining these three limit relations, we obtain
\[
\frac{\widehat{U}({n}/{k})({k\widehat{\nu
(S)}}/{(np)})^{{1}/{\widehat{\alpha
}}}}{U({n}/{k})({k\nu(S)}/{(np)})^{{1}/{\alpha }}}
\stackrel{\mathbb{P}}{\rightarrow} 1.\vadjust{\goodbreak}
\]
This and (\ref{inclu}) yields
that with probability tending to one, as $n \to\infty $,
\[
(1+\varepsilon )^2\widetilde{Q}_n\subset\widehat{Q}_n\subset
(1-\varepsilon )^2\widetilde{Q}_n.
\]
Then,
\[
\frac{P(\widetilde{Q}_n\triangle\widehat{Q}_n)}{p}
\leq\frac{1}{p}P\biggl(U\biggl(\frac{n}{k}\biggr)\biggl(\frac{k\nu(S)}{np}\biggr)^{
{1}/{\alpha
}}\bigl((1-\varepsilon )^2S\setminus(1+\varepsilon )^2S\bigr)\biggr),
\]
and, by (\ref{(4.32)}), the latter expression tends to
\begin{eqnarray*}
&&\nu\bigl((1-\varepsilon )^2S \setminus(1+\varepsilon
)^2S\bigr)/\nu(S)\\
&&\qquad =\nu\bigl((1-\varepsilon )^2S\bigr)/\nu(S) -\nu\bigl((1+\varepsilon )^2S\bigr)/\nu
(S)\\
&&\qquad =(1-\varepsilon )^{-2\alpha }-(1+\varepsilon )^{-2\alpha },
\end{eqnarray*}
which in
turn tends to 0, as $\varepsilon \downarrow0$.
\end{pf}
\begin{pf*}{Proof of Theorem~\ref{thm1}}
The result follows from
Propositions~\ref{prop1} and~\ref{prop3}.
\end{pf*}
\section*{Acknowledgments}
We thank two referees for many insightful comments that led to an
improved version of the paper. We are grateful to Kees Koedijk, Roger
Laeven, Ronald Mahieu and Chen Zhou for discussions of the financial
application.
|
2,877,628,090,209 | arxiv | \section{Introduction}
In non-critical strings of the type discussed in~\cite{aben},
one has a conformal field theory formulated in general in a
non-critical number of spatial dimensions with a central charge deficit $Q$.
This might either assume
discrete values, as in the minimal models discussed in~\cite{aben}, or
possibly vary continuously, as in models motivated by brane
world collisions~\cite{brany}. In the latter case, the central charge
of the corresponding world-sheet $\sigma$ model describing string excitations
on the brane and in the bulk space is proportional to some power of the
relative velocity of the moving models (assuming that the collisions are adiabatic, so that
perturbative string theory applies).
In general, non-equilibrium situations in string cosmology, such as those that
may well have characterized the early Universe, can be described~\cite{brany}
at large times long after the initial cosmic catastrophe that resulted in
the departure from equilibrium, within the framework of
Liouville strings~\cite{ddk}. The latter are strings described by world-sheet
$\sigma$-models propagating in non-conformal backgrounds of, say, graviton and dilaton fields,
that are dressed by an extra world-sheet field, the Liouville mode $\phi$,
in such a way that conformal invariance is restored. This construction enables
strings to propagate in a non-critical number of space-time dimensions.
It was argued in~\cite{brany} that in some \emph{supercitical}
models, i.e., world-sheet $\sigma$models with a
central charge \emph{surplus}: $-Q^2 \equiv (C-c^*)/3 > 0$
where $c^*$ is the critical central charge of the conformal theory.
the extra Liouville dimension, i.e., the zero mode of the world-sheetLiouville field $\phi$,
can be identified with the \emph{target time}. This identification
follows from dynamical arguments on the minimization of the
effective potential of the target-space-time effective
field theory, and is exemplified by, e.g., black-hole configurations.
Generic analyses~\cite{diamand} of cosmological models within this general framework
of Liouville cosmologies, which have been termed $Q$-cosmologies, reveals that the
asymptotic theory at large times corresponds to the conformal model of~\cite{aben}, with
a central charge given by the asymptotic constant value $Q_0$ of the central-charge deficit.
It should be noted that, in general, the central charge
of the Liouville cosmology is not a constant, but a time-dependent function,
$Q(t)$, whose form is found by solving the appropriate generalized conformal-invariance
conditions that describe the restoration of conformal invariance by the Liouville mode.
The cosmology of~\cite{aben} corresponds in target space to a linearly-expanding Universe.
However, the question arises how the geometry of the Universe evolves with time and,
in particular, whether and how this Universe exits from this expanding phase and reaches
an Minkowski space-time. The latter is the only realistic candidate for
a \emph{equilibrium} situation which may be reached
asymptotically in target time.
It was attempted in~\cite{hall} to visualize this evolving string Universe as a
world-sheet quantum Hall system, with the cosmologies of~\cite{aben}, that correspond to
various \emph{discrete} values of the central-charge deficit $Q$, being the
analogues of the conductivity plateaux of the Hall system. Transitions
between them, from one value $Q_1$ to another value $Q_2$ in, say, the discrete
series found in minimal models, would correspond to a non-conformal theory dressed
by the world-sheet Liouville mode.
According to~\cite{aben} therefore, the Universe would undergo a series of phase transitions
before reaching asymptotically the equilibrium Minkowski space-time that
corresponds to the $Q=0$ critical theory.
The question that then arises is how to describe such phase transitions
non-perturbatively on the world-sheet.
In ordinary field theory, the approach to a phase transition
is described by means of a renormalization-group flow.
An alterative to the conventional Wilsonian flow method was presented in~\cite{polonyi},
in which a mass parameter is relaxed from some high value, where the quantum corrections
are well controlled, down to small values. This procedure was applied initially
to $\phi^4$ field theory and then to QED and some $2+1$-dimensional models.
More recently, we applied this approach to string theory, imposing a fixed ultraviolet cutoff
$\Lambda$ on the world sheet, and using the Regge slope $\alpha^{'}$
as the control parameter~\cite{alexandre}.
In this way, we found a novel fixed point of the world-sheet
$\sigma$-model describing the bosonic string in cosmological graviton and dilaton
backgrounds, which is non-perturbative in $\alpha^{'}$
and describes a novel time-dependent string cosmology.
This novel fixed point is an infrared fixed point of the Wilsonian renormalization group,
and a marginal configuration of the alternative flow.
These theories remain conformal,
and one of the non-trivial tasks in~\cite{alexandre} was to argue that the new fixed
point respects world-sheet conformal invariance.
In this paper we
extend these results to Liouville theory, using as the control parameter of the
novel renormalization flow the central-charge deficit $Q$. It is known from the
original work on linearly-expanding cosmologies in~\cite{aben} that the
central charge induces mass shifts $\propto Q$ in the spectrum of
target-space excitations: there are tachyonic mass shifts, $\Delta m^2 = -|Q^2| < 0$ for
bosons when $-Q^2>0$~\footnote{We use units where $\alpha' = 1$. We note that fermion masses
do not acquire a $Q^2$ correction,
as discussed in \cite{aben}.}. In the case of initially massless states, this tachyonic shift
would imply tachyonic excitations in the spectrum, and hence instabilities.
On the other hand, its role in generating a mass gap makes $Q$ a suitable
candidate for controlling the quantum corrections. By treating it as variable,
we can discuss transitions among various linearly-expanding cosmologies,
and eventually the transition to Minkowski space as a fixed point of the
novel renormalization flow.
\section{Non-Perturbative Flows with Respect to the Central-Charge Deficit}
The bare action for the two-dimensional world-sheet $\sigma$ model for the
bosonic string is
\begin{equation}\label{liouvsmodel}
S=\int d^2\xi\left\{\frac{Q^2}{2}\partial_a\phi\partial^a\phi+\beta_Q R^{(2)}\phi+\mu^2P_B(\phi)e^\phi\right\},
\end{equation}
where $\beta_Q$ is a function of $Q^2=\frac{c^*-C}{3}$, $c^*=25$,
and $P_B(\phi)$ is a $Q^2$-independent
bare polynomial in the Liouville field $\phi$.
The effective action $\Gamma$, which is the generating functional for the
proper graphs, is defined in the Appendix. It describes the
corresponding quantum theory, and is labelled by the parameter $Q^2$.
The target-space Liouville field is {\it space-like} in when the
corresponding conformal theory is {\it subcritical}, i.e., is characterised by a central charge
{\it deficit}~\cite{ddk}, i.e., $Q^2 > 0$. On the other hand,
the target-space Liouville field is {\it time-like} in when the corresponding world-sheet theory is
{\it supercritical}, i.e., there is a central-charge {\it surplus}~\cite{aben}: $Q^2 < 0$.
It is the latter case that has been
employed previously~\cite{emn,diamand} to describe (non-equilibrium) string cosmologies,
which relax to equilibrium (critical-string) configurations asymptotically in target time,
the latter being identified with the zer mode of the
time-like Liouville field. In these cosmologies the initial central charge surplus may
be provided by some catastrophic cosmic event, e.g., the collision of brane worlds in the modern
version of string theory~\cite{diamand}.
From a world-sheet field-theory point of view, the subcritical string with
central charge $C < 1$ constitutes a
well-behaved theory, where functional computations can be performed, and
the critical (scaling) exponents of the theory are {\it real}~\cite{ddk}.
For the range $1 < C < 25$ of central charges there are {\it complex} scaling exponents, and the Liouville
theory is at {\it strong coupling}, which is not
well understood at present.
On the other hand, the supercritical Liouville theory $C > 25$, is characterised by a ghost-like field
$\phi$, since the kinetic term of the Liouville mode comes with the `wrong' (negative in our conventions) sign (c.f. (\ref{liouvsmodel})). However, in ths theory the critical exponents are also real, and in fact this regime can be thought of as the
analytical continuation of the region where $C < 1$, with the replacement $Q \to iQ$, with the Liouville scaling exponents $\alpha$ also undergoing a similar Wick rotation: $\alpha \to i\alpha$.
In this paper we shall present a novel way of quantising the Liouville theory, adapting a method developed previously in \cite{polonyi} for ordinary field theories. There, one identifies a
parameter (control parameter) in the theory, whose changes are governed by certain flow equations, which may be constructed by following standard (non-perturbative) functional methods. The resulting
flow describes the quantum-corrected behaviour of the theory in a non-perturbative way.
The main idea of this paper is to use the central charge deficit $Q^2$ of the
Liouville theory as an appropriate control parameter. We formulate the
flow equations first in the subcritical
case, which is well defined as a field theory, and then we continue analytically to the supercritical string case with $Q^2 < 0$.
We start our analysis at $Q^2>>1$, where
the theory is classical, since the bare Lagrangian is dominated by the kinetic term and therefore describes a free theory.
The decrease of $Q^2$ then induces the appearance of quantum fluctuations,
leading to the dressed theory. It is shown in the Appendix that it is possible to derive
an exact evolution equation for $\Gamma$ with $Q^2$, which is
\begin{equation}\label{evolG}
\dot\Gamma=\int d^2\xi\left\{\frac{1}{2}\partial_a\phi\partial^a\phi+\dot\beta_Q R^{(2)}\phi\right\}
+\frac{1}{2}\mbox{Tr}\left\{\frac{\partial}{\partial\xi_a}\frac{\partial}{\partial\zeta^a}
\left(\frac{\delta^2\Gamma}{\delta\phi_\xi\delta\phi_\zeta}\right)^{-1}\right\},
\end{equation}
where a dot denotes a derivative with respect to $Q^2$. In eq.(\ref{evolG}),
quantum fluctuations are contained in the trace on the right-hand side.
This trace needs a regulator, for which we use a fixed world-sheet cutoff $\Lambda$.
Any similarity of our evolution equation (\ref{evolG}) to the exact Wilsonian renormalization
equation is only apparent, since here we consider a {\it fixed} cutoff, and look at the
flows in $Q^2$. We emphasize that eq.(\ref{evolG}) is exact and corresponds to the
resummation of all loops, even though superficially it has the structure of a one-loop
correction. The reason for this is the fact that the trace
contains the dressed parameters, and not the bare ones: thus eq.(\ref{evolG}) is a self-consistent
partial differential equation for $\Gamma$, which describes the full quantum theory.
In order to obtain physical information from the evolution equation (\ref{evolG}), one has to assume a functional dependence of the effective action $\Gamma$.
Therefore, we consider the following Ansatz:
\begin{equation}\label{ansatz}
\Gamma=\int d^2\xi\left\{ \frac{Z_Q}{2}\partial_a\phi\partial^a\phi+\beta_Q R^{(2)}\phi +
\mu^2P_Q(\phi)e^\phi\right\},
\end{equation}
where $Z$ is a $Q^2$-dependent wave-function renormalization, and $P_Q(\phi)$ is a
$Q^2$-dependent function of $\phi$. The form (\ref{ansatz}) is dictated by conformal invariance.
Note that we do not expect quantum corrections for $\beta_Q$, since no term linear in $\phi$ is generated by the trace in eq.(\ref{evolG}).
It is shown in the Appendix that the Ansatz (\ref{ansatz}), inserted into eq.(\ref{evolG}),
leads to the following evolution equations:
\begin{eqnarray}\label{evoleqs}
\dot Z_Q&=&1\\
\dot P_Q(\phi)&=&-\frac{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}{8\pi Z_Q^2}
\ln\left(1+\frac{Z_Qe^{-\phi}\Lambda^2/\mu^2}{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}\right),
\nonumber
\end{eqnarray}
where a prime denotes a derivative with respect to $\phi$.
We observe that $Z$ remains classical and does not receive any quantum corrections.
Since the constant of integration in the evolution equation for $Z(Q)$
is absorbed into the critical value of the central charge, we find simply that $Z_Q=Q^2$
and the resulting evolution equation for $P$ is
\begin{equation}\label{evolP}
\dot P_Q(\phi)=-\frac{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}{8\pi Q^4}
\ln\left(1+\frac{Q^2e^{-\phi}\Lambda^2/\mu^2}{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}\right).
\end{equation}
We observe that there is only one exactly-marginal configuration, namely one with
$\dot P=0$, which must have $P+2P^{'}+P^{''}=0$. The solution for $P$ is then
\begin{equation}
P(\phi)=(C_1+C_2\phi)e^{-\phi},
\end{equation}
where $C_1,C_2$ are $Q^2$-independent constants. This solution corresponds to a linear potential
\begin{equation}\label{potfp}
\mu^2P(\phi)e^\phi=\mu^2(C_1+C_2\phi),
\end{equation}
which could have been expected, since this form does not generate quantum fluctuations,
and therefore should not depend on $Q^2$.
\section{Solution in the Case $P_B(\phi)=1$}
In the case where the bare potential term is $\mu^2 e^\phi$, it is known that the
effective potential is of the form $\mu^2_R\exp(g_R\phi)$, where $\mu^2_R$ and $g_R$
are renormalized parameters~\cite{Jackiw}. We indeed find a solution of (\ref{evolP})
if we consider the following Ansatz for the effective Liouville-mode potential $V(\phi)$:
\begin{equation}\label{ansatzP}
V(\phi) = \mu^2 P_Q(\phi)e^\phi, \qquad
P_Q(\phi)=\eta_Q\exp(\varepsilon_Q\phi),
\end{equation}
where $\eta_Q$ and $\varepsilon_Q$ are functions of $Q^2$. Since the limit $Q^2\to\infty$
corresponds to the classical theory, the corresponding limits for these functions are
$\eta_\infty=1$ and $\varepsilon_\infty=0$, which we use
as initial conditions when we integrate their evolution equations.
In order to check that the Ansatz (\ref{ansatzP}) is indeed correct, we consider
separately the two cases of large $Q^2 \gg 1$ and $Q^2\to 0$.
\subsection{Large $Q^2$}
If we insert the Ansatz (\ref{ansatzP}) into eq.(\ref{evolP}), we obtain
\begin{equation}\label{expLambda}
\dot\eta_Q+\eta_Q\dot\varepsilon_Q\phi=\frac{\eta_Q(1+\varepsilon_Q)^2}{8\pi Q^4}\left\{
-\ln\left(\frac{Q^2\Lambda^2}{\mu^2\eta_Q(1+\varepsilon_Q)^2}\right)+(1+\varepsilon_Q)\phi
+{\cal O}\left(\frac{\mu^2}{Q^2\Lambda^2}\right)\right\},
\end{equation}
where we need the condition $Q^2\Lambda^2>>\mu^2$ for the Ansatz (\ref{ansatzP}) to be
consistent. Indeed, after the expansion in $\mu^2/(Q^2\Lambda^2)$, one is left with a constant
and a term linear in $\phi$, which can then be identified with the left-hand side of
eq.(\ref{expLambda}), leading to
\begin{eqnarray}\label{epsilonqeq}
\dot\varepsilon_Q&=&\frac{(1+\varepsilon_Q)^3}{8\pi Q^4}\nonumber\\
\dot\eta_Q&=&\frac{\eta_Q(1+\varepsilon_Q)^2}{8\pi Q^4}
\left\{-\ln\left(\frac{Q^2\Lambda^2}{\mu^2(1+\varepsilon_Q)^2}\right)
+\ln(\eta_Q)\right\}.
\end{eqnarray}
The evolution equation for $\varepsilon_Q$ can easily be solved to yield
\begin{equation}\label{solepsilon}
1+\varepsilon_Q=\sqrt{\frac{4\pi Q^2}{4\pi Q^2+1}},
\end{equation}
and we stress again that this solution is not the result of a loop expansion, but is exact in the
framework of the Ansatz (\ref{ansatz}).
The solution (\ref{solepsilon}) leads to the following equation for $\eta_Q$:
\begin{equation}\label{evoleta}
\frac{\dot\eta_Q}{\eta_Q}=\frac{-1}{2Q^2(4\pi Q^2+1)}\left\{\ln\left(\frac{\Lambda^2}{\mu^2}\right)
+\ln\left(Q^2+\frac{1}{4\pi}\right)-\ln(\eta_Q)\right\},
\end{equation}
for which one can find an approximate solution if $Q^2>>1$, where $\eta_Q\simeq 1$.
We have then, for a {\it fixed} cutoff $\Lambda$, and keeping the dominant contributions,
\begin{equation}
\dot\eta_Q\simeq -\frac{\ln(Q^2)}{8\pi Q^4},
\end{equation}
which leads to the following dominant behaviour
\begin{equation}
\eta_Q\simeq 1+\frac{\ln(Q^2)}{8\pi Q^2}.
\end{equation}
In the limit where $Q^2>>1$ and from eq.(\ref{solepsilon}), we also have
$\varepsilon_Q\simeq -1/(8\pi Q^2)$, so that
the effective potential is finally
\begin{equation}\label{pqfinal}
V(\phi) = \mu^2P_Q(\phi)e^\phi\simeq\mu^2\left(1+\frac{\ln(Q^2)}{8\pi Q^2}\right)
\exp\left\{\left(1-\frac{1}{8\pi Q^2}\right)\phi\right\}~.
\end{equation}
\subsection{Limit $Q^2\to 0$}
In the limit where $Q^2\to 0$, the expansion (\ref{expLambda}) is not valid any more, and one
has to start from the original equation (\ref{evolP}). An expansion in $Q^2$ for a {\it fixed}
cutoff $\Lambda$ then gives
\begin{equation}\label{expQto0}
\dot\eta_Q+\eta_Q\dot\varepsilon_Q\phi=-\frac{\Lambda^2/\mu^2}{8\pi Q^2}\exp\left\{-(1+
\varepsilon_Q)\phi\right\} +{\cal O}(1).
\end{equation}
For the ansatz (\ref{ansatzP}) to be consistent, we consider an expansion in $\phi$ of the previous
equation, and identify the powers of $\phi$ to obtain
\begin{eqnarray}\label{epsilonqeq0}
\dot\varepsilon_Q&=&\frac{\Lambda^2/\mu^2}{8\pi Q^2}\frac{1+\varepsilon_Q}{\eta_Q}\nonumber\\
\dot\eta_Q&=&-\frac{\Lambda^2/\mu^2}{8\pi Q^2}.
\end{eqnarray}
These equations can easily be integrated to give
\begin{eqnarray}\label{limitpotqzero2}
1+\varepsilon_Q&\simeq &\left|\ln(Q^2)\right|^{-1}\nonumber\\
\eta_Q&\simeq &\frac{\Lambda^2/\mu^2}{8\pi}\left|\ln(Q^2)\right|,
\end{eqnarray}
where we have kept only the contributions that are dominant in $Q^2$.
Note that $1+\varepsilon_Q\to 0$, which is consistent with the expansion of the exponential function
appearing in eq.(\ref{expQto0}). Finally, the effective potential behaves as
\begin{equation}\label{limitpotqzero}
V(\phi) = \mu^2P_Q(\phi)e^\phi\simeq\frac{\Lambda^2}{8\pi}\left|\ln(Q^2)\right|
\exp\left\{\frac{\phi}{\left|\ln(Q^2)\right|}\right\}
\simeq\frac{\Lambda^2}{8\pi}\left|\ln(Q^2)\right|,
\end{equation}
and therefore goes to a (divergent) constant when $Q^2\to 0$. As a result, this limit consists
of a trivial theory, where the field $\phi$ neither propagates nor interacts. In this limit the
quantum fluctuations, from which $\varepsilon_Q$ is generated, are strong enough to cancel the
classical potential. This becomes visible in the present scheme because it is
non-perturbative.
\section{Conformal Invariance}
One of the most important properties of the Liouville field $\phi$ is the restoration of the conformal invariance of world-sheet vertex operators after Liouville dressing~\cite{ddk}, such that the Liouville-dressed world-sheet
theory, incorporating the extra dynamics of the Liouville mode $\phi$, is conformally invariant.
Before commencing our discussion, we recall that
it is customary~\cite{ddk} to renormalize the Liouville field so that it has a canonically-normalized
kinetic term:
\begin{eqnarray}
\phi \longrightarrow \hat\phi \equiv |Q| \phi .
\label{norm}
\end{eqnarray}
For a world-sheet ($\Sigma$) vertex operator $V_i$ that deforms a fixed-point theory with action $S^*$:
\begin{eqnarray}
S_{\rm deform} = S^* + g^i\int_\Sigma V_i ,
\label{deformed}
\end{eqnarray}
the Liouville-dressing procedure~\cite{ddk} is defined by coupling the Liouville mode $\phi$, with action $S \equiv S_L$ (\ref{liouvsmodel}), to (\ref{deformed}) as follows:
\begin{eqnarray} \label{liouvilledressing}
S_{\rm deform,Liouville} = S^* + S_L + g_i \int_\Sigma e^{\alpha_i\hat\phi} V_i ,
\end{eqnarray}
where we have used the canonically-normalized field $\hat\phi$ (\ref{norm}).
The Liouville anomalous dimension terms $\alpha_i$ are such that, if the
deformed subcritical theory has central-charge deficit $Q^2 >0 $, then the dressed deformation in
(\ref{liouvilledressing}) $e^{\alpha_i \hat\phi} V_i $
is conformally invariant, provided that,
\begin{eqnarray}
\alpha_i \left(\alpha_i + Q \right) = -(2 - \Delta_i) ~, \qquad Q^2 > 0 ~({\rm subcritical~strings}) ,
\label{cicond}
\end{eqnarray}
where $\Delta_i$ is the conformal (scaling) dimension of the operator $V_i$, and thus $\Delta_i -2$ is the scaling dimension. The relative signs are appropriate for the subcritical string case $Q^2 > 0$ of interest to us in this section, and are such that the Liouville dimension $\alpha_i$ and $Q$ are {\it real}.
The presence of the $Q$ term arises because of the appearance of the central charge deficit $Q$ in the world-sheet curvature term of the perturbative Liouville action~\cite{ddk}.
In the model at hand, the only deformation we considered explicitly was that implied by the identity operator on the world-sheet, namely the two-dimensional cosmological constant, which leads, in the quantum theory, to the effective Liouville potential term (\ref{ansatzP}). This corresponds to the case
with $\Delta_i =0$ in (\ref{cicond}). Moreover, in our (non-perturbative) quantum theory, the r\^ole of
the Liouville anomalous dimension is played by $(1 + \varepsilon _Q)/Q$, where the numerator is the exponent in (\ref{ansatzP}), whilst the r\^ole of the central charge deficit $Q$ in (\ref{cicond}), namely
the coefficient of the world-sheet curvature term in the normalized Liouville mode case $\hat\phi$,
is provided by the function $\beta_Q/Q$. Thus conformal invariance should be guaranteed provided
that the following relation holds:
\begin{eqnarray}
(1 + \varepsilon_Q) \left(1 + \varepsilon_Q + \beta_Q \right) = -2Q^2 \quad ~\Longrightarrow ~ \quad \beta_Q = -1 - \varepsilon_Q -\frac{2Q^2}{1 + \varepsilon _Q} .
\label{betaqepsilon}
\end{eqnarray}
As discussed in the Appendix and in previous sections, our quantization procedure determines
$\varepsilon_Q$ as a function of $Q$, so as to satisfy the appropriate flow equations
(\ref{epsilonqeq}) (and (\ref{epsilonqeq0}) for the $Q^2 \to 0^+$ case), assuming a specific form of
the function $Z=Q^2$, which receives no quantum corrections. Moreover, as we have seen, in our approach the function $\beta_Q$ (which is also not renormalized) is left undetermined.
Following the above discussion (c.f. (\ref{betaqepsilon})), the requirement of conformal invariance provides an extra constraint that determines the function
$\beta_Q$ in terms of $\varepsilon_Q$, with $Z=Q^2$.
It is worth checking the consistency of this approach in the conformal limit $Q^2 \to 0^+$, where one expects the Liouville theory to decouple.
Indeed, in such a limit, the expression for $1 + \varepsilon_Q$ is provided by
(\ref{limitpotqzero2}). From (\ref{betaqepsilon}), then, we derive to leading order as $Q^2 \to 0^+$:
\begin{eqnarray}\label{betaqeps}
\beta_Q \simeq -1 - \varepsilon_Q \simeq -\frac{1}{|\ln(Q^2)|} \to 0^- ~.
\end{eqnarray}
which is consistent with the decoupling of the Liouville mode in this limit, since each of the three terms
in the world-sheet action (\ref{liouvsmodel}) either vanishes $(Z, \beta_Q)$ or becomes an irrelevant (Liouville-independent) constant (as is the case with the two-dimensional cosmological constant term).
In a similar spirit, the limit $Q^2 \gg 1$ can also be studied analytically. To this end, we first notice that
the relation (\ref{betaqepsilon}) is generic and applies to all ranges of $Q^2$.
In the large-$Q^2$ case, $\varepsilon_Q \simeq - 1 / 8\pi Q^2$, and
\begin{eqnarray}\label{final2}
\beta_Q \simeq -2Q^2+ {\cal O}(1) < 0, \quad Q^2 \to +\infty .
\end{eqnarray}
We now remark that the central-charge term is not supposed to change sign during its
flow~\cite{aben,ddk}, i.e., a sub(super)critical theory should remain sub(super)critical until its reaches
an equilibrium point. From
(\ref{betaqeps}), (\ref{final2}) we observe that this expectation is compatible with the above
analysis, as in both limits $\beta_Q < 0$.
\section{Case with $Q^2 < 0$: Intepretation of the Liouville Mode as Target Time}
\label{sec:time}
As mentioned above, the region of central charges for which $Q^2 < 0$
can be treated by analytic continuation of the $C < 1$ case,
where formally $Q \to iQ$ and the Liouville scaling dimensions $\alpha \to i\alpha$. In this case, the exponents of the Liouville effective potential terms (\ref{ansatzP}), where - as we have discussed in the previous section- $1 + \varepsilon_Q$ plays the r\^ole of a Liouville scaling dimension
for the identity operator on the world-sheet, remain {\it real}.
From a target-space-time viewpoint, in this r{\' e}gime the
Liouville-mode is time-like, and thus
its world-sheet zero mode can be interpreted as the target time~\cite{emn,diamand}.
In this case, the effective potential term in the Liouville action corresponds in general to a
cosmological tachyonic-field instability. However, as we have seen in (\ref{limitpotqzero}),
in the limit $|Q^2| \to 0^+$ the effective potential term becomes a constant independent of the
Liouville field, so the instability disappears and the target-space theory is stabilized.
The remaining part of the Section addresses some subtleties in these arguments, that arise
because the target time is actually identified~\cite{emn} (up to a sign) with a
renormalized Liouville mode $\equiv |Q| \phi$, and this renormalization is singular in the limit $|Q^2| \to 0^+$.
As already mentioned, it is customary~\cite{ddk} to renormalize the Liouville field so that it has a canonically normalized
kinetic term.
It is in the normalized form $\hat\phi$ (\ref{norm}) that the properties of the Liouville mode as a field
restoring conformal symmetry in non-critical world-sheet $\sigma$-model theories are best
studied~\cite{ddk,aben}.
If we had used this normalization from the beginning, the only term in the two-dimensional
effective action depending explicitly on the control parameter $Q$ would have been that coupled
to the world-sheet curvature, which depends linearly on the normalized Liouville field, and thus
does not generate any quantum corrections. However, having derived the effective potential
(\ref{limitpotqzero}) above, we can now insert the correctly normalized Liouville mode and then
take the limit $|Q^2| \to 0^+$. In this case, the quantum-corrected potential,
expressed in terms of the normalized field $\hat\phi$, becomes:
\begin{equation}\label{limitpotqzeronorm}
\mu^2P_Q(\hat\phi)e^{\hat\phi}\simeq\frac{\Lambda^2}{8\pi}\left|\ln|Q^2|\right|
\exp\left\{\frac{\hat\phi}{\left|Q \ln|Q^2|\right|}\right\} .
\end{equation}
Notice first that, upon the above-mentioned complex
continuations $Q \to iQ$ and $(1 + \varepsilon_Q) \to i(1 + \varepsilon_Q)$ in order to discuss formally the supercritical $Q^2 < 0$ case, the exponent of the effective potential remains real.
We then see that the limit $|Q^2| \to 0^+$ leads to divergent terms in the branch
$\hat\phi \in \left(0,~+\infty)\right)$, while such terms become zero in the branch
$\hat\phi \in \left(-\infty,~0)\right)$.
As already mentioned, the quantity that is actually identified~\cite{emn} as
the target time $t$ in supercritical string theories with $Q^2 < 0$ is {\it minus}
the world-sheet zero mode, $\hat\phi$, so that
\begin{eqnarray}\label{minustime}
\hat\phi = - t~.
\end{eqnarray}
This identification can be derived by using conformal field theory on the world sheet, as
described briefly below~\footnote{It may also be imposed dynamically
in certain concrete examples of Liouville-time cosmologies involving colliding brane
worlds~\cite{gravanis}. In the latter case, the identification (\ref{minustime}) is enforced for
energetic reasons, specifically the minimization of the effective potential of the target-space theory.}.
This implies that, for the flow of cosmological time: $t \to + \infty$,
only the branch $\hat\phi \in (-\infty, ~ 0)$ is of physical relevance, which leads to a stable
target-space-time theory in the limit $|Q^2| \to 0^+$ of the full quantum theory, for the reasons
explained above. This target space stability, expressed via the disappearance of the tachyonic
modes and the vanishing of the tachyonic mass shifts $\Delta m^2=-|Q^2| < 0$ that characterize
the bosonic string states in~\cite{aben}, constitutes a physical argument in favour of the r\^ole of the
limit $|Q^2| \to 0^+$ as the final point of the flow with respect to the central charge in our approach.
For completeness, we review here briefly the derivation of the result (\ref{minustime}) from a
conformal-field-theory analysis. First of all, we note that even after quantum corrections, as our
analysis in Section 3 has shown, the effective potential
assumes the form (\ref{ansatzP}). From a world-sheet field-theory point
of view, this corresponds to a vertex operator of a Liouville-dressed cosmological constant term,
$V(z) =e^{\alpha\hat\phi}$, where $z$ is a complex world-sheet coordinate and
$\alpha (=\varepsilon _Q)$ is a constant, depending on the central-charge deficit $Q$, which plays
the r\^ole of the Liouville anomalous dimension~\cite{ddk}. More generally, one may consider
Liouville-dressed vertex operators $V_i^L \sim e^{\alpha_i\hat\phi} V_i$, where $\alpha_i$
is the corresponding Liouville anomalous dimension.
The N-point correlation functions of the world-sheet vertex operators $V_i$
can be evaluated by first performing the integration over the world-sheet Liouville zero mode. This
leads to expressions of the form:
\begin{equation}
<V_{i_1} \dots V_{i_N} >_\mu = \Gamma (-s) \mu ^s
<(\int d^2z \sqrt{{\hat \gamma }}e^{\alpha\hat\phi })^s
\tilde V_{i_1} \dots {\tilde V}_{i_N} >_{\mu =0} ,
\label{C12}
\end{equation}
where the ${\tilde V}_i$ have the Liouville zero mode removed, $\mu$ is a
scale related to the world-sheet cosmological constant,
and $s$ is the sum of the anomalous dimensions of the
$V_i~:~s=-\sum _{i=1}^{N} \frac{\alpha _i}{\alpha } - Q/\alpha$.
As it stands, the $\Gamma (-s)$ factor implies that (\ref{C12})
is ill-defined for $s=n^+ \in Z^+$. Such cases include physically interesting
Liouville models, such as those describing matter scattering off a two-dimensional
($s$-wave four-dimensional) string black hole~\cite{emn}, when it is excited to a `massive'
(topological) string state corresponding to a positive integer value for $s=n^+ \in {Z}^+$.
Similar divergent expressions are met in general Liouville theory when computing the
correlation functions by analytic continuation of the central charge of
the theory, so that the sums $s$ over Liouville anomalous dimensions acquire positive integer
values~\cite{goulian}. This also leads to ill-defined $\Gamma (-s)$ factors in the appropriate
analytically-continued correlators.
\begin{figure}[t]
\begin{center}
\epsfig{figure=97-58fig1.eps,width=0.3\linewidth}
\end{center}
\caption{ {\it The solid line is the
the Saalschutz contour in the complex
area ($A$) plane, which is used to
continue analytically the prefactor
$\Gamma (-s)$ for $ s \in Z^+$. It has been used in
conventional quantum field theory to relate dimensional
regularization to the Bogoliubov-Parasiuk-Hepp-Zimmermann
renormalization method. The dashed line denotes the
regularized contour, which avoids the ultraviolet
fixed point $A \rightarrow 0$, which is used in the closed time-like path
formalism.}}
\label{fig:ctp}
\end{figure}
Constraining the world-sheet area $A$ at a fixed value~\cite{ddk}, one can use the
following integral representation for $\Gamma (-s)$:
\begin{eqnarray}
\Gamma (-s)=\int dA e^{-A} A^{-s-1} ,
\label{integralA}
\end{eqnarray}
where $A$ is the covariant area of the world-sheet. In the
case $s=n^+ \in {Z}^+$ one can then regularize by analytic continuation,
replacing (\ref{integralA}) by an integral along the Saalschutz contour
shown in Fig. \ref{fig:ctp}~\cite{kogan2,emn}.
This is a well-known method of regularization
in conventional field theory, where integrals of
forms similar to (\ref{integralA}) appear in terms of
Feynman parameters.
A similar regularization was used to prove the equivalence
of the Bogolubov-Parasiuk-Hepp-Zimmerman renormalization prescription
with dimensional regularization in ordinary gauge field theories~\cite{BPZ}.
One result of such an analytic continuation is the
appearance of imaginary parts in the respective correlation functions,
which in our case are interpreted~\cite{kogan2,emn}
as renormalization-group instabilities of the system.
\begin{figure}[t]
\begin{center}
\epsfig{figure=bounce1.eps,width=0.3\linewidth}
\end{center}
\caption {Schematic repesentation
of the evolution of the world-sheet area as the renormalization-group scale moves along the
contour of Fig.~\ref{fig:ctp}.}
\label{fig:wsflow}
\end{figure}
Interpreting
the latter as an actual time flow, with the identification of the
(world-sheet) zero mode with the target time~\cite{emn},
we then interpret the contour of Fig.~\ref{fig:ctp} as implying
evolution of the world-sheet area in both
(negative and positive) directions of time as seen in
Fig.~\ref{fig:wsflow}, i.e.,
infrared fixed point $\to$ ultraviolet fixed point $\to$ infrared fixed point.
In each half of the world-sheet diagram of fig. \ref{fig:wsflow},
the Zamolodchikov $C$ theorem~\cite{zam}
tells us that we have an
irreversible Markov process.
This in turn implies a `bounce' interpretation
of the renormalization-group flow of Fig.~\ref{fig:wsflow}, in which
the infrared fixed point with large world-sheet area $A \to \infty$ is a `bounce' point,
similar to the corresponding picture in point-like field theory~\cite{coleman}.
Therefore, the physical flow of time $t$
is taken to be opposite to the conventional
renormalization-group flow, i.e., from the infrared
to the ultraviolet ($A \to 0$) fixed point on the world sheet.
In terms of the world-sheet zero mode of the Liouville field $\hat\phi_0$, we have
$\hat\phi_0 \sim {\rm ln}A \in \left(0,~-\infty \right)$. Our analysis in the previous Section
shows that the effective potential term (\ref{limitpotqzeronorm}) vanishes in the limit $Q^2 \to 0$,
so this limit corresponds to a stable target-space theory.
We stress once more that this is consistent with the disappearance (as $|Q^2| \to 0^+$)
of tachyonic instabilities in the target-space theory, as
manifested through tachyonic mass shifts $\Delta m^2 = -|Q^2| < 0$ of initially (i.e, before
Liouville dressing) massless
target-space excitations. Thus, the analysis of this paper
reinforces the previous arguments that the (zero mode of the) world-sheet Liouville mode may
be identified (up to a sign) with the target time.
\section{Summary and Perspectives}
We have demonstrated in this paper how a novel renormalization-group technique for controlling
quantum effects by relaxing a mass parameter can be used to obtain non-perturbative results for
non-critical string models. We have studied the behaviour of Liouville string theory as a
function of the departure from criticality, as parametrized by the central-charge deficit $Q$.
We have identified a renormalization-group fixed point in the limit $Q^2 \to 0^+$, in which the
dynamics of the Liouville field becomes trivial, as it neither propagates nor interacts, and
the target space-time is of Minkowski type (in the supercritical string case).
We have shown that the resulting theory is free of tachyonic instabilities in target space in the limit
$|Q^2| \to 0^+$. This analysis supports the previous identification~\cite{emn} of the (zero mode
of the) Liouville mode with the target time.
This approach may in the future be used to discuss the transitions between linear-dilaton cosmological
models with different values of $Q$, and ultimately the transition to an asymptotic state. It has
been shown previously that $Q$ corresponds to the vacuum energy in conventional
field-theoretical models of cosmological inflation~\cite{diamand,emn}. The transition from scalar field energy to
relativistic particles has bee studied extensively within that framework, and our approach
provides a framework for addressing such cosmological phase transitions in string theory.
Another area where this technique may be applied is the Quantum Hall effect (QHE). The
different values of $Q$ correspond to different Hall conductivity plateaux, and our approach
can be used to discuss transitions between these plateaux. The analogy between string
cosmology and black-hole physics, on the one hand, and the QHE, on the other hand, has
been advertised previously~\cite{hall}. The novel renormalization-group described here provides a
tool that can be used to quantify this relationship.
\section*{Acknowledgements}
The work of J.E. and N.E.M. was supported in part by the European Union through the Marie Curie Research and Training Network UniverseNet (MRTN-CT-2006-035863).
\section*{Appendix}
We review here the construction of the effective action $\Gamma$ and derive
the equation describing its evolution with $Q$. For reasons explained in the text, we restrict ourselves
to the subcritical string case $Q^2 \propto c^* -C > 0$. The supercritical string case $Q^2 < 0$ is treated formally by means of analytic continuation.
In terms of the microscopic field $\tilde\phi$, the bare action is
\begin{equation}
S=\int d^2\xi\left\{\frac{Q^2}{2}\partial_a\tilde\phi\partial^a\tilde\phi+\beta_Q R^{(2)}\tilde\phi
+\mu^2P_B(\phi)e^{\tilde\phi}\right\},
\end{equation}
The partition function, namely the functional of the source $j$, is defined as
\begin{equation}
{\cal Z}[j]=\int{\cal D}[\tilde\phi]\exp\left(-S-\int d^2\xi~j\tilde\phi\right),
\end{equation}
and is related to the functional $W$ that generates connected graphs by
\begin{equation}
W[j]=-\ln{\cal Z}[j].
\end{equation}
The classical field $\phi$ is defined by differentiation of $W$ with respect to the source $j$,
and we have
\begin{eqnarray}\label{diffW}
\frac{\delta W}{\delta j_\xi}&=&-\frac{1}{{\cal Z}}\frac{\partial {\cal Z}}{\partial j_\xi}
=\frac{<\tilde\phi_\xi>}{{\cal Z}}=\phi_\xi\nonumber\\
\frac{\delta^2 W}{\delta j_\xi\delta j_\zeta}&=&\phi_\xi\phi_\zeta-
\frac{<\tilde\phi_\xi\tilde\phi_\zeta>}{{\cal Z}},
\end{eqnarray}
where the quantum vacuum expectation value is
\begin{equation}
<\cdot\cdot\cdot>=\int{\cal D}[\tilde\phi](\cdot\cdot\cdot)\exp\left(-S-\int d^2\xi~j\tilde\phi\right).
\end{equation}
The effective action $\Gamma$, a functional of the classical field $\phi$, is introduced
as the Legendre transform of $W$:
\begin{equation}
\Gamma[\phi]=W[j]-\int j\phi,
\end{equation}
where the source $j$ has to be seen as a functional of $\phi$. The functional derivatives of $\Gamma$
are then
\begin{eqnarray}
\frac{\delta\Gamma}{\delta\phi_\xi}&=&-j_\xi\nonumber\\
\frac{\delta^2\Gamma}{\delta\phi_\xi\delta\phi_\zeta}&=&
-\left(\frac{\delta\phi_\xi}{\delta j_\zeta}\right)^{-1}=
-\left(\frac{\delta^2 W}{\delta j_\xi\delta j_\zeta}\right)^{-1}.
\end{eqnarray}
From eqs.(\ref{diffW}), the equation describing the evolution of $W$ with $Q^2$ is
\begin{eqnarray}
\dot W&=&\frac{1}{{\cal Z}}\int d^2\xi\int d^2\zeta\left\{
\frac{1}{2}\frac{\partial}{\partial\xi^a}\frac{\partial}{\partial\zeta_a}
\left<\tilde\phi_\xi\tilde\phi_\zeta\right>+\dot\beta_Q R^{(2)}\left<\tilde\phi_\xi\right>\right\}
\delta^{(2)}(\xi-\zeta)\nonumber\\
&=&\int d^2\xi \left\{\frac{1}{2}\partial_a\phi\partial^a\phi+\dot\beta_Q R^{(2)}\phi\right\}
-\frac{1}{2}\mbox{Tr}\left\{\frac{\partial}{\partial\xi_a}\frac{\partial}{\partial\zeta^a}
\left(\frac{\delta^2W}{\delta j_\xi\delta j_\zeta}\right)\right\}.
\end{eqnarray}
In order to find the evolution equation for $\Gamma$, one should remember that its independent
variables are $Q$ ad $\phi$, so that
\begin{equation}
\dot\Gamma=\dot W+\int d^2\xi~\frac{\delta W}{\delta j}\partial_Qj-\int d^2\xi~\partial_Qj\phi=\dot W.
\end{equation}
Using the previous results, finally we have
\begin{equation}\label{evolGApp}
\dot\Gamma=\int d^2\xi\left\{\frac{1}{2}\partial_a\phi\partial^a\phi+\dot\beta_Q R^{(2)}\phi\right\}
+\frac{1}{2}\mbox{Tr}\left\{\frac{\partial}{\partial\xi_a}\frac{\partial}{\partial\zeta^a}
\left(\frac{\delta^2\Gamma}{\delta\phi_\xi\delta\phi_\zeta}\right)^{-1}\right\}.
\end{equation}
\vspace{0.5cm}
In order to extract physical quantities from the evolution equation (\ref{evolGApp}),
we assume the following functional dependence of the effective action:
\begin{equation}
\Gamma=\int d^2\xi\left\{\frac{Z_Q}{2}\partial_a\phi\partial^a\phi+\beta_Q R^{(2)}\phi+
\mu^2P_Q(\phi)e^\phi\right\}.
\end{equation}
We have then
\begin{eqnarray}
\frac{\delta^2\Gamma}{\delta\phi_\xi\delta\phi_\zeta}&=&\left\{Z_Q\partial_a\partial^a+
U^{''}_Q(\phi)\right\} \delta^{(2)}(\xi-\zeta),\\
&&\mbox{where}~~~U_Q(\phi)=\mu^2P_Q(\phi)e^\phi,\nonumber
\end{eqnarray}
and a prime denotes a derivative with respect to $\phi$.
For the evolution of $P$ only, it would be enough to insert in the evolution equation (\ref{evolGApp})
a constant field $\phi_0$. But in order to derive the evolution of $Z_Q$, one needs a varying field and we
consider thus $\phi=\phi_0+2\rho\cos(k\xi)$, where $k$ is some fixed momentum. If ${\cal A}$ denotes the surface area of the world sheet, the effective action then reads
\begin{equation}\label{expGamma}
\Gamma={\cal A}\left(Z\rho^2 k^2+\beta_Q R^{(2)}\phi_0+U_Q(\phi_0)
+\frac{1}{2}\rho^2U^{''}_Q(\phi_0)+{\cal O}(\rho^3)\right),
\end{equation}
so that the evolution equation for $U$ is obtained by identifying the $k$-independent terms in
eq.(\ref{evolGApp}), and the evolution equation for $Z$ by identifying the
terms proportional to $\rho^2k^2$.
The second derivative of the effective action reads for this configuration $\phi$, in Fourier components,
\begin{eqnarray}
\frac{\delta^2\Gamma}{\delta\phi_p\delta\phi_q}
&=&\left(Z_Qp^2+U_Q^{''}(\phi_0)\right)(2\pi)^2\delta^{(2)}(p+q)\\
&&+\rho^2U^{'''}_Q(\phi_0)(2\pi)^2\left[\delta^{(2)}(p+q+k)+\delta^{(2)}(p+q-k)\right]+{\cal O}(\rho^3).\nonumber
\end{eqnarray}
The inverse of this matrix with components $p,q$ is computed using the following expansion
\begin{equation}\label{expansion}
(A+B)^{-1}=A^{-1}-A^{-1}BA^{-1}+A^{-1}BA^{-1}BA^{-1}+\cdot\cdot\cdot,
\end{equation}
where $A$ is a diagonal matrix with indices $p,q$, and $B$ is off-diagonal and
proportional to $\rho^2$.
In the previous expansion, the term linear in $A^{-1}$ is independent of $\rho,k$. It leads to the
evolution of $U$, and makes the following contribution to the trace which appears in
eq.(\ref{evolGApp}):
\begin{eqnarray}\label{quadratic}
&&{\cal A}\int\frac{d^2p}{(2\pi)^2}\frac{p^2}{Z_Qp^2+U^{''}_Q(\phi_0)}\nonumber\\
&=&{\cal A}\frac{\Lambda^2}{4\pi Z_Q}-{\cal A}\frac{U^{''}_Q(\phi_0)}
{4\pi Z^2}\ln\left(1+\frac{Z_Q\Lambda^2}{U^{''}_Q(\phi_0)}\right).
\end{eqnarray}
We note that the quadratic divergence is field-independent, and therefore is irrelevant. Also,
the term linear in $B$, which appears in the expansion (\ref{expansion}), has a vanishing trace
since it is off-diagonal.
The term quadratic in $B$ in the expansion (\ref{expansion}) contains a contribution that is
proportional to $\rho^2$ and independent of $k$, which does not bring any new information,
since it corresponds to the evolution of $U^{''}$, as can be seen from eq.(\ref{expGamma}).
It also contains a contribution proportional to $\rho^2 k^2$, which leads to the evolution of
$Z$. The corresponding trace is
\begin{eqnarray}
&&{\cal A}\rho^2\left[U^{'''}_Q(\phi_0)\right]^2
\int\frac{d^2p}{(2\pi)^2}\frac{4p^2}{\left(Z_Qp^2+U_Q^{''}(\phi_0)\right)^4}
\left(-Z_Qk^2+\frac{4Z_Q^2(kp)^2}{Z_Qp^2+U^{''}_Q(\phi_0)}\right)+{\cal O}(k^4)\nonumber\\
&=&{\cal A}\frac{\rho^2 k^2}{\pi Z_Q}\left[U^{'''}_Q(\phi_0)\right]^2\int_0^\infty dx
\left(\frac{-x}{\left(x+U_Q^{''}(\phi_0)\right)^4}+\frac{2x^2}{\left(x+U_Q^{''}(\phi_0)\right)^5}\right)+{\cal O}(k^4)\nonumber\\
&=&{\cal O}(k^4),
\end{eqnarray}
where we used the fact that, for any function $f(p^2)$,
\begin{equation}
\int\frac{d^2p}{(2\pi)^2}(kp)^2f(p^2)=\frac{k^2}{8\pi}\int d(p^2) p^2f(p^2).
\end{equation}
As a consequence, $Z$ does not receive quantum corrections.
Finally, the evolution equation for $P$ is found
from eqs.(\ref{evolGApp}), (\ref{expGamma}) and (\ref{quadratic})
where we disregard the field-independent quadratic divergence, to be
\begin{equation}
\dot P_Q(\phi)=-\frac{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}{8\pi Z_Q^2}
\ln\left(1+\frac{Z_Qe^{-\phi}\Lambda^2/\mu^2}{P_Q(\phi)+2P_Q^{'}(\phi)+P_Q^{''}(\phi)}\right).
\end{equation}
The reader can now see easily why the supercritical string case $Q^2 < 0$ presents
certain problems that can be treated by analytic continuation.
Considering the case $Q^2 < 0$ and a Euclidean world sheet metric, we have
\begin{equation}
\frac{\delta^2 S(\phi_0)}{\delta\phi(p)\delta\phi(q)}=
\left\{-|Q^2|(p_1^2+p_2^2)+\mu^2 e^{\phi_0}\right\}\delta^{(2)}(p+q).
\end{equation}
The propagator is the inverse of this, and hence cannot be integrated because of the
pole, whose presence is linked to the supercriticality of the string. This pole is not the usual one corresponding to a mass. Indeed, if one returns to a
Minkowski world-sheet metric, one obtains:
\begin{equation}\label{minkowsi}
\frac{\delta^2 S(\phi_0)}{\delta\phi(p)\delta\phi(q)}=
\left\{|Q^2|(p_0^2-p_1^2)+\mu^2 e^{\phi_0}\right\}\delta^{(2)}(p+q),
\end{equation}
where $p_0=ip_2$.
One should perform another `Wick rotation' on $p_1$ in order to treat the problem properly.
Formally, these issues are resolved simply by treating the $Q^2 < 0$ case in our method by the
above-mentioned complex continuation of both $Q \to iQ$ and the Liouville scaling exponents:
$\alpha =\frac{(1 + \varepsilon_Q)}{Q} \to i\alpha $.
|
2,877,628,090,210 | arxiv | \section*{Introduction}
Let $L$ be a finite lattice and $K[L]$ the polynomial ring over a field $K$ whose variables are the elements of $L.$ Let $I_L$ be the {\em join-meet
ideal} of $L,$ that is, the ideal of $K[L]$ which is generated by all the binomials of the form $f=ab-(a\wedge b)(a\vee b)$, where $a,b\in L$ are
incomparable elements. Of course one may ask whether algebraic properties of $I_L$ are related to the combinatorial properties of $L.$ $I_L$ is a
prime ideal if and only if $L$ is distributive as it was shown in \cite{H} and if $L$ is distributive, the Gr\"obner bases of $I_L$ with respect to
various monomial orders have been studied; see, for instance, \cite{H}, \cite{HH1}, \cite{AHH}, \cite{Q}. In the same hypothesis on $L,$ the toric
ring $K[L]/I_L$ is well understood; see \cite{H}, \cite{H1}, \cite{H2}, \cite{H3}.
Almost nothing is known about the join-meet ideal $I_L$ when $L$ is not distributive. In the present paper we focus on the join-meet ideals of modular
and non-distributive lattices. For basic properties of lattices, like distributivity and modularity, we refer the reader to the well known monographs
\cite{B} and \cite{St}.
It was conjectured in \cite{HH1} that, given a modular lattice $L$, for any monomial order $<$ on $K[L]$ the initial ideal $\ini_<(I_L)$ is not
squarefree, unless $L$ is distributive. We give a proof of this conjecture in Section~\ref{conjecture}. This result shows, in particular, that
for deciding whether a join-meet ideal $I_L$ of a modular and non-distributive lattice $L$ is radical one cannot use the known
statement that a polynomial ideal is radical if it has a squarefree initial ideal. Moreover, easy examples show that even if the lattice $L$
is rather closed to a distributive lattice, the ideal $I_L$ might not be radical; see Example~\ref{latticeN}. A general
characterization of radical join-meet ideals associated with modular non-distributive lattices seems to be difficult.
However, in Section~\ref{modular}, we find a class of modular non-distributive lattices $L$ whose join-meet ideal $I_L$ is radical. To prove this
property we intensively use the Gr\"obner basis theory.
For radical join-meet ideals, in Section~\ref{radsection}, we describe the minimal prime ideals. This description is used later, in Section~\ref{minsection}, to obtain a complete characterization of the minimal primes of the radical join-meet ideals studied in Section~\ref{modular}.
\section{The squarefree conjecture}
\label{conjecture}
Let $L$ be a finite lattice and $K[L]$ the polynomial ring over a field $K$ whose variables are the elements of $L.$ A binomial of $K[L]$ of the form
$f=ab-(a\wedge b)(a\vee b)$, where $a,b\in L$ are incomparable, is called a {\em basic binomial}. In some recent papers, the basic binomials are called Hibi relations.
\begin{Definition}
The {\em join-meet ideal} of $L$ is the ideal of $K[L]$ generated by the basic binomials, that is,
\[
I_L=(ab-(a\wedge b)(a\vee b): a,b\in L, a,b \text{ incomparable })\subset K[L].
\]
\end{Definition}
The join-meet ideal of a lattice was introduced in \cite{H}. For fundamental notions on lattices we refer to \cite{B} and \cite{St}.
The main result of this section answers positively a conjecture made in \cite{HH1}. We first need a preparatory result on modular and non-distributive lattices which might be known, but we include its proof since we could not find any reference.
\begin{Lemma}\label{small diamond}
Let $L$ be a modular non-distributive lattice. Then $L$ has a diamond sublattice $L^\prime$ such that
$\rank \max L^\prime - \rank \min L^\prime =2.$
\end{Lemma}
\begin{proof}
Let $\delta$ be a diamond of $L$ labeled as in Figure~\ref{small1} (i) of minimal rank, that is, $\rank e-\rank a=$ minimal.
\begin{figure}[bht]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-10.3,-2.5)(4,3.5)
\psline(-9,3)(-11,1)
\psline(-9,3)(-9,1)
\psline(-9,3)(-7,1)
\psline(-11,1)(-9,-1)
\psline(-9,1)(-9,-1)
\psline(-7,1)(-9,-1)
\rput(-9,3){$\bullet$}
\put(-9.1,3.3){$e$}
\rput(-11,1){$\bullet$}
\put(-11.6,0.9){$b$}
\rput(-9,1){$\bullet$}
\put(-8.6,0.9){$c$}
\rput(-7,1){$\bullet$}
\put(-6.7,1){$d$}
\rput(-9,-1){$\bullet$}
\put(-9.1,-1.6){$a$}
\rput(-13,-1){(i)}
\psline(1,3)(-1,1)
\psline(1,3)(1,1)
\psline(1,3)(3,2)
\psline(3,2)(3,0)
\psline(-1,1)(1,-1)
\psline(1,1)(1,-1)
\psline(3,0)(1,-1)
\rput(1,3){$\bullet$}
\put(1.1,3.3){$e$}
\rput(-1,1){$\bullet$}
\put(-1.6,0.9){$b$}
\rput(1,1){$\bullet$}
\put(1.4,0.9){$c$}
\rput(3,2){$\bullet$}
\put(3.3,2){$d$}
\rput(1,-1){$\bullet$}
\put(0.9,-1.6){$a$}
\rput(3,0){$\bullet$}
\put(3.3,0){$f$}
\rput(5,-1){(ii)}
\end{pspicture}
\end{center}
\caption{}\label{small1}
\end{figure}
We show that $\rank a-\rank e=2.$ Let us assume that $\rank e > \rank a +2.$ By duality, we may assume, for instance, that $\rank d > \rank a +1,$
that is, there exists $f\in L$ such that $a < f < d.$ Then we get the lattice displayed in Figure~\ref{small1} (ii) where $c\wedge f=c\wedge d=a$ and
$c\vee f \leq c\vee d=e.$ If $c\vee f=e,$ then $L$ has a pentagon subblattice (with the elements $a,c,f,d,e$), which is impossible since $L$ is modular.
Therefore, we must have $c\vee f < e. $
\begin{figure}[bht]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-5.3,-2.5)(4,3.5)
\psline(6,3)(4,1)
\psline(6,3)(6,1)
\psline(6,3)(8,1)
\psline(4,1)(6,-1)
\psline(6,1)(6,-1)
\psline(6,1)(6,-1)
\psline(8,1)(6,-1)
\rput(6,3){$\bullet$}
\put(5.9,3.3){$c\vee f$}
\rput(4,1){$\bullet$}
\put(0,0.9){$b\wedge(c\vee f)$}
\rput(6,1){$\bullet$}
\put(6.4,0.9){$c$}
\rput(8,1){$\bullet$}
\put(8.3,1){$f$}
\rput(6,-1){$\bullet$}
\put(5.9,-1.6){$a$}
\rput(-13,-1){(i)}
\psline(-9,3)(-11,1)
\psline(-9,3)(-7,2)
\psline(-7,2)(-7,0)
\psline(-11,1)(-9,-1)
\psline(-7,0)(-9,-1)
\rput(-9,3){$\bullet$}
\put(-8.9,3.3){$e$}
\rput(-11,1){$\bullet$}
\put(-11.6,0.9){$b$}
\rput(-7,2){$\bullet$}
\put(-6.7,2){$c\vee f$}
\rput(-9,-1){$\bullet$}
\put(-9.1,-1.6){$a$}
\rput(-7,0){$\bullet$}
\put(-6.7,0){$c$}
\rput(2,-1){(ii)}
\end{pspicture}
\end{center}
\caption{}\label{small2}
\end{figure}
We now look at the lattice with the elements $a,b,c,c\vee f,$ and $e.$ Here we have $b\vee(c\vee f)=(b\vee c)\vee f=e\vee f=e$ and
$b\wedge (c\vee f)\geq a.$ If $b\wedge (c\vee f)=a$ we get again a pentagon sublattice of $L$; see Figure~\ref{small2} (i). Since $L$ is
modular, we must have $b\wedge (c\vee f) > a.$ We look at the lattice with elements $a,c,f,b\wedge (c\vee f),$ and $c\vee f$; see Figure~\ref{small2}
(ii). The following relations hold:
\[
c\wedge(b\wedge (c\vee f))=(c\wedge b)\wedge (c\vee f)=a,
\]
and
\[
c\vee(b\wedge (c\vee f))=(c\vee b)\wedge (c\vee f)=e\wedge (c\vee f)=c\vee f,
\]
the first equality in the latter relation being true by modularity. Moreover, we have
\[
f\wedge (b\wedge(c\vee f))=(f\wedge b)\wedge (c\vee f)=a\wedge (c\vee f)=a,
\]
and
\[
f\vee (b\wedge (c\vee f))=(f\vee b)\wedge (c\vee f),
\]
again by modularity, and, thus,
\[
f\vee (b\wedge (c\vee f))\leq c\vee f.
\]
If $f\vee (b\wedge (c\vee f))= c\vee f,$ then we get a diamond sublattice of $L$ as in Figure~\ref{small2} (ii) of smaller rank than $\delta$, which is impossible by our assumption. Hence we must have
\[
(f\vee b)\wedge (c\vee f) < c\vee f.
\]
\begin{figure}[bht]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-5.3,-2.5)(4,3.5)
\psline(6,3)(4,1)
\psline(6,3)(6,1)
\psline(6,3)(8,1)
\psline(4,1)(6,-1)
\psline(6,1)(6,-1)
\psline(6,1)(6,-1)
\psline(8,1)(6,-1)
\rput(6,3){$\bullet$}
\put(5.9,3.3){$(c\vee f)\wedge(b\vee f)$}
\rput(4,1){$\bullet$}
\put(0,1.4){$c\wedge(b\vee f)$}
\rput(6,1){$\bullet$}
\put(6.2,0.2){$(c\vee f)\wedge b$}
\rput(8,1){$\bullet$}
\put(8.3,1.5){$f$}
\rput(6,-1){$\bullet$}
\put(5.9,-1.6){$a$}
\rput(-13,-1){(i)}
\psline(-9,3)(-11,1)
\psline(-9,3)(-7,2)
\psline(-7,2)(-7,0)
\psline(-11,1)(-9,-1)
\psline(-7,0)(-9,-1)
\rput(-9,3){$\bullet$}
\put(-8.9,3.3){$c\vee f$}
\rput(-11,1){$\bullet$}
\put(-11.6,0.9){$c$}
\rput(-7,2){$\bullet$}
\put(-6.7,2){$(c\vee f)\wedge(b\vee f)$}
\rput(-9,-1){$\bullet$}
\put(-9.1,-1.6){$a$}
\rput(-7,0){$\bullet$}
\put(-6.7,0){$(c\vee f)\wedge b$}
\rput(2,-1){(ii)}
\end{pspicture}
\end{center}
\caption{}\label{small3}
\end{figure}
Let us consider now the lattice with the elements $a,c, (c\vee f)\wedge b, f\vee(b\wedge(c\vee f))=(c\vee f)\wedge(b\vee f)$, and
$c\vee f.$ The following equalities hold:
\[
((c\vee f)\wedge b)\wedge c=(c\vee f)\wedge(b\wedge c)=a,
\]
and, by modularity,
\[
c\vee(b\wedge(c\vee f))=(c\vee b)\wedge (c\vee f)=c\vee f.
\]
Next, we have:
\[
c\vee(f\vee(b\wedge(c\vee f)))=(c\vee f)\vee(b\wedge (c\vee f))=c\vee f.
\]
Therefore, if $c\wedge((c\vee f)\wedge(b\vee f))=c\wedge (b\vee f)=a$, then $L$ has a pentagon sublattice; see Figure~\ref{small3} (i). Hence we must
have \[
c\wedge(b\vee f)>a.
\]
Finally, we look at the lattice with the elements $a,c\wedge(b\vee f), (c\vee f)\wedge b, f,$ and $(c\vee f)\wedge(b\vee f).$
The following equalities hold:
\[
f\wedge(c\wedge(b\vee f))=(c\wedge f)\wedge (b\vee f)=a\wedge(b\vee f)=a,
\]
and
\[
f\vee (c\wedge(b\vee f))=(f\vee c)\wedge (b\vee f) \text{ (by modularity) }.
\]
Next,
\[
f\wedge(b\wedge(c\vee f))=(f\wedge b)\wedge (c\vee f)=a\wedge(c\vee f)=a,
\]
and
\[
f\vee(b\wedge(c\vee f))=(f\vee b)\wedge (c\vee f) \text{ (by modularity) }.
\]
We also have:
\[
(c\wedge(b\vee f))\wedge ((c\vee f)\wedge b)=(b\wedge c)\wedge(b\vee f)\wedge (c\vee f)=a
\]
and, by applying modularity,
\[
(c\wedge(b\vee f))\vee ((c\vee f)\wedge b)=((c\wedge(b\vee f))\vee b)\wedge(c\vee f)
\]
\[
=((b\vee c)\wedge(b\vee f))\wedge(c\vee f)=e\wedge(b\vee f)\wedge(c\vee f)=(b\vee f)\wedge(c\vee f).
\]
Consequently, we have got another diamond sublattice of $L$ (see Figure~\ref{small3} (ii)) with a smaller rank than $\delta$, again a contradiction.
\end{proof}
In the proof of the next theorem we use some arguments which are taken from the proof of \cite[Theorem 1.1]{HH1}, but we include them for the convenience of the reader.
\begin{Theorem}\label{HH conjecture}
Let $L$ be a modular non-distributive lattice. Then, for any monomial order $<$ on $K[L]$, the initial ideal $\ini_<(I_L)$ is not squarefree.
\end{Theorem}
\begin{proof}
By Lemma~\ref{small diamond}, $L$ has a sublattice $L^\prime$ with $a=\min L^\prime$, $e=\max L^\prime$ such that $\rank e-\rank a=2.$ Let
$b_1,b_2,\ldots,b_k\in L$, $k\geq 3,$ be the elements of $L$ such that for any $1\leq i<j\leq n,$ $b_i\vee b_j=e$ and $b_i\wedge b_j=a.$
Therefore, we have the following relations in $I_L:$ $b_ib_j-ae$ for $1\leq i< j\leq k.$
Let $<$ be an arbitrary monomial order on $K[L]$. We may assume that, with respect to this order, we have $b_1>\cdots >b_k.$ We are going to show
that
$\ini_<(I_L)$ is not squarefree. We have to analyze the following two cases.
Case 1. Assume that $ae< b_i b_j$ for any $1\leq i< j\leq k.$ Let $b=b_k$ and consider the binomial $f=ab^2e-a^2e^2$ which, by the proof of
\cite[Theorem 1.1]{HH1}, belongs to $I_L.$ Let us assume that $\ini_<(I_L)$ is squarefree. Then, since $f\in I_L,$ we must have $abe\in \ini_<(I_L)$,
hence, following the arguments of the proof of \cite[Theorem 1.1]{HH1}, there exists a binomial $g=abe-u\in I_L$ where $u=\ell mn$ with
$\ell, m,n\in L$, all of them in the interval $[a,e]$ of $L,$ and, in addition, with $\ini_<(g)=abe.$ Also, from the arguments of the cited proof, it
follows that at least two of the variables $\ell, m,n$ are distinct. Indeed, let
\begin{equation}\label{exprofg}
g=\sum_{i=1}^N x_i(v_i-w_i)
\end{equation}
where each $x_i$ is a variable and
$v_i-w_i$ is a basic binomial of $I_L$ such that $x_1v_1=abe$, $x_iw_i=x_{i+1}v_{i+1}$ for $1\leq i <N,$ and $x_Nw_N=u.$ Then each variable that
appears in the binomial $x_i(v_i-w_i)$ must belong to the interval $[a,e]$ of $L.$ This is true since for any basic binomial $v-w,$ one has
$\supp(v)\subset [a,e]$ if and and only if $\supp(w)\subset [a,e].$ In particular, $x_Nw_N=u$ is of the form $u=\ell mn$ with $\ell, m, n\in [a,e]$ and,
by (\ref{exprofg}), at least two of $\ell, m, n$ are distinct.
Moreover, by (\ref{exprofg}), it also follows
that $\rank a+\rank b+\rank e =\rank \ell +\rank m +\rank n.$ Since in $L^\prime$ we have $\rank e-\rank a=2,$ it follows that
\begin{equation}\label{eqrk}
\rank \ell +\rank m +\rank n= 3 \rank a +3.
\end{equation}
Of course we may assume that $\rank \ell \geq \rank m\geq \rank n. $ Let us suppose that $\rank n> \rank a.$ Then, by using equation~(\ref{eqrk}),
we obtain $\rank \ell =\rank m=\rank n=\rank a +1,$ hence $\ell, m,n \in \{b_1,b_2,\ldots,b_k\}$. It follows that $g=abe-b_ib_jb_p$ for some
$i,j,p\in \{1,2,\ldots,k\}$ with at least two of them distinct. Let us assume that $i\neq j.$ Then, since $ae< b_ib_j$ and $b\leq b_p$, we get a
contradiction to the fact that $\ini_<(g)=abe$.
Let now $\rank n=\rank a.$ This implies that $\rank \ell + \rank m=2 \rank a +3,$ which leads to the conclusion that $\rank \ell=\rank a +2$ and
$\rank m=\rank a+1.$ Therefore, we get $n=a, \ell=e,$ and $m=b_i$ for some $1\leq i\leq k.$ We then have $g=abe-ab_i e$ which is impossible since
obviously $abe\leq ab_i e$ by the choice of $b.$
Hence, in Case 1, $\ini_<(I_L)$ is not squarefree.
Case 2. There exist $1\leq i <j \leq k$ such that $ae >b_ib_j.$ Let $bd$ be the smallest monomial among all the monomials $b_ib_j, 1\leq i < j \leq k.$
In particular, it follows that $ae> bd.$ We first claim that $b^2d-bd^2\in I_L.$ Indeed. one may easily check the following identity:
\[
b^2d-bd^2=(b-d)(bd-ae)-b(cd-ae)+d(bc-ae),
\]
where $c$ is an arbitrary variable in $\{b_1,\ldots,b_k\}\setminus \{b,d\}.$ Let us assume that $\ini_<(I_L)$ is squarefree. Then we have
$bd\in \ini_<(I_L)$. This implies that there exists a binomial $g=bd-\ell m\in I_L$ with $bd\in\ini_<(I_L).$ Since $ae> bd,$ we cannot have $\ell m=ae.$
Therefore, $g=bd-b_ib_j$ for some $1\leq i< j\leq n,$ which is again impossible by our choice of the monomial $bd$.
\end{proof}
\section{Radical join-meet ideals of finite lattices}
\label{radsection}
In this section we describe the associated primes of a radical join-meet ideal of a finite lattice.
\begin{Proposition}\label{general}
Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let $I\subset S$ be a binomial ideal, that is, an ideal which is generated by differences of two monomials. If $I$ is a radical ideal, then:
\begin{itemize}
\item [(a)] $I: (\prod_{i=1}^n x_i)^\infty=I: \prod_{i=1}^n x_i.$
\item [(b)] $I: \prod_{i=1}^n x_i$ is a prime ideal.
\end{itemize}
\end{Proposition}
\begin{proof}
(a). Let $\Min^\ast(I)$ be the set of all prime ideals of $I$ which contain no variable. Then
\[
I: \prod_{i=1}^n x_i=\bigcap_{P\in \Min(I)}(P:\prod_{i=1}^n x_i)=\bigcap_{P\in \Min^\ast(I)}P=\bigcap_{P\in \Min(I)}(P:(\prod_{i=1}^n x_i)^\infty)=I: (\prod_{i=1}^n x_i)^\infty.
\]
(b). By \cite{ES} or \cite{OP}, $I: \prod_{i=1}^n x_i$ is a lattice ideal, let us say $I_{\mathcal L}$ where ${\mathcal L}\subset {\NZQ Z}^n$ is a lattice.
By \cite[Theorem 2.1]{ES}, it is enough to show
that ${\mathcal L}$ is saturated, in other words, if ${\bold x}^{m{\bold a}}-{\bold x}^{m{\bold b}}\in I: \prod_{i=1}^n x_i$ for some positive integer $m,$ then
${\bold x}^{\bold a}-{\bold x}^{\bold b}\in I: \prod_{i=1}^n x_i.$
The proof depends on the characteristic of the field. Let us first assume that $\chara K=0.$ Since, by the proof of (a), we have
$I: \prod_{i=1}^n x_i=\bigcap_{P\in \Min^\ast (I)}P,$ we get ${\bold x}^{m{\bold a}}-{\bold x}^{m{\bold b}}\in P$ for any prime ideal $P\in \Min^\ast(I).$ Since $P$ does not
contain any variable, it follows that the polynomial $g={\bold x}^{(m-1){\bold a}}+\ldots +{\bold x}^{(m-1){\bold b}}\not \in P$ since $g(1,\ldots,1)=m\neq 0,$ hence
${\bold x}^{\bold a}-{\bold x}^{\bold b}\in P$ for any
$P\in \Min^\ast(I).$ Therefore, we obtain ${\bold x}^{\bold a}-{\bold x}^{\bold b}\in I: \prod_{i=1}^n x_i.$
A similar proof works in positive characteristic. Indeed, let $p>0$ be the characteristic of the field and let $m=p^t q$ for some non-negative integer
$t$ and some positive integer $q$ such that $(p,q)=1$. Then
\[
{\bold x}^{m{\bold a}}-{\bold x}^{m{\bold b}}=({\bold x}^{q{\bold a}}-{\bold x}^{q{\bold b}})^{p^t}=({\bold x}^{\bold a}-{\bold x}^{\bold b})^{p^t}({\bold x}^{(q-1){\bold a}}+\cdots + {\bold x}^{(q-1){\bold b}})^{p^t}\in P
\]
for all $P\in \Min^\ast(I)$. Let $h=({\bold x}^{(q-1){\bold a}}+\cdots + {\bold x}^{(q-1){\bold b}})^{p^t}={\bold x}^{(q-1){\bold a} p^t}+\cdots + {\bold x}^{(q-1){\bold b} p^t}.$ Then $h(1,\ldots,1)=q\neq 0.$ It follows, by using the same argument as in the zero characteristic, that $({\bold x}^{\bold a}-{\bold x}^{\bold b})^{p^t}\in P$ and thus ${\bold x}^{\bold a}-{\bold x}^{\bold b}\in P$ for every $P\in \Min^\ast(I)$. This implies that ${\bold x}^{\bold a}-{\bold x}^{\bold b}\in I: \prod_{i=1}^n x_i.$
\end{proof}
Now we are going to characterize the associated primes of a radical join-meet ideal of a finite lattice. We first need the following
\begin{Definition}
Let $L$ be a lattice and $A$ a subset of $L.$ $A$ is called {\em admissible} if it is empty or it is non-empty and has the following property: for any
basic binomial $ab-cd$
of $I_L$, if $a\in A$ or $b\in A$, then $c\in A$ or $d\in A.$
\end{Definition}
In other words, the set $A$ is admissible if and only if, for any basic binomial, either $A$ "covers" both monomials of the binomial or none of them.
Of course, the empty set and $L$ are admissible sets for $I_L.$
\begin{Remark}{\em
Let $A$ be an admissible set for $I_L$. We set $L_A=L\setminus A.$ Then $L_A$ is a sublattice of $L$ with respect to the order induced from $L.$
Indeed, let $a,b\in L_A$ be two incomparable elements. Since $A$ is admissible, it follows that $a\vee b$ and $a\wedge b$ do not belong to $A.$
}
\end{Remark}
\begin{Proposition}\label{radsubset}
Let $I_L$ be a radical ideal. Then, for any admissible set, the ideal $I_{L_A}$ is radical.
\end{Proposition}
\begin{proof}
Assume that there exists $A\subset L$ such that $I_{L_A}$ is not radical, hence there exists a polynomial $f\in K[\{a: a\in L\setminus A\}]$ such
that $f\in \sqrt{I_{L_A}}\setminus I_{L_A}.$ Then obviously $f\in \sqrt{I_L}.$ We claim that $f\not\in I_L$ which shows that $I_L$ is not radical, a contradiction. Let us assume that $f\in I_L.$ Then we may write
\[
f=\sum_{a,b\not\in A}h_{ab}(ab-(a\wedge b)(a\vee b)) + \sum_{a\in A \text{ or }b\in A}h_{ab}(ab-(a\wedge b)(a\vee b))
\]
for some polynomials $h_{ab}\in K[L].$ We map to zero all the variables of $A.$ In this way, since $A$ is admissible, it follows that the second sum in the above formula vanishes while in the first sum, all the basic binomials survive. Therefore, $f\in I_{L_A},$ a contradiction.
\end{proof}
\begin{Remark}{\em
We are going to see in Example~\ref{exampleR} that the radical property does
not pass from a lattice to any of its proper sublattices.
}
\end{Remark}
For an admissible set $A\subset L,$ we set $$P_A(L)=I_{L_A}:\prod_{a\not\in A}a+ (a : a\in A).$$ If $I_L$ is a radical ideal, then $I_{L_A}$ is a radical ideal by Proposition \ref{radsubset}, and, by Proposition~\ref{general}, it follows that $I_{L_A}:\prod_{a\not\in A}a$ is prime. Thus $P_A(L)$ is a prime ideal for any admissible set $A$ if $I_L$ is a radical ideal. Obviously, $P_A(L)\supset I_L$ for any admissible set $A.$
\begin{Theorem}\label{intersection}
Let $L$ be a lattice such that $I_L$ is a radical ideal. Then
\[
I_L=\bigcap\limits_{A\subset L\atop A\text{ admissible }}P_A(L).
\]
\end{Theorem}
\begin{proof}
It is enough to show that any minimal prime ideal of $I_L$ is of the form $P_A(L)$ for some admissible set $A\subset L.$
Let $P$ be a minimal prime of
$I_L$ and $A=\{a: a\in P\}$. If $A=\emptyset,$ that is, $P$ does not contain any variable, then $P\supset I_L:\prod_{a\in L}a\supset I_L.$
Since, by Proposition~\ref{general}, $I_L:\prod_{a\in L}a$ is a prime ideal, we obtain $P=P_\emptyset(L).$
Now let $A$ be nonempty. We claim that $A$ is admissible. Indeed, let $ab-cd$ be a basic binomial such that $a\in A.$ It follows that
$cd \in P$, which implies that $c\in A$ or $d\in A.$ We show that $P=P_A(L).$ Indeed, since $P\supset I_L$ and $P\supset (a: a\in A)$, we also
have $P\supset I_L+(a:a\in A)=I_{L_A}+(a:a\in A).$ It follows that $P\supset (I_{L_A}+(a:a\in A)):\prod_{a\notin A}a=P_A(L).$ Since $P$ is
minimal over $I$, we must have $P=P_A(L).$
\end{proof}
\begin{Proposition}\label{mincond}
Let $I_L$ be radical. Then for two admissible sets $A,B\subset L$, we have $P_A(L)\subsetneq P_B(L)$ if and only if
\[
A\subsetneq B
\text{ and } I_{L_A}:\prod_{a\not\in A}a\subset I_{L_B}:\prod_{b\not\in B}b +(b: b\in B\setminus A).
\]
\end{Proposition}
\begin{proof}
Let $A\subset B$. Then $P_A(L)\subset P_B(L)$ if and only if $$I_{L_A}:\prod_{a\not\in A}a=P_A(L)/ (a: a\in A)\subset P_B(L)/(a: a\in A)$$
$$=I_{L_B}:\prod_{b\not\in B}b +(b: b\in B\setminus A).$$
\end{proof}
The following example illustrates Theorem~\ref{intersection} and Proposition~\ref{mincond}.
\begin{Example}{\em
Let $Q$ be the lattice of Figure~\ref{latticeQ}. The Gr\"obner basis of $I_Q$ with respect to the lexicographic order induced by $a>b>\cdots> g$ is
$\{ae-bc,ag-cf,bg-ef,cd-cf,de-ef\}.$ Thus, $\ini_<(I_Q)$ is squarefree which implies that $I_Q$ is a radical ideal and we may apply Theorem~\ref{intersection} and Proposition~\ref{mincond} to determine the minimal primes of $I_Q.$
\begin{figure}[hbt]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-1.3,-5.5)(1,2)
\psline(0,-5)(2,-3)
\psline(0,-5)(-2,-3)
\psline(2,-3)(0,-1)
\psline(-2,-3)(-3,-2)
\psline(0,-1)(-1.5,1.5)
\psline(-3,0)(-1.5,1.5)
\psline(-2,-3)(0,-1)
\psline(-3,-2)(-3,0)
\rput(0,-5){$\bullet$}
\put(-0.8,-5){$a$}
\rput(2,-3){$\bullet$}
\put(2.3,-3){$c$}
\rput(-2,-3){$\bullet$}
\put(-2.8,-3.2){$b$}
\rput(0,-1){$\bullet$}
\put(0.3,-1){$e$}
\rput(-3,-2){$\bullet$}
\put(-3.8,-2){$d$}
\rput(-3,0){$\bullet$}
\put(-3.8,0){$f$}
\rput(-1.5,1.5){$\bullet$}
\put(-2.3,1.5){$g$}
\end{pspicture}
\end{center}
\caption{}\label{latticeQ}
\end{figure}
One easily sees that
\[
I_Q:\prod_{x\in Q}x\supset J=(ae-bc,ag-cf,bg-ef,d-f)\supset I_Q.
\]
But $K[a,b,c,d,e,f,g]/J\cong K[a,b,c,d,e,g]/(ae-bc,ag-cd,bg-de)$, and the latter
quotient ring is a domain. Therefore, $J$ is a prime ideal. Moreover, $I_Q:\prod_{x\in Q}x=J=P_\emptyset (Q).$ The other minimal primes
of $I$ are $(a,b,c,e)$ and $(c,e,g)$, that is, $I=J\cap(a,b,c,e)\cap(c,e,g).$ Note that, for instance, the set $A=\{g,d,f\}$ is an admissible set,
but the corresponding prime ideal $P_A(Q)$ is not a minimal prime of $I_Q$ since $P_A(Q)\supsetneq P_\emptyset(Q).$
}
\end{Example}
\section{Join-meet ideals of modular non-distributive lattices }
\label{modular}
It is well known that, given an ideal $I$ of a polynomial ring $S$ over a field, if $\ini_<(I)$ is radical for some monomial order $<$ on $S,$ then
the ideal $I$ is radical as well; see \cite[Proposition 3.3.7]{HHbook} or \cite[Lemma 6.51]{EH} for an alternative proof. This gives also a procedure
to show that a polynomial ideal is radical. However, there are radical polynomial ideals whose initial ideals are always non-radical. For such ideals
one has to use other kind of arguments to prove the radical property.
In this section we mainly study a class of modular non-distributive lattices whose join-meet ideals are radical. Before beginning our study, let us look at the next
\begin{Example}\label{latticeN}{\em
Let $N$ be the lattice of rank $4$ of Figure~\ref{FigN}. This is rather a simple example of a modular non-distributive lattice. We "included" only
one diamond into a distributive lattice with $8$ elements. However, as we are going to show, the join-meet ideal of lattice $N$ is not radical.
\begin{figure}[hbt]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-25.3,-2.5)(4,5)
\psline(-9,3)(-11,1)
\psline(-9,3)(-9,1)
\psline(-9,3)(-7,1)
\psline(-11,1)(-9,-1)
\psline(-9,1)(-9,-1)
\psline(-7,1)(-9,-1)
\psline(-9,3)(-11,5)
\psline(-13,3)(-11,5)
\psline(-13,3)(-11,1)
\psline(-11,1)(-13,-1)
\psline(-13,-1)(-11,-3)
\psline(-11,-3)(-9,-1)
\rput(-9,3){$\bullet$}
\put(-9.1,3.3){$h$}
\rput(-11,1){$\bullet$}
\put(-11.6,0.9){$d$}
\rput(-9,1){$\bullet$}
\put(-8.6,0.9){$e$}
\rput(-7,1){$\bullet$}
\put(-6.7,1){$f$}
\rput(-9,-1){$\bullet$}
\put(-9.1,-1.6){$c$}
\rput(-11,-3){$\bullet$}
\put(-11,-3.5){$a$}
\rput(-13,-1){$\bullet$}
\put(-13.5,-1){$b$}
\rput(-13,3){$\bullet$}
\put(-13.5,3){$g$}
\rput(-11,5){$\bullet$}
\put(-11,5.5){$\ell$}
\end{pspicture}
\end{center}
\caption{}\label{FigN}
\end{figure}
We claim that $a\ell g(d-f)^2\in I_L,$ which implies that $(a\ell g(d-f))^2\in I_L,$ therefore, $a\ell g(d-f)\in \sqrt{I_L}.$ Indeed, one may easily see that
\[
a\ell g(d-f)^2=a\ell gd^2-2a\ell gdf + a \ell gf^2 \equiv ag^2hd-ag^2h f -agf(gh-\ell f)
\]
\[
\equiv ag^2h(d-f)-agf\ell ( d- f)\equiv ag^2h(d-f)-a\ell^2 c(d-f) \mod I_L.
\]
On the other hand, $ah(d-f)\in I_L$ and $\ell c(d-f)\in I_L$. One may easily check this. For instance, for the first membership, we may use the
following identity:
\[
ah(d-f)=b(de-ch)+(f-d)(be- ah)-b(ef-ch).
\]
Thus, $a\ell g(d-f)\in \sqrt{I_L}.$ The Gr\"obner basis of $I_L$ with respect to reverse
lexicographic order contains, apart of the basic binomials of $L$, the following binomials: $ce\ell-cf\ell, cd\ell-cf\ell,ceh-cfh,aeh-afh,
cdh-cfh,adh-afh,cf^2\ell-c^2h\ell,ad^2\ell-ach\ell,cf^2h-c^2h^2,af^2h-ach^2.$ Thus, $\ini_<(a\ell gd-a\ell gf)\not\in \ini_<(I_L)$ which implies that
$a\ell g(d-f)\not\in I_L$.
}
\end{Example}
Therefore, the following question arises. Is there a class of distributive lattices such that by "including" just one small diamond one may
get a radical joint-meet ideal for the new lattice? We are going to answer this question in the next theorem.
Let $D$ be the distributive lattice of the divisors of $2\cdot 3^n$ for some integer $n\geq 1$ with the elements labeled as in Figure~\ref{chain} (a). For every $1\leq k\leq n-1,$ we denote by $L_k$ the lattice of Figure~\ref{chain} (b).
\begin{figure}[hbt]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-1,-11)(5,2)
\psline(-9,1)(-11,-1)
\psline(-9,1)(-7,-1)
\psline(-9,-3)(-7,-1)
\psline(-11,-1)(-9,-3)
\psline(-5,-3)(-7,-5)
\psline(-7,-5)(-5,-7)
\psline(-5,-7)(-3,-5)
\psline(-3,-5)(-5,-3)
\psline(-3,-9)(-1,-7)
\psline(-1,-7)(1,-9)
\psline(1,-9)(-1,-11)
\psline(-1,-11)(-3,-9)
\psline[linestyle=dotted](-9,-3)(-7,-5)
\psline[linestyle=dotted](-7,-1)(-5,-3)
\psline[linestyle=dotted](-5,-7)(-3,-9)
\psline[linestyle=dotted](-3,-5)(-1,-7)
\rput(-9,1){$\bullet$}
\put(-8.5,1){$y_n$}
\rput(-11,-1){$\bullet$}
\put(-12.5,-1){$x_n$}
\rput(-7,-1){$\bullet$}
\put(-6.5,-1){$y_{n-1}$}
\rput(-9,-3){$\bullet$}
\put(-11,-3){$x_{n-1}$}
\rput(-5,-3){$\bullet$}
\put(-4.5,-3){$y_{k+1}$}
\rput(-7,-5){$\bullet$}
\put(-9,-5){$x_{k+1}$}
\rput(-3,-5){$\bullet$}
\put(-2.5,-5){$y_k$}
\rput(-5,-7){$\bullet$}
\put(-6.5,-7){$x_k$}
\rput(-1,-7){$\bullet$}
\put(-0.5,-7){$y_2$}
\rput(-3,-9){$\bullet$}
\put(-4.5,-9){$x_2$}
\rput(1,-9){$\bullet$}
\put(1.5,-9){$y_1$}
\rput(-1,-11){$\bullet$}
\put(-2.5,-11){$x_1$}
\rput(-7,-9){(a)}
\psline(6,1)(4,-1)
\psline(6,1)(8,-1)
\psline(6,-3)(8,-1)
\psline(4,-1)(6,-3)
\psline(10,-3)(8,-5)
\psline(8,-5)(10,-7)
\psline(10,-7)(12,-5)
\psline(12,-5)(10,-3)
\psline(12,-9)(14,-7)
\psline(14,-7)(16,-9)
\psline(16,-9)(14,-11)
\psline(14,-11)(12,-9)
\psline[linestyle=dotted](6,-3)(8,-5)
\psline[linestyle=dotted](8,-1)(10,-3)
\psline[linestyle=dotted](10,-7)(12,-9)
\psline[linestyle=dotted](12,-5)(14,-7)
\psline(10,-3)(10,-5)
\psline(10,-5)(10,-7)
\rput(6,1){$\bullet$}
\put(6.5,1){$y_n$}
\rput(4,-1){$\bullet$}
\put(2.5,-1){$x_n$}
\rput(8,-1){$\bullet$}
\put(8.5,-1){$y_{n-1}$}
\rput(6,-3){$\bullet$}
\put(4,-3){$x_{n-1}$}
\rput(10,-3){$\bullet$}
\put(10.5,-3){$y_{k+1}$}
\rput(8,-5){$\bullet$}
\put(6,-5){$x_{k+1}$}
\rput(12,-5){$\bullet$}
\put(12.5,-5){$y_k$}
\rput(10,-7){$\bullet$}
\put(8.5,-7){$x_k$}
\rput(14,-7){$\bullet$}
\put(14.5,-7){$y_2$}
\rput(12,-9){$\bullet$}
\put(10.5,-9){$x_2$}
\rput(16,-9){$\bullet$}
\put(16.5,-9){$y_1$}
\rput(14,-11){$\bullet$}
\put(12.5,-11){$x_1$}
\rput(10,-5){$\bullet$}
\put(10.3,-5){$z$}
\rput(8,-9){(b)}
\end{pspicture}
\end{center}
\caption{}\label{chain}
\end{figure}
Before stating our first preparatory result, we need to introduce some notation. For $1\leq k\leq n-1,$ let
\[
p_k=x_{k+1}z-y_k z; r_k=y_k^2z-y_kz^2; g_i=x_iy_{k+1}-y_iz, \text{ for }1\leq i <k;
\]
\[
h_j=x_ky_j-x_jz, \text{ for }k+1\leq j \leq n;
f_{ij}=x_jy_i-x_iy_j, \text{ for } 1\leq i < j\leq n, j\neq k+1, i\neq k;
\]
\[
f_{i,k+1}=x_{k+1}y_i-y_iz, \text{ for }1\leq i\leq k; f_{kj}=x_jy_k-x_jz, \text{ for } j> k+1;
\]
\[
p_{ij}=x_ix_{k+1}y_j-x_iy_jz, t_{ij}=x_iy_ky_j-x_iy_jz, \text{ for } 1\leq i< k< k+1 < j\leq n,
\]
and
\[
q_{ik}= y_iy_kz-y_iz^2, \text{ for }1\leq i <k.
\]
\begin{Lemma}\label{GBofI}
The set
\[
{\mathcal G}=\{p_k,r_k\}\cup \{g_i,q_{ik} : 1\leq i<k\}\cup \{h_j: k+1\leq j\leq n\}\cup\{f_{ij}: 1\leq i < j\leq n\}
\]
\[
\cup \{p_{ij}, t_{ij}: 1\leq i<k<k+1 < j\leq n\}
\]
is a Gr\"obner basis of $I=I_{L_k}$ with respect to the reverse lexicographic order induced by $x_1>\cdots >x_n> y_1>\cdots> y_n>z.$
In particular, it follows that $\ini_<(I)$ is generated by the following set of monomials:
\[
{\mathcal M}=\{x_jy_i: 1\leq i< j\leq n\}\cup\{x_iy_{k+1}: 1\leq i< k\}\cup\{x_ky_j: k+1\leq j\leq n\}
\]
\[
\cup\{x_ix_{k+1}y_j,x_iy_ky_j: 1\leq i< k< k+1 < j\leq n\}\cup\{y_iy_kz: 1\leq i<k\}\cup\{x_{k+1}z,y_k^2z\}.
\]
\end{Lemma}
\begin{proof}
We first note that ${\mathcal G}$ is a generating set of $I$ and next one applies Buchberger's criterion, that is, one checks that all the $S$-polynomials
of the pairs $(f,g)\in {\mathcal G}\times {\mathcal G}$ reduce to zero modulo ${\mathcal G}.$ Note that for many pairs $(f,g)\in {\mathcal G}\times {\mathcal G}$ the checks are superfluous since
the initial monomials $\ini_<(f)$ and $\ini_<(g)$ are relatively prime. Moreover, in order to eliminate many checks, one may use the following known
fact. If $f,g$ are two polynomials with $\ini_<(f)$ and $\ini_<(g)$ relatively prime, then, for any monomials $u,v$ the $S$-polynomial $S(uf,vg)$
reduces to zero modulo $uf$ and $vg.$
\end{proof}
\begin{Theorem}\label{radical}
For every $1\leq k\leq n-1,$ the join-meet ideal $I_{L_k}$ is radical.
\end{Theorem}
The proof of this theorem has several steps which are shown in the following lemmas, but the basic idea of the proof is very simple. We
actually show that one may decompose $I$ as an intersection of two radical ideals, namely $I=(I,x_{k+1}-y_k)\cap (I,z),$ hence $I$ itself is a
radical ideal.
\begin{Lemma}\label{intersect}
Let $1\leq k\leq n-1 $ and $I=I_{L_k}.$ Then $I=(I,x_{k+1}-y_k)\cap (I,z).$
\end{Lemma}
\begin{proof}
The inclusion $I\subset (I,x_{k+1}-y_k)\cap (I,z)$ is obvious. For getting the equality we show that
\begin{equation}\label{eqint}
\ini_<(I,x_{k+1}-y_k)\cap \ini_<(I,z)\subset \ini_<(I).
\end{equation}
This will imply that $\ini_<((I,x_{k+1}-y_k)\cap (I,z))\subset \ini_<(I),$ thus,
\[
\ini_<(I)=\ini_<((I,x_{k+1}-y_k)\cap (I,z))
\]
which leads to the desired statement.
We know the generators of $\ini_<(I)$ from Lemma~\ref{GBofI}. We now compute the Gr\"obner bases of $(I,z)$ and $(I,x_{k+1}-y_k)$ with respect to the
reverse lexicographic order induced by $x_1>\cdots >x_n> y_1>\cdots >y_n>z$.
By using the Gr\"obner basis of $I,$ one easily sees that $(I,z)$ is generated by the binomials $f_{ij}=x_jy_i-x_iy_j$ where $1\leq i<j\leq n$ and
$j\neq k+1$, $i\neq k$ and by the following set of monomials: $\{z\}\cup\{x_iy_{k+1}: 1\leq i<k\}\cup\{x_ky_j: k+1\leq j\leq n\}\cup\{x_{k+1}y_i:
1\leq i\leq k\}\cup\{x_jy_k: j> k+1\}\cup \{x_ix_{k+1}y_j, x_iy_ky_j: 1\leq i<k<k+1<j\leq n\}.$ By using Buchberger's criterion, one immediately
checks that the above set of generators of $(I,z)$ is a Gr\"obner basis of $(I,z)$. Consequently,
\[
G(\ini_<(I,z))=(G(\ini_<(I)\setminus\{x_{k+1}z,y_kz^2,y_iy_kz:1\leq i<k\})\cup\{z\}
\]
which implies that
\begin{equation}\label{eqini1}
\ini_<(I,z)=(\ini_<(I),z).
\end{equation}
Here we used the notation $G(J)$ for the minimal set of monomial generators of the monomial ideal $J.$
By using the Gr\"obner basis of $I$ it follows that the ideal $(I,x_{k+1}-y_k)$ is generated by the binomials $x_{k+1}-y_k, g_i, 1\leq i<k, h_j, k+1\leq j\leq n,
f_{ij}, 1\leq i<j\leq n, j\neq k+1,i\neq k,
f_{i,k+1}^\prime=y_iy_k-y_iz=f_{i,k+1}-y_i(x_{k+1}-z), 1\leq i\leq k, f_{kj}, j> k+1,r_k,$ and $p_{ij}^\prime=t_{ij}=x_iy_jy_k-x_iy_jz, 1\leq i<k<k+1<j\leq n,$ since $q_{ik}=zf_{i,k+1}^\prime$. Buchberger's criterion applied to this set of generators shows that they form a Gr\"obner basis of $(I,x_{k+1}-y_k).$ Moreover, we obtain
\[
G(\ini_<(I,x_{k+1}-y_k))=(G(\ini_<(I)\setminus(\{x_{k+1}z, x_{k+1}y_i: 1\leq i\leq k \}\cup\{y_iy_kz: 1\leq i<k\}))
\]
\[
\cup\{x_{k+1},y_iy_k: 1\leq i\leq k\}.
\]
therefore, we get the following equality:
\begin{equation}\label{eqini2}
\ini_<(I,x_{k+1}-y_k)=(\ini_<(I),x_{k+1},y_1y_k,\ldots,y_{k-1}y_k,y_k^2).
\end{equation}
By using the relations (\ref{eqini1}) and (\ref{eqini2}), we get
\[
\ini_<(I,x_{k+1}-y_k)\cap \ini_<(I,z)=(\ini_<(I),x_{k+1}z, y_1y_kz,\ldots,y_k^2z)\subset \ini_<(I).
\]
\end{proof}
From the above proof we may also derive the following
\begin{Corollary}\label{corsquarefree}
$(I,z)$ is a radical ideal.
\end{Corollary}
\begin{proof}
By (\ref{eqini1}), we have $\ini_<(I,z)=(\ini_<(I),z)$. Since $\ini_<(I)$ has only one non-squarefree generator, namely $y_k^2z$ which is "killed" by
$z,$ it follows that $\ini_<(I,z)$ is square free and, consequently, $(I,z)$ is a radical ideal.
\end{proof}
The last step in the proof of Theorem~\ref{radical} is shown in the following
\begin{Lemma}\label{sqfree2}
The ideal $(I,x_{k+1}-y_k)$ is radical.
\end{Lemma}
\begin{proof}
We show that $(I,x_{k+1}-y_k)$ has a squarefree initial ideal with respect to the lexicographic order induced by
$z> x_1>\cdots >x_n>y_1>\cdots >y_n.$ We recall from the proof of Lemma~\ref{intersect} that $(I,x_{k+1}-y_k)$ is generated by
$x_{k+1}-y_k, g_i, 1\leq i<k, h_j, k+1\leq j\leq n, f_{ij}, 1\leq i<j\leq n, j\neq k+1,i\neq k,
f_{i,k+1}^\prime=y_iy_k-y_iz, 1\leq i\leq k, f_{kj}, j> k+1,r_k,$ and $p_{ij}^\prime=t_{ij}=x_iy_jy_k-x_iy_jz, 1\leq i<k<k+1<j\leq n.$ In this
generating set, the generators $r_k$ and $p_{ij}^\prime$ are redundant. Indeed, $r_k=z f_{k,k+1}^\prime$ and $p_{ij}^\prime=(y_k-z)f_{ij}
-x_j f_{i,k+1}^\prime$ for any $1\leq i< k< k+1 < j\leq n.$ Moreover, for every $1\leq i<k$ we may replace the generator $g_i$ by
$g_i^\prime=x_iy_{k+1}-y_iy_k=f_{i,k+1}^\prime- g_i.$ Finally, for $j> k+1$ we may replace the generator $f_{kj}$ by $x_ky_j-x_jy_k=f_{kj}-h_j.$
Therefore, $(I,x_{k+1}-y_k)$ is generated by the following binomials: $x_{k+1}-y_k, g_i^\prime=x_iy_{k+1}-y_iy_k$ for $1\leq i<k,$ $h_j=zx_j-x_ky_j$
for $k+1\leq j\leq n$, $f_{i,k+1}^\prime=zy_i-y_iy_k$ for $1\leq i\leq k$, and $f_{ij}=x_iy_j-x_jy_i$ for $1\leq i< j\leq n$ with $j\neq k+1.$ By trivial
calculations one may check that this set of generators is a Gr\"obner basis of $(I,x_{k+1}-y_k)$ with respect to the lexicographic order induced by
$z>x_1>\cdots >x_n>y_1>\cdots >y_n.$ Since all these generators have squarefree initial monomials, it follows that the initial ideal of $(I,x_{k+1}-y_k)$ is squarefree and, thus, $(I,x_{k+1}-y_k)$ is a radical ideal.
\end{proof}
We end this section with a few comments. Going back to our Example~\ref{latticeN}, by applying Theorem~\ref{radical}, we see that every proper
sublattice $N^\prime$ of $N$ has a radical join-meet ideal although $I_N$ is not radical. The following example shows that the radical property does
not pass from a lattice to any of its proper sublattices.
\begin{Example}\label{exampleR} {\em
Let $R$ be the lattice of Figure~\ref{latticeR}.
\begin{figure}[hbt]
\begin{center}
\psset{unit=0.5cm}
\begin{pspicture}(-25.3,-2.5)(4,5)
\psline(-9,3)(-11,1)
\psline(-9,3)(-9,1)
\psline(-9,3)(-7,1)
\psline(-11,1)(-9,-1)
\psline(-9,1)(-9,-1)
\psline(-7,1)(-9,-1)
\psline(-9,3)(-11,5)
\psline(-13,3)(-11,5)
\psline(-13,3)(-11,1)
\psline(-11,1)(-13,-1)
\psline(-13,-1)(-11,-3)
\psline(-11,-3)(-9,-1)
\psline(-15,1)(-13,-1)
\psline(-15,1)(-13,3)
\rput(-9,3){$\bullet$}
\rput(-11,1){$\bullet$}
\rput(-9,1){$\bullet$}
\rput(-7,1){$\bullet$}
\rput(-9,-1){$\bullet$}
\rput(-11,-3){$\bullet$}
\rput(-13,-1){$\bullet$}
\rput(-13,3){$\bullet$}
\rput(-11,5){$\bullet$}
\rput(-15,1){$\bullet$}
\end{pspicture}
\end{center}
\caption{}\label{latticeR}
\end{figure}
One may check with Singular \cite{GPS} that $I_R$ is a radical ideal. However the ideal $I_N$ attached to its proper sublattice $N$ is not radical, as we have seen in Example~\ref{latticeN}.
}
\end{Example}
\section{The minimal primes of the join-meet ideal of $L_k$}
\label{minsection}
In this section we apply the results of Section~\ref{radsection} to determine explicitly the minimal primes of the ideals $I_{L_k}$ for
$1\leq k\leq n-1.$ We recall that we denoted by $D$ the distributive lattice displayed in Figure~\ref{chain} (a), and by $L_k$ the
lattice displayed in Figure~\ref{chain} (b). We denote by $D_k$ the sublattice of $D$ with the elements $x_i,y_i, 1\leq i\leq k,$ and by $D_k^\prime$ the sublattice of $D$ with the elements $x_{i},y_i, k+1\leq i\leq n.$
Before stating the main theorem of this section, we need to prove a preparatory result.
\begin{Lemma}\label{ufff}
For any $1\leq k\leq n-1,$ the ideal $(I_D,x_{k+1}-y_k)$ is prime.
\end{Lemma}
\begin{proof}
It is enogh to show that $(I_D,x_2-y_1)$ is a prime ideal since by an appropriate change of variables, we may map the ideal $(I_D,x_2-y_1)$ into
$(I_D,x_{k+1}-y_k).$
Let $f_{ij}=x_iy_j-x_jy_i,$ $1\leq i < j\leq n$ the generators of $I_D.$ By \cite[Theorem 2.2]{HH1}, $\{f_{ij}: 1\leq i< j\leq n\}$ is a Gr\"obner
basis of $I_D$ with respect to any monomial order. Actually, if $\ini_<f_{ij}$ and $\ini_<f_{k\ell}$ are not relatively prime, then the
$S$-polynomial of the pair $(f_{ij},f_{k\ell})$ may be expressed as
\begin{equation}\label{spoly}
S(f_{ij},f_{k\ell})=z f_{pq}
\end{equation}
for some variable $z\in K[D]$ and $1\leq p< q\leq n.$
Let $<$ be an arbitrary monomial order on $K[D].$ For any $1\leq i < j\leq n,$ we denote by $g_{ij}$ the reduction of $f_{ij}$ modulo $x_2-y_1$. More
precisely, $g_{ij}$ is obtained from $f_{ij}$ by replacing $x_2$ by $y_1$ if $x_2>y_1$ or $y_1 $ by $x_2$ if $y_1> x_2.$ Since
$\{f_{ij}:1\leq i < j\leq n\}$ is a Gr\"obner basis of $I_D$ with respect to $<$, it follows that the set ${\mathcal G}=\{g_{ij}:1\leq i < j\leq
n\}\cup\{x_2-y_1\}$ is a Gr\"obner basis of $(I_D,x_2-y_1)$ with respect to $<.$ This is essentially due to equation (\ref{spoly}). In particular,
${\mathcal G}$ is a Gr\"obner basis of $(I_D,x_2-y_1)$ with respect to the lexicographic order induced by $x_1>\cdots > x_n> y_1 >\cdots > y_n$. In this case
it follows that the initial ideal of $(I_D,x_2-y_1)$ is generated by the following squarefree monomials: $x_2,x_iy_j$ for $i,j\neq 2$, $x_1y_2$, and
$x_jy_2$ for $2<j\leq n.$ This shows that $(I_D,x_2-y_1)$ is a radical ideal.
On the other hand, by applying \cite[Lemma 12.1]{S}, it follows that all the variables are regular on $(I_D,x_2-y_1)$, which implies that
$(I_D,x_2-y_1): \prod_{1\leq i\leq n}x_i\prod_{1\leq j\leq n}y_j=(I_D,x_2-y_1).$ Finally, by applying Proposition~\ref{general}, we get the desired conclusion.
\end{proof}
\begin{Theorem}\label{minprimes}
Let $1\leq k\leq n-1$ and $I=I_{L_k}$ the join-meet ideal of the lattice $L_k.$ The minimal primes of $I$ are the followings:
\[
P=(I, z-x_{k+1},z-y_k), P_1=(z,x_1,\ldots,x_n), P_1^\prime=(z,y_1,\ldots,y_n),
\]
\[
P_2=(z,x_1,\ldots,x_k,y_1,\ldots,y_k)+I_{D_k^\prime}, P_2^\prime=(z,x_{k+1},\ldots,x_n, y_{k+1},\ldots,y_n) + I_{D_k},
\]
\[
P_3=(x_1,\ldots,x_n,y_1,\ldots,y_k), P_3^\prime=(y_1,\ldots,y_n,x_{k+1},\ldots,x_n).
\]
\end{Theorem}
\begin{proof}
By Theorem~\ref{intersection}, since $I$ is a radical ideal, we know that any minimal prime of $I$ is of the form $P_A(L_k)$ where
$A$ is an admissible set of $I.$
Let $P=P_{\emptyset}(L_k).$ Then $P=I: (z\prod_{1\leq i\leq n}x_i\prod_{1\leq j\leq n}y_j).$ We obviously have
\begin{equation}\label{eqstar}
P\supset (I,z-x_{k+1},x-y_k)\supset I.
\end{equation}
On the other hand,
\[
\frac{K[L_k]}{(I,z-x_{k+1},x-y_k)}\cong \frac{K[D]}{(I_D,x_{k+1}-y_k)}.
\]
Since, by Lemma~\ref{ufff}, $(I_D,x_{k+1}-y_k)$ is a prime ideal, it follows that $(I,z-x_{k+1},x-y_k)$ is a prime ideal as well. Therefore, since $P$
is a minimal prime of $I,$ by using (\ref{eqstar}), we must have $P=(I,z-x_{k+1},x-y_k).$
Now we look at the minimal primes which correspond to non-empty admissible sets. Let $A$ be such an admissible set and assume first that $z\in A.$
If $y_\ell\not\in A$ for every $1\leq \ell\leq n,$ then, by using the basic binomials $zy_i-x_iy_{k+1}$ for $i\leq k$ and $x_ky_j-x_jy_k$ for $j\geq
k+1$, it follows that $P\supset (z,x_1,\ldots,x_n)\supset I,$ hence, we get $P_A(L_k)=(z,x_1,\ldots,x_n)=P_1.$ Since the dual lattice of $L_k$ has
obviously the same relation ideal, it follows that $P_1^\prime$ is the minimal prime which correspond to the admissible set $A$ which contains $z$
and does not contain any of the variables $x_i, i=1,\ldots,n$.
Now we consider an admissible set $A$ which contains $z$ and has the property that there exist $1\leq i,j\leq n$ such that $x_i,y_j\not\in A.$ If
$i\neq j,$ then, since $x_iy_j-x_jy_i$ is a basic binomial, it follows that $x_j,y_i\not\in A.$ Therefore, we may assume that there exists
$1\leq i\leq n$ such that $x_i,y_i\not\in A.$ Let us suppose that $x_k,y_k\not\in A$. From the relations $x_jz-x_ky_j$ we get $y_j\in A$ for $j\geq
k+1$
and, next, from the relations $x_jy_k-x_ky_j$, we get $x_j\in A$ for $j\geq k+ 1$. Thus, in this case, $P_A(L_k)\supset P_2^\prime\supset I$. But
$P_2^\prime$ is
obviously a prime ideal, therefore, $P_A(L)=P_2^\prime.$ The dual situation correspond to $x_{k+1},y_{k+1}\not\in A,$ and in this case one gets
$P_A(L_k)=P_2.$ It remains to consider $x_k,y_k,x_{k+1},y_{k+1}\in A.$ Then it follows that $P_A(L_k)\supsetneq P$ which implies that $P_A(L_k)$ is
not a minimal prime.
We still need to identify the minimal primes which correspond to non-empty admissible sets $A$ which do not contain $z.$ Let $A$ be such that
$z\not\in A$ and $P_A(L_k)$ is a minimal prime of $I.$ Since $zy_k-x_ky_{k+1}, zx_{k+1}-x_ky_{k+1}, y_kx_{k+1}-x_ky_{k+1}\in I\subset P_A(L_k)$, we
get $z(y_k-x_{k+1})\in P_A(L_k),$ hence $y_k-x_{k+1}\in P_A(L)$, and $x_{k+1}(z-y_k)\in P_A(L)$. If $x_{k+1}\not\in P_A(L)$, it follows that
$z-x_{k+1}\in P_A(L).$ But this further implies that $P_A(L_k)\supsetneq P,$ hence $P_A(L_k)$ is not a minimal prime. Consequently,
$x_{k+1}\in A,$ and, next, $y_k\in P_A(L_k).$ By using again the basic binomial $ y_kx_{k+1}-x_ky_{k+1},$ we obtain $x_k\in A$ or $y_{k+1}\in A.$
We analyze the following cases.
Case 1. $x_k\in A$ and $y_{k+1}\not\in A.$ By using the relations $x_jz-x_ky_j$ for $j> k+1$, we get $x_j\in A$ for $j>k+1.$ Similarly, by using
the basic binomials $y_{k+1}x_i-y_ix_{k+1}$ for $i< k,$ we get $x_i\in A$ for all $i<k.$ Therefore, we have $x_i\in A$ for all $i=1,\ldots,n$. By
using the basic binomials $zy_i-x_iy_{k+1}$ for $i< k,$ we also get $y_i\in A$. Then we have actually proved that $P_A(L_k)\supset (x_1,\ldots,x_n,y_1,\ldots,y_k)=P_3\supset I$. Since $P_A(L_k)$ is a minimal prime of $I,$ we must have $P_A(L_k)=P_3.$
Case 2. $x_k\not\in A$ and $y_{k+1}\in A.$ This is the dual of the above case and leads to the conclusion that $P_A(L_k)=P_3^\prime.$
Case 3. Let $x_k,y_{k+1}\in A.$ From the relations $zy_i-x_iy_{k+1}$ for $i<k,$ and $x_jz-x_ky_j$ for $j> k$, we obtain
$y_i\in A$ for $i<k$, and $x_j\in A$ for $j>k.$ If there exists $i<k$ such that $x_i\not\in A,$ by using the relations $x_iy_j-x_jy_i$ for $j>k+1$, we get $y_j\in A$ for all $j>k+1.$ In this case it follows that $A\supset \{y_1,\ldots,y_n,x_k,\ldots,x_n\}$ and $P_A(L_k)\supsetneq P_3^\prime,$ hence $P_A(L_k)$ is not a minimal prime, contradiction. In other words, Case 3 does not hold, and this completes the proof.
\end{proof}
\begin{Corollary}\label{dim}
The join-meet ideal$I_{L_k}$ is not unmixed and $\dim(K[L_k]/I_{L_k})=n.$
\end{Corollary}
\begin{proof}
It is known (see \cite{H}), that if ${\mathcal D}$ is a distributive lattice, then $\dim(K[{\mathcal D}]/I_{{\mathcal D}})$ is equal to the number of the join irreducible
elements of ${\mathcal D}$ plus $1.$ Therefore, we get
\[
\dim(K[L_k]/P)=n=\dim(K[L_k]/P_1)=\dim(K[L_k]/P_1^\prime),
\]
\[
\dim(K[L_k]/P_2)=n-k, \dim(K[L_k]/P_2^\prime)=k,
\]
\[
\dim(K[L_k]/P_3)=n-k+1, \dim(K[L_k]/P_3^\prime)=k+1.
\] The above equalities yield the desired statements.
\end{proof}
\medskip
|
2,877,628,090,211 | arxiv |
\section{Introduction}
Reinforcement learning (RL) is a popular framework for the control of autonomous agents, due to its ability to autonomously derive control policies without assuming that the dynamics of the environment are known \citep{Sutton1998,bertsekas1996neuro}.
Despite the many remarkable successes in different applications of this type of approaches \citep{nian2020review}, two key problems for RL algorithms remain to be solved: (i) potentially {\em long} learning times and (ii) the lack of convergence or performance guarantees (important for example in safety-critical applications) during learning \citep{berkenkamp2017safe,pfeiffer2018reinforced}.
To overcome these problems, a possible solution, particularly for control applications of RL, is to adopt model-based solutions where the learning agent derives and refines a data-driven model of the environment during the learning process. Examples in the literature include \cite{deisenroth2011pilco,kurutach2018model} among many others.
However, in many control applications, some equation-based models of the environment are often available, even though they might not be accurate enough to allow for an entirely model-based solution of the control problem. When classical RL is used, such approximate or partial models of the environment are often discarded in favour of a completely model-free approach.
In this paper, we investigate the possibility of embedding a feedback control law synthesized by using a partial model of the environment to assist the learning process.
In the same spirit as human-assisted learning strategies where data collected from {\em humans} in the loop are exploited to enhance the learning process \citep{lien2009interactive,suppakun2020coaching,zhan2021human,nguyen2019apprenticeship}, here we propose the use of a {\em feedback controller} in the loop with the aim of steering the learning phase, reducing the amount of data samples required and improving the ability of learned policies to achieve a given control goal.
The contributions of this paper can be summarized as follows: (1) we propose a novel algorithm that leverages the use of a feedback controller in the loop to make the RL process more data efficient; (2) we present both a deterministic and a probabilistic approach to implement the strategy above and decide when assistance from the control tutor (the feedback controller in the loop) is invoked during the learning; (3) by using a set of aptly defined metrics, we compare the performance of the novel approaches with those of a classical RL algorithm both from a learning and a control perspective by using the inverted pendulum benchmark implementation from OpenAI Gym~\citep{brockman2016openai}.
We wish to emphasize that, to our knowledge, this is the first analysis of this type for algorithms where learning is assisted via a feedback controller. Our results convincingly show that such ``control-tutored'' learning approaches require fewer data samples and/or obtain higher rewards, while achieving smaller errors in control regulation tasks.
\section{Related Work}
Several solutions in the existing literature aim at combining control theoretic strategies with reinforcement learning to solve control problems.
In particular, various approaches combine RL with model predictive control (MPC).
For instance, in \cite{rathi2020driving}, a MPC is used to decide the action when the state of the system to control is in a certain region, while the action taken from a $Q$-table is used otherwise; the table being updated after every action.
The use of a (linear) MPC strategy is again suggested in \cite{zanon2020safe}, where a reinforcement learning module can vary the parameters of the cost function and refine the available model of the system to control.
Other solutions combining control strategies with RL include those in \cite{abbeel2006using}, where a policy gradient algorithm is adopted which uses preexisting knowledge of the system dynamics in the form of an approximate Markov decision process; or that presented in \cite{li2021safe}, where a \emph{reference action governor} is used to enforce safety constraints (in the sense of restricting the state space to admissible regions). In so doing, the action is decided via an optimization problem that penalizes deviations from the action suggested by a RL strategy, making these approaches a valuable solution to achieve safe RL.
A strategy similar in spirit to the CTQL we propose here is contained in \cite{argerich2020tutor4rl}, where a Deep $Q$-Network is extended with the policy having a probability to take an action dictated by an ``expert'', which can solve the control problem, to improve data efficiency.
However, differently from \cite{argerich2020tutor4rl}, in our CTQL algorithm, we consider the ``expert'' to be a feedback control law, that if deployed on its own would be unable to achieve the control goal.
Also, note that contrary to previous approaches, e.g. \cite{deisenroth2011pilco}, where an approximation for the system dynamics is learned during the control steps, here we assume to possess and exploit some information on the environment model before simulations so as to be able to derive some feedback control law to be used to assist the learning phase.
An earlier preliminary version of the CTQL was recently presented in \cite{de2020tutoring}.
\section{Mathematical Preliminaries}
\paragraph{Notation.}
Sets are denoted by calligraphic capital characters and random variables are denoted via capital letters. For example, $X$ is a random variable and we denote its realization by $x$.
The probability density (mass) function of the continuous (discrete) random variable $X$ is denoted by $p(x)$ and we use the notation $x\sim p(x)$ to denote the sampling of a random variable from its probability function.
For both continuous and discrete random variables, we always consider the situation where the support of $p(x)$ is compact;
$\R{rand}(\C{A})$ denotes the uniform distribution over the set $\C{A}$.
The expectation of a function, say $h(\cdot)$, of $X$ is defined as
$\BB{E}_p[h(X)] \coloneqq \int h(X)p(x) \R{d}x$, when this is continuous; if $X$ is discrete, we have $\BB{E}_p[h(X)] \coloneqq \sum h(x)p(x)$.
In both cases, the integral/sum is taken on the support of $p(x)$, and we might omit $p$ in $\BB{E}_p$ when there is no ambiguity.
We denote by $\norm{\cdot}$ the Euclidean norm.
\paragraph{Problem set-up.}
We consider a discrete time dynamical system affected by noise, of the form
\begin{equation}\label{eq:dynamical_system}
X_{k+1} = f_k(X_k, U_k, W_k),
\quad x_0 = \tilde{x}_0,
\end{equation}
where $k \in \BB{N}_{\ge 0}$ is discrete time,
$X_k \in \C{X}$ is the {state} of the system at time $k$, with
$\C{X}$ being the state space,
$\tilde{x}_0 \in \C{X}$ is the initial condition,
$U_k \in \C{U}$ is the {control input} (or {action}) and $\C{U}$ is the set of feasible inputs. Also, $W_k$ is a random variable representing {noise} and $f_k : \C{X} \times \C{U} \times \C{W} \rightarrow \C{X}$ is the system's {dynamics}.
Following e.g. \cite{matni2019self,recht2019tour}, given this set-up, we consider the problem of learning a plan of actions {$\pi_1, \dots, \pi_{N-1}$} to solve the following finite-horizon optimization problem:
\begin{subequations}\label{eq:rl_problem_objective}
\begin{align}
\max_{\pi_1,\ldots,\pi_{N-1}}& \ \ \BB{E}[J^{\bar{\pi}}],\\
\text{s.t.}
& \ \ X_{k+1} = f_k(X_k,U_k,W_k),
\quad k \in \{ 1, \dots, N-1 \},\\
&\ \ U_k = \pi_k(X_{k}),
\quad k \in \{ 0, \dots, N-1 \},\\
&\ \ x_0 \ \text{given},
\end{align}
\end{subequations}
where the time horizon is between $0$ and $N$. In (\ref{eq:rl_problem_objective}) the cost is set as the expectation of the \emph{objective function}
\begin{equation} \label{eq:objective}
J^{\bar{\pi}} = r_N(X_{N}) + \sum_{k=1}^{N-1} r_k(X_{k}, X_{k-1}, U_{k-1}),
\end{equation}
with $r_k : \C{X} \times \C{X} \times \C{U} \rightarrow \BB{R}$ and $r_N:\C{X} \rightarrow \BB{R}$ being the \emph{rewards} received, at each $k$, by the agent. In what follows,
whenever we assume a function or quantity is stationary, we drop the subscript $k$ in the notation.
We observe that in many RL scenarios, even if the system dynamics $f_1, \dots, f_{N-1}$ are not perfectly known, some partial knowledge about the plant (from e.g. first-principles) might be available and encoded in some mathematical model of the plant. We propose that this limited information can be exploited to design a feedback control law (or control tutor) that can be used to assist and drive the learning process towards the solution of a control problem of interest, reducing the learning times and improving the control performance. In particular, the control tutor can be invoked under certain circumstances during the learning stage to suggest actions that the agent can take as an alternative to those computed using a more traditional approach, e.g. obtained by reading the $Q$-table.
\section{Control Tutored Reinforcement Learning}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figures/ctrl_scheme.pdf}
\caption{Schematic of the Control-Tutored Reinforcement Learning (CTRL) framework.}
\label{fig:ctrl_scheme}
\end{figure}
We start by assuming that we have an estimate of $f$, say $\hat{f} : \C{X} \times \C{U} \rightarrow \C{X}$, so that the dynamics of system \eqref{eq:dynamical_system} is rewritten as
$f(x,u,w) = \hat{f}(x, u) + \delta (x, u, w)$, $\forall x \in \C{X}, \forall u \in \C{U}, \forall w \in \C{W}$,
where $\delta : \C{X} \times \C{U} \times \C{W} \rightarrow \C{X}$ describes the effect of unknown terms in the dynamics and/or of noise on the system's dynamics. We term the (possibly, model-based) policy obtained by considering only $\hat{f}$ as the \emph{control tutor policy}, and denote it by $\pi^{\R{c}} : \C{X} \rightarrow \C{U}$.
The architecture of the \emph{Control Tutored Reinforcement Learning} (CTRL) \citep{de2020tutoring} is schematically shown in Figure \ref{fig:ctrl_scheme}.
The figure highlights the presence of a switching condition $\zeta$ that orchestrates, at each $k$, the use of either a policy coming from a RL algorithm or the tutor policy. The result is the following switching policy used for learning:
\begin{equation}\label{eq:ctrl}
\pi(x) =
\begin{dcases}
\pi^{\R{rl}}(x), & \text{if } \zeta \ \text{is true,}\\
\pi^{\R{c}}(x), & \text{otherwise},
\end{dcases}
\end{equation}
where $\zeta$ is a Boolean function (that might depend on time, previous states, etc.) and $\pi^{\R{rl}}$ is the policy of a RL algorithm.
For concreteness, we now provide a simple expression for the control tutor policy $\pi^{\R{c}}$.
First, let $\bar{\C{U}} \supseteq \C{U}$ ($\bar{\C{U}}$ might be a continuous set whose discretization yields $\C{U}$); then, from $\hat{f}$ we can design a feedback control strategy $v : \C{X} \rightarrow \bar{\C{U}}$.
At this point, from $v$, letting $\epsilon^{\R{c}} \in (0, 1)$, and $\forall x \in \C{X}$, we take the \emph{control tutor policy} in \eqref{eq:ctrl} as
\begin{equation}\label{eq:tutor_policy}
\pi^\R{c}(x) = \begin{dcases}
\arg \min\limits_{ u \in \C{U}} \norm{v(x) - u},
& \text{with probability } 1 - \epsilon^{\R{c}},\\
u \sim \R{rand}(\C{U}),
& \text{with probability } \epsilon^{\R{c}},\\
\end{dcases}
\end{equation}
On the other hand, for the reinforcement learning policy $\pi^{\R{rl}}$ in \eqref{eq:ctrl}, we adopt an $\epsilon$-greedy Q-learning solution, i.e.,
\begin{equation}\label{eq:rl_policy}
\pi^{\R{rl}}(x_k) = \begin{dcases}
\arg \max_{u\in \C{U}}Q_k(x_k,u),
& \text{with probability } 1 - \epsilon^{\R{rl}},\\
u \sim \R{rand}(\C{U}),
& \text{with probability } \epsilon^{\R{rl}},\\
\end{dcases}
\end{equation}
where $\epsilon^{\R{rl}} \in (0, 1)$ and $Q_k : \C{X} \times \C{U} \rightarrow \BB{R}$ is the well-known \emph{state-action value function} \citep{Sutton1998, bertsekas1996neuro} at time $k$.
At time $k$, once an action is selected from either $\pi^{\R{rl}}$ or $\pi^{\R{c}}$, the corresponding reward is obtained and used to update the $Q$-table according to the law
\begin{equation}\label{eq:Q_update}
Q_{k+1}(x_{k},u_{k}) = (1-\alpha)Q_k(x_{k},u_{k}) + \alpha[r(x_{k+1}, x_{k}, u_{k}) + \gamma \max\limits_{u \in \C{U}}Q_k(x_{k+1},u)],
\end{equation}
where $\alpha \in (0, 1]$ is the \emph{learning rate} and $\gamma \in (0, 1]$ is the \emph{discount factor}.
The remaining term to be defined in \eqref{eq:ctrl} is $\zeta$.
In the following, we present two alternative choices for $\zeta$ that result into two different algorithms.
\subsection{Control-Tutored Q-Learning}\label{sec:ctql}
This first algorithm based on the CTRL framework is the \emph{Control-Tutored Q-learning} (CTQL), which was first presented in \cite{de2020tutoring}.
This algorithm uses a reward with a specific structure.
In particular, let $x^* \in \C{X}$ be a \emph{goal state},
and $d(x) \coloneqq \norm{x - x^*}^2$.
Moreover, letting $\theta \in \BB{R}_{>0}$ and $\bar{\rho} \in \BB{R}_{>0}$, we let the prize function
\begin{equation}\label{eq:prize_reward}
\rho(x) = \begin{dcases}
\bar \rho, & \text{if } \norm{x - x^*} < \theta, \\
0, & \text{otherwise},
\end{dcases}
\end{equation}
and the reward $r_k$ in \eqref{eq:objective} is given as
\begin{equation}\label{eq:reward_ctql}
r(X_{k}, X_{k-1}, U_{k-1}) = d(X_{k-1}) - d(X_{k}) + \rho(X_{k}),
\quad k=1,\ldots,N-1,
\end{equation}
with $r_N(X_N) = 0$.
The switching criterion $\zeta$ in \eqref{eq:ctrl} depends on the current state $x_k$, where
\begin{equation}\label{eq:zeta_ctql}
\zeta \ \text{is} \begin{cases}
\text{true}, & \text{if}\ \max\limits_{u \in \C{U}} Q_k(x_k,u) > 0,\\
\text{false}, & \text{otherwise}.
\end{cases}
\end{equation}
Additionally, $\forall x \in \C{X}, \forall u \in \C{U}$, we initialize $Q_0(x,u) = 0$.
Thus, in the first phase of learning, when limited information about the environment is available, the control tutor policy $\pi^{\R{c}}$ drives the learning process.
Then, gradually, as the values of the $Q$-table are updated using \eqref{eq:Q_update}, the reinforcement learning policy $\pi^{\R{rl}}$ is preferred.
\subsection{Probabilistic Control-Tutored Q-Learning}
Although we found the CTQL to have better performance with respect to the classical Q-learning in certain scenarios (see Section \ref{sec:comparison_learning_performance}), the reward \eqref{eq:reward_ctql} does not satisfy the hypotheses used in the classical proof of convergence used for the Q-learning (see, e.g., \citep{bertsekas1996neuro}), as it is not either non-negative or non-positive.
Moreover, we verified that
the CTQL fails when the reward function is changed or not chosen appropriately.
Therefore, we propose next a simpler probabilistic-based choice for the Boolean condition
$\zeta$ in \eqref{eq:ctrl}. We name the resulting algorithm as \emph{probabilistic Control Tutored Learning} (pCTQL), where by removing the constraints on the reward function required by the deterministic approach, we can use a more standard reward function (compliant with e.g. the classic QL proofs).
In particular, letting $\beta \in [0, 1]$,
\begin{equation}\label{eq:zeta_pctql}
\zeta \ \text{is} \begin{cases}
\text{true}, & \text{with probability } \beta,\\
\text{false}, & \text{otherwise},
\end{cases}
\end{equation}
the pCTQL policy is defined as
\begin{equation}\label{eq:policy_pctql}
\pi(x) = \begin{dcases}
\arg \max_{u\in \C{U}}Q_k(x,u),
& \text{with probability } \beta (1 - \epsilon^{\R{rl}}),\\
\arg \min\limits_{ u \in \C{U}} \norm{v(x) - u},
& \text{with probability } \omega \coloneqq (1 - \beta) (1 - \epsilon^{\R{c}}),\\
u \sim \R{rand}(\C{U}),
& \text{otherwise},\\
\end{dcases}
\end{equation}
Note that it is also possible to introduce a dependency of the probability $\beta$ on the current state, time, or other quantities.
\section{Metrics}
Here we define several metrics to characterize and compare quantitatively the performance of different control algorithms. Each numerical simulation is run in $S \in \BB{N}_{> 0}$ independent sessions.
Each session is composed of $E \in \BB{N}_{> 0}$ episodes: the learned quantities (e.g., $Q$-table) are carried over from one episode to the next, and re-initialized at each session.
Each episode consists of a simulation of $N \in \BB{N}_{> 0}$ time steps.
We let $J_e^{\pi}$ be the cumulative reward (as given in \eqref{eq:objective}) obtained in episode $e$.
Moreover, we let the \emph{goal condition} be a Boolean proposition that assesses whether the control goal can be considered as having been achieved in an episode (the specific form of the goal condition depends on the task at hand).
We define the following three metrics to assess the learning performance.
\begin{definition}[Learning metrics]\label{def:learning_metrics}
(i) The \emph{average cumulative reward} is
$J_{\R{avg}}^{\pi} \coloneqq \frac{1}{E} \sum_{e = 1}^E J_e^{\pi}$.
(ii) The \emph{terminal episode} $E_{\R{t}}$ is the smallest episode such that the goal condition is satisfied for all $e \in \{E_{\R{t}}- 30, \dots, E_{\R{t}}\}$.
(iii) The \emph{average cumulative reward after terminal episode} is
$J_{\R{avg}, \R{t}}^{\pi} \coloneqq \frac{1}{E_{\R{t}}} \sum_{e = E_{\R{t}}}^E J_e^{\pi}$.
\end{definition}
$J_{\R{avg}}^{\pi}$ is a common metric typically used in RL \citep{duan2016benchmarking, wang2019benchmarking}; $E_{\R{t}}$ is used to assess when the learning phase might be considered concluded, and thus to evaluate data efficiency; $J_{\R{avg}, \R{t}}^{\pi}$ describes how performing the controller is, in terms of rewards, once training is completed.
Next, we define two metrics inspired by those commonly used in control theory to assess the transient and steady-state performance of an algorithm.
Let again $x^*$ be a goal state, let $\eta \in \BB{R}_{\ge 0}$, $N^- \in \BB{N}_{>0}$ with $N^- < N$, and let the goal condition be true if
\begin{equation}\label{eq:goal_condition}
\exists \bar{k} \in [0, N^-] :
\norm{x_k - x^*} \le \eta,
\quad \forall k \in [ \bar{k}, N].
\end{equation}
\begin{definition}[Control metrics]\label{def:control_metrics}
(i) In an episode, the \emph{settling time} $k_{\R{g}}$ is the smallest value of $\bar{k}$ that fulfills \eqref{eq:goal_condition}.
(ii) The \emph{steady state error} is $e_{\R{g}} \coloneqq \frac{1}{N - k_{\R{g}} + 1} \sum_{k = k_{\R{g}}}^{N}
\norm{x - x_{\R{g}}}$.
\end{definition}
\section{Benchmark Description}
\subsection{Control Problem}
As a benchmark problem to compare the performance of the proposed algorithms, we consider the problem of stabilizing a pendulum in its inverted position, provided by the OpenAI Gym framework \citep{brockman2016openai, GYM}.
This problem is particularly representative for two reasons.
(i) As the upward position is unstable and the the system dynamics is nonlinear, this problem is typically used in control theory as a test for new control strategies \citep{Khalil:1173048}.
(ii) We will select a linear feedback controller ($v$ in \eqref{eq:tutor_policy}), which by itself cannot stabilize the pendulum.
This means that any benefit observed when using CTQL and pCTQL will be due to the combination of the reinforcement learning policy and the model-based one, and not just the latter.
\paragraph{Environment.}
The pendulum is a rigid rod of length $l = 1 \ \R{m}$, with a homogeneous distribution of mass $m = 1 \ \R{kg}$;
its moment of inertia is $I = ml^2/3$ and it is affected by gravity, with acceleration $g$.
We let $x_k = [x_{1,k} \ \ x_{2,k}]^{\mathsf{T}}$, where $x_{1,k}$ and $x_{2,k}$ are the angular position and angular velocity of the pendulum, respectively; $x_{k,1} = 0$ corresponds to the unstable vertical position.
The control input $u_k$ is a torque applied to the pendulum.
The discrete-time dynamics is obtained by discretizing the continuous-time dynamics with a sampling time $T = 0.05 \ \R{s}$ using the forward Euler method.
Unless noted otherwise, the initial condition is the downward stable position $\tilde{x}_0 = [\pi \ \ 0]^{\mathsf{T}}$.
\paragraph{State and control spaces.}
The spaces for states and control variable are bounded, so that
$x_{k} \in \left[ -{\pi}, {\pi} \right] \times \left[-8, 8 \right]$, and
$u_k \in [-2, 2]$.
Additionally, both spaces are discretized as follows.
Concerning $x_{1, k}$,
the interval $\left[ -\pi, -\frac{\pi}{9} \right]$ is discretized into 8 equally spaced values,
$\left( -\frac{\pi}{9}, -\frac{\pi}{36} \right]$ into 7 values, and
$\left( -\frac{\pi}{36}, 0 \right]$ into 5 values;
$\left[ 0, \pi \right]$ is discretized in an analogous fashion.
Concerning $x_{2, k}$,
$\left[ - 8, -1 \right]$ is discretized into 10 values, and
$\left( - 1, 0 \right]$ into 9 values
(analogously for $\left[ 0, 8 \right]$).
Concerning $u_k$,
$\left[ - 2, -0.2 \right]$ is discretized into 9 values, and
$\left( -0.2, 0 \right]$ into 4 values
(analogously for $\left[ 0, 2 \right]$.
\paragraph{Iterations.}
For each set-up, we run $S = 10$ sessions and average the results.
For each session, we run $E = 10000$ episodes, composed of $N = 400$ time steps.
\paragraph{Goal and rewards.}
The objective is to stabilize the pendulum in its upward position, $x^* = [0 \ \ 0]^{\mathsf{T}}$.
Concerning the goal condition in \eqref{eq:goal_condition}, we take $N^- = 300$ and $\eta = 0.05 x_{\R{max}}$, where $x_{\R{max}} \coloneqq \norm{[\pi \ \ 8]}$.
This goal is encoded in two reward functions.
The first one is
\begin{equation}\label{eq:reward_distance}
r^{\R{a}}(X_{k}, X_{k-1}, U_{k-1}) = d(X_{k-1}) - d(X_{k}) + \rho(X_{k}),
\end{equation}
where $d(x) \coloneqq x_{1}^2 + 0.1 x_{2}^2$, and
$\rho$ was given in \eqref{eq:prize_reward}, with $\bar{\rho} = 5$ and $\theta = 0.05$.
The second reward function we will consider is the standard Gym reward, i.e.,
\begin{equation}\label{eq:reward_gym}
r^{\R{g}}(X_{k}, X_{k-1}, U_{k-1}) = X_{1,k}^2 + 0.1 X_{2,k}^2 + 0.001 U_{k-1}^2.
\end{equation}
\paragraph{Hyperparameters.}
In \eqref{eq:Q_update}, we take $\gamma = 0.97$ and $\alpha = \left( 1+\frac{e}{1000} \right)^{-\frac{1}{2}}$, where $e$ is the current episode \citep{even2003learning}, so that the learning rate decays approximately from $0.7$ to
$0.3$, over $10000$ episodes.
In \eqref{eq:tutor_policy} and \eqref{eq:rl_policy}, we take $\epsilon^{\R{c}} = \epsilon^{\R{rl}} = 0.03$.
Concerning $\beta$ in \eqref{eq:zeta_pctql}, we tested $ \beta \in \{ 0.9990 , 0.9948, 0.9897, 0.9485, 0.8969 \}$, which approximately corresponds to $\omega \in \{ 0.001, 0.005, 0.010, 0.050, 0.100 \}$ in \eqref{eq:policy_pctql}.
\paragraph{Feedback control law.}
We assume we have partial information on the pendulum dynamics, in the form of an approximate dynamics $\hat{f}$.
In particular, $\hat{f}$ is the linear dynamics that is topologically equivalent to the nonlinear dynamics of the pendulum, close to the origin $[0 \ \ 0]^{\mathsf{T}}$ (also the goal state).
Namely,
$\hat{f}\left(x_{k}, v_{k}\right) = A x_{k} + B v_{k}$,
where
$A = \left[ \begin{smallmatrix}
0 & 1 + T \\
3 T g /2 l & 1
\end{smallmatrix} \right]$
and
$B = \left[ \begin{smallmatrix}
0 \\
T/I
\end{smallmatrix} \right]$
.
From $\hat{f}$, we synthesize the linear controller $v_k = -Kx_k$, where $K = [5.83 \ \ 1.83]^{\mathsf{T}}$.
This controller can locally stabilize the pendulum in its inverted position from nearby initial conditions, and is obtained, for the sake of simplicity, via a pole placement technique, assigning poles
to have an acceptable settling time.
Note that this controller if used on its own is in unable to swing up the pendulum from its downward asymptotically stable position.
\section{Comparison of Learning Performance}
\label{sec:comparison_learning_performance}
\paragraph{Case of reward (\ref{eq:reward_distance}).}
First, we compare Q-Learning, CTQL and pCTQL with different values of $\omega$, when using reward \eqref{eq:reward_distance}.
The results are reported in Figures
\ref{fig:scenario_1}.(a)--(b), and \ref{fig:learning_metrics}.(a)--(c), containing the cumulative reward per episode $J_e^\pi$,
the frequency with which the control tutor is used, and the learning metrics (Definition \ref{def:learning_metrics}), respectively.
For the sake of clarity, in Figure \ref{fig:scenario_1} the results of the pCTQL were only plotted for $\omega = 0.01$, as we found that value to give the best performance overall.
From Figure \ref{fig:learning_metrics}.(a), comparing CTQL and pCTQL to Q-learning, we observe that $E_{\R{t}}$---a measure of data efficiency---is smaller (by a statistically significant margin) for the CTQL and for the pCTQL with $\omega = 0.05$; on the other hand, the pCTQL with other values of $\omega$ are on par with the Q-learning.
This fact is also visible in Figure \ref{fig:scenario_1}.(a), as the reward curves of pCTQL and CTQL grow earlier than that of Q-learning, and in Figure \ref{fig:learning_metrics}.(b), showing that the average rewards $J_{\R{avg}}^{\pi}$ are higher for CTQL and pCTQL.
Finally, Figure \ref{fig:scenario_1}.(b) shows that CTQL uses the control tutor policy more in the beginning, and progressively less as episodes are completed.
\paragraph{Case of reward (\ref{eq:reward_gym})}
We also compared the performance of Q-learning and pCTQL when using reward \eqref{eq:reward_gym}; the results are portrayed in Figures
\ref{fig:learning_metrics}.(d)--(f).
We see that pCTQL with $\omega = 0.01$ is comparable to Q-learning in terms of learning time ($E_{\R{t}}$), yet obtains a larger average reward ($J_{\R{avg}}^{\pi}$) and average reward after terminal episode ($J_{\R{avg}, \R{t}}^{\pi}$), confirming the effectiveness of a control tutor-based architecture, even when the reward has a structure different from \eqref{eq:reward_ctql}.
\begin{figure}[t]
\centering
\stackunder[1pt]{\includegraphics[width=.48\linewidth]{figures/scenario1_reward.pdf}}{}
\label{fig:scenario_1_reward}
\stackunder[1pt]{\includegraphics[width=.48\linewidth]{figures/scenario1_tutor.pdf}}{}
\label{fig:scenario_1-tutor}
\caption{(a) Cumulative reward per episode $J^\pi_e$, obtained with reward \eqref{eq:reward_distance}.
(b) Percentage of steps the control-tutor policy $\pi^{\R{c}}$ was used in each episode.
In both (a) and (b) the solid curves are the mean of the results of $S$ sessions; for readability, the curves are averaged with a moving average of 100 samples (taken on the right); shaded areas corresponds to the means plus or minus the standard deviations.}
\label{fig:scenario_1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/learning_metrics.pdf}
\caption{(a), (b), (c): $E_{\R{t}}$, $J_{\R{avg}}^{\pi}$, and $J_{\R{avg}, \R{t}}^{\pi}$, respectively (Definition \ref{def:learning_metrics}), with reward \eqref{eq:reward_distance}.
(d), (e), (f): analogously but with reward \eqref{eq:reward_gym}.
pCTQL is reported with different values of $\omega$, expressed in percentage.
The means and standard deviations of $S$ sessions are portrayed.
Values that are statistically significantly different from those of the Q-learning are in bold (according to a Welch’s t-test with $p$-value less than $0.05$ \cite{welch1947generalization}).}
\label{fig:learning_metrics}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/validation_metrics.pdf}
\caption{(a), (b): $k_{\R{g}}$ and $e_{\R{g}} / x_{\R{max}}$, respectively (Definition \ref{def:control_metrics}), with reward \eqref{eq:reward_distance}, under nominal conditions.
(c), (d): analogous, but with reward \eqref{eq:reward_gym}.}
\label{fig:validation_metrics}
\end{figure}
\section{Comparison of Control Performance}
\paragraph{Nominal conditions.}
We also compared the algorithms in terms of their control performance at the end of the learning stage, using the metrics given in Definition \ref{def:control_metrics}.
The results, using both rewards \eqref{eq:reward_distance} and \eqref{eq:reward_gym} are shown in Figure \ref{fig:validation_metrics}.
Firstly, from Figure \ref{fig:validation_metrics}.(a),(c) we
show that, as it is desirable, the differences in settling time of pCTQL and CTQL with respect to Q-learning are not statistically significant.
Secondly, when using reward \ref{eq:reward_distance} we observed that the CTQL achieves the best (lowest) steady state error, whereas when using reward \ref{eq:reward_gym} (Figure \ref{fig:validation_metrics}.(d)), the smallest error is given by the pCTQL with $\omega = 0.01$.
\paragraph{Perturbed conditions.}
To test the robustness of the learned control strategies to changes in the environment, we generated $1000$ set-ups, by varying the initial conditions randomly (with a uniform distribution) in the state and control spaces, and varying the mass $m$ and length $l$ of the pendulum by $\pm 5\%$ of their nominal values, and using the Latin hypercube method \citep{loh1996latin}. The results, not portrayed here for the sake of brevity, show that, as is desirable, we obtain similar settling times for all the algorithms. Also, concerning the steady state error, when using reward \eqref{eq:reward_distance}, for all the algorithms, performance remain centered around that obtained under nominal conditions. Differently, when using reward \eqref{eq:reward_gym}, pCTQL displays a larger error when compared to that obtained under nominal condition (which was however lower than that of Q-learning), whereas Q-learning retains the same mean.
\section{Conclusions}
We presented a deterministic and a probabilistic Control-Tutored Q-learning strategy, that integrate a feedback control law synthesized on a partial model of the plant within a Q-learning framework to render the learning process faster and improving the performance of the learnt policies in achieving a control goal of interest.
We compared the control-tutored strategies with a classical Q-learning approach using the inverted pendulum stabilization benchmark from OpenAI Gym as a representative control problem.
We found that, when compared to Q-learning, CTQL requires fewer data samples and has a larger average reward, while pCTQL yields higher rewards with a comparable number of data samples; moreover, both CTQL and pCTQL yield lower regulation error when certain reward functions are used. Our numerical results show that both from a learning and a control viewpoint using a control-tutored learning approach might be beneficial.
The next step is the derivation of proofs of convergence for the control-tutored algorithms presented in this paper.
Also, we wish to uncover and formally characterize the relationships among the specific choice of the reward function, the performance of the algorithms and the approximate system dynamics needed to synthesize the control tutor.
We wish to emphasize that embedding a control tutor in the loop could be used to render more efficient learning strategies other than $Q$-learning. This will also be the subject of future investigation.
\newpage
|
2,877,628,090,212 | arxiv | \section{Prelude}
Legal arguments extensively involve text and speech. For any statement to be generated in the legal context, we require a comprehensive knowledge of prior information regarding the case, the arguments that have previously been made as well as external knowledge and facts. This makes the industry heavily data-driven, and consequently, an interesting domain for the application of machine learning models to automate and improve contextual coherence of arguments.
\vspace{3 mm}
\\
The application of machine learning in the legal industry could imply categorizing certain types of documents or generating plausible arguments to help lawyers with their cases. With efficient algorithms, lawyers can avoid repetitive work and instead focus on complex, higher-value analysis to solve their clients’ legal problems, resulting in a substantial saving of time and effort. This redefines the scope of what lawyers and firms can achieve, allowing them to take on cases which would have been too time-consuming or too expensive for the client if they were to be conducted manually.
\vspace{3 mm}
\\
Since law is essentially expressed in vernacular, Natural Language Processing (NLP) is a crucial component in understanding and prediction of information and contexts. NLP aids in solving computational problems like information retrieval, information extraction, speech recognition and question-answering. One of the most efficient techniques to achieve accurate prediction of a sequence of words is Neural Machine Translation, which has subsequently led to the development of the domain of Transfer Learning.
\section{Transfer Learning}
There are several drawbacks when using algorithms that perform supervised learning (SL), unsupervised learning (UL) or reinforcement learning (RL), especially when it comes to robustness to new/unknown inputs. Supervised learning is not generalizable as it breaks down when we do not have sufficient labeled data to train a reliable model. Unsupervised learning is generalizable only when a stable prior distribution exists and it too shows fragility when we are presented with an outlier. Although reinforcement learning is generalizable, it is computationally intensive since it addresses the task of selecting actions to maximize the reward function through state observation and interaction with the environment.
\vspace{3 mm}
\\
Transfer Learning [11] is an attempt to use one supervised learning model to work on another related setting with minimal re-training. It allows us to deal with scenarios where a model can be trained using a similar "pre-trained" one by leveraging the existing labeled data of some related task or domain. This gained knowledge is stored for solving the source task in the source domain and we apply it to our problem of interest. For example: in our case we are dealing with language modeling on a legal corpus which is also equivalently a valid English text dataset. Hence, two models that show breadth-level similarity could have a common parent domain and due to this advantage, transfer learning is witnessing a sharp rise in its usage across different applications.
\vspace{3 mm}
\\
Transfer Learning is a key aspect of this project since the generation of legal arguments involves learning from prior actions to make more informed and coherent statements. In the NLP domain, transfer learning entails making use of correlations between words generated previously with the context currently under discussion. This has enabled prediction algorithms to become more accurate and efficient.
\section{Problem Statement}
It has always been challenging to generate coherent long-form text. Even with the breakthroughs in Neural Machine Translation algorithms in achieving local dependencies within sentences, we still have a long way to go before we can fully capture global dependencies within a conversation. The concept of 'attention' introduced in [21, 22] brought a fresh perspective to the field of language modeling. This concept has been extensively studied in intra-sentence framework but in recent times there have been a few instances of focused research directed towards conversational coherence [5, 6].
\vspace{3mm}
\\
Our main contribution in this paper is to build a dialog agent by proposing a unique architecture that ensures intra-sentence as well as inter-sentence coherence and cohesion. Such a model can prove useful in many industry-based as well as everyday tasks. In this paper, we propose a use-case for such a model in the legal industry, which is highly text-data intensive.
\section{Preliminaries}
It is imperative to understand the historical evolution of similar ideas in the sphere of natural language processing to make sense of our extension. We will briskly step through some of the recent advancements before taking a look at the attention revolution ushered in by the transformer model.
\subsection{Early Work}
Natural Language Understanding (NLU) had its origins in techniques that did not use any explicit hypothesis classes in their respective SL settings. Latent Semantic Analysis (LSA) [1, 2], for instance, operates on extracting dominant tokens from a corpus of documents by working with the Term Frequency-Inverse Document Frequency (TF-IDF) matrix. The $(i, j)^{th}$ entry of this matrix represents the term frequency of token $i$ in the document $j$ normalized by the term frequency of token $i$ across all documents. Upon performing a low-rank approximation (via the Eckart-Young Theorem) to this matrix, one is able to construct low dimensional representations of these "document" column vectors thereby allowing the extraction of "latent semantics" of the corpus in a computationally efficient manner. With LSA one could, in principle, perform sentence classification, but it would clearly fail if the query text to be classified did not contain any of the dominant tokens.
\vspace{3 mm}
\\
Latent Dirichlet Allocation (LDA) [3, 4] aims to improve on this deficiency by modeling the interaction between the tokens from a probabilistic viewpoint. It works by constructing "topics" for each document: wherein each topic consists of several such dominant tokens. Words in the document can now be seen as mixture models of such topics, and the weights of these mixtures are sampled from a separate multinomial distribution. It derives its name from the fact that the parameter set for this multinomial distribution is actually derived from a Dirichlet distribution, thus generating these multivariate parameter vectors efficiently.
\vspace{3 mm}
\\
These techniques are robust due to their independence from the requirement of working within a hypothesis framework. By utilizing the Bayesian mixture model setting in general, they come very close to theoretical guarantees of expected performance but, due to the very same architectural generality, are not suitable for fine-tuned/context-based NLU tasks. For example: LDA would be successful in retrieving closest documents matching a certain token or a sentence, but it would not tell us whether the said token is used in a positive or negative connotation.
\subsection{Recurrent Neural Networks}
While Recurrent Neural Networks themselves have been around well before the ascent of language modeling [7], the real impetus for their usage in tasks involving language translation and understanding stemmed from the construction of an Encoder-Decoder system [8]. The encoder RNN would process input tokens sequentially and concomitantly update its hidden state variable until the separator token is issued. The decoder then uses this final hidden encoder state vector to generate output tokens that maximize the conditional probability i.e., the likelihood of the source given a specific translated sentence.
\begin{figure}[!htb]
\includegraphics[scale=0.65]{EncDec.PNG}
\caption{The Encoder-Decoder RNN Architecture}
\end{figure}
The authors of [8] subsequently propose an upgrade to the RNN cells by introducing a "gate" inside each cell that allows the $i^{th}$ cell to choose from one of the two actions: (a) update the hidden state vector coming from the $(i-1)^{st}$ cell or (b) to \textit{forget} this vector and instead reset the state back to $h_0$. This extension goes by the name of the Gated Recurrence Unit (GRU). It allows the standard RNN architecture to adapt to selective memory and makes it more agile towards handling irrelevant tokens. However, it has been definitively proven [9, 10] that GRUs significantly under-perform when compared to their more elaborate predecessor, the LSTM cell, which we now turn to.
\subsection{Long Short-Term Memory}
The LSTM model was introduced prior to the encoder-decoder model as a multi-gate variant of the GRU. The original idea [12] started off as a "Constant-Error-Carousel" system which was designed to mitigate the impending training issues [13, 14] during backward propagation (backprop) in simple RNN/GRU based models. In short, the backprop step while training RNNs starts involving factors in the gradient that contain the weight parameters raised to the power of the memory span ($\delta t$) and depending on whether the parameter is less or greater than 1, the gradient can vanish or explode exponentially with $\delta t$. Due to the large number of such factors, this problem becomes difficult to resolve via clipping/projecting the gradients back onto a fixed domain every time such a violation takes place during automatic differentiation.
\vspace{3 mm}
\\
LSTMs solve the problem of gradient divergence by circulating the error inside the cell (hence the "carousel") using multiple stages of hidden state gating. The GRU was actually based off this initial design, wherein the authors of [8] decided to keep only the "forget" gate in the end to make the overall architecture nimble and easy to train. The template for the LSTM cell used in Figure 2 was taken from the highly recommended blog article by Chris Olah [15].
\vspace{3 mm}
\\
\begin{figure}[!htb]
\includegraphics[scale=0.4]{LSTM.PNG}
\caption{The LSTM State Machine}
\end{figure}
This cell structure is one of the many variants of the original model devised by Hochreiter and Schmidhuber, and [16, 17] provide another variant that allow these gates to "peep" into the processing of the other gates for more intricate dynamics. Another excellent resource especially for text generation using LSTMs is by Karpathy [18] which demonstrates the diverse applicability of these ideas: from Shakespearean playwriting to automated mathematical proof generation in \LaTeX. However, since the building blocks continue to be RNN centric, even LSTMs have the issue of divergent gradients [14]: they do not fix the problem but simply make it less likely by delaying it through error containment. Furthermore, due to the multi-gate internals which themselves comprise of smaller RNNs, training on a large document corpus by sequentially feeding in inputs is slow and impractical.
\section{Attention \& Transformers}
Notwithstanding these problems, adding further control to enable long range dependency makes LSTMs a good contender for contextual argument generation, but it faces a serious functional hazard when presented with sentences that have syntactic ambiguities. Examples of such situations, which (often unintentionally) involve multiple interpretations of a given sentence are found very frequently in literature. Take the following example that demonstrates the "Dangling Modifier" ambiguity [19]
\vspace{3 mm}
\\
\textsc{\textbf{Leafing through the pages}, the book appeared to be much more than what it initially seemed to be.}
\vspace{3 mm}
\\
For the human reader, the phrase in bold clearly refers to the narrator's action with the book as he describes its contents, but for an algorithm it may as well have been the book leafing through its own pages since it is the only noun phrase visible in the sentence. Things become complex if we have multiple noun phrases around an ambigious pronoun:
\vspace{3 mm}
\\
\textsc{He peered through the \textbf{pet door} to look at the poor \textbf{dog} and saw that \textbf{it} was visibly shivering.}
\vspace{3 mm}
\\
While the sentence may seem far too obvious from a human perspective on understanding: the dog was visibly shivering as the narrator looked inside through the pet flap, the algorithm has the propensity to correlate the pronoun "it" with the "pet door" as well, thereby wrongly concluding that the petflap was shivering instead. The takeaway here is that, not only is the causal structure of the token placement an important facet, but also establishing correlation between their relative locations becomes crucial when deriving context. It is interesting to see that this implies a significant loss of "exchangeability" when the token ordering is seen as a joint density, and thus prevents the use of the theorem due to Bruno de Finetti [20] that gave the license to probabilistic techniques such as LDA to be seen as functionally useful.
\vspace{3 mm}
\\
This idea of learning these correlation structures is known in the NMT community as "self-attention" [21] and it was effectively used to tackle the issue of long sentence parsing and translation in classic encoder-decoder RNN settings. However, the landmark paper [22] due to Vaswani et. al., shows that comparable results can be achieved by only learning these correlation structures and nothing else. That is, training based on these attention scores alone is enough for observing comparable levels of contextual relevance (thereby justifying the title of [22], which we think is cool). The block diagram of a Transformer is awfully similar to that of the LSTM and RNN based translation systems, excecpt that it now tracks all the encoder hidden states with symmetric attention span, involves positional encoding to track token locations, and uses a "masked" attention unit on the decoder side to perform translation in one go:
\begin{figure}[!htb]
\includegraphics[scale=0.6]{Transformer.PNG}
\caption{The Transformer}
\end{figure}
\\
The attention unit that is proposed in [22] is very intuitive in terms of how it is able to establish the dominant neighboring tokens for a given keyword. Given a token we learn a dictionary of nearby token configurations that index the corresponding likelihood probabilities/correlation coefficients. The token whose attention span is to be generated is given as a vector embedding $\bar{Q}$ to the following function:
$$\mathcal{A} = \sigma(\bar{Q}\hat{K}^T)\bar{V}$$
Here $\hat{K}$ represents the "token configuration key" matrix and $\bar{V}$ represents the vector of likelihoods. There is a stark difference between the way these attention units behave during encoding and decoding, in the sense that during encoding we can look both ways due to complete information (the "symmetric" unit: thus enabling bi-directionality) but during next token generation we mask the right half of the sentence while producing newer tokens. We have provided an infinitely rudimentary exposition of what is actually a vast subject and we refer the reader to [22, 23, 24] for finer details (such as Multi-Head Attention (MHA) and so on).
\section{The Pre-Training Revolution: BERT \& GPT}
The development of transformer architecture has led to the evolution of many state-of-the-art language models in the natural language processing domain. BERT, which stands for Bidirectional Encoder Representation for Transformer [32], is one such revolutionary piece of technology that marks a new era in the field of Natural Language Understanding. It is made by stacking transformer encoder blocks on top of each other. BERT was an important development which managed to combine the bidirectional conditioning of each word that ELMo [31] earlier presented, along with the benefits of a fine-tunable, pre-trained transformer model.
\vspace{3mm}
\\
The paper [32] presented two model sizes for BERT, namely BERT Base and BERT Large. The comparison between these models is presented in the following table. These models can be used for many language understanding tasks like machine translation, sentence classification, question-answering, etc. The input to any of these models is a sequence of words padded with input (token, segment and position) embeddings. Once the input is ready and is received by the first encoder block, 'self-attention' is applied to it and the result is passed on to the feedforward network. The output of the first encoder block is then handed off to the next encoder block and the process repeats. It is important to note here that BERT does not predict the next word, rather it uses Masked Language Model (MLM) to predict random words in a particular sentence by taking into account both the right and left side context within a sentence.
\vspace{3 mm}
\\
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{Model } & \multicolumn{1}{l|}{\textbf{Encoder Block Count}} & \multicolumn{1}{l|}{\textbf{Hidden Units in FF Network}} & \multicolumn{1}{l|}{\textbf{Attention Heads}} \\ \hline
BERT Base & 12 & 768 & 12 \\ \hline
BERT Large & 24 & 1024 & 16 \\ \hline
\end{tabular}
\end{center}
The next innovation that followed was using only the decoder blocks of the transformer which led to the development of GPT, which stands for Generative Pre-trained Transformer [33]. GPT-1, 2 and 3 are different models based on the size of the text corpus it was trained on. Unlike BERT, these models can generate entire sentences and hence prove to be a powerful tool for language generation. GPT generates next token from a sequence of tokens in an unsupervised manner. It is auto-regressive in nature, as the newly generated token is added to the initial input sentence and the whole sentence is again provided as an input to the model. While BERT uses MLM to mask words within the entire sentence, the self-attention layer of the decoder blocks in GPT masks future words and takes into account only the past and present tokens. The processing of each token is quite similar to how it is done in BERT encoder blocks, i.e, the input includes the positional embeddings, then passes through the self-attention layer in the decoder and finally the resulting vector is passed on to the next decoder block.
BERT and GPT are heavily pre-trained, general-purpose NLP models that bring transfer learning to the masses by allowing them to fine tune these models to their context. We have leveraged these models in our architecture to build our dialog agent in legal context.
\vspace{3 mm}
\\
\section{HumBERT}
Our architecture is inspired from the original transformer hypothesis [22] wherein we propose to expand the self-attention concept, which is currently restricted to tokens, to the larger goal of correlation in between consecutive sequences of tokens, or more generally, dialogues.
\begin{figure}[!htb]
\includegraphics[scale=0.45]{ourarch.PNG}
\caption{HumBERT Subcontext Transformer}
\end{figure}
\newpage
We believe that HumBERT can be rightfully considered as a "subcontext" driven transformer wherein it achieves three fundamental tasks:
\begin{itemize}
\item \textbf{\underline{Seek}}:
\begin{itemize}
\item Given an initial query $q_0$, the "JudgeBERT" unit $\mathcal{J}$ is activated to return which broad case file it finds $q_0$ to be most relevant to exist in.
\item JudgeBERT is nothing but the HuggingFace BERT Base model [29] wrapped in a shallow soft-max classifier that gives us a $K$ dimensional logit vector output, where $K = ||\mu||$ and $\mu$ is the total legal corpus used for training tasks [28].
\item We simply select the largest logit entry index and parse it as the case identifier $\mathcal{C}\in \mu$.
\end{itemize}
\item \textbf{\underline{Read}}:
\begin{itemize}
\item HumBERT Vision is the encoder section of the open source "Sentence Transformer" model by Reimers and Gurevych [25]. We will refer to it as $\mathcal{B}(q, \mathcal{C})$ henceforth.
\item It is initialized on the GPU and is trained to recognize "entailment" [26] sequences in our case file. That is, if we split $\mathcal{C}$ into the sentence set $\mathcal{S} = \{S_0, S_1, ..., S_M\}$ and enforce causal ordering on $\mathcal{S}$, we consider $S_{j+1}$ to entail $S_j$ and therefore insert $S_j \to S_{j+1}$ "entailment sequences" in our training set. Obviously, for $\mathcal{C}$ we will have $M$ such sequences.
\item Once trained on $\mathcal{C}$, we construct the vector embedding of $\mathcal{S}$ derived by mapping the encoder on each sentence of the case $\bar{\mathcal{S}} \leftarrow \mathcal{B}(\mathcal{S}, \mathcal{C})$. We also construct the embedding for the query $\bar{q}_0 \leftarrow \mathcal{B}(q_0, \mathcal{C})$
\item We then compute the cosine similarity [27] array $CS_j= <\bar{q}_0, \bar{S}_j>$ and find the location $j*$ which maximizes the correlation. The sentential neighborhood of $S_{j*}$ (which we call $\mathcal{N}_{j*}$) effectively represents the "subcontext" we have been alluding to thus far.
\end{itemize}
\item \textbf{\underline{Reply}}:
\begin{itemize}
\item HumBERT Speech comprises of two sub-units: the conversational attention unit a.k.a the "history"/"memory" unit $\mathcal{H}$ and a response generation engine $\mathcal{G}$ which is essentially GPT-2 pretrained on $\mathcal{C}$.
\item Once the pretraining is done, $\mathcal{G}$ processes the subcontext $\mathcal{N}_{j*}$ by using it as the seed sentence to generate $P$ contextually relevant "arguments" $\{a_1, a_2, ...., a_P\}$ .
\item $\mathcal{H}$ initializes itself once every conversation and holds the last $R$ "arguments" $\{h_1, h_2, ..., h_R\}$ that have been exchanged so far. Of course, when the conversation begins $\mathcal{H} = \{h_1 = q_0\}$.
\item For each $a_l$ we calculate the average historical correlation
$$\rho_l = \frac{1}{R}(\sum_{k=1}^R <h_k, a_l>)$$
\item We pick the argument that has the maximal average correlation and produce the reply $q_1$ to the original query $q_0$. We also update $\mathcal{H} \leftarrow \{q_0, q_1\}$.
\end{itemize}
\end{itemize}
Once $q_1$ is generated, we can cache and reuse the Vision and Speech modules $\mathcal{B}, \mathcal{G}$ to continue conversing without any additional training steps required. The following representative equations describe the iteration at a future step given human response $s_k$:
\vspace{2 mm}
\\
$$\mathcal{H} \leftarrow \mathcal{H} + \{s_k\}$$
$$j* \leftarrow arg\, max_{j = 1, 2, ..., M} <\mathcal{B}(s_k, \mathcal{C}), \bar{S}_j>$$
$$\mathcal{A} = \{a_1, a_2, ..., a_P\} \leftarrow \mathcal{G}(\mathcal{N}_{j*})$$
$$q_{k} \leftarrow max_{a_l \in \mathcal{A}}[\frac{1}{R}\sum_{h \in \mathcal{H}}<h, a_l>]$$
$$\mathcal{H} \leftarrow \mathcal{H} + \{q_k\}$$
\newpage
While our historical cache based conversational relevance scheme is primitive, it is significantly different from the approaches of both [5] and [6]: the former uses a unidirectional transformer and a policy based approach to select the apt response and the latter is suited for intent classification coupled with its matured entailment strategy. We hope to investigate better conversation patterns with different combinations of the free parameters $M, P, R$.
\section{Results}
We have trained JudgeBERT on select cases spread over 40 years of the proceedings of the Supreme Court of India [30]. We validated our model on some specimen initialization texts and our results are encouraging:
\vspace{3 mm}
\\
\textbf{\underline{Example 1}}
\vspace{1 mm}
\\
\textbf{Human: } \texttt{case involving an unjustified amount of paddy in a godown}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{The petitioner denied the correctness of the allegation that he had hoarded rice in an unauthorised fashion. That he took it under his thumb and took it on the spur of the moment was proved by the facts and the petitioners' testimony.}
\vspace{2 mm}
\\
\textbf{Human: } \texttt{where was this paddy and rice being smuggled to?}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{were smuggling rice to Meghalaya for earning undue profit? We must now find out. The answer was simple.}
\vspace{2 mm}
\\
\textbf{Human: } \texttt{what did the case simplify as a "technical error"?}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{In this case at least two grounds are vague, one ground is found to be false and of the remaining in one there is no explanation and in the other there is a lame excuse that the driver of the truck did not furnish the full information. He made his complaint and the court did not, in fact, accept him's evidence in all of them.}
\vspace{2 mm}
\\
\textbf{Human: } \texttt{which previous case did the attorney general and the court rely their decision upon?}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{v. the State of West Bengal, a Bench of five Judges of this Court explained the distinction between the vague grounds and irrelevant grounds and said that they do not stand on the same footing.}
\vspace{3 mm}
\\
\textbf{\underline{Example 2}}
\vspace{1 mm}
\\
\textbf{Human: } \texttt{cashewnut is not considered a foodstuff}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{When cashewnut is exported, it is exported as a foodstuff. And a foodstuff cannot, for example, be consumed by anyone who cannot produce food from them, and consequently, there could not be the same effect on the production of the food which is taken from them.}
\vspace{2 mm}
\\
\textbf{Human: } \texttt{how is a foodstuff defined}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{Therefore, 'foodstuff' need not necessarily mean only the final food product which is consumed. This would mean a total of 10 g/day or more. For example, a daily 2 g meal is equivalent to 2 g of sugar per day.}
\vspace{2 mm}
\\
\textbf{Human: } \texttt{how is an essential commodity defined?}
\vspace{2 mm}
\\
\textbf{HumBERT: } \texttt{What can be looked upon more of an essential commodity than both rice and paddy? It's a question that requires no further investigation. Since we can see all the other components in the rice they are much finer than either the rice or the paddy itself.}
\section{Statistics}
\begin{tabular}{|c|c|}
\hline
\textbf{Metric} & \textbf{Value} \\ \hline
Legal Corpus Size & 1,383,674 Lines (39,665,578 Words) \\ \hline
Training Time for JudgeBERT & 21 Hours (5.5 Hours X 4 Epochs) \\ \hline
Size of JudgeBERT Model & 446 MB \\ \hline
JudgeBERT Online Resource Usage & $\sim$1 GeForce GTX 1080 Ti GPU, 500MB, Loading Time: 9.8s \\ \hline
JudgeBERT Case Retrieval Time & \textless{}1s \\ \hline
HumBERT Vision Online Resource Usage & $\sim$1 GeForce GTX 1080 Ti GPU, 1.2GB, \textless 5 Minutes \\ \hline
HumBERT Speech Online Resource Usage & $\sim$2 GeForce GTX 1080 Ti GPUs, 1.2GB, \textless 5 Minutes \\ \hline
Average Query Response Time & 32 seconds \\ \hline
\end{tabular}
\section{Future Work}
While we were successful in multi-text generation, the question remains to validate the sentence as well as paragraph level relevance. We are looking at incorporating architectural concepts of coherence and cohesion based neural discriminators as elaborated by Cho et. al., in [34]. Simultaneously, our aim is to look into the concept of one-shot or few-shot learning to move closer towards human-like language pattern representations.
\section*{References}
\medskip
\small
[1] Dumais, S.T., Latent semantic analysis. \textit{Annual Review of Information Science and Technology}, Vol. 38, pp. 188-230. doi:10.1002/aris.1440380105, 2004.
[2] Deerwester, S., Dumais, S. T., Furnas, G. W., Landauer, T. K., Harshman, R., Indexing By Latent Semantic Analysis, \textit{Journal of the American Society For Information Science}, 41, 391-407, 1990.
[3] Blei, D. M., Ng, Andrew., Jordan, M. I., Latent Dirichlet Allocation, \textit{Journal of Machine Learning Research}, Vol 3, pp. 993-1022, 2003.
[4] Hofmann T., Learning the Similarity of Documents : an information-geometric approach to document retrieval and categorization, \textit{Advances in Neural Information Processing Systems 12}, pp-914-920, MIT Press, 2000.
[5] Vlasov V., Mosig J. E. M., Nichol A., Dialogue Transformers, \textit{arXiv:1910.00486}, May 2020.
[6] Henderson M., Casanueva I., Mrk\v si\'c N., Su P. H., Tsung-Hsien W., Vuli\'c I., ConveRT: Efficient and Accurate Conversational Representations from Transformers,\textit{ arXiv:1911.03688}, April 2020.
[7] Rumelhart, D., Hinton, G., Williams, R., Learning representations by back-propagating errors, \textit{Nature} 323, 533–536 (1986). https://doi.org/10.1038/323533a0.
[8] Cho K., van Merrienb\"oer B., Gulcehre C., Bahdanau D., Bougares F., Schwenk H., Bengio Y., Learning Phrase Representations using RNN Encoder–Decoder for Statistical Machine Translation,\textit{ arXiv:1406.1078v3}, 2014.
[9] Weiss, G., Goldberg, Y., Yahav, E., On the Practical Computational Power of Finite Precision RNNs for Language Recognition, \textit{arXiv:1805.04908}, 2018.
[10] Britz, D., Goldie, A., Luong, Minh-Thang., Le, Q., Massive Exploration of Neural Machine Translation Architectures. \textit{arXiv:1703.03906}, 2018.
[11] Pratt L., Discriminability-Based Transfer between Neural Networks. \textit{In Advances in Neural Information Processing Systems 5, [NIPS Conference]}. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, pp. 204–211, 1992.
[12] Hochreiter S., Schmidhuber J., Long Short-Term Memory, \textit{Neural Computation}, 9(8), pp 1735-1780, 1997.
[13] Pascanu R., Mikolov T., Bengio Y., On the difficulty of training Recurrent Nueral networks, \textit{arXiv:1211.5063v2}, 2013.
[14] \textit{How does LSTM prevent the vanishing gradient problem?}, Cross Validated Stack Exchange, https://stats.stackexchange.com/q/263956.
[15] Olah C., Understanding LSTM Networks, \textit{Chris Olah's Personal Blog}, https://colah.github.io/posts/2015-08-Understanding-LSTMs/, August 2015.
[16] Greff K., Srivastava R., Koutník J., Steunebrink B., Schmidhuber J., LSTM: A Search Space Odyssey, \textit{arXiv:1503.04069}, 2017.
[17] Graves A., Schmidhuber J., Framewise phoneme classification with bidirectional LSTM and other neural network architectures, \textit{Neural Networks}, Vol. 18(5–6), pp. 602–610, July 2005.
[18] Karpathy A., The Unreasonable Effectiveness of Recurrent Neural Networks, \textit{Andrej Karpathy Blog}, http://karpathy.github.io/2015/05/21/rnn-effectiveness/, 2015.
[19] McArthur, T., The dangling modifier or participle, \textit{The Oxford Companion to the English Language}, pp. 752-753. Oxford University Press, 1992.
[20] Diaconis, P., Freedman, D., Finite exchangeable sequences, \textit{Annals of Probability}, Vol. 8 (4), pp. 745–764, 1980.
[21] Bahdanau, D., Cho, K., and Bengio, Y., Neural Machine Translation by Jointly Learning to Align and Translate, \textit{http://arxiv.org/abs/1409.0473}, 2014.
[22] Vaswani A., Shazeer N., Parmar N., Uszkoreit J., Jones L., Gomez A., Kaiser L., Polousukhin I., Attention Is All You Need, \textit{Advances In Neural Information Processing Systems 30}, pp. 5998-6008, http://papers.nips.cc/paper/7181-attention-is-all-you-need.pdf, 2017.
[23] Kaiser L., Attentional Neural Network Models Masterclass, \textit{YouTube}, https://www.youtube.com/watch?v=rBCqOTEfxvg, October 2017.
[24] Alammar J., The Illustrated Transformer, \textit{Visualizing machine learning one concept at a time.}, http://jalammar.github.io/illustrated-transformer/, July 2018.
[25] Reimers, N., Gurevych, I., Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks, \textit{Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing}, 2019.
[26] Bowman S., Angeli G., Potts C., and Manning C., A large annotated corpus for learning natural language inference, \textit{In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing (EMNLP)}, 2015.
[27] \texttt{scipy.spatial.distance.cosine}, \textit{SciPy Documentation}, https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.spatial.distance.cosine.html.
[28] McCormick C., BERT Fine-Tuning Tutorial with PyTorch, \textit{Personal Blog}, https://mccormickml.com/2019/07/22/BERT-fine-tuning/, 2019.
[29] Wolf T., Debut L., Sanh V., Chaumond J., Delangue C., Moi A., Cistac P., Rault T., Louf R., Funtowicz M., Brew J., HuggingFace's Transformers: State-of-the-art Natural Language Processing, \textit{http://arxiv.org/abs/1910.03771}, 2019.
[30] The Judgements Information System, \textit{Supreme Court of India}, www.judis.nic.in, 1950-1990.
[31] Peters, M.E., Neumann, M., Iyyer, M., Gardner, M., Clark, C., Lee, K., Zettlemoyer, L., Deep contextualized word representations. \textit{http://arxiv.org/abs/1802.05365}, 2018.
[32] Devlin J., Chang M., Lee K., Toutanova K., BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding, \textit{https://arxiv.org/abs/1810.04805}, 2018.
[33] Brown, T. B., Mann, B., Ryder, N., Subbiah, M., Kaplan, J., Dhariwal, P., Agarwal, S., Language Models are Few-Shot Learners, \textit{http://arXiv.org/abs/2005.14165}, 2020.
[34] Cho, W. S., Zhang, P., Zhang, Y., Li, X., Galley, M., Brockett, C., Wang, M., Gao,J., Towards coherent and cohesive long-form text generation, \textit{https://arxiv.org/abs/1811.00511}, 2019.
\end{document} |
2,877,628,090,213 | arxiv | \section{}
In recent papers \cite{Hill1,Hill2, Hill3} we have computed the effect of a coherent
oscillating axion dark matter field, via the electromagnetic anomaly, upon
the magnetic moment of an electron, or arbitrary magnetic multi-pole source.
Figure (1) has been computed in several
ways and the results are consistent, nontrivial, and have potentially
interesting physical and observational implications.
This can be viewed as a scattering amplitude for the coherent cosmic axion
field on a heavy, static, magnetic dipole moment, with conversion to an outgoing photon
or classical radiation field. We find, however, that
this leads to the consistent
interpretation that the electron behaves as though it has
acquired an ``effective oscillating
electric dipole moment'' (OEDM) in the background oscillating cosmic axion field,
which then acts as a source for electric dipole radiation.
In ref.\cite{Flambaum}, however,
it is claimed that the results of the analysis \cite{Hill1,Hill2} are wrong.
The authors actually claim that the Feynman
diagram of Fig.(1) ``when properly computed'' vanishes.
We emphatically disagree with the conclusions of Flambaum, \etal
We show that they have made assumptions that lead them to compute
a vanishing total divergence. Indeed, we previously computed the
full effective action for a stationary
electron in an arbitrary gauge, \cite{Hill1,Hill2}. One can
readily see that it contains the Flambaum \etal result in their special limit,
where indeed it reduces to a vanishing total divergence.
However, the full amplitude is nonvanishing and physical, and the Flambaum \etal limit
is irrelevant and misses the physics.
\vspace{0.1in}
Let us first review the situation. In the simplest case,
we consider the comoving cosmic axion field
$a(t)/f_a =\theta(t) = \theta_0\cos(m_a t)$, in the
limit of a stationary, non-recoiling electron (this is the relevant limit
since the axion mass $m_a << m_e$). From Fig.(1)
we obtain the following
effective interaction, written in terms of nonrelativistic two-component
spinors \cite{Hill1}:
\beq
\label{one}
\int d^4x\; g_a\;\mu_{Bohr}{\theta}(t) \psi^\dagger\vec{\sigma}\psi \cdot \vec{E}
\eeq
This result is a contact term and is computed in radiation gauge, where the electric field
is $\vec{E}=-\partial_t \vec{A}$ for vector potential $\vec{A}$
and $\vec{\nabla}\cdot\vec{A}=0$. In momentum space it takes the form
$ g_a m_a\;\mu_{Bohr}\theta_0 \psi^\dagger\vec{\sigma}\psi \cdot \vec{\epsilon} $
where $\vec{\epsilon} $ is the photon polarization. Clearly the amplitude
vanishes in the limit $m_a\rightarrow 0$. The $m_a$ factor is absorbed into
$\vec{E}=-\partial_t \vec{A}$ in writing eq.(\ref{one}).
\begin{figure}[tbp]
\vskip0.1in
\begin{center}
\includegraphics[width=4cm, height=4cm]{feynman1.eps}
\end{center}
\caption{The dotted vertex is the axion-anomaly, $\theta F\widetilde{F}$,
and the solid vertical line is the electron. The
electron--photon vertex is the magnetic moment of the electron. The incoming
axion with 4-momentum $(m_a,\vec{0})$ absorbs a spacelike photon of 4-momentum
$(0, \vec{k})$ with $|\vec{k}|=m_a$ to produce an outgoing photon
of momentum $\sim (m_a, \vec{k})$. The electron barely recoils,
since $m_e >> m_a$..}
\label{figure_back}
\end{figure}
Given the form of this result, we interpret this
as an effective, induced OEDM for the electron. We claim this
result is general, and the interaction produces
electric dipole radiation from
any static magnetic moment immersed in, and
absorbing energy from, the oscillating cosmic axion field.
Indeed, since the result follows from a tree-diagram, it
can be demonstrated classically by a straightforward manipulation of
Maxwell's equations, \cite{Hill3}. The
radiation is formally that of an oscillating (Hertzian)
electric dipole, with outgoing electric field polarization
aligned in the direction of the magnetic moment, and thus apparently violating CP.
The emitted power by a free electron, in a spin-up to spin-up
transition, is (for a derivation see section IV.B of \cite{Hill2}):
\bea
\label{two}
P & =& \frac{1}{12\pi} (g_{a}\theta_0\; \mu_{Bohr})^2 m^4_a
\eea
This result is equivalent to that obtained from the classical Maxwell equations
for a fixed classical magnetic moment $\vec{m} = 2\mu_{Bohr}(\vec{s}/2)$ with a
spin unit-vector $\vec{s}$
\cite{Hill2,Hill3}.
More generally, we have computed Fig.(1) in an arbitrary gauge for the
background electric field, \cite{Hill1, Hill2}.
We obtained in the static
$ \vec{P}(x)= \mu_{Bohr}\psi^\dagger\vec{\sigma}\psi $ limit:
\beq
\label{three}
S=g\int d^{4}x\; \theta (t)\left( \vec{P}\cdot
\vec{E}+\vec{\nabla }\cdot \vec{P}\left(
\frac{1}{\vec{\nabla }^{2}}\right) \vec{\nabla }\cdot
\vec{E}\right)
\eeq
This result
differs from the radiation
gauge result eq.(\ref{one}) by the appearance of the nonlocal term.
Such nonlocal terms occur in electrodynamics when certain gauge choices are
specified, as in the case of the ``transverse current,'' (see below
and \cite{Jackson}).
Here, $\frac{1}{\vec{\nabla }^{2}}$
is a static Green's function,
\ie,
\beq
\label{four}
A(x)
\frac{1}{ \vec{\nabla }^{2} } B(x)
=
\int d^{4}y\; A(x)\frac{\delta(x_0 -y_0) }{ 4\pi | \vec{x}-\vec{y}| }B(y)
\eeq
In an arbitrary gauge,
$\vec{E}=\vec{\nabla }\varphi
-\partial _{t}\vec{A}$, after integrations by parts,
the action of eq.(\ref{three})
takes the form:
\bea
\label{five}
&& S =g\int d^{4}x\; \theta (t)\vec{\nabla }\cdot (
\vec{P}\varphi)
\nonumber \\
& &\!\!\!\!\!\!\!\! +g\int d^{4}x\; \partial _{t}\theta
(t)\left( \vec{P}\cdot \vec{A}+\vec{
\nabla }\cdot \vec{P}\left( \frac{1}{\vec{\nabla }^{2}}
\right) \vec{\nabla }\cdot \vec{A}\right)
\eea
This result is indeed gauge invariant as can be checked explicitly,
as it is just a rewrite of the manifestly gauge invariant eq.(\ref{three}).
If there are no surface terms we can drop the
first term on the {\em rhs} which is a total divergence,
and with $\vec{\nabla }\cdot \vec{A}=0$
(radiation gauge; this follows from $\vec{\nabla }\cdot \vec{E}=0$
upon integrating by parts in time) the result reduces back to eq.(\ref{one}).
It should be noted that the first term on the {\em rhs} of
eq.(\ref{three}) or eq.(\ref{five}) actually represents a force exerted
upon the OEDM by an applied oscillating $\vec{E}$,
hence there is potentially more physics here than dipole radiation.
\vspace{0.1in}
We can now see several
flaws with the Flambaum \etal analysis. They have ``properly computed''
this result in the particular case $\vec{A}=0$ and
$A_0=\varphi\neq 0$. In this case we see that only the first term
will be formally nonzero in eq.(\ref{five}), but that term
is just a spatial total divergence, and
hence it contributes nothing to the physics. A total divergence
is zero in momentum space and the Feynman
diagram of Fig.(1) then yields zero.
Moreover, Flambaum \etal claim that this
is a ``gauge choice.'' But this is, in fact, {\em a physics choice} since
one cannot generally make $\vec{A}$ vanish by a gauge transformation.
Furthermore, a time dependent $A_0=\varphi $ {\em necessarily requires a nonzero $\vec{A}$
by equations of motion} as we show in the discussion below
eq.(\ref{eight}).
Therefore, Flambaum \etal, by
using only a Coulomb potential to probe a
dynamical time dependent radiating source, are forcing the external
field to be static and thus obtain
a false null result by Fourier mismatch, as well as total divergence.
Finally, their result is consistent with our result in taking
the pure Coulomb or static limit, but it is
our result which they are attacking!
\vspace{0.1in}
The many conceptual errors and discrepancies of Flambaum \etal with our results seem to stem
from a faulty definition which they claim
to be valid for any EDM. They state:
\vspace{0.1in}
``(1) The EDM of an elementary particle is defined by the linear energy shift that it produces
through its interaction with an applied {\em static} electric field: $\delta\epsilon = −\vec{d}\cdot\vec{E}$.
As we show explicitly,
the interaction of an electron with an applied {\em static} electric field, in the presence of the axion
electromagnetic anomaly, in the lowest order does not produce an energy shift in the limit
$v/c \rightarrow 0$. This implies that no electron EDM is generated by this mechanism in the same
limit.''
\vspace{0.1in}
While this definition may be applicable to a static EDM, as in an
introductory course in electromagnetism,
{\em it is inapplicable to an intrinsically time dependent one.}
With an OEDM we are dealing
with a dynamical situation and must resort to a
more general definition, phrased in the context
of an action.
We should define the EDM or OEDM of any
object as a covariant action of the form:
\beq
\label{six}
S = g\int d^4 x\; S_{\mu\nu}(x) F^{\mu\nu}(x)
\eeq
where $S_{\mu\nu}$ is an antisymmetric odd parity dipole density
(\eg, $S_{\mu\nu} \sim \bar{\psi}\sigma_{\mu\nu} \gamma^5\psi$
for a relativisitic particle).
For concreteness, let us consider the case
of the axion induced neutron OEDM. The neutron OEDM
is believed to arise in QCD from instantons. It is being sought in a
proposed experiment (see ref.\cite{budkher} and references therein).
In the common rest frame of the neutron and axion,
the OEDM action of eq.(\ref{six}) reduces to:
\beq
\label{seven}
S=g\int d^{4}x\; \theta (t) \vec{P}\cdot
\vec{E}(t)
\eeq
where $ \vec{P}(x)=(e/m_N) \psi^\dagger \vec{\sigma}\psi(x)$
is the dipole spin density,
written in terms of two-component spinors.
$ \vec{P}(x)$ is localized in space
and static (time independent),
and the oscillating aspect of the EDM comes from the axion $\theta(t)$.
Note that a non-recoiling neutron is the kinematically favored limit,
\eg, as in Fig.(1). The neutron (or electron) is very heavy
compared to the axion, and like a truck being hit by a ping-pong ball
can only
acquire an insignificant kinetic energy. Therefore,
the radiated photon must carry off the full energy of the incident axion,
with a 4-momentum of $(m_a, \vec{k})$, and $|\vec{k}|=m_a$
(and the exchange photon 4-momentum
is spacelike, $(0, \vec{k})$).
Clearly, for a constant background
electric field the actions of eqs.(\ref{one},\ref{seven}) average to zero.
The radiated photon is necessarily
time dependent with frequency $m_a$, as
will be the case for any OEDM.
In the case of a radiation gauge photon, we have $A_0=0$ and
a non-zero $\vec{A}$ with $\vec{\nabla}\cdot \vec{A}=0$. In this case our action
for the neutron OEDM is indistinguishable from the OEDM of the electron
of eq.(\ref{one}). Both
require a time dependent $\vec{E}$,
and are $\propto \partial_t\theta(t)$ upon integration
by parts in time.
\vspace{0.1in}
Our result of eq.(\ref{one}), induced by the axion-QED anomaly,
has also been attacked by several other individuals
for violating the Adler
decoupling of the axion. The decoupling limit corresponds to $m_a\rightarrow 0$
and it superficially appears that eq.(\ref{one}) does not vanish in this limit as decoupling would
dictate (of course, it came from the momentum-space
result that was obviously $\propto m_a$, and this appears explictly
in ref\cite{Hill1}).
However, in refs.\cite{Hill2,Hill3} the issue
of the axion decoupling is studied in detail, and it is
found to be somewhat subtle in general.
In fact, eq.(\ref{one}) displays the same
behavior as the anomaly itself. The anomaly, in a constant
$\vec{B}$ field, can be written either in a manifestly
gauge invariant
form $\propto \theta(t) \vec{E}\cdot \vec{B}$ or in a manifestly decoupling form
$\propto \partial_t (\theta(t)) \vec{A}\cdot \vec{B}$ where
$\vec{E}=-\partial_t \vec{A}$
in a radiation gauge. It is not possible to display simultaneously
the manifest decoupling, and gauge invariance. Likewise, in the static electron
limit eq.(\ref{one}) can be written as:
\beq
\label{eight}
\int d^4x\; g\;\mu_{Bohr}\partial_t{\theta}(t) \psi^\dagger\vec{\sigma}\psi \cdot\vec{A}
\eeq
where $\vec{A}$ is the vector potential. Here we see manifest decoupling, but
an expression written in terms of a vector potential. More generally
the result
in an arbitrary gauge with recoil can be derived and displays the same
behavior.
The decoupling is actually subtle and beautiful. One can see this explicitly in
the eqs.(56,57) of ref.\cite{Hill2} for the near-zone radiation field
(and in eqs.(44) for the RF cavity)
and in the classical analysis of \cite{Hill3}. The decoupling is actually
occuring in the spatial structure of the nearzone radiation field
(or RF cavity modes). These vanish as $m_a^2$
due to a ``magic cancellation:'' the static magnetic dipole field,
which multiplies $\theta(t)$, does not radiate and
cancels, in the $m_a\rightarrow 0$ limit, against the outgoing radiation field
which is retarded and proportional to $\theta(t-r/c)$, leaving
terms of order $m_a^2$. This implies that here there is no ``Witten effect,''
whereby a constant induced electric dipole would remain
in the $\theta\rightarrow$ constant
limit: the
would-be Witten term cancels against the retarded outgoing radiation field
in the near-zone.
In the end the radiated power is $\propto m_a^4$,
and axion decoupling is certainly working as it should.
Such radiation is physically
interesting, and may be detectable in experiment \cite{Hill2}.
\vspace{0.1in}
Let us consider the problem of allowing $A_0$ to be time dependent while
trying to maintain $\vec{A}=0$.
$A_0$ is a non-propagating
field and cannot represent a physical out-going on-shell photon. The equation
of motion for $A_0$ is $\vec{\nabla}^2 A_0 =-\rho(x)$, where $\rho(x)$ is a
charge density. If we want to allow time dependent $A_0$, then
$\nabla^2 \partial_0 A_0 = -\partial_0 \rho(x,t)$,
but from current
conservation we have $\partial_0 \rho(x)=\nabla\cdot\vec{j}$
where $\vec{j}$ is the 3-current.
Hence, we have $\partial_0 A_0 = -(1/\vec{\nabla}^2 )\vec{\nabla}\cdot \vec{j} $.
This means that if $A_0$ is to be time dependent, then there must
necessarily be a 3-current, hence there is a source
for the vector potential, $\vec{A}$, and we cannot maintain $\vec{A}=0$.
Let us impose
the condition $\vec{\nabla}\cdot\vec{A}=0$. $\vec{A}$
satisfies $ (\partial_0^2-\nabla^2) \vec{A}
-\vec{\nabla}\partial_0 A_0= \vec{j}$ ( \ie,
$\partial_\mu F^{\mu i} = j^i$ ).
This is often written as
$ (\partial_0^2-\nabla^2) \vec{A}
= \vec{j}_T $
where $\vec{j}_T $ is the ``transverse current'' \cite{Jackson}.
Upon eliminating $\partial_0 A_0$, the transverse current
takes the nonlocal form
$\vec{j}_T = \vec{j}-\vec{\nabla}(1/\nabla^2) \vec{\nabla}\cdot \vec{j} $.
Thus, introducing $A_0$ time dependence requires a nonzero vector potential,
and its source is essentially nonlocal.
The nonlocal
term we obtained in eq.(\ref{three})
is the analogue of the transverse current \cite{Hill2}.
As stated above, the calculation in Flambaum, \etal, was restricted
to a 4-vector potential of the pure Coulomb form, $A_\mu=(A_0, \vec{0})$
\ie, $\vec{E} = \vec{\nabla }A_0$. This is {\em not a gauge choice}, since
a general 4-vector potential, $A_\mu(x,t)$, cannot be brought to the pure timelike
form by a gauge transformation, and if $\vec{A}=0$ then $A_0$
must be static in time. Thus a pure Coulomb potential cannot probe
an OEDM since the action averages to zero in time.
\vspace{0.1in}
In conclusion,
Ref.\cite{Flambaum} has argued that Fig.(1) is zero. However, they have
made specific assumptions that enforce
a static electric field configuration, and end up computing a total
spatial divergence which is automatically null.
From this they argue that there can be
no induced effective OEDM for the electron.
However, they have not considered the case of a time dependent
radiation field, or even a homogeneous field that has a Fourier time component
matched to the oscillation frequency of the axion.
The diagram of Fig.(1)
represents real physics, and can be interpreted as the effective action
of an induced electron OEDM, interacting with a coherent oscillating axion field.
It produces
electric $N$-pole radiation emanating from any magnetic $N$-pole placed
in the oscillating cosmic axion field. This can be
seen in various quantum computations at various levels
of detail \cite{Hill1,Hill2}, or directly from Maxwell's equations \cite{Hill3}.
The emission of electric
dipole radiation from magnets
could form a basis for broadband radiative detectors for cosmic
axions.
These conclusions have certainly not been falsified
by the authors of ref.\cite{Flambaum}.
\vspace{0.1in}
I thank, for discussions,
Bill Bardeen, Aaron Chou,
Graham Ross, Arkady Vainshtein, and various members of the Fermilab
axion search and breakfast groups.
This work was done at Fermilab, operated by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
\vspace{0.5in}
|
2,877,628,090,214 | arxiv | \section{Introduction} \label{sec:intro}
The Galilean satellites of Jupiter -- namely, Io, Europa, Ganymede, and Callisto -- were discovered by Galileo Galilei in the 17th century. They were the first objects to be observed orbiting another body than the Earth and Sun. Historically, their importance lies in the fact that it was one of the first observational evidence supporting the Copernican view of the Solar System. Early mathematical studies of the Galilean satellites motion around Jupiter were also crucial to promote the development of celestial mechanics and our understanding of resonances in the Solar System and beyond it \citep{Jo78}. Today -- 400 years after their discovery -- the origins of the Galilean satellites remain an intense topic of debate.
Space-mission explorations, ground and space-based observations have provided a series of important constraints on Galilean satellites' formation models. The low orbital eccentricities and inclinations of these satellites and their common direction of rotation around Jupiter suggest formation in a thin common disk, in a process similar to the formation of planets around a star \citep{Lu82}. Indeed, observations \citep{Pi19,Ch19} and numerical simulations \citep{Lu99,Kl99} suggest that the gas and dust not yet accreted to the envelope of a growing gas giant planet may give rise to a circumplanetary disc (hereafter referred as CPD). The formation of a circumplanetary disk is particularly possible if the temperature at the planet's envelope surface is not above a threshold ($\sim$ 2000~K) to prevent the planet's envelope from contracting and material in-falling into a fairly thin disk around its equator \citep{Wa10,Sz16}.
A growing gas giant planet eventually opens a gap in the gaseous circumstellar disk \citep[hereafter referred as CSD;][]{Li86a,Li86b}. Numerical simulations show that the CPD is fed by a fraction of the gas from the CSD that enters the planet's Hill radius~\citep{Lu99,Kl99} due to the meridional circulation of gas in the gap’s vicinity~\citep{Ta12,Mo14,Sz14,Sc20}. This theoretical result has been recently supported by observations of CPDs~\citep{isellaetal19,Te19}. In this work, we use N-body numerical simulations to model the formation of the Galilean satellites in a gaseous circumplanetary disk around Jupiter. Our model includes the effects of pebble accretion, gas-driven migration, tidal damping of eccentricity and inclination, and gas drag. Before presenting the very details of our model and results, we briefly discuss the physical and orbital properties of the Galilean satellites and also review existing models accounting for the origins of the Galilean satellites.
\subsection{Physical and orbital properties of the Galilean system}
\begin{table*}
\centering
\caption{Physical and orbital parameters of the Galilean satellites. From left-to-right the columns are semi-major axis ($a$), eccentricity ($e$), inclination ($I$), mass ($M$), radius ($R$), the water-ice mass fraction (ice), and bulk density ($d$) of the Galilean satellites \citep{Sc04,Og12} \label{tab:intro}}
\begin{tabular}{lcccccccc} \hline \hline
& $a$ (${\rm R_J}$) & $e$ ($10^{-3})$ & $I$ (deg) & $M$ ($10^{-5}~{\rm M_J}$) & $R$ (km) & \% ice & $d$ (g/cm$^3$) \\ \hline
Io & 5.9 & 4.1 & 0.04 & 4.70 & 1822 & 0 & 3.53 \\
Europa & 9.4 & 10.0 & 0.47 & 2.53 & 1565 & 8 & 2.99 \\
Ganymede & 15.0 & 1.5 & 0.19 & 7.80 & 2631 & 45 & 1.94 \\
Callisto & 26.4 & 7.0 & 0.28 & 5.69 & 2410 & 56 & 1.83 \\ \hline
\end{tabular}
\end{table*}
The Galilean satellites form a dynamically compact system. The innermost satellite -- Io -- orbits Jupiter at about $\sim$6~${\rm R_J}$ while the outermost one -- Callisto -- is at $\sim$26~${\rm R_J}$, where ${\rm R_J}$ is the physical radius of Jupiter. All these satellites have almost circular and coplanar orbits. Table~\ref{tab:intro} summarizes the physical and orbital parameters of the Galilean satellites.
The Galilean satellites system forms an intricate chain of orbital resonances. Io and Europa evolve in a 2:1 mean-motion resonance (MMR), where Io completes two orbits around Jupiter while Europa completes one. Europa and Ganymede are also in a 2:1 MMR. These resonances are associated with the characteristic resonant angles $2\lambda_E-\lambda_I-\varpi_I$, $2\lambda_E-\lambda_I-\varpi_E$, and $2\lambda_G-\lambda_E-\varpi_E$, where $\varpi$ followed by a subscript label denotes a specific satellite \citep{Pe99}. These 2-body MMRs have low-amplitude libration, and they can be combined into an associated resonant angle $\phi_{I,E,G}$ that librates around $180^{\circ}$ with a small amplitude of $0.03^{\circ}$. $\phi_{I,E,G}$ defines the so-called Laplace resonance and it is given as \citep{Gr77}
\begin{equation}
\phi_{I,E,G}=2\lambda_G-3\lambda_E+\lambda_I, \label{laplacian}
\end{equation}
where $\lambda$ is the mean anomaly, and the subscripts $_I$, $_E$, and $_G$ refer to the satellites Io, Europa, and Ganymede, respectively.
Unlike the three innermost Galilean satellites, Callisto is not locked in a first-order mean motion resonant configuration with any other satellite. This is an important constraint to formation and also dynamical evolution models.
Formation models are also constrained by the satellites' composition. Table \ref{tab:intro} shows that the water-ice content (density) of these satellites increases (decreases) with their orbital distance to Jupiter. Io is virtually dry, Europa carries about 8\% of its mass as water-ice, Ganymede and Callisto are water-ice rich bodies with $\sim$40-50\% water-ice mass content.
The low water-ice contents of Io and Europa have been originally interpreted as evidence of formation in hot regions of the CPD disk, probably mostly inside the disk snowline -- the location in the Jovian circumplanetary disk where water condensates as ice \citep{Lu82}. Their compositions have been used as strong constraints on several formation models \citep{Mo03a,Mo03b,Ca09,Ro17,Sh19}. However, it has been more recently proposed that Io and Europa may have formed mostly from water-ice rich material (similar to Ganymede and Callisto) and lost (most of) their water. The energy deposited by the accretion of solids combined with warm temperatures in the inner parts of the CPD can lead to the formation of surface water oceans and rich water-vapor atmospheres. These water reservoirs are then rapidly lost via hydrodynamic escape, partially (or fully) drying the two innermost satellites \citep{Bi20}. Additional water loss may be driven by giant impacts during their formation and enhanced tidal heating effects caused by the Laplacian resonance \citep{Dw13,Ha20}. Different from Io and Europa, the high concentrations of water-ice in Ganymede and Callisto suggest formation in cold and volatile-rich environments of the CPD \citep{Sc04}, very likely outside the disk snowline where very limited water loss via hydrodynamic escape took place, if any at all.
Formation models have also attempted to account for expected differences in the internal structures of the Galilean satellites. Different levels of core-mantle segregation are typically associated to different accretion timescales \citep{Mo03a,Mo03b,Ca02,Ca06,Ca09,Sa10,Mi16,Ro17,Sh19}. Io, Europa, and Ganymede are most likely fully differentiated bodies, with well distinct metallic cores and silicate mantles \citep{Sc04}. Unlikely, Callisto has been thought to be at most only partially differentiated \citep{Sc04}. The weak evidence of endogenic activity on Callisto's surface found by the Galileo mission in combination with frustrated detection of a core magnetic field by magnetic observations suggest limited differentiation \citep{Jo20}. This implies that Callisto had a relatively late and protracted accretion phase, potentially completed after the extinction of short-lived radioactive nuclei \citep[e.g. $^{\rm 26}$Al,][]{Ba08,Ba10} -- the most likely source of heating to cause large scale ice melting and core segregation. In this school of thought, it has been proposed that Callisto formed no earlier than $\sim$3~Myr after the calcium–aluminum-rich inclusions \citep{Mc06}. Io, Europa, and Ganymede should have formed earlier than that to account for their differentiated state. However, the interior structure of Callisto is still debated. It has been more recently suggested that even in the scenario of late/protracted formation, it could be very difficult to suppress differentiation of Callisto due to density gradients trapping heat generated by the decay of long-lived radioisotopes \citep{OR14} during the Solar System history. The presence of non-hydrostatic pressure gradients in Callisto could also allow a complete differentiated state~\citep{Ga13}. So, if Callisto is (fully) differentiated or not remains unclear \citep{Ro20}.
Finally, it is also important to note in Table \ref{tab:intro} that the masses of the Galilean satellites do not show any clear correlation with their orbital distance to Jupiter (e.g., no radial mass ranking), which is an important constraint to formation models \citep{Cr12}. In the next section, we review Galilean satellite formation models to motivate this work.
\subsection{The minimum mass sub-nebula model (MMSN model)}
Based on the minimum mass solar nebula model for the Solar System \citep{We77,Ha81}, \cite{Lu82} and \cite{Mo03a,Mo03b} proposed a modified version of this scenario applied to the Galilean satellites system. In these models, solids in the CPD are assumed to be in form of km-sized satellitesimals (following previous studies \citep{Mo03a,Mo03b} we use the term ``satellitesimals'' to refer to km-sized objects, precursors of the satellites in the CPD) that grow by mutual collisions up to the Galilean satellites masses. In most of these simulations, satellites are formed in very short timescales, typically of about $10^2$-$10^3$~years, which would suggest that they all should have differentiated interiors (or similar internal structure). This has been one of the issues raised against the MMSN model because several models considered that Callisto is at most only partially differentiated \citep[e.g.][]{Ca02}. Regardless if Callisto is or not differentiated in reality, other inconsistencies between the model results and the real satellite system exist.
In addition, the masses and orbits of the simulated satellites poorly match the Galilean ones (e.g. some simulations show satellite systems with radial mass ranking).
\cite{Mi16} analyzed the formation of the Galilean satellites considering different MMSN model scenarios. They invoked a semi-analytical model to simulate migration and growth including also the effects of an inner cavity in the gas disk. This disk feature was mostly neglected in previous studies but it is crucial to avoid the dramatic loss of solids via gas drag and gas driven migration \citep{Mi16} in the disk. When a body reaches the inner disk cavity its drift and migration stop which allows other bodies migrating/drifting inwards to be captured in MMRs. This process tends to repeat and leads to the formation of a resonant chain anchored at the disk inner edge \citep[analogues to the formation of super-Earth systems; see][]{Iz17,Iz19}. The authors verified that the final period-ratio of adjacent satellites in their simulated systems better reproduce the Galilean system if the migration timescale is increased, relative to those used in \cite{Mo03a,Mo03b}. However, by invoking longer migration timescales to avoid the mentioned issue, the masses of their simulated satellites did not provide a reasonable match to the Galilean satellites~\citep{Mi16}. Finally, the semi-analytical treatment invoked in \cite{Mi16} did not allow them to precisely model the secular and resonant interaction of adjacent satellite pairs neither their growth via giant impacts.
\cite{Mo18} explored the MMSN model using N-body simulations starting from a population of satellites-embryos that are allowed to type-I migrate and grow via giant impacts. Some of their simulations were successful in producing four satellites starting from a population of $\sim$20 satellite-embryos and many relatively smaller satellitesimals, but their simulations fail to match the final masses and orbital configuration of the Galilean satellites. In their best Galilean system analogue, the innermost satellite is at 17~${\rm R_J}$ whereas Io is at 5.9~${\rm R_J}$ (see Table \ref{tab:intro}). Their simulations also show that when several satellites reach the disk inner edge, forming a long-resonant chain, they all get engulfed by Jupiter. It is not clear why this phenomenon takes place in their simulations. If satellites are successively pushed inside the disk inner cavity and eventually collide with Jupiter one-by-one, one would expect that at least one satellite should survive anchored at the disk inner edge at the end of this process.
\subsection{The classic gas-starved disk model (GSD model)}
One of the major differences between the MMSN disk models and the gas-starved disk model (GSD model) is that the latter invokes CPDs that are orders of magnitude lower-mass than MMSN disks. The GSD model is probably one of the most successful early models for the origins of the Galilean satellites \citep{Ca02,Ca06,Ca09}. However, this model requires to be revisited because our paradigm of planet formation has evolved significantly in the last 10 years.
The GSD model is built on the assumption that a semi-steady gas flows from the circumstellar disk to Jupiter's circumplanetary disk simultaneously delivers gas and solid material to the CPD with roughly solar dust-to-gas ratio composition.
The original GSD model invokes that all dust delivered to the CPD by gas in-fall coagulates and grows into satellitesimals of masses of about $5\times 10^{-7}~{\rm M_J}$. Then, satellitesimals grow to satellites by mutual collisions~\citep{Ca02,Ca06,Ca09}.
Although very appealing when initially proposed, this idea presents some conflicts with our current understanding of planet formation. Recent simulations show that a 10-20 Earth-mass planet, gravitationally interacting with the gas disk, creates a pressure bump outside its orbit \citep{La14,Bi18} that prevents sufficiently large dust grains in CSD from being delivered to the planet's circumplanetary disk. So, in fact, the dust-to-gas ratio in Jupiter's CPD incoming gas is expected to be lower than that in the sun's CSD, perhaps several orders of magnitude lower than the solar value \citep{Ro18,weber2018characterizing}. This view is also supported by mass-independent isotopic anomalies measured in carbonaceous and non-carbonaceous meteorites. The isotopic differences between these two classes of meteorites have been interpreted as evidence of very efficient separation of the inner and outer Solar System pebble reservoirs, potentially caused by Jupiter's formation \citep[see][]{Kr17,brasser2020partitioning}. The filtering of pebbles promoted by Jupiter would have also affected the abundance of pebbles in its own CPD \citep{Ro20}. This is a critical issue because it challenges the in-situ formation of satellites(imals) in the CPD. Satellitesimal formation via streaming instability \citep{Yo07,Si16,Ar16,Dr18} -- the favorite scenario to explain how mm-cm size dust grains grow to km-sized objects -- requires dust-to-gas ratio of at least a few percent \citep[e.g.][]{Ya17}.
Finally, the origins of satellitesimals in the CPD is probably more easily explained via capture of planetesimals or fragments (produced in planetesimal-planetesimal collisions) from the CSD. Planetesimals or fragments on eccentric orbits around the Sun may eventually cross the orbit of the growing Jupiter \citep[e.g.][]{Ra17} and get temporarily or even permanently captured in the CPD~\citep{Es06,Ca09}. This is possible because gas drag dissipative effects act to damp the orbits of these objects when they travel across the CPD~\citep{Ad76,Es06,Mo10,Fu13,DA15,Su16,Su17}. Planetesimals traveling across the CPD are also ablated and this mechanism is probably the main source of pebbles (mm-cm-sized dust grains) to the CPD \citep{Es06,Es09,Mo10,Fu13,DA15,Su16,Su17,Ro20}. The total mass in planetesimals/fragments captured and pebbles created via this process depends on planetesimals/fragments sizes, the total mass in planetesimals/fragments, and gas density in the giant's planet region which are not strongly constrained \citep[e.g.][]{Ra17,Ro20}. Nevertheless, this scenario is very appealing because it invokes a single mechanism to explain the origins of pebbles and satellitesimals in the CPD.
\subsection{More Recent Models}
Galilean satellites' formation models have also invoked gas-drag assisted accretion of millimeter and centimeter size pebbles to account for their origin~\citep{Ro17,Sh19,Ro20}. This regime of growth is popularly known as pebble accretion \citep{Or10,La12,La14,Mo15,Le15,Bi18}
One of the key advantages of invoking pebble accretion to explain the formation of the Galilean system is that the growth of satellitesimals up to the masses of these satellites does not necessarily require a large (N$>$500-1000) population of satellitesimals to exist in the CPD -- as assumed in traditional models \citep{Mo03a,Mo03b,Ca02,Ca06,Ca09}. As already discussed, the in-situ formation of satellitesimals in the CPD may be problematic. However, if at least a handful of sufficiently massive satellitesimals exist in the CPD, pebble accretion may be efficient in forming a system with a few relatively massive final satellites.
\cite{Sh19} studied the growth of the Galilean satellites via pebble accretion. Their simulations include the effects of gas drag and type-I migration. They invoke integrated pebble fluxes in the CPD of about $1.5\times 10^{-3}~{\rm M_J}$. \cite{Sh19} set the CPD's inner edge at Io's current position (but see also \cite{Sa10,Mi16,Mo18}). \cite{Sh19} performed simulations considering initially 4 satellitesimals in the CPD. Satellitesimals are individually inserted in the CPD at different times, to mimic planetesimal capture from the CSD. They admittedly fine-tuned their simulations to produce systems that match well the masses, orbits, and water ice fractions of the Galilean satellites. Although this is an interesting approach, giving the large number of free parameters in the model, one of the key caveats of their scenario is that it is built on semi-analytical calculations rather than in N-body numerical simulations. Their simulations do not account for the gravitational interaction between satellites as they accrete pebbles and migrate \citep[see also][]{Ci18}. The efficiency of pebble accretion is strongly dependent on the orbital parameters of the growing satellites ~\cite[e.g.][]{levisonetal15b}. Thus, they can not precisely assess the final architecture of their systems. One of the questions that remains unanswered is whether or not an initial number of satellitesimals larger than four is also successful in reproducing the Galilean system. This is one of the questions we try to answer in this paper. We also advance to the reader that all our simulations starting initially with 4 satellitesimals produced less than 4 satellites at the end.
In a recent study, \cite{Ba20} proposed that the Galilean satellites' formation occurred in a vertically-fed CPD disk that spreads viscously outwards (the vertically averaged radial gas velocity is $v_r >0$ everywhere in the CPD). In their model, satellitesimals in the CPD grow to satellites in an oligarchic growth fashion -- via satellitesimal-satellitesimal collisions -- rather than via pebble accretion. Numerical simulations in \cite{Ba20} start from planetesimals with masses of $4\times 10^{-7}~{\rm M_J}$ (4 times more massive than the initial masses considered in this paper). The authors find that in their disk model pebble accretion can be simply neglected, which is not necessarily the case for other disk models \citep[e.g.][]{Ro17,Ro20}. Their model successfully explains some characteristics of the Galilean system, as the overall masses of the Galilean satellites and the Laplacian resonance, however, it remains to be demonstrated that the total dust reservoir assumed in their model can in fact settle into the CPD disk mid-plane to promote efficient satellitesimal formation via some sort of gravitational-hydrodynamic instability \citep{Ba20}.
In this work, we use N-body numerical simulations to model the formation of the Galilean satellites in a GSD-style circumplanetary disk. Motivated by previous studies, the flux of pebbles assumed in our simulations is consistent with pebble fluxes estimated via ablation of planetesimals entering the circumplanetary disk \citep{Ro20}. The initial total number of satellitesimals in the CPD is not strongly constrained, so in our simulations, we test 4, 30, and 50 satellitesimals. Our study represents a further step towards the understanding of the origins of the Galilean satellites because previous studies modeling their formation via pebble accretion have typically invoked simple semi-analytical models that neglect the mutual interaction of the satellites when they grow and migrate in the disk. Here, we self-consistently model the growth and mutual dynamical interaction of satellitesimals allowing also for growth via giant collisions.
The model that we propose here has the very same basic ingredients invoked in models for the formation of the so-called close-in super-Earths and giant planets around other stars \citep[e.g.][]{Iz19,Bi19,La19} -- namely pebble accretion and migration. Typical close-in super-Earths have masses of $\sim10 ^{-5}~{\rm M_{star}}$. Interestingly, the mass ratio of individual Galilean satellites and Jupiter is also $\sim10^{-5}$. The resonant dynamical architecture of the Galilean satellites also recalls that of some super-Earths systems \citep[e.g. Kepler-223 system;][]{Mib16}. So if one can explain the formation of these both types of systems via the same processes it would be reassuring.
The structure of this paper is as follows. In Section~\ref{sec:methods} we describe the methods used in this work. In Section~\ref{sec:simulation} we describe our simulations, and in Section~\ref{sec:results} we present the main results. In Section~\ref{sec:faketides} we analyze the long-term evolution of our systems. We discuss our results and model in Section~\ref{sec:discussion}. Finally, we summarize our main findings in Section~\ref{sec:conclusions}.
\section{Methods}\label{sec:methods}
Our numerical simulations were performed using an adapted version of the MERCURY package \citep{Ch99} including artificial forces to mimic the effects of the gas disk. These forces are: 1) gas drag; 2) type I migration, and eccentricity and inclination damping (Section \ref{subsec:ge}). Our pebble accretion prescription is described in details in Section \ref{subsec:pa}. Satellitesimals are allowed to grow via pebble accretion and collisions. Collisions are modeled as perfect merging events that conserve mass and linear momentum.
\subsection{Circumplanetary disk model}\label{subsec:cdm}
We assume that as Jupiter grows via runaway gas accretion, opens a deep gap in the circumstellar disk, and a disk mostly composed by gas forms around its equator. The circumplanetary disk is continually supplied by the in-fall of material from the CSD. Assuming a semi-steady flow of gas and balance between the in-fall of material from the CSD and the mass accretion rate onto Jupiter, \cite{Ca02} obtained that the radial surface density of gas is given by
\begin{equation}
\Sigma_g(r)=\frac{\dot{\rm M}_{\rm g}}{3\pi\nu(r)}\left\{\begin{array}{ll} 1-\frac{4}{5}\sqrt{\frac{R_c}{R_d}}-\frac{1}{5}\left(\frac{r}{R_c}\right)^2 & \textrm{for}~r\leq R_c \\ \frac{4}{5}\sqrt{\frac{R_c}{r}}-\frac{4}{5}\sqrt{\frac{R_c}{R_d}} & \textrm{for}~r>R_c,\end{array}\right.
\end{equation}
where $R_c$ and $R_d$ are the centrifugal and outer radius of the disk, respectively, $\dot{M}_g$ the mass in-fall rate, and $\nu$ the turbulent viscosity. Note that our disk model is qualitatively consistent with the delivery of gas to the CPD via meridional circulation of gas near the planet's gap \citep{Ta12,Mo14,Sz14,Sc20,Ba20}. In our model, we assume that the gas is deposited into the CPD midplane at around the centrifugal radius ($R_c$), and then spreads viscously~\citep{Ca02}. $R_c$ corresponds to the location in the CPD where the angular momentum of the inflowing material is equal to the Keplerian angular momentum. $R_c$ is treated as a free parameter in our model~\cite[see also][]{Ba20}.
The mass in-fall rate from the CSD on the CPD in our simulations was set as \citep{Sa10,Ro17}
\begin{equation}
\dot{\rm M}_{\rm g}=10^{-7}e^{-\frac{t}{\tau_d}}~{\rm M_J}{\rm /yr},
\end{equation}
where $t$ is the time and $\tau_d= 1.0$~Myr is the disk decay timescale \citep{Ca02}. For simplicity, in all our simulations, we neglect Jupiter's growth via gas accretion and set its mass as the current one.
We set the centrifugal radius at the fixed distance $R_c=26~{\rm R_J}$ \citep{Ro17} and the outer edge of the disk at $R_d=150~{\rm R_J}$, based on the results of hydrodynamic simulations \citep{Ta12}. The interaction of the CPD with Jupiter's magnetic field tends to slow down the planet's rotation and promotes the formation of an inner disk cavity. \cite{Ba18} found that magnetic effects dominate the gas dynamics in the inner regions of the CPD up to $4-5~{\rm R_J}$. Motivated by this result, we follow \cite{Iz17} and impose a disk inner edge at $R_i=5~{\rm R_J}$ \citep{Ba20} in our CPD by re-scaling the gas surface density by
\begin{equation}
\mathcal{R}=\tanh\left(\frac{r-R_i}{0.05R_i}\right)
\end{equation}
We assumed the standard $\alpha$-viscosity prescription to represent the disk viscosity \citep{Sh73}
\begin{equation}
\nu=\alpha_zc_sH_g,
\end{equation}
where $\alpha_z=10^{-3}$ \citep{Ro17} is the coefficient of turbulent viscosity, $c_s$ is the isothermal sound speed and $H_g$ the gas scale height ($H_g=c_s/\Omega_k$, where $\Omega_k$ is the keplerian orbital frequency). For a CPD in hydrostatic equilibrium, the sound speed relates to the disk temperature $T(r)$ as $c_s^2=2.56\times 10^{23}~g^{-1}~k_bT$, where $k_b$ is the Boltzmann constant \citep{Ha81}. The snowline is initially located at $r\sim 14.5~{\rm R_J}$ and the radial temperature profile is given by \citep{Ro17}
\begin{equation}
T=225\left(\frac{r}{10~{\rm R}_{\rm J}}\right)^{-3/4}K \label{eq:Tprofile}
\end{equation}
Figure~\ref{fig:hvrad} shows the CPD aspect ratio ($h_g=H_g/r$, solid line) and ratio between the vertically averaged gas radial and keplerian velocities ($v_r/v_k$, dotted line). In our disk model, the gas inside $\sim R_c$ flows inwards whereas gas outside $\sim R_c$ flows outwards \citep[see also][]{Ba20}, as one can note in Figure~\ref{fig:hvrad}.
\begin{figure}
\subfigure[]{\includegraphics[width=\columnwidth]{figures/hrvrad.png} \label{fig:hvrad}}
\subfigure[]{\includegraphics[width=\columnwidth]{figures/sigmapeb.png} \label{fig:sigmapeb}}
\subfigure[]{\includegraphics[width=\columnwidth]{figures/effqt.png} \label{fig:e2d3d}}
\caption{(a) CPD aspect ratio ($h_g=H_g/r$, solid line) and vertically averaged gas radial velocity normalized by the keplerian velocity ($v_r/v_k$, dotted line); (b) pebble surface density (solid lines) and Stokes number (dotted lines) as a function of the distance to the planet. Each color shows different times: 0~Myr (black), 0.5~Myr (blue), 1.0~Myr (green), and 2.0~Myr (orange); (c) Threshold curves of 2D and 3D pebble accretion regimes for different satellite masses and Stokes numbers. The different colored lines correspond to different locations of the disk: 5~${\rm R_J}$ (red), 20~${\rm R_J}$ (gray), 50~${\rm R_J}$ (navy blue), and 100~${\rm R_J}$ (purple). The region above (below) of each curve corresponds to the region where the 3D (2D) accretion efficiency is higher than the 2D (3D) one (see Eq.~\ref{eftotal}). The initial pebble flux is $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr. The integrated pebble flux over time is $10^{-3}$~M$\mathrm{_J}$. \label{fig:disk}}
\end{figure}
\subsection{Gas effects}\label{subsec:ge}
Our simulations start with satellitesimals of masses of $m\sim 10^{-7}~{\rm M_J}$ and bulk density of ${\rm 2~g/cm^3}$. We have verified that in our disk setup gas-drag plays an important role in the dynamics of satellitesimals of these sizes \cite[see also][]{Mi16}. Thus, our model includes both the effects of gas drag and satellitesimal-disk gravitational interactions. We now describe how we model these effects.
\subsubsection{Gas Drag}
The CPD around Jupiter rotates at sub-Keplerian speed because it is pressure supported. Small satellitesimals in the CPD orbiting at keplerian speed feel an strong headwind, lose energy, and tend to spiral inwards. The azimuthal gas disk velocity is given by
\begin{align}
v_g=(1-\eta)v_k, \label{velgas}
\end{align}
where $v_k=\Omega_kr$ is the Keplerian velocity and ${\rm \eta}$ characterizes the sub-Keplerian velocity of the gas disk. It is given by
\begin{equation}
\eta=-\frac{h_g^2}{2}\frac{\partial \ln{c_s^2\rho_g}}{\partial \ln{r}},
\end{equation}\label{eq:eta}
Setting $\rho_g=\int_{-\infty}^\infty \Sigma_g dz$, the gas volumetric density $\rho_g$ is given by \citep{We77}
\begin{equation}
\rho_g=\frac{1}{\sqrt{2\pi}}\frac{\Sigma_g}{H_g}e^{-z^2/2H_g^2}
\end{equation}
where $z$ is the reference-frame vertical component.
The gas-drag acceleration on a body of radius $R_s$ and bulk density $\rho_s$ is given by \citep{Ad76}
\begin{equation}
\vec{a}_{gd}=-\frac{3}{8}\frac{C_d\rho_g v_{rel}}{\rho_sR_s}\vec{v}_{rel}
\end{equation}
where $C_d$ is the drag coefficient, $\vec{v}_{rel}$ is velocity of the body with respect to the gas.
The drag coefficient computes the intensity of the interaction between the gas and satellitesimal, and it is given by \citep{Br07}
\begin{equation}
C_d=\left\{ \begin{array}{ll} 2 & \mathcal{M}\geq 1\\ 0.44+1.56\mathcal{M}^2 & \mathcal{M}<1,~Re\geq 10^3\\ \frac{24(1+0.15Re^{0.687})}{Re} & \mathcal{M}<1,~Re<10^3, \end{array}\right.
\end{equation}
where $\mathcal{M}=v_{rel}/c_s$ is the Mach number and $Re$ the Reynolds number written as
\begin{equation}
Re\approx 2.66\times 10^8\rho_g R_s\mathcal{M}.
\end{equation}
\subsubsection{Type-I migration}
Sufficiently large satellitesimals interact gravitationally with the CPD launching spiral density waves that transport angular momentum. This interaction tends to promote eccentricity and inclination damping of orbits and radial migration. Satellites with masses of the order of the Galilean satellites ($m\sim 10^{-5}~{\rm M_J}$) or lower mass are not expected to open gaps in our CPD disk and migrate in the type-I regime \citep{Ca06}. The total type-I migration torque consists of contributions from the Lindblad and Co-rotational torques \citep{Pa10,Pa11}.
The surface density and temperature gradient profiles, $x$ and $\beta$, respectively, are given by
\begin{equation}
x=-\frac{\partial\ln{\Sigma_g}}{\partial\ln{r}}
\end{equation}
and
\begin{equation}
\beta=-\frac{\partial\ln{T}}{\partial\ln{r}}.
\end{equation}
The scaling torque at the satellite's location is given by \citep{Cr08}
\begin{equation}
\Gamma_0=\left(\frac{q_s}{h_g}\right)^2\Sigma_gr^4\Omega_k^2,
\end{equation}
where $q_s=m/{\rm M_J}$ is the satellite-Jupiter mass ratio.
The Lindblad torque felt by a satellite on circular and coplanar orbit is parameterized as \citep{Pa10,Pa11}
\begin{equation}
\Gamma_L=(-2.5-1.5\beta+0.1x)\frac{\Gamma_0}{\gamma_{\textrm{eff}}},
\end{equation}
where $\gamma_{\textrm{eff}}$ is the effective adiabatic index defined by \citep{Pa10,Pa11}
\begin{equation}
\gamma_{\textrm{eff}}=\frac{2\gamma Q}{\gamma Q+\frac{1}{2}\sqrt{2\psi+2\gamma^2 Q^2-2}},
\end{equation}
\begin{equation}
\psi=\sqrt{(\gamma^2 Q^2+1)^2-16Q^2(\gamma-1)},
\end{equation}
and
\begin{equation}
Q=\frac{2\chi}{3h_g^3r^2\Omega_k}
\end{equation}
where $\gamma$ is the adiabatic index assumed as $\gamma=1.4$ in this work. The thermal diffusion coefficient depends on the opacity $\kappa$ and is given by \citep{Pa10,Pa11}
\begin{equation}
\chi=\frac{16\gamma(\gamma-1)k_BT^4}{3\kappa\rho_g^2(h_gr)^2\Omega_k^2}.
\end{equation}
The CPD disk opacity is calculated as in \cite{belllin94}.
A satellite on circular and coplanar orbit experiences a corotation torque that reads as \citep{Pa10,Pa11}
\begin{equation}
\begin{split}
\Gamma_C=&\left[\frac{7.9\xi}{\gamma_{\textrm{eff}}}F(p_\nu)G(p_\nu)+0.7\left(\frac{3}{2}-x\right)(1-K(p_\nu))+\right.\\
&+\left(2.2\xi-\frac{1.4\xi}{\gamma_{\textrm{eff}}}\right)F(p_\nu)F(p_\chi)\sqrt{G(p_\nu)G(p_\chi)}\\
&+\left.1.1\left(\frac{3}{2}-x\right)\sqrt{(1-K(p_\nu))(1-K(p_\chi))}\right]\frac{\Gamma_0}{\gamma_{\textrm{eff}}},
\end{split}
\end{equation}
where $\xi=\beta-(\gamma-1)x$, and $p_\nu$ and $p_\chi$ are parameters that measure the viscous and thermal saturations of the co-orbital torque, respectively. $F$, $G$ and $K$ are functions of these parameters.
The parameters $p_\nu$ and $p_\chi$ are given by \citep{Pa10,Pa11}
\begin{equation}
p_\nu=\frac{2}{3}\sqrt{\frac{r^2\Omega_k}{2\pi\nu}\left(\frac{1.1}{\gamma_{\textrm{eff}}^{1/4}}\sqrt{\frac{q_s}{h_g}}\right)^3}
\end{equation}
and
\begin{equation}
p_\chi=\frac{2}{3}\sqrt{\frac{r^2\Omega_k}{2\pi\chi}\left(\frac{1.1}{\gamma_{\textrm{eff}}^{1/4}}\sqrt{\frac{q_s}{h_g}}\right)^3}
\end{equation}
The functions $F$, $G$ e $K$ are \citep{Pa10,Pa11}
\begin{equation}
F(p)=\frac{1}{1+\left(\frac{p}{1.3}\right)^2},
\end{equation}
\begin{equation}
G(p)=\left\{ \begin{array}{ll} \frac{16}{25}\left(\frac{45\pi}{8}^{3/4}p^{3/2}\right) & p<\sqrt{\frac{8}{45\pi}}\\
1-\frac{9}{25}\left(\frac{8}{45\pi}^{4/3}p^{-8/3}\right) & p\geq\sqrt{\frac{8}{45\pi}} \end{array}\right.,
\end{equation}
and
\begin{equation}
K(p)=\left\{ \begin{array}{ll} \frac{16}{25}\left(\frac{45\pi}{28}^{3/4}p^{3/2}\right) & p<\sqrt{\frac{28}{45\pi}}\\
1-\frac{9}{25}\left(\frac{28}{45\pi}^{4/3}p^{-8/3}\right) & p\geq\sqrt{\frac{28}{45\pi}} \end{array}\right.
\end{equation}
The torques $\Gamma_L$ and $\Gamma_C$ need to be modified to account for satellitesimals/satellites on eccentric or inclined orbits. The Lindblad torque is reduced by factor $\Delta_L$ and the corotation torque by $\Delta_C$ \citep{Cr08,Co14,Fe13} given by
\begin{equation}
\begin{split}
\Delta_L=&\left\{P_e+\frac{P_e}{|P_e|}\left[0.07\left(\frac{i_s}{h_g}\right)+0.085\left(\frac{i_s}{h_g}\right)^4\right.\right.\\
&\left.\left.-0.08\left(\frac{e_s}{h_g}\right)\left(\frac{i_s}{h_g}\right)^2\right]\right\}^{-1}
\end{split}
\end{equation}
and
\begin{equation}
\Delta_C=\textrm{exp}\left(-\frac{e_s}{0.5h_g+0.01}\right)\left[1-\tanh{\left(\frac{i_s}{h_g}\right)}\right]
\end{equation}
where
\begin{equation}
P_e=\frac{1+\left(\frac{e_s}{2.25h_g}\right)^{1.2}+\left(\frac{e_s}{2.84h_g}\right)^{6}}{1-\left(\frac{e_s}{2.02h_g}\right)^{4}}
\end{equation}
Finally, we can write the total torque associated to the type I migration as
\begin{equation}
\Gamma=\Delta_L\Gamma_L+\Delta_C\Gamma_C
\end{equation}
and the acceleration $\vec{a}_m$ of a satellite with angular momentum $L$ is \citep{Cr08}
\begin{equation}
\vec{a}_m=\frac{\vec{v}}{L}\Gamma
\end{equation}
The gas disk also damps eccentricity and inclination of sufficiently massive bodies on timescales given by \citep{Cr08}
\begin{equation}
\begin{split}
t_e=&\frac{t_{wv}}{0.780}\left[1-0.14\left(\frac{e_s}{h_g}\right)^2+0.06\left(\frac{e_s}{h_g}\right)^3\right.\\
&\left.+0.18\left(\frac{e_s}{h_g}\right)\left(\frac{i_s}{h_g}\right)^2\right]
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
t_i=&\frac{t_{wv}}{0.544}\left[1-0.30\left(\frac{i_s}{h_g}\right)^2+0.24\left(\frac{i_s}{h_g}\right)^3\right.\\
&\left.+0.14\left(\frac{e_s}{h_g}\right)^2\left(\frac{i_s}{h_g}\right)\right]
\end{split}
\end{equation}
where $t_{\rm wv}$ is the wave timescale \citep{Cr08}
\begin{equation}
t_{\rm wv}=\left(\frac{M*}{q_s\Sigma_ga_s^2}\right)\frac{h_g^4}{\Omega_k},
\end{equation}
$a_s$ is the satellite/satellitesimal semi-major axis.
The accelerations experienced by the bodies due to the eccentricity and inclination damping are \citep{Cr08}
\begin{equation}
\vec{a}_e=-2\frac{(\vec{v}\cdot\vec{r})\vec{r}}{r^2t_e}
\end{equation}
and
\begin{equation}
\vec{a}_i=-2\frac{(\vec{v}\cdot\hat{z})\hat{z}}{t_i}
\end{equation}
\subsection{Pebble Accretion}\label{subsec:pa}
The pebble surface density $\Sigma_p$ in our model is \citep{Sh19}
\begin{equation}
\Sigma_p=\frac{\dot{\rm M}_{\rm p}}{4\pi \tau_s\eta v_kr}(1+\tau_s^2),
\end{equation}
where $\tau_s$ is the Stokes number and $\dot{\rm M}_{\rm p}$ is the pebble mass flux given by $\dot{\rm M}_{\rm p}=\dot{\rm M}_{\rm p0}e^{-\frac{t}{\tau_d}}$ where $\dot{\rm M}_{\rm p0}$ is a free scaling parameter that we take to vary between $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr and $5\times 10^{-9}~{\rm M_J}$/yr. Thus the integrated pebble fluxes in our simulations vary between $8\times 10^{-4}~{\rm M_J}$ and $4\times 10^{-3}~{\rm M_J}$. These values are consistent with the pebble fluxes estimated via ablation of planetesimals \citep{Ro20}.
The Stokes number of a pebble with a physical radius $R$ and bulk density $\rho$ is given by \citep{La12}
\begin{equation}
\tau_s=\frac{\sqrt{2\pi}\rho R}{\Sigma_g}
\end{equation}
In this work, we assumed a bi-modal population of pebbles in the CPD.
In our CPD the snowline is at $\sim 14.5~{\rm R_J}$. Pebbles outside the snowline are assumed to have sizes of $R=1.0$~cm and $\rho=2.0$~g/cm$^3$. As pebbles drift inwards via gas drag they eventually cross the CPD snowline. At this location, we reduce the pebble flux by a factor of 2 to account for the sublimation of the ice-pebble component. Ice pebbles sublimate at the snowline releasing small silicate dust grains. To account for this effect, pebbles inside the snowline have sizes of $R=0.1$~cm and $\rho=5.5$~g/cm$^3$. Figure~\ref{fig:sigmapeb} shows the pebble surface density evolution and the Stokes number in function of the radial distance, in units of $R_{\rm J}$.
The pebble accretion rate is a product of the pebble flux and the total accretion efficiency $\epsilon$ \citep{ida16}. Following \cite{Li18} and \cite{Or18}, the 2D accretion efficiency in the settling regime is given by
\begin{equation}
\epsilon_{2d,set}=0.32\sqrt{\frac{q_s}{\tau_s\eta^2}\left(\frac{\Delta V}{v_k}\right)}f_{\rm set}
\end{equation}
where $\Delta V$ is the relative velocity between pebble and satellite and $f_{\rm set}$ is a function that fits well the transition between different accretion regimes.
The relative velocity of a satellitesimal on a circular orbit and a pebble is \citep{Li18}
\begin{equation}
V_{\rm cir}=\left[1+5.7\left(\frac{q_s\tau_s}{\eta^3}\right)\right]^{-1}\eta v_k+0.52(q_s\tau_s)^{1/3}v_k
\end{equation}
while the relative velocity of satellite on a non-circular but coplanar orbits reads \citep{Li18}
\begin{equation}
V_{\rm ecc}=0.76e_sv_k
\end{equation}
One can write the relative velocity between a pebble and a satellite in both regimes as \citep{Li18,Or18}
\begin{equation}
\Delta V_y=\textrm{max}(V_{\rm cir},V_{\rm ecc})
\end{equation}
The relative velocity between a pebble and a satellite on a slightly inclined orbit can be written as $\Delta V_z=0.68i_sv_k$. Finally, the total relative velocity becomes \citep{Li18,Or18}
\begin{equation}
\Delta V=\sqrt{\Delta V_y^2+\Delta V_z^2}
\end{equation}
The transition function $f_{\rm set}$ is given by \citep{Or18}
\begin{equation}
\begin{split}
f_{\rm set}=&\exp\left[-0.5\left(\frac{\Delta V_y^2}{V^2_*}+\frac{\Delta V_z^2}{V_*^2+0.33\sigma_{pz}^2}\right)\right]\\
&\times\frac{V_*}{\sqrt{V_*^2+0.33\sigma_{pz}^2}}
\end{split}
\end{equation}
where $V_*$ is the transition velocity
\begin{equation}
V_*=\sqrt[3]{\frac{q_s}{\tau_s}}v_k
\end{equation}
and $\sigma_{pz}$ is the turbulent velocity in the z-direction \citep{Yo07}:
\begin{equation}
\sigma_{pz}=\sqrt{\frac{\alpha_z}{1+\tau_s}}\left(1+\frac{\tau_s}{1+\tau_s}\right)^{-1/2}h_gv_k
\end{equation}
The pebble disk aspect ratio is given by \citep{Yo07}
\begin{equation}
h_p=\sqrt{\frac{\alpha_z}{\alpha_z+\tau_s}}\left(1+\frac{\tau_s}{1+\tau_s}\right)^{-1/2}h_g,
\end{equation}
and the pebble volume density is \citep{Or18}
\begin{equation}
\rho_p=\frac{\Sigma_p}{\sqrt{2\pi}rh_p}
\end{equation}
According to \cite{Li18} and \cite{Or18}, the tridimensional pebble accretion efficiency in the settling regime is
\begin{equation}
\epsilon_{\rm 3d,set}=0.39\frac{q_s}{\eta h_{\textrm{eff}}}f_{\rm set}^2
\end{equation}
where $h_{\textrm{eff}}$ is the effective aspect ratio of pebbles in relation to a satellite in inclined orbit \citep{Or18}
\begin{equation}
h_{\textrm{eff}}=\sqrt{h_p^2+\frac{\pi i_s^2}{2}\left[1-\exp\left(-\frac{i_s}{2h_p}\right)\right]}
\end{equation}
The accretion efficiency in the 2D and 3D ballistic regimes are \citep{Li18,Or18}
\begin{equation}
\epsilon_{\rm 2d,bal}=\frac{R_s}{2\pi\tau_s\eta r}\sqrt{\frac{2q_s r}{R_s}\left(\frac{\Delta V}{v_k}\right)^2}(1-f_{\rm set})
\end{equation}
and
\begin{equation}
\epsilon_{\rm 3d,bal}=\frac{1}{4\sqrt{2\pi}\eta\tau_sh_p}\left(2q_s\frac{v_k}{\Delta V}\frac{R_s}{r}+\frac{R_s^2}{r^2}\frac{\Delta V}{v_k}\right)(1-f_{\rm set}^2)
\end{equation}
Finally, the total pebble accretion efficiency is
\begin{equation}
\epsilon=\frac{f_{\rm set}}{\sqrt{\epsilon_{\rm 2d,set}^{-2}+\epsilon_{\rm 3d,set}^{-2}}}+\frac{1-f_{\rm set}}{\sqrt{\epsilon_{\rm 2d,bal}^{-2}+\epsilon_{\rm 3d,bal}^{-2}}} \label{eftotal}
\end{equation}
Figure~\ref{fig:e2d3d} shows the curves $\epsilon_{2d}=\epsilon_{3d}$ for different Stokes number and satellite's masses. Each curve samples the accretion regime at one specific location of the disk. Regions above each curve corresponds to regions where the 3D accretion rate is higher than the 2D rate. Below each curve, it is the opposite.
When a satellite reaches the isolation mass M$_{\rm iso}$, its Hill radius becomes greater than the disk height creating a pressure bump that deflects gas and pebbles. If the satellite reaches M$_{\rm iso}$ the pebble accretion breaks off and the satellite grow only by impacts.
The pebble isolation mass in ${\rm M_J}$ is given by \citep{At18}
\begin{equation}
\begin{split}
{\rm M_{iso}}=&h_g^3\sqrt{37.3\alpha_z+0.01}\left\{1+0.2\left(\frac{\sqrt{\alpha_z}}{h_g}\sqrt{\frac{1}{\tau_s^2}+4}\right)^{0.7}\right\} \label{miso}
\end{split}
\end{equation}
\section{Simulations}\label{sec:simulation}
We have performed 120 simulations considering different pebble fluxes and the initial number of satellitesimals. The initial number of satellitesimals in the CPD is poorly constrained and it may also increase in time if additional planetesimals are captured from the CSD as the system evolves \citep{Mo10,Ro20}. We decided to neglect the capture of new planetesimals after our simulation starts by assuming that only sufficiently early captured planetesimals would successively grow by pebble accretion. We have performed simulations starting with 4, 30, and 50 satellitesimals in the CPD. While it would be ideal to systematically test the effects of different initial number of satellitesimals in our model, our N-body simulations are computationally expensive what limits our approach. Our simulations starting with 4 satellitesimals are designed to test the scenario proposed by \cite{Sh19} using self-consistent N-body simulations. Satellitesimals are initially distributed between 20~${\rm R_J}$ and 120~${\rm R_J}$. Their initial masses are set $10^{-7}~{\rm M_J}$ and bulk densities $\rho=2.0$~g/cm$^3$ which are consistent with the typical masses/sizes of planetesimals formed via streaming instability \citep[$R\sim250$~km,][]{Jo14a,Ar16,Si16a}. Satellitesimals are initially separated from each other by 5 to 10 mutual Hill radii $R_H$ \citep[e.g.][]{Ko00,Kr14}
\begin{equation}
R_H=\frac{a_i+a_j}{2}\left(\frac{2\times 10^{-7}}{3}\right)^{1/3}
\end{equation}
where $a_i$ and $a_j$ are the semi-major axes of a pair of adjacent satellitesimals. They are set initially on nearly circular and coplanar orbits ($e\leq 10^{-4}$ and $I\leq 10^{-4}$). Other angular orbital elements are randomly and uniformly selected between 0 and 360 deg.
Our simulations are numerically integrated for 2~Myr, considering the gas disk effects. In a few cases, to evaluate the long-term dynamical stability of our final systems, we extended our simulations up to $\sim$10~Myr, assuming that the gaseous circumplanetary disk dissipates at 2~Myr. For simplicity, we relate the pebble flux in our simulations to the gas accretion flow in the CPD \citep[e.g.][]{Sh19}. We performed simulations with $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr, $1.5\times 10^{-9}~{\rm M_J}$/yr, $3\times 10^{-9}~{\rm M_J}$/yr, and $5\times 10^{-9}~{\rm M_J}$/yr. The integrated pebble flux from the lowest pebbles flux to the highest one are $8\times 10^{-4}~{\rm M_J}$, $10^{-3}~{\rm M_J}$, $3\times 10^{-3}~{\rm M_J}$, and $4\times 10^{-3}~{\rm M_J}$.
In these simulations, we have neglected the evolution of the CPD's temperature, but the gas surface dissipates exponentially with an e-fold timescale of 1~Myr.
\subsection{Constraining our model}\label{sec:constraints}
Our model is strongly constrained by key features of the Galilean satellites system as the number of satellites, orbital configuration, final masses, and compositions. To evaluate how our simulations match the Galilean satellites system, we define a list of relatively generous constraints. A Galilean system analogue must satisfy the following conditions:
\begin{itemize}
\item[{\bf i})] the final system must contain at least four satellites;
\item[{\bf ii})] the two innermost satellite-pairs must be locked in a 2:1 MMR;
\item[{\bf iii})] the individual masses of all satellites must be between 0.8${\rm M_E}$ and 1.2${\rm M_G}$, where ${\rm M_E}$ and ${\rm M_G}$ are the masses of Europa and Ganymede, respectively;
\item[{\bf iv})] the two outermost satellites must have water-ice rich compositions ($>$0.3 water mass fraction).
\end{itemize}
Although Io is extremely water-ice depleted today and Europa contains only $\sim$8\% of its mass as water-ice (Table \ref{tab:intro}), we do not consider these estimates to be strong constraints to our model, because it is possible that these satellites formed with water richer compositions and lost all/most of their water \citep{Bi20}. Also, we stress that Callisto is not locked in a first-order mean motion resonance with Ganymede today. We also do not consider this observation a critical constrain to our model because Callisto may have left the resonance chain via divergent migration due to Jupiter-satellites tidal interactions \citep{Fu16,downey2020inclination}. We will further discuss these two issues later in the paper.
\section{Results} \label{sec:results}
We start by presenting the results of our sets of simulations considering initially 4 satellitesimals in the CPD, inspired by simulations of \cite{Sh19}. In \cite{Sh19}, satellitesimals are assumed to ``appear'' in the CPD successively, at very specific times. In our simulations, they are assumed to appear simultaneously at the beginning of the simulation in arbitrarily selected positions. Note that, in our case, the four satellitesimals start about 20~${\rm R_J}$ from each other, which roughly mimics their approach. By disposing our 4 satellitesimals initially fairly apart from each other, we avoid early collisions and allow them to grow at least one order of magnitude in mass before they start to strongly interact with each other (which may affect the efficiency of pebble accretion).
Figure~\ref{fig:4sat} shows the evolution of one of our representative systems. In this simulation, we set $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr. Figure~\ref{fig:4sat} shows that satellitesimals first grow by pebble accretion and start to migrate inwards. When they reach the inner edge of the disk a dynamical instability takes place and leads to a collision. At the end of this simulation, at 2~Myr, 3 satellites survive with individual masses of the order of $10^{-5}~{\rm M_J}$. The innermost and outermost satellite pairs are locked in a compact 5:4 and 3:2 MMR, respectively. The orbital eccentricities of satellites at the end of our simulations are between 0.02 and 0.05, which are larger than those of the Galilean satellites.
\begin{figure}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Ysma1.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Yecc1.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Yinc1.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Ymass1.png}}
\caption{{\bf From top to bottom:} Evolution of the semi-major axes, orbital eccentricities, inclinations, and mass of satellitesimals in a simulation starting with four satellitesimals and $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr. Satellitesimals initially grow via pebble accretion and migrate inwards. When they reach the inner edge of the disk locate at about $5~{\rm R_J}$, the innermost satellite pair collides at $\sim1.2$~Myr forming a system with only three final satellites. The blue dot-dashed line in the top panel shows the evolution of the snowline location in the disk.\label{fig:4sat} }
\end{figure}
We verified that each one of our 40 numerical simulations starting with 4 satellitesimals shows at least one collision during the gas disk phase, typically when satellites approach the disk inner edge. This suggests that more than four satellitesimals are required to explain the Galilean system. This is in conflict with the results of \cite{Sh19}, and these simulations violate our constraint i). In fact, even the final period ratio of the satellites in our simulations does not agree with those \cite{Sh19} have found. They have assumed that migrating satellites are successively captured in the 2:1 MMR when the migration timescale is longer than a critical timescale \citep{Og13}. The critical timescale criteria used in the semi-analytical model of \cite{Sh19} does not fully account for the eccentricity/inclination evolution of the satellites due to secular and resonant interactions. The capture in MMR also depends on the resonance order and mass-ratio of the migrating satellites \citep{2015MNRAS.451.2589B}. For instance, when the inner satellite is less massive than the outer one (e.g. Europa and Ganymede), the 2:1MMR can be skipped even when the adiabatic criteria for capture is attended \citep{2015MNRAS.451.2589B}. Thus, the criteria for capture in resonance assumed by \cite{Sh19} is in fact too simplistic. It is just a good proxy to infer (non-)capture in mean motion resonance if the eccentricities of the satellites can be neglected \citep{Og13}, which we show here to not be the case.
We performed some simple simulations considering the very disk model of \cite{Sh19}. This specific set of simulations starts with an Io-mass satellite fully formed and residing slightly outside the disk inner cavity, and an Europa-mass satellite initially placed at 15~${\rm R_J}$. These satellites are released to migrate inwards -- but they are not allowed to accrete pebbles. Our goal here is just compare the final resonant architecture of these satellites in our simulations and that predicted by the approach invoked in \cite{Sh19}. To conduct this set of experiments, we assume that satellites migrate with constant migration timescales, that do not vary with the distance to Jupiter. Figure~\ref{fig:mig} shows these results. In the vertical axis of Figure~\ref{fig:mig}, we show the migration timescale and in the horizontal axis the final period ratio of the satellites. Critical migration timescales that lead to capture in 2:1 and 3:2~MMR used in \cite{Sh19} are shown as solid black and red lines, respectively. The dot-dashed black and red lines show the migration timescales that lead to capture in 2:1 and 3:2 MMR when orbital eccentricities are taken into account \citep{Go14}. We use the eccentricity of the satellites in our simulations as input to calculate the latter timescales. The black dots show the results of our numerical simulations which agree very well with those of \cite{Go14}. The difference observed between our results and those of \cite{Sh19} is caused by the increase in the eccentricity of the satellites when they approach each other, which breaks down the validity of their criteria for capture in resonance. We have found that the timescale predicted to lead to capture in 2:1 MMR in simulations of \cite{Sh19} in fact tends to lead to collisions.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/mig.png}
\caption{Correlation between final period ratio and migration timescale in simulations considering only two satellites with masses analogues of those of Io and Europa. The black dots represent the results of our numerical simulations, and horizontal lines are estimated critical timescales that lead to capture into 3:2 MMR (black line) and 2:1 MMR (red lines). The solid lines show the critical timescales inferred by Ogihara \& Kobayashi (2013) and the dashed lines show the critical timescales given by Goldreich \& Schlichting (2014a). We have found that migration timescales shorter than $\sim$130~yrs tend to lead to dynamical instabilities and collisions of the satellites. \label{fig:mig} }
\end{figure}
\subsection{Effects of the pebble flux} \label{subsec:parameters}
In this section, we present the results of our simulations starting with 30 and 50 satellitesimals and compare the effects of different pebble fluxes. Figure~\ref{fig:number} shows the final satellite systems produced in our simulations at the end of the gas disk phase ($\sim$2~Myr). Simulations starting with 30 satellitesimals are shown on the left panels, and simulations starting with 50 satellitesimals are presented on the right panels. The horizontal axis of each panel shows the final semi-major axis and the vertical axis shows the mass of satellites. Each panel shows the results of 10 different simulations, where in each simulation, satellitesimals start with slightly different orbital parameters. Colorful dot-dashed lines connecting different points (symbols) show different satellite systems. The black filled circles show the real Galilean satellite system for reference. For presentation purposes, in Figure \ref{fig:number}, we have re-scaled the position of the satellites by a factor of order of unity to make the position of the innermost satellite in our simulations to correspond to the distance of Io to Jupiter ($a_{\rm in}$). At the top of each panel, we show the fraction of systems that produced a given final number of satellites.
\begin{figure*}
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N700H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N700H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N1000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N1000H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N2000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N2000H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N3000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/N3000H50.png} }
\caption{Final satellite systems produced in simulations starting with different initial number of satellitesimals and pebble fluxes. The left and right side panels show the results of simulations starting with 30 and 50 satellitesimals, respectively. From top-to-bottom the panels show the results of simulations with different pebble fluxes: a) and b) $\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr, c) and d) $\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr, e) and f) $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr, and g) and h) $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr. The lines connecting different points (symbols) show satellites in a same system. The fraction of simulations that produce two (green dot-dash lines), three (red dot-dash), four (blue dot-dash), and five satellites (orange dot-dash) are given at the top of each panel. The black solid line shows the real Galilean system. The horizontal pink lines correspond to 0.8${\rm M_E}$ and 1.2${\rm M_G}$ (see constraint iii)). Note that we have re-scaled the position of the satellites by a factor of order of unity to make the position of the innermost satellite in each of our simulations to correspond to the distance of Io to Jupiter ($a_{\rm in}$).\label{fig:number}}.
\end{figure*}
\begin{figure*}
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R700H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R700H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R1000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R1000H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R2000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R2000H50.png} }
\centering
\subfigure[$\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr and 30 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R3000H30.png} }
\quad \quad \quad
\subfigure[$\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr and 50 satellitesimals]{\includegraphics[width=0.8\columnwidth]{figures/R3000H50.png} }
\caption{Period ratio distribution of adjacent satellite-pairs. Each panel shows the results of 10 simulations. The left and right side panels show the results of simulations starting with 30 and 50 satellitesimals, respectively. From top-to-bottom the panels show the results of simulations with different pebble fluxes: a) and b) $\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr, c) and d) $\dot{\rm M}_{\rm p0}=3\times 10^{-9}~{\rm M_J}$/yr, e) and f) $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr, and g) and h) $\dot{\rm M}_{\rm p0}=10^{-9}~{\rm M_J}$/yr.\label{fig:period}}
\end{figure*}
Figure \ref{fig:number} shows that when the pebble flux decreases, from top to bottom, the final masses of the satellites also decrease. Final masses of satellites in simulations starting with 30 and 50 satellitesimals, and same pebble fluxes, are not dramatically different from each other. However, both sets of simulations show that an increase in the pebble flux by a factor of $\gtrsim$2 (from $\dot{\rm M}_{\rm p0}=1.5 \times 10^{-9}~{\rm M_J}$/yr to $\dot{\rm M}_{\rm p0}=5\times 10^{-9}~{\rm M_J}$/yr) is enough to change the final structure of our satellite systems from systems where satellites have masses fairly similar to those of the Galilean satellites to systems where satellites have systematically larger masses. Our overall best match to masses Galilean satellite system comes from simulations with $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr. The time-integrated pebble flux in this latter case corresponds to $\sim10^{-3}{\rm M_J}$. Lower pebble fluxes ($\dot{\rm M}_{\rm p0}= 10^{-9}~{\rm M_J}$/yr) produce too low-mass satellites, also inconsistent with the real system. Our simulations that best reproduce the mass of the Galilean satellites typically produce between 3 and 5 satellites. The efficiency of pebbles accretion in these systems is about 9\%, which is similar to efficiencies calculated by \cite{Ro20} ($\sim$10\%).
Figure \ref{fig:period} shows the final period ratio distribution of adjacent satellites in our simulations. Again, the left and right panels show the results of simulations starting with 30 and 50 satellitesimals, respectively. The two top panels in this figure a) and b) -- which correspond to the highest pebble flux -- show that systems with more massive satellites tend to have dynamically less compact systems (e.g. a larger fraction of pairs with period ratio $\geq2$). Our sets of simulations that better match the masses of the Galilean satellites (Figure \ref{fig:period}; panels e) and f) ) typically produce $\sim$10-20\% of satellite pairs locked in 2:1 MMR. The majority of satellite pairs in our systems is locked in more compact resonant configurations.
Figure~\ref{fig:triangular} shows the final semi-major axis versus the mass of satellites in selected systems that satisfy all constraints defined in Section \ref{sec:constraints}. As in Figure~\ref{fig:number}, each satellite is shown by a colorful point (symbol) -- where lines connect satellites in a same system. The system shown in red represents our nominal analogue that will be further discussed later in the paper. The real Galilean system is shown in black. The three innermost vertical dotted lines mark the respective positions of the three innermost Galilean satellites. The outermost vertical dotted line shows the location of Callisto if it was also locked in a 2:1 mean motion resonance with Ganymede. The horizontal pink lines show the limiting masses defined in constraint iii) of Section \ref{sec:constraints}. These systems match fairly well the masses and resonant configurations of the Galilean satellite system. However, the outermost satellite in our simulations is always locked in a 2:1 MMR with the second outermost one. As discussed before, Callisto is not locked in resonance with Ganymede. We do not consider this to be a critical issue for our model because Callisto may have left the resonance chain via divergent migration due to tidal-interaction of satellites with Jupiter \citep{Fu16,downey2020inclination}.
Figure~\ref{fig:triangular} also shows that in our best analogues, the mass of the innermost satellite is typically larger than that of the second innermost one, which is also true in the Galilean system. However, the masses of our Ganymede-analogues are typically lower than that of the real satellite. Finally, one of the important result of our model is that the radial-mass distribution of our satellites do not present any radial mass-ranking (e.g. satellites' mass increases with their distance to Jupiter) or triangular-mass distribution (e.g. the innermost and outermost satellites are the less massive). These are typical issues found in previous models \cite[e.g.][]{Mo03a,Mo03b,Cr12}.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/mass.png}
\caption{Final masses and semi-major axes of satellites produced in our best-case simulations. In all cases, only 4 satellites are formed. For presentation purposes, we have re-scaled the position of the satellites by a factor of order of unity to make the position of the innermost satellite in our simulations to coincide with the distance of Io to Jupiter ($a_{\rm in}$). The real Galilean system is given by the black line with a circle and the vertical dotted lines give the location of the 8:4:2:1 resonant chain and the pink vertical lines give the limits on mass of our third constraint (0.8${\rm M_E}$ and 1.2${\rm M_G}$). Our final systems show no radial mass ranking. The system represented by the green solid line shows a co-orbital satellite with the third innermost satellite.\label{fig:triangular}}
\end{figure}
\subsection{The Dynamical Architecture of our Systems}
In this section, we present the dynamical evolution of some of our best Galilean system analogues. Figure~\ref{fig:system3} shows a simulation with $\dot{\rm M}_{\rm p0}= 1.5\times 10^{-9}~{\rm M_J}$/yr, and starting with 50 satellitesimals. From top to bottom, the panels show the temporal evolution of semi-major axis, orbital eccentricity, orbital inclination, and masses of satellitesimals/satellites. The colorful horizontal dotted lines in the bottom panel match the color of one of the analogues, and they show the mass that each analogue should have to exactly match the mass of its corresponding Galilean satellite. Satellitesimals first grow by pebble accretion and start to migrate inwards. The innermost satellitesimal reaches the inner edge and stop migrating. As additional satellitesimals converge to the inner edge of the disk, orbital eccentricities and inclinations start to increase, which leads to scattering events and collisions. Finally, the satellitesimals converge into a resonant configuration anchored at the disk inner edge. In this case, all adjacent satellite pairs are evolving in a 2:1 MMR. Their final orbits typically exhibit very low orbital inclinations ($\sim$0.01~deg). However, the final orbital eccentricities of our satellites are modestly high ($\sim$0.05-0.2). Table \ref{tab:intro} shows that the orbital eccentricities of the Galilean satellites are much lower than those of analogues in Figure \ref{fig:system3}. More importantly, this is a trend present in all our best analogues. In the next section, we attempt to address this issue by slightly changing our disk model.
\begin{figure}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Ysma3.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Yecc3.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Yinc3.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/Ymass3.png}}
\caption{{\bf From to to bottom:} Evolution of semi-major axes, orbital eccentricities, orbital inclinations, and masses of satellites in a simulation with $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and starting with 50 satellitesimals. The dot-dashed line in the top panel shows the snowline location, and horizontal lines in the bottom panel show the masses of the real Galilean satellites. These horizontal line matches the color of the analogues to indicate the mass they should have to be a perfect match. All satellite pairs evolve in a 2:1 MMR.\label{fig:system3}}
\end{figure}
\subsection{Simulations with a more realistic cooling CPD} \label{subsec:nominal}
In all our previous simulations, we have neglected the evolution of the disk temperature. However, as the disk evolves, it looses masses and gets colder and thin \citep[e.g.][]{Sz16}. In this section we replace Eq. \ref{eq:Tprofile}, representing our original CPD temperature profile, by the following new temperature profile \citep{Sa10,Ro17}
\begin{equation}
T_{\rm new}=225\left(\frac{r}{10~{\rm R}_{\rm J}}\right)^{-3/4}e^{-\frac{t}{4\tau_d}}~K \label{eq:Tprofilenew},
\end{equation}
where $t$ represents the time, and $\tau_d$=1~Myr.
In our simulations where the disk temperature does not evolve in time, the CPD aspect ratio at 15~R$_{\rm J}$ (Ganymede's distance to Jupiter) is $\sim0.06-0.07$. In our new disk setup, the disk aspect ratio at 15~R$_{\rm J}$ also at the end of the gas disk phase is $\sim0.04$. We have conducted 40 new simulations, considering that the CPD temperature decays in time as Eq.~\ref{eq:Tprofilenew}.
Figure~\ref{fig:nom2} shows the dynamical evolution of one of our good analogues (see Section \ref{sec:constraints}). The gas in the CPD dissipates at 2~Myr and we follow the long-term dynamical evolution of these satellites up to 10~Myr. Figure~\ref{fig:nom2} shows, from top to bottom, the evolution of semi-major axis, eccentricity, inclination, and masses of the satellitesimals. This simulation starts with 50 satellitesimals. The evolution of satellitesimals in this case is remarkably similar to that of simulations where the disk temperature is kept fixed in time. The combined pebble accretion efficiency of the system is about 13\%, which is also similar to our other simulations with the exponential decay of the disk temperature. At the end of the gas disk phase, at 2~Myr, the satellites form a resonant chain anchored at the disk inner edge with all-three satellite pairs evolving in a 2:1 MMR. The final orbital eccentricities of satellites are lower than those observed in Figure \ref{fig:system3}, as expected. However, the orbital eccentricities of our analogues are still too high compared to those of the Galilean system.
We have found that the orbital eccentricities of satellites in simulations where the disk cools down in time are overall lower than those where the disk temperature is kept constant. This is not a surprising result because the equilibrium eccentricity of capture in resonance scales as $e_{\rm eq}\sim h_g$~\citep{goldreichschlichting14,deckbatygin15,pichierietal18}. This is in agreement with the results of our simulations (see Figure~\ref{fig:4sat} and \ref{fig:system3}). However, the observed eccentricities of our final satellites are still too high compared to those of the real satellites today (Figure \ref{fig:cumulative}). One could conjecture that this is because the disk aspect ratio of our new disk is still not low enough. However, in order to have a disk aspect ratio of $\leq$0.01-0.001 at $<$15~$R_{\rm J}$ -- which could more easily lead to final satellites with eccentricities similar to those of Galilean satellites -- it would require a CPD with gas temperature of about $\leq$5~K at 15~$R_{\rm J}$, which is unrealistic \citep[e.g.][]{Sz16}. So we suggest that in order to damp the orbital eccentricities of our analogues down to the level of the Galilean satellites, a different mechanism should be invoked. We will conduct new simulations and discuss a possible solution to this issue in Section \ref{sec:faketides}.
\begin{figure}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/YsmaGD.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/YeccGD.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/YincGD.png}}
\subfigure[]{\includegraphics[width=0.95\columnwidth]{figures/YmassGD.png}}
\caption{{\bf From top to bottom:} Temporal evolution of semi-major axis, orbital eccentricity, inclination, and mass of satellitesimals in a simulation starting with 50 satellitesimals. As the disk cools down, the snowline moves inwards as shown by the blue dotted line in the top panel. The horizontal lines in the bottom panel show the masses of the real Galilean satellites. These horizontal line matches the color of the analogues to indicate the mass they should have to be a perfect match. All satellite adjacent pairs are involved in a 2:1 MMR, forming a resonant chain. The gas disk dissipates at 2~Myr and the system remains dynamically stable up to 10~Myr. \label{fig:nom2}}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/cumulative.png}
\caption{Eccentricity distribution of our final satellites in simulations with $\dot{\rm M}_{\rm p0}=1.5\times 10^{-9}~{\rm M_J}$/yr and 50 satellitesimals. The blue curve shows the case where the disk temperature is kept constant during the gas disk phase. The red line shows the case where the disk temperature decays exponentially in time (see Eq. \ref{eq:Tprofilenew}). \label{fig:cumulative}}
\end{figure}
\subsection{The Water-Ice mass fraction of our satellites}
In this section, we analyze the water mass fraction of the satellites in our best Galilean system analogues. Figure~\ref{fig:compilation} shows three of our best analogues. Each line shows one satellite system and the horizontal axis shows its distance from Jupiter. The color-coded dots show individual satellites, where the color represents their water-ice fraction and dot sizes scale linearly with mass. The real Galilean system is shown at the bottom, for reference. The orbital eccentricity of each satellite is represented by the horizontal red bars showing the variation in the planetocentric distance over the semi-major axis ($R_{\rm J}$). In all these systems, adjacent satellite pairs evolve in a 2:1 MMR.
It is clear in Figure \ref{fig:compilation} that the water-ice fraction of the innermost satellites in our simulations are significantly higher than those of Io and Europa. However, in all our systems the two outermost satellites have water-ice rich compositions similar to those of Ganymede and Callisto. This is a trend observed in all our simulations. In this work, we assume that pebbles inside the snowline are 0.1~cm size silicate particles and those outside are larger, 1~cm size icy-particles. Therefore, pebbles outside the snowline are far more efficiently accreted by the satellitesimals than those inside it. Consequently, satellitesimals beyond the snowline grow faster than those inside it and tend to starve the innermost satellitesimals by reducing the flux of pebbles that they receive. As distant more massive objects migrate inwards they collide with lower-mass satellites growing by the accretion of silicate pebbles. Typically, the innermost satellite in our simulations is a satellitesimal that -- on its way to the inner edge of the disk -- has collided with several satellitesimals growing inside the snowline. This tends to affect its final water-mass-fraction as observed in Figure \ref{fig:compilation}. However, in none of our simulations, our innermost analogue is as dry as Io or Europa. We have performed a limited number of simulations where we increase the sizes of the pebbles inside the snowline by a factor of 3, to attempt to accelerate the growth of the innermost satellites, but none of these simulations produce good analogues satisfying the i)-iv) conditions of Section \ref{sec:constraints}.
As discussed before, it is not clear if Io and Europa were born with water-ice poor compositions and then lost (most/all) their water-content after formation \citep{Bi20} or formed with their current compositions. So if Io and Europa formed by pebble accretion and with water-poor compositions they must have formed very early, probably much earlier than Ganymede and Callisto, and at a time where satellitesimals existed (or were able to efficiently grow via pebble accretion) only well inside the disk snowline.
On the other hand, if Io and Europa lost their water via hydrodynamic escape \citep{Bi20} the masses of the innermost satellites in our simulations should be reduced by a factor of 1.2-1.7 to account for the water-ice component loss. We do not simulate this effect here, but if this is the case our innermost satellite would still have mass satisfying constraint iii) in most of our analogues. This is not true for the second innermost satellites in our analogues (Figure \ref{fig:compilation}). But, as some of our simulations do produce Europa analogues twice as massive as the current Europa (Figure \ref{fig:compilation}) our lack of a good match in our analogues' sample is probably a consequence of the stochasticity of these simulations and small number statistics.
Finally, the top system in Figure \ref{fig:compilation} shows that the third innermost analogue shares its orbit with a co-orbital satellite. Note that in our system analogues, Ganymede analogues are typically not as massive as the real Ganymede (but masses typically agree within a factor of $\sim$2). A potential future collision of these co-orbital satellites could result in a satellite with final mass even closer to that of Ganymede.
\begin{figure*}
\includegraphics[width=1.8\columnwidth]{figures/nominais.png}
\caption{Galilean system analogues at the end of the gas disk phase (2~Myr). Each line shows a satellite system produced in our simulations. Individual satellites are represented by color-coded dots. Dot's size scales linearly with the satellite's mass and color represents its water-ice fraction. The horizontal axis shows satellites' orbital semi-major axis. Orbital eccentricities are represented by the horizontal red bars showing the variation in heliocentric distance over semi-major axis ($R_{\rm J}$). The three systems presented here are also shown in Figure~\ref{fig:triangular}.\label{fig:compilation}}
\end{figure*}
\section{Mimicking the long-term dynamical evolution of our Galilean system analogues}\label{sec:faketides}
In our simulations, when the gas in the CPD dissipates, at 2~Myr, orbital eccentricity damping and gas-driven migration cease. Our simulations are numerically integrated at most for additional 8~Myr in a gas-free environment where satellites and Jupiter gravitationally interact as point-mass particles. However, in reality, the long-term dynamical evolution of the Galilean satellites is also modulated by tidal interactions with Jupiter and other satellites \citep{Pe02,Fu16,lari2020long}. Tidal effects tend to increase the angular momentum of the satellites -- that migrate outwards because the Galilean satellites are outside the centrifugal radius of Jupiter -- and decrease their orbital eccentricities \citep{Gr73}. The resonant configuration of the Galilean satellites is expected to enhance the effects of planet-satellite tidal dissipation \citep{lari2020long}.
In the theory of dynamical tides, if each satellite has a different effective tidal quality factor $Q$ due to resonance locking between moons and internal oscillation modes of Jupiter (see Eq.~\ref{fuller} and \ref{k2Q}), tidal migration can be divergent~\citep{Fu16}. To estimate the potential effects of the resonance locking on the long-term dynamical evolution of our system analogues, we conduct some additional numerical simulations where we use a simple prescription to crudely mimic the effects of tides on our satellite systems. Our main goal here is to test if the three innermost satellites remain in resonance when subject to tidal dissipation forces and to infer the level of dynamical excitation of our final systems.
The migration timescale of a satellite via dynamical tides is given by \citep{Fu16,nimmo2018,downey2020inclination}:
\begin{equation}
\frac{1}{t_a}=\frac{|\dot{a}|}{a}=\frac{2}{3}\frac{1}{t_\alpha}\left(\frac{\Omega}{n}-1\right), \label{fuller}
\end{equation}
with
\begin{equation}
\frac{k_2}{Q}=\frac{1}{3nq_s}\left(\frac{a}{\rm R_J}\right)^5\frac{1}{t_a}, \label{k2Q}
\end{equation}
where $\Omega$ and $t_\alpha$ are Jupiter's rotation frequency and internal oscillation modes timescale and $k_2$ is the Love number. Astrometric observations \citep{valery2009} suggest that three innermost Galilean satellites migrate outwards with timescales of 20~Gyr, which translates to $t_\alpha$ equal to 44, 101, and 217~Gyr, from the Io to Ganymede \citep{downey2020inclination}. There is no estimate of the current tidal migration timescale of Callisto via astrometric observations~\citep{downey2020inclination}. So we use equation~\ref{fuller} to estimate Callisto's current tidal migration timescale assuming that: i) Callisto is initially locked in a 2:1 MMR with Ganymede; ii) Ganymede is initially at 13~$R_{\rm J}$ (consistent with our Analogue 1); iii) Callisto migrates to its current position over the age of the Solar System; iv) the three innermost satellites remain locked in the Laplace resonance. We obtained for Callisto a $t_a=12$~Gyr ($t_\alpha=318$~Gyr). Figure~\ref{fig:edampSMA} shows the estimated position of the satellites as a function of time, given by equation~\ref{fuller}. The solid lines show the semi-major axis evolution of each satellite. The green dot-dashed line shows the position of the 2:1~MMR with Ganymede (red solid line). As one can see, at the end of the simulation at 4.57~Gyr, Callisto migrates outwards faster than the 2:1~MMR with Ganymede moves outwards, leaving the resonance configuration. The three innermost satellites, on the other hand, remain locked in the Laplace resonance as we will show later. With the migration timescales of all satellites in hand, we can now perform N-body numerical simulations to probe the long-term dynamical evolution of our analogues.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/sma_tides.png}
\caption{Temporal evolution of the semimajor axis of the Galilean satellites over the Solar System age as estimated via resonance locking theory (Fuller et al 2016). The values of $t_\alpha$ for Io, Europa, Ganymede, and Callisto are 44, 101, 217, and 318~Gyr, respectively. The dot-dashed line corresponds to the evolution of the 2:1 MMR with Ganymede.\label{fig:edampSMA}}
\end{figure}
To perform our simulations mimicking the effects of dynamical tides, we use the final orbital configuration of the system at the end of the gas disk phase (Analogue~1) and we consider the subsequent evolution of the system in a gas-free scenario. Due to the high computational cost of these simulations, we do not numerically integrate our system for the entire age of the Solar System, but only for 20~Myr (which requires 3 weeks of CPU-time in a regular desktop). We have assumed for the three innermost satellites $t_a=20$~Gyr, as suggested by observations~\citep{valery2009}, and $t_a=12$~Gyr for Callisto as discussed before. When performing our N-body simulations we also apply eccentricity damping to the satellites, which is assumed to correlate with the tidal migration timescale via the factor $S$ given as :
\begin{equation}
S=\frac{1/t_e}{1/t_a}=\frac{|\dot{e}/e|}{|\dot{a}/a|}
\end{equation}
here we test $S=10^5$ and $S=10^6$. These values are based on the ratio between semimajor axis and eccentricity variation timescales for Galilean satellites in classical tidal theory \citep{goldreich1966q,zhang2004,lainey2017new}.
\begin{figure}
\subfigure[]{\includegraphics[width=\columnwidth]{figures/adamp1e5.png}}
\subfigure[]{\includegraphics[width=\columnwidth]{figures/edamp1e5.png}}
\caption{a) Semi-major axis and b) eccentricity of the satellites in Analogue 1 for $S=10^5$. Each solid line shows the evolution of one satellite. Dot-dashed horizontal lines show the orbital eccentricities of the Galilean satellites for reference.\label{fig:edampF}}
\end{figure}
Figure~\ref{fig:edampF} shows the semi-major axes and orbital eccentricities' evolution of satellites in Analogue~1 during 20~Myr. Our analogue satellites migrate outwards very slowly and the system remains dynamically stable. The bottom panel of Figure~\ref{fig:edampF} shows that the orbital eccentricities of our analogues are damped to values consistent with those of the real Galilean satellites in a relatively short timescale. The Figure~\ref{fig:edampF} corresponds to $S=10^5$. Our simulation where $S=10^6$ resulted in an even lower level of orbital excitation at 20~Myr.
Finally, in Figure \ref{fig:resangF}, we analyze the behavior of the resonant angles characterizing our resonant chain in the same simulation of Figure~\ref{fig:edampF}. From top to bottom, the first three panels show the resonant angles associated with the 2:1 MMR. These plots show that the innermost and second innermost satellite pairs remain locked in a 2:1 MMR when we mimic the tidal dissipation effects on eccentricity. This is indicated by the reduction -- relative to the beginning of the simulation -- in the libration amplitudes of the resonant angles ($\phi$) which librate around zero with small amplitude after the simulation timespan. However, in the case of the outermost satellite pair, the amplitudes of libration of the associated resonance angles gradually increase until they start to circulate, dissolving the resonant configuration, without breaking the resonant configuration of the other pairs. The bottom panel shows that the amplitude of the resonant angle associated with the Laplacian resonant slightly increases at $\sim 8$~Myr -- the moment when the outermost satellite leaves the resonance with the second outermost satellite -- but the three innermost satellites remain locked in this configuration. Therefore, our results also suggest that the Galilean satellite system is a primordial resonant chain, where Callisto was once in resonance with Ganymede but left this configuration via divergent migration due to dynamical tides \citep{Fu16,downey2020inclination,lari2020long,Ha20,Durante2020,Idini2021}. Of course, a complete validation of this result may require self-consistent simulations modeling tidal planet-satellite dissipation effects but this is beyond the scope of this paper.
\begin{figure}
\subfigure[$\phi=2\lambda_2-\lambda_1-\varpi_1-180^{\circ}$ (red line) and $\phi=2\lambda_2-\lambda_1-\varpi_2$ (blue line)]{\includegraphics[width=\columnwidth]{figures/resso1e5.png}}
\subfigure[$\phi=2\lambda_3-\lambda_2-\varpi_2-180^{\circ}$~(red~line)~and~$\phi=2\lambda_3-\lambda_2-\varpi_3$~(blue~line)]{\includegraphics[width=\columnwidth]{figures/resso2e5.png}}
\subfigure[$\phi=2\lambda_4-\lambda_3-\varpi_3-180^{\circ}$ (red line) and $\phi=2\lambda_4-\lambda_3-\varpi_4$ (blue line)]{\includegraphics[width=\columnwidth]{figures/resso3e5.png}}
\subfigure[$\phi=2\lambda_3-3\lambda_2+\lambda_1$]{\includegraphics[width=\columnwidth]{figures/resso4e5.png}}
\caption{From top to bottom, the first three panels show the evolution of resonant angles associated with the 2:1 MMR of different satellite pairs. The bottom panels shows the resonant angle associated with the Laplacian resonance characterized by the three innermost satellites. The labels 1, 2, 3, and 4 corresponds to the innermost, second innermost, third innermost, and outermost satellite in our simulation, respectively. All these resonant angles librate at the beginning of the simulation, which corresponds to the the end of the gas disk phase. The orbital eccentricity of the innermost satellite is damped in a timescale $t_{e}=t_{a}/S$, where $S=10^5$ (see Figure \ref{fig:edampF}), to mimic the effects of tidal dissipation. \label{fig:resangF}}
\end{figure}
\section{Discussion}\label{sec:discussion}
We have assumed a fully formed Jupiter from the beginning of our simulations. We believe this is a fair approximation because Jupiter's CPD and the Galilean satellites mostly likely started to form during the final phase of Jupiter's growth. \cite{Sz16} showed that the characteristics of a circumplanetary disk are mainly determined by the temperature at the planet's location -- which is driven by the gas accretion rate onto the planet -- and have a weak dependency on the planet mass. In our model, the total flux of gas from the CSD to the CPD is about $\sim0.1~{\rm M_J}$. So, even if one assumes that all gas entering the CPD is accreted by Jupiter, it should have no less than 90\% of its current mass at the beginning of our simulations. Thus, we do not expect this small difference in the planet's mass than we have considered here to change the quality of our results. One potential impact could be seen in the migration timescale of the satellites. However, the migration timescale depends also on CPD's model, as surface density and aspect ratio. As we have performed simulations considering different dissipation modes for the gas disk we consider our main findings to be fairly robust against this issue.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/impact.png}
\caption{ Normalized impact velocity (${\rm v_{imp}/v_{esc}}$) as a function of the impact angle in simulations that produced Galilean satellite analogues. The color-coding shows the predicted outcome following Kokubo \& Genda (2010) and Genda et al. (2012). Blue (red) dots correspond to the impacts that fall in the merging (hit-and-run) regime. 85\% of the impacts in these selected simulation qualify as perfect merging events. \label{fig:impact}}
\end{figure}
Our treatment of collisions is also simplistic. All of our collisions are modeled as perfect merging events that conserve mass and linear momentum. To evaluate whether or not our assumption is adequate, we have analyzed the expected outcome of collisions in our simulations following \cite{2010ApJ...714L..21K} and \cite{2012ApJ...744..137G}. We have found that about 85\% of our collisions qualify as perfect merging events, and only 15\% of our simulations fall into the hit-and-run regime (see Figure~\ref{fig:impact}). Impact velocities in our simulations are very low because collisions happen during the gas disk phase when satellites have low orbital eccentricities and inclinations due to gas-tidal damping and drag. These results show that our treatment of impacts is fairly appropriate to study the formation of the Galilean satellites.
For simplicity, our simulations started with satellitesimals distributed initially between 20 and 120~${\rm R_J}$. We have argued that these objects were most likely captured from the CSD \citep{Zh12} rather than having been born in-situ. However, we do not model the capture of planetesimals from the CSD in the CPD in this work. The CPD's location where planetesimals from the CSD are preferentially captured depends on planetesimals sizes, orbits, gas surface density, and ablation degree when they enter the CPD \citep{Fu13,Ro20}. Simulations from \cite{Ro20} suggest the total time necessary to Jupiter capture the total mass in satellitesimals assumed at the beginning of our simulations would be only $\sim 10^3-10^4$~years.
Our best Galilean satellite analogues have typically four satellites as the real Galilean system. However, Jupiter hosts a complex system of moons. Jupiter hosts four satellites with sizes of tens of kilometers inside Io's orbit and a large set of regular satellites of a few kilometers in size outside Callisto's orbit. Here we speculate that these two populations of objects represent fragments, late captured, or leftover satellitesimals that were too small to efficiently grow by pebble accretion. This hypothesis is under investigation.
Our model is honestly simplified and we do not plead to definitively explain the formation of the Galilean satellites. However, it is astonishing that using the same physical processes invoked in models to explain the formation of planets in the Solar System and around other stars \cite[e.g.][]{levisonetal15b,Iz19}, our simulations can still form satellites systems that resemble fairly well the Galilean moons. Our results also show the importance of future studies to provide firmer constraints on the original composition of Io and Europa. Were they born with water-rich compositions or not?
\section{Conclusions}\label{sec:conclusions}
We have performed a suite of N-body numerical simulations to study the formation of the Galilean satellites in a gas-starved disk scenario \citep{Ca02,Ca06,Ca09,Sa10} invoking pebble accretion and migration. Our simulations start considering an initial population of $\sim$200~km-size satellitesimals in the circumplanetary disk that are envisioned to be planetesimals captured from the sun's natal disk. We have performed simulations testing different initial number of satellitesimals in the CPD and pebble fluxes. Our pebbles fluxes are consistent with those estimated via the ablation of planetesimals in the circumplanetary disk \citep{Ro20}. Our best Galilean system analogues were produced in simulations where the time-integrated pebble flux is about $10^{-3}~{\rm M_J}$. In these simulations, satellitesimals grow via pebble accretion and migrate to the disk inner edge where they stop migrating at the inner edge trap. When they approach this location, dynamical instabilities and orbital crossing promote further growth via impacts. Our simulations typically produced between 3 and 5 final satellites anchored at the disk inner edge, forming a resonant chain. Their masses match relatively well those of the Galilean satellites. All our system analogues show 3 pairs of satellites locked in a 2:1 mean motion resonance. Thus, we propose that Callisto -- the outermost Galilean satellite -- was originally locked in resonance with Ganymede but left this primordial configuration via divergent migration due to tidal dissipative effects \citep{Fu16,downey2020inclination}. We also proposed that the orbital eccentricities of the Galilean satellites were much higher in the past and were damped to their current values via tidal dissipation without destroying the resonant configuration of the innermost satellites \citep{Fu16,downey2020inclination,lari2020long,Ha20,Durante2020,Idini2021}. Finally, we proposed that the Galilean system represents a primordial resonant chain that did not become unstable after the circumplanetary gas disk dispersal. Thus the formation path of the Galilean system is probably similar to that of systems of close-in super-Earths around stars, as the Kepler-223 \citep{mills2016resonant}, TRAPPIST-1 \citep{gillon2017seven}, and TOI-178 \citep{Leleu21} systems~\citep{Iz17,Iz19}.
Our simulations do not reproduce the current low water-ice fractions of Io and Europa, and require efficient water loss via hydrodynamic escape \citep{Bi20} to occur to match their current bulk compositions. If efficient water loss via hydrodynamic took place it is expected that Europa should have developed a higher deuterium-to-hydrogen ratio compared with Ganymede and Callisto \citep{Bi20}. This prediction may be tested in the future via in-situ measurements by the Europa Clipper spacecraft or infrared spectroscopic observations \citep{Bi20}. In all our simulations, the two outermost satellites have water-ice rich compositions similar to Ganymede and Callisto. Our results suggest that if Io and Europa were born water-ice depleted, they should have formed much earlier than Ganymede and Callisto and well inside the CPD's snowline. Additional constraints on Io and Europa are now crucial to constrain Galilean satellites' formation models.
\section*{Acknowledgements}
We thank the referee, Konstantin Batygin, for carefully reading our paper and providing valuable comments and suggestions. G.M. thanks FAPESP for financial support via grant 2018/23568-6.
S.M.G.W. thanks FAPESP (2016/24561-0), CNPq (309714/2016-8; 313043/2020-5) and Capes for the financial support.
A.I thanks financial support from FAPESP (2016/12686-2; 2016/19556-7) and CNPq (313998/2018-3) during the initial preparation of this work. A.~I. also thanks financial support via NASA grant 80NSSC18K0828 during the final preparation and submission of this work.
Research carried out using the computational resources of the Center for Mathematical Sciences Applied to Industry (CeMEAI) funded by FAPESP (grant 2013/07375-0).
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
2,877,628,090,215 | arxiv | \section{Introduction}
The analysis and interpretation of ongoing and future neutrino oscillation
experiments strongly rely on the nuclear modeling for describing the
interaction of neutrinos and anti-neutrinos with the detector.
Moreover, neutrino-nucleus scattering has recently become a matter of debate
in connection with the possibility of extracting information on the nucleon
axial mass.
Specifically, the data on muon neutrino charged-current quasielastic (CCQE)
cross sections obtained by the MiniBooNE
collaboration~\cite{AguilarArevalo:2010zc} are substantially underestimated by
the Relativistic Fermi Gas (RFG) prediction.
This has been ascribed either to effects in the elementary
neutrino-nucleon interaction, or to nuclear effects.
The most poorly known ingredient of the single nucleon
cross section is the cutoff parameter $M_A$
employed in the dipole prescription for the axial form factor of the nucleon,
which can be extracted from $\nu$ and $\overline\nu$ scattering off hydrogen
and deuterium and from charged pion electroproduction.
If $M_A$ is kept as a free parameter in the
RFG calculation, a best fit of the MiniBooNE data yields a value of
the order of 1.35 GeV/c$^2$, much larger than the average value
$M_A\simeq 1.026 \pm 0.021$ GeV/c$^2$ extracted from the (anti)neutrino world
data~\cite{Bernard:2001rs}. This should be taken more
as an indication of incompleteness of the theoretical description of
the data based upon the RFG, rather than as a true indication for a larger
axial mass.
Indeed it is well-known from comparisons with electron scattering data
that the RFG model is too crude to account for the nuclear dynamics.
Hence it is crucial to explore more sophisticated nuclear models
before drawing conclusions on the value of $M_A$.
Several calculations have been recently performed and applied to neutrino reactions.
These include, besides the approach that will be presented here,
models based on
nuclear spectral functions~\cite{Barbaro:1996vd,Benhar:2005dj,Benhar:2006nr,Benhar:2009wi,Ankowski:2010yh,Benhar:2010nx,Juszczak:2010ve,Ankowski:2011dc},
relativistic independent particle models~\cite{Alberico:1997vh,Alberico:1997rm,Alberico:1998qw},
relativistic Green function approaches~\cite{Meucci:2003cv,Meucci:2004ip,Meucci:2006ir,Meucci:2011pi,Meucci:2011vd},
models including NN correlations~\cite{Antonov:2006md,Antonov:2007vd,Ivanov:2008ng},
coupled-channel transport models~\cite{Leitner:2006ww,Leitner:2006sp,Buss:2007ar,Leitner:2010kp}, RPA calculations~\cite{Nieves:2004wx,Nieves:2005rq,Valverde:2006zn} and models including multinucleon knock-out~\cite{Martini:2009uj,Martini:2010ex,Nieves:2011pp,Nieves:2011yp}.
The difference between the predictions of the above models can be large due to
the different treatment of both initial and final state interactions.
As a general trend, the models based on impulse approximation, where the
neutrino is supposed to scatter off a single nucleon inside the nucleus,
tend to underestimate the MiniBooNE data, while a sizable increase of the
cross section is obtained when two-particle-two-hole (2p-2h) mechanisms are
included in the calculations. Furthermore, a recent calculation performed
within the relativistic Green function (RGF) framework has shown that at this
kinematics the results strongly depend on the phenomenological optical
potential used to describe the final state interaction between the ejected
nucleon and the residual nucleus~\cite{Meucci:2011vd}.
With an appropriate choice of the optical potential the RGF model
can reproduce the MiniBooNE data without the need of modifying the axial mass
(see Giusti's contribution to this volume~\cite{Giusti:2011ar}).
The kinematics of the MiniBooNE experiment, where the neutrino flux spans
a wide range of energies reaching values as high as 3 GeV, demands relativity as an essential ingredient. This is illustrated in Fig.~1, where the
relativistic and non-relativistic Fermi gas results for the
CCQE double differential cross section of 1 GeV muon neutrinos on $^{12}C$ are
shown as a function of the outgoing muon momentum and for two values of the
muon scattering angle. The relativistic effects, which affect both the
kinematics and the dynamics of the problem, have been shown to be relevant
even at moderate momentum and energy transfers~\cite{Amaro:1998ta,Amaro:2002mj}.
\begin{figure}[h]
\begin{minipage}{38pc}
\includegraphics[height=10pc]{fignr1b.eps}
\hspace{3pc}
\includegraphics[height=10pc]{fignr2b.eps}
\caption{\label{fig:nr} $\nu_\mu$CCQE
double differential cross sections on $^{12}C$
displayed versus the outgoing muon momentum for non-relativistic (NRFG) and relativistic (RFG) Fermi gas.}
\end{minipage}\hspace{2pc
\end{figure}
Hence in our approach we try to retain as much as possible
the relativistic aspects of the problems.
In spite of its simplicity, the RFG has the merit of incorporating
an exact relativistic treatment, fulfilling the fundamental properties of
Lorentz covariance and gauge invariance. However, it badly fails to reproduce the electron
scattering data, in particular when it is compared with the Rosenbluth-separated
longitudinal and transverse responses.
Comparison with electron scattering data must be
a guiding principle in selecting reliable models for neutrino reactions.
A strong constraint in this connection is represented
by the ``superscaling'' analysis of the world inclusive $(e,e')$
data: in Refs.~\cite{Day:1990mf,Jourdan:1996ut,Donnelly:1998xg,Donnelly:1999sw,
Maieron:2001it} it has been proved that, for sufficiently large momentum
transfers, the reduced cross section (namely the double differential
cross section divided by the appropriate single nucleon factors),
when represented versus the scaling variable $\psi$~\cite{Alberico:1988bv},
is largely independent of the momentum transfer (first-kind scaling) and of
the nuclear target (second-kind scaling). The simultaneous occurrence of the
two kinds of scaling is called susperscaling.
Moreover, from the experimental longitudinal response a
phenomenological quasielastic scaling function has been extracted that shows a
clear asymmetry with respect to the quasielastic peak (QEP)
with a long tail extended to
positive values of the scaling variable, i.e., larger energy transfers.
On the contrary the RFG model, as well as most models based on impulse
approximation, give a symmetric superscaling function with a maximum value
20-30\% higher than the data~\cite{Barbaro:2006me}.
In this contribution, after recalling the basic formalism for CCQE reactions
and their connection with electron scattering,
we shall illustrate two models which provide good agreement with the
above properties of electron scattering data: one of them, the relativistic
mean field (RMF) model,
comes from microscopic many-body theory, the other, the
superscaling approximation (SuSA) model, is
extracted from $(e,e')$ phenomenology.
We shall then include the contribution of 2p-2h excitations in the SuSA
model and finally compare our results with the MiniBooNE double differential,
single differential and total cross sections.
Most of the results which will be presented are contained in
Refs.~\cite{Amaro:2010sd} and \cite{Amaro:2011qb}.
\section{Formalism}
Charged current quasielastic
muonic neutrino scattering $(\nu_\mu,\mu^{-})$ off nuclei
is very closely related to quasielastic inclusive electron scattering $(e,e')$.
However two major differences occur between the two processes:
\begin{enumerate}
\item in the former case the probe interacts with the nucleus via the weak
force, in the latter the interaction is (dominantly) electromagnetic.
While the weak vector
current is related to the electromagnetic one via the CVC theorem,
the axial current gives rise to a more complex structure of the cross sections,
with new response functions which cannot be related to the electromagnetic ones.
As a consequence, while in electron scattering the double-differential cross
section can be expressed in terms of two response functions, longitudinal and
transverse with respect to three-momentum carried by the virtual photon, for
the CCQE process it can be written according to a Rosenbluth-like decomposition
as~\cite{Amaro:2004bs}
\begin{equation}
\left[ \frac{d^2\sigma}{dT_\mu d\cos\theta} \right]_{E_\nu} =
\sigma_0 \left[ {\hat V}_{L} R_L
+ {\hat V}_T R_T + {\hat V}_{T^\prime} R_{T^\prime} \right] ,
\label{eq:d2s}
\end{equation}
where $T_\mu$ and $\theta$ are the muon kinetic energy and scattering angle,
$E_\nu$ is the incident neutrino energy, $\sigma_0$ is the elementary
cross section, ${\hat V}_i$ are kinematic
factors and $R_i$ are the nuclear response functions,
the indices $L,T,T^\prime$ referring to longitudinal, transverse,
transverse-axial, components of the nuclear current, respectively.
The expression (\ref{eq:d2s}) is formally analogous to the inclusive electron
scattering case, but: {\it i)} the ``longitudinal'' response $R_L$
takes contributions from the charge (0) and longitudinal (3) components of the
nuclear weak current, which, at variance with the electromagnetic case,
are not related to each other by current conservation, {\it ii)} $R_L$ and $R_T$
have both ``VV'' and ``AA'' components (stemming from the product of two vector
or axial currents, respectively),
{\it iii)} a new
response, $R_{T^\prime}$, arises from the interference between the axial and vector
parts of the weak nuclear current.
In Fig.~2 we show the separate contributions of the three responses in
(\ref{eq:d2s}), evaluated
in the RFG model for the $^{12}C$ target nucleus, for two different values of
the scattering angle.
It can be seen that in the forward direction the transverse response dominates
over the longitudinal and transverse-axial ones, whereas at higher angles the
$L$-component becomes negligible and the $T^\prime$ and $T$ responses are almost
equal. This cancellation has important consequences on antineutrino-nucleus
scattering, where the response $R_{T^\prime}$ has opposite sign.
\begin{figure}[h]
\begin{minipage}{38pc}
\includegraphics[width=16pc]{cos095_sep_nodata.eps}
\hspace{3pc}
\includegraphics[width=16pc]{cos005_sep_nodata.eps}
\caption{\label{fig:sep} Separate contributions of the RFG longitudinal $(L)$,
transverse $(T)$ and axial-vector interference $(T^\prime)$ responses to the
double differential $\nu_\mu$CCQE cross sections displayed versus the muon kinetic
energy at two different angles. The neutrino energy is averaged over the
MiniBooNE flux and the axial mass parameter has the standard value.
}
\end{minipage}\hspace{2pc
\end{figure}
\item In $(e,e')$ experiments the energy of the electron is well-known, and therefore the detection of the outgoing electron univoquely determines
the energy and momentum transferred to the nucleus.
In neutrino experiments the neutrino energy is not known, but distributed over a range of
values (for MiniBooNE from 0 to 3 GeV with an average value
of about 0.8 GeV).
The cross section must then be evaluated
as an average over the experimental flux $\Phi(E_\nu)$
\begin{equation}
\frac{d^2\sigma}{dT_\mu d\cos\theta}
= \frac{1}{\Phi_{tot}} \int
\left[ \frac{d^2\sigma}{dT_\mu d\cos\theta} \right]_{E_\nu}
\Phi(E_\nu) dE_\nu \ ,
\label{eq:fluxint}
\end{equation}
which may require to
account for effects not included in models devised for quasi-free scattering.
This is, for instance, the situation at the most forward scattering angles,
where a significant contribution in the cross section comes from
very low-lying excitations in nuclei~\cite{Amaro:2010sd}, as illustrated in Fig.~3:
here the double differential cross section is evaluated in the SuSA model
(see later) at the MiniBooNE kinematics and the lowest angular bin and compared
with the result obtained by excluding the energy transfers lower than 50 MeV
from the flux-integral (\ref{eq:fluxint}).
\begin{figure}[h]
\begin{minipage}{17pc}
\includegraphics[width=17pc]{cutomega_cos095.eps}
\end{minipage}\hspace{3pc
\begin{minipage}{17pc}
\caption{\label{fig:cut}(Color online) Solid lines (red online):
flux-integrated $\nu_\mu$CCQE cross sections on $^{12}C$ calculated in the SuSA
model for a specific bin of scattering angle. Dashed lines (green online):
a lower cut $\omega=50$ MeV is set in the integral over the neutrino flux.}
\end{minipage
\end{figure}
It appears that at these angles 30-40\% of the cross section corresponds
to very low energy transfers, where collective effects dominate.
Moreover, processes involving meson exchange currents (MEC), which can excite
both one-particle-one-hole (1p1h) and two-particle-two-hole (2p-2h) states
via the exchange of a virtual meson,
should also be taken into account, since they lead to final states where no
pions are present, classified as ``quasielastic'' in the MiniBooNE
experiment.
\end{enumerate}
\section{Models}
In this Section we briefly outline the main ingredients of the RMF and SuSA
model and
we illustrate our calculation of the contribution of 2p-2h meson
exchange currents.
\subsection{RMF}
In the RMF model
a fully relativistic description of both the kinematics and the
dynamics of the process is incorporated.
Details on the RMF model applied to inclusive QE electron and CCQE neutrino
reactions can be found in Refs.~\cite{Caballero:2006wi,Caballero:2007tz,Caballero:2005sj,Amaro:2006tf,Maieron:2003df,Amaro:2006if}.
Here we simply recall that the weak response functions are given by
taking the appropriate components of the weak hadronic tensor,
constructed from the single-nucleon current matrix elements
\begin{equation}
\langle J_W^\mu\rangle = \int d{\bf r}\overline{\phi}_F({\bf r})\hat{J}_W^\mu({\bf r})\phi_B({\bf r})
\, ,
\end{equation}
where $\phi_B$ and $\phi_F$ are relativistic bound-state and
scattering wave functions, respectively, and $\hat{J}_W^\mu$ is
the relativistic one-body current operator modeling the coupling
between the virtual $W$-boson and a nucleon. The bound
nucleon states are described as self-consistent Dirac-Hartree
solutions, derived by using a Lagrangian containing $\sigma$, $\omega$ and
$\rho$
mesons~\cite{Horowitz,Serot,Sharma:1993it}. The outgoing nucleon
wave function is computed by using the same relativistic mean
field (scalar and vector energy-independent potentials) employed in the
initial state and incorporates the final state interactions (FSI)
between the ejected proton and the residual nucleus.
The RMF model successfully
reproduces the scaling behaviour of inclusive QE $(e,e')$ processes
and, more importantly, it gives rise to a
superscaling function with a significant asymmetry, namely, in
complete accord with data~\cite{Caballero:2006wi,Caballero:2007tz}.
This is a peculiar property associated to the consistent treatment of initial
and final state interactions. It has been shown
in Refs.~\cite{Caballero:2006wi,Caballero:2007tz} that other versions of the RMF
model, which deal with the FSI through a real relativistic optical potential, are not
capable of reproducing the asymmetry of the scaling function.
Moreover, contrary to most other models based on impulse approximation,
where scaling of the
zeroth kind - namely the equality of the longitudinal and transverse scaling functions - occurs, the RMF model provides $L$ and $T$ scaling
functions which differ by typically $20\%$, the T one being larger.
This agrees with the analysis \cite{Donnelly:1999sw} of
the existing $L/T$ separated data, which has shown that, after
removing inelastic contributions and two-particle-emission effects,
the purely nucleonic transverse scaling function
is significantly larger than the longitudinal one.
\subsection{SuSA}
The SuSA model is based on the phenomenological superscaling function
extracted from the world data on quasielastic electron
scattering~\cite{Jourdan:1996ut}. The model has been extended to the
$\Delta$-resonance region in Ref.~\cite{Amaro:2004bs} and to neutral current
scattering in Ref.~\cite{Amaro:2006pr}, but here we restrict our attention
to the quasielastic charged current case.
Assuming the scaling function $f$ extracted from $(e,e')$ data
to be universal, i.e., valid for electromagnetic and weak interactions,
in~\cite{Amaro:2004bs,Amaro:2005dn} CCQE neutrino-nucleus cross
sections have been evaluated by multiplying $f$ by the corresponding elementary
weak cross section.
Thus in the SuSA approach all
the nuclear responses in (\ref{eq:d2s}) are expressed as follows
\begin{equation}
R_K = N \frac{2 E_F}{k_F q} U_K(q,\omega) f(\psi) \ \ , \ \ \ (K=L,T,T^\prime)
\end{equation}
where $U_K$ are the elementary lepton-nucleon responses, $E_F$ and $k_F$ are
the Fermi energy and momentum, $N$ is the number of nucleons (neutrons
in the $\nu_\mu$CCQE case) and $f(\psi)$ in the universal superscaling function,
depending only on the scaling variable $\psi$~\cite{Alberico:1988bv}.
The SuSA approach
provides nuclear-model-independent neutrino-nucleus cross sections
and reproduces the longitudinal electron data by construction.
However, its reliability rests on some basic assumptions.
First, it assumes that the scaling function - extracted from {\it longitudinal}
$(e,e')$ data - is appropriate
for all of the weak responses involved in neutrino
scattering (charge-charge, charge-longitudinal,
longitudinal-longitudinal, transverse and axial), and is independent
of the vector or axial nature of the nuclear current entering the
hadronic tensor. In particular
it assumes the equality of the longitudinal and transverse
scaling functions (scaling of the zeroth kind), which, as mentioned before,
has been shown to be violated both by experiment and by some microscopic models,
for example relativistic mean field theory.
Second, the charged-current neutrino responses are purely isovector, whereas the
electromagnetic ones contain both isoscalar and isovector
components and the former involve axial-vector as well as vector responses.
One then has to invoke a further kind of scaling, namely
the independence of the scaling function of the choice of isospin
channel --- so-called scaling of the third kind.
Finally, the SuSA approach neglects violations of scaling of first and second kinds. These are known to be important at energies above the QE peak and to reside mainly in the transverse channel, being associated to effects which go
beyond the
impulse approximation: inelastic scattering, meson-exchange currents
and the associated correlations needed to conserve the vector current.
The inclusion of these contributions in the SuSA model is discussed
in the next paragraph.
\subsection{2p-2h MEC}
Meson exchange currents are two-body currents carried by a virtual meson
which is exchanged between two nucleons in the nucleus. They are represented by
the diagrams in Fig.~\ref{fig:mec}, where the external lines correspond to
the virtual boson ($\gamma$ or $W$) and the dashed lines to the exchanged
meson: in our approach we only consider the pion, which is believed to give the
dominant contribution in the quasielastic regime. The thick lines in diagrams
(d)-(g) represent the propagation of a $\Delta$-resonance.
The explicit relativistic expressions for the current matrix elements can be found, e.g., in Ref.~\cite{Amaro:2010iu}.
\begin{figure}[h]
\includegraphics[width=25pc]{diagmec.eps}\hspace{2pc
\begin{minipage}{11pc}
\vspace{-5pc
\caption{\label{fig:mec} Two-body meson-exchange currents.
(a) and (b): ``contact'', or ``seagull'' diagram;
(c): ``pion-in-flight'' diagram;
(d)-(g): ``$\Delta$-MEC'' diagram.}
\end{minipage}
\end{figure}
Being two-body currents, the MEC can excite both one-particle one-hole
(1p-1h) and two-particle two-hole (2p-2h) states.
In the 1p-1h sector,
MEC studies of electromagnetic $(e,e^\prime)$
process have been performed for low-to-intermediate momentum transfers
(see, {\it e.g.}, \cite{Alberico:1989aja,Amaro:2002mj,Amaro:2003yd,Amaro:2009dd}),
showing a small reduction of the total response at the quasielastic peak,
mainly due to diagrams involving the electroexcitation of the
$\Delta$ resonance.
However in a perturbative scheme where all the diagrams containing one and only
one pionic line are retained, the MEC are not the only diagrams arising,
but pionic correlation contributions, where the virtual boson is attached to
one of the two interacting nucleons, should also be considered.
These are represented by the same diagrams as in Fig.~\ref{fig:mec}(d)-(g), where now the thick lines are nucleon propagators. Only when all the diagrams
are taken into account gauge invariance is fulfilled and the full two-body
current is conserved.
Correlation diagrams have been shown to roughly compensate the pure MEC
contribution~\cite{Alberico:1989aja,Amaro:2002mj,Amaro:2003yd,Amaro:2009dd},
so that in first approximation we can neglect the 1p-1h sector
and restrict our attention to 2p-2h final states.
The contribution to the inclusive electron scattering cross section
arising from two-nucleon emission via meson exchange current interactions
was first calculated in the Fermi gas model in
Refs.~\cite{Donnelly:1978xa,VanOrden:1980tg}, where sizable effects
were found at large energy transfers.
In these references a non-relativistic reduction of the
currents was performed, while fully relativistic calculations have been
developed more recently in Refs.~\cite{Dekker:1994yc,De Pace:2003xu,Amaro:2010iu}. It
has been found that the MEC give a significant positive contribution
which leads to a partial filling of the ``dip'' between the quasielastic peak
and the analogous peak associated with the excitation of the $\Delta$ resonance.
Moreover, the MEC
have been shown to break scaling of both first and second
kinds~\cite{De Pace:2004cr}.
Here we use the fully relativistic model of \cite{De Pace:2003xu}, where all the MEC many-body diagrams containing two pionic
lines that contribute to the electromagnetic 2p-2h transverse response
were taken into account.
Similar results for the 2p-2h MEC were obtained in
Ref.~\cite{Amaro:2010iu}, where the correlation diagrams were also included.
In order to apply the model to neutrino scattering,
we observe that in lowest order the 2p2h sector is not directly reachable
for the axial-vector matrix elements. Hence the MEC affect only
the transverse polar vector response, $R_T^{VV}$. Note that, at variance with the 1p-1h sector, where the contribution of the
MEC diagrams originates from the interference between 1-body and 2-body
amplitudes and has therefore no definite sign (in fact it turns out to be
negative due to the dominance of the diagrams involving the $\Delta$),
the 2p-2h contribution of MEC to the nuclear responses is the square of an
amplitude, hence it is positive by definition.
Therefore the net effect of 2p-2h MEC to neutrino scattering is to increase
the transverse vector response function, as will be illustrated in the next section.
\section{Results}
In this Section we present the predictions of the above models and their
comparison with the MiniBooNE data. More results can be found in
Refs.~\cite{Amaro:2010sd} and \cite{Amaro:2011qb}.
In Figs.~\ref{fig:cos} and \ref{fig:tmu} the flux-integrated
double-differential cross
section per target nucleon for the $\nu_\mu$CCQE process on
$^{12}$C is evaluated for the three nuclear models above described:
the RMF model and
the SuSA approach with and without the contribution of 2p-2h MEC.
In Fig.~\ref{fig:cos} the cross sections are displayed versus the muon
kinetic energy $T_\mu$ at fixed scattering energy $\theta$, in Fig.~\ref{fig:tmu}
they are displayed versus $\cos\theta$ at fixed $T_\mu$.
It appears that the SuSA predictions systematically underestimate the
experimental cross section, the discrepancy being larger at high scattering
angle and low muon kinetic energy.
The inclusion of 2p-2h MEC tends to improve the agreement with the data at low angles,
but it is not sufficient to account for the discrepancy at higher angles.
The RMF calculation, which, as already mentioned, incorporates violations of
scaling of the zeroth kind with a substantial enhancement of the vector
transverse response, yields cross sections which are in general larger than
the SuSA ones.
In particular, in the region close to the peak in the cross section,
the RMF result becomes larger than the one obtained with SuSA+MEC. Furthermore,
the RMF does better than SuSA in fitting the shape of the experimental curves
versus both the scattering angle and the muon energy:
this is partly due to the fact that the RMF is better describing the
low-energy excitation region whereas the SuSA model has no predictive power at very low angles, where the cross section is dominated by low excitation energies and the superscaling ideas are not supposed to apply.
Concerning the SuSA+MEC results, a possible explanation of the theory/data
disagreement is the fact that,
as already mentioned, a fully consistent treatment of two-body currents
should take into account not only the genuine MEC contributions, but also
the correlation diagrams that are necessary in order to preserve the gauge invariance of the theory. This, however, is not an easy task because
in an infinite system like the RFG the correlation diagrams give rise to
divergences which need to be regularized~\cite{Amaro:2010iu}. The divergences
arise from a double pole in some of the diagrams, associated to the presence
of on-shell nucleon propagators. Different prescriptions have been used in the
literature in order to overcome this problem~\cite{Alberico:1983zg,Alberico:1990fc,Gil:1997bm,Amaro:2010iu},
leading to a substantial model-dependence of the results.
In particular in Ref.~\cite{Amaro:2010iu} the divergence has been cured by introducing a
parameter $\epsilon$ which accounts for the finite size of the nucleus (and therefore
the finite time of propagation of a nucleon inside the nucleus) and the $\epsilon$-dependence
of the contribution of correlation diagrams has been explored. The study has shown that
for reasonable values of the parameter the correlations add to the pure MEC in the
high-energy tail and are roughly of the same order of magnitude, but now contributing
to both the longitudinal and the transverse channels. The inclusion of these terms in neutrino
reactions is in progress~\cite{Amaro:wip} and is expected to give a further enhancement of the
cross sections.
\begin{figure}[h]
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_c085.epsi}
\end{minipage}\hspace{2pc
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_c065.epsi}
\end{minipage}\hspace{2pc
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_c045.epsi}
\end{minipage}
\caption{\label{fig:cos} Flux-integrated double-differential cross
section per target nucleon for the $\nu_\mu$ CCQE process on
$^{12}$C evaluated in the RMF (red) model and in the SuSA approach
with (blue line) and without (green line) the contribution of MEC and
displayed versus the muon kinetic energy $T_\mu$ for three specific bins
of the scattering angle.
The data are from MiniBooNE~\cite{AguilarArevalo:2010zc}.}
\end{figure}
\begin{figure}[h]
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_t025.epsi}
\end{minipage}\hspace{2pc
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_t045.epsi}
\end{minipage}
\begin{minipage}{12pc}
\includegraphics[width=12pc]{RMF_t065.epsi}
\end{minipage}
\caption{\label{fig:tmu} Flux-integrated double-differential cross
section per target nucleon for the $\nu_\mu$ CCQE process on
$^{12}$C evaluated in the RMF (red) model and in the SuSA approach
with (blue line) and without (green line) the contribution of MEC and
displayed versus the muon scattering angle for three bins of the
muon kinetic energy $T_\mu$.
The data are from MiniBooNE~\cite{AguilarArevalo:2010zc}.
}
\end{figure}
The single
differential cross sections with respect to the muon kinetic
energy and scattering angle, respectively, are presented
in Figs.~\ref{fig:csvst} and \ref{fig:csvscos}, where the relativistic Fermi
gas result is also shown for comparison:
again it appears that the RMF gives slightly higher cross sections
than SuSA, due to the $L/T$ unbalance, but both models still underestimate
the data for
most kinematics. The inclusion of 2p-2h excitations leads to a good agreement
with the data at high $T_\mu$, but strength is still missing at the lower muon
kinetic energies (namely higher energy transfers) and higher angles.
\begin{figure}[h]
\begin{minipage}{17pc}
\includegraphics[width=17pc]{dsdcos.epsi}
\caption{\label{fig:csvst}(Color online)
Flux-averaged
$\nu_\mu$CCQE cross section on $^{12}C$
integrated over the scattering angle and
displayed versus the muon kinetic energy.
The data are from MiniBooNE~\cite{AguilarArevalo:2010zc}.
}
\end{minipage}\hspace{2pc
\begin{minipage}{17pc}
\includegraphics[width=17pc]{dsdt.epsi}
\caption{\label{fig:csvscos}(Color online)
Flux-averaged
$\nu_\mu$CCQE cross section on $^{12}C$
integrated over the muon kinetic energy
and displayed versus the scattering angle.
The data are from MiniBooNE~\cite{AguilarArevalo:2010zc}.
}
\end{minipage}
\end{figure}
Finally, in Fig.~\ref{fig:csvsenu} the total
(namely integrated over over all muon scattering angles and energies)
CCQE cross section per neutron is displayed versus the neutrino energy
and compared with the experimental
flux-unfolded data. Besides the models above discussed, we show
for comparison also the results of the relativistic mean field model
when the final state interactions are ignored (denoted as RPWIA - relativistic
plane wave impulse approximation)
or described through a real optical potential (denoted as rROP).
Note that the discrepancies between the various models, observed in
Figs.~\ref{fig:cos} and \ref{fig:tmu},
tend to be washed out by the integration, yielding very similar
results for the models that include FSI (SuSA, RMF and rROP), all of
them giving a lower total cross section than the models without FSI
(RFG and RPWIA). On the other hand the SuSA+MEC curve, while being
closer to the data at high neutrino energies, has a somewhat
different shape with respect to the other models, in qualitative
agreement with the relativistic calculation of \cite{Nieves:2011pp}.
It should be noted, however, that the result is affected by an
uncertainty of about 5\% associated with the treatment of the
2p-2h contribution at low momentum transfers and that pionic correlations
are not included.
\begin{figure}[h]
\begin{minipage}{17pc}
\includegraphics[width=17pc]{sigma.epsi}\hspace{2pc
\end{minipage}\hspace{3pc
\begin{minipage}{17pc}\caption{\label{fig:csvsenu}(Color online)
Total CCQE cross section
per neutron versus the neutrino energy. The curves corresponding to different
nuclear models are compared with the flux unfolded
MiniBooNE data~\cite{AguilarArevalo:2010zc}.}
\end{minipage}
\end{figure}
\section{Conclusions}
Two different relativistic models, one (SuSA)
phenomenological and the other (RMF) microscopic, have been applied to the study
of charged-current quasielastic neutrino scattering and
the impact of 2p-2h meson exchange currents on the cross sections has been
investigated.
The results can be summarized as follows:
\begin{enumerate}
\item
Both the SuSA and the RMF models, in contrast with the
relativistic Fermi gas, are fitting with good accuracy the longitudinal
quasielastic electron scattering response at intermediate to high energy and
momentum transfer.
The SuSA and RMF models give very similar results for the integrated neutrino
cross section and both substantially under-predict the MiniBooNE experimental
data. However the comparison with the double differential experimental
cross section reveals some differences between the two models, which are washed
out by the integration. Indeed the RMF, although being lower than the data,
reproduces better the slopes of the cross section versus the muon energy and
scattering angle. This is essentially due to the enhancement of the transverse
response, which arises from the self-consistent mean field approach of RMF
(in particular from the consistent treatment of initial and final state
interactions) and is absent in the superscaling approach.
\item
In relativistic
or semi-relativistic models final state interactions have been shown to play
an essential role for reproducing the shape and size of the electromagnetic
response~\cite{Maieron:2003df,Caballero:2006wi,Amaro:2006if}
and cannot be neglected, in our scheme, in the study of neutrino interactions.
The effect of final state interactions in the SuSA and RMF models is to
lower the cross section, giving
a discrepancy with the data larger than the RFG.
\item
In the transverse channel, the analysis of $(e,e')$ data points to the
importance of meson-exchange currents which, through the excitation of
two-particle-two-hole states, are partially responsible of filling
the ``dip'' region between the QE and $\Delta$ peaks. The 2p-2h MEC can be
even more relevant in the CCQE process, where ``quasielastic'' implies simply that
no pions are present in the final state but, due to the large energy region
spanned by the neutrino flux, processes involving the exchange of virtual
pions can give a sizable contribution.
In fact the inclusion of 2p-2h MEC
contributions yields larger cross sections and accordingly better
agreement with the data, although the theoretical curves still lie
below the data at high angles and low muon energy.
It should be stressed, however, that the present calculation, though exact and
fully relativistic, is incomplete. In order to preserve gauge invariance
the full two-body current,
including not only the MEC but also the corresponding correlation diagrams, must
be included. These have recently
been shown to yield a sizable contribution at high
energies in $(e,e')$ scattering~\cite{Amaro:2010iu} and are likely to
improve the agreement of our models with the MiniBooNE data.
\item
In all our calculations the standard value $M_A=$1.03 GeV/c$^2$ has been used.
It has been suggested that a larger value of the axial mass (1.35 GeV/c$^2$)
would eliminate the disagreement with the data. However the fit was done
using a RFG analysis, and more sophisticated nuclear models must be explored
before drawing conclusions on the actual value of the axial mass.
For instance in Ref.~\cite{Bodek:2011ps} it is shown that the MiniBooNE
data can as well be fitted by effectively incorporating some nuclear effects
in the magnetic form factor of the bound nucleon, without changing the axial
mass.
Although our scope here is not to extract a value for the axial mass of
the nucleon, but rather to understand which nuclear effects are
effectively accounted for by a large axial cutoff parameter, let us
mention that a best fit of the RMF and SuSA results to the MiniBooNE
experimental cross section gives an effective axial mass
$M_A^{\rm eff}\simeq$ 1.5 GeV/c$^2$
and values in the range $1.35<M_A^{\rm eff}<1.65$ GeV/c$^2$ yield
results compatible with the MiniBooNE data within the experimental errors.
A similar analysis in the model including the 2p-2h contribution will be
possible only when the above mentioned correlation diagrams will be
consistently evaluated~\cite{Amaro:wip}.
\end{enumerate}
\section*{Acknowledgments}
{I would like to thank
J.E. Amaro, J.A. Caballero, T.W. Donnelly, J.M. Udias and C.W. Williamson for
the fruitful collaboration which lead to the results reported in this
contribution.
}
\section*{References}
|
2,877,628,090,216 | arxiv | \section*{Introduction}
Seals and gaskets are crucial components in many hydraulic systems
such as water taps, pipes, pumps, or valves\cite{Flitney07}.
Their main function is to prevent undesired or uncontrolled leakage of
gases or fluids from one region to another.
Despite their importance, attempts to estimate leak rates of seal systems
from first principles succeeded for the first time only less than a decade
ago\cite{Persson08JPCM,Lorenz09EPL,Lorenz10EPJE}.
Earlier treatments could not accurately predict the distribution of microscopic
interfacial separations in a mechanical contact, which is needed for the
fluid-mechanics aspect of the problem. Persson's contact mechanics
theory\cite{Persson01,Persson05JPCM} provides this information and computes
the leakage in terms of Bruggeman's effective-medium
approximation\cite{Bruggeman35} to Reynolds equation\cite{Reynolds86}.
In a previous publication\cite{Dapp12PRL}, we demonstrated that Persson's contact
mechanics theory combined with a slightly modified version of Bruggeman's
effective-medium approximation reproduced almost perfectly the results of computer
simulations, in which an ideally well-defined leakage problem was solved to high
numerical precision.
The favourable comparison of theory and simulations benefited to some degree from
fortuitous error cancellation: Persson theory slightly underestimates the rate at
which (mean) gaps diminish with increasing load,
which almost exactly compensates the minor
overestimation of leakage in Bruggeman's approximation.
Persson's treatment is therefore certainly accurate enough to explain why leakage
through interfaces decreases roughly exponentially with the mechanical
load\cite{Lorenz09EPL,Lorenz10EPJE,Armand64} pressing two (elastic) bodies
against each other, where at least one of them has a
self-affine rough surface (see the method section).
Although Persson theory has proven successful in describing leakage over a broad
parameter range, one cannot expect it to hold near the sealing transition.
One reason is that mean-field theories like Bruggeman's, which is part of
Persson's approach to leakage, are known to fail near critical points, even if
they perform quite well outside the critical region\cite{Kirkpatrick73}.
Alternative, percolation-theory-based treatments of
leakage\cite{Bottiglione09,Bottiglione09TI} or related approaches assuming that
most of the fluid pressure drops near a single, narrow constriction (or a
two-dimensional network of constrictions)\cite{Lorenz09EPL} also risk to fail
in the vicinity of the sealing transition.
This is because length and width of an isolated
constriction show different scaling with the applied load\cite{Dapp15EPL} in
contrast to assumptions made in the respective theories.
In this work, we investigate the fluid leakage through a mechano-hydraulic interface
by means of computer simulations.
In contrast to previous studies\cite{Dapp12PRL,Vallet09a,Vallet09b} our focus lies
on calculating leakage between randomly rough bodies near the percolation threshold.
A particular motivation to revisit the problem stems from our observation that
local details, such as the presence or absence of adhesion between the surfaces,
affect the conductance exponent of isolated constrictions\cite{Dapp15EPL}.
It remains unclear if or to what degree the critical behaviour (evaluated near but
not too close to the percolation threshold) is determined by the disorder at large
length scales, which is usually considered central in percolation
theory\cite{Stauffer91}.
\section*{Results and Discussion}
\subsection*{Adhesion-free sealing transition in the continuum limit}
We begin the analysis of leakage near the percolation threshold by simulating our
``default model''.
It is based on approximations that are commonly made to study either the
contact-mechanics or the fluid-mechanics aspects of our leakage problem:
self-similar surface roughness, linearly elastic bodies, small surface slopes,
and absence of adhesion between the surfaces.
Fluid flow through the interface is treated in terms of the Reynolds equation.
Some of the approximations of our default model are relaxed below.
More details are given in the method section.
Leakage flow for our default system is shown in Fig.~\ref{fig:defaultSystem}
for different reduced loads $1-L/L_{\rm c}$,
where $L$ is the absolute load squeezing the surfaces together and $L_{\rm c}$
is the critical load, defined as the largest load at which at least one fluid
channel still percolates from the right to the left side of the interface.
Our data is based on different surfaces, which are produced with identical
stochastic rules but different random seeds.
To enhance sampling, we also considered inverted and 90$^\circ$ rotated surfaces.
All realisations show similar behaviour:
For very small loads, the current decreases very quickly
with increasing load before the dependence becomes roughly exponential.
At $1-L/L_{\rm c} = O(10\%)$, a crossover to a powerlaw ensues
\begin{equation}
j \propto (1-L/L_{\rm c})^\beta,
\label{eq:criticalEq}
\end{equation}
where $\beta$ is the conductance exponent.
The value of $\beta$ deduced from the data is consistent with the one we identified
for isolated,
single-wavelength
constrictions\cite{Dapp15EPL}, i.e., $\beta=69/20$.
This value is much greater than typical conductance exponents for seemingly
related percolation problems such as the two-dimensional
random (on/off) resistance network, for which $\beta = 1$\cite{Webman77}.
Surprisingly, Bruggeman's effective medium approach predicts the current
quite accurately for most random surface realisations investigated in this study,
even close to the percolation point and in all cases does it find cross-over
loads within roughly 10\% percent at which the exponential load-current relation
ceases to be valid.
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\linewidth]{fig01.eps}
\caption{\label{fig:defaultSystem}
A--D:
Visualisation of the fluid flow through a microscopically rough contact at different
loads, as indicated in panel E.
Black colour marks regions that do not belong to the percolating fluid channel.
Blue and green colours indicate the fluid pressure, which drops from one (blue)
on the left-hand side of the interface to zero (green) on the right-hand side.
Red and yellow indicate the absolute value of the fluid current density.
E main panel: Double-logarithmic representation of the mean leakage current $j$
as a function of the reduced load $1-L/L_{\rm c}$.
Differently coloured symbols represent different random realisations of the surface
roughness.
Data is shifted vertically (by as much as a factor of 10) to superimpose in the
critical region.
In the inset of panel E, the dimensionless load $L/L_{\rm c}$ is plotted linearly
and the current is now normalised (shifting factors $\lesssim 2$) such that it
superimposes in the domain where it decreases exponentially with load.
Red lines show the predictions of effective medium theory, modified such
that it reproduces the exact critical load for a given random realisation
(see Ref.\citenum{Dapp12PRL}).
}
\end{figure*}
To rationalise how the mean fluid flow develops as a function of load,
it is instructive to visualise the spatially resolved fluid pressure and current
density for a particular random realisation.
This is done in Fig.~\ref{fig:defaultSystem}~A--D.
In the domain where flow decreases exponentially with load,
the fluid pressure drops in a quasi-discrete fashion at a number of constrictions.
These constrictions are distributed seemingly randomly throughout the interface
thereby roughly mimicking the conditions assumed in the derivation of
Bruggeman's effective-medium theory.
Once the fluid pressure drops predominantly at a single constriction,
see Fig.~~\ref{fig:defaultSystem}~C and D,
mean-field theory may still be correct, albeit only incidentally.
In the language of percolation theory, all current now goes through one
hot bond.
In contrast to assumptions commonly made for random disorder,
the resistivity assigned to individual points
is not discrete but it changes continuously with the control parameter and
eventually diverges at the critical point.
In our case, the control parameter is the load, while in most percolation models
it would be the probability with which a bond (or a vertex point or an individual
point in a continuous domain)
would be assigned a (fixed) finite or infinite resistance.
Our system can be characterised as having correlated disorder\cite{Schrenk13}
(if the gap is large at a given position, then the gap is also large nearby)
and at the same time long-range interactions\cite{Grassberger13}
(the elastic Green's functions in real space decay with $1/r$).
Apparently, the way in which these two ingredients are combined here
turns percolation of seals into a local problem such that it is not possible to assign
a (unique) universality class to the leakage problem, even if the stochastic
properties of the problem are fully defined.
We note that neither changing the Hurst exponent nor increasing system size alters
the observed behaviour in a qualitative fashion.
While increasing the range over which the surface spectrum is self similar can and
does affect the low-load flow quite dramatically, the critical region does
not appear to be affected, at least not for practically relevant spectra,
in which self-similarity is rarely observed for more than five or six decades
in wavelength.
In all cases we find that critical behaviour, i.e., the range of loads in which
equation~(\ref{eq:criticalEq}) holds to within a few ten percents, starts to set
in at roughly 0.8 to 0.9 times the critical load.
We substantiated these claims by running additional
simulations for $H = 0.3$,
by extending the ratio of roll-off wavelength $\lambda_{\rm r}$ and short
wavelength cutoff $\lambda_{\rm s}$ from 64 to 256 (and within
Persson theory to $10^7$),
and by extending the ratio of system size ${\cal L}$ and $\lambda_{\rm r}$
from 2 to 16.
\subsubsection*{Size-dependence of the critical regime}
In the critical regime, the pressure drops predominantly at a single constriction.
One might argue that large systems have a smaller critical regime, because significant
pressure drops can then occur at several constrictions.
We therefore analysed how the size of the critical regime depends on the system size.
For this, we changed the ratios $\epsilon_{\rm t} = {\cal L}/\lambda_{\rm r}$ and
$\epsilon_{\rm f} = \lambda_{\rm r}/\lambda_{\rm s}$.
Here, we may associate $\epsilon_{\rm t} \to 0$ with the thermodynamic limit
and $\epsilon_{\rm f} \to 0$ with the fractal limit.
In real applications, the true (mathematical) limits have no significance, which is why
we content ourselves with projections of our results to more realistic values of
$\epsilon_{\rm t} = \mathcal{O}(10^{-1})$ and $\epsilon_{\rm f} = \mathcal{O}(10^{-5})$.
Figure~\ref{fig:sizeDependence}A reveals that changing $\epsilon_{\rm t}$ does not
have a sufficiently strong systematic effect to dominate the fluctuations between
different random realisations, at least not when changing $\epsilon_{\rm t}$ by a
factor of four, i.e., for our three choices of $\epsilon_{\rm t}$ there is not even
a monotonic trend.
We note that we carried out a disorder average for $\epsilon_{\rm t} = 1/2$ over
16 different realisations but considered the large system $\epsilon_{\rm t} = 1/8$
sufficiently self-averaging.
Since $\epsilon_{\rm t}$ is never a very small number in practice, we conclude that
the critical leakage regime in real applications should not be much reduced in size
compared to the calculations presented here.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{fig02.eps}
\caption{\label{fig:sizeDependence}
Size-dependence of the critical regime. {\bf Panel A:} Fluid current $j$ as a
function of reduced load for different ratios of system size and rolloff wavelength
${\cal L}/\lambda_{\rm r}$. The current is shifted vertically to superimpose the data
close to the percolation threshold. Other dimensionless quotients are kept constant,
e.g., $\lambda_{\rm r}/\lambda_{\rm s} = 64$ and $\lambda_{\rm s}/a = 16$.
{\bf Panel B:} Fluid current $j$ in Persson theory as a function of the normalised
load $L/L_{\rm c}$ for different ratios of roll-off wavelength and short-wavelength
cutoffs, $\lambda_{\rm r}/\lambda_{\rm s}$. As the normalising factor for $j$, we
chose the current that one would obtain if the gap $g$ were set to $\lambda_{\rm s}$
everywhere.
}
\end{figure}
We also changed $\epsilon_{\rm f}$ and trends on the size of the critical regime were again
difficult to ascertain, due to large statistical scatter.
We therefore considered realistic values for $\epsilon_{\rm f}$ within Persson
theory, where we proceed for the calculation of the gap distribution function
as in Ref.~\citenum{Almqvist11},
and present our results on the mean flow for $\epsilon_{\rm f} = 10^{-2}$
and $10^{-5}$ as well as a rather small value (irrelevant for practical
applications) of $10^{-7}$,
see Fig.~\ref{fig:sizeDependence}B.
The results for the analysed values of $\epsilon_{\rm t}$
are very close at loads approaching the critical load.
While the critical region is slightly reduced for small values of
$\epsilon_{\rm f}$, it is clearly revealed that decreasing $\epsilon_{\rm f}$ below
$10^{-5}$ has only marginal effects at loads exceeding $L_{\rm c}/2$.
This could have been expected from the following argument:
decreasing $\epsilon_{\rm f}$ corresponds to adding roughness at long wavelengths.
Since the effective elastic compliance decreases with the inverse wavelength,
this extra roughness is immediately accommodated by the elastic seal.
We conclude that from a mathematical point of view the expected flow or conductivity has a
fractal limit, which is only (approximately) reached in practice for loads not
too small compared to the critical load.
\subsection*{Critical leakage for negative slip lengths}
Conductance exponents in percolation theory frequently turn out to be universal,
that is, they remain unaltered when details of a model change.
In view of this finding, we explore whether the conductance exponent
also remains unaltered with small alterations to the default leakage model.
One simple modification is to assume that the fluid flow velocity does not
extrapolate to zero precisely at the walls (so-called stick condition) but
already a distance $d_0$ before the wall.
In fluid dynamics, $d_0$ is called a (negative) slip length.
In a recent work\cite{SchnyderEtAl2015}, softening the fluid-obstacle repulsion,
in effect using a positive sliplength, suppressed the expected,
universal critical behaviour for 2D particle transport through porous media, albeit
for non-correlated obstacles.
In the present context, one could argue that a negative slip length accounts
for the finite size of particles to lowest order:
the fluid particles can only penetrate gaps with a height greater than $2\,d_0$.
The local fluid conductivity now scales with $\{g(x,y)-2d_0\}^3$ rather than
with $g^3(x,y)$, where $g(x,y)$ is the gap as deduced from the contact-mechanics
calculation at an in-plane coordinate $(x,y)$.
To analyse the effect of finite, negative slip lengths, we solve the
Reynolds equation for the same gap topography as before, but using the
just-described conductivity.
We choose $d_0$ to be a small fraction of the root-mean-square height $\bar{h}$ of the
rough substrate. We varied this fraction by a factor of 1000 without a qualitative
change of the observations.
Figure~\ref{fig:slipLength} reveals that the flow is not affected far from
the percolation point.
However, the conductance exponent now appears to be $\beta = 3$.
This value can be readily explained:
in the present model, the constriction (which is located around the point
where the substrate height has a saddle point) is not yet fully closed
when it appears closed for the fluid.
Thus, the point is only critical for the flow but not for the contact
mechanics.
This means that near the critical load $L_{\rm c}$, the true height of the
gap at the saddle point, $g_0(L)$, the true length of the constriction $l_0(L)$,
and the true width of the constriction $w_0(L)$ are all ``simple functions'' of the
load, which each can be expanded into a Taylor series according to
$f(L) = f(L_{\rm c}) + (L-L_{\rm c}) f'(L_{\rm c})$.
The same quantities, as perceived by the fluid, e.g., the effective local height,
or the effective width of a constriction,
have similar functional dependencies as the true height, however, different offsets.
In fact, all offsets $f(L_{\rm c})$ for the effective quantities can be set to zero,
since height, width, and length of the constriction --- ``as seen by the fluid'' ---
are all zero at the critical point.
Since the resistivity
of the constriction scales as the inverse
third power of the effective gap, linear with the length of the constriction
(as in a serial coupling of resistors) and with the inverse width of the
constriction (as in a parallel coupling of resistors), the fluid resistance
of the constriction follows
\begin{equation}
R(L) \propto \frac{l_0(L)}{w_0(L) g_0^3(L)},
\label{eq:consRes}
\end{equation}
where the proportionality factor is linear in the viscosity of the
fluid and also depends on the geometry of the constriction.
Inserting our Taylor series approximations for effective height, width, and length of
the constriction into Eq.~(\ref{eq:consRes}) then reveals that $R(L) \propto
(1-L/L_{\rm c})^{-3}$ implying $\beta = 3$.
As discussed in a previous paper\cite{Dapp15EPL}, the case of zero slip length is
more complicated, because $g_0(L)$, $l_0(L)$, and $w_0(L)$ all approach
zero as non-integer powerlaws of the reduced load.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{fig03.eps}
\caption{\label{fig:slipLength}
Leakage current $j$ as a function of the reduced load $1-L/L_{\rm c}$ for a system
with a negative slip length for the fluid flow.
The same random realisation is studied as in Fig.~\ref{fig:defaultSystem}A--D,
where zero slip-length flow is considered. The data is smoothed to remove scatter
and shifted vertically to yield a current of one at an infinitesimally small
load.
}
\end{figure}
\subsection*{Critical flow through adhesive interfaces}
We now turn our attention to surfaces that attract each other via adhesive forces.
The fluid flow has stick boundary conditions again.
In a previous work\cite{Dapp15EPL}, we found, for the case of isolated
constrictions and short-range adhesion, that constrictions closed discontinuously.
Long-range adhesion was not considered explicitly as it reduces to a simple adhesive
offset force for the investigated single-wavelength isolated constriction (see the
method section).
The range of adhesion is quantified by a dimensionless number called the
Tabor coefficient $\mu_{\rm T}$\cite{Tabor77}.
Its use is best known in the context of single-asperity contacts, but the concept
extends to randomly-rough, self-affine surfaces\cite{Persson14JCP}.
Except for prefactors, which can be chosen at will\cite{Muser14Beil}, it is defined
as~$\mu_{\rm T} = R_{\rm c}^{1/3} (\vert \gamma_0 \vert/E^*)^{2/3}/z_0$.
Here, $z_0$ is a characteristic length scale of the interaction,
$E^*$ is the effective elastic contact modulus, and
$\gamma_0$ is the surface energy.
$R_{\rm c}$ is the radius of curvature for a Hertzian contact geometry,
or a measure for the inverse surface curvature:
$1/R_{\rm c}^2 \equiv
\sum_{\vec{q}} (q_x^4+q_y^4) C(\vec{q})/2$
where $C(\vec{q})$ is the (discrete) wavevector-dependent height spectrum
defined in the method section.
Figure~\ref{fig:adhesion} shows that the critical leakage current sensitively
depends on the adhesive range.
Long-range adhesion yields a similar dependence of the current on the
reduced load as the non-adhesive case.
However, at a Tabor coefficient around $\mu_{\rm T}=1$, the leakage-load dependence
starts to show a different powerlaw near the sealing transition.
Specifically, for $\mu_{\rm T} \gtrsim 1$,
load regimes with an apparent conductance exponent of $\beta\approx 1$ occur.
We thus have the second example for a change of conductance exponent of macroscopic,
or at least mesoscopic, response functions due to small changes in the model.
Below, we demonstrate that the observed crossover is also present in an
isolated constriction and thus not due to the multi-scale topology of the
percolating channel.
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\linewidth]{fig04.eps}
\caption{\label{fig:adhesion}
A: Leakage current $j$ as a function of the reduced load $1-L/L_{\rm c}$
for adhesive contacts differing in their Hurst roughness exponent $H$ and
Tabor coefficient $\mu_{\rm T}$.
Data is shifted vertically to superimpose in the critical regions.
B and E show the true contact near the percolation threshold for one random
realisation with $H = 0.8$, in the case of short-range ($\mu_{\rm T} = 2$) and
long-range ($\mu_{\rm T} = 1/4$) adhesion, respectively.
C and F are the corresponding flow patterns at loads indicated in panel~A.
D and G are high-resolution zooms into the critical constriction.
}
\end{figure*}
An interesting aspect of Fig.~\ref{fig:adhesion} is that the coarse features
of the contact area look almost identical near the sealing transition, even
in the two extremes of no adhesion and short-range adhesion.
However, the contact lines look much smoother and less fractal for short-range
adhesion (E) than for no or long-range adhesion (B).
This difference in local contact features ultimately accounts for the different
behaviour near the percolation threshold.
\subsubsection*{Flow through isolated, adhesive constrictions}
We now address the question of whether the cross-over of exponents presented in
Fig.~\ref{fig:adhesion} is due to the multi-scale roughness of the surfaces or
originates from the properties of an isolated constriction.
Towards this end, we revisit the contact mechanics of single-wavelength roughness,
in particular that of a square saddle point (for details see Ref.~\citenum{Dapp15EPL}).
However, here we do not only consider short-range adhesion as in our precedent study on
isolated constrictions, but also allow for medium- or long-range adhesion.
Figure~{\ref{fig:isolated}} shows that for $\mu_{\rm T} \lesssim 1$ the scaling of the
current on the load changes near the
percolation point, from the $\beta=3.45$ behaviour also seen in the non-adhesive case,
towards a scaling with $\beta=1$.
For $\mu_{\rm T} = 1$, the latter regime is rather narrow and the leakage quickly
becomes similar to that of non-adhesive surfaces as the sealing transition is
approached.
In a narrow range of $\mu_{\rm T} \gtrsim 1$, the ``new'' scaling is valid
over more than one decade. For $\mu_{\rm T} \gtrsim 2$, there seems
to be a discontinuous drop of finite to zero conductance of the critical
junction. The critical constriction snaps shut before
scaling
can be observed. With decreasing range of the adhesive potential,
this point of adhesive instability is moved to smaller loads, and away from the
critical load.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{fig05.eps}
\caption{\label{fig:isolated}
Leakage current $j$ as a function of the reduced load $1-L/L_{\rm c}$ for an
isolated, single-wavelength constriction, for different Tabor parameters.
}
\end{figure}
\section*{Conclusions}
From the three leakage models analysed in this study, it has become clear that
seal systems are unlikely to belong to a (unique) universality class
of percolation theory, even if one could consider flow through a seal as a paradigm
percolation problem:
when adding small alterations to our default model, in which common
approximations of lubrication theory are made, we observe qualitative changes
in the transition between finite and zero conductance.
In fact, it appears as though the default model represents a multi-critical point,
because the conductance exponent changes when an arbitrarily-small negative slip
length is introduced and/or the transition changes from continuous to discontinuous
when short-range adhesion between the surfaces is introduced.
In practice, many additional complications can and in general will affect the leakage
problem, most notably the formation of fluid capillaries, clogging by
contaminating particles, as well as viscoelastic deformation and ageing of the
sealing material.
These complications are likely to correspond to relevant perturbations affecting
the nature of the percolation transition as well.
While it is certainly possible to predict leakage over a broad pressure range from
statistical properties of the surfaces alone, it appears impossible to do so accurately
when the load exceeds 80\% or 90\% of the critical load, above which no open channel
percolates from one side of the interface to the other side.
At such high loads, most of the pressure pushing the fluid through the interface
drops at a single constriction.
The (local) properties of this constriction then determine the behaviour of the
whole system.
This is one reason why it is impossible to predict with great accuracy how strongly
we have to tighten a water tap to make it stop dripping and also how it starts dripping
once we loosen it again.
\section*{Methods}
\subsection*{GFMD}
The contact mechanics treatment and its description is in large parts identical
to that presented in our study on isolated constrictions\cite{Dapp15EPL}:
We assume linear elasticity and the small-slope approximation, so that the roughness
can be mapped to a rigid substrate and the elastic compliance to a flat counter
body without loss of generality.
The effective contact modulus is used to define the unit of pressure,
i.e., $E^* = 1$.
Elasticity is treated with Green's function molecular dynamics
(GFMD)\cite{Campana06} and the continuum version of the stress-displacement
relation in Fourier space,
$\tilde{\sigma}(\vec{q}) = qE^* \tilde{u}(\vec{q})/2$,
where $\vec{q}$ is a wave vector and $q$ its magnitude.
Simulations are run in a force-controlled fashion.
After the external load has changed by a small amount, all degrees of
freedom are relaxed until convergence is attained.
Two surfaces interact with a hard-wall constraint, i.e., they are not allowed
to overlap.
In addition, we assume a finite-range surface energy
$\gamma = -\gamma_0 \exp\{-g/z_0\}$,
where $g=g(x,y)$ is the local gap or interfacial separation.
When adhesion is switched on, we chose $\gamma_0$ such that the
contact area roughly doubles at a given load compared to the
adhesion-free case.
The linear size of the solids is denoted by ${\cal L}$.
Periodic boundary conditions are employed within the $xy$-plane so that
the local height in real space can be written as a Fourier sum
$h(\vec{r}) = \sum_{\vec{q}} \tilde{h}(\vec{q})
\exp(i\vec{q}\cdot\vec{r})$.
Most technical and natural surfaces are self-affine rough, e.g.,
ground steel, asphalt, human skin, and sandblasted plexiglass show
self-similar height spectra over a broad range with Hurst roughness exponent
of $H \approx 0.8$~\cite{Persson14TL}. The exponent states that
the root-mean-square deviation of the height increases as $\Delta h \propto
\Delta r^H$, where $\Delta r$ is the in-plane distance from a given point on
the surface. An ideal random walk corresponds to $H=0.5$. A larger value of
$H$ indicates that height spectra increase in magnitude at long wavelengths
relative to short wavelengths. Sometimes $3-H$ is called the fractal
dimension of the surface.
Pertaining to the single-wavelength $\lambda$ constrictions discussed in
Ref.~\citenum{Dapp15EPL} and in the appendix, we note that their equilibrium
height-profile can be generated, for example, from
$h(x,y) \equiv 2 + \cos(2\pi x/\lambda) + \cos(2\pi y/\lambda)$.
The roughness spectra $C({\bf q}) = \langle \vert \tilde{h}(\vec{q}) \vert^2 \rangle$
is constant for wave vectors of magnitude
$2\pi/{\cal L} \le q < 2\pi/\lambda_{\rm r}$, where
$\lambda_{\rm r}$ is called the rolloff wavelength.
For wave vectors of magnitude
$2\pi/\lambda_{\rm r} \le q < 2\pi/\lambda_{\rm s}$,
the spectra are power laws according to $C({\bf q}) \propto q^{-2(1+H)}$.
A typical setup can be characterised by the following dimensionless numbers:
$H = 0.8$,
${\cal L}/\lambda_{\rm r} = 2$,
$\lambda_{\rm r}/\lambda_{\rm s} = 64$, and
$\lambda_{\rm s}/a = 64$.
When approaching the percolation threshold or when treating short-range
adhesion, we further increase the ratio
of $\lambda_{\rm s}/a$ while keeping the random realisation of the surface
profile the same.
To determine fluctuations of the fluid conductance at small loads, we
decrease $\lambda_{\rm s}/a$ but take much larger ratios for
${\cal L}/\lambda_{\rm r}$ and $\lambda_{\rm r}/\lambda_{\rm s}$.
The largest GFMD calculation presented in this work consisted of
$2^{15}\times 2^{15} \approx 1\times 10^9$ discretisation points.
\subsection*{Reynolds solver}
For the fluid-mechanics-related calculations, we assume a reservoir of liquid
on the left side of the system ($x=0$), while the other, right side is a sink
for said liquid, with a liquid pressure of zero.
In the transverse direction, the system is treated as periodic, in order to
minimise finite-size effects.
The local fluid conductivity in the Reynolds equation scales with the
third power of the local gap as seen by the fluid.
Like the contact-mechanics aspects, all solution strategies including their
descriptions are in large parts identical to those presented in our study
on isolated constrictions\cite{Dapp15EPL}:
We use the \verb+hypre+ package\cite{FalgoutEtAl2006} to solve the sparse linear system that
the discretised Reynolds equation can be expressed as.
We employ the solvers supplied with \verb+hypre+ using the CG (conjugate
gradient), or GMRES (generalised minimal residual) methods\cite{SaadSchultz1986},
each preconditioned using the PFMG method, which is a parallel
semicoarsening multigrid solver\cite{AshbyFalgout1996}.
Our in-house code is MPI-parallelised and uses HDF5 for I/O.
The fluid pressure and its gradients are assumed small enough to not deform
the walls.
We verified
that our results for the conductance exponent do not depend on this
approximation, by including the coupling of the fluid pressure (up to 30\% of the
external mechanical pressure near the percolation threshold) to the wall for an
isolated (adhesionless, zero-slip length) constriction.
We merely observed an increase in
the percolation load, a shift in the location of the critical constriction,
and a reduction of symmetry of the gap and contact line profiles.
Coupling to GFMD is done through an iterative perturbation treatment, in which
the Reynolds output is fed back into the contact mechanics calculation.
\section*{Acknowledgements}
We gratefully acknowledge computing time on JUROPA and JUQUEEN at the
J\"ulich Supercomputing Centre as well as valuable discussions with Bo Persson.
\section*{Additional information}
\textbf{Competing financial interests:} The authors declare no
competing financial interests.
\section*{Author contributions}
The authors contributed to the current article as follows. M.H.M. developed the concept of the study.
W.B.D. and M.H.M. designed the contact mechanics code.
W.B.D. wrote the Reynolds solver, ran the simulations, and performed the
analysis.
W.B.D. and M.H.M. wrote the manuscript.
|
2,877,628,090,217 | arxiv |
\section{Compositionality}\label{sec:background}
\subsection{Compositional Generalization}
\ld{a rough first pass; will condense and make more precise once the story is more coherent! see also \href{https://docs.google.com/document/d/1uuIPmM_bShv7eti9J4V_Qlo64a6y0wryHR5QLbdBfmc/edit}{this}}
Basically, compositional generalization is the ability to generalize to new linguistic structures using compositional principles. For humans, this is understood as the algebraic capability to understand and produce a potentially infinite number of novel linguistic expressions by dynamically recombining known elements \citep{chomsky-1957-syntactic,fodor-pylyshyn-1988-connectionism,fodor-lepore-2002-compositionality}. For language models, the task is a semantic parsing task: a language model is expected to construct a semantic representation of a given English sentence. Importantly, the training and evaluation sets for such a task systematically differ such that success on the evaluation set requires out-of-distribution generalization \citep{kim-linzen-2020-cogs}.
It has been argued that the lack of compositionality is one reason why modern neural networks require huge amounts of data to induce correct generalizations, in stark contrast to humans \citep{lake-baroni-2018-generalization}. Robust compositionality requires a theory of syntax, a theory of what meanings are, and a theory of the functions that allow meanings to combine. For example, human grammar accommodates the semantic difference between arguments (\textit{\textbf{The girl} laughs}), restrictive modifiers (\textit{The professor \textbf{of physics} laughs}), and non-restrictive modifiers (\textit{The professor \textbf{under the umbrella} laughs}) with distinct syntax. This reflects the understanding that, in a compositional grammar, both the syntax and semantics are algebras, and there is a homomorphism that maps elements of the syntactic algebra onto elements of the semantic algebra \cite{partee2008compositionality,liu-etal-2021-learning}.
Much current work on compositional generalization follows the distinction of \citet{kim-linzen-2020-cogs} between \textit{lexical generalization}, a novel combination of a familiar primitive and a familiar structure, and \textit{structural generalization}, a novel combination of two familiar structures such that the sentence structure itself is novel. For example, while simple object to subject NP substitution is considered lexical, CP or PP recursion is considered structural. Though this is a helpful distinction for parsing and evaluation purposes, it glosses over the nuances of compositionality such as PP attachment preference and ambiguity, and word selectional preference based on features like animacy (limitations \citeauthor{kim-linzen-2020-cogs} note themselves). For example, in a sentence such as \textit{The hedgehog loves the girl next to the bear with its whole heart}, the PP attachments are unclear: while the first PP has a slight reading preference for modifying \textit{the girl}, the second PP had a preferred reading for modifying the verb. The most salient reading for a human speaker will depend both on lexical features and context, elements that current tasks dedicated to compositional generalization lack. \ld{K\&L also note the need for constraints on generalization}
\subsection{COGS dataset \citep{kim-linzen-2020-cogs}}\label{subsec:cogsdataset}
\citet{kim-linzen-2020-cogs} introduced the COGS\footnote{
\url{https://github.com/najoungkim/COGS}.
}
dataset, a benchmark for compositional generalization, with (sentence, logical form) pairs, e.g.\footnote{
examples slightly modified for ease of presentation.
}:
\begin{exe}
\ex\label{ex:boywanted} The boy wanted to go. \\
\lform{* boy($x_1$) ; want.agent($x_2, x_1$) $\land$ want.xcomp($x_2, x_4$) \\ $\land$ go.agent($x_4, x_1$)}
\ex\label{ex:other} Ava was lended a cookie in a bottle. \\
\lform{lend.recipient($x_2,$ Ava) \\ $\land$ lend.theme($x_2, x_4$) \\ $\land$ cookie($x_4$) \\ $\land$ cookie.nmod.in($x_4, x_7$) \\ $\land$ bottle($x_7$)}
\ex\label{ex:cprec} Ava said that Ben declared that Claire slept. \\
\lform{say.agent($x_1,$ Ava) \\ $\land$ say.ccomp($x_1, x_4$) \\ $\land$ declare.agent($x_4,$ Ben) \\ $\land$ declare.ccomp($x_4, x_7$) \\ $\land$ sleep.agent($x_7,$ Claire)}
\ex\label{ex:prim_touch} touch \\
\lform{$\lambda a. \lambda b. \lambda e.$ touch.agent($e, b$) $\land$ touch.theme($e, a$)}
\end{exe}
The English declarative sentences on the input side
are mapped to a logical form
which are post-processed forms of \citet{reddy-etal-2017-universal}. Note that they are not grounded in any database.
Except for \sortof{primitives} (e.g.~\cref{ex:prim_touch}) which will be discussed below, logical forms consist of a conjunction of terms and a possibly empty list of terms prefixed to this conjunction using iota operators ($\iota$ written as $*$).
A term consist of a predicate name (e.g.~\lform{boy}, \lform{want.agent} or \lform{cookie.nmod.in})
and one or two arguments which are variables (e.g. $x_1, x_2, \dots$ or $a,b,e$ for primitives respectively) or proper names like \lform{Ava}.
The subscript of a $x_i$ variable reveals the alignment to the $i$th input token (0-indexed).
The iota terms are only used for definite NPs like \phrase{the boy} (cf.~\cref{ex:boywanted}).\footnote{
compare to the indefinite NP \phrase{a cookie} in \cref{ex:other}
which is part of the normal conjunction and not the iota prefix.
}
Note that the terms are sorted based on the input token position of their arguments.
This is especially relevant for exact match evaluation.\\
There are also \emph{primitives} in the training data which represent the \sortof{meaning} of one word in isolation (cf. \cref{ex:prim_touch}).
For more details we refer the reader to their paper.
On the generalization set the models are required to make 21 types of generalizations at once.\footnote{
unlike SCAN of \citet{lake-baroni-2018-generalization}
with separate sets for each generalization type
}
This includes
generalizing from a primitive (e.g. \word{shark}) to its usage in context (e.g. \word{a chief heard the shark}), usage of nouns in subject position that were only observed as objects in training (and vice versa),
generalizing to higher CP/PP recursion depths, or generalizing to another verb argument structure (e.g. active vs passive).
For a full list we refer to their paper.
\citet[§5.2.1]{kim-linzen-2020-cogs} distinguish between lexical and more structural generalizations\footnote{
\todo{is structural generalization synonymous to \citet{liu-etal-2021-learning}'s \emph{algebraic recombination}?}
}.
Most of the 21 generalization types in COGS address lexical generalizations:
interpreting a word in structure only seen with other words, e.g. using a noun in a grammatical role it hasn't been seen with. The 3 structural generalization types in COGS are not tied to specific words, but require generalization to a novel combination of familiar structures, such as deeper recursion (right-branching PP or CP) or observing a PP modifier at a new grammatical role (object PP to subject PP).
Since the lexical generalization types are overrepresented in COGS, models targeting only or mostly these will look much better at \emph{compositionality} than they actually are:
\ignore{The large pretrained seq2seq model BART achieved an exact match score of 68 to 71\todo{check numbers}\pw{this will be presented in later sections!}, and}
\citet[Table 2]{akyurek-andreas-2021-lexicon} report that even an LSTM with a copy function can achieve 66\% exact match accuracy and \citet{akyurek-andreas-2021-lexicon}'s more elaborate approaches all focusing on learning a good lexicon worked very well on the lexical cases but failed at structural generalization.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.42\textwidth}
\centering
\hbox{\tiny
\input{figs/tree_to_subj_pp}
}
\caption{object PP to subject PP}\label{fig:structural-generalization:toSubjPP}
\end{subfigure}
\begin{subfigure}[b]{0.56\textwidth}
\centering
\hbox{\tiny
\input{figs/tree_pp_recursion}
}
\caption{PP recursion}\label{fig:structural-generalization:PPrecursion}
\end{subfigure}
\caption{Structural generalization in COGS. \todo{get spacing right}}
\label{fig:structural-generalization}
\end{figure*}
\subsection{On actual compositionality}\label{sec:actual_compos}
\subsubsection{Error analysis}\label{subsec:erroranalysis}
\todo{do we have time/space for error analysis?}
Similar to what is reported for the seq2seq models in \citet{kim-linzen-2020-cogs}, if it doesn't succeed the AM parser also seems to be very close to the solution in some cases (note that exact match is a very strict metric that doesn't reward partial successes) but fails exactly at the output position where the generalization should take effect.\todo{example needed}. \todo{same for BART?}
\subsection{Citations}
\input{secs/conclusion}
\section{Parsing COGS with the AM parser}\label{sec:amparsing4cogs}
\subsection{The AM parser}
To better understand how compositional models perform on compositional generalization, we adapt the broad-coverage AM parser to COGS.
The AM parser \cite{groschwitz-etal-2018-amr} is a compositional
semantic parser that learns to map sentences to graphs. It was the
first semantic parser to perform with high accuracy across all major
graphbanks \citep{lindemann-etal-2019-compositional} and can achieve
very high parsing speeds \citep{lindemann-etal-2020-fast}. Thus, though not yet tested on synthetic generalization sets, the AM parser exhibits the ability to handle natural language and related generalizations in the wild.
Instead of predicting the graph directly, the AM parser first predicts
a graph fragment for each token in the sentence and a (semantic)
dependency tree that connects them. This is illustrated in
\cref{fig:am}a; words that do not contribute to the sentence
meaning are tagged with $\bot$. This dependency tree is then
evaluated deterministically into a graph (\cref{fig:am}b) using
the operations of the \emph{AM algebra}. The ``Apply'' operation fills
an argument slot of a graph (drawn in red) by inserting the root node (drawn with a bold outline)
of another graph into this slot; for instance, this is how the \app{s}
operation inserts the ``boy'' node into the ARG0 of ``want''. The
``Modify'' operation attaches a modifier to a node; this is how the
\modify{m} operation attaches the ``manner-sound'' graph to the ``sleep''
node. The dependency tree captures how the meaning of the
sentence can be compositionally obtained from the meanings of the
words.
AM parsing is done by combining a neural dependency
parser with a neural tagger for predicting the graph fragments. We follow \citet{lindemann-etal-2019-compositional} and rely on the
dependency parsing model of \citet{kiperwasser-goldberg-2016-simple},
which scores each dependency edge by feeding neural representations for the
two tokens to an MLP.
We follow the setup of \citet{groschwitz-etal-2021-learning}, which
does not require explicit annotations with AM dependency trees, to
train the parser.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figs/am}
\caption{(a) AM dependency tree with (b) its value.}
\label{fig:am}
\end{figure}
\subsection{AM parsing for COGS}\label{subsec:cogs2graph}
We apply the AM parser to COGS by converting the semantic
representations in COGS to graphs. The conversion is illustrated in
\cref{fig:cogsgraph}.
\begin{figure}[t]
\centering
\lform{\scriptsize
*table($x_9$); see.agent($x_1,$ Ava) $\land$ see.theme($x_1, x_3$) $\land$ ball($x_3$) $\land$ ball.nmod.in($x_3, x_6$) $\land$ bowl($x_6$) $\land$ bowl.nmod.on($x_6,x_9$)
}\\[8pt]
\includegraphics[width=0.9\columnwidth]{./figs/cogs-conversion-example-ak}
\caption{
Logical form to graph conversion for \sentence{Ava saw a ball in a bowl on the table} (cf.~\cref{tab:cogssamples}c).
}\label{fig:cogsgraph}
\end{figure}
Given a logical form of COGS, we create a graph that has one node for
each variable $x_i$ and each constant (e.g. \lform{Ava}). If a
variable appears as the first argument of an atom of the
form $\lform{pred}.\lform{arg}(x,y)$, we assign it the node label
$\lform{pred}$ in the graph. We also add an edge from $x$ to $y$ with
label $\lform{arg}$.
E.g.~\lform{see.agent($x_1,$ Ava)} turns into an \graphedge{agent} edge from \graphnode{see} to \graphnode{Ava}.
Each \emph{iota term}
\lform{*noun($x_{\mathrm{noun}}$)} is treated as an edge from a fresh
node with label ``the'' to $x_{\text{noun}}$. Preposition meaning
\lform{bowl.nmod.on($x_6,x_9$)}
is
represented as a node (labeled \graphnode{on}) with outgoing edges to the
two arguments/nouns (\graphedge{nmod.op1} to \word{bowl},
\graphedge{nmod.op2} to \word{table}).
By encoding the logical form as a graph, we lose the ordering of the
conjuncts. The \sortof{correct} order is restored in postprocessing.
More details and graph conversion examples are in
\cref{sec:details_cogs2graph}.
\section{Compositional Generalization}\label{sec:background}
Compositional generalization is the ability to determine the meaning
of unseen sentences using compositional principles. Humans can
understand and produce a potentially infinite number of novel
linguistic expressions by dynamically recombining known elements
\citep{chomsky-1957-syntactic,fodor-pylyshyn-1988-connectionism,fodor-lepore-2002-compositionality}.
For semantic parsers, compositional generalization tasks
systematically vary language use between the training and the
generalization set; as such, the system must recombine parts
of multiple training instances to predict the meaning of a single test
instance.
COGS \cite{kim-linzen-2020-cogs} is a synthetic semantic parsing
dataset in which English sentences must be mapped to logic-based
meaning representations. It distinguishes 21 \emph{generalization
types}, each of which requires generalizing from training instances
to test instances in a particular systematic and
linguistically-informed way. We follow the authors and distinguish two
classes of generalization types; we further comment on a third class
based on data from model performance.
\input{tables/cogs_examples}
\emph{Lexical generalization} involves recombining known grammatical
structures with words that were not observed in these particular
structures in training. An example is the generalization type
``subject to object (common)'' (\cref{tab:cogssamples}a), in which a
common noun (``hedgehog'') is only seen as a subject in training,
whereas it is only used as on object in the generalization testset.
Note that the syntactic structure at generalization time (e.g.\ that
of a transitive sentence) was already observed in training. On the
semantics side, the meaning representations are identical, except
for replacing some constants and quantifiers and renaming some
variables.
Thus, lexical generalization in COGS amounts to
learning how to fill fixed templates.
\input{figs/structual_gen_tree_figure}
By contrast, \emph{structural generalization} involves generalizing to
linguistic structures that were not seen in training (cf.\
\cref{tab:cogssamples}c,d). Examples are the generalization types ``PP
recursion'', where training instances contain prepositional phrases of
depth up to two and generalization instances have PPs of depth 3--12;
and ``object PP to subject PP'', where PPs modify only objects in
training and only subjects at test time. These structural changes are
illustrated in
\cref{fig:structural-generalization}.
A third class we observe involves
generalizing to \emph{object usage of proper nouns} (\cref{tab:cogssamples}b).
Though technically a subset of lexical generalization, this subgroup is harder than types of the same class (cf.~\cref{subsec:first:experiments}); we report and discuss it separately.
Lexical generalization captures a very limited fragment of
compositionality, in that it only requires to fill a fixed number of
slots with new values. The key point about compositionality in
semantics is that language is infinitely productive, and humans can
assign meaning to new grammatical structures based on finite
experience. Assigning meaning to unseen structures is exercised only
by structural types. This distinction is borne out in model
performance (\cref{subsec:first:experiments}): while lexical
generalization can be handled by many neural architectures,
structural generalization requires parsing architectures aware of
complex sentence structure.
\section{Discussion}\label{sec:discussion}
\paragraph{Compositional generalization requires compositional parsers.}
\Cref{tab:selected_gentype_eval} paints a clear picture: compositional
generalization in COGS can be solved by semantic parsers that have
compositionality built in, but seq2seq models perform poorly on
structural generalization. This remains true even for seq2seq models
that are known to perform well on semantic parsing, for syntactic
rather than semantic generalization, and for seq2seq models that are
biased towards learning structure-aware representations by
incorporating information about syntax. Obviously, statements about
entire classes of models must be made with care. But when despite the
best efforts of an active research community \emph{all} seq2seq models
underperform the compositional models, that seems like rather strong
evidence.
Our results are surprising, in that seq2seq models have been shown
through probing tasks to learn some linguistic structure, both with
respect to syntax \cite{blevins-etal-2018-deep} and semantics
\cite{tenney-etal-2019-bert}. At the same time, as mentioned above,
seq2seq models like BART perform very well on broad-coverage tasks
such as AMR parsing. It is an interesting question for future research
to reconcile the ability of seq2seq models to learn soft structural
information with their apparent difficulties in exploiting this
ability to generalize structurally; perhaps their ability to learn
structure rests on the variety of structures observed in
broad-coverage training sets, but not in COGS.
\paragraph{Focus on structural generalization.}
Our experiments indicate that \genclass{Struct} is consistently harder than
\genclass{Prop} and \genclass{Lex} with respect to generalization accuracy. Not only
is \genclass{Lex}\ essentially a solved problem; but as we discussed in
Section~\ref{sec:background}, the infinitely productive nature of full
compositionality is only captured by structural types of
generalization. Compositionality is not just about using new and
similar words in known structures (slot filling), but also about
building new, acceptable structures based on known ones.
When papers only report the mean accuracy of a system across all
generalization types, the accuracy on the 16 lexical generalization
types overshadows the accuracy on the three structural generalization
types. The overall accuracy can make systems look more capable of
compositional generalization than they really are.
Future work on compositional generalization will benefit from (i)
reporting the accuracy on structural generalization tasks separately
and (ii) expanding datasets that test compositional generalization to
include more types of structural generalization. We hope to offer such
a dataset in future research.
\paragraph{What's so difficult about objPP to subjPP?}
``ObjPP to subjPP'' is the most challenging generalization type across
all models. It is illuminating to investigate the errors that happen
here, as they differ across models.
Table~\ref{tab:erranalysis} shows typical errors of BART and the AM
parser. The AM parser chooses to use the most recent simple NP (``the
house'') as the agent of ``scream'' and then attaches ``the baby on a
tray'' in some random place. By contrast, BART analyzes the sentence
as ``the baby screamed the tray on the house'', preferring to reuse the pattern for object-PP sentences even if the
intransitive verb does not license it. BART also displays an unawareness
of word order that is reminiscent of the difficulties that seq2seq
models otherwise face in relating syntax to word order
\citep{DBLP:journals/tacl/McCoyFL20}.
We see from both examples that ``objPP to\linebreak subjPP'' involves major
structural changes to the formula that must be grounded in both
lexical (verb valency) and structural (word order)
information. Developing a model that learns to do this with perfect
accuracy remains an interesting challenge.
\begin{table}[t]
\centering
\footnotesize
\begin{tabularx}{\linewidth}{lX} \toprule
Gold & \lform{*baby($x_1$); *house($x_7$); baby.nmod.on($x_1, x_4$) $\land$ tray($x_4$) $\land$ tray.nmod.in($x_4, x_7$) $\land$ scream.agent($x_8, x_1$)} \\
BART & \lform{*baby($x_1$); *house($x_7$); scream.agent($\textcolor{red}{x_2}, x_1$) $\land$ scream.\textcolor{red}{theme}($\textcolor{red}{x_2}, x_4$) $\land$ tray($x_4$) $\land$ tray.nmod.in($x_4, x_7$)} \\
AM & \lform{*baby($x_1$); *house($x_7$); baby.nmod.on($x_1, x_4$) $\land$ tray($x_4$) $\land$ tray.nmod.in($x_4, \textcolor{red}{x_8}$) $\land$ scream.agent($x_8 , \textcolor{red}{x_7}$)} \\ \bottomrule
\end{tabularx}
\caption{Error analysis for the sentence \sentence{The baby on a tray in the house screamed}.}\label{tab:erranalysis}
\end{table}
\section{Sequence-to-sequence model on COGS} \label{sec:seq2seq}
\subsection{Syntactic generalization}\label{subsec:seq2seq:syntactic_gen}
We first explore hypothesis \ref{it:syngen} by training BART as a syntactic parser on COGS.
The syntactic annotations are obtained from original PCFG grammar used to generate COGS \cite{kim-linzen-2020-cogs} with NLTK.
Similar to the semantic case, BART takes as input plain text but now outputs linearized constituent parse trees rather than semantic meaning representations.
To make comparison with a structured syntactic parser, we also experimented with a structured constituency parser of \citet{kitaev-klein-2018-constituency}, which consists of a self-attention encoder and a chart decoder.
Results are in Table \ref{tab:syntax generalization}.
\begin{table}[tbh]
\centering
\begin{tabular}{llrrr}
\toprule
& & struct & non-struct & All\\ \midrule
\multirow{2}*{label} & BART & 1.70\% & 86.99\% & 74.81\%\\
~ & benepar & 99.03\% & 85.87\% & 87.75\% \\
unlabel & BART & 5.60\% & 94.06\% & 77.14\% \\
\bottomrule
\end{tabular}
\caption{Exact match accuracy results for syntactic parsing on generalization set of COGS.
\textit{benepar} denotes the parser from \citet{kitaev-klein-2018-constituency}. \textit{label}/\textit{unlabel} denotes we train the model to output labeled/unlabeled linearized constituent tree.}
\label{tab:syntax generalization}
\end{table}
From Table \ref{tab:syntax generalization}, we can observe the difference between structured and seq2seq parser mainly lies in structural generalization types. BART completely fails on structural generalization types (5.6\%).
Remember that in the unlabeled case, the model only need to output original bracket-annotated source tokens (e.g. \lform{A rose} $\to$ \lform{((A)(rose))}).
This suggests that BART cannot even recognize correct syntactic structures in structural generalization types, and thus it is not surprising it cannot solve same types in the semantic level.
On the other hand, a structured syntactic parser (e.g. benepar) performs well on such generalization types (99.03\%), which explains why structured semantic parsers generally perform well on structural generalization types.
\todo{add analysis for non-struct types?}
\todo{more discussion on structure vs seq2seq?}
\todo{add attention visualization if we have space?}
\subsection{Compositional generalization from correct syntax}\label{subsec:seq2seq:semantic_gen_with_syntax}
Given that BART cannot learn syntactic generalization, it is natural to ask whether sequence-to-sequence model can do compositional generalization given the correct syntactic structure (cf. hypothesis \ref{it:semgen_with_syn}).
In this paper we test this with three popular methods described below to incorporate syntax information into our model. For all methods we use labeled constituent parse tree as syntactic annotations.
\paragraph{Syntax-enhanced Input} We feed parsed text into the encoder instead of plain text (e.g. \lform{A rose} $\to$ \lform{(NP ({Det} A) ({N} rose))}).
This method is similar to \cite{li-etal-2017-modeling, currey-heafield-2019-incorporating}, but here we directly make the decoder attend to all source tokens. Here syntactic tokens are used to augment feature representations from encoder.
\paragraph{Multi-task Learning} We also experiment with multi-task learning setting by inserting a special token in the end of each source sentence (e.g. \lform{A rose} $\to$ \lform{A rose <syn> | A rose <sem>}).
The model is trained to generate targets with corresponding formalism (e.g. syntax parse tree or semantic meaning representation) given the special token.
This method is mostly used in natural language generation tasks \cite{sennrich-etal-2016-controlling, currey-heafield-2019-incorporating}. Here we use constituent parsing loss as a regularized loss for semantic parsing task.
\paragraph{Self-attention Mask} \citet{kim-etal-2021-improving} shows that providing sequential encoder with structure hints helps compositional generalization ability on classification task. Here we also test with their method and use syntax as a mask for self-attention layers in encoder.
Specifically, each token now is only allowed to attend to its head, children and itself in the parse tree during encoding.
As suggested by \citet{kim-etal-2021-improving}, this method does not \textit{add} any attention capability but remove non-structured attention edges.
\begin{table}[tbh]
\centering
\begin{tabular}{lrrr}
\toprule
& struct & non-struct & All\\ \midrule
BART & 4.20\% & 89.90\% & 77.61\% \\
+syntax input & 6.30\% & 91.47\% & 79.34\% \\
+MTL & 2.60\% & 90.58\% & 78.01\% \\
+AttnMask & 4.54\% & 90.05\% & 77.83\% \\
\bottomrule
\end{tabular}
\caption{
Exact match accuracy results for semantic parsing on generalization set of COGS.
}\label{tab:semantic generalization with syntax}
\end{table}
Results are shown in Table \ref{tab:semantic generalization with syntax}.
We can observe all methods only gives very limited improvements, especially for structured generalization types. Multi-task learning have been shown useful in low-resource scenarios in machine translation \cite{currey-heafield-2019-incorporating} but not in our case.
We consider this is because compositional generalization is different from generalizing from limited data in that the former requires learning syntax first and then interpreting syntactically combined lexical tokens into semantics.
Thus it may not be enough to only make the model output syntax and semantics together. Modeling the causal relation from syntactic to semantic output may address this.
\todo{better description?}
Self-attention mask is very useful in \cite{kim-etal-2021-improving} on a classification task for compositional generalization but poorly on COGS.
We consider this comes from the difficulty of generation task compared with classification in \cite{kim-etal-2021-improving}. In their classification task, a source sentence is concatenated with a target meaning representation to be classified as a positive or negative sample.
In our generation task, the model need not only encode syntactic information(e.g. self-attention in encoder), but attend to correct tokens for each decoding step(e.g. cross-attention in decoder).
Also, \cite{kim-etal-2021-improving} combine self-attention mask with entity cross link (e.g. allow attention between aligned entities in sentence and meaning representation), which we do not use in our task.
There are other methods exploiting syntax in downstream tasks \cite{zhang-etal-2019-syntax-enhanced, glavas-vulic-2021-supervised} such as encoding syntax as dense embedding features.
Here We leave better exploitation of syntax for compositional generalization for future works.
\section{Conclusion}\label{sec:conclusion}
We have shown that compositional semantic parsers systematically
outperform recent seq2seq models on structural generalization in
COGS. While both BART and the AM parser support accurate
broad-coverage semantic parsing, we find that BART struggles with
structural compositional generalization as much as other seq2seq
models, whereas the compositional AM parser
achieves state-of-the-art generalization accuracy on COGS.
These results suggests that even powerful seq2seq
models lack a structural bias that is required to generalize across
linguistic structures as humans do. This lack of bias is not limited
to semantics; our findings indicate that seq2seq models struggle just
as hard to learn syntactic generalizations that are easy for
structure-aware models. Given that all recent models are accurate on
most generalization types, we suggest focusing future evaluations on a
model's accuracy on structural generalization types, and perhaps
extend COGS to a corpus that offers a greater variety of these.
\section{Related Work}\label{sec:related_work}
\citet{kim-linzen-2020-cogs} demonstrate that simple
seq2seq models (LSTMs and Transformers) struggle with all
generalization types in COGS. Subsequent work with novel seq2seq
architectures achieve a much higher mean accuracy on the COGS
generalization set \citep{akyurek-andreas-2021-lexicon,csordas-etal-2021-devil,
conklin-etal-2021-meta,tay-etal-2021-pretrained,orhan-2021-compgen,
zheng-lapata-2021-disentangled}, but their accuracy on the
generalization set still lags more than ten points behind that on the
in-domain test set.
COGS can also be addressed with \emph{compositional models}, which
directly model linguistic structure and implement the Principle of
Compositionality. The LeAR model of \citet{liu-etal-2021-learning}
achieves a generalization accuracy of 98\%, outperforming all known
seq2seq models by at least ten points. LeAR also sets new states of
the art on CFQ and Geoquery, but has not been demonstrated to be
applicable to broad-coverage semantic parsing.
Compositional semantic parsers for other tasks include the AM parser
\citep{groschwitz-etal-2018-amr,lindemann-etal-2020-fast}
(\cref{sec:amparsing4cogs}) and SpanBasedSP
\citep{herzig-berant-2021-span-based}. The AM parser has been shown
to achieve high accuracy and parsing speed on broad-coverage semantic
parsing datasets such as the AMRBank. SpanBasedSP parses Geoquery,
SCAN, and CLOSURE accurately through unsupervised training of a
span-based chart parser. \citet{shaw-etal-2021-compositional} combine
quasi-synchronous context-free grammars with the T5 language model to
obtain even higher accuracies on Geoquery, demonstrating some
generalization from easy training examples to hard test instances.
Structural generalization has also been probed in syntactic parsing tasks.
\citet{linzen2016assessing} define
a number-prediction task that requires learning syntactic
structure and find that LSTMs perform with some success;
however, \citet{kuncoro-etal-2018-lstms} find that structure-aware
RNNGs perform this task more accurately.
\citet{DBLP:journals/tacl/McCoyFL20} found that hierarchical representations are necessary for human-like syntactic generalizations on a question formation
task, which seq2seq models cannot learn.
\section{The role of syntax}
\label{sec:analysis-extensions}
Our finding that seq2seq models perform so poorly on structural
generalization in COGS begs the question: Is there anything special
about the meaning representations in COGS that makes structural
generalization hard, or would seq2seq models struggle similarly on
other target representations for these generalization types? Do
seq2seq models have a specific weakness regarding semantic
compositionality? Or is it because they systematically lack a bias
that would help them generalize over structure in language? In this
section, we investigate these questions by recasting COGS as a
syntactic corpus.
\subsection{Syntactic generalization}\label{subsec:seq2seq:syntactic_gen}
We obtain a syntactic annotation for each instance in COGS from the
(unambiguous) original PCFG grammar used to generate COGS (cf.\
\cref{fig:structural-generalization}). We replace the very
fine-grained non-terminals (e.g.\ \lform{NP\_animate\_dobj\_noPP}) of
the original PCFG with more general ones (e.g.\ \lform{NP}) and remove
duplicate rules (e.g.\ \lform{NP}$\to$\lform{NP}) resulting from this.
We train BART on predicting linearized constituency trees from the
input strings. For comparison, we also train the Neural Berkeley
Parser \citep{kitaev-klein-2018-constituency} on COGS syntax
(``Benepar'' in the tables). This parser consists of a self-attention
encoder and a chart decoder. It is therefore \emph{structure-aware},
in that it explicitly models tree structures; this is the analogue of
a compositional parser for semantics.
Results are shown in the two bottom rows of
\cref{tab:selected_gentype_eval}. We find the same pattern as in the
semantic parsing case: the seq2seq model does well on \genclass{Prop} and
\genclass{Lex}, but struggles with \genclass{Struct}. The structure-aware Berkeley
parser handles all three generalization types well. Thus, the
difficulties that seq2seq models have on structural generalization on
COGS are not limited to semantics: rather, they seem to be a general
limitation in the ability of seq2seq models to learn linguistic
structure from structurally simple examples and use it
productively. Not only does compositional generalization require
compositional parsers; structural generalization in semantics or
syntax seems to require parsers which are aware of that structure.
\subsection{Compositional generalization from correct
syntax}\label{subsec:seq2seq:semantic_gen_with_syntax}
But perhaps the poor performance of seq2seq semantic parsers on
\genclass{Struct}\ is caused \emph{only} by their inability to learn to
generalize syntactically? Would their accuracy catch up with that of
compositional models if we gave them access to syntax?
We retrained BART on predicting semantic representations, but
instead of feeding it the raw sentence, we provide as input the
linearized gold constituency tree (``\lform{(NP ({Det} a) ({N} rose))}''),
both for training and inference.
This method is similar to
\citet{li-etal-2017-modeling} and \citet{currey-heafield-2019-incorporating},
but we allow attention over special tokens such as ``\lform{(}'' during decoding.
We report the results as ``BART+syn'' in \cref{tab:results} and
\cref{tab:selected_gentype_eval}; the overall accuracy increases by
3.2\% over BART. This
is mostly because providing the syntax tree allows BART to generalize
correctly on \genclass{Prop}.
However, \genclass{Struct} remains out
of reach for BART+syn, confirming the deep difficulty of
structural generalization for seq2seq models.
We also explored other ways to inform BART with syntax, through
multi-task learning \cite{sennrich-etal-2016-controlling,
currey-heafield-2019-incorporating} and syntax-based masking in the
self-attention encoder \cite{kim-etal-2021-improving}. Neither method
substantially improved the accuracy of BART on the COGS generalization
set (+1.4\% and +2.1\% overall accuracy, respectively). More
detailed results are in \cref{sec:detailed_evaluation_results}.
\section{Introduction}\label{sec:introduction}
Compositionality is a fundamental principle of natural language
semantics: ``The meaning of a whole [expression] is a function of the
meanings of the parts and of the way they are syntactically combined''
\citep{partee-1984-compositionality}.
A growing body of research focuses on
\emph{compositional generalization}, the ability of a
semantic parser to
combine known linguistic elements in novel structures in ways akin to humans. For example,
observing the meanings of \textit{``The hedgehog ate a cake''} and \textit{``A baby
liked the penguin,''} can a model predict the meaning of \textit{``A baby liked
the hedgehog''}?
Dynamic, compositional recombination helps explain efficient human language learning and usage, and investigating whether NLP models make use of the same property offers important insight into their behavior.
Current research on compositional generalization shows
the task to be challenging and complex. Such research centers around a
number of corpora designed specifically for the task, including SCAN
\citep{lake-baroni-2018-generalization} and CFQ
\citep{keysers-etal-2020-measuring}. We focus on COGS \cite{kim-linzen-2020-cogs}, a synthetic
semantic parsing corpus of English whose test set consists of 21
\emph{generalization types} such as the example above
(\cref{sec:background}). Kim and Linzen report that simple
sequence-to-sequence (seq2seq) models such as LSTMs and Transformers
struggle with many of their generalization types, achieving an overall
highest accuracy on the generalization set of 35\%.
Subsequent work
has improved accuracy on the COGS generalization set considerably
\citep{tay-etal-2021-pretrained,akyurek-andreas-2021-lexicon,conklin-etal-2021-meta,csordas-etal-2021-devil,orhan-2021-compgen,zheng-lapata-2021-disentangled},
but the accuracy of even the best seq2seq models remains below 88\%.
By contrast, \citet{liu-etal-2021-learning} report an accuracy of
98\%, using an algebraic model that implements compositionality
(\cref{sec:related_work}).
Here, we investigate whether this difference in compositional
generalization accuracy is incidental, or whether there is a
systematic difference between seq2seq models and models that are
guided by compositional principles and aware of complex
structure. Comparisons between entire classes of models must be made
with care. Thus in order to make claims about the class of
compositional models, we first work out a second compositional model
for COGS (in addition to Liu et al.'s). We apply the AM
parser \cite{groschwitz-etal-2021-learning}, a compositional semantic
parser which can parse a variety of graphbanks fast and accurately
\cite{lindemann-etal-2020-fast}, to COGS after minimal adaptations
(\cref{sec:amparsing4cogs}). The AM parser achieves a generalization
accuracy above 98\%, making it the first semantic parser shown to
perform accurately on both COGS
and broad-coverage semantic parsing.
We then
compare these two
compositional models to all published seq2seq models for COGS. We find
that the difference in generalization accuracy can be attributed
specifically to \emph{structural} types of compositional
generalization, which require the parser to generalize to novel
syntactic structures that were not observed in training. While the
compositional parsers achieve excellent accuracy on these
generalization types, all known seq2seq models perform very poorly,
with accuracies close to zero. This is even true for BART
\cite{lewis-etal-2020-bart-acl}, which we apply to COGS for the first
time; this is surprising because BART achieves very high accuracy on
broad-coverage semantic parsing tasks \cite{bevilacqua-etal-2021-one}. We conclude that seq2seq
models, as a class, seem to have a weakness with regard to structural
generalization that compositional models overcome
(Section~\ref{subsec:first:experiments}).
Finally, we investigate the role of syntax in compositional
generalization (\cref{sec:analysis-extensions}). We show that parsers
which explicitly model syntactic tree structures can easily learn
structural generalization when trained to predict syntax trees on
COGS, whereas BART again performs poorly.
BART does
not learn structural generalization even if we enrich its input with
syntactic information. Thus, the poor performance of seq2seq models on
structural generalization is not specifically due to representational
choices in COGS, or even to the specific compositional demands of
semantic parsing; structural generalization requires
structure-aware models.
We discuss implications for future work on compositional
generalization in \cref{sec:discussion}. All code will be made
publicly available upon acceptance.
\newcommand{\ignore}[1]{}
\ignore{
Our contributions: \todo{to do}
\begin{itemize}
\item we outline a transformation of COGS logical forms to graphs such that the AM parser can work with them (§\ref{sec:cogs2graph}).
\item we show how COGS can be solved with a compositional graph-based parser (with little supervision). Whereas BART \cite{lewis-etal-2020-bart-acl}, a pretrained encoder-decoder network, fails to do so despite the big pretraining dataset (§\ref{sec:experiments}).
\item by that we demonstrate that the AM parser is indeed compositional and can generalize compositionally.
\end{itemize}
Outline of the paper:
the compositional parser and the COGS dataset (§\ref{sec:background}),
the conversion from logical forms to graphs (§\ref{sec:amparsing4cogs}),
experiments, results and error analysis (§\ref{sec:experiments}),
discussion (§\ref{sec:discussion}),
related work (§\ref{sec:related_work}),
§\ref{sec:conclusion} concludes.
}
\section{Experiments on COGS}\label{subsec:first:experiments}
With two compositional models available on COGS, we can now compare
compositional semantic parsers, as a class, to seq2seq models, as
a class, on compositional generalization in COGS.
\subsection{Experimental setup}\label{subsec:first:setup}
We follow standard COGS practice and evaluate all models on both the
(in-distribution) test set and the generalization set. In addition to
the regular COGS training set (\sortof{train}) of 24,155 training
instances, we also report numbers for models trained on the extended
training set \sortof{train100}, of 39,500 instances
\citep[Appendix E.2]{kim-linzen-2020-cogs}. The \sortof{train100}
set extends \sortof{train} with 100 copies of each exposure
example. For instance, for the generalization instance in
Table~\ref{tab:cogssamples}a, \sortof{train100} will contain 100 different
sentences in which ``the/a hedgehog'' appears as subject (rather than
just one in `train').
We report exact match accuracies, averaged across 5 training runs,
along with their standard deviations.
\input{tables/semanticcogs_main_eval_small}
\input{tables/few_gentype_eval_transposed}
\paragraph{Sequence-to-sequence models.} \label{subsec:seq2seq:setup}
We train BART \cite{lewis-etal-2020-bart-acl} as a semantic parser on
COGS. This is a strong representative of the family of seq2seq models,
as a slightly extended form of BART \cite{bevilacqua-etal-2021-one}
set a new state of the art on
semantic
parsing on the AMR corpus \cite{banarescu-etal-2013-abstract}.
To apply BART on COGS, we directly fine-tune the pretrained
\textit{bart-base} model on it with the corresponding tokenizer. Training details are described in \cref{sec:training_details_bart}.
We also report results for all other published seq2seq models for COGS
\citep{kim-linzen-2020-cogs,conklin-etal-2021-meta,csordas-etal-2021-devil,akyurek-andreas-2021-lexicon,tay-etal-2021-pretrained,orhan-2021-compgen,zheng-lapata-2021-disentangled}.
We retrained some of these models on train100 to measure the impact of
the training set.
\paragraph{Compositional models.}
We train the AM parser on the COGS graph corpus
(cf.~\cref{subsec:cogs2graph}) and copied most hyperparameter values
from \citet{groschwitz-etal-2021-learning}'s training setup for AMR to make overfitting to COGS less likely; details are described in
\cref{sec:training_details_amparser}.
The AM parser either receives pretrained word embeddings from BERT
\citep{devlin-etal-2019-bert} (\sortof{\amBert}) or learns embeddings
from the COGS data only (\sortof{\amToken}). We run the training
algorithm with up to three argument slots to enable the analysis of
ditransitive verbs. For evaluation, we revert the graph conversion to
reconstruct the logical forms.
\footnotetext{All LeAR numbers are based on our reproduction of
their COGS evaluation; they report an accuracy of 97.7.}
For PP recursion, COGS
eliminates potential PP attachment ambiguities and
assumes that each PP modifies the noun immediately to its left.
We hypothesize that explicit distance information between tokens could
help the AM parser learn this regularity:
Instead of passing only the representations of the
potential parent and child node to the edge-scoring model, we also
pass an encoding of their relative distance in the string
\cite{vaswani-etal-2017-attention}, yielding the AM parser models with the ``+\dist'' suffix.
Finally, we report evaluation results for LeAR, the compositional COGS parser of
\citet{liu-etal-2021-learning}.
\subsection{Results}\label{subsec:results}
The results are summarized in \Cref{tab:results}.
\paragraph{Compositional outperforms seq2seq.}
While all models achieve near-perfect accuracy on the in-distribution
test sets, we find that when trained on \sortof{train100}, all
compositional models outperform all seq2seq models on the
generalization set, by a wide margin. This includes the very strong
BART baseline, which holds the state of the art in broad-coverage
parsing for AMR.
LeAR even achieves its near-perfect accuracy when trained on `train',
and outperforms all seq2seq models trained on either dataset. See
below for a detailed discussion of the AM parser.
\paragraph{Performance by generalization type.}
To understand this result more clearly, we break down the accuracy by
generalization type. This analysis is shown in
\cref{tab:selected_gentype_eval}. We will explain ``BART+syn'' in
\cref{subsec:seq2seq:semantic_gen_with_syntax} and the ``syntax''
rows in \cref{subsec:seq2seq:syntactic_gen}. We compare the
compositional models against all seq2seq models that report these
fine-grained numbers or for which they were easy to reproduce (see
\cref{sec:training_details_bart,sec:detailed_evaluation_results} for details).
The results group neatly with the three classes of generalization types
outlined in \cref{sec:background}: \textsc{Lex}, \textsc{Struct}, and \textsc{Prop}. All recent models achieve
near-perfect accuracy on each of the 16 lexical generalization
types. On structural generalization types, seq2seq models achieve very
low accuracies, whereas the compositional parsers (\amBert+\dist and LeAR)
are still very accurate. The proper-noun object cases are somewhere in
the middle, with the seq2seq models reporting middling numbers.
\paragraph{Depth generalization (recursion).}
There is a particularly pronounced difference between compositional
and seq2seq models on the two ``recursion'' generalization types (cf.\
\cref{fig:structural-generalization:PPrecursion}). In these cases,
the training data contains examples up to depth two and the
generalization data has depths 3--12. \Cref{fig:depths_plot} shows
the accuracy of several models on PP recursion in detail. As we see,
the accuracy of BART (even when informed by syntax, cf.\
\cref{subsec:seq2seq:semantic_gen_with_syntax}) degrades
quickly with recursion depth. By contrast, both LeAR and \amBert+\dist
maintain their high accuracy across all recursion depths. This
suggests that they learn the correct structural generalizations even
from training observations of limited depth.
\begin{figure}
\centering
\includegraphics[scale=.4,trim={5mm 5mm 40mm 0},clip]{figs/recursion_depth.pdf}
\caption{Influence of PP recursion depth on overall PP depth
generalization accuracy.
}\label{fig:depths_plot}
\end{figure}
\paragraph{Effect of distance encoding for AM parser.}
As illustrated in \cref{fig:depths_plot}, the accuracy of the
unmodified AM parser without the distance feature degrades with
increasing PP recursion depth. An error analysis showed that this is
because the AM parser is uncertain about the attachment of PPs in the
middle of the string, confirming our hypothesis that it does not learn
the idiosyncratic treatment of PPs in COGS (always attach
low). Adding the distance feature solves this problem.
There is an interesting asymmetry between the behavior of the AM
parser on PP recursion and CP recursion, which nests sentential
complements within each other (``Emma said that Noah knew that the cat
danced''): The accuracy of the unmodified AM parser is stable across
recursion depths for CP recursion, and the distance feature is only
needed for PPs. This can be explained by the way in which the AM
parser learns to incorporate PPs and CPs into the dependency tree: it
uses \app{} edges to combine verbs with CPs, which ensures that only a
single CP can be combined with each sentence-embedding verb. By
contrast, each NP can be modified by an arbitrary number of PPs using
\modify{} edges. Thus a confusion over attachment is only possible for
PPs, not CPs.
\paragraph{Effect of training regime.}
Parsers on COGS are traditionally not allowed any pretraining
\cite{kim-linzen-2020-cogs}, in order to judge their ability to
generalize from limited observations. We see in the
experiments above that the use of pretrained word embeddings
helps the AM parser achieve accuracy parity with LeAR, but is not
needed to outperform all seq2seq models on `train100'.
Training on \sortof{train100} helps the AM parser more than any other
model in \cref{tab:results}. The difference between its accuracy on
`train' and `train100' is due to lexical issues: we found that when
trained on `train', the AM parser typically predicts the correct
delexicalized formulas and then inserts an incorrect but related
constant or predicate symbol (e.g.\ \word{Emma} instead of
\word{Charlie} in Table~\ref{tab:cogssamples}b). Trained on `train',
\amBert+\dist achieves a mean accuracy on \genclass{Struct}\ of 89.6 (compared
to 92.3 for `train100'), whereas the mean accuracy on \genclass{Lex}\ drops to
76. Even without BERT and trained on `train', \amToken+\dist gets 74.6 on
\genclass{Struct}, drastically outperforming the seq2seq models
(Appendix~\ref{sec:detailed_evaluation_results}).
\section{On appendices in general}
\section{COGS dataset statistics}\label{sec:cogsstatistics}
The COGS dataset
contains English declarative sentences mapped with logical forms. It was created by \citet{kim-linzen-2020-cogs} and is publicly available at \url{https://github.com/najoungkim/COGS} (MIT license).
We use the version from April 2nd 2021 \href{https://github.com/najoungkim/COGS/tree/6f663835897945e94fd330c8cbebbdc494fbb690}{commit \texttt{6f66383}} and use the dataset as-is (no datapoints excluded or changed, use their data set splits), except for the AM parser for which we conduct the logical form to graph preprocessing described in \cref{subsec:cogs2graph}.\\
The normal training set (\sortof{train}) consists of 24,155 samples (24k in distribution, 143 primitives, 12 exposure examples), the dev and test set both contain 3k in distribution samples each.
Primitives and exposure examples contain \sortof{lexical trigger words} necessary for all but the three structural generalization types: these lexical trigger words each appear only once and in one sample in the whole training set.
Primitives are one-word sentences, therefore presenting word-meaning mapping without context of a sentence (necessary for the types Primitive to *). In contrast, exposure examples are full sentences e.g.~for the subject to object (common noun) generalization this sentence contains \word{hedgehog} as the subject. In the generalization set this word appears in 1k samples, but in a different syntactic configuration compared to the exposure example (e.g.~\word{hedgehog} in object position).
There is also an additional larger training set (\sortof{train100}) with 39,500 samples containing the lexical trigger words in 100 samples each, instead of just in one sample.
The out-of-distribution generalization set contains 21k samples, 1k per generalization type.
\section{Training details of the AM parser}
\label{sec:training_details_amparser}
\textit{The corresponding code will be made publicly available upon acceptance.}
\paragraph{Hyperparameters.}
For the AM parser, we mostly copied the hyperparameter values from the AMR experiments of \citet{groschwitz-etal-2021-learning}. This should help against overfitting on COGS, but we also note that hyperparameter tuning for compositional generalization datasets can be difficult anyways since one can typically easily achieve perfect scores on an in-doman dev set.
Copied values include for instance the number of epochs (60 due to supervised loss for edge existence and lexical labels), the batch size, the number and dimensionality of neural network layers and not using early stopping (but selecting best model based on per epoch evaluation metric on the dev set).
Choosing 3 sources has worked well on other datasets \citep{groschwitz-etal-2021-learning} and we adopt this hyperparameter choice. We note that with ditransitive verbs (i.e. verbs requiring NPs filling agent, theme, and recipient roles) present in COGS we need at least three sources anyway to account for these.
\paragraph{Deviations from \citet{groschwitz-etal-2021-learning}'s settings.}
For training on train (but not train100), we set the vocabulary threshold from 7 down to 1 to account for the fact that the lexical generalizations rely on a single occurrence of a word in the training data (on train100 we keep 7 as a threshold since the trigger words occur 100 times in there).
Furthermore, the COGS dataset doesn't have part-of-speech, lemma or named-entity annotations, so we just don't use embeddings for these.
For the word embeddings we either use BERT-Large-uncased \citep{devlin-etal-2019-bert} or learn embeddings from the dataset only (embedding dimension 1024, same as for the BERT model).
We also decreased the learning rate from 0.001 to 0.0001: we observed that the learning curves are still converging very quickly and hypothesize that COGS training set might also be easier than the AMR one used in \citet{groschwitz-etal-2021-learning}.\\
Unlike them we didn't use the fixed-tree decoder (described in \citealt{groschwitz-etal-2018-amr}), but opted for the projective A* decoder \citep[§4.2]{lindemann-etal-2020-fast}: in pre-experiments this showed better results. In addition, it makes comparison to related work (such as LeAR by \citet{liu-etal-2021-learning}) easier which uses only projective latent trees.
We also use supervised loss for edge existence and lexical labels: we can use supervised loss for both as they do not depend on the source names to be learnt. In preliminary experiments this yielded better results than using the automaton-based loss for them too. The supervised loss wasn't described in \citet{groschwitz-etal-2021-learning}, but already implemented in their code base and they note there that the effect on performance was mixed in their experiments (similar for SDP, worse for AMR).
\paragraph{Relative distance encoding.}
For the relative distance encodings we added to the dependency edge existence scoring, we used sine-cosine interleaved encoding function introduced by \citet[§3.5]{vaswani-etal-2017-attention} and as input to it use the relative distance $dist(i,j)=i-j$ between sentence positions $i$ and $j$. We use a dimensionality of 64 for the distance encodings ($d_{model}$ in \citet{vaswani-etal-2017-attention} is 512). These distance encodings are then concatenated together with the BiLSTM representations for possible heads and dependents used in the standard \citet{kiperwasser-goldberg-2016-simple} edge scoring model. This constitutes the input to the MLP emitting a score for each token pair. In other words, for each token pair $\langle i,j\rangle$ the MLP has to decide edge existence based on the representations of the tokens at positions $i$ and $j$, and an encoding of the relative distance $dist(i,j)=i-j$.
These models have the suffix \sortof{\dist} in the tables.
\paragraph{Runtimes.}
Training the AM parser took 5 to 7 hours on train with 60 epochs and 6 to 9.5 hours on train100. In general, training with BERT took longer than without, same holds for adding relative distance encodings.
Inference with a trained model on the full 21k generalization samples took about 15 minutes using the Astar decoder with the \sortof{ignore aware} heuristic.
All AM parser experiments were performed using Intel Xeon E5-2687W v3 10-core processors at 3.10Ghz and 256GB RAM, and MSI Nvidia Titan-X (2015) GPU cards (12GB).
\paragraph{Number of parameters.}
For their models, \citet{kim-linzen-2020-cogs} tried to keep the number of parameters comparable (9.5 to 11 million) and therefore rule out model capacity as a confound. The number of trainable parameters of the AM parser model used is 10.7 to 11.5million (lower one is with BERT, higher without. Impact of relative distance encoding is rather minimal: $<17$k), so the improved performance is not just due to a higher number of parameters.
\paragraph{Dev set performance.}
As usual for compositional generalization datasets, it is relatively easy to get (near) perfect results on the (in domain) dev/test sets. We observed this too: all AM parser models had an exact match score of at least 99.9 on the dev set and at least 99.8 on the (in distribution) test set.
\paragraph{Evaluation procedure.}
Unfortunately, \citet{kim-linzen-2020-cogs} didn't provide a separate evaluation script.
As a main evaluation metric they use (string) exact match accuracy on the logical forms which we adopt.
Note that this requires models to learn the \sortof{correct} order of conjuncts: even if a logically equivalent form with a different order of conjuncts would be predicted, string exact match would count it as a failure.
In lack of an official evaluation script we implemented our own evaluation script to compute exact match.
\section{Training details of Seq2seq} \label{sec:training_details_bart}
\paragraph{Hyperparameters.} We use the same hyperparameter setting for BART on both syntactic and semantic experiments.
We use \textit{bart-base}\footnote{\url{https://huggingface.co/facebook/bart-base}} model in all our experiments. Our batch size is 64.
We use Adam optimizer \cite{kingma-ba-2015-adam} with learning rate 1e-4 and
gradient accumulation steps 8. Loss averaged over tokens is used as the validation metric
for early stopping following \citet{kim-linzen-2020-cogs}. During
inference, we use beam search with beam size 4.
\paragraph{Dev set performance.} The exact match accuracy is at least 99.6 for both dev set and (in-distribution) test set in all experiments.
\paragraph{Other details.} Training took 4 hours for BART with about 80 epochs on train and 5 hours with about 50 epochs on train100. Inference on generalization set took about 1 hour. All BART experiments were run on Tesla V100 GPU cards (32GB). The number of parameters in our BART model is 140 million.
\paragraph{Syntactic annotations.} To obtain syntactic annotations, we use NLTK\footnote{\url{https://www.nltk.org/}} to parse each sentence in COGS with PCFG grammar generating COGS. In our experiments, we found this parsing process did not yield any ambiguous tree. The original PCFG grammar contains rules such as \lform{NP$\to$NP\_animate\_dobj\_noPP}. We replace such fine-grained nonterminals (e.g.\ \lform{NP\_animate\_dobj\_noPP}) with general nonterminals (e.g.\ \lform{NP}). This results in duplicate patterns (e.g.\ \lform{NP$\to$NP}) and we further remove such patterns from the output tree.
\paragraph{Results from other papers.} \citet{conklin-etal-2021-meta}\footnote{\url{https://github.com/berlino/tensor2struct-public}}, \citet{akyurek-andreas-2021-lexicon}\footnote{\url{https://github.com/ekinakyurek/lexical}}, \citet{csordas-etal-2021-devil}\footnote{\url{https://github.com/robertcsordas/transformer_generalization}} and \citet{tay-etal-2021-pretrained} did not report performance of their model on train100 set. To report these numbers, we additionally use their published code to train their model on train100 for 5 runs. We use seed 6-10 for \citet{conklin-etal-2021-meta} and random number seeds for \citet{csordas-etal-2021-devil}, following their default setting. We use their default configuration file for their best model to set the hyperparameters. \citet{tay-etal-2021-pretrained}, did not publish their code so we did not report that. \citet{orhan-2021-compgen}\footnote{\url{https://github.com/eminorhan/parsing-transformers}} and \citet{zheng-lapata-2021-disentangled} are the two most recently published seq2seq approaches. Both did not provide numbers for train100 training and because of their recency we weren't able to run their models on the train100 set so far. We thus only report their published results for train set.
\section{Detailed evaluation results}\label{sec:detailed_evaluation_results}
The main results are summarized in the main paper in \cref{subsec:results} with \cref{tab:results} and \cref{tab:selected_gentype_eval}.
Here we present
AM parser (\cref{tab:detailed_eval_accuracy}),
LeAR (\cref{tab:detailed_eval_accuracy_lear}) and BART (\cref{tab:detailed_eval_accuracy_seq2seq})
performance for each of COGS' 21 generalization types separately with the usual mean and standard deviation of 5 runs.
For descriptions of the generalization types we refer to \citet[][§3 and Fig.~1]{kim-linzen-2020-cogs}.
\paragraph{On accuracy computation for LeAR.}
We observed that the LeAR model skips 22 sentences in the generalization set due to out-of-vocabulary tokens.\footnote{The words \word{gardner} and \word{monastery} occur zero times in the train set, but in total in 22 sentences of the generalization set. The majority (15) of these appear in PP recursion samples.}
We do include these sentences in the accuracy computation (as failures) for the generalization set.
The published LeAR code does not convert its internally used representation back to logical forms, therefore we evaluate on the logical forms like it is done for other models, but have to rely on accuracy computation done in the LeAR code for the internal representation.
Furthermore we would like to note that--based on inspecting the published code\footnote{\url{https://github.com/thousfeet/LEAR} }--, LeAR made the preprocessing choice to ignore the contribution of the definite determiner, basically treating indefinite and definite NPs equally, resulting in a big conjunction without any iota (\sortof{\lform{*}}) prefixes.
\paragraph{On model numbers copied from other papers.}
\citet{kim-linzen-2020-cogs} provide three baseline models, among which the Transformer model reached the best performance on train and train100. Per generalization type results can be found in their Appendix F (Table 5 on page 9105) from which we report the Transformer model numbers.\\
The strongest model of \citet{akyurek-andreas-2021-lexicon} is actually \sortof{Lex:Simple:Soft} (cf. their Table~5) with a generalization accuracy of 83\% (also reported in our \cref{tab:results}), whereas their Lex:Simple model lags 1 point behind. For the latter, but not for the former, the authors provide per generalization type output in their accompanying GitHub repository as part of a \href{https://github.com/ekinakyurek/lexical/blob/e7a44e19d23a1d99726cd76c5cd88f56ca586653/analyze.ipynb}{jupyter notebook}. Therefore numbers in \cref{tab:selected_gentype_eval} are for Lex:Simple, not Lex:Simple:Soft.\\% see Out[98]
We picked the best performing model of \citet{orhan-2021-compgen}: According to their Table 2 the \verb+t5-3b mt5_xl+ model shows the best generalization performance (84.6\% average accuracy). From the accompanying GitHub repository\footnote{\url{https://github.com/eminorhan/parsing-transformers}} we copy the model's results, specifically we average over the 5 runs of the model \href{https://github.com/eminorhan/parsing-transformers/tree/9887632a348f9d2e3b010f86a7931691a0faf044/results/3b/cogs_mt5/epochs_10}{3b-cogs-mt5-epochs10 (commit \texttt{04a2508})}. We note that other models reported in \citet{orhan-2021-compgen} showed the same performance pattern with respect to our three generalization classes \genclass{Lex}, \genclass{Prop}, and \genclass{Struct}.\\
For \citet{zheng-lapata-2021-disentangled}, our reported number is slightly different from the original paper. This is because we asked the authors for detailed results and they provide us with their newest results averaged over 5 runs.
\paragraph{Abbreviations in the tables.}
\sortof{Subj} means \sortof{subject},
\sortof{Obj} means \sortof{object},
\sortof{Prim} means \sortof{primitive},
\sortof{Infin. arg} means \sortof{infinitival argument},
\sortof{ObjmodPP to SubjmodPP} means \sortof{object-modifying PP to subject-modifying PP},
\sortof{ObjOTrans.} means \sortof{object omitted transitive},
\sortof{trans.} means \sortof{transitive},
\sortof{unacc} means \sortof{unaccusative},
\sortof{Dobj} means \sortof{Double Object}.
\input{tables/detailed_eval_accuracy}
\input{tables/detailed_eval_accuracy_lear}
\input{tables/detailed_eval_seq2seq}
\section{Additional information on COGS to graph conversions}\label{sec:details_cogs2graph}
This is a more detailed explanation of the COGS logical form to graph conversion described in \cref{subsec:cogs2graph}
based on four additional example sentences:
\begin{exe}
\ex\label{ex:boywanted} The boy wanted to go. \\
\lform{*boy($x_1$); want.agent($x_2, x_1$) $\land$ want.xcomp($x_2, x_4$) \\ $\land$ go.agent($x_4, x_1$)}
\ex\label{ex:other} Ava was lended a cookie in a bottle. \\
\lform{lend.recipient($x_2,$ Ava) \\ $\land$ lend.theme($x_2, x_4$) \\ $\land$ cookie($x_4$) \\ $\land$ cookie.nmod.in($x_4, x_7$) \\ $\land$ bottle($x_7$)}
\ex\label{ex:cprec} Ava said that Ben declared that Claire slept. \\
\lform{say.agent($x_1,$ Ava) \\ $\land$ say.ccomp($x_1, x_4$) \\ $\land$ declare.agent($x_4,$ Ben) \\ $\land$ declare.ccomp($x_4, x_7$) \\ $\land$ sleep.agent($x_7,$ Claire)}
\ex\label{ex:prim_touch} touch \\
\lform{$\lambda a. \lambda b. \lambda e.$ touch.agent($e, b$) $\land$ touch.theme($e, a$)}
\end{exe}
The first of these is used as the main example for now. Its graph conversion can be found in \cref{fig:cogsgraph-boy}.
\begin{figure}[t]
\centering
\input{./figs/theBoyWantedToGo}
\caption{
Logical form to graph conversion for \sentence{The boy wanted to go} (cf.~\cref{ex:boywanted}).
For illustration only we use node names (the part before the `/') to outline the token alignment.
}\label{fig:cogsgraph-boy}
\end{figure}
\paragraph{Basic ideas.}
\emph{Arguments} of predicates (variables like $x_i$ or proper names like \lform{Ava}) are translated to nodes.
The first part of each predicate name (e.g.~\lform{boy}, \lform{want}, \lform{go}) is the lemma of the token pointed to by the first argument (e.g.~$x_1, x_2, x_4$), we strip this lemma (\sortof{delexialize}) from the predicate and insert it as the node label of the first argument (post-processing reverses this).\\
\emph{Binary predicates} (i.e.~terms with 2 arguments) are translated into edges, pointing from their first to their second argument, e.g.~\lform{want.agent($x_2, x_1$)} is converted to an \graphedge{agent} edge from node $x_2$ (the \graphnode{want} node) to node $x_1$.
Because of the delexicalization described above, there are only 8 different edge labels: \graphedge{agent}, \graphedge{theme}, \graphedge{recipient}, \graphedge{xcomp}, \graphedge{ccomp}, \graphedge{iota} and 2 preposition-introduced edges described below.\\
For \emph{unary predicates} like \lform{boy($x_1$)} the delexicalization already suffices, so we don't add any edge (in lack of a proper target node). We restore unary predicates during postprocessing for nodes with no outgoing edges.\\
Each \emph{iota term} \lform{*noun($x_{\text{noun}}$);} is treated as if it was a conjunction of the noun meaning (i.e.~\lform{noun($x_{\text{noun}}$)}) and \sortof{definite determiner meaning} binary predicate \lform{the.iota($x_{\text{the}}, x_{\text{noun}}$)}.\\
The AM parser further requires one node to be the \emph{root node}.
For non-primitives we select it heuristically as the node with no incoming edges (excluding preposition and determiner nodes).
\paragraph{Prepositions.}
Instead of being treated as an edge as the above would suggest,
we \sortof{reify} them,
so each preposition becomes a node of the graph with outgoing \graphedge{nmod} edges to the modified NP and the argument NP. So for \word{cookie in the bottle} (cf. \cref{ex:other} and \cref{fig:lendedCookie}) we create a node with label \graphedge{in} and draw an outgoing \graphedge{nmod.op1} edge to the \graphnode{cookie}-node and an \graphedge{nmod.op2} edge to the \graphnode{bottle}-node.
\paragraph{Alignments.}
For training the AM parser additionally needs \emph{alignments} of the nodes to the input tokens.
Luckily all $x_i$ nodes naturally provide alignments (alignment to $i$th input token).
For proper names we simply align them to the first occurrence in the sentence\footnote{
this works because it seems that a name never appears more than once within a sentence.
Names in the logical forms also seem to be ordered based on their token position.
}, the special determiner node is aligned to the token preceding the corresponding $x_{noun}$.\footnote{
we can do so because there are --beyond \word{the} and \word{a}-- no pre-nominal modifiers like adjectives in this dataset.
}
The edges are implicitly aligned by the blob heuristics, which are pretty simple here; every edge
belongs to the blob of the node it originates from.
\paragraph{Primitives.}
For primitive examples (e.g.~\word{touch} \cref{ex:prim_touch}) we mostly follow the same procedure. Unlike non-primitives, however, their resulting graph \emph{can} have open sources beyond the root node,
e.g.~\word{touch} would have sources at the nodes $b$ and $a$ (incoming \sortof{agent} or \sortof{theme} edge respectively).
These nodes can receive any source out of the three available (\srcr{S0},\srcr{S1},\srcr{S2})\footnote{with the restriction that different nodes should have different sources to prevent the nodes from being merged. Also we don't consider non-empty type requests for these nodes here.}, so the tree automaton build as part of \citet{groschwitz-etal-2021-learning}'s method would allow any combination of source names for the unfilled \sortof{arguments}.
Because there is only one input token, the alignment is trivial.
In fact, primitives quite closely resemble the \sortof{supertags} of the AM parser.\\
Note that by encoding the logical form as a graph we get rid of the ordering of the conjuncts. The \sortof{correct} order (crucial for exact match evaluation) is restored during postprocessing.
The graph conversion for \cref{ex:boywanted} was already presented in \cref{fig:cogsgraph-boy}.
For the other three examples \crefrange{ex:other}{ex:prim_touch}, we present the graph conversions in \cref{fig:moreexamples}.
\begin{figure}[tbh]
\centering
\begin{subfigure}[b]{\columnwidth}
\input{figs/avaWasLendedCookieIn_reified}
\caption{See also \cref{ex:other}.}\label{fig:lendedCookie}
\end{subfigure}
\begin{subfigure}[b]{\columnwidth}
\input{figs/cpRecursionExample}
\caption{See also \cref{ex:cprec}.}\label{fig:cprecconversion}
\end{subfigure}
\begin{subfigure}[b]{.5\columnwidth}
\input{figs/primitiveTouch}
\caption{See also \cref{ex:prim_touch}.}\label{fig:prim_touch}
\end{subfigure}
\caption{
Results of the logical form to graph conversion for \crefrange{ex:other}{ex:prim_touch}.
Actually for (\subref{fig:prim_touch}) the tree automaton contained all possible source name combinations for nodes $a$ and $b$, not just $\langle$\textcolor{red}{\src{s0}},\textcolor{red}{\src{s1}}$\rangle$.
}\label{fig:moreexamples}
\end{figure}
|
2,877,628,090,218 | arxiv | \section{Introduction}
\label{sec:intro}
\noindent Let $u$ be a nonnegative function on ${\mathbb R}^n$
that vanishes at infinity.
Many geometric inequalities
relate $u$ with its symmetric decreasing rearrangement,
$u^\star$.
The {\em P\'olya-Szeg\H{o} inequality} states that
\begin{equation}\label{PS}
\| \nabla u^\star \|_p\le \| \nabla u\|_p
\end{equation}
for every $1\le p\le \infty$ such that the distributional gradient
$|\nabla u|$ lies in $L^p$;
in particular, $|\nabla u^\star|$ again lies in $L^p$~\cite{PS}.
For $p=1$, this reduces to the
isoperimetric inequality, and for $p=\infty$, it follows from
the fact that symmetric decreasing rearrangement
improves the modulus of continuity.
{
Inequality~\eqref{PS} has been extended in various directions.
It holds for general convex Dirichlet-type functionals~\cite{BZ, Ta97},
on the larger space of functions that are locally of bounded
variation~\cite{CiFu1}, and with other symmetrizations in place
of the symmetric decreasing rearrangement~\cite{AFLT, Ka, B, Br1}.
The functionals that satisfy general P\'olya-Szeg\H{o} inequalities
have been fully characterized; they are
known to include all rearrangement-invariant norms~\cite[Theorem 1.2]{CiFe}.}
In this paper, we study functions that
produce equality in \eqref{PS} for some $p$ with $1<p<\infty$.
Such a function will be called an {\em extremal} of the inequality.
Extremals of \eqref{PS}
were first analyzed by Brothers and Ziemer in 1988~\cite{BZ}.
Clearly, every translate of a symmetric decreasing function
is an extremal. In the converse direction,
the level sets of extremals must
be balls, but they need not be concentric. For example, a
function whose graph consists of a small cone stacked
on the frustrum of a large cone is an extremal, regardless of
the precise position of the smaller cone on the plateau.
Brothers and Ziemer discovered that a
similar phenomenon can occur even for functions without plateaus.
Under the assumption that the distribution function
of $u$ is absolutely continuous, they proved that the only extremals
are translates of $u^\star$. Otherwise, there exist
extremals that are equimeasurable to,
but not translates of~$u^\star$.
The condition that the distribution function
be absolutely continuous is equivalent to requiring
that the set of non-trivial critical points of $u^\star$
has measure zero. What can be said about extremals
where this set has positive measure?
In 2006, Cianchi and Fusco proved that every
extremal of \eqref{PS} whose support has finite measure
satisfies
\begin{equation}
\label{eq:CF}
\|u-u^\star\circ\tau\|_1 \le
L_n ||\nabla u||_p \cdot
\lambda_n( {\rm supp}\, u)^{\frac{1}{p'}+\frac{2n-1}{2n^2}}
\,\lambda_n(C)^{\frac{1}{2n^2}}
\end{equation}
for a suitable translation $\tau$~\cite[Theorem 1.1]{CiFu2}.
Here, $L_n$ is a constant that depends
on the dimension,
$\lambda_n$ is the $n$-dimensional Lebesgue measure,
$p'=p/(p\!-\!1)$ is the H\"older dual exponent of $p$,
and $C$ is the set of critical points defined
in Theorem~\ref{main} below.
Our goal is to simplify the
analysis and construct explicit bounds for extremals whose
support need not have finite measure.
The results of Brothers-Ziemer and Cianchi-Fusco apply
to certain convex Dirichlet-type functionals that will be
described below, and to the more general functionals
treated in~\cite{CiFe}, which need not be in integral form.
They remain valid --- after adjusting the constants ---
for the convex rearrangement, which replaces
the level sets of $u$ by suitably scaled copies
of a {centrally symmetric} convex body $B\subset{\mathbb R}^n$.
For the sake of simplicity, we
focus on the classical case of the $L^p$-norm of the gradient
with $1<p<\infty$, leaving the discussion of more
general functionals for the last section of the paper.
\begin{Theorem}\label{main} Let $u$ be a nonnegative function on ${\mathbb R}^n$
that vanishes at infinity and whose distributional gradient
lies in $L^p$ for some $1<p<\infty$,
and let $u^\star$ be its symmetric decreasing rearrangement.
If
$$
\| \nabla u \|_p =\| \nabla u^\star \|_p\,,
$$
then there exists a translation $\tau$ such that
\begin{equation}
\label{eq:main}
\| u-u^\star\circ \tau\|_q\le K_n^{1/q} \, ||u||_q^{1/n'}\,
||u\mathcal X_C||_q^{1/n}\,
\end{equation}
for every $q\ge 1$ with $u\in L^q$. Here, $K_n=2\omega_{n-1}/\omega_n$, and
\begin{equation*}
C= \bigl\{x\in {\mathbb R}^n :
0< u(x)< {\rm ess} \sup u\>\> \mbox{and}\>\> |\nabla u(x)| = 0\bigr\}\,.
\end{equation*}
\end{Theorem}
\smallskip
The translation $\tau$ is chosen to align the graphs of
$u^\star\circ \tau$ {and} $u$ at the top.
The value of the constant is given by
$K_n=\bigl(\int_0^{\pi/2} \cos^n\theta\, d\theta\bigr)^{-1}
\sim (2n/\pi)^{1/2}$.
The set $C$ consists of the non-trivial critical points of $u$,
except for the possible plateau at height $ {\rm ess} \sup u$.
For the conclusion of the theorem, we can equivalently
replace the function $u$ by $u^\star$ on the right
hand side of \eqref{eq:main} and in the definition of $C$.
Indeed, if $u\in W^{1,1}_{loc}$ and $S$ is the set
where the singular part of the
distribution function is concentrated, then
$C\supset u^{-1}(S)$ in general, and equality holds
if $u$ is an extremal.
If $C$ has finite measure,
there is a simpler estimate in terms of
its volume radius.
\begin{Theorem}\label{finite}
Under the assumptions of Theorem~\ref{main}, if
$\lambda_n(C)<\infty$ then there exists a translation $\tau$ such that
\begin{equation}\label{eq:finite}
|| u-u^\star\circ\tau ||_p\le
||\nabla u||_p\cdot \left(\frac{\lambda_n(C)}{\omega_n}\right)^{\frac1n}.
\end{equation}
\end{Theorem}
For $1<p<n$, a natural choice for $q$ in Theorem~\ref{main}
is the Sobolev exponent
$p^*=np/(n\!-\!p)$, for which the
right hand side of \eqref{eq:main} is bounded by the
Sobolev inequality. Interpolating
with \eqref{eq:finite} yields $L^q$-bounds
for $p<q<p^*$, provided that $C$ has finite measure.
For $p>n$, there is a corresponding bound in $L^\infty$.
\begin{Theorem}\label{Morrey}
Under the assumptions of Theorem \ref{main}, if $p>n$
and $\lambda_n(C)<\infty$ then
there exists a translation $\tau$ such that
$$
||u-u^\star\circ \tau||_\infty \le
M_{n,p} \,
\| \nabla u\|_p\cdot
\left(\frac{\lambda_n(C)}{\omega_n}\right)^{\frac1n-\frac1p}\,,
$$
where $M_{n,p}$ is the Morrey constant.
\end{Theorem}
\smallskip
We briefly describe the relation
with the literature.
Theorem~\ref{main} contains the result of Brothers and Ziemer,
because the right hand side of \eqref{eq:main}
vanishes when the distribution function of $u$ is absolutely continuous.
Similarly, Theorem~\ref{finite} contains the bound of Cianchi and
Fusco. To see this, apply
H\"older's inequality on the left hand side of
\eqref{eq:finite} and use that
$C\subset {\rm supp}\, u$ on the right to obtain \eqref{eq:CF} with
$L_n=2^{1/p'}\omega_n^{-1/n}$.
We will show below that the proof of Theorem~\ref{main}
also implies~\eqref{eq:CF}.
However, Theorems~\ref{finite} and~\ref{Morrey} do
not seem to follow directly from Theorem~\ref{main}.
{
\medskip\noindent{\em Acknowledgments.}
Research for this paper was supported in part
by an NSERC Discovery Grant and a GNAMPA Project.
}
\section{Outline of the proof}
Brothers and Ziemer characterized extremals as follows.
If $u$ satisfies the assumptions of Theorem~\ref{main},
then its level sets are balls,
\begin{equation}\label{eq:balls}
\{u>t\}= \xi_t + \{u^\star>t\}
\end{equation}
(up to sets of Lebesgue measure zero). Furthermore,
the gradient is equidistributed on level sets,
\begin{equation}\label{eq:grad}
|\nabla u(x)|_{\rfloor \partial \{u>t\}}
= |\nabla u^\star |_{\rfloor \partial \{u^\star>t\}}
\end{equation}
for $\mathcal H^{n-1}$-almost every $x\in\partial \{u>t\}$
and almost every $t\in (0, {\rm ess} \sup u)$.
This equidistribution property is a consequence of the strict
convexity of the function $t\to t^p$. All later work
on the problem relies on this characterization.
A more delicate issue is to prove that the level
sets are {\em concentric} balls
if the distribution function of $u$ is absolutely
continuous. Brothers and Ziemer, starting from
\eqref{eq:balls}, express $u^\star$ in terms of $u$ as
$u^\star=u\circ T$ and study the regularity of the transformation
$T$ under the assumption of the {\em continuity} of the
distribution function. The crucial point is the
evaluation of $\nabla u^\star$, which
requires a non-standard chain rule
because $u$ is just a Sobolev function.
The {\em absolute
continuity} of the distribution function is needed to
deduce that $T$ is a translation of the identity map
from \eqref{eq:grad} and the fact that
$\nabla u^\star (x)=(DT(x))^t \,\nabla u (T(x))$.
In the last ten years, several new proofs of these
results have appeared.
In~\cite{FV1}, the authors reverse the approach of Brothers
and Ziemer and express
$u$ in terms of $u^\star$ as $u(x)= u^\star (T(x))$.
That leads to an easier case
of the chain rule, because $u^\star$ is essentially
a function of a single variable.
Finally, the conclusion is obtained by a gradient-flow argument.
Since this last part of the proof relies
on the uniform convexity and smoothness properties
of the Euclidean norm, the authors later
developed yet another geometric argument to treat
rearrangements with respect to arbitrary
norms in ${\mathbb R}^n$~\cite{FV2}.
Their method was subsequently used by
Cianchi and Fusco in~\cite{CiFu2}.
The argument in ~\cite{FV2} proceeds as follows.
Let $\xi_t$ denote the center of the ball $\{u>t\}$,
and let $R(x)$ be the function that assigns
to each point $x\in{\mathbb R}^n$ the radius
of the ball $\{u>u(x)\}$. The key
observation is that for all $s,t\in (0, {\rm ess} \sup u)$
there exists a pair of points
$x\in\partial\{u>s\}$ and $y\in\partial\{u>t\}$
such that
\begin{equation} \label{eq:xi-Phi}
|\xi_s-\xi_t| =
|R(x)-R(y)| -|x-y|\,,
\end{equation}
see Fig.~\ref{fig:key}.
If the distribution function of $u$ is absolutely
continuous, then
$R$ is Lipschitz continuous and
$|\nabla R|\equiv 1$.
It follows that $\xi_t$ is constant, proving that
$u$ is a translate of $u^\star$.
\begin{figure} [htbp]
\centering
\includegraphics{KEY.pdf}
\caption{\small Two circles ordered by inclusion.
The difference between the radii can be expressed
as the sum of their distance $|x-y|$ and the distance
of the centers $|\xi_s-\xi_t|$, see Eq.~\eqref{eq:xi-Phi}.}
\label{fig:key}
\end{figure}
If the distribution function of $u$ is not absolutely
continuous, then $R$ is of bounded variation.
In~\cite{CiFu2}, the
distribution function of $u$ is approximated by
an absolutely continuous function. Instead, we
approximate $u$ by functions whose distribution functions
have jumps, but no singular continuous part. In Section~\ref{sec:balls},
we analyze the variation of $R$
for such functions and derive the bound
\begin{equation}
\label{eq:var-xi}
|\xi_s-\xi_t|\le \left( \frac{1}{\omega_n}\mu^s ((s,t])\right)^{\frac1n}\,,
\end{equation}
where $\mu^s$ is the singular part of the measure
associated with the distribution function of $u$.
It follows that the total variation of $\xi$ is
bounded by the volume radius of $C$,
\begin{equation}
\label{eq:TV-xi}
||D\xi|| \le \left(\frac{\lambda_n(C)}{\omega_n}\right)^{\frac1n}\,.
\end{equation}
In Section~\ref{sec:main}, we show that this
implies the main results.
The final Section~\ref{sec:Dirichlet}
is dedicated to convex Dirichlet functionals.
Before turning to the technical part,
observe that the characterization
of extremals given by Brothers and Ziemer
does not depend on the value of $p$.
If $u$ is an extremal for {\em some} $1<p< \infty$,
then \eqref{eq:balls}-\eqref{eq:grad} imply by
the coarea formula that $u$ produces equality
in \eqref{PS} for {\em every} $1\le p\le \infty$.
In fact, much more is true. According
to~\cite[Theorem 1.7]{CiFe}, there is a wide class of
functionals satisfying a suitable strict monotonicity condition
that have the same family of extremals, which all satisfy
the conclusions of Theorem~\ref{main}-\ref{Morrey}
(as well as those of Corollaries~\ref{cor:main}-\ref{cor:finite}
with $V=\emptyset$).
\section{Notation and preliminary results}
\label{sec:def}
We work on ${\mathbb R}^n$, equipped with the
standard Euclidean norm $|\cdot|$ and
Lebesgue measure $\lambda_n$.
Let $u$ be a nonnegative measurable function on~${\mathbb R}^n$.
We always assume that $u$ {\it vanishes at infinity}, in the
sense that its level sets $\{x\in{\mathbb R}^n: u(x)>t\}$ have finite
measure for all $t>0$. Its {\it distribution function}
is given by
\begin{equation*}
F(t)=\lambda_n (\{x\in{\mathbb R}^n : |u(x)|>t\})\,, \qquad t>0\,.
\end{equation*}
The {\it symmetric decreasing rearrangement} of $u$ is defined by
\begin{equation}
\label{eq:def-ustar}
u^\star (x)=\sup\{t>0: F(t)>\omega_n|x|^n\}\qquad x\in{\mathbb R}^n\,;
\end{equation}
it is the unique radially decreasing
function that is equimeasurable to $u$ and lower semicontinuous.
The following construction removes
a collection of horizontal slices from the graphs of $u$ and $u^\star$
(see Fig.~\ref{fig:approx}).
Given a finite or countable union of intervals
$I\subset {\mathbb R}_+$,
set
\begin{equation}
\label{eq:slice}
f(t)=\lambda_1([0,t]\setminus I)\,.
\end{equation}
Then $f\circ u$ vanishes at infinity, and
$(f\circ u)^\star = f \circ u^\star$. If $u\in \mathop{W^{1,1}_{\rm loc}}$,
then $f\circ u\in W^{1,1}_{\rm loc}$, and
$$
\nabla (f\circ u)(x) = \mathcal X_{\{u(x)\not \in I\}} \nabla u(x)
$$
almost everywhere on ${\mathbb R}^n$ (see~\cite[Corollary 6.18]{LL}).
\begin{figure}[htbp]
\centering%
\includegraphics{BefAft.pdf}
\caption{\small Removing a horizontal slice from the
graph of $u$, see Eq.~\eqref{eq:slice}.
The right panel shows
$f\circ u$ with $I=[0,a)\cup(b,c)\cup (d,\infty)$.
}
\label{fig:approx}
\end{figure}
Many useful quantities can be expressed in terms of distribution
functions. For any absolutely continuous
function $\Psi$ on ${\mathbb R}_+$ with $\Psi(0)=0$,
there is the {\em layer-cake representation}
$$
\int \Psi(u)\, dx = \int_0^\infty F(t)\,\Psi'(t)\, dt\,.
$$
We note for later use that
\begin{equation}
\label{eq:L1-dist}
||u-v||_1= \int_0^\infty \lambda_n(\{u>t\} \bigtriangleup
\{v>t\})\, dt\,
\end{equation}
for any pair of nonnegative integrable functions
(here {$\bigtriangleup$}
stands for the symmetric difference of sets).
The following two lemmas provide similar formulas
for other convex functions of $|u-v|$.
\begin{lemma}
\label{lem:Psi-1}
Let $\Psi$ be a convex function on ${\mathbb R}_+$
with $\Psi(0)=\Psi'(0)=0$, let $\nu$ be the measure
that represents its second distributional derivative,
and let $u,v$ be nonnegative measurable functions. Then
\begin{align}
\notag \int \Psi(|u-v|)\, dx
& =\int_0^\infty\!\!\int_0^t
\bigl[
\lambda_n(\{u>t\}\setminus \{v>t\!-\!s\})\\
\label{eq:Psi-dist-1}
& \qquad\qquad +\lambda_n(\{v>t\}\setminus\{u>t-s\})\bigr]
\, d\nu(s) dt\,.
\end{align}
\end{lemma}
\begin{proof} We use that
\begin{align*}
\Psi(b-a) = \int_a^b \!\!\int_0^{t-a}\, d\nu(s)dt
= (\nu\!\times\!\lambda_1)
\bigl(\{(s,t): a+s<t<b\}\bigr)\,
\end{align*}
for $b>a>0$, and split the integral according to the sign of $u-v$,
$$
\int \Psi(|u-v|) \, dx = \int_{\{v>u\}} \Psi(v-u)\, dx
+\int_{\{u>v\}} \Psi(u-v)\, dx \,.
$$
For the first integral on the right, Fubini's theorem
gives
\begin{align*}
\int_{\{v>u\}} \Psi(v-u)\, dx
&= (\nu\!\times\! \lambda_1\!\times\!\lambda_n)
\bigl(\{
(s,t,x): u(x)+s <t<v(x)\}\bigr)\\
&=\int_0^\infty\!\!\int_0^t \lambda_n\bigl(\{v>t\}\setminus \{u>t-s\}\bigr)\,
d\nu(s)dt\,.
\end{align*}
Treating the second integral in the same way, we
arrive at the claimed identity.
\end{proof}
\begin{lemma}
\label{lem:Psi-2}
Let $\Psi$ be a convex function on ${\mathbb R}_+$
with $\Psi(0)=0$,
and let $u,v$ be nonnegative measurable functions. Then
$$
\int \Psi(|u-v|)\, dx
\le \int_0^\infty \lambda_n(\{u>t\} \bigtriangleup
\{v>t\})\, \Psi'(t)\, dt\,.
$$
\end{lemma}
\begin{proof} Since the claim holds for {\em linear}
functions $\Psi$ by \eqref{eq:L1-dist}, we may assume, by
replacing $\Psi(t)$ with $\Psi(t)-t\Psi'(0_+)$ that
$\Psi'(0)=0$. Then we can apply Lemma~\ref{lem:Psi-1}.
The right hand side of \eqref{eq:Psi-dist-1} increases if
we set $s=0$ in the integrand. We then evaluate
the inner integral, using that $\nu(0,t)=\Psi'(t)$ for almost every $t$.
\end{proof}
\smallskip
Since the distribution function of $u$ is monotonically decreasing, it
defines a Borel measure on ${\mathbb R}_+$ by
\begin{equation}\label{mon}
\mu ((a,b])=F(a)-F(b) =
\lambda_n\bigl(u^{-1}(a,b]\bigr)\,.
\end{equation}
If $u$ is essentially bounded, we restrict this
measure to the interval $(0, {\rm ess} \sup u)$, neglecting
plateaus at $t=0$ and $t= {\rm ess} \sup u$.
Consider the Lebesgue-Radon-Nikodym decomposition
$\mu=\mu^{ac}+\mu^s$, where $\mu^{ac}\ll \lambda_1$
and $\mu^s\perp \lambda_1$. This gives rise to a decomposition
of the distribution function
$F=F^{ac}+F^s$, where $F^s(t)=\mu^s(t, {\rm ess} \sup u)$
and $F^{ac}$ is absolutely continuous.
By the Fundamental Theorem of Calculus, the density of
$\mu^{ac}$ is given by
the classical derivative
\begin{equation}\label{eq:F-prime}
-F'(t)=\frac{{\rm Per}\,\left (\{u^\star >t\}\right )}
{|\nabla u^\star |_{\rfloor \partial \{u^\star>t\}}}\,,
\end{equation}
and the derivative of $F^{s}$ vanishes almost everywhere.
The singular part of the measure is given by
$$
\mu^s((a,b]) = F^s(a)-F^s(b)=\lambda_n\bigl(\{x\in (u^\star)^{-1}\left ((a,b]\right ):
\nabla u^\star(x)=0\}\bigr)\,,
$$
it is supported on the set of singular values
$$
S = \bigl\{t\in (0, {\rm ess} \sup u):
F\ \mbox{is not differentiable at}\ t\bigr\}\,.
$$
Since $u^\star$ is a monotone function of the single radial variable,
the set $S$ has measure zero.
Clearly $\mu$ is absolutely continuous if and only if $(u^\star)^{-1}(S )$
has measure zero.
Since the gradient of $u$ vanishes almost everywhere on
$u^{-1}(S)$~\cite[Theorem 6.19]{LL},
its follows that $C\supset u^{-1}(S)$ up
to a set of measure zero, and
\begin{equation}\label{eq:coarea}
\mu^s \left ( (a,b]\right ) \le
\lambda_n\bigl(\{x\in u^{-1}\left ((a,b]\right ):
\nabla u (x)=0\}\bigr)\,.
\end{equation}
{Note} that strict inequality can occur: when equality holds
in \eqref{eq:coarea} $u$ is said to be {\em{coarea regular}}
\cite{AlLi}. The next result shows that every extremal
is coarea regular (see also \cite{B, CiFu1}), and allows
us to interpret $F^s(t)$ as the distribution
function of the restriction of $u$ to $C$.
\begin{lemma} \label{lem:C-S}
Under the assumptions of Theorem~\ref{main},
$C = u^{-1}(S)$ up to a set of measure zero.
Furthermore,
$$
F^s(t) =
\lambda_n(\{x\in C: u(x)>t\}) \,.
$$
\end{lemma}
\begin{proof}
Since $C\supset u^{-1}(S)$, we have only to prove the reverse inclusion.
The coarea formula and the characterization
of extremals in \eqref{eq:balls} and \eqref{eq:grad}
show that
\begin{align*}
\lambda_n(\{x\not\inC: u(x)>t\})
&= \int_t^\infty \int_{\partial\{u>t\}}
|\nabla u|^{-1} d\mathcal H^{n-1} dt\\
&=\int_t^\infty |\nabla u^\star|^{-1} {\rm Per}\,(\{u^\star>t\})\, dt\\
&= \lambda_n(\{x: \nabla u^\star(x)\ne 0, u^\star(x)>t\})\,.
\end{align*}
Since $u$ and $u^\star$ are equimeasurable, it follows that
\begin{align*}
\lambda_n(\{x\in C: u(x)>t\})
&= \lambda_n(\{x: \nabla u^\star(x)=0, u^\star(x)>t\})\\
&= F^s(t)\,.
\end{align*}
\vskip -1\baselineskip
\end{proof}
\smallskip
In general, $\mu^s$ can be further decomposed
into a sum of (at most) countably many point masses
that correspond to plateaus of $u$,
and a singular continuous component.
However, one can always
approximate $u$ by functions
whose distribution function has no singular continuous part.
\begin{lemma} (Approximation)
\label{lem:approx}
Let $u\in \mathop{W^{1,1}_{\rm loc}}$ be a nonnegative function
that vanishes at infinity.
There exists an increasing sequence of functions
$u_m\in\mathop{W^{1,1}_{\rm loc}}$ and a decreasing sequence
of sets $C_m\subset {\mathbb R}^n$ with
$$
\lim u_m=u\,, \qquad \bigcap C_m=u^{-1}(S)\,,
$$
such that $u_m$ is bounded and supported on a set of
finite measure, each level set $\{u_m>t\}$ is also a
level set of $u$, the distribution function
of $u_m$ has no singular continuous part, and
$$
\nabla u_m = \mathcal X_{C_m}\nabla u\,,\qquad (m\ge 1)\,.
$$
\end{lemma}
\begin{proof}
Since $\mu^s$ is a regular Borel measure,
there exists a decreasing sequence of open sets $S_m$
containing $S$ such that
$$
\lim \mu(S_m\cap (t, {\rm ess} \sup u)) =\mu^s(t, {\rm ess} \sup u)
$$
for all $t>0$. If $u$ is unbounded or the support of $u$ does
not have finite measure, we ask
that $S_m\supset [0,1/m)\cup (m,\infty)$.
Set $f_m(t) = \lambda_1([0,t]\setminusS_m)$, let $u_m=f_m\circ u$,
and let $C_m=u^{-1}(S_m)$.
Then $u_m$ is bounded and supported on a set of finite measure.
Moreover, since $S_m$ is open, it is the union
of (at most) countably many disjoint intervals.
Therefore
$$
\nabla u_m(x) = \mathcal X_{u(x)\not\in S_m}\nabla u(x)\,
$$
for almost every $x$.
By construction, $u_m$ increases
monotonically to $u$, and $|\nabla u_m|$ increases
to $|\nabla u|$. For each connected component $(a,b)$ of $S_m$,
the distribution function
of $u_m$ has a jump of size $F(a)-F(b)$, corresponding
to a plateau of $u_m^\star$. Since $S_m\supset S$,
the distribution function of $u_m$ has no singular
continuous component.
\end{proof}
\smallskip
We will also consider functions on ${\mathbb R}^n$
that do not lie in $\mathop{W^{1,1}_{\rm loc}}$ but in the larger space
$\mathop{BV_{\rm loc}}$. A function
$u$ is locally of bounded variation, if its
distributional derivative is represented by
a vector-valued Radon measure, $[Du]$.
We denote by $|Du|$ the corresponding variation measure,
and by $||Du||=|Du|({\mathbb R}^n)$ its total variation.
The variation measure has Lebesgue-Radon-Nikodym representation
$[Du] = [D^{ac}u] + [D^su]$, where the
absolutely continuous component has density $\nabla u$, and $[D^su]$
is supported on a set of Lebesgue measure zero.
We will always use the {\em precise representative}
of $u$ that agrees
with its Lebesgue density limit at every point where
it exists.
For more information about $\mathop{BV_{\rm loc}}$, we refer the reader to
to~\cite{EG,AFP}.
If $u$ is a nonnegative function in $\mathop{BV_{\rm loc}}$ that vanishes at
infinity, then $u^\star\in \mathop{BV_{\rm loc}}$. Since $u^\star$ is a monotone
function of the radius,
its singular continuous component
is supported on $(u^\star)^{-1}(\{t\in (0, {\rm ess} \sup u):F'(t)=0\})$.
\section{Properties of extremals}
\label{sec:balls}
Throughout this section,
we assume that $u$ is an extremal for the P\'olya-Szeg\H{o}
inequality~\eqref{PS}. The goal is to prove
the bounds on the variation of $\xi$ in \eqref{eq:var-xi}
and \eqref{eq:TV-xi}.
Let $R$ be the function that assigns to each
point $x\in{\mathbb R}^n$ the radius of the level set of $u$
at height $u(x)$,
\begin{equation*}
R(x)=\left(\frac{1}{\omega_n} F(u(x))\right)^{\frac{1}{n}}.
\end{equation*}
Since $u^\star$ is a radial function, we can write
$u=u^\star\circ T$, where $T(x)=R(x) \cdot x/|x|$,
as in~\cite{FV1,FV2}.
The next two lemmas provide a bound on $|R(x)-R(y)|$.
\begin{lemma} \label{lem:grad-Phi}
Under the assumptions of Theorem~\ref{main},
if the support of $u$ has finite measure then
$R$ is of bounded variation.
The absolutely continuous part of its variation
has density $\nabla R$, where
\begin{equation}\label{eq:grad-Phi}
|\nabla R|(x)=\begin{cases}
1 & \mbox{if}\ u(x)\not \in S
\>\mbox{and}\ 0<u(x)< {\rm ess} \sup u\,,\\
0&\mbox{otherwise}\,, \end{cases}
\end{equation}
for almost every $x$.
\end{lemma}
\begin{proof} The total variation of $R$, {given by}
$$
||DR||=\int_0^\infty {\rm Per}\,(\{R< t\})\, dt\,,
$$
is finite, because its value is bounded
by the radius of the support of $u$
and its (sub-) level sets $\{R<t\}$
are smaller balls.
The distributional derivative of $R$ is represented by
the vector-valued measure
$[DR]$. For its absolutely continuous component,
the chain rule yields on $u^{-1}((0, {\rm ess} \sup u)\setminus S)$
\begin{equation*}
\nabla R (x) = \frac{F'(u(x))}{n\omega_n^{1/n}F(u(x))^{1/n'}}
\, \nabla u(x) =
-\frac{ \nabla u(x)}{|\nabla u^\star |_{\rfloor \{u^\star=u(x)\}}}\,,
\end{equation*}
see \cite[Eqs.(3.8)-(3.10)]{FV2}.
Here, $F'(t)$ is the classical derivative of $F$, and we have used
\eqref{eq:F-prime} in the second step.
Since $u$ is an extremal, we see from \eqref{eq:grad}
that the denominator agrees with
$|\nabla u(x)|$, and therefore $|\nabla R|=1$
almost everywhere on $u^{-1}\bigl((0, {\rm ess} \sup u)\setminusS\bigr)$.
Since $\lambda_1(S)=0$,
the gradient vanishes almost everywhere on $u^{-1}(S)$.
\end{proof}
\smallskip
\begin{lemma}\label{lem:lip} Under the
assumptions of Theorem~\ref{main},
\begin{equation}\label{eq:lip}
|R(x)-R(y)|\le |x-y|+
\left(\frac{1}{\omega_n}
\bigl(\mu^s((u(x), u(y)]\right)^{\frac1n}
\end{equation} for almost every $x,y$
with $u(x)<u(y)$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:approx},
it suffices to consider functions $u$
whose support has finite measure and whose
distribution function has no
singular continuous component. Let
$$
r(\theta )=R(\theta x+(1-\theta)y)\,,\qquad 0\le \theta\le 1
$$
be the restriction
of $R$ to the line segment that
joins $y$ with $x$. Since
we choose for $R$ its precise representative
in $\mathop{BV_{\rm loc}}$, the restriction is of bounded variation
and the chain rule holds for almost every choice of
$x,y$ and almost every $\theta$~\cite[Theorem 3.107]{AFP}.
By Lemma~\ref{lem:grad-Phi}, we have
$$
|r'(\theta)| = |\langle \nabla R( \theta x+(1-\theta)y), x-y \rangle|
\le 1\,,
$$
and obtain for the absolutely continuous part
$$
r^{ac}(1)-r^{ac}(0)\le |x-y|\,.
$$
\begin{figure}[htbp]
\centering%
\includegraphics{twicenone.pdf}
\caption{\small The line segment from
$y$ to $x$
crosses the boundary of a higher level set either twice
({\em left}) or never ({\em right)}.}
\label{fig:line}
\end{figure}
For the singular part, recall that $u(x)<u(y)$,
and thus $R(x)>R(y)$.
The line segment enters each level set $\{u>t\}$
with $t\in (u(x),u(y)]$ exactly once;
the boundary of a level set outside this range is
crossed either twice, in opposite directions,
or not at all, see Fig.~\ref{fig:line}.
When the line segment
enters $\{u>t\}$
for some $t\in S$, then $R$ experiences
a positive jump of size
$\omega_n^{-1/n}\bigl(F(t_-)^{1/n} -F(t)^{1/n}\bigr)$;
the jump is reversed upon exit. Since $S$ is countable,
this yields
\begin{align*}
r^s(1)-r^s(0)
&= \omega_n^{-\frac1n}\sum_{t\in S\cap (u(x),u(y)]}
\bigl(F(t_-)^{\frac1n}- F(t)^{\frac{1}{n}}\bigr)\\
&= \omega_n^{-\frac1n}
\sum_{t\inS\cap(u(x),u(y)]}\int_{F(t_-)}^{F(t)}
\frac{1}{n} s^{\frac{1}{n}-1}\, ds\\
&\le
\biggl(\frac{1}{\omega_n}\sum_{t\in S\cap (u(x),u(y)]}
\bigl(F(t_-)-F(t)\bigr)\biggr)^{\frac1n}\,.
\end{align*}
We have used that the intervals $(F(t_-), F(t))$ are disjoint
and that the function $s\mapsto s^{1-1/n}$
is decreasing to move the domain
of integration to the origin. By definition, the last sum equals
$\mu^s\bigl((u(x),u(y)]\bigr)$.
The claim follows by adding the inequalities
for $r^{ac}$ and $r^s$.
\end{proof}
\smallskip
\begin{proof} [Proof of \eqref{eq:var-xi}-\eqref{eq:TV-xi}]\
Insert Lemma~\ref{lem:lip} into \eqref{eq:xi-Phi}
to obtain the bound on $|\xi_s-\xi_t|$. The bound
on the total variation follows by maximizing over
$s,t$ and using Lemma~\ref{lem:C-S}.
\end{proof}
\smallskip
We have used
Lemmas~\ref{lem:grad-Phi} and \ref{lem:lip}
to show that the total variation of $\xi$ is bounded by
$(||D^sF||/\omega_n)^{1/n}$.
The proof of Lemma~\ref{lem:lip} yields
the somewhat stronger statement that
\begin{equation}
||D\xi||\le
\Bigl\|D^s\left(\frac{F}{\omega_n}\right)^{1/n}\Bigr\|\,.
\end{equation}
A similar computation as in Lemma~\ref{lem:lip}
shows that the singular part of the variation of $R$ is
bounded by the measure of the set
of critical points,
\begin{equation}
||D^sR||\le \mu^s((0, {\rm ess} \sup u)) + \inf_{t< {\rm ess} \sup u} F(t)\,.
\end{equation}
Here, the last term on the right
represents the possible plateau at the top, which does
not contribute to the variation of $\xi$.
\section{Proof of the main results}
\label{sec:main}
\begin{proof}[Proof of Theorem \ref{main}]
Since $u$ is an extremal, each level set $\{u>t\}$ is a ball
centered at $\xi_t$. Let $\xi_\infty$ be the center of the ball
$$
\bigcap_{t\in (0, {\rm ess} \sup u)}\,\{u>t\}
$$
(which may consist of a single point),
and consider the translation $\tau(x)=x-\xi_\infty$.
By \eqref{eq:xi-Phi} and Lemma~\ref{lem:lip},
the distance between the level sets $\{u>t\}$
and $\{u^\star\circ\tau>t\}$ is bounded by
$$
|\xi_t-\xi_\infty|\le \left(\frac{F^s(t)}{\omega_n}\right)^{\frac1n}\,.
$$
Since the symmetric difference between
two balls of equal radius in ${\mathbb R}^n$ satisfies
\begin{equation}
\label{eq:symmdiff}
\lambda_n((\xi+B) \bigtriangleup (\eta +B) )\le
2\omega_{n-1}\left(\frac{\lambda_n(B)}{\omega_n}\right)^{\frac{1}{n'}}
\cdot |\xi-\eta| \,,
\end{equation}
it follows that
$$
\lambda_n(\{u>t\}\bigtriangleup \{u^\star\circ\tau>t\})
\le K_n \,F(t)^{\frac{1}{n'}}\,F^s(t)^{\frac1n}\,,
$$
where {$K_n=2\omega_{n-1}/\omega_n$}.
From Lemma~\ref{lem:Psi-2} with $\Psi(t)=t^q$, we deduce
\begin{align}\label{key<}
\notag
||u-u^\star \circ \tau||_q^q
&\le K_n\int_0^\infty F (t)^{\frac{1}{n'}}\,
F^s(t) ^{\frac1n}\, qt^{q-1}dt\\
&\le K_n \left (\int_0^\infty F(t) \,qt^{q-1}dt\right )^{\frac{1}{n'}}
\left (\int_0^\infty F^s(t)\,qt^{q-1}\, dt\right )^{\frac1n}\\
\notag &= K_n\, ||u||_q^{q/n'} \, \|u\mathcal X_{C}\|_q^{q/n}\,.
\end{align}
We have applied
H\"older's inequality with exponents $1/n'$ and $1/n$,
and used Lemma~\ref{lem:C-S} to interpret
$F^s$ as the distribution function of $u\mathcal X_C$.
\end{proof}
\smallskip
The basic estimate in
the first line of \eqref{key<} can be used to derive
other bounds on $u-u^\star\circ\tau$, for example
\begin{equation}
\frac{||u-u^\star \circ \tau||^q_q}{\| u\|^q_q}\le
K_n\,
\sup_{0<t< {\rm ess} \sup u}\left (\frac{\lambda_n (C\cap \{u>t\})}{\lambda_n
(\{u >t\})}\right )^{\frac 1 {n}}\,.
\end{equation}
The ratio on the right hand side can be viewed
as the density of $C$ in the level set $\{u>t\}$.
It is always strictly less than one, because $C$ does not contain
the possible plateau at $ {\rm ess} \sup u$.
Alternately, we can interpret $n\omega_n^{1/n}F(t)^{1/n'}$ as the
perimeter of the level set $\{u>t\}$ and apply the coarea formula
to \eqref{key<} to obtain
$$
||u-u^\star \circ \tau||_q^q \le K_n \int F^s(u)^{\frac 1 n} |\nabla u^q|\, dx\,;
$$
if $C$ has finite measure, this implies \eqref{eq:finite}
with $p=1$.
\begin{proof}[Proof of Theorem~\ref{finite}]
Let $\xi_\infty$ and $\tau$ be as
in the proof of Theorem~\ref{main}, and consider
Lemma~\ref{lem:Psi-1} with $\Psi(t)=t^p$.
Since the intersection between any pair of balls
decreases with the distance of their centers,
it follows that
$$
||u-u^\star\circ\tau||_p^p
\le ||u^\star -u^\star\circ \tilde \tau||_p^p
\le ||\nabla u^\star||_p\cdot ||D\xi||
\,,
$$
where $\tilde \tau(x)=x- ||D\xi|| \, w$ for some unit
vector $w$. The bound on the total variation of $\xi$
in \eqref{eq:TV-xi} yields the claim.
\end{proof}
\smallskip
\begin{proof}[Proof of Theorem~\ref{Morrey}]
Let $\xi_\infty$ and $\tau$ be
as in the proof of Theorem~\ref{main}.
Since $u(x)=u^\star(x-\xi_{u(x)})$ and
$(u^\star\circ\tau)(x)=u^\star(x-\xi_\infty)$,
Morrey's inequality says that
$$
|u(x)-(u^\star\circ \tau)(x)|
\le M_{n,p} \, ||\nabla u||_p \cdot
|\xi_{u(x)}-\xi_\infty|^{1-\frac{n}{p}}\,.
$$
The claim follows with \eqref{eq:TV-xi}.
\end{proof}
\section{Dirichlet-type functionals on $\mathop{BV_{\rm loc}}$}
\label{sec:Dirichlet}
At last, we turn to more general convex gradient functionals.
A {\em Young function} is a nonnegative, nondecreasing
convex function $\Phi$ on ${\mathbb R}_+$ with $\Phi(0)=0$.
The {\em Dirichlet functional} associated with this
Young function is defined by
$$
{\mathscr F}(u) = \int \Phi(|\nabla u|)\, dx\,,
$$
provided that the distributional gradient of $u$ is
locally integrable.
If $\Phi$ grows linearly at infinity, the functional
is extended to $\mathop{BV_{\rm loc}}$ by
$$
{\mathscr F}(u) = \int \Phi(|\nabla u|)\, dx + \phi \, ||D^s u||\,,
$$
where $\phi =\lim_{t \to\infty} \Phi(t)/t$,
and $||D^s u||$ is the singular part of the total
variation. Then
${\mathscr F}(u)$ is always well-defined but it may
take the value $+\infty$.
In this setting, the
P\'olya-Szeg\H{o} inequality says that
\begin{equation}
\label{eq:PS-D}
{\mathscr F}(u^\star)\le {\mathscr F}(u)\,
\end{equation}
for all $u\in\mathop{BV_{\rm loc}}$~\cite{CiFu1}.
\begin{corollary} \label{cor:main} Let ${\mathscr F}$ be a
Dirichlet functional on ${\mathbb R}^n$ given by a strictly increasing
Young function $\Phi$. Let $u\in\mathop{BV_{\rm loc}}$
be a nonnegative function that vanishes
at infinity, and let $u^\star$ be its symmetric decreasing
rearrangement. If
$$
{\mathscr F}(u)={\mathscr F}(u^\star)<\infty\,,
$$
then there exists a translation $\tau$ such that
$$
\int \Psi(|u-u^\star\circ \tau|)\, dx
\le K_n\,\left(\int \Psi(u)\, dx\right)^{1/n'}
\left(\int_{C_\Phi} \Psi(u)\, dx\right)^{1/n}\,
$$
for every Young function $\Psi$ such that $\Psi\circ u$ is integrable.
Here,
$$
C_\Phi =\left
\{x\in{\mathbb R}^n: 0< u(x)< {\rm ess} \sup u, |\nabla u(x)|\in \{0\}\cup V\right\}\,,
$$
and $V$ is the maximal open subset of ${\mathbb R}_+$ such that
$\Phi$ is affine on each connected component of $V$.
The constant is given by $K_n=2\omega_{n-1}/ \omega_n$.
\end{corollary}
Note that the conclusion depends on the Young function
$\Phi$ only through the set $V$; in particular,
{
if $\Phi$ is {\em strictly} convex then
$V=\emptyset$ and $C_\Phi=C$.}
\begin{proof} [Proof of Corollary~\ref{cor:main}]
Cianchi and Fusco established in~\cite{CiFu2} that
\eqref{eq:balls} holds under the given assumptions,
i.e., the level sets of extremals are balls.
Moreover, \eqref{eq:grad} holds for a.e. $t\in(0, {\rm ess} \sup u)$
such that $|\nabla u^\star|_{\rfloor \partial \{u^\star>t\}}\not \in V$,
i.e., $|\nabla u|$ is equidistributed on $\partial\{u>t\}$
for $\mathcal H^{n-1}$-almost every $x\in\partial \{u>t\}$.
Let
$$
S_\Phi= S\cup \{t>0: F'(t)=0\} \cup \{t>0:
|\nabla u^\star|_{\rfloor \partial \{u^\star>t\}}\in V\}\,,
$$
and set $f(t)=\lambda_1((0,t)\setminusS_\Phi)$. Then
$f\circ u^\star$ is absolutely continuous,
its distribution function has no singular continuous
component,
and both $f\circ u^\star$ and $u^\star-f\circ u^\star$
are radially decreasing functions
that vanish at infinity. By definition,
$C_\Phi= u^{-1}(S_\Phi)$. Since
$$
\nabla (f\circ u) = \mathcal X_{C_\Phi} \nabla u\,,
$$
the set of critical points of $f\circ u$ is given by $C_\Phi$.
Furthermore,
$$
{\mathscr F}(u)={\mathscr F}(f\circ u) + {\mathscr F}(u\!-\!f\circ u)\,,
$$
and correspondingly for $u^\star$.
The P\'olya-Szeg\H{o} inequality holds for each
summand, and therefore
$f\circ u$ and $u-f\circ u$ must be extremals.
In particular, $f\circ u$ satisfies \eqref{eq:var-xi}.
Since every level set $\{u>t\}$
with $t\not\in S_\Phi$ is also a level set
of $f\circ u$, it follows that
\begin{equation}
\label{eq:var-xi-D}
|\xi_t-\xi_\infty |\le
\left(\frac{F^s_\Phi(t)}{\omega_n} \right)^{\frac1n}\,,
\end{equation}
where
$$
F^s_\Phi(t) = \lambda_n(\{x\inC_\Phi: u(x)>t\})\,.
$$
By Lemma~\ref{lem:Psi-2} and H\"older's inequality,
\begin{align*}
\int \Psi(|u-u^\star\circ\tau|)\, dx &\le
K_n \int_0^\infty F(t)^{\frac{1}{n'}} \,
F^s_\Phi(t)^{\frac1n}\,
\Psi'(t)\, dt\\
&\hskip -1.5cm
\le K_n \!\left(\int_0^\infty F(t) \Psi'(t)\, dt\right)^{\frac{1}{n'}}\!
\left(\int_0^\infty F^s_\Phi(t)\,\Psi'(t)\, dt\right)^{\frac1n}.
\end{align*}
Using the layer-cake principle, we recognize the
integrals {in the last line} as
$\int \Psi(u)\, dx$ and $\int_{C_\Phi} \Psi(u)\, dx$.
As in Theorems~\ref{main}-\ref{Morrey}, we
can equivalently replace $u$ with $u^\star$ in
these integrals and in the definition of $C_\Phi$.
\end{proof}
\smallskip
If $u$ is supported on a set of
finite measure, then
we can use \eqref{eq:Psi-2} with $\Psi(t)=t$
and apply Jensen's inequality once more to conclude that
\begin{align*}
||u-u^\star\circ\tau||_1 &
\le ||\nabla u||_1 \cdot \left(\frac{\lambda_n(C_\Phi)}
{\omega_n}\right)^{\frac1n}\\
&\le
\Phi^{-1} \left(\frac{{\mathscr F}(u)}{\lambda_n( {\rm supp}\, u)}\right)
\cdot \lambda_n( {\rm supp}\, u)
\left(\frac{\lambda_n(C_\Phi)}{\omega_n} \right)^{\frac1n}\,.
\end{align*}
Since $C_\Phi\subset {\rm supp}\, u$, this
implies~\cite[Theorem 1.1]{CiFu2}.
For $\Phi(t)=t^p$, we recover \eqref{eq:CF} with $L_n=\omega_n^{-1/n}$.
\begin{corollary}
\label{cor:finite}
Under the assumptions of
Corollary~\ref{cor:main}, if $C_\Phi$ has finite measure,
then there exists a translation $\tau$ such that
$$
\int \Phi\left(|u-u^\star \circ \tau|\cdot \left(\frac{\lambda_n(C_\Phi)}
{\omega_n}\right)^{-\frac1n}\right)\, dx \le {\mathscr F}(u)\,.
$$
\end{corollary}
\begin{proof} Let $\xi_\infty$ and
$\tau$ be as in the proof of Theorem~\ref{main}.
We will show that
\begin{equation}
\label{eq:Psi-2}
\int\Psi(|u-u^\star\circ\tau|)
\, dx
\le \int \Psi\left(|\nabla u|\cdot ||D\xi||\right)\, dx
\end{equation}
for every Young function $\Psi$ such
that the right hand side is finite.
We then set $\Psi(t)=\Phi(t /||D\xi||)$, and use that
$$
||D\xi|| \le \left(\frac{\lambda_n(C_\Phi)}{\omega_n}\right)^{\frac1n}
$$
by \eqref{eq:var-xi-D}.
For \eqref{eq:Psi-2}, we combine \eqref{eq:L1-dist} with Lemma~\ref{lem:Psi-1}
and argue as in the proof of Theorem~\ref{finite}
that the integral on the left hand side increases if
$u$ is replaced by $u^\star\circ\tilde\tau$,
where $\tilde \tau(x)=x-||D\xi||\,w$ for
some unit vector $w$. Since
$$
u^\star(x)-u^\star\circ\tilde\tau(x)
= \int_0^1
\langle
\nabla u^\star(x+\theta \, ||D\xi||\,w), ||D\xi||\, w\rangle\, d\theta\,,
$$
Jensen's inequality implies that
\begin{align*}
\int \Psi(|u^\star-u^\star\circ\tilde \tau|)\, dx
&\le \int \Psi\left(
\int_0^1\bigl|\nabla u^\star(x+\theta \, ||D\xi||\, w)\bigr|\, d\theta
\cdot ||D\xi||
\!\right) dx\\
& \le \int \Psi\left(|\nabla u^\star|\cdot||D\xi||\right)\, dx \\
&\le \int \Psi\left(|\nabla u|\cdot ||D\xi||\right)\, dx\,.
\end{align*}
The last step holds by the P\'olya-Szeg\H{o} inequality in
\eqref{eq:PS-D}.
\end{proof}
|
2,877,628,090,219 | arxiv | \section{Introduction}
Several years ago, I wrote some lectures\cite{Donoghue:1995cz} on the application of effective field theory (EFT)
to general relativity. Because those lectures were given at an advanced school on effective field theory, most
of the students were well versed about the effective field theory side, and the point was to show how general
relativity matches standard effective field theory practice. In contrast, the present lectures were delivered at
a school where the students primarily knew general relativity, but knew less about effective field theory. So
the written version here will focus more on the effective field theory side, and can be viewed as complementary
to previous lectures. There will be some repetition and updating of portions of the previous manuscript, but the
two can potentially be used together\footnote{The reader can also consult my original paper on the subject\cite{Donoghue:1994dn}
or the review by Cliff Burgess\cite{Burgess:2003jk}, and of course Steven
Weinberg's thoughts on effective field theory\cite{Weinberg:2009bg} are always interesting.}.
There has been a great deal of good work combining general relativity and field theory. In the
early days the focus was on the high energy behavior of the theory, especially on the divergences.
However, this is the portion of gravity theory that we are most unsure about - the conventional
expectation is that gravity needs to be modified beyond the Planck scale. What effective field
theory does is to shift the focus to low energy where we reliably know the content of general
relativity. It provides a well-defined framework for organizing quantum calculations and
understanding which aspects are reliable and which are not. In this sense, it may provide the
maximum content of quantum general relativity until/unless we are able to solve the mystery of Planck scale physics.
Effective field theory has added something important to the understanding of quantum gravity.
One can find thousands of statements in the literature to the effect that ``general relativity and
quantum mechanics are incompatible''. These are completely outdated and no longer relevant. Effective
field theory shows that general relativity and quantum mechanics work together perfectly normally
over a range of scales and curvatures, including those relevant for the world that we see around us.
However, effective field theories are only valid over some range of scales. General relativity
certainly does have problematic issues at extreme scales. There are important problems which
the effective field theory does not solve because they are beyond its range of validity. However,
this means that the issue of quantum gravity is not what we thought it to be. Rather than a fundamental
incompatibility of quantum mechanics and gravity, we are in the more familiar situation of needing a
more complete theory beyond the range of their combined applicability. The usual marriage
of general relativity and quantum mechanics is fine at ordinary energies, but we now seek to uncover
the modifications that must be present in more extreme conditions. This is the modern view of the
problem of quantum gravity, and it represents progress over the outdated view of the past.
\section{Effective field theory in general}
Let us start by asking how {\em any} quantum mechanical calculation can be reliable.
Quantum perturbation theory instructs us to sum over {\em all} intermediate states of {\em all} energies.
However, because physics is an experimental science, we do not know all the states that exist at high
energy and we do not know what the interactions are there either. So why doesn't this lack of knowledge
get in the way of us making good predictions? It is not because energy denominators cut off the high
energy portion - in fact phase space favors high energy. It is not because the
transition matrix elements are small. In fact, one can argue that all processes are sensitive to
the highest energy at some order in perturbation theory.
The answer is related to the uncertainty principle. The effects of the highest energies correspond
to such short distances that they are effectively local when viewed at low energy. As such they are
identical to some local term in the Lagrangian that we use to define our low energy theory. These terms
in the Lagrangian come with coupling constants or masses as coefficients, and then the effect of the highest
energy goes into measured value of these parameters.
This is a well-known story for divergences, but is true for finite effects also,
and would be important for quantum predictions even if the ultimate theory had no divergences.
In contrast, low energy effects of massless (or very light) particles are not local. Examples include
the photon propagator or two photon exchange potentials. Intermediate states that go on-shell clearly
propagate long distances. Sometimes loop diagrams of light particles can have both short and long distance
contributions within the same diagram. We will see that we can separate these, because the short distant
part looks like a local effect and we can catalog all the local effects. The long distance portions of
processes can be separated from the short distance physics.
Effective field theory is then the procedure for describing the long-distance physics of the light
particles that are active at low energy. It can be applied usefully in both cases where we know the
full theory or when we do not. In both situations the effects of heavy particles and interactions
(known or unknown) are described by a local effective Lagrangian, and we treat the light particles
with a full field theoretic treatment, including loops, renormalization, etc.
\section{Constructing an explicit effective field theory - the sigma model}
Perhaps the best way to understand the procedures of effective field theory is to make an explicit construction
of one from a renormalizable theory\footnote{The treatment of this section follows that of
our book\cite{Donoghue:1992dd}, to which the reader is referred for more information.}. This can be done
using the linear sigma model. Moreover the resulting effective field theory has a form with many similarities to general relativity.
Let us start with the bosonic sector of the linear sigma model\footnote{The full linear sigma
model also includes coupling to a fermion doublet, but that aspect is not needed for our purpose.}
\begin{eqnarray}
{\cal L} &=& \frac12
\partial_\mu \vec{\pi}\cdot \partial^\mu \vec{\pi} + \frac12
\partial_\mu\sigma\partial^\mu \sigma \nonumber \\
&+& {1\over 2} \mu^2 \left( \sigma^2 + {\vec{\pi}}^2\right) -
{\lambda\over 4} \left( \sigma^2 + \vec{\pi}^2\right)^2 \ \ .
\end{eqnarray}
with four fields $\sigma,~ \pi_1,~\pi_2,~\pi_3 $ and an obvious invariance of rotations
among these fields. This will be referred to as "the full theory". The invariance can be
described by $SU(2)_l\times SU(2)_R$ symmetry, with the transformation being
\begin{equation}
\sigma + i \vec{\tau} \cdot \vec{\pi} \to V_L \left(\sigma + i \vec{\tau }\cdot \vec{\pi} \right)V^\dagger_R
\end{equation}
where $V_{L,R}$ are $2\times 2$ elements of $SU(2)_{L,R}$, and $\tau^i$ are the Pauli $SU(2)$ matrices.
This theory is the well-known example of spontaneous symmetry breaking with the ground state
being obtained for $<\sigma >=v =\sqrt{\mu^2/\lambda}$. The $\sigma$ field picks up a
mass $m_\sigma^2 =2\mu^2$, and the $\vec{ \pi}$ fields are massless Goldstone bosons.
After the shift $\sigma =v + \tilde{\sigma}$, the theory becomes
\begin{eqnarray}
{\cal L} &=& \frac12
\partial_\mu \vec{\pi}\cdot \partial^\mu \vec{\pi} + \frac12\left[
\partial_\mu\tilde{\sigma}\partial^\mu \tilde{\sigma} +m_\sigma^2 \tilde{\sigma}^2\right]\nonumber \\
&-& \lambda v\tilde{ \sigma}(\tilde{\sigma}^2 + \vec{\pi}^2 )-
{\lambda\over 4} \left( \tilde{\sigma}^2 + \vec{\pi}^2\right)^2 \ \
\label{shifted}
\end{eqnarray}
The only dimensional parameter in this theory is $v$ or equivalently $\mu= \sqrt{\lambda} v$ or $m_\sigma =\sqrt{2}\mu$,
all of which carry the dimension of a mass. For readers concerned with gravity, you should think of
this as ``the Planck mass'' of the theory. Consider it to be very large, well beyond any energy that
one can reach with using just the massless $\vec{\pi}$ fields.
At low energy, well below ``the Planck mass'', all that you will see are the pions and you
will seek a Lagrangian describing their interactions without involving the $\sigma$. This is not
obvious looking at Eq. \ref{shifted}, but can be done. That result is
\begin{equation}
{\cal L}_{\rm eff} = {v^2\over 4}
{\rm Tr} \left(\partial_\mu U\partial^\mu U^\dagger\right)
\label{Leff}
\end{equation}
where
\begin{equation}
U = \exp [ \frac{i\vec{\tau}\cdot \vec{\pi}}{v}] \ \ .
\label{U}
\end{equation}
This of course looks very different. It is a ``non-renormalizable'' Lagrangian where the
exponential generates non-linear interactions to all orders of the pion field. It also
comes with two derivatives acting on the fields, so that all matrix elements are proportional
to the energy or momentum of the fields squared. In contrast, the original sigma model has
non-derivative interactions such as the $\lambda {\mathbf \pi}^4$ coupling. As we will see
in more detail, these features of the effective theory are shared with general relativity.
\subsection{Example of the equivalence}
Before I describe the derivation of the effective theory, let us look at an explicit calculation
in order to see the nature of the approximation involved. If we calculate a process
such as $\pi^+\pi^0\to \pi^+\pi^0$ scattering, in the full theory one finds both a
direct pion coupling and also a diagram that involves $\sigma $ exchange which arises
from the trilinear coupling in Eq. \ref{shifted}, as shown in Fig. 1.
\begin{figure}
$\begin{array} {c@{\hspace{0.01 in}}c} \multicolumn{1}{l}{} &
\multicolumn{1}{l}{} \\
{\resizebox{2.5in}{!}{\includegraphics{Tree1EFT2.eps}}}&
~~~~~~~~{\resizebox{2.5in}{!}{\includegraphics{Tree2EFT2.eps}}} \\
\end{array}$
\caption{Diagrams for $\pi -\pi$ scattering}
\end{figure}
This results in a cancelation such that the matrix element is
\begin{eqnarray}
i{\cal M}_{\pi^+ \pi^0\to \pi^+\pi^0} = - 2i\lambda
+ \left( -
2i\lambda v\right)^2 {i\over q^2 - m^2_\sigma}
= - 2i\lambda\left[ 1 + {2\lambda v^2\over q^2 - 2\lambda v^2}\right] \\
= +i\left[\frac{q^2}{v^2} + \frac{q^4}{v^2m_\sigma^2}+ \ldots\right] \ \ .
\label{full tree}
\end{eqnarray}
Here $q^2 =(p_+-p_+')^2 =t$ is the momentum transfer\footnote{For future use, the reader is reminded of the
Mandelstam variables $s= (p_++p_0)^2,~t=(p_+-p_+')^2=q^2,~ u=(p_0-p_+')^2$.} and I remind the reader that $m_\sigma^2 =2\lambda v^2$.
Despite the full theory
having only polynomial interactions without derivatives, the result ends up proportional to $q^2$. Here
we also see the nature of the energy expansion. The correction to the leading term is
suppressed by two powers of the sigma mass (i.e. ``the Planck mass'').
In the effective theory one proceeds by expanding the exponential in Eq. \ref{Leff} to order $\pi^4$ and
taking the matrix element. Because the Lagrangian has two derivatives, the result will automatically be of order $q^2$. We find
\begin{equation}
i{\cal M}_{\pi^+ \pi^0\to \pi^+\pi^0} = {iq^2\over v^2}
\end{equation}
which of course is exactly the same as the first term in the scattering amplitude of the
full theory. Moreover, exploration of other matrix elements will show that {\em all} amplitudes
will agree for the full theory and the effective theory at this order in the energy expansion.
This is a very non-trivial fact. We will see how one obtains the next order term (and more) soon.
\subsection{Construction of the effective theory}
Lets build the effective theory starting with the original linear sigma model.
Instead of the redefinition of the field $\sigma = v +\tilde{\sigma}$, let us consider the renaming
\begin{equation}
\sigma + i \vec{\tau} \cdot \vec{\pi} = (v+ \sigma') \exp [ \frac{i\vec{\tau}\cdot\vec{\pi}'}{v}] = v +\sigma' +i\vec{\tau} \cdot \vec{\pi}' +\ldots
\end{equation}
This change of fields from $(\sigma, \vec{\pi})$ to $(\sigma', \vec{\pi'})$ is simply a renaming of the fields
which by general principles of field theory will not change the physical amplitudes. Using the new
fields (although quickly dropping the primes for notational convenience) the full original linear sigma model can be rewritten without
any approximation as
\begin{equation}
{\cal L}_{\rm eff} = \frac14 (v+\sigma)^2 {\rm Tr} \left(\partial_\mu U\partial^\mu U^\dagger\right) + {\cal L}(\sigma)
\label{alternate}
\end{equation}
where $U$ is the exponential of the pion fields of Eq. \ref{U} and
\begin{equation}
{\cal L}(\sigma) = \frac12\left[
\partial_\mu{\sigma}\partial^\mu {\sigma} +m_\sigma^2 {\sigma}^2\right]
- \lambda v \sigma^3 -\frac{\lambda}{4} \sigma^4 \ \ .
\end{equation}
This alternate form also describes a heavy field $\sigma$ with a set of self interactions and
with couplings to the massless pions. The symmetry of the original
theory $\sigma + i \vec{\tau} \cdot \vec{\pi} \to V_L \left(\sigma + i \vec{\tau} \cdot \vec{\pi} \right)V^\dagger_R$ is still
manifest as $U \to V_L U V^\dagger_R,~~\sigma ' \to \sigma '$.
This form {\em looks} non-renormalizable, but is not - it is really just the same theory
as the original form.
To transition to the effective field theory, we note that the $\sigma$ lives at "the Planck scale"
and therefore is inaccessible at low energy. Only virtual effects of the $\sigma$ can have any influence
on low energy physics. By inspection we see that the coupling of the heavy particle to the pions involves
two derivatives (i.e. powers of the energy), so that exchange of the $\sigma$ will involve at least four
powers of the energy. (We will treat loops soon.) So it appears fairly obvious that the leading effect - accurate to two derivative order -
is simply to neglect the exchange of the $\sigma$, directly leading to the effective Lagrangian of Eq. \ref{Leff}.
To confirm and extend this, it is useful to calculate explicitly the effect of $\sigma$ exchange.
The effect of this is pictured schematically in Fig 2, where the $\times$ represents
the "current" $\sim v {\rm Tr} \left(\partial_\mu U\partial^\mu U^\dagger\right) /2$. This diagram
yields the modification to the effective Lagrangian
\begin{equation}
{\cal L}_{\rm eff} = {v^2\over 4}
{\rm Tr} \left(\partial_\mu U\partial^\mu U^\dagger\right)+ \frac{v^2}{8 m_\sigma^2}[{\rm Tr} \left(\partial_\mu U\partial^\mu U^\dagger\right)]^2
\label{L4}
\end{equation}
As expected, the modification involves four derivatives. Moreover, if we take the
matrix element for the scattering amplitude calculated in the previous section,
we recover exactly the $q^4$ term found in the full theory, Eq. \ref{full tree}. Using the effective Lagrangian framework
we are able to match, order by order in the energy expansion, the results of the full theory.
\begin{figure}
\includegraphics[height=.2\textheight]{IntoutEFT2.eps}
\caption{The tree-level effect of the exchange of a heavy scalar, $\sigma$. The $\times$ represents a vertex involving pions, as described in the text.}
\end{figure}
\subsection{Loops and renormalization}
The real power of effective field theory comes when we treat it as a full field theory - including loop diagrams -
rather than just a set of effective Lagrangians. The idea is that we "integrate out" the heavy field, leaving a
result depending on the light fields only. In path integral notation this means
\begin{eqnarray}
{\cal Z } &=& \int [d \pi][d\sigma ] e^{i \int d^4x {\cal L}(\sigma , \pi) } \nonumber \\
&=& \int [d{\pi}] e^{i \int d^4x {\cal L}_{eff}( \pi) }
\end{eqnarray}
In practice, it means that we treat loop diagrams involving the $\sigma$ as well as tree diagrams.
But in loops also, the heavy particle does not propagate far, so
even the effects of heavy particle in loops can be represented by local Lagrangians.
The key to determining the low energy effective field theory is to {\em match} the predictions of
the full theory to that of the effective theory. To be sure, the effective field theory has an
incorrect high energy behavior. This can be seen even within specific diagrams. For example consider the diagram of
Fig. 3a which occurs within the full theory. This diagram is finite, although pretty complicated. When treated as an
effective theory we approximate the sigma propagator as a constant, shrinking the effect of sigma exchange to a local
vertex. This leads to the diagram of Fig. 3b, which is clearly divergent\footnote{The Taylor expansion of the propagator
will lead to increasingly divergent terms also.}. So new divergences will occur in the effective field theory
treatment which are not present in the full theory.
\begin{figure}
$\begin{array} {c@{\hspace{0.01 in}}c} \multicolumn{1}{l}{} &
\multicolumn{1}{l}{} \\
{\resizebox{2.5in}{!}{\includegraphics{boxEFT2.eps}}}&
~~~~~~~~{\resizebox{2.5in}{!}{\includegraphics{pinchedboxEFT2.eps}}} \\
\end{array}$
\caption{a) A finite box diagram which occurs in the full theory, b) A bubble diagram which occurs
in the effective theory in the situation when the
propagator of the heavy $\sigma$ has been shrunken to a point. }
\end{figure}
However, the low energy behavior of Fig. 3a and Fig 3b will be similar. When the loop momentum is
very small, the approximation of sigma exchange as a local vertex will be appropriate, and the pions will propagate long
distances. This will be manifest in non-analytic behaviors such as $\ln (-s)$ dependence at low energy\footnote{The $-i\pi$ which follows
from $\ln (-s)= \ln s -i\pi$ for $s>0$ accounts for the discontinuity from on-shell intermediate states. }. If the reader
has not yet struggled with the complexity of box diagrams, it would be a good exercise to look up the
box diagram for Fig 3a\cite{Denner:1991qq} and verify that the $\ln (-s)$ behavior is identical
(in the low energy limit) to that of the much simpler diagram Fig 3b.
In order to deal with the general divergences of the effective theory we need to
have the most general local Lagrangian. With the symmetry of the theory being $U\to V_LUV^\dagger_R$ the
general form with up to four derivatives is
\begin{equation}
{\cal L}_{eff} = \frac{M_P^2}{4} ~{\rm Tr}[\partial_\mu U \partial^\mu U^\dagger] + {\cal \ell}_1\left({\rm Tr}[\partial_\mu U \partial^\mu
U^\dagger] \right)^2 +{\cal \ell}_2{\rm Tr}[\partial_\mu U \partial_\nu U^\dagger] {\rm Tr}[\partial^\mu U \partial^\nu U^\dagger]
\end{equation}
where ${\cal \ell}_i$ are constant coefficients. We saw at tree level that
\begin{equation}
{\cal \ell}_1 =\frac{v^2}{8m_\sigma^2}~,~~~~ {\cal \ell}_2 =0
\end{equation}
The divergences of the effective theory will be local and will go into the renormalization of these coefficients.
The comparison of the full theory and the effective theory can be carried out directly
for the reaction $\pi^+ \pi^0\to \pi^+\pi^0$ . The dimensionally
regularized result for the full theory is quite complicated - it is given in \cite{Manohar:2008tc}.
However, the expansion of the full theory at low energy in terms of renormalized parameters is relatively simple\cite{GL}
\begin{eqnarray}
{\cal M}_{full}&=& {t\over v^2} +\left[{1\over m_\sigma^2 v^2}-{11\over 96\pi^2v^4}\right]t^2 \cr
&-&{1\over 144\pi^2 v^4}[s(s-u) +u(u-s)] \cr
&-&{1\over 96\pi^2 v^4}\left[3t^2 \ln {-t\over m_\sigma^2}+s(s-u)\ln {-s\over m_\sigma^2}+u(u-s)\ln {-u\over m_\sigma^2}\right]
\end{eqnarray}
One calculates the same reaction in the effective theory, which clearly does not know about
the existence of the $\sigma$. The result \cite{GL, Lehmann:1972kv} has a very similar form,
\begin{eqnarray}
{\cal M}_{eff}&=& {t\over v^2} +\left[8\ell_1^r+2\ell_2^r+{5\over 192\pi^2}\right]{t^2\over v^4} \\
&+&\left[2\ell_2^r+{7\over 576\pi^2}\right][s(s-u) +u(u-s)]/v^4 \\
&-&{1\over 96\pi^2 v^4}\left[3t^2 \ln {-t\over \mu^2}+s(s-u)\ln {-s\over \mu^2}+u(u-s)\ln {-u\over \mu^2}\right]
\end{eqnarray}
where we have defined the renormalized parameters
\begin{eqnarray}
\ell_1^r &=&\ell_1 +{1\over 384\pi^2 }\left[{2\over 4-d} - \gamma + \ln 4\pi \right] \\
\ell_2^r &=& \ell_2 +{1\over 192\pi^2 }\left[{2\over 4-d} - \gamma + \ln 4\pi \right]
\end{eqnarray}
At this stage we can match the two theories, providing identical scattering amplitudes to this order, through the choice
\begin{eqnarray}
\ell_1^r &=& {v^2\over 8 m_\sigma^2} +{1\over 384\pi^2}\left[\ln {m_\sigma^2\over \mu^2} -{35\over 6}\right] \\
\ell_2^r &=& {1\over 192\pi^2 }\left[\ln {m_\sigma^2\over \mu^2} -{11\over 6}\right]
\end{eqnarray}
Loops have modified the result of tree level matching by a finite amount. We have not only obtained a more precise matching,
we also have generated important kinematic dependence, particularly the logarithms, in the scattering amplitude. The logarithms
do not involve the chiral coefficients $\ell_i$ because the logs follow from the long-distance portion of loops while the chiral
coefficients are explicitly short-distance. This is part of the evidence for a separation of long and short distances.
We have seen that we can renormalize a "nonrenormalizable" theory. The divergences are local and can be absorbed into
parameters of a local Lagrangian. Moreover, the predictions of the full theory can be reproduced even when
using only the light degrees of freedom, as long as one chooses the coefficients of the effective lagrangian
appropriately. This holds for {\it all} observables. This can be demonstrated using a
background field method \cite{GL, Donoghue:1992dd, GL2}. Once the matching is done, other processes can be
calculated using the effective theory without the need to match again for each process. The total effect of
the heavy particle, both tree diagrams
and loops, has been reduced to a few numbers in the Lagrangian, which we have deduced
from matching conditions to a given order in an expansion in the energy.
\subsection{Power counting and the energy expansion}
We saw that the result of the one loop calculation involved effects that carried two extra factors
of the momenta $q^2$. It is relatively easy to realize that this is a general result. The only dimensional
parameter in the effective theory is $v$, and one can count the powers of $v$ that enter into loop diagrams.
These enter in the denominator because the exponential is expanded in $\tau \cdot \pi/v$. In dimensional regularlization
there is no scale to provide any competing powers in the renormalization
procedure\footnote{There is an arbitrary scale $\mu$ which enters only logarithmically.}. Therefore the
external momenta $q^2, ~s, ~t, ~u$ must enter to compensate for the powers of $1/v^n$, leading to an expansion in $q^2/v^2$ or
more appropriately $q^2/(4\pi v)^2$.
This energy expansion is a second key feature of EFT techniques. Predictions are ordered in an
expansion of powers of the light energy scales over the high energy scale, ``the Planck scale'',
of the full theory. For the chiral theory, Weinberg provided a compact theorem showing how higher
loop diagrams always lead to higher energy dependence\cite{Weinberg:1978kz}. A similar power counting occurs in general relativity.
The power counting allows an efficient approach to loops in gravity also because general
relativity has a power counting behavior similar to the chiral case described here.
The coupling constant of gravity, Newton's constant $G$, carries the dimension of the
inverse Planck mass squared, $G\sim 1/M_P^2$. This means that loops carry extra powers of $G$
(just like the extra factors of $1/v^2$ of the chiral case) and this is compensated for by
factors of the low energy scale in the numerator, leading to an efficient energy expansion.
\subsection{Effective field theory in action}
Lest the above description sound somewhat formal, I should point out that the chiral effective
field theory is actively and widely used in phenomenological applications for QCD. Of course,
the linear sigma model is not the same as QCD. However, if the mass of the up and down quarks
were zero, QCD would also have an exact $SU(2)_L \times SU(2)_R$ symmetry which is dynamically
broken, with massless pions as the Goldstone bosons. The effective Lagrangian of the theory would
have the same symmetry as the sigma model and so the general structure of the effective Lagrangian
would be the same. The identification of the vector and axial currents allows the identification of $v$ with
the pion decay constant $v=F_\pi = 92.4$~MeV measured in the decay $\pi \to \mu \nu$. The chiral
coefficients $\ell_1,~\ell_2$ are relatively different from those of the sigma model. In QCD they are
difficult to calculate from first principles, but as measured by experiment they are of order $10^{-3}$ and
their relative sizes reveal the influence of the vector meson $\rho(770)$ rather than of a scalar $\sigma$\cite{Donoghue:1988ed}.
The chiral symmetry of QCD also has small explicit breaking from the up and down quark masses, leading to a non-zero pion mass.
This symmetry breaking can also be treated perturbatively as an element in the energy expansion. The result is chiral perturbation
theory, a lively interplay of energies, masses, loop diagrams, experimental measurements that makes for a fine demonstration of the
practical power of effective field theory. The reader is referred to our book \cite{Donoghue:1992dd} for a more complete
pedagogic development of the subject.
There also are very many more uses of effective field theory throughout physics. Some of these have slightly
different issues and techniques, and the above is not a complete introduction to all aspects of effective field
theory. However, all EFTs share the common features of 1) using only the active degrees of freedom and interactions
relevant for the energy that one is working at, and 2) organizing the results in some form of an energy expansion.
The example of the linear sigma model is particularly well suited for the introduction to the gravitational theory.
\subsection{Summary of general procedures}
In preparation for the discussion of the effective field theory for general relativity, let me summarize the
techniques for an EFT in the situation when we do not know the full theory. In contrast with the linear sigma
model, we then cannot match the EFT to the full theory. However EFT techniques can still be useful and predictive.
1) One starts by identifying the low energy particles and the symmetries governing their interaction.
2) This information allows one to construct the most general effective Lagrangian using only these particles
and consistent with the symmetry. This Lagrangian can be ordered in an energy expansion in terms of increasing dimensions of the
operators involved. In contrast with the stringent constraints of "renormalizable field theory", one allows operators with
dimension greater than four. Renormalizable theories can be thought of as a subclass of EFTs where the operators of
dimension greater than four are not (yet) needed.
3) The operator(s) of lowest dimension provide the leading interactions at low energy. One can quantize the
fields and identify propagators in the usual way.
4) These interactions can now be used in a full field theoretic treatment. Loop diagrams can be computed.
Because the divergences will be local they can be absorbed into the renormalization of terms in the local effective Lagrangian.
5) Because, by assumption, the full theory is not known, the coefficients of various terms in the effective
Lagrangian cannot be predicted. However, in principle they can be measured experimentally. In this case,
these coefficients are not predictions of the effective field theory treatment - they represent information
about the full theory. However, once measured they can be used in multiple processes.
6) Predictions can be made. The goal is to obtain predictions from long distance (low energy) physics. Because
we know that short distance physics is local, it is equivalent to the local terms in the Lagrangian, and
everything left over is predictive. This most visible involves nonanalytic terms in momentum space, such
as the logarithmic terms in the scattering amplitude above. These are obviously not from a local Lagrangian
and correspond to long distance propagation. The logs can also pick up imaginary parts when their arguments
are negative - the imaginary parts are indicative of on-shell intermediate states. Analytic terms,
such as the $s^2/16\pi^2 v^2$ terms in the scattering amplitude above, can also be predictive when
they can be separated from the Lagrangian parameters, such as $\ell_i$, through measurement of the parameters in other reactions.
\section{Gravity as an effective field theory}
In the case of gravity, we assume that there is {\em some} well defined full theory of gravity that yields
general relativity as the low energy limit. We do not need to know what it is, but we take as experimental
fact that it is well behaved, with no significant instabilities or run-away solutions when dealing with
low curvature situations such as our own. Other particles and interactions - the Standard Model - can be
added, but are not crucial to the EFT treatment. Basically, we assume that there is a limit of the full
theory that looks like the world that we see around us.
If the low energy limit is general relativity, we know that the relevant degrees of freedom are massless
gravitons, which are excitations of the metric. The symmetry is general coordinate invariance. In constructing
the energy expansion of the effective Lagrangian, one must pay attention to the number of derivatives. The connection
\begin{equation}
\Gamma^\lambda_{\alpha\beta} = \frac{g^{\lambda\sigma}}{2}\left[\partial_\alpha g_{\beta\sigma}+\partial_\beta g_{\alpha\sigma}
- \partial_\sigma g_{\alpha\beta}\right]
\end{equation}
has one derivative of the metric, while the curvatures such as
\begin{equation}
R_{\mu\nu} = \partial_\mu \Gamma^\lambda_{\nu\lambda}-\partial_\lambda\Gamma^\lambda_{\mu\nu} -\Gamma^\sigma_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
-\Gamma^\sigma_{\mu\nu}\Gamma^\lambda_{\lambda\sigma}
\end{equation}
have two. The various contractions of the Riemann tensor are coordinate invariant. These features
determine the nature of the energy expansion of the the action for general relativity
\begin{equation}\label{gravL}
S_{grav}=\int d^4x\, \sqrt{g}\, \left[\Lambda +{2\over{\kappa^2}}\, R+c_1\, R^2 +
c_2\, R_{\mu \nu} R^{\mu \nu} + \ldots + {\cal L}_{matter}\,\right]
\end{equation}
Here the terms have zero, two and four derivatives respectively.
Following our EFT script, we turn to experiment to determine the parameters of this
Lagrangian. The first term, the cosmological constant, appears to be non-zero but it
is so tiny that it is not relevant on ordinary scales. The EFT treatment does not say
anything novel about the smallness of the cosmological constant - it is treated simply
as an experimental fact. The next term is the Einstein action, with coefficient determined from Newton's constant $\kappa^2 =32\pi G$.
This is the usual starting place for a treatment of general relativity. The curvature squared terms
yield effects that are tiny on normal scales if the coefficients $c_{1,2}$ are of order unity. In
fact these are bounded by experiment \cite{Stelle:1977ry} to be less than $10^{+74}$ - the ridiculous
weakness of this constraint illustrates just how irrelevant these terms are for normal physics. So we
see that general coordinate invariance allows a simple energy expansion.
At times people worry that the presence of curvature squared terms in the action will lead to instabilities or pathological behavior.
Such potential problems have been shown to only occur at scales beyond the Planck scale \cite{Simon:1990ic} where yet
higher order terms are also equally important. This is not a flaw of the effective field theory, which holds only
below the Planck scale. Given the assumption of a well-behaved full theory of gravity, there is no aspect of the
effective theory that needs to display a pathology.
\subsection{Quantization and renormalization}
The quantization of general relativity is rather like that of Yang-Mills theory. There are subtle features connected
with the gauge invariance, so that only physical degrees of freedom count in loops. Feynman\cite{Feynman:1963ax}, and
then DeWitt\cite{DeWitt:1967ub}, did this successfully in the 1960's, introducing gauge fixing and then ghost fields to
cancel off the unphysical graviton states. The background field method employed by 'tHooft and Veltman\cite{'tHooft:1974bx}
was also a beautiful step forward. It allows the expansion about a background metric ($\bar{g}_{\mu\nu}$) and explicitly
preserves the symmetries of general relativity. It is then clear that quantization does not spoil general covariance and
that all quantum effects respect this symmetry. The fluctuation of the metric around the background is the graviton
\begin{equation}
g_{\mu\nu}(x) = \bar{g}_{\mu\nu}(x)+ \kappa h_{\mu\nu}(x)
\end{equation}
and the action can be expanded in powers of $h_{\mu\nu}(x)$ (with corresponding powers of $\kappa$). The Feynman rules
after gauge fixing and the addition of ghosts have been given in several places\cite{Donoghue:1995cz, Donoghue:1994dn, 'tHooft:1974bx}
and need not be repeated here. They are unremarkable aside from the complexity of the tensor indices involved.
Renormalization also proceeds straightforwardly. As advertised, the divergences are local, with the one loop effect being
equivalent to\cite{'tHooft:1974bx}
\begin{equation}
\Delta {\cal L} =\frac{1}{16\pi^2}\frac{2}{4-d} \left[\frac{1}{120}R^2 + \frac{7}{20} R_{\mu\nu}R^{\mu\nu}\right]
\end{equation}
The divergence can then be easily absorbed into renormalized values of the coefficients $c_{1,2}$. The fact
that these occur at the order of four derivatives can be seen by counting powers of $\kappa$ and is completely
equivalent to the energy expansion of the linear sigma model described above.
Pure gravity is one-loop finite. This is because the equations of motion for pure gravity (not including the
cosmological constant) is $R_{\mu\nu}=0$, so that both of the divergent counter terms vanish when treated as a
perturbation. This is an interesting and useful fact, although in the real world it does not imply any special
finiteness to the theory because in the presence of matter the counterterms are physically relevant.
\subsection{Predictions of the effective field theory}
Perhaps the most elementary prediction of quantum general relativity is the scattering of
two gravitons. This was worked out to one-loop by Dunbar and Norridge \cite{Dunbar:1994bn}. The form is
\begin{eqnarray}\label{eq:2}
{\cal A}(++;++) & = &{i\over4}\,{\kappa^2 s^3 \over t u}\left(1 + \frac{\kappa^2~s~t~u }
{4(4\pi)^{2-\epsilon}}\,
\frac{\Gamma^2(1-\epsilon)\Gamma(1+\epsilon)}
{\Gamma(1-2\epsilon)}\, ~\times
\right.\nonumber \\
&&\left.\hspace{-0em}\times\left[\rule{0pt}{4.5ex}\right.
\frac{2}{\epsilon}\left(
\frac{\ln(-u)}{st}\,+\,\frac{\ln(-t)}{su}\,+\,\frac{\ln(-s)}{tu}
\right)+\,\frac{1}{s^2}\,f\left(\frac{-t}{s},\frac{-u}{s}\right)\right.
\nonumber\\
&&\left.\hspace{1.4em}
+2\,\left(\frac{\ln(-u)\ln(-s)}{su}\,+\,\frac{\ln(-t)\ln(-s)}{tu}\,+\,
\frac{\ln(-t)\ln(-s)}{ts}\right)
\left.\rule{0pt}{4.5ex}\right]\right)\nonumber \\
{\cal A}(++;--) & = & -i\,{\kappa^4 \over 30720
\pi^2}
\left( s^2+t^2 + u^2 \right) \nonumber\\
{\cal A}(++;+-) & = & -{1 \over 3}
{\cal A}(++;--)
\end{eqnarray}
where
\begin{eqnarray}\label{eq:f}
f\left(\frac{-t}{s},\frac{-u}{s}\right)&=&
\frac{(t+2u)(2t+u)\left(2t^4+2t^3u-t^2u^2+2tu^3+2u^4\right)}
{s^6}
\left(\ln^2\frac{t}{u}+\pi^2\right)\nonumber\\&&
+\frac{(t-u)\left(341t^4+1609t^3u+2566t^2u^2+1609tu^3+
341u^4\right)}
{30s^5}\ln\frac{t}{u}\nonumber\\&&
+\frac{1922t^4+9143t^3u+14622t^2u^2+9143tu^3+1922u^4}
{180s^4},
\end{eqnarray}
and where the $+,~ -$ refer to the graviton helicities. Here ${\cal A}(++;++)$ shows the nature of the energy expansion for general
relativity most clearly. It is
the only amplitude with a tree level matrix element - the others all vanish at tree level. The tree amplitude is corrected by terms
at the next order in the energy expansion, i.e. by factors of order $\kappa^2 ({\rm Energy})^2$ relative to the leading contribution.
Note also the nonanalytic logarithms. Another
interesting feature, despite the presence of $1/\epsilon$ terms in the formulas, is that these results are finite without any
unknown parameters. Because the counterterms vanish in pure gravity, as noted above,
the scattering amplitudes cannot depend on the coefficients of the higher order terms.
The $1/\epsilon$ terms have been shown to be totally infrared in origin, and are canceled
as usual by the inclusion of gravitational bremsstrahlung \cite{Donoghue:1996mt}, as would be
expected from general principles\cite{Weinberg:1965nx}.
This result is a beautiful ``low energy theorem'' of quantum gravity. No matter what
the ultimate ultraviolet completion of the gravitational theory, the scattering process must have this form and
only this form, with no free parameters, as long as the full theory limits to general relativity at low energy.
The leading quantum correction to the gravitational potential is also a low energy theorem independent of
the ultimate high energy theory. The result for the potential of gravitational scattering of two heavy masses is\cite{potential, kk}
\begin{equation}
V(r) = -{GMm \over r} \left[ 1 + 3 {G(M+m)\over rc^2} + \frac{41}{10\pi}
{G\hbar\over r^2 c^3}\right] \ \ .
\label{potential}
\end{equation}
where the last term is the quantum correction and the term preceding that is the classical
post-Newtonian correction. Let me explain how this is calculated and why it is reliable.
We already know the form of the quantum correction from dimensional analysis, as the unique
dimensionless parameter linear in $\hbar$ and linear in $G$ is $G\hbar/r^2c^3$. The classical post-Newtonian correction
is also a well-known dimensionless combination, without $\hbar$. Fourier transforming tells us that the corresponding
results in momentum space are
\begin{equation}
\frac{1}{r} \sim \frac{1}{q^2} , ~~~~~~~\frac{1}{r^2} \sim \frac{1}{q^2} \times \sqrt{q^2},~~~~~~~\frac{1}{r^3}
\sim \frac{1}{q^2}\times q^2\ln q^2, ~~~~~~~ \delta^3(\mathbf{x}) \sim \frac{1}{q^2}\times q^2
\end{equation}
So here we see the nature of the energy expansion. The leading potential comes from one-graviton exchange with
the $1/q^2$ coming from the massless propagator. The corrections come from loop diagrams - all the one-loop
diagrams that can contribute to the scattering of two masses. The kinematic dependence of the loops then brings
in nonanalytic corrections of the form $Gm\sqrt{q^2}$, $Gq^2 \ln q^2$, as well as analytic terms $Gq^2$. As shown
above in the chiral scattering result, the analytic terms include the effects of the next order
Lagrangian (the $c_i$ coefficients from Eq. \ref{gravL}) and the divergences.
However, since $\frac{1}{q^2}\times q^2 \sim {\rm constant}$, the Fourier transforms of the
analytic term appears as a delta function in position space, and these terms do not lead to any
long-distance modifications of the potential. The power-law corrections come from the non-analytic
terms, with the logarithm generating the quantum correction. This standard reasoning of effective field
theory allows us to know in advance that the quantum correction will be finite and free from unknown parameters.
The actual calculation is pretty standard, although it is notationally complicated because of all the
tensor indices for graviton couplings. After a series of partial calculations, possible mistakes and
alternative definitions\cite{otherpotentials}, the result appears solid with two groups in
agreement\footnote{I also know of some unpublished confirmations of the same result.}\cite{potential,kk}. However,
the result itself is interesting mainly because of the understanding of why it is calculable. In this calculation
we see the compatibility of general relativity and quantum mechanics at low energy, and have separated off the unknown
high-energy portion of the theory. The magnitude of the correction is far too small to be seen - a correction of $10^{-40}$
at a distance of one fermi. There are no free parameters that we can adjust to change this fact. However, in some ways
this can be pitched as a positive. When we do perturbation theory, the calculations are the most reliable when the
corrections are small. The gravitational quantum correction is the smallest perturbative correction
of all our fundamental theories. So instead of
general relativity being the worst quantum theory as is normally advertised, perhaps it should be considered the best!
It is also worth commenting on the classical correction in Eq. \ref{potential}\footnote{The classical correction was previously known to be
calculable using field theory methods \cite{Iwasaki:1971vb, Gupta:1980zu}.}. This comes from a one-loop
calculation, which surprises some who know the supposed theorem that the loop expansion is an
expansion in $\hbar$. However, that theorem is not really a theorem, and this is one of several
counterexamples\cite{Holstein:2004dn}. The usual proof neglects to note that there also can be
a factor of $\hbar$ within the Lagrangian, as mass terms carry a factor of $m/\hbar$. This is
manifest within the calculation in the square-root nonanalytic term as an extra factor $\sqrt{m^2/k^2\hbar^2}$.
This is perhaps a good place to note that a powerful {\em classical} EFT treatment of gravity has been
developed by Goldberger, Rothstein and their collaborators\cite{Goldberger:2004jt}. This requires a further
development of EFT methods to separate out the relevant components within the graviton field itself. Their
work provides a systematic treatment of the classical bound state and gravitational radiation problems.
It is now a useful component of the gravity wave community, presently ahead of conventional methods in
the prediction of gravity waves of binaries with spin.
Another calculation that shows how EFT methods work is the quantum correction to the Reissner-Nordstom
and Kerr-Newman metrics (for charged objects without and with spin), which takes the form\cite{Donoghue:2001qc} in harmonic gauge
\begin{eqnarray}
g_{00}&=& 1-{2Gm\over r}+{G\alpha\over r^2}
- {8G\alpha\hbar \over 3\pi mr^3} +\ldots\nonumber\\
g_{0i}&=&({2G\over r^3}-{G\alpha\over
mr^4}+{2G\alpha\hbar\over \pi m^2r^5})(\vec{S}\times\vec{r})_i\nonumber\\
g_{ij}&=&-\delta_{ij}-\delta_{ij}{2Gm\over r}+G\alpha{r_ir_j\over
r^2}+ {4G\alpha\hbar\over 3\pi mr^3}\left({r_ir_j\over
r^2}-\delta_{ij}\right)+\ldots
\end{eqnarray}
Here the quantum correction comes from photon loops, and gravity is treated classically. But the techniques
are the same. One finds that the classical correction again comes from the square-root non-analytic term in
momentum space. Physically we can identify exactly what causes this correction. It comes from the electric
field which surrounds the charged object. The electric field is non-local, falling with $r^{-2}$ and it carries energy.
Gravity couples to this energy and precisely\cite{Donoghue:2001qc} reproduces the classical correction in the metric.
At tree-level in field theory one sees only the point charged particle, and
the photon loop diagram then is needed to describe the energy in
the electric field surrounding the charged object. The quantum correction again comes from the logarithm in momentum space.
A calculation of Hawking radiation that appears solidly in the spirit of effective field theory is by
Burgess and Hambli\cite{Hambli:1995pp}. They study a scalar propagator in a low curvature region outside
the black hole and regularize with a high energy cutoff. The flux from the black hole can calculated from
the propagator and appears insensitive to the cutoff. However, EFT cannot address questions about the end
state of black hole evaporation - this concerns a situation beyond the region of validity of the effective field theory.
\section{Issues in the gravitational effective field theory}
Effective field theories are expected to have limits. While it is logically conceivable that we could find some
innovative method that allows one to extend general relativity to all energies, this is generally viewed as
unlikely. More typically, effective field theories get modified by new particles and new interactions as one
goes to high energy. The archetypical example is QCD, where the pions of the effective theory get replaced by
the quarks and gluons of QCD at high energy. The most common expectation is that something similar happens for
general relativity. String theory would be a consistent example for general relativity, although there may be
other ways to form an ultraviolet completion of the gravitational interactions. It would be lovely to understand
the nature of this high energy modification. However, since physics is an experimental science and the Planck scale
appears to be frustratingly out of reach of experiment, it is not clear that we will have reliable evidence for the
nature of the ultimate theory in the foreseeable future. The competition between divergent proposals for the ultimate
theory, which in principle would be a scientific discussion if experiment could keep up, may remain unresolved in our
lifetimes without the input of experiment.
However, it does remain interesting to explore the limits of the effective theory. Potentially these
limits can help us understand when new physics
must enter. One obvious case is at the Planck energy. Scattering amplitudes are all proportional to powers of $GE^2$, with
\begin{equation}
{\cal M}= {\cal M}_0 [1 + a G E^2 +b G E^2 \ln E^2+.... ]
\end{equation}
This clearly leads to problems at the Planck scale, where the energy expansion breaks down.
However, there may also be other limits to the validity of the effective theory. I have argued
elsewhere\cite{Donoghue:1994dn, Donoghue:2009mn} that the extreme infrared of general relativity
may pose problems that other effective theories do not face. This is because gravitational
effects may build up - the integrated curvature may be large even in the local curvature is not.
Such effects are manifest as horizons and singularities. Horizons by themselves are not expected to
be problematic - in a local neighborhood nothing special need be evident. But the long distance
relations of horizons to spatial infinity clearly brings in problematic features for black holes.
Moreover, singularities pose problems for the effective theory in the long distance limit. The
singularity itself is not the problem - we know that the effective theory breaks down when the
curvature is large. But even for the long distance theory, propagating past a singularity poses
some difficulties. We do not know the fate of the modes that flow into the singularity. Perhaps
this can be solved by treating the location of the singularity as an external source. In chiral
perturbation theory, this is done for baryons, which arguably appear as solitons in the effective
theory which are found only beyond the limits of the effective theory, yet also serve as a source
for the Goldstone bosons. Perhaps singular regions can be treated as localized sources of gravitons.
However, even if singularities can be isolated and tamed, there are issues of the relation of
localized regions of small curvature, where the effective theory is demonstrably valid, to far
distant regions where the integrated curvature is large. This conflict is at the heart of black
hole paradoxes, where the horizon is well behave locally but is problematic when defined by its relation to spatial infinity.
\section{Gravitational corrections and running coupling constants}
Let me also address what effective field theory has to say about the gravitational corrections
to the running of coupling constants, including that of gravity itself. This subject has had
a confusing recent history, and EFT is useful in sorting out the issues\footnote{The comments of
this section continue to assume that the cosmological constant is small enough to be neglected.
In the presence of a cosmological constant the story is different and there can be genuine contributions to running couplings\cite{Toms:2009vd}}.
Despite having a pre-history\cite{Kiritsis:1994ta},
recent activity stems from the work of Robinson and Wilczek\cite{Robinson:2005fj}, who suggested that
the beta function of a gauge theory could have the form
\begin{equation}
\beta (e, E) = \frac{b_0}{(4\pi)^2}e^3 + a_0 e\kappa^2E^2
\end{equation}
and calculated $a_0$ to be negative. While this correction is tiny for most energies,
the negative sign suggests that all couplings could be asymptotically free if naively
extrapolated past the Planck scale. Subsequent work by several authors, all using
dimensional regularization, found that the gravitational correction to the running coupling
vanishes\cite{dimreg}. Further work, including some of the same authors, using variations of
cutoff ($\Lambda$) regularization then found that it does run\cite{cutoff}, where the cutoff
plays the role of the energy, i.e. $\beta \sim G\Lambda^2$. Papers trying to
clarify this muddle include\cite{ad1, ad2, Toms:2011zza, Ellis:2010rw}. My treatment
here most naturally follows the ones of my collaborators and myself\cite{ad1,ad2}
In renormalizable field theories, the use of a running coupling is both useful and universal.
It is useful because it sums up a set of quantum corrections which potentially could have large
logarithmic factors. It is universal because the logarithmic factors are tied to the renormalization
of the charge, and hence enter the same way in all processes. Specifically, when one employs
dimensional regularization the charge renormalization and the $\ln \mu^2$ factors always enter the same way, because of the expansion
of $(\mu^2)^\epsilon/\epsilon$. For example, in perturbation theory photon exchange at high energy involves
\begin{equation}
\frac{e^2}{q^2} =\frac{e_0^2}{q^2}\left[1+ \frac{\alpha}{3\pi^2}\left(\frac{2}{4-d} - \ln\frac{-q^2}{\mu^2}+...\right)\right]
\end{equation}
The $\ln \mu^2$ dependence follows the renormalization of the charge, and dimensionally the $\ln q^2$ dependence has to accompany
the
$\ln \mu^2 $ dependence. When this is turned into a running coupling, the energy dependence always appears in a universal
fashion because the charge is renormalized the same way in all processes.
In addition, with logarithmic running the running coupling constant is crossing symmetric
since $\ln q^2$ has the same value, up to an imaginary part due to on-shell intermediate states,
for $q^2$ either positive or negative. This lets the running coupling constant be applicable in
both the direct channel and in the crossed channel for a given type of reaction.
For gravitational corrections in the perturbative regime, effective field theory explains why each
of the features (useful, universal, crossing symmetric) no longer holds. Let me discuss them individually.
We have seen that gravitational corrections are expansions in $Gq^2$ with potentially extra
logarithms. Because in different reactions $q^2$ can take on either sign, the gravitational
correction will not have the same sign under crossing. For example if we calculated the
gravitational correction to a process such as $f+ \bar{f} \to f'+\bar{f}'$, for two different
flavors $f,~f'$, within QED, one would find a correction of the form
\begin{equation}
{\cal M} \sim \frac{e^2}{s}[1 + bGs]
\end{equation}
with $s=(p_1+p_2)^2>0$ and $b$ is a constant to be calculated. However, if one studies the
related crossed process $f +f' \to f+f'$ one would have
\begin{equation}
{\cal M} \sim \frac{e^2}{t}[1 + bGt]
\end{equation}
with $t=(p_1-p_3)^2<0$ and $b$ is the same constant. If one tried to absorb the quantum correction
into a running coupling, it would be an increasing function of energy in the one process and a
decreasing function of the energy in the other process. Moreover, other processes such as the single
flavor case $f+ \bar{f} \to f+\bar{f}$, would involve both $s$ and $t$ variables within the same reaction.
A universal energy dependent running coupling with quadratic four-momentum dependence cannot account for the features of crossing.
Effective field theory also explains why gravitational corrections are not a universal factor correcting
the coupling. We have seen in the direct calculations above and in the discussion of the energy expansion
that gravitational corrections do not renormalize the original operator, but generate divergences that are
two factors of the energy (derivatives) higher. This of course is due to the dimensional coupling constant.
For gravity itself, after starting with the Einstein action $R$ we found that divergences were proportional
to $R^2$ and $R_{\mu\nu}R^{\mu\nu}$. For the QED case, the higher order operators could be
\begin{equation}
\bar{\psi}\gamma_\mu\partial_\nu \psi \partial^\mu A^\nu,~~~~~~~~~~~~\bar{\psi}\gamma_\mu\psi \partial^2 A^\mu, ~~~~~~~~~~~~\partial^\mu F^{\lambda\nu}\partial_\lambda F_{\mu\nu}, ~~~~
~~~~~~~~\bar{\psi}\gamma_\mu\psi \bar{\psi}\gamma^\mu\psi ~~.
\end{equation}
These operators are related to each other by the equations of motion, but there are two points associated
with this observation. One is that, in contrast to renormalizable field theories, we are
not renormalizing the original coupling so that we do not have any expectation of universality. But in addition,
different processes involve different combinations of the higher order operators, so that the renormalization
that takes place is intrinsically different for different processes. Different reactions will involve
different factors. It is for this reason that use of the renormalization group in effective field theory,
which is in fact a well studied subject\cite{Weinberg:1978kz, rg}, does not involve the running of the basic coupling, but instead
is limited to predicting logarithmic factors associated with the higher order operators.
So we have seen that effective field theory, in the region where we have control over the calculation,
does not lead to gravitational corrections to the original running couplings. We might still ask if there might
nevertheless be some useful definition that plays such a role. This logically could occur if one
repackaged some of the effects of the higher order operators {\em as if} they were the original
operator - i.e. some sort of truncation of the operator basis. However, here one can simply calculate
gravitational corrections to various processes to see if some useful definition emerges. The answer is
decidedly negative\cite{ad2}. The numerical factors involved in different reactions vary by large factors and come with both signs.
In the perturbative regime there is no useful and universal definition of running $G(q^2)$ nor gravitational
corrections to running $e(q^2)$, and there is a kinematic crossing obstacle to any conceivable proposal.
Far too often in the literature, a dimensional cutoff ($\Lambda$) is employed to incorrectly conclude that
it contributes to a running coupling. For example, one can use cutoff and find that the original coupling
{\em is} renormalized. For example for gravity or QED
\begin{equation}
G = G_0[1+cG_0\Lambda^2], ~~~~~~~e^2= e_0^2[1+dG\Lambda^2]
\end{equation}
for some constants $c, ~d$. If one then says the magic phrase ``Wilsonian'', one might think that
this then defines a running coupling $G(\Lambda)$ or $e(\Lambda)$. However, this is not a running coupling -
it is just renormalization. The quadratic $\Lambda$ dependence disappears into the renormalized value
of $G$ or $e$ at low energy\cite{ad2, Toms:2011zza}. Once you measure this value there is no remaining
quadratic $\Lambda$ dependence, and there is no energy dependence that tracks the $G\Lambda^2$ behavior.
These observations pose questions for the sub-field of Asymptotic Safety\cite{Weinberg:1980gg, asymsafety}.
In this area, on considers a Euclidean coupling $g=Gk_E^2$, where $k_E$ is a Euclidean momenta, and attempts
to find a ultraviolet fixed point for $g$. Since the operator basis expands to operators of increasingly high
number of derivatives, this involves a truncation of the operator basis. Given such a Euclidean truncation,
one can by construction define a running coupling, and one does find a UV fixed point well beyond the Planck
scale. However, the question remains whether this implies anything useful for real processes in Lorentzian gravity.
One would think that one should find evidence of such a running coupling in the perturbative region where we have
control over the calculation. However, we have seen the reasons why this does not occur. The conflict certainly deserves further study.
\section{Summary}
Effective field theory is a well-developed framework for isolating the quantum effects of low energy particles and
interactions, even if these interactions by themselves fall outside of a complete renormalizable field theory. It
works well with general relativity applied over ordinary curvatures and energy scales. The effective field theory
has limits to its validity, most notably it is limited to scales below the Planck energy, and does not resolve all
of the issues of quantum gravity. However, effective field theory has shown that general relativity and quantum
mechanics do in fact go together fine at ordinary scales where both are valid. GR behaves like an ordinary field
theory over those scales. This is important progress. We still have work to do in order to understand
gravity and the other interactions at extreme scales.
\begin{theacknowledgments}
This material was presented at the Sixth International School on Field Theory and Gravitation, Petropolis Brazil, April 2012.
I thank the organizers and my fellow participants for an interesting and informative school. This work is supported in
part by the U.S. National Science Foundation grant PHY-0855119
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
2,877,628,090,220 | arxiv | \section{Introduction}
Applying a surveillance scheme to monitor the stability of dispersion (homogeneity, scale or other related notions)
is a common task used in industry to maintain, for example, the repeatability level of gauges,
the uniformity of certain entities over time or space, the risk level of some financial asset,
the stability of the variance underlying the control limits of a mean control chart and so forth.
To provide an explicit example, we look at a scanning electron microscope (SEM) at a semiconductor company, where
a battery of daily measurements is executed for the sake of repeatability monitoring.
Typically, well-defined features (lines, spaces and so on) on a wafer
are measured $n=5$ times, and the resulting
\begin{figure}[hbt]
\centering
\includegraphics[width=.6\textwidth]{s2_v1}
\caption{Shewhart $S$ chart for monitoring the short-term repeatability of a scanning electron microscope (SEM);
sample size $n=5$; ordinate scale $nm$; EWMA ($\lambda=0.2$, blue) added.}\label{fig:01}
\end{figure}
sample standard deviation is recorded on a Shewhart $S$ chart --- see Figure~\ref{fig:01}.
Hence, it is not surprising that Shewhart, cumulative sum (CUSUM) and
exponentially weighted moving average (EWMA)
variance control charts are presented in popular textbooks, including
\cite{Mont:2009}, pages 259, 414 and 426, respectively, and \cite{Qiu:2013}, pages 74, 146 and 198, respectively.
Here, we wish to investigate methods for calibrating EWMA schemes based on the sample variance
$S^2$ when the in-control (IC) level of the variance must be estimated based on a preliminary sample (phase I) of the IC data.
The EWMA control chart was introduced by \cite{Robe:1959} and gained much attention with and after \cite{Luca:Sacc:1990a}.
The initial works on using EWMA charts for dispersion monitoring include
\cite{Wort:Ring:1971}, \cite{Swee:1986}, \cite{Doma:Patc:1991},
\cite{Crow:Hami:1992a}, \cite{MacG:Harr:1993} and \cite{Chan:Gan:1994} --- see \cite{Knot:2005b, Knot:2010a} for more details.
With regard to EWMA charts,
\cite{Jone:Cham:Rigd:2001} started the analysis of using estimated parameters instead of merely assuming known ones.
It became, for control charts in general, an important topic in the statistical process control (SPC) literature over the course of the last 20 years, and
\cite{Jens:Jone:Cham:Wood:2006} and \cite{Psar:Vyni:Cast:2014} have provided detailed surveys on this subject.
Typically, it is assumed that the parameters are estimated through $m$ phase I samples, each of size $n$, which means that nearly all the performance measures used for control charts become uncertain.
For example, the well-known average run length (ARL; expected number of samples until the chart signals) becomes a random variable.
Two frameworks are commonly used to deal with this additional uncertainty.
In the first framework, an unconditional form is calculated by applying a total probability mechanism. We will refer to this form as the unconditional ARL --- others use notions such as the marginal or mean ARL.
Below, we will illustrate that controlling the unconditional IC ARL induces some puzzling side effects.
The second framework started with \cite{Albe:Kall:2001, Albe:Kall:2004a, Albe:Kall:2004b},
who considered probabilistic bounds for performance measures, such as the conditional ARL.
More recent contributions from, for instance,
\cite{Capi:Masa:2010b},
\cite{Jone:Stei:2011}
and \cite{Gand:Kval:2013}, have stimulated
a series of additional publications employing this approach.
A popular motivation for this guarantee-a-minimum IC ARL is that it incorporates appropriately
the so-called practitioner-to-practitioner variability.
This framework has been discussed extensively for Shewhart charts monitoring the normal variance in
\cite{Eppr:Lour:Chak:2015},
\cite{Guo:Wang:2017},
\cite{Goed:EtAl:2017},
\cite{Fara:Wood:Heuc:2015},
\cite{Fara:Heuc:Sani:2018},
\cite{Apar:Mosq:Eppr:2018},
and \cite{Jardi:Sarm:Chak:Eppr:2019}.
Most of the latter works derived numerical procedures for calculating the limit adjustments, which
is much more difficult for CUSUM and EWMA control charts, where Monte Carlo simulations (bootstrapping for
the phase I dataset) are typically used. Thus, it is not surprising that nothing has been published on
monitoring the normal variance for CUSUM and EWMA charts as of yet.
Moreover, there are further problems with this framework. First, it is truly difficult to communicate the
probabilistic bound for the random (conditional) IC ARL to a practitioner.
Second, the modified limits are commonly quite wide, resulting in a prolongation of the detection delays.
Therefore, \cite{Capi:Masa:2020} proposed to re-estimate the modification regularly during phase II
to tighten the limits. Third, the calculations of the actual modifications for CUSUM and EWMA charts
are involved and time consuming. While the last problem will be probably be solved soon, the other problems
persist. Hence, we propose a different approach. We widen the limits of an EWMA $S^2$ chart by assuring a
certain unconditional IC run length (RL) quantile. Later, we will see that
aiming at an unconditional IC RL quantile leads to a widening of the limits, whereas deploying the
unconditional IC ARL can tighten them.
Contrary to the case of the Shewhart variance chart, there are only a few contributions dealing with EWMA variance
charts under parameter uncertainty, namely, \cite{Mara:Cast:2009}, and more recently, \cite{Zwet:Scho:Does:2015} and \cite{Zwet:Ajad:2019}.
All together control the unconditional IC ARL.
\cite{Mara:Cast:2009} investigated EWMA charts utilizing $\ln S^2$.
From their unconditional out-of-control (OOC) ARL results we pick a few, in order
to discuss what we call the unconditional ARL puzzle.
In their Table 2, unconditional OOC ARL numbers for an upper EWMA $\ln S^2$ with an
unconditional IC ARL of 370.4 were given. We provide these numbers in Table~\ref{Tab:01},
for $\lambda = 0.01$, $n = 4$ and $m \in \{10, 20, 40, 80\}$ as well as $m = \infty$ (known parameter case).
\begin{table}[hbt]
\centering
\begin{tabular}{@{}rcrrrrcr@{}} \toprule
$m$ & \phantom{a} & 10 & 20 & 40 & 80 & \phantom{a} & $\infty$ \\ \midrule
ARL & & 13.1 & 16.3 & 18.9 & 20.5 & & 22.6 \\ \bottomrule
\end{tabular}
\caption{Side effects of using the unconditional ARL as a calibration target (IC 370.4):
Unconditional OOC ARL values (standard deviation increased by 20\%) for several
phase I sizes, $m$; sample size $n=5$; EWMA $\ln S^2$ chart with $\lambda = 0.01$.}\label{Tab:01}
\end{table}
These results include a non-remediable issue, namely, the favorable ARL values for
small $m$ suggest that small phase I samples should be utilized.
In other words, the more reliable the estimate of the unknown $\sigma_0^2$ (including in the known parameter case),
the longer one has to wait to detect this specific increase, that is, controlling the unconditional IC ARL
tightens the limits substantially, resulting in this uncommon improvement in the detection behavior. However, this
tightening greatly increases the probability of early false alarms. The heavy tail of the
unconditional IC RL distribution enlarges the corresponding mean (the unconditional IC ARL), while the
probability of low RL values becomes larger at the same time.
Later on, we will discuss this issue in more detail.
\cite{Zwet:Scho:Does:2015} utilized the unconditional ARL to adjust their limits and obtained
similar OOC ARL anomalies. Neither paper discussed these patterns.
Yet, \cite{Chak:2007} indicated that focusing on the unconditional ARL is dangerous.
In sum, controlling the unconditional IC ARL is not the way to go.
It should be added that the ARL paradigm has been criticized apart from studying the
estimation uncertainty influence on control charts limits.
For example, \cite{Yash:1985b} wrote:
\textit{``Though ARL is probably meaningful in the off-target situation, it can be
highly misleading when the on-target case is under study (primarily because the set of possible CUSUM paths includes `too many' extremely `short' members)''}.
For a more recent critique, see \cite{Mei:2008} or \cite{Kuhn:Mand:Taim:2019} and the references therein.
\cite{Yash:1985b}
also reported (for the competing CUSUM control chart):
\textit{``In general, the user of a CUSUM scheme probably feels uneasy about specifying a particular ARL for the on-target situation; what he typically wants is that the scheme will not generate a false alarm within a certain period of time (say, a shift) with probability of at least, say, 0.99.''}. This last passage refers to our design principle.
In sum, we study and propose two key features:
(i) A novel control chart design rule for incorporating estimation uncertainty that uses neither the misleading unconditional ARL nor the too conservative guarantee-a-minimum conditional ARL.
We control the unconditional false alarm probability via an unconditional IC RL quantile.
(ii)We utilize a numerical procedure that is more accurate than the Markov chain approximation \citep{Mara:Cast:2009}
and much quicker than Monte Carlo-based procedures \citep{Zwet:Scho:Does:2015}.
The paper proceeds as follows:
In Section \ref{sec:model}, we introduce the EWMA $S^2$ chart in detail and illustrate the peculiarities that emerge when
some estimate of the IC level is simply plugged in. In addition, we elucidate the deceptive
concept of adjusting the unconditional IC ARL.
Afterwards, in Section \ref{sec:algo},
we describe our novel approach and the numerical algorithm used to obtain the unconditional RL quantiles.
Eventually, in Section \ref{sec:comp}, we use this machinery to study the impact of the actual EWMA design (smoothing constant)
and of the phase I size $m$ on both the resulting control limit modification and the detection performance.
In the last section, we present our concluding discussion.
\section{EWMA $S^2$ under in-control level uncertainty} \label{sec:model}
EWMA schemes utilizing the sample variance $S^2$ are one type of EWMA chart monitoring dispersion.
Competitors include $\ln S^2$, which is used in \cite{Crow:Hami:1992a};
$S$, as in, for example, \cite{Mitt:Stem:Tewe:1998};
the sample range $R$, which is found in \cite{Ng:Case:1989}; and
$a + b \ln ( S^2 + c )$, as in \cite{Cast:2005}.
Note that all these papers, including ours, consider normally distributed data.
There are several reasons to prefer $S^2$.
First, it is an unbiased estimator of the variance. Second, EWMA $S^2$ frequently exhibits the best detection performance -- refer to
\cite{Knot:2005b, Knot:2010a}. Third, the calculation is more feasible if all the estimation and monitoring is done with $S^2$.
Now, let $\{X_{ij}\}$ be a sequence of subgroups of independent and normally distributed data.
Each subgroup $i$ consists of $n>1$ observations $X_{i1},\ldots,X_{in}$.
As usual, we assume that the phase I data come from a stable process and that the variance change occurs at the beginning of the monitoring period or never.
Calculating the running sample variance $S_i^2$, $i=1, 2,\ldots$,
\begin{equation*}
S_i^2 = \frac{1}{n-1} \sum_{j=1}^n \big( X_{ij} - \bar X_i \big)^2 \;\;,\quad \bar X_i = \frac{1}{n} \sum_{j=1}^n X_{ij} \,,
\end{equation*}
we feed the EWMA iteration sequence in the usual way:
\begin{equation*}
Z_i = (1-\lambda) Z_{i-1} + \lambda S_i^2 \;\;,\quad Z_0 = z_0 = \sigma_0^2 \,.
\end{equation*}
The EWMA smoothing constant, $\lambda$, is in the
interval $(0,1]$ and controls the detection sensitivity. The EWMA sequence $\{Z_i\}$ is initialized with
the IC variance level, $\sigma_0^2$. Here, we must estimate $\sigma_0^2$ anyway.
We want to detect increases or two-sided changes in the variance level.
Hence, the following (alarm) stopping times are utilized:
\begin{align*}
L_\text{upper} & = \min \left\{ i\ge 1: Z_i > c_u \right\} \,, \\
L_\text{two} & = \min \left\{ i\ge 1: Z_i > c_u \text{ or } Z_i < c_l \right\} \,.
\end{align*}
Note that introducing a lower reflection barrier to the upper scheme would diminish the inertia effects --- see \cite{Wood:Mahm:2005}
for more details. It would, however, also increase the complexity and dismantle the rolling estimate feature of the plain EWMA sequence $Z_i$.
Moreover, the inertia effect is less pronounced for a $S^2$-based chart with the intrinsic lower limit 0 compared to
a mean chart, which would be unbounded from below.
Therefore, we prefer the simpler design without a lower barrier.
Typically, the control limits are chosen to provide a pre-defined IC ARL, for example, by aiming for $E(L) = 500$.
In the case of a known IC variance $\sigma_0^2$, there is a rich body of literature on calculating the ARL and solving the inverse task of determining control limits for
a given IC ARL value. In this paper, we use algorithms from \cite{Knot:2005b, Knot:2007} to compute the ARL, RL quantiles,
and RL distribution for an EWMA $S^2$ control chart. The related \texttt{R} package \texttt{spc}
offers functions that make this calculation easy.
We will start with a typical situation: Sample size $n=5$, EWMA constant $\lambda=0.1$ and target IC ARL $500$.
This setup calls for thresholds $c_l = 0.6259$ and $c_u=1.5496$ for
the two-sided EWMA alarm design, and the threshold $c_u =1.4781$ for the upper EWMA alarm design.
For the former, we decided to use an ARL-unbiased design.
This notion was introduced by \cite{Pign:Acos:Rao:1995} and \cite{Acos:Pign:2000}, but the
phenomenon was discussed earlier in \cite{Uhlm:1982} and \cite{Krum:1992} (both in German), in \cite{Cham:Lowr:1994} and, presumably, in further publications.
For more details, refer to the more recent \cite{Knot:2010a} and \cite{Knot:Mora:2015}.
In a nutshell, ARL-unbiased designs render the ARL maximum at the IC level; in this case, $\sigma_0^2$.
For the unknown parameter case,
\cite{Guo:Wang:2017} provided results recently for ARL-unbiased Shewhart $S^2$ charts while
guaranteeing a minimum conditional IC ARL.
In this paper, we use a phase I reference dataset consisting of $m$ samples of size $n$ and
build the estimate, that is, the pooled sample variance
\begin{equation}
\hat\sigma_0^2 = \frac{1}{m} \sum_{i=1}^m s_i^2 \label{s2pooled} \,,
\end{equation}
where $s_i^2$ denotes the sample variance of the pre-run sample $i = 1, 2, \ldots, m$.
For a discussion on appropriate estimators of the unknown $\sigma_0^2$,
we refer to \cite{Mahm:Hend:Eppr:Wood:2010}, \cite{Zwet:Scho:Does:2015} and \cite{Sale:Mahm:Jone:Zwet:Wood:2015}.
Here, we focus on the above ``natural'' estimator because it is unbiased (no further corrections are needed) and
its distribution is readily available, that is, a $\chi^2$ distribution with $m\times (n-1)$ degrees of freedom.
\cite{Zwet:Scho:Does:2015} mentioned that
\textit{“under in-control data the EWMA control charts show similar performance across all estimators,''}
where ``in-control'' refers to an uncontaminated phase I.
Replacing this “natural” estimator with a more robust \citep{Zwet:Scho:Does:2015}
or otherwise more suitable estimator does not change the framework described below, but doing so
makes the calculations more complicated. We want to emphasize, however, that all our theoretical and numerical results make
use of the pooled variance estimator \eqref{s2pooled}.
To use the aforementioned estimate means that the observed $X_{ij}$ are standardized, resulting in $\tilde{X}_{ij} = X_{ij}/\hat{\sigma}_0$.
In consequence, we run an EWMA chart design for $\sigma_0^2 = 1 = z_0$ with the limits mentioned above.
Now, we wish to study the impact on the unconditional cumulative distribution function (CDF) $P(L\le l)$ depending on the phase I sample size $m$.
Note that $P(L\le l)$ covers two sources of uncertainty: (i) phase I estimation, and (ii) phase II monitoring.
Applying the numerical algorithms described in the next section,
in Figure~\ref{fig:02}, we illustrate this CDF for several phase I sizes assuming a
known $\sigma_0^2$ during setup.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_02_a_steps} &
\includegraphics[width=.52\textwidth]{fig_02_b_steps}
\end{tabular}
%
\caption{Unconditional IC RL CDF for the EWMA ($\lambda=0.1$) $S^2$ ($n=5$), phase I size $m$, unadjusted case.}\label{fig:02}
\end{figure}
The graphs in the two-sided case feature simple patterns, namely, the smaller the phase I sample size $m$, the higher the probability that a false alarm
is flagged by $l$ for any $l \in \{1,2,\ldots\}$. In the case of the upper chart, this relation remains valid for early values
of $l \le 350$ (roughly the original RL median) only. For large values, it is
reversed. For small phase I sample sizes, such as $m \le 50$, the unconditional IC RL distribution has heavy tails.
For instance, the unconditional probability $P(L > 10^5)$ is roughly 0.1 and 0.02 for $m = 10$ and $m=30$, respectively.
Given that, for example, \cite{Zwet:Scho:Does:2015} truncate their Monte Carlo simulations at $l = 30\,000$, some potential
problems with the unconditional IC ARL and even some IC RL quantiles might be hidden.
Nonetheless, using the unconditional IC ARL to adjust the limits is dangerous because the particular tails distort the expectation and
explains the peculiar numbers in Table~\ref{Tab:01} taken from \cite{Mara:Cast:2009}. Hence, calibrating upper EWMA variance charts
by aiming for a certain unconditional IC ARL is misleading.
\section{Numerical algorithm and actual adjustment} \label{sec:algo}
We want to adjust the EWMA control limits $c_u$ and $c_l$ (just the two-sided case) in order to achieve
\begin{equation}
P( L \le \bar{l} ) = \alpha \label{eq:design_rule} \,,
\end{equation}
where $P()$ is the unconditional IC CDF of the RL $L$.
In other words, we alter the limits so that $\bar{l}$ becomes the unconditional IC RL $\alpha$ quantile.
Recall that there is a direct link between the ARL and an RL quantile of order $\alpha$ for Shewhart charts with known IC parameters:
RL$_\alpha = \big\lceil \ln(1-\alpha)/\ln\big(1-1/\text{ARL}\big)\big\rceil$\,.
This simple formula remains approximately valid for EWMA $S^2$ charts in the IC case if $\sigma_0^2$ is known:
The median RL (MRL) is equal to $348$ and $349 \approx 347 = \lceil \ln(0.5)/\ln\big(1-1/500\big)\rceil$ with an ARL $ = 500$ for the upper and
two-sided EWMA $S^2$ charts, respectively, in Figure~\ref{fig:02}. However, this simple relationship is lost,
if we deal with the unconditional IC CDF. In the beginning of Section~\ref{sec:comp}, we provide some illustrations
of this phenomenon. This behavior is not surprising because the unconditional IC CDF is very different from the
simple geometric distribution we exploited for the Shewhart chart RL statement.
Using our rule \eqref{eq:design_rule}, we tackle the problems observed in Figure~\ref{fig:02} directly.
Appropriate choices of $\alpha$ are those that are smaller than 0.5 (we will use $\alpha = 0.25$), while $\bar l$ could be either
derived from the ARL and RL quantile relationship for known $\sigma_0^2$ or set manually, as in $\bar l = 1\,000$, which is used
for a typical control chart in practice. Next, we develop
an algorithm to calculate the unconditional CDF and use the result to solve the implicit
function \eqref{eq:design_rule} numerically with the secant rule.
For calculating the unconditional CDF, we adhere to \cite{Wald:1986a}, who proposed the idea for
EWMA control charts being used to monitor a normal mean with known IC parameters.
We start with the upper chart, which only requires an adjustment to its upper limit $c_u$.
Let $p_l(z) = P(L>l\mid Z_0=z)$ denote the survival function (SF) of the RL for known $\sigma_0^2$ and starting values $Z_0 = z$.
Adding further arguments, such as the actual variance $\sigma^2$ and control limit $c_u$, we produce
its unconditional version
\begin{equation}
p_{l,\text{unc.}}(z; \sigma^2, c_u) = \int_0^\infty f_{\hat\sigma_0^2}(s^2) p_l(z; \sigma^2 / s^2, c_u) \,ds^2
\quad,\; l = 1, 2, \ldots \label{eq:tSF}
\end{equation}
The formula \eqref{eq:tSF} is related to (16) in \cite{Jone:Cham:Rigd:2001}. Note that only one integral is needed and that we consider the SF
instead of the probability mass function of the RL $L$.
To increase the computational speed for \eqref{eq:tSF}, the geometric tail behaviors
\citep{Wald:1986a}
of $p_l(z; \sigma^2/s^2_i, c_u)$ at each quadrature node $s^2_i$ are
(for large $l$) exploited individually. Unfortunately, it is lacking for $p_{l,\text{unc.}}(z; \sigma^2, c_u)$,
as has been mentioned previously in, for example, \cite{Psar:Vyni:Cast:2014}.
The density $f_{\hat\sigma_0^2}()$ is roughly the probability density function (PDF) of a chi-square distribution (multiply the degrees of freedom).
The numerical implementation for $p_l()$ is taken from \cite{Knot:2007}.
Because its presentation is not easily accessible, we provide some necessary details here.
We begin with the transition (from $z_0$ to $z$) density of the EWMA $S^2$ sequence as follows:
\begin{equation*}
\delta(z_0,z) = \frac{1}{\lambda}\, f_{\chi^2;n-1}\!\left(\frac{n-1}{\sigma^2} \left[\frac{z-(1-\lambda) z_0}{\lambda}\right]\right) \frac{n-1}{\sigma^2} \,,
\end{equation*}
where $f_{\chi^2;n-1}()$ and $F_{\chi^2;n-1}()$ denote the PDF and CDF, respectively,
of a chi-square distribution with $n-1$ degrees of freedom. Then we obtain
the following recursions for the SF $p_l(z; \ldots) := p_l(z; \sigma^2 / s^2, c_u)$:
\begin{align}
p_1(z_0;\ldots) &
= \int_{(1-\lambda) z_0}^{c_u} \delta(z_0,z)\,dz = F_{\chi^2;n-1}\!\left(\frac{n-1}{\sigma^2} \left[\frac{c_u-(1-\lambda) z_0}{\lambda}\right]\right)
\,, \label{eq:seq1} \\
p_l(z_0;\ldots) &
= \int_{(1-\lambda) z_0}^{c_u} p_{l-1}(z;\ldots)\,\delta(z_0,z)\,dz \quad,\; l=2,3,\ldots \,. \label{eq:seql}
\end{align}
A common approach to approximating the integral recursions is to replace the integrals by quadratures
\citep{Jone:Cham:Rigd:2001}. Fixed quadrature grids, however, have lower integral limit issues since this limit, $(1-\lambda) z_0$,
depends on the argument $p_l(z_0;\ldots)$, see \cite{Knot:2005b} for a more thorough analysis.
Another idea is to apply a collocation type of procedure, as in \cite{Knot:2007}, \cite{Shu:Huan:Sub:Tsui:2013} and \cite{Huan:Shu:Jian:Tsui:2013},
who transferred the collocation principle from the ARL integral equation in \cite{Knot:2005b} to integral recursions.
Essentially, for every $l = 2, 3, \ldots$, we approximate
\begin{equation*}
p_l(z;\ldots) \approx \sum_{s=1}^N g_{ls} \, T_s^*(z) \,,
\end{equation*}
with $N$ suitably shifted Chebyshev polynomials $T_s^*(z)$, $s = 1, 2, \ldots, N$ (the $T_s()$ are the unit versions),
\begin{align*}
T_s^*(z) & = T_{s-1} \big( (2z-c_u)/c_u \big) \quad,\; z \in [0,c_u] \,, \\
T_s(z) & = \cos\big( s\,\arccos(z) \big) \quad,\;z \in[-1,1] \,.
\end{align*}
Then we pick $N$ nodes $z_r$ defined as (roots of $T_N(z)$ shifted to the interval $[0,c_u]$)
\begin{equation*}
z_r = \frac{c_u}{2} \left[1+\cos\left( \frac{(2\,i-1)\,\pi}{2\,N} \right)\right] \quad,\; r = 1, 2, \ldots, N \,,
\end{equation*}
and consider the following recursion on the grid $\{z_r\}$, $l = 2, 3, \ldots,$
\begin{equation*}
\sum_{s=1}^N g_{ls} \, T_s(z_r) =
\sum_{s=1}^N g_{l-1,s} \int_{(1-\lambda) z_r}^{c_u} T_s(z) \delta(z_r,z)\,dz \,.
\end{equation*}
These definite integrals must be determined numerically.
Because they do not depend on $l$ (only on $s$ and $r$),
we calculate them once and store them in an $N\times N$ matrix. Using this matrix and the starting vector
$\bm{g}_1 = (g_{11}, g_{12}, \ldots, g_{1N})^\prime$ derived from \eqref{eq:seq1}, we build a numerical approximation
for \eqref{eq:seql} that provides a highly accurate numerical presentation of the SF $p_l(z_0;\ldots)$ used in
\eqref{eq:tSF} to determine the unconditional CDF of the RL $L$.
The resulting SF $p_{l,\text{unc.}}(z_0; \sigma^2, c_u)$ is implemented in the \texttt{R} package \texttt{spc} as the function
\texttt{sewma.sf.prerun(l, lambda, 0, cu, sigma, n-1, m*(n-1), hs=z0, sided="upper")}, see the Appendix for an example of an application.
In Figure~\ref{fig:mchain}, we compare the approximation accuracies of the collocation and the Markov chain framework.
\begin{figure}[htb]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize unconditional IC SF, i.\,e. $p_{l,\text{unc.}}(1; 1, c_u) = 1 - \alpha = 0.75$& \footnotesize unconditional IC ARL \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_mchain_uncond_sf_n5} &
\includegraphics[width=.52\textwidth]{fig_mchain_uncond_arl_n5}
\end{tabular}
\caption{Approximation accuracies of the Markov chain and collocation
for EWMA ($\lambda=0.1$) $S^2$ ($n=5$), phase I size $m = 50$, $P_\text{IC}(L\le 10^3) = 0.25$.}\label{fig:mchain}
\end{figure}
We investigate the EWMA $S^2$ with $\lambda=0.1$ and sample size $n = 5$ and set $\bar{l}=10^3$ and $\alpha = 0.25$, resulting in $c_u = 1.719846$.
The integral in \eqref{eq:tSF} is approximated by the Gau\ss{}-Legendre quadrature with 60 nodes,
while replacing the upper limit $\infty$ by 1.773 ($1 - 10^{-10}$ quantile of a chi-square distribution
with $m\times(n-1) = 200$ degrees of freedom divided by 200).
This integral is deployed for the SF $p_{l,\text{unc.}}(z_0; \sigma^2, c_u)$ and the unconditional ARL as well.
The matrix dimension $N$ indicates the size of the collocation basis (see above) and the number of
transient states of the Markov chain.
From the two figures, we conclude that collocation with $N = 50$ yields much higher accuracy than
the Markov chain with $N = 500$. For calculating $p_{l,\text{unc.}}(1; 1, c_u)$,
collocation needs about 1 second for $N = 50$, while the Markov chain approximation
requires 3, 6, 15 and 22 seconds for $N = 200$, $300$, $400$ and $500$, respectively.
Eventually, we want to solve \eqref{eq:design_rule} as an implicit, continuous function of the upper limit $c_u$
numerically by executing a secant rule-type algorithm. The starting values will be slightly increased limits from the
known parameter case.
Turning to the two-sided case, we face additional problems. First, the numerical procedure (collocation proceeds ``piece-wise'' now) becomes
more involved and therefore more time consuming.
The general idea follows what has been outlined above, so we will skip the details \citep[for an elaborated description see][]{Knot:2005b}.
The second problem is that we must now determine two limits, $c_l$ and $c_u$, without an intrinsic symmetric limit design, unlike what
we encountered for monitoring the normal mean. Hence, we must either deploy the symmetric design $\sigma_0^2 \pm c$ by simply ignoring
the asymmetric behavior of EWMA $S^2$ or enforce something similar to the ARL-unbiased design used in the known-parameter case.
We try two concepts: (i) make the unconditional $P_\sigma(L\le \bar{l})$
minimal in $\sigma=\sigma_0=1$ (with no loss of generality) --- denoted henceforth as the
``unbiased'' version, and (ii) perform (i) for the known parameter case (much faster) and expand the resulting limits $(c_l^\infty, c_u^\infty)$
by incorporating the correction $\xi > 1$ via $c_l = c_l^\infty / \xi$ and $c_u = c_u^\infty\cdot \xi$ so that we achieve
the unconditional $P_\text{IC}(L\le \bar{l}) = \alpha$ --- we label this method as the ``quasi-unbiased'' method.
\begin{figure}[htb]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize unconditional $P_\sigma(L\le 10^3)$ & \footnotesize unconditional ARL \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_unb_cdf} &
\includegraphics[width=.52\textwidth]{fig_unb_arl}
\end{tabular}
\caption{Judging ``unbiasedness'' in the two-sided case: The unconditional $P_\sigma(L\le 10^3)$
and ARL as functions of the actual standard deviation $\sigma$
for EWMA ($\lambda=0.1$) $S^2$ ($n=5$), phase I size $m = 50$, $P_\text{IC}(L\le 10^3) = 0.25$.}\label{fig:two_sided}
\end{figure}
All three approaches
are illustrated in the following example: $\lambda = 0.1$, $\bar l = 10^3$, $\alpha = 0.25$, $n = 5$ and $m = 50$.
In all cases, we determine the new limits numerically by essentially applying the secant rule
(a more sophisticated implementation is the function \texttt{uniroot()} in \texttt{R}).
The resulting limits are
$(0.280153, 1.719847)$ (half width $c = 0.719847$),
$(0.528670, 1.824855)$ and
$(0.526394, 1.817301)$ (correction factor $\xi=1.065821$ applied to
$(c_l^\infty, c_u^\infty)=(0.561042, 1.705071)$) for the
symmetric, unbiased and quasi-unbiased approaches, respectively.
In Figure~\ref{fig:two_sided},
we illustrate the resulting profiles for $P(L\le 10^3)$ and the ARL as functions of the actual standard deviation $\sigma$.
For both, we deployed the unconditional distribution. Note that the simple symmetric design exhibits profiles (SF and ARL)
that are far from being unbiased. Moreover, the unconditional OOC ARL is very large for $\sigma < \sigma_0 = 1$ (the
lower limit is much smaller than those of the two competitors).
Hence, from this point forward, we will drop the symmetric limit design. The more sophisticated procedures feature rather equal profiles.
In the sequel, we will apply the unbiased approach to be on the safe side. However, because it needs considerably more computing time
than the quasi-unbiased scheme, we recommend the latter for daily practice.
We should note that the large unconditional ARL values are the result of the special setup utilized here. For instance,
when $\sigma_0^2$ is known, we observe an IC ARL of about
$3\,450$, which is then inflated to about
$6\,800$ by two sources: the widened limits and enlarged tails of the unconditional RL distribution. To achieve smaller
values, $\bar l = 10^3$ should be decreased or $\alpha = 0.25$ should be increased.
It should be noted that that using $\sigma_0 = 1$ does not violate the generality of our results.
Hence, $\sigma$ will refer to the standardized version $\sigma_0 = 1$. For example, $\sigma = 1.2$ means that
the OOC standard deviation is 20\% larger than its (unknown) IC counterpart.
Finally, we should emphasize that the unconditional $\alpha=0.25$ RL quantile $\bar{l} = 10^3$
differs substantially from measures such as ``$\text{\textit{Percentile}}_\text{marginal}$'' \citep{Zhan:Mega:Wood:2014},
where the RL quantile for known $\sigma_0^2$ replaces $p_l(z;\ldots)$ in \eqref{eq:tSF}.
This weighted average over all conditional RL quantiles is much larger. For example, we
obtain 244\,325 for $\alpha=0.25$.
The exception is the unconditional ARL. To calculate it, we could utilize either
\eqref{eq:tSF} and plug in the conditional means
or sum up $p_{l,\text{unc.}}(z;\ldots)$ over all $l$, which is just the expectation of the unconditional
RL distribution.
It remains somewhat unclear what exactly is being measured with $\text{\textit{Percentile}}_\text{marginal}$.
After deriving these quite involved algorithms, we use them to illustrate the dependence of $c_u$ on the phase I size $m$.
\begin{figure}[hbt]
\centering
\includegraphics[width=.55\textwidth]{fig_03}
\caption{Modified control limits for $P_\text{IC}(L\le 10^3) = 0.25$, upper and two-sided EWMA ($\lambda=0.1$) $S^2$ ($n=5$),
$m$ varies.}\label{fig:03}
\end{figure}
Utilizing our setup with $\bar{l} = 1\,000$, $\alpha = 0.25$ and EWMA's $\lambda = 0.1$,
we start with the limits for known $\sigma_0^2$ as a benchmark ---
$c_u = 1.6453$ and $(c_l=0.5610, c_u=1.7051)$ for the upper and two-sided cases, respectively.
For realistic values of phase I sample sizes $m$ between 10 and 1\,000, we obtain widened limits, as can be seen in Figure~\ref{fig:03}.
From the profiles, we conclude that the widening is less pronounced than might be expected. From sizes $m=50$ on, the resulting limits on the
control chart device in use would not really differ from the ideal case in which $\sigma_0^2$ is known.
Applying these new control limits changes the CDF profiles from those in Figure~\ref{fig:02}
to the ones presented in Figure~\ref{fig:04}.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_04_a_steps} &
\includegraphics[width=.52\textwidth]{fig_04_b_steps}
\end{tabular}
\caption{Unconditional IC RL CDF for EWMA ($\lambda=0.1$) $S^2$ ($n=5$), phase I size $m$, $P_\text{IC}(L\le 10^3) = 0.25$.}\label{fig:04}
\end{figure}
All profiles go through the point $(\bar{l},\alpha)$ by construction, of course. However, we observe that the smaller the phase size $m$, the more likely the very early false alarms.
Widening the limits allows poor false alarm levels to be dealt with. However, the behavior in the OOC case
has deteriorated. Using the limits from $P_\text{IC}(L\le 1\,000) = 0.25$, we show
the unconditional CDFs for selected OOC cases
($\sigma_1 \in \{0.8, 1.2\}$)
in Figure~\ref{fig:05}.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_05_a_steps} &
\includegraphics[width=.52\textwidth]{fig_05_b_steps}
\end{tabular}
%
\caption{Unconditional OOC RL CDFs for EWMA ($\lambda=0.1$) $S^2$ ($n=5$), phase I size $m$,
$P_\text{IC}(L\le 10^3) = 0.25$.}\label{fig:05}
\end{figure}
Note the poor detection behavior for smaller values of $m$. For $m < 50$, it is possible that
the variance change will remain undetected over the entire planned monitoring time span ($\bar{l} = 1\,000$ observations). It is even worse for the two-sided case.
Based on the profiles in Figure~\ref{fig:05}, we would recommend phase I sizes of at least 100.
For more details, we refer the reader to the next section.
In order to provide some more familiar representations and at least get an idea of the detection speed, we add
some unconditional ARL values to the OOC case in Figure~\ref{fig:06}.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.52\textwidth]{fig_06_a_mod} &
\includegraphics[width=.52\textwidth]{fig_06_b_mod}
\end{tabular}
\caption{Unconditional OOC ARLs for EWMA ($\lambda=0.1$) $S^2$ ($n=5$) vs. phase I size $m$,
$P_\text{IC}(L\le 10^3) = 0.25$.}\label{fig:06}
\end{figure}
The differences in the benchmark case are considerably large for $m < 100$ and become negligible only for $m > 200$. Hence, there is an
obvious price to pay if we account for the phase I estimation uncertainty when calibrating the chart.
Because the false alarm behavior is really important for practical control charting in industry,
the calibration strategy utilizing $P(L \le \bar{l}) = \alpha$ seems to be passable despite these side effects.
Note that the even more conservative approach of guaranteeing a minimum conditional IC ARL
yields substantially larger unconditional OOC ARL results.
\section{Sensitivity and competition} \label{sec:comp}
To reconcile the common IC ARL user to this method, we investigate the selection of the monitoring horizon $\bar l$ and false alarm probability $\alpha$
for a given IC ARL of, for example, 500 and its impact on the actual adjustment of the control limits accounting for
the estimation uncertainty. To begin with, we set $\alpha = 0.5$ to
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.52\textwidth]{arl_equivalent_upper} &
\includegraphics[width=.52\textwidth]{arl_equivalent_two_upper}
\end{tabular}
\caption{Choice of $(\bar l, \alpha)$ within $\alpha = P(L\le \bar l)$ and $E(L) = 500$ (all for known $\sigma_0^2$)
and its impact on the $c_U$ modification to secure $\alpha = P(L\le \bar l)$ in the case of unknown $\sigma_0^2$, which will be estimated
with a $m=50 \times n=5$ phase I sample.}\label{fig:07}
\end{figure}
ensure that the IC median run length (MRL) 348 (349 in the two-sided case) is achieved. From Figure~\ref{fig:07}(a) , we conclude that
focusing on the unconditional ARL yields the smallest $c_U$, followed by simply utilizing the $c_U$ value for known $\sigma_0^2$ and,
finally, the unconditional MRL (median RL) design. Obviously, downsizing the upper limit seems to be counter-intuitive and results in more false alarms
than intended. The slight increase of $c_U$ from the known $\sigma_0^2$ case to the MRL- conserving approach offers a cautious
and effective way of dealing with the estimation uncertainty. By changing $\alpha$ (or $\bar l$), we can see that for all $\alpha < 0.56$, the
modified $c_U$ is larger than for known $\sigma_0^2$.
In addition, decreasing $\alpha$ (and $\bar{l}$, accordingly) increases $c_U$ further (except for very small $\alpha$).
Of course, proper choices of $\alpha$ are 0.5 or smaller.
In the two-sided case, all designs securing some unconditional measure widen the original limits. In Figure~\ref{fig:07}(b),
we plot only the upper value $c_U$. For $\alpha < 0.75$, the unconditional RL quantiles induce wider limits than the unconditional
ARL design. In summary, deciding on a reasonable combination $(\bar l, \alpha)$ provides plausible and effective limit adjustments
that can overcome the estimation uncertainty distortions.
Next, we wish to compare the detection behaviors of various values of the smoothing constant $\lambda \in \{0.05, 0.1, 0.2, 0.3\}$.
In all cases, we calibrate the schemes to ensure that $P_\text{IC}(L\le 10^3)=0.25$.
Again, we consider samples of size $n = 5$ and a phase I study of size $m = 50$.
In Table~\ref{Tab:02} and Table~\ref{Tab:03}, we provide some (unconditional) ARL values
\begin{table}[hbt]
\centering
\begin{tabular}{cccccc} \toprule
$\lambda$ & 0.05 & 0.1 & 0.2 & 0.3 & 1$_\text{\tiny Shewhart}$ \\ \toprule
\multicolumn{6}{c}{known $\sigma_0^2$} \\ \midrule
%
$c_u$ & 1.3995 & 1.6453 & 2.0690 & 2.4653 & 5.3026 \\ \midrule
$E_1(L)$ & 3\,444 & 3\,461 & 3\,470 & 3\,473 & 3\,476 \\
$E_{1.2}(L)$ & 32.9 & 38.4 & 55.9 & 76.1 & 188.8 \\
$E_{1.5}(L)$ & 8.75 & 8.05 & 8.24 & 9.11 & 19.5 \\ \midrule
%
\multicolumn{6}{c}{phase I with $m=50$} \\ \midrule
%
$c_u$ & 1.4680 & 1.7198 & 2.1538 & 2.5596 & 5.4654 \\ \midrule
$E_1(L)$ & $> 5\times 10^9$ & $>8\times 10^5$ & 47\,128 & 21\,477 & 8\,091 \\
$E_{1.2}(L)$ & 70.4 & 84.8 & 119.2 & 151.8 & 293.4 \\
$E_{1.5}(L)$ & 10.6 & 9.52 & 9.79 & 11.0 & 24.0 \\ \bottomrule
\end{tabular}
\caption{Unconditional ARL results for various values of $\lambda$ and $P_\text{IC}(L\le 10^3)=0.25$, upper chart.}\label{Tab:02}
\end{table}
for the upper and the two-sided designs, respectively.
To judge phase I‘s influence on the uncertainty, we compare the unconditional ARL numbers with the
initial ones for a known IC variance. Interestingly, the
new IC ARL results are very large but decline with increasing $\lambda$.
Similar patterns can be observed in the OOC case, where for $\sigma = 1.2$, the ARL numbers are
doubled. For the medium size increase, that is, $\sigma = 1.5$, the change is much smaller.
Note that the order between the different EWMA designs remains stable, that is, $\lambda = 0.05$ is the best
for $\sigma = 1.2$, while $\lambda = 0.1$ works best with $\sigma = 1.5$.
The Shewhart $S^2$ chart ARL results are added, which are considerably larger than all EWMA ones.
Turning to the two-sided designs, we observe some slight differences. Most notably, the IC ARL values do not explode.
\begin{table}[hbt]
\centering
\begin{tabular}{cccccc} \toprule
$\lambda$ & 0.05 & 0.1 & 0.2 & 0.3 & 1$_\text{\tiny Shewhart}$ \\ \toprule
\multicolumn{6}{c}{known $\sigma_0^2$} \\ \toprule
$c_l, c_u$ & 0.6825, 1.4377 & 0.5610, 1.7051 & 0.4146, 2.1721 & 0.3200, 2.6159 & 0.0112, 6.3542 \\ \midrule
$E_{0.5}(L)$ & 11.3 & 8.91 & 7.35 & 6.99 & 264 \\
$E_{0.8}(L)$ & 35.3 & 40.5 & 68.0 & 113 & 1\,671 \\
$E_1(L)$ & 3\,425 & 3\,453 & 3\,462 & 3\,467 & 3\,465 \\
$E_{1.2}(L)$ & 38.9 & 49.0 & 81.1 & 121 & 639.5 \\
$E_{1.5}(L)$ & 9.64 & 8.96 & 9.53 & 11.1 & 42.6 \\ \midrule
%
\multicolumn{6}{c}{phase I with $m=50$} \\ \midrule
%
$c_l, c_u$ & 0.6377, 1.5488 & 0.5287, 1.8249 & 0.3947, 2.3077 & 0.3067, 2.7669 & 0.0111, 6.6824 \\ \midrule
$E_{0.5}(L)$ & 13.5 & 10.0 & 8.03 & 7.60 & 277 \\
$E_{0.8}(L)$ & 75.4 & 93.3 & 151 & 222 & 1\,752 \\
$E_1(L)$ & 11\,240 & 6\,803 & 4\,961 & 4\,386 & 3\,500 \\
$E_{1.2}(L)$ & 140 & 173 & 251 & 328 & 1\,094 \\
$E_{1.5}(L)$ & 12.9 & 11.5 & 12.4 & 14.8 & 62.6 \\ \bottomrule
\end{tabular}
\caption{Unconditional ARL results for various values of $\lambda$ and $P_\text{IC}(L\le 10^3)=0.25$, two-sided chart.}\label{Tab:03}
\end{table}
Again, the OOC ARL results are tripled ($1.2$) and doubled ($0.8$) for small changes,
while the increases are quite small for larger variance changes ($0.5, 1.5$).
For the simple Shewhart chart, the unconditional ARL values nearly coincide with
their known $\sigma_0^2$ counterparts (increasing $\sigma$ only).
For the upper and two-sided designs, the patterns in the detection ranking
remain stable.
For small shifts, for example, the EWMA chart with $\lambda = 0.05$ exhibits the best detection
performance for known and unknown IC variances.
It should be stated that the Shewhart ARL performance is much worse than the EWMA ARL performance for the changes considered
here.
Following the focus of this paper, we now examine the CDF profiles. Beginning with the IC versions for both designs in Figure~\ref{fig:08},
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize upper & \footnotesize two-sided \\[-2ex]
\includegraphics[width=.5\textwidth]{fig_08_a_steps} &
\includegraphics[width=.5\textwidth]{fig_08_b_steps}
\end{tabular}
\caption{IC CDFs of the RL $L$, $P_\text{IC}(L\le 10^3)=0.25$, EWMA (various $\lambda$) $S^2$ ($n=5$), $m=50$ phase I samples.}\label{fig:08}
\end{figure}
we conclude that for $l \le \bar l = 1\,000$, the profiles look very similar. The $\lambda = 0.05$ line lies above all the others for these $l$,
which changes for $l > \bar l$, where it features the lowest values. All other curves follow according their $\lambda$ values, that is, the
larger the $\lambda$, the lower the $l \le \bar l$ and the higher the $l > \bar l$.
Comparing the results for the upper and two-sided EWMA designs, we observe much steeper developments for the latter ones.
In summary, we conclude that for the interesting part, namely, $l \le \bar l$, the IC behavior of $P(L\le l)$
for all considered $\lambda$ values and for both design types does not really differ.
Turning to the OOC case, we start with the upper EWMA chart and two different possible new $\sigma$ values,
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize $\sigma_1=1.2$ & \footnotesize $\sigma_1=1.5$ \\[-2ex]
\includegraphics[width=.5\textwidth]{fig_09_a_steps} &
\includegraphics[width=.5\textwidth]{fig_09_b_steps}
\end{tabular}
\caption{OOC CDFs of the RL $L$, $P_\text{IC}(L\le 10^3)=0.25$, upper EWMA (various $\lambda$) $S^2$ ($n=5$), $m=50$ phase I samples.}\label{fig:09}
\end{figure}
namely, the small change $\sigma_1 = 1.2$ and the medium one $\sigma_1 = 1.5$. Examining Figure~\ref{fig:09}, we observe the following stylized facts.
The larger change is detected by $l \le 100$ with probability one, while for $\sigma_1 = 1.2$, we need the whole time span, that is, $l \le 1\,000$.
Recall the corresponding expected values in Table~\ref{Tab:02}, which are roughly 10 for $\sigma_1 = 1.5$ for all EWMA designs,
while for $\sigma_1 = 1.2$, they range from 70 for $\lambda=0.05$ to 150 for the largest $\lambda < 1$.
Moreover, the order between these $\lambda$ values defined by
their ARL values is reflected by the $P(L\le l)$ profiles. Similar patterns can be recognized for the two-sided designs in Figure~\ref{fig:10},
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\footnotesize $\sigma_1=1.2$ & \footnotesize $\sigma_1=1.5$ \\[-2ex]
\includegraphics[width=.5\textwidth]{fig_10_a_steps} &
\includegraphics[width=.5\textwidth]{fig_10_b_steps} \\
\footnotesize $\sigma_1=0.8$ & \footnotesize $\sigma_1=0.5$ \\[-2ex]
\includegraphics[width=.5\textwidth]{fig_10_c_steps} &
\includegraphics[width=.5\textwidth]{fig_10_d_steps}
\end{tabular}
\caption{OOC CDFs of the RL $L$, $P_\text{IC}(L\le 10^3)=0.25$, two-sided EWMA (various $\lambda$) $S^2$ ($n=5$), $m=50$ phase I samples.}\label{fig:10}
\end{figure}
where we included the results for decreased variances. Not surprisingly, the detection performance for $\sigma_1 \in\{1.2,1.5\}$
is weaker compared to the upper chart profiles in Figure~\ref{fig:09}. However, detecting decreases of the same relative order proceeds more
quickly.
After some first glimpses of the impact of the phase 1 sample size, namely, $m$, on the magnitude of
the limit modification in Figure~\ref{fig:03}, the resulting unconditional OOC ARLs in Figure~\ref{fig:06}
and the snapshots in Tables~\ref{Tab:02} and \ref{Tab:03},
some additional details will be provided here to develop recommendations regarding some lower bound
for $m$ and the choice of the smoothing constant $\lambda$. We start with the upper design
and look at our selection of EWMA smoothing constants $\lambda \in \{0.05, 0.1, 0.2, 0.3\}$.
In the following Figure~\ref{fig:11}, the $c_u$ vs. phase I size $m$ profiles are provided in two
ways. First, the raw $c_u$ limits are presented, demonstrating the typical behavior of
decreasing values if $\lambda \downarrow$ or $m \uparrow$. Note that the change from $\lambda = 0.3$
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\scriptsize (a) raw size & \scriptsize (b) relative change to known parameter case \\[-1ex]
\includegraphics[width=.5\textwidth]{fig_11_a_mod} &
\includegraphics[width=.5\textwidth]{fig_11_b_mod} \\
\end{tabular}
\caption{Modified $c_u$ needed to achieve $P_\text{IC}(L\le 10^3)=0.25$, upper EWMA (various $\lambda$) $S^2$ ($n=5$), phase I size $m=15, 16, \ldots, 1200$.}\label{fig:11}
\end{figure}
to the Shewhart case ($\lambda = 1$) is really pronounced. However, the amount of widening in the
control chart's continuation region done
to cope with estimating the IC value of the variance is not large.
Compared to the known parameter case, illustrated in Figure~\ref{fig:11}(b), we must increase the
original $c_u$ by 5 to 10\% (along the $\lambda$ range) for small $m < 50$ and by less than 3\% for $m > 100$.
The relative amount of change decreases with increasing $\lambda$.
Similar results can be observed for the two-sided case.
Next, we consider just the relative changes plotted in Figure~\ref{fig:12}.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\scriptsize (a) lower limit & \scriptsize (b) upper limit \\[-1ex]
\includegraphics[width=.5\textwidth]{fig_12_a_mod} &
\includegraphics[width=.5\textwidth]{fig_12_b_mod} \\
\end{tabular}
\caption{Ratios of the modified control limits to the original ones (known IC variance),
$P_\text{IC}(L\le 10^3)=0.25$, two-sided EWMA (various $\lambda$) $S^2$ ($n=5$), phase I size $m=15, 16, \ldots, 1200$.}\label{fig:12}
\end{figure}
Except for the lower limit in the Shewhart chart, which is driven by the small sample size $n = 5$ creating
difficulties while detecting variance decreases and is typically very close to zero,
the profiles do not really differ from those for the one-sided case.
Not surprisingly, the adjustment needed is larger than for the one-sided design.
Overall, the widening of the control chart limits is about 10\% or smaller for $m \ge 30$.
Even some crude rule of thumb for selected values of $m$ could be derived, such as widen the limits by
10, 8, 5, 3, 2 and 1\% for $m = $ 20, 30, 50, 100, 200 and 400 for the phase I samples, respectively.
In conclusion, utilizing the $P(L\le \bar l)=\alpha$ design yields moderate changes to the control
chart limits. To identify a minimum $m$ rule or a $\lambda$ guideline, we now consider
the unconditional OOC ARL for two magnitudes of change.
Starting with the upper chart,
we present the unconditional OOC ARL values for
$\sigma_1 = 1.2$ and $\sigma_1 = 1.5$ in Figure~\ref{fig:13}
for the previously considered configurations.
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\scriptsize (a) $\sigma_1 = 1.2$ & \scriptsize (b) $\sigma_1 = 1.5$ \\[-1ex]
\includegraphics[width=.5\textwidth]{fig_13_a_mod} &
\includegraphics[width=.5\textwidth]{fig_13_b_mod} \\
\end{tabular}
\caption{Unconditional OOC ARLs of the upper EWMA (various $\lambda$) $S^2$ ($n=5$) charts,
$P_\text{IC}(L\le 10^3) = 0.25$, phase I size $m=15, 16, \ldots, 1200$.}\label{fig:13}
\end{figure}
In Figure~\ref{fig:13}(a), we detect two segments in the ARL profiles. For small values of $m < 50$,
we observe huge ARL values for all considered values of $\lambda$. In addition, the smaller the $\lambda$, the steeper
the curves, which completely changes the order of the analyzed control chart designs.
Given these patterns, it can be concluded that when attempting to detect small changes with an EWMA $S^2$ chart,
phase I samples with $m\ge 50$ are definitely needed. Namely, avoiding too many false alarms for small $m < 50$
leads inevitably to the delayed detection of small changes. Things look much better for the medium-sized change
$\sigma_1 = 1.5$ in Figure~\ref{fig:13}(b), where for all $\lambda$ and roughly all $m$,
the adjustment of the upper limit $c_u$ only mildly distorts the unconditional OOC ARL.
Moreover, the popular choices $\lambda = 0.1$ and $= 0.2$ produce overall decent ARL levels, indicating
that a reasonable approach would be to recommend these two values, in general. Turning now to the
two-sided case in Figure~\ref{fig:14},
\begin{figure}[hbt]
\renewcommand{\tabcolsep}{-.7ex}
\begin{tabular}{cc}
\scriptsize (a) $\sigma_1 = 1.2$ & \scriptsize (b) $\sigma_1 = 1.5$ \\[-1ex]
\includegraphics[width=.5\textwidth]{fig_14_a_mod} &
\includegraphics[width=.5\textwidth]{fig_14_b_mod} \\
\scriptsize (c) $\sigma_1 = 0.8$ & \scriptsize (d) $\sigma_1 = 0.5$ \\[-1ex]
\includegraphics[width=.5\textwidth]{fig_14_c_mod} &
\includegraphics[width=.5\textwidth]{fig_14_d_mod} \\
\end{tabular}
\caption{Unconditional OOC ARLs of the two-sided EWMA (various $\lambda$) $S^2$ ($n=5$) charts,
$P_\text{IC}(L\le 10^3) = 0.25$, phase I size $m=15, 16, \ldots, 1200$.}\label{fig:14}
\end{figure}
we confirm the judgments made for the upper schemes, where again small changes create problems for
$m <50$. The good news is that for the control chart user who is interested in detecting medium-sized and large
changes, the proposed adjustments of the control limits do not destroy the ability of the applied
EWMA charts to detect these changes. If flagging smaller changes is of concern, a larger phase I sample size is needed,
that is, $m \ge 50$, to obtain a detection performance that is comparable to that of the known parameter case.
It should be noted that the ARL values for the Shewhart chart are almost always too large to be displayed in
Figures~\ref{fig:13} and \ref{fig:14}.
The one and only exception in Figure~\ref{fig:13}(a) emphasizes that detecting a small variance increase
in presence of an unknown IC variance level is difficult if only $m \le 40$ observations are available
to estimate the latter value.
\section{Conclusions}
In order to control the false alarm behavior of EWMA $S^2$ charts used for monitoring a normal variance,
we proposed an approach that widens the limits in a balanced way. The resulting control
chart design exhibits reasonable false alarm behavior while still being able to detect
medium-sized and large changes. To detect small changes, the phase I sample size must be increased
to $m \ge 50$ to achieve a performance that is comparable to the known parameter case.
Moreover, we believe that the notion that we are calibrating for a certain
false alarm probability $\alpha$ within a given number of control chart values (the chart horizon $\bar l$)
is easier to communicate to the statistical process monitoring community than declaring that one guarantees with
probability $1 - \alpha$
that the random conditional IC ARL is at least some nominal value, which corresponds more or less directly to $\bar l$
anyway. In addition, we recommend that $\lambda = 0.1$ or $= 0.2$ be used for setting up
a reasonable EWMA $S^2$ chart.
Finally, it should be noted that we have prepared an \texttt{R}
package (available from CRAN: \url{https://cran.r-project.org/}) that contains the functions needed to calculate the unconditional RL quantiles
and ARL values as well as the control charts limits (including their adjustments for a given phase I size $m$).
Some examples are given in the Appendix.
Moreover, the shiny app \url{https://kassandra.hsu-hh.de/apps/knoth/s2ewmaP/} provides a more convenient access.
\bibliographystyle{unsrtnat}
|
2,877,628,090,221 | arxiv | \section{Introduction}
This paper investigates the problem of multi-terminal channel coding
for relayless networks with the general message access structure
shown in Fig.~\ref{fig:channel}.
Multi-terminal channels include
broadcast channels~\cite{C72}\cite{GP80}\cite{IO05}\cite{LKP11}\cite{M79}\cite{ITW13},
multiple-access channels~\cite{A71}\cite{A74}\cite{EC80}\cite{H79}\cite{H98}\cite{L72}\cite{SHANNON61}\cite{SW73MAC},
and interference channels~\cite{A74}\cite{CMGE08}\cite{HK81}\cite{JXG08}\cite{ICC}.
The contribution of this paper is the introduction
of codes for this type of network
by using constrained-random-number generators,
which are the basic building blocks for the construction
of the both encoders and decoders.
Sparse matrices (with logarithmic column degree) are available
for code construction.
The construction includes
the case when all messages are private \cite{CRNG-MULTI}
and the case when all encoders have access to all common messages \cite{ICC}.
It should be noted that
there is an unsupported case in which previous constructions \cite{ICC}\cite{CRNG-MULTI} cannot be applied directly.
It is shown that the multi-letter characterized capacity region
of this network is achievable with this code.
This capacity region is specified in terms of entropy functions
and provides an alternative to the region derived
in~\cite{CRNG-MULTI}\cite{SV06}.
It should be noted that,
when random variables are assumed to be stationary and memoryless,
our region provides the best known single-letter characterized
achievable regions
for general stationary memoryless channels,
where the rate-splitting technique is unnecessary~\cite{ITW13}\cite{ICC}.
\begin{figure}[ht]
\begin{center}
\unitlength 0.52mm
\begin{picture}(157,85)(-5,10)
\put(-6,90){\makebox(0,0){$M^{(n)}_1$}}
\put(-6,66){\makebox(0,0){$M^{(n)}_2$}}
\put(-6,38){\makebox(0,0){$\vdots$}}
\put(-6,6){\makebox(0,0){$M^{(n)}_{|\mathcal{S}|}$}}
\put(0,90){\vector(20,-11){20}}
\put(0,90){\vector(20,-70){20}}
\put(0,66){\vector(20,12){20}}
\put(0,66){\vector(20,-14){20}}
\put(0,6){\vector(20,14){20}}
\put(0,6){\vector(20,71){20}}
\put(27,92){\makebox(0,0){Encoders}}
\put(20,68){\framebox(14,20){$\Phi^{(n)}_1$}}
\put(20,42){\framebox(14,20){$\Phi^{(n)}_2$}}
\put(27,38){\makebox(0,0){$\vdots$}}
\put(20,10){\framebox(14,20){$\Phi^{(n)}_{|\mathcal{I}|}$}}
\put(34,78){\vector(1,0){10}}
\put(49,78){\makebox(0,0){$X^n_1$}}
\put(54,78){\vector(1,0){10}}
\put(34,52){\vector(1,0){10}}
\put(49,52){\makebox(0,0){$X^n_2$}}
\put(54,52){\vector(1,0){10}}
\put(49,38){\makebox(0,0){$\vdots$}}
\put(34,20){\vector(1,0){10}}
\put(49,20){\makebox(0,0){$X^n_{|\mathcal{I}|}$}}
\put(54,20){\vector(1,0){10}}
\put(64,10){\framebox(24,78){\small $\mu_{Y^n_{\mathcal{J}}|X^n_{\mathcal{I}}}$}}
\put(127,92){\makebox(0,0){Decoders}}
\put(88,78){\vector(1,0){10}}
\put(104,78){\makebox(0,0){$Y^n_1$}}
\put(110,78){\vector(1,0){10}}
\put(120,68){\framebox(14,20){$\Psi^{(n)}_1$}}
\put(88,52){\vector(1,0){10}}
\put(104,52){\makebox(0,0){$Y^n_2$}}
\put(110,52){\vector(1,0){10}}
\put(120,42){\framebox(14,20){$\Psi^{(n)}_2$}}
\put(127,38){\makebox(0,0){$\vdots$}}
\put(88,20){\vector(1,0){10}}
\put(104,20){\makebox(0,0){$Y^n_{|\mathcal{J}|}$}}
\put(110,20){\vector(1,0){10}}
\put(120,10){\framebox(14,20){$\Psi^{(n)}_{|\mathcal{J}|}$}}
\put(134,78){\vector(1,0){10}}
\put(153,78){\makebox(0,0){$\widehat{M}^{(n)}_{\mathcal{D}(1)}$}}
\put(134,52){\vector(1,0){10}}
\put(153,52){\makebox(0,0){$\widehat{M}^{(n)}_{\mathcal{D}(2)}$}}
\put(153,38){\makebox(0,0){$\vdots$}}
\put(134,20){\vector(1,0){10}}
\put(153,20){\makebox(0,0){$\widehat{M}^{(n)}_{\mathcal{D}(|\mathcal{J}|)}$}}
\end{picture}
\end{center}
\caption{Multi-terminal Channel Coding}
\label{fig:channel}
\end{figure}
Throughout this paper, we use the following definitions and notations.
When $\mathcal{U}$ is a set and $\mathcal{V}_u$ is also a set for each $u\in\mathcal{U}$,
we use the notation $\mathcal{V}_{\mathcal{U}}\equiv\operatornamewithlimits{\text{\Large $\times$}}_{u\in\mathcal{U}}\mathcal{V}_u$.
We use the notation $v_{\mathcal{U}}\equiv\{v_u\}_{u\in\mathcal{U}}\in\mathcal{V}_{\mathcal{U}}$
to represent the sequence of elements
(e.g.\ sequences, random variables, functions)
$v_u$ with index $u\in\mathcal{U}$.
We use the notation $|\mathcal{U}|$ to represent the cardinality of $\mathcal{U}$.
Let $2^{\mathcal{U}}$ be the power set of $\mathcal{U}$.
Let $\mathcal{I}$ be the index set of channel inputs,
and $\mathcal{J}$ be the index set of channel outputs.
Then, a general channel is characterized
by sequence $\{\mu_{Y^n_{\mathcal{J}}|X^n_{\mathcal{I}}}\}_{n=1}^{\infty}$
of conditional distributions,
where $n\in\mathbb{N}$ is the block length of the channel input,
$X^n_{\mathcal{I}}\equiv\{X^n_i\}_{i\in\mathcal{I}}$
is the set of random variables of multiple channel inputs,
and $Y^n_{\mathcal{J}}\equiv\{Y^n_j\}_{j\in\mathcal{J}}$
is the set of random variables of multiple channel outputs.
For each $i\in\mathcal{I}$ and $n\in\mathbb{N}$,
let $\mathcal{X}_i^n$ be the alphabet of random variable
$X_i^n\equiv(X_{i,1},\ldots,X_{i,n})$,
where we assume that
$\mathcal{X}_i^n$ is the $n$-dimensional Cartesian product of finite set $\mathcal{X}_i$
and $X_{i,k}\in\mathcal{X}_i$ for all $k\in\{1,\ldots,n\}$.
For each $j\in\mathcal{J}$ and $n\in\mathbb{N}$,
let $\mathcal{Y}_j^n$ be the alphabet of random variable $Y_j^n$.
It should be noted that
$\mathcal{Y}_j^n$ is allowed to be an infinite/continuous set
and it is unnecessary to assume that
$\mathcal{Y}_j^n$ is the $n$-dimensional Cartesian product of $\mathcal{Y}_j$.
For example, we can assume that $\mathcal{Y}^n_j\equiv\bigcup_{n=0}^{\infty}\mathcal{X}_i^n$,
which is the set of all finite length sequences with alphabet $\mathcal{X}_i$,
to describe insertion-deletion-substitution channels.
We use notations $Y_j^n$ and $\mathcal{Y}_j^n$
to consider the stationary memoryless case.
Let $\mathcal{S}$ be the index set of multiple messages.
For each $s\in\mathcal{S}$ and $n\in\mathbb{N}$,
let $M^{(n)}_s$ be a random variable of Message $s$
corresponding to the uniform distribution on alphabet $\mathcal{M}^{(n)}_s$.
We assume that $\{M^{(n)}_s\}_{s\in\mathcal{S}}$ are mutually independent.
We consider the situation that
each encoder has access to some of the messages in $\{M^{(n)}_s\}_{s\in\mathcal{S}}$,
where some messages are common to some encoders.
The definition of the general message access structure
between messages and encoders is introduced in Section~\ref{sec:access}.
\section{Message Access Structure}
\label{sec:access}
This section introduces the message access structure.
{\em (Message) access structure} $\mathcal{A}$ is a subset of $\mathcal{S}\times\mathcal{I}$,
where member $(s,i)\in\mathcal{A}$ indicates that
Encoder $i$ has access to Message~$s$.
It should be noted that $(\mathcal{S},\mathcal{I},\mathcal{A})$ forms a directed bipartite graph,
where $(s,i)\in\mathcal{A}$ corresponds to the arc (directed edge) $s\to i$.
For each $s\in\mathcal{S}$, let $\mathcal{I}(s)$ be the set of all indices of encoders
that have access to Message $s$, where $\mathcal{I}(s)$ is defined as
\begin{equation*}
\mathcal{I}(s)\equiv\{i\in\mathcal{I}: (s,i)\in\mathcal{A}\}.
\end{equation*}
For each $i\in\mathcal{I}$, let $\mathcal{S}(i)$ be the set of all indices of the messages
to which Encoder $i$ has access, where $\mathcal{S}(i)$ is defined as
\begin{equation*}
\mathcal{S}(i)\equiv\{s\in\mathcal{S}: (s,i)\in\mathcal{A}\}.
\end{equation*}
We have the fact that $i\in\mathcal{I}(s)$ is equivalent to $s\in\mathcal{S}(i)$.
For a given $\mathcal{I}'\in2^{\mathcal{I}}$,
we refer to the set of encoders whose index belongs to $\mathcal{I}'$
as {\em Encoders $\mathcal{I}'$}.
Let $\mathcal{S}(\mathcal{I}')$ be the index set of messages common to Encoders $\mathcal{I}'$,
where $\mathcal{S}(\mathcal{I}')$ is defined as
\begin{equation}
\mathcal{S}(\mathcal{I}')
\equiv
\{
s\in\mathcal{S}: \mathcal{I}(s)=\mathcal{I}'
\}.
\label{eq:SI}
\end{equation}
We define $\mathfrak{I}$ as
\begin{equation*}
\mathfrak{I}
\equiv
\{\mathcal{I}'\in 2^{\mathcal{I}}: \mathcal{S}(\mathcal{I}')\neq\emptyset\}.
\end{equation*}
Here, let us introduce a few examples.
\begin{example}[Broadcast channel with a common message]
\label{ex:1}
The access structure
of a broadcast channel with a common message (Fig.~\ref{fig:example1})
can be written as
\begin{align*}
\mathcal{S}&\equiv\{1,2,12\}
\\
\mathcal{I}&\equiv\{1\}
\\
\mathcal{A}&\equiv\{(1,1),(2,1),(12,1)\},
\end{align*}
where
Encoder $1$ has access to Messages $1$, $2$, and $12$,
Message $12$ is reproduced by Decoders $1$ and $2$,
and Message $i$ is reproduced by Decoder $i$ for each $i\in\{1,2\}$.
We have
\begin{align*}
\mathcal{S}(1)&\equiv\{1,2,12\}
\\
\mathcal{I}(1)&\equiv\{1\}
\\
\mathcal{I}(2)&\equiv\{1\}
\\
\mathcal{I}(12)&\equiv\{1\}
\\
\mathfrak{I}&\equiv\{\{1\}\}
\\
\mathcal{S}(\{1\})&\equiv\{1,2,12\}.
\end{align*}
\end{example}
\begin{figure}[ht]
\begin{center}
\unitlength 0.52mm
\begin{picture}(57,53)(-5,10)
\put(0,46){\makebox(0,0)[r]{$M^{(n)}_1$}}
\put(0,33){\makebox(0,0)[r]{$M^{(n)}_{12}$}}
\put(0,20){\makebox(0,0)[r]{$M^{(n)}_2$}}
\put(0,46){\vector(20,-12){20}}
\put(0,33){\vector(20,0){20}}
\put(0,20){\vector(20,12){20}}
\put(27,48){\makebox(0,0){Encoder}}
\put(20,23){\framebox(14,20){$\Phi^{(n)}_1$}}
\put(34,33){\vector(1,0){10}}
\put(49,33){\makebox(0,0){$X^n_1$}}
\end{picture}
\end{center}
\caption{Access Structure of Example \ref{ex:1}}
\label{fig:example1}
\end{figure}
\begin{example}[Two-input multiple access channel with a common message]
\label{ex:2}
The access structure of a two-input multiple-access channel
with a common message (Fig.~\ref{fig:example2})
can be written as
\begin{align*}
\mathcal{S}&\equiv\{1,2,12\}
\\
\mathcal{I}&\equiv\{1,2\}
\\
\mathcal{A}&\equiv\{(1,1),(2,2),(12,1),(12,2)\},
\end{align*}
where Message $12$ is a common message for Encoders $1$ and $2$,
and Message $i$ is a private message for Encoder $i$ for each $i\in\{1,2\}$.
In other words, Encoder $i$ has access to Messages $i$ and $12$
for each $i\in\{1,2\}$.
This access structure is the same as that of
the two-user interference channel with a common message \cite{JXG08},
where Decoder $i$ reproduces Messages $i$ and $12$ for each $i\in\{1,2\}$.
We have
\begin{align*}
\mathcal{S}(1)&\equiv\{1,12\}
\\
\mathcal{S}(2)&\equiv\{2,12\}
\\
\mathcal{I}(1)&\equiv\{1\}
\\
\mathcal{I}(2)&\equiv\{2\}
\\
\mathcal{I}(12)&\equiv\{1,2\}
\\
\mathfrak{I}
&\equiv
\{
\{1,2\},
\{1\},
\{2\}
\}
\\
\mathcal{S}(\{1,2\})
&\equiv
\{12\}
\\
\mathcal{S}(\{1\})
&\equiv
\{1\}
\\
\mathcal{S}(\{2\})
&\equiv
\{2\}.
\end{align*}
\end{example}
\begin{figure}[ht]
\begin{center}
\unitlength 0.52mm
\begin{picture}(57,65)(-5,10)
\put(0,46){\makebox(0,0)[r]{$M^{(n)}_1$}}
\put(0,33){\makebox(0,0)[r]{$M^{(n)}_{12}$}}
\put(0,20){\makebox(0,0)[r]{$M^{(n)}_2$}}
\put(0,46){\vector(1,0){20}}
\put(0,33){\vector(20,12){20}}
\put(0,33){\vector(20,-12){20}}
\put(0,20){\vector(1,0){20}}
\put(27,60){\makebox(0,0){Encoders}}
\put(20,36){\framebox(14,20){$\Phi^{(n)}_1$}}
\put(20,10){\framebox(14,20){$\Phi^{(n)}_2$}}
\put(34,46){\vector(1,0){10}}
\put(49,46){\makebox(0,0){$X^n_1$}}
\put(34,20){\vector(1,0){10}}
\put(49,20){\makebox(0,0){$X^n_2$}}
\end{picture}
\end{center}
\caption{Access Structure of Example \ref{ex:2}}
\label{fig:example2}
\end{figure}
\begin{example}
\label{ex:3}
Here, we introduce an access structure
of a multiple-access channel with three inputs
(Fig.~\ref{fig:example3});
it is written as
\begin{align*}
\mathcal{S}&\equiv\{1,3,12,23,123\}
\\
\mathcal{I}&\equiv\{1,2,3\}
\\
\mathcal{A}&\equiv\lrb{
\begin{aligned}
&(1,1),(3,3),
\\
&(12,1),(12,2),(23,2),(23,3),
\\
&(123,1),(123,2),(123,3)
\end{aligned}
},
\end{align*}
where Message $i$ is a private message for Encoder $i$
for each $i\in\{1,3\}$,
Message $ij$ is a common message to Encoders $i$ and $j$
for each two-digit indexes $ij\in\{12,23\}$,
and Message $123$ is a common message to Encoders $1$, $2$, and $3$.
In other words,
Encoder $1$ has access to Messages $1$, $12$, and $123$,
Encoder $2$ has access to Messages $12$, $23$, and $123$,
and Encoder $3$ has access to Messages $3$, $23$, and $123$.
It should be noted that
there are partially-common messages $12$ and $23$,
that do not appear in two-input multiple access channels.
We have
\begin{align*}
\mathcal{S}(1)
&\equiv\{1,12,123\}
\\
\mathcal{S}(2)
&\equiv\{12,23,123\}
\\
\mathcal{S}(3)
&\equiv\{3,23,123\}
\\
\mathcal{I}(1)
&\equiv\{1\}
\\
\mathcal{I}(3)
&\equiv\{3\}
\\
\mathcal{I}(12)
&\equiv\{1,2\}
\\
\mathcal{I}(23)
&\equiv\{2,3\}
\\
\mathcal{I}(123)
&\equiv\{1,2,3\}
\\
\mathfrak{I}
&\equiv
\{
\{1,2,3\},
\{1,2\},
\{2,3\},
\{1\},
\{3\}
\}
\\
\mathcal{S}(\{1,2,3\})
&\equiv
\{123\}
\\
\mathcal{S}(\{1,2\})
&\equiv
\{12\}
\\
\mathcal{S}(\{2,3\})
&\equiv
\{23\}
\\
\mathcal{S}(\{1\})
&\equiv
\{1\}
\\
\mathcal{S}(\{3\})
&\equiv
\{3\}.
\end{align*}
\end{example}
\begin{figure}[ht]
\begin{center}
\unitlength 0.52mm
\begin{picture}(57,85)(-5,10)
\put(0,84){\makebox(0,0)[r]{$M^{(n)}_1$}}
\put(0,65){\makebox(0,0)[r]{$M^{(n)}_{12}$}}
\put(0,46){\makebox(0,0)[r]{$M^{(n)}_{123}$}}
\put(0,27){\makebox(0,0)[r]{$M^{(n)}_{23}$}}
\put(0,8){\makebox(0,0)[r]{$M^{(n)}_3$}}
\put(0,84){\vector(20,-11){20}}
\put(0,65){\vector(20,7){20}}
\put(0,46){\vector(20,25){20}}
\put(0,65){\vector(20,-18){20}}
\put(0,46){\vector(1,0){20}}
\put(0,27){\vector(20,18){20}}
\put(0,46){\vector(20,-25){20}}
\put(0,27){\vector(20,-7){20}}
\put(0,8){\vector(20,11){20}}
\put(27,86){\makebox(0,0){Encoders}}
\put(20,62){\framebox(14,20){$\Phi^{(n)}_1$}}
\put(20,36){\framebox(14,20){$\Phi^{(n)}_2$}}
\put(20,10){\framebox(14,20){$\Phi^{(n)}_3$}}
\put(34,72){\vector(1,0){10}}
\put(49,72){\makebox(0,0){$X^n_1$}}
\put(34,46){\vector(1,0){10}}
\put(49,46){\makebox(0,0){$X^n_2$}}
\put(34,20){\vector(1,0){10}}
\put(49,20){\makebox(0,0){$X^n_3$}}
\end{picture}
\end{center}
\caption{Access Structure of Example \ref{ex:3}}
\label{fig:example3}
\end{figure}
From the following two lemmas,
we have the fact that $\{\mathcal{S}(\mathcal{I}')\}_{\mathcal{I}'\in\mathfrak{I}}$ forms a partition of $\mathcal{S}$.
\begin{lem}
\label{lem:union}
\begin{equation*}
\bigcup_{\mathcal{I}'\in\mathfrak{I}}\mathcal{S}(\mathcal{I}')=\mathcal{S}.
\end{equation*}
\end{lem}
\begin{IEEEproof}
Since $\bigcup_{\mathcal{I}'\in\mathfrak{I}}\mathcal{S}(\mathcal{I}')\subset\mathcal{S}$ is trivial,
it is sufficient to show that $\mathcal{S}\subset\bigcup_{\mathcal{I}'\in\mathfrak{I}}\mathcal{S}(\mathcal{I}')$.
Assume that $s\in\mathcal{S}$ and $\mathcal{I}'\equiv\mathcal{I}(s)$.
Then we have $s\in\mathcal{S}(\mathcal{I}(s))=\{s':\mathcal{I}(s')=\mathcal{I}(s)\}$.
This implies that $\mathcal{S}(\mathcal{I}(s))\neq\emptyset$ and $\mathcal{I}'=\mathcal{I}(s)\in\mathfrak{I}$.
Then we have $s\in\bigcup_{\mathcal{I}'\in\mathfrak{I}}\mathcal{S}(\mathcal{I}')$
and $\mathcal{S}\subset\bigcup_{\mathcal{I}'\in\mathfrak{I}}\mathcal{S}(\mathcal{I}')$.
\end{IEEEproof}
\begin{lem}
\label{lem:disjoint}
For any $\mathcal{I}'$ and $\mathcal{I}''$ satisfying $\mathcal{I}'\neq\mathcal{I}''$,
we have
\begin{equation*}
\mathcal{S}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}'')=\emptyset.
\end{equation*}
\end{lem}
\begin{IEEEproof}
We show the lemma by contradiction.
Assume that $\mathcal{I}'\neq\mathcal{I}''$ and $\mathcal{S}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}'')\neq\emptyset$.
From $\mathcal{S}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}'')\neq\emptyset$,
there is $s\in\mathcal{S}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}'')$ satisfying $\mathcal{I}(s)=\mathcal{I}'$ and $\mathcal{I}(s)=\mathcal{I}''$.
Then we have $\mathcal{I}'=\mathcal{I}''$, which contradicts $\mathcal{I}'\neq\mathcal{I}''$.
\end{IEEEproof}
Let $\mathring{\cS}(\mathcal{I}')$ be defined as
\begin{equation}
\mathring{\cS}(\mathcal{I}')
\equiv
\bigcup_{\substack{
\mathcal{I}''\in\mathfrak{I}:
\\
\mathcal{I}'\subsetneq\mathcal{I}''
}}
\mathcal{S}(\mathcal{I}'').
\label{eq:oSI}
\end{equation}
Then we have the following lemmas.
\begin{lem}
\label{lem:disjoint-SIk-oSIk}
For any $\mathcal{I}'\in\mathfrak{I}$, we have
\begin{equation*}
\mathcal{S}(\mathcal{I}')\cap\mathring{\cS}(\mathcal{I}')=\emptyset.
\end{equation*}
\end{lem}
\begin{IEEEproof}
The lemma is shown immediately from Lemma~\ref{lem:disjoint}.
\end{IEEEproof}
\begin{lem}
\label{lem:cap-Si}
For any $\mathcal{I}'\in\mathfrak{I}$, we have
\begin{equation*}
\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)
=
\bigcup_{\substack{
\mathcal{I}''\in\mathfrak{I}:
\\
\mathcal{I}'\subset\mathcal{I}''
}}
\mathcal{S}(\mathcal{I}'')
=
\mathring{\cS}(\mathcal{I}')\cup\mathcal{S}(\mathcal{I}').
\end{equation*}
\end{lem}
\begin{IEEEproof}
The relation
$\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')=\mathring{\cS}(\mathcal{I}')\cup\mathcal{S}(\mathcal{I}')$
is shown immediately from the definition of $\mathring{\cS}(\mathcal{I}')$.
We show below the relations
$\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)\subset\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')$
and
$\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')\subset\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)$; together they imply
$\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)=\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')$.
First, assume that $s\in\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)$ and $\mathcal{I}''\equiv\mathcal{I}(s)$.
Then we have the fact that $s\in\mathcal{S}(i)$ for all $i\in\mathcal{I}'$.
Since $s\in\mathcal{S}(i)$ implies $i\in\mathcal{I}(s)$,
we have the fact that $\mathcal{I}'\subset\mathcal{I}(s)=\mathcal{I}''$.
Since $s\in\mathcal{S}(\mathcal{I}(s))=\{s':\mathcal{I}(s')=\mathcal{I}(s)\}$,
we have $\mathcal{S}(\mathcal{I}(s))\neq\emptyset$ and $\mathcal{I}''=\mathcal{I}(s)\in\mathfrak{I}$.
Then we have $s\in\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')$,
which implies
$\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)\subset\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')$.
Next, assume that $s\in\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}'\subset\mathcal{I}''}\mathcal{S}(\mathcal{I}'')$.
Then there is $\mathcal{I}''$ such that $\mathcal{I}'\subset\mathcal{I}''$ and $s\in\mathcal{S}(\mathcal{I}'')$.
Since $s\in\mathcal{S}(\mathcal{I}'')$ implies $\mathcal{I}'\subset\mathcal{I}''=\mathcal{I}(s)$,
we have $i\in\mathcal{I}(s)$ and $s\in\mathcal{S}(i)$ for all $i\in\mathcal{I}'$.
Then we have $s\in\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)$,
which implies
$\bigcup_{\mathcal{I}''\in\mathfrak{I}:\mathcal{I}''\supset\mathcal{I}'}\mathcal{S}(\mathcal{I}'')\subset\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)$.
\end{IEEEproof}
\begin{lem}
\label{lem:subsetSi}
For any $i\in\mathcal{I}'$,
we have
\begin{align}
\mathcal{S}(\mathcal{I}')
&\subset
\mathcal{S}(i)
\label{eq:SIsupSi}
\\
\mathring{\cS}(\mathcal{I}')
&\subset
\mathcal{S}(i).
\label{eq:SsupIsupSi}
\end{align}
The above relations imply that
Encoder $i$ has access to the set of messages
$M^{(n)}_{\mathcal{S}(\mathcal{I}')}\equiv\{M^{(n)}_s\}_{s\in\mathcal{S}(\mathcal{I}')}$
and $M^{(n)}_{\mathring{\cS}(\mathcal{I}')}\equiv\{M^{(n)}_s\}_{s\in\mathring{\cS}(\mathcal{I}')}$.
\end{lem}
\begin{IEEEproof}
The lemma is shown immediately from Lemma \ref{lem:cap-Si}.
\end{IEEEproof}
\begin{lem}
\label{lem:unionSIk=Si}
For any $i\in\mathcal{I}$, we have
\begin{equation*}
\bigcup_{\substack{
\mathcal{I}'\in\mathfrak{I}:
\\
i\in\mathcal{I}'
}}
\mathcal{S}(\mathcal{I}')=\mathcal{S}(i).
\end{equation*}
\end{lem}
\begin{IEEEproof}
First, we show $\bigcup_{\mathcal{I}'\in\mathfrak{I}: i\in\mathcal{I}'}\mathcal{S}(\mathcal{I}')\subset\mathcal{S}(i)$.
Let $s\in\bigcup_{\mathcal{I}'\in\mathfrak{I}: i\in\mathcal{I}'}\mathcal{S}(\mathcal{I}')$.
Then there is $\mathcal{I}'\in\mathfrak{I}$ such that $i\in\mathcal{I}'$ and $s\in\mathcal{S}(\mathcal{I}')$.
From $s\in\mathcal{S}(\mathcal{I}')$, we have $\mathcal{I}(s)=\mathcal{I}'$.
Since $i\in\mathcal{I}'$ implies $i\in\mathcal{I}(s)$, we have $s\in\mathcal{S}(i)$.
Next, we show $\mathcal{S}(i)\subset\bigcup_{\mathcal{I}'\in\mathfrak{I}: i\in\mathcal{I}'}\mathcal{S}(\mathcal{I}')$.
Let $s\in\mathcal{S}(i)$ and $\mathcal{I}'\equiv\mathcal{I}(s)$.
Then we have $i\in\mathcal{I}(s)=\mathcal{I}'$.
Since $s\in\mathcal{S}(\mathcal{I}(s))=\{s': \mathcal{I}(s')=\mathcal{I}(s)\}$,
we have $\mathcal{S}(\mathcal{I}(s))\neq\emptyset$ and $\mathcal{I}'=\mathcal{I}(s)\in\mathfrak{I}$.
Then we have $s\in\bigcup_{\mathcal{I}'\in\mathfrak{I}: i\in\mathcal{I}'}\mathcal{S}(\mathcal{I}')$.
From the above two facts, we have the lemma.
\end{IEEEproof}
In subsequent sections,
we assume that all elements in $\mathfrak{I}\equiv\{\mathcal{I}_1,\mathcal{I}_2,\ldots,\mathcal{I}_{|\mathfrak{I}|}\}$
are sorted in a linear extension of the reversed partial ordering,
which yields the following property:
$\mathcal{I}_k\subsetneq\mathcal{I}_{k'}$ implies $k'<k$
for all $k,k'\in\{1,2,\ldots,|\mathfrak{I}|\}$.
In Examples \ref{ex:1}--\ref{ex:3},
all elements of $\mathfrak{I}$ are sorted in this order.
An algorithm for computing the linear extension
is described in Appendix \ref{sec:tsort}.
From (\ref{eq:oSI}), we have
\begin{align}
\mathring{\cS}(\mathcal{I}_k)
=
\bigcup_{\substack{
k'\in\{1,\ldots,k-1\}:
\\
\mathcal{I}_k\subsetneq\mathcal{I}_{k'}
}}
\mathcal{S}(\mathcal{I}_{k'}).
\label{eq:ocSIk}
\end{align}
\section{Capacity Region}
This section introduces
the definition of a multi-letter characterized capacity region
for a general multiple-input-multiple-output channel coding \cite{CRNG-MULTI}.
Let $\mathrm{P}(\cdot)$ denote the probability of an event.
For each $i\in\mathcal{I}$,
Encoder $i$ generates channel input $X^n_i$
from the set of messages $M^{(n)}_{\mathcal{S}(i)}\equiv\{M^{(n)}_s\}_{s\in\mathcal{S}(i)}$.
Decoder $j$ receives channel output $Y^n_j$
and reproduces the set of messages
$\widehat{M}^{(n)}_{\mathcal{D}(j)}\equiv\{\widehat{M}^{(n)}_{j,s}\}_{s\in\mathcal{D}(j)}$,
where $\mathcal{D}(j)$ is the index set of messages reproduced by Decoder $j$,
and $\widehat{M}^{(n)}_{j,s}$ is the reproduction by Decoder $j$
corresponding to Message $s$.
Let $\widehat{M}^{(n)}_{\mathcal{D}(\mathcal{J})}\equiv\{\widehat{M}^{(n)}_{j,s}\}_{j\in\mathcal{J},s\in\mathcal{D}(j)}$.
Then the joint distribution of
$(M^{(n)}_{\mathcal{S}},X_{\mathcal{I}}^n,Y_{\mathcal{J}}^n,\widehat{M}^{(n)}_{\mathcal{D}(\mathcal{J})})$ is given as
\begin{align*}
&
\mu_{M^{(n)}_{\mathcal{S}}X_{\mathcal{I}}^nY_{\mathcal{J}}^n\widehat{M}^{(n)}_{\mathcal{D}(\mathcal{J})}}(
\boldsymbol{m}_{\mathcal{S}},\boldsymbol{x}_{\mathcal{I}},\boldsymbol{y}_{\mathcal{J}},\widehat{\mm}_{\mathcal{D}(\mathcal{J})}
)
\notag
\\*
&
=
\lrB{\prod_{j\in\mathcal{J}}\mu_{\widehat{M}^{(n)}_{\mathcal{D}(j)}|Y_j^n}(\widehat{\mm}_{\mathcal{D}(j)}|\boldsymbol{y}_{j})}
\mu_{Y_{\mathcal{J}}^n|X_{\mathcal{I}}^n}(\boldsymbol{y}_{\mathcal{J}}|\boldsymbol{x}_{\mathcal{I}})
\lrB{\prod_{i\in\mathcal{I}}\mu_{X_i^n|M^{(n)}_{\mathcal{S}(i)}}(\boldsymbol{x}_i|\boldsymbol{m}_{\mathcal{S}(i)})}
\lrB{\prod_{s\in\mathcal{S}}\frac 1{|\mathcal{M}^{(n)}_s|}}.
\end{align*}
We expect that, with probability close to $1$, $\widehat{M}^{(n)}_{j,s}=M^{(n)}_s$
for all $j\in\mathcal{J}$ and $s\in\mathcal{D}(j)$ if $n$ is sufficiently large.
We call rate vector $\{R_s\}_{s\in\mathcal{S}}$ {\em achievable}
if there is a (possibly stochastic) code
$\{(\{\Phi^{(n)}_i\}_{i\in\mathcal{I}},\{\Psi^{(n)}_j\}_{j\in\mathcal{J}})\}_{n=1}^{\infty}$
consisting of encoders $\Phi^{(n)}_i:\mathcal{M}^{(n)}_{\mathcal{S}(i)}\to\mathcal{X}^n_i$
and decoders $\Psi^{(n)}_j:\mathcal{Y}^n_j\to\mathcal{M}^{(n)}_{\mathcal{D}(j)}$ such that
\begin{gather}
\liminf_{n\to\infty}\frac{\log_2 |\mathcal{M}^{(n)}_s|}n\geq R_s
\quad\text{for all}\ s\in\mathcal{S}
\label{eq:ROP-rate}
\\
\lim_{n\to\infty}
\mathrm{P}\lrsb{
\widehat{M}^{(n)}_{j,s}\neq M^{(n)}_s\ \text{for some}\ j\in\mathcal{J}\ \text{and}\ s\in\mathcal{D}(j)
}
=0,
\label{eq:ROP-error}
\end{gather}
where $X_i^n\equiv\Phi^{(n)}_i(M^{(n)}_{\mathcal{S}(i)})$
and $\widehat{M}^{(n)}_{\mathcal{D}(j)}\equiv\Psi^{(n)}_j(Y_j^n)$.
Capacity region $\R_{\mathrm{OP}}$ is defined as
the closure of the set of all achievable rate vectors.
In the following, we use the information spectrum method
introduced in \cite{HAN}, and we do not assume conditions such as consistency,
stationarity, and ergodicity.
For sequence $\{\mu_{U_nV_n}\}_{n=1}^{\infty}$ of
joint probability distributions corresponding to
$(\boldsymbol{U},\boldsymbol{V})\equiv\{(U_n,V_n)\}_{n=1}^{\infty}$,
$\underline{H}(\boldsymbol{U}|\boldsymbol{V})$ denotes the spectral conditional inf-entropy rate
and $\overline{H}(\boldsymbol{U}|\boldsymbol{V})$ denotes the spectral conditional sup-entropy rate.
Formal definitions are given in Appendix~\ref{sec:ispec}.
Let $Z^n_{\mathcal{S}}\equiv\{Z^n_s\}_{s\in\mathcal{S}}$
be the random variables subject to the distribution defined as
\begin{align}
p_{Z^n_{\mathcal{S}}}(\boldsymbol{z}_{\mathcal{S}})
&=
\prod_{\substack{
\mathcal{I}'\in\mathfrak{I}
}}
p_{Z^n_{\mathcal{S}(\mathcal{I}')}|Z^n_{\mathring{\cS}(\mathcal{I}')}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')}),
\label{eq:jointZ}
\end{align}
where
\begin{align*}
p_{Z^n_{\mathcal{S}(\mathcal{I}')}|Z^n_{\mathring{\cS}(\mathcal{I}')}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')})
&\equiv
p_{Z^n_{\mathcal{S}(\mathcal{I}')}}(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')})
\quad\text{if}\ \mathring{\cS}(\mathcal{I}')=\emptyset.
\end{align*}
The alphabet of $Z^n_s$ is denoted by $\mathcal{Z}^n_s$ for each $s\in\mathcal{S}$.
It should be noted that it is unnecessary to assume
that $\mathcal{Z}_s^n$ is the $n$-dimensional Cartesian product of $\mathcal{Z}_s$.
We use notations $Z_s^n$ and $\mathcal{Z}_s^n$ to consider
the stationary memoryless case.
Let $\R_{\mathrm{IT}}$ be defined as the set of all $\{R_s\}_{s\in\mathcal{S}}$
satisfying the condition that
there are a set of general sources $\boldsymbol{Z}_{\mathcal{S}}\equiv\{\boldsymbol{Z}_s\}_{s\in\mathcal{S}}$
and a set of numbers $\{r_s\}_{s\in\mathcal{S}}$ such that
\begin{align}
R_s
&\geq
0
\label{eq:RIT-R}
\\
\sum_{s\in\mathcal{S}'}
[R_s+r_s]
&\leq
\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}(\mathcal{I}')})
\label{eq:RIT-rR}
\\
\sum_{s\in\mathcal{D}'}r_s
&\geq
\overline{H}(\boldsymbol{Z}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{Z}_{\mathcal{D}(j)\setminus\mathcal{D}'})
\label{eq:RIT-r}
\end{align}
for all $(\mathcal{I}',\mathcal{S}',j,\mathcal{D}')$ satisfying
$\mathcal{I}'\in\mathfrak{I}$, $\emptyset\neq\mathcal{S}'\subset\mathcal{S}(\mathcal{I}')$,
$j\in\mathcal{J}$, $\emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)$,
where the joint distribution of $(Z^n_{\mathcal{S}},X^n_{\mathcal{I}},Y_{\mathcal{J}}^n)$ is given as
\begin{align}
\mu_{Z_{\mathcal{S}}^nX_{\mathcal{I}}^nY_{\mathcal{J}}^n}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{x}_{\mathcal{I}},\boldsymbol{y}_{\mathcal{J}})
&=
\mu_{Y_{\mathcal{J}}^n|X_{\mathcal{I}}^n}(\boldsymbol{y}_{\mathcal{J}}|\boldsymbol{x}_{\mathcal{I}})
\lrB{\prod_{i\in\mathcal{I}}\mu_{X_i^n|Z^n_{\mathcal{S}(i)}}(\boldsymbol{x}_i|\boldsymbol{z}_{\mathcal{S}(i)})}
\mu_{Z^n_{\mathcal{S}}}(\boldsymbol{z}_{\mathcal{S}})
\label{eq:jointXYZ}
\end{align}
by using $\mu_{Z^n_{\mathcal{S}}}$ defined by (\ref{eq:jointZ}).
It should be noted that we can eliminate auxiliary variables
$\{r_s\}_{s\in\mathcal{S}}$ by applying the Fourier-Motzkin method \cite[Appendix D]{EK11}.
We have the following theorem.
The proof of $\R_{\mathrm{OP}}\subset\R_{\mathrm{IT}}$ is given in Section \ref{sec:converse}.
For the proof of $\R_{\mathrm{OP}}\supset\R_{\mathrm{IT}}$,
we construct a code in Section \ref{sec:channel-code}.
\begin{thm}
\label{thm:ROP=RIT}
\begin{equation*}
\R_{\mathrm{OP}}=\R_{\mathrm{IT}}.
\end{equation*}
\end{thm}
\begin{rem}
From (\ref{eq:jointZ})
and Lemma \ref{lem:independent} in Appendix \ref{sec:ispec},
we have the relation
$\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}(\mathcal{I}')})=\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathcal{S}\setminus\mathcal{S}(\mathcal{I}')})$
and the condition (\ref{eq:RIT-rR})
can be replaced by
\begin{equation*}
\sum_{s\in\mathcal{S}'}
[R_s+r_s]
\leq
\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathcal{S}\setminus\mathcal{S}(\mathcal{I}')}).
\end{equation*}
\end{rem}
\begin{rem}
When channels and auxiliary sources are stationary and memoryless,
we can replace $\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}(\mathcal{I}')})$ by $H(Z_{\mathcal{S}'}|Z_{\mathring{\cS}(\mathcal{I}')})$
and $\overline{H}(\boldsymbol{Z}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{Z}_{\mathcal{D}(j)\setminus\mathcal{D}'})$
by $H(Z_{\mathcal{D}'}|Y_j,Z_{\mathcal{D}(j)\setminus\mathcal{D}'})$
to obtain a single-letter characterized {\em achievable} region.
By considering an extension of the problem (super problem \cite{ICC})
with zero rate auxiliary messages, which is analogous to introducing
auxiliary random variables,
we can obtain a possible extension of the single-letter
characterized achievable region,
where the specific cases are given in \cite{ITW13}\cite{ICC}\cite{CRNG-MULTI}.
In \cite{CRNG-MULTI},
we find a multi-letter characterized capacity region
by using the reduction technique introduced
in \cite[Problems 14.22-14.24]{CK11}\cite{H79}.
It should be noted here that
the characterization presented in this paper
provides a potentially larger achievable region
when channels and auxiliary sources are restricted to being stationary and memoryless.
\end{rem}
\section{Joint Distribution Consistent with Access Structure}
Before proving Theorem \ref{thm:ROP=RIT},
we investigate the possible joint distribution
of random variables consistent with access structure~$\mathcal{A}$.
Let $\mathcal{I}'$ be a subset of $\mathcal{I}$.
We refer to $Z^n_{\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)}$
as {\em common sources for Encoders $\mathcal{I}'$},
to $Z^n_{\mathcal{S}(\mathcal{I}')}$ as {\em private sources for Encoders~$\mathcal{I}'$},
and to $Z^n_{\lrB{\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)}\cap\mathcal{S}(\mathcal{I}')^{\complement}}$
as {\em public sources for Encoders $\mathcal{I}'$}.
Here, let us consider the following problem.
Encoder $i$ has access to the set of random variables
$Z^n_{\mathcal{S}(i)}\equiv\{Z^n_s\}_{s\in\mathcal{S}(i)}$,
where $Z^n_s$ corresponds to message $M^{(n)}_s$
but random variables $\{Z^n_s\}_{s\in\mathcal{S}}$ are allowed to
be correlated on condition that
private sources for Encoders $\mathcal{I}'$
are allowed to be correlated with other sources
only through their public sources for all $\mathcal{I}'\in\mathfrak{I}$;
that is, they satisfy the following Markov relation
\begin{equation}
Z^n_{\lrB{\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)}^{\complement}}
\leftrightarrow Z^n_{\lrB{\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)}\cap\mathcal{S}(\mathcal{I}')^{\complement}}
\leftrightarrow Z^n_{\mathcal{S}(\mathcal{I}')}
\label{eq:markov}
\end{equation}
for all $\mathcal{I}'\in\mathfrak{I}$.
It should be noted that
common sources for Encoders $\mathcal{I}'$ are allowed to be correlated.
Other encoders may have access to the public sources for Encoders $\mathcal{I}'$
but do not have access to the private sources for Encoders $\mathcal{I}'$.
In this situation,
we specify the joint distribution of $\{Z^n_s\}_{s\in\mathcal{S}}$.
The following lemma solves this problem.
\begin{lem}
\label{lem:joint}
The following two statements are equivalent.
\begin{itemize}
\item
Joint source $Z^n_{\mathcal{S}}$ satisfies
(\ref{eq:markov}) for all $\mathcal{I}'\in\mathfrak{I}$.
\item
The joint distribution $\mu_{Z^n_{\mathcal{S}}}$ of $Z^n_{\mathcal{S}}$
is given by (\ref{eq:jointZ}).
\end{itemize}
\end{lem}
\begin{IEEEproof}
We have
\begin{align}
\lrB{\bigcap_{i\in\mathcal{I}'}\mathcal{S}(i)}\cap\mathcal{S}(\mathcal{I}')^{\complement}
&=
\lrB{\mathcal{S}(\mathcal{I}')\cup\mathring{\cS}(\mathcal{I}')}\cap\mathcal{S}(\mathcal{I}')^{\complement}
\notag
\\
&=
\lrB{\mathcal{S}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}')^{\complement}}
\cup
\lrB{\mathring{\cS}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}')^{\complement}}
\notag
\\
&=
\mathring{\cS}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}')^{\complement}
\notag
\\
&=
\lrB{\mathring{\cS}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}')^{\complement}}
\cup\lrB{\mathring{\cS}(\mathcal{I}')\cap\mathcal{S}(\mathcal{I}')}
\notag
\\
&=
\mathring{\cS}(\mathcal{I}'),
\label{eq:ZocSI'}
\end{align}
where the first equality comes from Lemma \ref{lem:cap-Si},
and the fourth equality comes from Lemma \ref{lem:disjoint-SIk-oSIk}.
Then we have the fact that the second statement implies the first statement.
In the following, we show that
the first statement implies the second statement.
From Lemma \ref{lem:cap-Si} and (\ref{eq:ZocSI'}),
we have the fact that condition (\ref{eq:markov}) is equivalent to
\begin{equation}
Z^n_{\lrB{\mathcal{S}(\mathcal{I}')\cup\mathring{\cS}(\mathcal{I}')}^{\complement}}
\leftrightarrow Z^n_{\mathring{\cS}(\mathcal{I}')}
\leftrightarrow Z^n_{\mathcal{S}(\mathcal{I}')}.
\label{eq:markov-ocSI'}
\end{equation}
Then we have
\begin{align}
\mu_{Z^n_{\mathcal{S}}}
(\boldsymbol{z}_{\mathcal{S}})
&=
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{Z^n_{\mathcal{S}(\mathcal{I}_k)}|Z^n_{\bigcup_{k'=1}^{k-1}\mathcal{S}(\mathcal{I}_{k'})}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_k)}|\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_{k'})})
\notag
\\
&=
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{Z^n_{\mathcal{S}(\mathcal{I}_k)}|Z^n_{\mathring{\cS}(\mathcal{I}_k)}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_k)}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}_k)})
\notag
\\
&=
\prod_{\mathcal{I}'\in\mathfrak{I}}
\mu_{Z^n_{\mathcal{S}(\mathcal{I}')}|Z^n_{\mathring{\cS}(\mathcal{I}')}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')}),
\end{align}
where $\{\mathcal{I}_k\}_{k=1}^{|\mathfrak{I}|}$
is defined at the end of Section \ref{sec:access},
the first equality comes from Lemmas \ref{lem:union} and \ref{lem:disjoint},
and the second equality comes from
(\ref{eq:ocSIk}) and (\ref{eq:markov-ocSI'}).
\end{IEEEproof}
\section{Proof of the Converse}
\label{sec:converse}
In the following, we prove $\R_{\mathrm{OP}}\subset\R_{\mathrm{IT}}$.
Assume that $\{R_s\}_{s\in\mathcal{S}}\in\R_{\mathrm{OP}}$ and let $r_s\equiv0$ for each $s\in\mathcal{S}$.
Then there is a code
$\{(\{\Phi^{(n)}_i\}_{i\in\mathcal{I}},\{\Psi^{(n)}_j\}_{j\in\mathcal{J}})\}_{n=1}^{\infty}$
that satisfies (\ref{eq:ROP-rate}) and (\ref{eq:ROP-error})
for all $i\in\mathcal{I}$ and $j\in\mathcal{J}$.
For $j\in\mathcal{J}$ and $\mathcal{D}'\subset\mathcal{D}(j)$,
let $\Psi^{(n)}_{j,\mathcal{D}'}(Y^n_j)$
be the projection of $\Psi^{(n)}_j(Y^n_j)$ on $\mathcal{M}^{(n)}_{\mathcal{D}'}$.
Then we have
\begin{equation*}
\lim_{n\to\infty} P(\Psi^{(n)}_{j,\mathcal{D}'}(Y_j^n)\neq M^{(n)}_{\mathcal{D}'})=0
\end{equation*}
from (\ref{eq:ROP-error}).
From Lemmas~\ref{lem:oH(U|V)>oH(U|VW)} and \ref{lem:fano}
in Appendix \ref{sec:ispec},
we have
\begin{align}
\overline{H}(\boldsymbol{M}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{M}_{\mathcal{D}(j)\setminus\mathcal{D}'})
&\leq
\overline{H}(\boldsymbol{M}_{\mathcal{D}'}|\boldsymbol{Y}_j)
\notag
\\
&=0.
\label{eq:HMgYM>=0}
\end{align}
From (\ref{eq:HMgYM>=0})
and $\overline{H}(\boldsymbol{M}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{M}_{\mathcal{D}(j)\setminus\mathcal{D}'})\geq 0$,
we have
\begin{equation*}
\overline{H}(\boldsymbol{M}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{M}_{\mathcal{D}(j)\setminus\mathcal{D}'})=0.
\end{equation*}
Then it is clear that
\begin{align}
\sum_{s\in\mathcal{D}'}r_s\geq \overline{H}(\boldsymbol{M}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{M}_{\mathcal{D}(j)\setminus\mathcal{D}'})
\label{eq:proof-converse-r}
\end{align}
for all $(j,\mathcal{D}')$ satisfying $j\in\mathcal{J}$ and $\emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)$.
Assume that $\mathcal{I}'\in\mathfrak{I}$ and $\mathcal{S}'\subset\mathcal{S}(\mathcal{I}')$.
Since the distribution $\mu_{M^{(n)}_{\mathcal{S}'}}$
is uniform on $\mathcal{M}^{(n)}_{\mathcal{S}'}$,
we have the fact that
\begin{align}
\frac1n\log_2\frac 1{\mu_{M^{(n)}_{\mathcal{S}'}}(\boldsymbol{m}_{\mathcal{S}'})}
&=
\frac1n\log_2|\mathcal{M}^{(n)}_{\mathcal{S}'}|
\notag
\\
&\geq
\liminf_{n\to\infty}\frac 1n \log_2|\mathcal{M}^{(n)}_{\mathcal{S}'}|-\delta
\end{align}
for all $\boldsymbol{m}_{\mathcal{S}'}\in\mathcal{M}^{(n)}_{\mathcal{S}'}$, $\delta>0$,
and all sufficiently large $n$.
This implies that
\begin{align}
\lim_{n\to\infty}
\mathrm{P}\lrsb{
\frac1n\log_2\frac 1{\mu_{M^{(n)}_{\mathcal{S}'}}(M^{(n)}_{\mathcal{S}'})}
<\liminf_{n\to\infty}\frac 1n \log_2|\mathcal{M}^{(n)}_{\mathcal{S}'}|-\delta
}
&=
0.
\label{eq:proof-converse-PM}
\end{align}
Let $\boldsymbol{M}_{\mathcal{S}'}\equiv\{M^{(n)}_{\mathcal{S}'}\}_{n=1}^{\infty}$ be a general source.
Then we have
\begin{align}
\liminf_{n\to\infty}\frac 1n \log_2|\mathcal{M}^{(n)}_{\mathcal{S}'}|-\delta
&\leq
\underline{H}(\boldsymbol{M}_{\mathcal{S}'})
\notag
\\
&=
\underline{H}(\boldsymbol{M}_{\mathcal{S}'}|\boldsymbol{M}_{\mathring{\cS}(\mathcal{I}')}),
\label{eq:proof-converse-uHM}
\end{align}
where the inequality comes from (\ref{eq:proof-converse-PM})
and the definition of $\underline{H}(\boldsymbol{M}_{\mathcal{S}'})$ in Appendix \ref{sec:ispec},
and the equality comes from
Lemma \ref{lem:independent} in Appendix \ref{sec:ispec}
and the fact that Lemma \ref{lem:disjoint-SIk-oSIk}
implies that $M^{(n)}_{\mathcal{S}'}$ and $M^{(n)}_{\mathring{\cS}(\mathcal{I}')}$ are independent.
We have
\begin{align}
\sum_{s\in\mathcal{S}'}
[R_s+r_s]
&=
\sum_{s\in\mathcal{S}'}
R_s
\notag
\\
&\leq
\liminf_{n\to\infty}\frac{\log_2|\mathcal{M}^{(n)}_{\mathcal{S}'}|}n
\notag
\\
&\leq
\underline{H}(\boldsymbol{M}_{\mathcal{S}'}|\boldsymbol{M}_{\mathring{\cS}(\mathcal{I}')})+\delta,
\end{align}
where the equality comes from the fact that $r_s=0$ for all $s\in\mathcal{S}$,
the first inequality comes from (\ref{eq:ROP-rate}),
and the second inequality comes from (\ref{eq:proof-converse-uHM}).
By letting $\delta\to0$, we have
\begin{align}
\sum_{s\in\mathcal{S}'}
[R_s+r_s]
&\leq
\underline{H}(\boldsymbol{M}_{\mathcal{S}'}|\boldsymbol{M}_{\mathring{\cS}(\mathcal{I}')})
\label{eq:proof-converse-rR}
\end{align}
for all $(\mathcal{I}',\mathcal{S}')$
satisfying $\mathcal{I}'\in\mathfrak{I}$ and $\emptyset\neq\mathcal{S}'\subset\mathcal{S}(\mathcal{I}')$.
Let $\boldsymbol{Z}_s\equiv\boldsymbol{M}_s$ for each $s\in\mathcal{S}$
and $\boldsymbol{X}_i\equiv\{\Phi^{(n)}_i(M^{(n)}_{\mathcal{S}(i)})\}_{n=1}^{\infty}$
for each $i\in\mathcal{I}$.
From Lemma \ref{lem:joint}
and the fact that $Z^n_{\mathcal{S}}$ satisfies (\ref{eq:markov}) for all $\mathcal{I}'\in\mathfrak{I}$,
the joint distribution of $(Z^n_{\mathcal{S}},X^n_{\mathcal{I}},Y_{\mathcal{J}}^n)$ is
allowed to be given by (\ref{eq:jointZ}) and (\ref{eq:jointXYZ}).
Then, from (\ref{eq:proof-converse-r}) and (\ref{eq:proof-converse-rR}),
we have $\{R_s\}_{s\in\mathcal{S}}\in\R_{\mathrm{IT}}$, which implies $\R_{\mathrm{OP}}\subset\R_{\mathrm{IT}}$.
\hfill\IEEEQED
\section{Construction of Channel Code}
\label{sec:channel-code}
This section introduces a channel code based on the idea drawn from
\cite{CRNG}\cite{HASH}\cite{HASH-BC}\cite{HASH-MAC}\cite{CRNG-CHANNEL};
a similar idea is found in \cite[Theorem 14.3]{CK11}\cite{YAG12}.
For each $s\in\mathcal{S}$, let us introduce a set $\mathcal{C}^{(n)}_s$
and two functions $f_s:\mathcal{Z}^n_s\to\mathcal{C}^{(n)}_s$ and $g_s:\mathcal{Z}^n_s\to\mathcal{M}^{(n)}_s$,
where the dependence of $f_s$ and $g_s$ on $n$ is omitted.
We can use sparse matrices as functions $f_s$ and $g_s$ by assuming that
$\mathcal{Z}_s^n$, $\mathcal{C}^{(n)}_s$, and $\mathcal{M}^{(n)}_s$ are linear spaces
on the same finite field.
For a given $\mathcal{S}'\subset\mathcal{S}$ and $f_{\mathcal{S}'}\equiv\{f_s\}_{s\in\mathcal{S}'}$,
$g_{\mathcal{S}'}\equiv\{g_s\}_{s\in\mathcal{S}'}$, $\boldsymbol{c}_{\mathcal{S}'}\equiv\{\boldsymbol{c}_s\}_{s\in\mathcal{S}'}$,
$\boldsymbol{m}_{\mathcal{S}'}\equiv\{\boldsymbol{m}_s\}_{s\in\mathcal{S}'}$,
let
\begin{align*}
\mathfrak{C}_{f_{\mathcal{S}'}}(\boldsymbol{c}_{\mathcal{S}'})
&\equiv
\{\boldsymbol{z}_{\mathcal{S}'}: f_s(\boldsymbol{z}_s)=\boldsymbol{c}_s\ \text{for all}\ s\in\mathcal{S}'\}
\\
\mathfrak{C}_{(f,g)_{\mathcal{S}'}}(\boldsymbol{c}_{\mathcal{S}'},\boldsymbol{m}_{\mathcal{S}'})
&\equiv
\{\boldsymbol{z}_{\mathcal{S}'}: f_s(\boldsymbol{z}_s)=\boldsymbol{c}_s, g_s(\boldsymbol{z}_s)=\boldsymbol{m}_s\ \text{for all}\ s\in\mathcal{S}'\},
\end{align*}
where $\boldsymbol{z}_{\mathcal{S}'}\equiv\{\boldsymbol{z}_s\}_{s\in\mathcal{S}'}$
and $(f,g)_{\mathcal{S}'}(\boldsymbol{z})\equiv\{(f_s(\boldsymbol{z}),g_s(\boldsymbol{z}))\}_{s\in\mathcal{S}'}$.
We define $\chi(\mathrm{S})$ as
\begin{gather*}
\chi(\mathrm{S})
\equiv
\begin{cases}
1
&\text{if statement $\mathrm{S}$ is true}
\\
0
&\text{if statement $\mathrm{S}$ is false}.
\end{cases}
\end{gather*}
We fix two sets of functions $f_{\mathcal{S}}$ and $g_{\mathcal{S}}$,
and a set of vectors $\boldsymbol{c}_{\mathcal{S}}$
such that they are available for constructing encoders and decoders.
For each $i\in\mathcal{I}$,
Encoder $i$ uses $f_{\mathcal{S}(i)}$, $g_{\mathcal{S}(i)}$, and $\boldsymbol{c}_{\mathcal{S}(i)}$.
For each $j\in\mathcal{J}$,
Decoder $j$ uses $f_{\mathcal{D}(j)}$, $g_{\mathcal{D}(j)}$, and $\boldsymbol{c}_{\mathcal{D}(j)}$.
We fix the probability distribution
$\mu_{Z^n_{\mathcal{S}}}\equiv\{\mu_{Z^n_s}\}_{s\in\mathcal{S}}$ given by (\ref{eq:jointZ})
and conditional probability distributions
$\{\mu_{X_i^n|Z^n_{\mathcal{S}(i)}}\}_{i\in\mathcal{I}}$.
We define a constrained-random-number generator for encoder use.
For given $\mathcal{I}'\in\mathfrak{I}$,
$\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')}$, $\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')}$, and $\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')}$,
let $\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}')}$ be a random variable corresponding to the distribution
\begin{align}
&
\mu_{\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}')}|\widetilde{Z}^n_{\mathring{\cS}(\mathcal{I}')}C^{(n)}_{\mathcal{S}(\mathcal{I}')}M^{(n)}_{\mathcal{S}(\mathcal{I}')}}
(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')},\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')},\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')})
\notag
\\*
&\equiv
\frac{
\mu_{Z^n_{\mathcal{S}(\mathcal{I}')}|Z^n_{\mathring{\cS}(\mathcal{I}')}}(\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')})
\chi(
\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}\in\mathfrak{C}_{(f,g)_{\mathcal{S}(\mathcal{I}')}}(\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')},\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')})
)
}{
\mu_{Z^n_{\mathcal{S}(\mathcal{I}')}|Z^n_{\mathring{\cS}(\mathcal{I}')}}
(
\mathfrak{C}_{(f,g)_{\mathcal{S}(\mathcal{I}')}}(\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')},\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')})
|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')}
)
},
\label{eq:tZI}
\end{align}
where $\{\mathring{\cS}(\mathcal{I}')\}_{I'\in\mathfrak{I}}$
is obtained before encoding by employing Algorithm \ref{alg:ocSIk}.
We assume that the constrained-random number generator outputs the same $\boldsymbol{z}_{\mathcal{S}(\mathcal{I}')}$ to all encoders that have access to message $\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')}$ for a given $\mathcal{I}'\in\mathfrak{I}$.
Then Encoder $i$ generates $\boldsymbol{z}_{\mathcal{S}(i)}$ by using Algorithm \ref{alg:zSi}
based on Lemma \ref{lem:unionSIk=Si} and (\ref{eq:jointZ}),
where (\ref{eq:ocSIk}) implies that
Encoder $i$ has obtained $\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}_k)}$
at Line 2 of Algorithm \ref{alg:zSi}.
We define encoding function $\Phi^{(n)}_i:\mathcal{M}^{(n)}_{\mathcal{S}(i)}\to\mathcal{X}_i^n$ as
\begin{equation*}
\Phi^{(n)}_i(\boldsymbol{m}_{\mathcal{S}(i)})
\equiv
W^n_i(\widetilde{Z}^n_{\mathcal{S}(i)}),
\end{equation*}
where the encoder claims an error when
$\mu_{Z_{\mathcal{S}(\mathcal{I}')}^n|Z_{\mathring{\cS}(\mathcal{I}')}^n}
(\mathfrak{C}_{(f,g)_{\mathcal{S}(\mathcal{I}')}}(\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')},\boldsymbol{m}_{\mathcal{S}(\mathcal{I}')})|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}')})=0$
and $W^n_i$ is the channel corresponding to
the conditional probability distribution $\mu_{X^n_i|Z^n_{\mathcal{S}(i)}}$.
The flow of vectors is illustrated in Fig.\ \ref{fig:encoder}.
\begin{rem}
By using the interval algorithm introduced in \cite{CRNG},
encoders can share the same output of
a given constrained-random-number generator
by sharing a fixed real number belonging to $[0,1]$.
\end{rem}
\begin{algorithm}[t]
\caption{Construction of $\{\mathring{\cS}(\mathcal{I}')\}_{I'\in\mathfrak{I}}$}
\hspace*{\algorithmicindent}\textbf{Input:}
List $\mathfrak{I}\equiv\{\mathcal{I}_k\}_{k=1}^{|\mathfrak{I}|}$,
which is sorted so that $\mathcal{I}_k\subsetneq\mathcal{I}_{k'}$ implies $k'<k$
for all $k,k'\in\{1,2,\ldots,|\mathfrak{I}|\}$.
\\
\hspace*{\algorithmicindent}\phantom{\textbf{Input:}}
List $\{\mathcal{S}(\mathcal{I}_k)\}_{k=1}^{|\mathfrak{I}|}$.
\\
\hspace*{\algorithmicindent}\textbf{Output:}
List $\{\mathring{\cS}(\mathcal{I}_k)\}_{k=1}^{|\mathfrak{I}|}$.
\label{alg:ocSIk}
\begin{algorithmic}[1]
\For{$k\in\{1,\ldots,|\mathfrak{I}|\}$}
\State $\mathring{\cS}(\mathcal{I}_k)\leftarrow\emptyset$
\For{$k'\in\{1,\ldots,k-1\}$}
\IIf{$\mathcal{I}_{k'}\supsetneq\mathcal{I}_k$}
$\mathring{\cS}(\mathcal{I}_k)\leftarrow\mathring{\cS}(\mathcal{I}_k)\cup\mathcal{S}(\mathcal{I}_{k'})$.
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[t]
\caption{Generation of $\boldsymbol{z}_{\mathcal{S}(i)}$}
\hspace*{\algorithmicindent}\textbf{Input:}
Lists $\{\mathcal{S}(\mathcal{I}_k)\}_{k=1}^{|\mathfrak{I}|}$, $\{\mathring{\cS}(\mathcal{I}_k)\}_{k=1}^{|\mathfrak{I}|}$,
$\boldsymbol{c}_{\mathcal{S}(i)}\equiv\{\boldsymbol{c}_s\}_{s\in\mathcal{S}(i)}$,
and $\boldsymbol{m}_{\mathcal{S}(i)}\equiv\{\boldsymbol{m}_s\}_{s\in\mathcal{S}(i)}$.
\\
\hspace*{\algorithmicindent}\textbf{Output:}
Vectors $\boldsymbol{z}_{\mathcal{S}(i)}\equiv\{\boldsymbol{z}_s\}_{s\in\mathcal{S}(i)}$.
\label{alg:zSi}
\begin{algorithmic}[1]
\For{$k\in\{1,\ldots,|\mathfrak{I}|\}$}
\IIf{$i\in\mathcal{I}_k$}
generate
$\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_k)}$
subject to the distribution
$\mu_{\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}_k)}|Z^n_{\mathring{\cS}(\mathcal{I}_k)}C^{(n)}_{\mathcal{S}(\mathcal{I}_k)}M^{(n)}_{\mathcal{S}(\mathcal{I}_k)}}
(\cdot|\boldsymbol{z}_{\mathring{\cS}(\mathcal{I}_k)},\boldsymbol{c}_{\mathcal{S}(\mathcal{I}_k)},\boldsymbol{m}_{\mathcal{S}(\mathcal{I}_k)})$.
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{figure}
\begin{center}
\unitlength 0.77mm
\begin{picture}(240,95)(0,0)
\put(126,85){\makebox(0,0){Encoder $\Phi_i$}}
\put(22,63){\makebox(0,0)[r]{$\boldsymbol{m}_{\mathcal{S}(\mathcal{I}_{k_1})}$}}
\put(22,41){\makebox(0,0)[r]{$\boldsymbol{m}_{\mathcal{S}(\mathcal{I}_{k_2})}$}}
\put(15,30){\makebox(0,0){$\vdots$}}
\put(22,11){\makebox(0,0)[r]{$\boldsymbol{m}_{\mathcal{S}(\mathcal{I}_{k_{|\mathcal{K}(i)|}})}$}}
\put(51,69){\makebox(0,0)[r]{$\boldsymbol{c}_{\mathcal{S}(\mathcal{I}_{k_1})}$}}
\put(52,70){\vector(1,0){8}}
\put(97,47){\makebox(0,0)[r]{$\boldsymbol{c}_{\mathcal{S}(\mathcal{I}_{k_2})}$}}
\put(98,48){\vector(1,0){8}}
\put(139,17){\makebox(0,0)[r]{$\boldsymbol{c}_{\mathcal{S}(\mathcal{I}_{k_{|\mathcal{K}(i)|}})}$}}
\put(140,18){\vector(1,0){8}}
\put(22,64){\vector(1,0){38}}
\put(22,42){\vector(1,0){84}}
\put(22,12){\vector(1,0){126}}
\put(60,58){\framebox(22,18){$\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}_{k_1})}$}}
\put(82,68){\vector(1,0){8}}
\put(98,67){\makebox(0,0){$\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_{k_1})}$}}
\put(106,68){\vector(1,0){94}}
\put(117,68){\vector(0,-1){14}}
\put(106,36){\framebox(22,18){$\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}_{k_2})}$}}
\put(128,46){\vector(1,0){8}}
\put(144,45){\makebox(0,0){$\boldsymbol{z}_{\mathcal{S}(\mathcal{I}_{k_2})}$}}
\put(152,46){\vector(1,0){48}}
\put(136,31){\makebox(0,0){$\ddots$}}
\put(166,68){\vector(0,-1){44}}
\put(162,46){\vector(0,-1){22}}
\put(158,31){\makebox(0,0)[r]{$\ldots$}}
\put(148,6){\framebox(22,18){$\widetilde{Z}^n_{\mathcal{S}(\mathcal{I}_{k_{|\mathcal{K}(i)|}})}$}}
\put(170,16){\vector(1,0){8}}
\put(188,15){\makebox(0,0){$\boldsymbol{z}_{\mathcal{S}_{\mathcal{I}_{k_{|\mathcal{K}(i)|}}}}$}}
\put(192,16){\vector(1,0){8}}
\put(200,6){\framebox(14,70){$W^n_i$}}
\put(214,44){\vector(1,0){16}}
\put(231,44){\makebox(0,0)[l]{$\boldsymbol{x}_i$}}
\put(30,0){\framebox(192,82){}}
\end{picture}
\end{center}
\caption{Construction of Encoder $i$:
It is assumed that
$\mathcal{K}(i)\equiv\{k: i\in\mathcal{I}_k\}\equiv\{k_1,k_2,\ldots,k_{|\mathcal{K}(i)|}\}$
satisfies $k_1<k_2<\cdots<k_{|\mathcal{K}(i)|}$.
Arrows from $\boldsymbol{z}_{\mathcal{S}(I_{k'})}$ to $\widetilde{Z}_{\mathcal{S}(I_k)}$
are ignored when $\mathcal{I}_k\subsetneq\mathcal{I}_{k'}$ is not satisfied.}
\label{fig:encoder}
\end{figure}
We define a constrained-random-number generator used by Decoder $j$.
For each $j\in\mathcal{J}$, Decoder $j$ generates
$\widehat{\zz}_{\mathcal{D}(j)}\equiv\{\widehat{\zz}_s\}_{s\in\mathcal{D}(j)}$
by using a constrained-random-number generator whose distribution is given as
\begin{align}
\mu_{\widehat{Z}_{\mathcal{D}(j)}^n|C^{(n)}_{\mathcal{D}(j)}Y_j^n}(\widehat{\zz}_{\mathcal{D}(j)}|\boldsymbol{c}_{\mathcal{D}(j)},\boldsymbol{y}_j)
&\equiv
\frac{
\mu_{Z_{\mathcal{D}(j)}^n|Y_j^n}(\widehat{\zz}_{\mathcal{D}(j)}|\boldsymbol{y}_j)
\chi(f_{\mathcal{D}(j)}(\widehat{\zz}_{\mathcal{D}(j)})=\boldsymbol{c}_{\mathcal{D}_j})
}{
\mu_{Z_{\mathcal{D}(j)}^n|Y_j^n}(\mathfrak{C}_{f_{\mathcal{D}(j)}}(\boldsymbol{c}_{\mathcal{D}(j)})|\boldsymbol{y}_j)
}
\label{eq:decoder}
\end{align}
for given vector $\boldsymbol{c}_{\mathcal{D}(j)}$ and side information $\boldsymbol{y}_j\in\mathcal{Y}_j^n$,
where $f_{\mathcal{D}(j)}(\widehat{\zz}_{\mathcal{D}(j)})\equiv\lrb{f_s(\widehat{\zz}_s)}_{s\in\mathcal{D}(j)}$.
We define the decoding function $\Psi^{(n)}_j:\mathcal{Y}_j^n\to\mathcal{M}^{(n)}_{\mathcal{D}(j)}$ as
\begin{equation*}
\Psi^{(n)}_j(\boldsymbol{y}_j)
\equiv
\{g_s(\widehat{\zz}_{j,s})\}_{s\in\mathcal{D}(j)}.
\end{equation*}
The flow of vectors is illustrated in Fig.\ \ref{fig:decoder}.
It should be noted here that $\widehat{Z}^n_{\mathcal{D}(j)}$ is analogous
to the decoder reproducing the output
$\boldsymbol{z}_{\mathcal{D}(j)}\equiv\{\boldsymbol{z}_s\}_{s\in\mathcal{D}(j)}$ of correlated sources,
where $\boldsymbol{c}_s\equiv f_s(\boldsymbol{z}_s)$ corresponds
to the codeword by using encoding function $f_s$.
It should be noted that, for sources that are memoryless,
the tractable approximation algorithms
for a constrained-random-number generator
summarized in~\cite{SDECODING} can be used;
the maximum a posteriori probability decoder
is optimal but may be intractable.
\begin{figure}
\begin{center}
\unitlength 0.78mm
\begin{picture}(126,100)(0,4)
\put(67,95){\makebox(0,0){Decoder $\Phi_j$}}
\put(116,77){\makebox(0,0)[l]{$\widehat{\mm}_{j,s_1}$}}
\put(116,57){\makebox(0,0)[l]{$\widehat{\mm}_{j,s_2}$}}
\put(120,46){\makebox(0,0){$\vdots$}}
\put(116,31){\makebox(0,0)[l]{$\widehat{\mm}_{j,s_{|\mathcal{D}(j)|}}$}}
\put(100,78){\vector(1,0){16}}
\put(100,58){\vector(1,0){16}}
\put(100,32){\vector(1,0){16}}
\put(84,70){\framebox(16,16){$g_{s_1}$}}
\put(84,50){\framebox(16,16){$g_{s_2}$}}
\put(91,46){\makebox(0,0){$\vdots$}}
\put(84,24){\framebox(16,16){$g_{s_{|\mathcal{D}(j)|}}$}}
\put(50,78){\vector(1,0){8}}
\put(50,58){\vector(1,0){8}}
\put(50,32){\vector(1,0){8}}
\put(68,78){\makebox(0,0){$\widehat{\zz}_{j,s_1}$}}
\put(68,58){\makebox(0,0){$\widehat{\zz}_{j,s_2}$}}
\put(68,46){\makebox(0,0){$\vdots$}}
\put(68,32){\makebox(0,0){$\widehat{\zz}_{j,s_{|\mathcal{D}(j)|}}$}}
\put(76,78){\vector(1,0){8}}
\put(76,58){\vector(1,0){8}}
\put(76,32){\vector(1,0){8}}
\put(34,10){\framebox(16,76){$\widehat{Z}^n_{\mathcal{D}(j)}$}}
\put(18,78){\vector(1,0){16}}
\put(18,58){\vector(1,0){16}}
\put(18,32){\vector(1,0){16}}
\put(18,18){\vector(1,0){16}}
\put(18,78){\makebox(0,0)[r]{$\boldsymbol{c}_{s_1}$}}
\put(18,58){\makebox(0,0)[r]{$\boldsymbol{c}_{s_2}$}}
\put(18,32){\makebox(0,0)[r]{$\boldsymbol{c}_{s_{|\mathcal{D}(j)|}}$}}
\put(18,18){\makebox(0,0)[r]{$\boldsymbol{y}_j$}}
\put(26,4){\framebox(82,88){}}
\end{picture}
\end{center}
\caption{Construction of Decoder $j$:
It is assumed that $\mathcal{D}(j)\equiv\{s_1,s_2,\ldots,s_{|\mathcal{D}(j)|}\}$.}
\label{fig:decoder}
\end{figure}
Let
\begin{align*}
r_s
&\equiv
\frac{\log|\mathcal{C}^{(n)}_s|}n
\\
R_s
&\equiv
\frac{\log|\mathcal{M}^{(n)}_s|}n,
\end{align*}
where $R_s$ represents the rate of Message $s$.
Let $\widehat{M}^{(n)}_{\mathcal{D}(j)}\equiv \Psi^{(n)}_j(Y^n_j)$
and $\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})$ be the error probability defined as
\begin{equation}
\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})
\equiv
\mathrm{P}\lrsb{
\begin{aligned}
&\widehat{M}^{(n)}_{j,s}\neq M^{(n)}_s\ \text{for some}\ j\in\mathcal{J}, s\in\mathcal{D}(j),
\\
&\text{or}
\ \mu_{Z_{\mathcal{S}(\mathcal{I}')}^n|Z_{\mathring{\cS}(\mathcal{I}')}^n}
(\mathfrak{C}_{(f,g)_{\mathcal{S}(\mathcal{I}')}}(\boldsymbol{c}_{\mathcal{S}(\mathcal{I}')},M^{(n)}_{\mathcal{S}(\mathcal{I}')})|Z^n_{\mathring{\cS}(\mathcal{I}')})=0
\ \text{for some}\ \mathcal{I}'\in\mathfrak{I}
\end{aligned}
}.
\label{eq:error}
\end{equation}
The following theorem,
implies the achievability part, $\R_{\mathrm{IT}}\subset\R_{\mathrm{OP}}$,
of Theorem \ref{thm:ROP=RIT}.
The proof is given in
Section~\ref{sec:proof-channel}.
\begin{thm}
\label{thm:channel}
For a given access structure $(\mathcal{S},\mathcal{I},\mathcal{A})$,
let us define $\mathfrak{I}$, $\mathcal{S}(\mathcal{I}')$, and $\mathring{\cS}(\mathcal{I}')$
as described in Section~\ref{sec:access}.
Let us assume that
$\{(r_s,R_s)\}_{s\in\mathcal{S}}$ satisfies
\begin{align}
R_s
&\geq
0
\label{eq:rate-positive}
\\
\sum_{s\in\mathcal{S}'}[R_s+r_s]
&<
\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}(\mathcal{I}')})
\label{eq:rate-encoder}
\\
\sum_{s\in\mathcal{D}'}r_s
&>
\overline{H}(\boldsymbol{Z}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{Z}_{\mathcal{D}(j)\setminus\mathcal{D}'})
\label{eq:rate-decoder}
\end{align}
for all $(\mathcal{I}',\mathcal{S}',j,\mathcal{D}')$ satisfying
$\mathcal{I}'\in\mathfrak{I}$, $\emptyset\neq\mathcal{S}'\subset\mathcal{S}(\mathcal{I}')$,
$j\in\mathcal{J}$, and
$\emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)$,
where the joint distribution of $(Z^n_{\mathcal{S}},X^n_{\mathcal{I}},Y^n_{\mathcal{J}})$
is given by (\ref{eq:jointZ}) and (\ref{eq:jointXYZ}).
Then for all $\delta>0$ and all sufficiently large $n$
there are functions (sparse matrices) $f_{\mathcal{S}}$, $g_{\mathcal{S}}$,
and a set of vectors $\boldsymbol{c}_{\mathcal{S}}$ such that
$\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})\leq\delta$.
\end{thm}
\begin{rem}
It should be noted that
for specific $\mu_{Z^n_{\mathcal{S}}X_{\mathcal{I}}^nY_{\mathcal{J}}^n}$ and $\{R_s\}_{s\in\mathcal{S}}$
we can find $\{r_s\}_{s\in\mathcal{S}}$
satisfying (\ref{eq:rate-encoder}) and (\ref{eq:rate-decoder})
by employing linear programming whenever they exist.
\end{rem}
\section{Proof of Theorem~\ref{thm:channel}}
\label{sec:proof-channel}
In the following,
we omit the dependence of $C$, $M$, $X$, $Y$, and $Z$ on $n$.
Let us assume that
ensembles $(\mathcal{F}_s,p_{F_s})$ and $(\mathcal{G}_s,p_{G_s})$,
where their dependence on $n$ is omitted,
have the hash property ((\ref{eq:hash}) described in Appendix \ref{sec:hash})
for every $s\in\mathcal{S}$.
For each $s\in\mathcal{S}$, let
\begin{align*}
\mathcal{C}_s
&\equiv
\mathrm{Im}\mathcal{F}_s
\\
&\equiv
\bigcup_{f\in\mathcal{F}}\{f(\boldsymbol{z}): \boldsymbol{z}\in\mathcal{Z}^n\}
\\
\mathcal{M}_s
&\equiv
\mathrm{Im}\mathcal{G}_s
\\
&\equiv
\bigcup_{g\in\mathcal{G}}\{g(\boldsymbol{z}): \boldsymbol{z}\in\mathcal{Z}^n\},
\end{align*}
where we omit the dependence of $\mathcal{C}_s$ and $\mathcal{M}_s$ on $n$.
We use the fact without notice that
\begin{equation*}
\{\mathfrak{C}_{(f,g)_{\mathcal{S}'}}(\boldsymbol{c}_{\mathcal{S}'},\boldsymbol{m}_{\mathcal{S}'})\}_{\boldsymbol{c}_{\mathcal{S}'}
\in\mathcal{C}_{\mathcal{S}'},\boldsymbol{m}_{\mathcal{S}'}\in\mathcal{M}_{\mathcal{S}'}}
\end{equation*}
forms a partition of $\mathcal{Z}_{\mathcal{S}'}^n$ for a given $\mathcal{S}'\subset\mathcal{S}$.
For a given $k\in\{1,2,\ldots,|\mathfrak{I}|\}$,
let us define
\begin{align*}
\mathcal{S}_k
&\equiv
\mathcal{S}(\mathcal{I}_k)
\\
\mathring{\cS}_k
&\equiv
\mathring{\cS}(\mathcal{I}_k)
\\
\mathcal{S}^k
&\equiv
\bigcup_{k'=1}^k
\mathcal{S}(\mathcal{I}_{k'}),
\end{align*}
where we $\{\mathcal{I}_k\}_{k=1}^{|\mathfrak{I}|}$ is defined
at the end of Section \ref{sec:access}.
We use the fact without notice that
$\{\mathcal{S}_k\}_{k=1}^{|\mathfrak{I}|}$ forms a partition of $\mathcal{S}$,
where it is shown by Lemmas \ref{lem:union} and \ref{lem:disjoint}.
We use the fact without notice that $\mathcal{S}_k$ and $\mathring{\cS}_k$ are disjoint,
where it is shown by Lemma~\ref{lem:disjoint-SIk-oSIk}.
Since (\ref{eq:ocSIk}) implies
\begin{align}
\mathring{\cS}_k
&\subset
\bigcup_{k'=1}^{k-1}
\mathcal{S}(\mathcal{I}_{k'})
\notag
\\
&=
\mathcal{S}^{k-1},
\end{align}
we have
\begin{equation}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}
=
\mu_{Z_{\mathcal{S}_k}|Z_{\mathring{\cS}_k}}
\label{eq:mu_ZSk|ZoSk}
\end{equation}
from (\ref{eq:jointZ}).
Let
\begin{align*}
\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
&\equiv
\lrb{
\boldsymbol{m}_{\mathcal{S}_k}:
\mu_{Z_{\mathcal{S}_k}|Z_{\mathring{\cS}_k}}
(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathring{\cS}_k}
)=0
\ \text{for some}
\ \boldsymbol{z}_{\mathring{\cS}_k}\in\mathcal{Z}^n_{\mathring{\cS}_k}
}
\\
\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})
&\equiv
\lrb{
\boldsymbol{m}_{\mathcal{S}}:
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
\ \text{for some}\ k\in\{1,\ldots,|\mathfrak{I}|\}
}
\\
\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
&\equiv
\lrb{
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}:
g_s(\widehat{\zz}_{j,s})\neq \boldsymbol{m}_s
\ \text{for some}\ j\in\mathcal{J}, s\in\mathcal{D}(j)
},
\end{align*}
where $\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\equiv\{\widehat{\zz}_{j,s}\}_{j\in\mathcal{J},s\in\mathcal{D}(j)}$.
It follows that error probability
$\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})$ is evaluated as
\begin{align}
\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})
&\leq
\sum_{
\boldsymbol{m}_{\mathcal{S}}\in\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})
}
\prod_{s\in\mathcal{S}}
\frac1{
|\mathcal{M}_s|
}
\notag
\\*
&\quad
+
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\notin\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}}),
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}),
\boldsymbol{x}_{\mathcal{I}}\in\mathcal{X}^n_{\mathcal{I}},
\\
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
}}
\lrB{
\prod_{j\in\mathcal{J}}
\mu_{\widehat{Z}_{\mathcal{D}(j)}|C_{\mathcal{D}(j)Y_j}}(\widehat{\zz}_{\mathcal{D}(j)}|\boldsymbol{c}_{\mathcal{D}(j)},\boldsymbol{y}_j)
}
\mu_{Y_{\mathcal{J}}|X_{\mathcal{I}}}(\boldsymbol{y}_{\mathcal{J}}|\boldsymbol{x}_{\mathcal{I}})
\notag
\\*
&\qquad\cdot
\lrB{
\prod_{i\in\mathcal{I}}
\mu_{X_i|Z_{\mathcal{S}(i)}}(\boldsymbol{x}_i|\boldsymbol{z}_{\mathcal{S}(i)})
}
\lrB{
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{\widetilde{Z}_{\mathcal{S}_k}|\widetilde{Z}_{\mathring{\cS}_k}C_{\mathcal{S}_k}M_{\mathcal{S}_k}}(
\boldsymbol{z}_{\mathcal{S}_k}
|\boldsymbol{z}_{\mathring{\cS}_k},\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k}
)
}
\lrB{
\prod_{s\in\mathcal{S}}\frac1{|\mathcal{M}_s|}
},
\label{eq:channel-error}
\end{align}
where the first term on the right hand side
corresponds to the encoding error probability
and the second term on the right hand side
corresponds to the decoding error probability.
By using the union bound, the first term on the right hand side of the equality
in (\ref{eq:channel-error}) is evaluated as
\begin{equation*}
\text{[the first term of (\ref{eq:channel-error})]}
\leq
\sum_{k=1}^{|\mathfrak{I}|}
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
}
\frac1{|\mathcal{M}_{\mathcal{S}_k}|}.
\label{eq:channel-error-1}
\end{equation*}
The second term on the right hand side of the equality
in (\ref{eq:channel-error}) is evaluated as
\begin{align}
&
\text{[the second term of (\ref{eq:channel-error})]}
\notag
\\*
&=
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\notin\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}}),
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}),
\\
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
}}
\frac{
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|\boldsymbol{c}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
}{
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
|\mathcal{M}_{\mathcal{S}_k}|
}
\notag
\\
&\leq
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\notin\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}}),
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}),
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
}}
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|\boldsymbol{c}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\notag
\\*
&\qquad\cdot
|\mathcal{C}_{\mathcal{S}}|
\lrB{
\lrbar{
\frac1{
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}})
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-1
}
+
1
}
\notag
\\
&\leq
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\notin\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}}),
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
}}
\mu_{Z_{\mathcal{S}}}(\boldsymbol{z}_{\mathcal{S}})
\lrbar{
\prod_{k=1}^{|\mathfrak{I}|}
\frac1{
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}})
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-1
}
\notag
\\*
&\quad
+
\sum_{j\in\mathcal{J}}
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\in\mathcal{M}_{\mathcal{S}},
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}),
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{Z}^n_{\mathcal{D}(\mathcal{J})}:
\\
\widehat{\zz}_{\mathcal{D}(j)}\neq \boldsymbol{z}_{\mathcal{D}(j)}
}}
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|\boldsymbol{c}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}}),
\label{eq:channel-error-2}
\end{align}
where the equality comes from (\ref{eq:tZI}), (\ref{eq:mu_ZSk|ZoSk}),
and the relations
\begin{align*}
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|\boldsymbol{c}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
&\equiv
\prod_{j\in\mathcal{J}}
\mu_{\widehat{Z}_{\mathcal{D}(j)}|C_{\mathcal{D}(j)}Y_j}(\widehat{\zz}_{\mathcal{D}(j)}|\boldsymbol{c}_{\mathcal{D}(j)},\boldsymbol{y}_j)
\\
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
&=
\sum_{\boldsymbol{x}_{\mathcal{I}}\in\mathcal{X}^n_{\mathcal{I}}}
\mu_{Y_{\mathcal{J}}|X_{\mathcal{I}}}(\boldsymbol{y}_{\mathcal{J}}|\boldsymbol{x}_{\mathcal{I}})
\lrB{
\prod_{i\in\mathcal{I}}
\mu_{X_i|Z_{\mathcal{S}(i)}}(\boldsymbol{x}_i|\boldsymbol{z}_{\mathcal{S}(i)})
}
\lrB{
\prod_{k=1}^{|\mathfrak{I}|}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathring{\cS}_k}}(\boldsymbol{z}_{\mathcal{S}_k}|\boldsymbol{z}_{\mathring{\cS}_k})
}.
\end{align*}
The first inequality comes from the triangular inequality,
and the second inequality comes from the relation
\begin{equation*}
\sum_{
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})
}
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|\boldsymbol{c}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\leq
\mu_{Z_{\mathcal{S}}}(\boldsymbol{z}_{\mathcal{S}})
\end{equation*}
and the union bound with the fact that
$\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{E}(g_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}})$
implies $\widehat{\zz}_{\mathcal{D}(j)}\neq\boldsymbol{z}_{\mathcal{D}(j)}$ for some $j\in\mathcal{J}$.
The first term on the right hand side of (\ref{eq:channel-error-2})
is evaluated as
\begin{align}
&
[\text{the first term of (\ref{eq:channel-error-2})}]
\notag
\\*
&\leq
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\in\mathcal{M}_{\mathcal{S}},
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}}}(\boldsymbol{c}_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}):
\\
\boldsymbol{m}_{\mathcal{S}_k}\notin\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
\ \text{for all}\ k\in\{1,\ldots,|\mathfrak{I}|\}
}}
\mu_{Z_{\mathcal{S}}}(\boldsymbol{z}_{\mathcal{S}})
\sum_{k=1}^{|\mathfrak{I}|}
\lrbar{
\frac1{
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}})
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-1
}
\notag
\\*
&\quad\cdot
\prod_{k'=k+1}^{|\mathfrak{I}|}
\frac1{
\mu_{Z_{\mathcal{S}_{k'}}|Z_{\mathcal{S}^{k'-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_{k'}}}(\boldsymbol{c}_{\mathcal{S}_{k'}},\boldsymbol{m}_{\mathcal{S}_{k'}})
|\boldsymbol{z}_{\mathcal{S}^{k'-1}})
|\mathcal{C}_{\mathcal{S}_{k'}}|
|\mathcal{M}_{\mathcal{S}_{k'}}|
}
\notag
\\
&=
\sum_{k=1}^{|\mathfrak{I}|}
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}^{k-1}}\in\mathcal{M}_{\mathcal{S}^{k-1}},
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{(f,g)_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}},\boldsymbol{m}_{\mathcal{S}^{k-1}}):
\\
\boldsymbol{m}_{\mathcal{S}_{k'}}\notin\mathcal{E}((f,g)_{\mathcal{S}_{k'}},\boldsymbol{c}_{\mathcal{S}_{k'}})
\ \text{for all}\ k'\in\{1,\ldots,k-1\}
}}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\notag
\\*
&\quad\cdot
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}_k}\notin\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
\\
\boldsymbol{z}_{\mathcal{S}_k}\in\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k}),
}}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}
(\boldsymbol{z}_{\mathcal{S}_k}|\boldsymbol{z}_{\mathcal{S}^{k-1}})
\lrbar{
\frac1{
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}})
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-1
}
\notag
\\*
&\quad\cdot
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}_{k+1}}\notin\mathcal{E}((f,g)_{\mathcal{S}_{k+1}},\boldsymbol{c}_{\mathcal{S}_{k+1}})
\\
\boldsymbol{z}_{\mathcal{S}_{k+1}}
\in\mathfrak{C}_{(f,g)_{\mathcal{S}_{k+1}}}(\boldsymbol{c}_{\mathcal{S}_{k+1}},\boldsymbol{m}_{\mathcal{S}_{k+1}})
}}
\frac{
\mu_{Z_{\mathcal{S}_{k+1}}|Z_{\mathcal{S}^k}}(
\boldsymbol{z}_{\mathcal{S}_{k+1}}|\boldsymbol{z}_{\mathcal{S}^k}
)
}{
\mu_{Z_{\mathcal{S}_{k+1}}|Z_{\mathcal{S}^k}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_{k+1}}}(\boldsymbol{c}_{\mathcal{S}_{k+1}},\boldsymbol{m}_{\mathcal{S}_{k+1}})
|\boldsymbol{z}_{\mathcal{S}^k})
|\mathcal{C}_{\mathcal{S}_{k+1}}|
|\mathcal{M}_{\mathcal{S}_{k+1}}|
}
\notag
\\*
&\ \quad\vdots
\notag
\\*
&\quad\cdot
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}(|\mathfrak{I}|)}\notin\mathcal{E}((f,g)_{\mathcal{S}_{|\mathfrak{I}|}},\boldsymbol{c}_{\mathcal{S}_{|\mathfrak{I}|}})
\\
\boldsymbol{z}_{\mathcal{S}_{|\mathfrak{I}|}}
\in\mathfrak{C}_{(f,g)_{\mathcal{S}_{|\mathfrak{I}|}}}(\boldsymbol{c}_{\mathcal{S}_{|\mathfrak{I}|}},\boldsymbol{m}_{\mathcal{S}_{|\mathfrak{I}|}})
}}
\frac{
\mu_{Z_{\mathcal{S}_{|\mathfrak{I}|}}|Z_{\mathcal{S}^{|\mathfrak{I}|-1}}}(
\boldsymbol{z}_{\mathcal{S}_{|\mathfrak{I}|}}|\boldsymbol{z}_{\mathcal{S}^{|\mathfrak{I}|-1}}
)
}{
\mu_{Z_{\mathcal{S}_{|\mathfrak{I}|}}|Z_{\mathcal{S}^{|\mathfrak{I}|-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_{|\mathfrak{I}|}}}(\boldsymbol{c}_{\mathcal{S}_{|\mathfrak{I}|}},\boldsymbol{m}_{\mathcal{S}_{|\mathfrak{I}|}})
|\boldsymbol{z}_{\mathcal{S}^{|\mathfrak{I}|-1}})
|\mathcal{C}_{\mathcal{S}_{|\mathfrak{I}|}}|
|\mathcal{M}_{\mathcal{S}_{|\mathfrak{I}|}}|
}
\notag
\\
&\leq
\sum_{k=1}^{|\mathfrak{I}|}
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{z}_{\mathcal{S}^{k-1}}
\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\notag
\\*
&\quad\cdot
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}_k}\notin\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
}}
\lrbar{
\frac1{
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
}
\prod_{k'=k+1}^{|\mathfrak{I}|}
\frac1{
|\mathcal{C}_{\mathcal{S}_{k'}}|
}
\notag
\\
&=
\sum_{k=1}^{|\mathfrak{I}|}
\lrB{
\prod_{k'=1}^k
|\mathcal{C}_{\mathcal{S}_{k'}}|
}
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{M}_{\mathcal{S}_k}
}
\lrbar{
\frac1{
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
}
\notag
\\*
&\quad
-
\sum_{k=1}^{|\mathfrak{I}|}
\lrB{
\prod_{k'=1}^{k-1}
|\mathcal{C}_{\mathcal{S}(k')}|
}
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})
}
\frac1{
|\mathcal{M}_{\mathcal{S}_k}|
},
\label{eq:channel-error-2-1}
\end{align}
where the first inequality comes from
Lemma \ref{lem:diff-prod} in Appendix \ref{sec:proof-lemma}
and the fact that
$\boldsymbol{m}_{\mathcal{S}}\notin\mathcal{E}((f,g)_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})$
iff
$\boldsymbol{m}_{\mathcal{S}_k}\notin\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})$
for all $k\in\{1,\ldots,|\mathfrak{I}|\}$;
the second inequality comes from the fact that
$\boldsymbol{m}_{\mathcal{S}_{k'}}\notin\mathcal{E}((f,g)_{\mathcal{S}_{k'}},\boldsymbol{c}_{\mathcal{S}_{k'}})$ implies
\begin{align}
\sum_{
\boldsymbol{z}_{\mathcal{S}_{k'}}
\in\mathfrak{C}_{(f,g)_{\mathcal{S}_{k'}}}(\boldsymbol{c}_{\mathcal{S}_{k'}},\boldsymbol{m}_{\mathcal{S}_{k'}})
}
\mu_{Z_{\mathcal{S}_{k'}}|Z_{\mathcal{S}^{k'-1}}}(
\boldsymbol{z}_{\mathcal{S}_{k'}}|\boldsymbol{z}_{\mathcal{S}^{k'-1}}
)
&=
\mu_{Z_{\mathcal{S}_{k'}}|Z_{\mathcal{S}^{k'-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_{k'}}}(\boldsymbol{c}_{\mathcal{S}_{k'}},\boldsymbol{m}_{\mathcal{S}_{k'}})
|\boldsymbol{z}_{\mathcal{S}^{k'-1}})
\notag
\\
&>0
\end{align}
for all $k'\in\{|\mathfrak{I}|,|\mathfrak{I}|-1,\ldots,k\}$
and
\begin{align}
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}_{k'}}\notin\mathcal{E}((f,g)_{\mathcal{S}_{k'}},\boldsymbol{c}_{\mathcal{S}_{k'}})
\\
\boldsymbol{z}_{\mathcal{S}_{k'}}
\in\mathfrak{C}_{(f,g)_{\mathcal{S}_{k'}}}(\boldsymbol{c}_{\mathcal{S}_{k'}},\boldsymbol{m}_{\mathcal{S}_{k'}})
}}
\frac{
\mu_{Z_{\mathcal{S}_{k'}}|Z_{\mathcal{S}^{k'-1}}}(
\boldsymbol{z}_{\mathcal{S}_{k'}}|\boldsymbol{z}_{\mathcal{S}^{k'-1}}
)
}{
\mu_{Z_{\mathcal{S}_{k'}}|Z_{\mathcal{S}^{k'-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_{k'}}}(\boldsymbol{c}_{\mathcal{S}_{k'}},\boldsymbol{m}_{\mathcal{S}_{k'}})
|\boldsymbol{z}_{\mathcal{S}^{k'-1}})
|\mathcal{M}_{\mathcal{S}_{k'}}|
}
&=
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}_{k'}}\notin\mathcal{E}((f,g)_{\mathcal{S}_{k'}},\boldsymbol{c}_{\mathcal{S}_{k'}})
}}
\frac1{
|\mathcal{M}_{\mathcal{S}_{k'}}|
}
\notag
\\
&\leq
1
\end{align}
for all $k'\in\{|\mathfrak{I}|,|\mathfrak{I}|-1,\ldots,k+1\}$.
The last equality comes from the fact that
$\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((f,g)_{\mathcal{S}_k},\boldsymbol{c}_{\mathcal{S}_k})$
implies
\begin{equation*}
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(f,g)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
=
0.
\end{equation*}
Let $C_{\mathcal{S}}\equiv\{C_s\}_{s\in\mathcal{S}}$ be the set of random variables
corresponding to the uniform distribution on $\mathcal{C}_{\mathcal{S}}$.
We have
\begin{align}
&
E_{(F,G)_{\mathcal{S}}C_{\mathcal{S}}}\lrB{
\mathrm{Error}(F_{\mathcal{S}},G_{\mathcal{S}},C_{\mathcal{S}})
}
\notag
\\*
&\leq
E_{(F,G)_{\mathcal{S}}C_{\mathcal{S}}}\left[
\sum_{k=1}^{|\mathfrak{I}|}
\lrB{
\prod_{k'=1}^k
|\mathcal{C}_{\mathcal{S}_{k'}}|
}
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{F_{\mathcal{S}^{k-1}}}(C_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\right.
\notag
\\*
&\qquad\cdot
\left.
\vphantom{
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{F_{\mathcal{S}^{k-1}}}(C_{\mathcal{S}^{k-1}})
}
}
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{M}_{\mathcal{S}_k}
}
\lrbar{
\frac{
1
}{
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(F,G)_{\mathcal{S}_k}}(C_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
}
\right]
\notag
\\*
&\quad
+
E_{(F,G)_{\mathcal{S}}C_{\mathcal{S}}}\lrB{
\sum_{j\in\mathcal{J}}
|\mathcal{C}_{\mathcal{S}}|
\sum_{\substack{
\boldsymbol{m}_{\mathcal{S}}\in\mathcal{M}_{\mathcal{S}},
\boldsymbol{z}_{\mathcal{S}}\in\mathfrak{C}_{(F,G)_{\mathcal{S}}}(C_{\mathcal{S}},\boldsymbol{m}_{\mathcal{S}}),
\\
\boldsymbol{y}_{\mathcal{J}}\in\mathcal{Y}^n_{\mathcal{J}},
\widehat{\zz}_{\mathcal{D}(\mathcal{J})}\in\mathcal{Z}^n_{\mathcal{D}(\mathcal{J})}:
\\
\widehat{\zz}_{\mathcal{D}(j)}\neq \boldsymbol{z}_{\mathcal{D}(j)}
}}
\mu_{\widehat{Z}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}}Y_{\mathcal{J}}}(\widehat{\zz}_{\mathcal{D}(\mathcal{J})}|C_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
\mu_{Z_{\mathcal{S}}\boldsymbol{Y}_{\mathcal{J}}}(\boldsymbol{z}_{\mathcal{S}},\boldsymbol{y}_{\mathcal{J}})
}
\notag
\\
&=
\sum_{k=1}^{|\mathfrak{I}|}
E_{(F,G)_{\mathcal{S}_k}}\left[
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathcal{Z}^n_{\mathcal{S}^{k-1}}
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\sum_{
\boldsymbol{c}_{\mathcal{S}_k}\in\mathcal{C}_{\mathcal{S}_k},
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{M}_{\mathcal{S}_k}
}
\lrbar{
\frac{
1
}{
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-
\mu_{Z_{\mathcal{S}_k}|Z_{\mathcal{S}^{k-1}}}(
\mathfrak{C}_{(F,G)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathcal{S}^{k-1}}
)
}
\right]
\notag
\\*
&\quad
+
\sum_{j\in\mathcal{J}}
E_{F_{\mathcal{D}(j)}}\lrB{
\sum_{\substack{
\boldsymbol{c}_{\mathcal{D}(j)}\in\mathcal{C}_{\mathcal{D}(j)},
\boldsymbol{z}_{\mathcal{D}(j)}\in\mathfrak{C}_{F_{\mathcal{D}(j)}}(\boldsymbol{c}_{\mathcal{D}(j)}),
\boldsymbol{y}_j\in\mathcal{Y}^n_j,
\widehat{\zz}_{\mathcal{D}(j)}\in\mathcal{Z}^n_{\mathcal{D}(j)}:
\\
\widehat{\zz}_{\mathcal{D}(j)}\neq \boldsymbol{z}_{\mathcal{D}(j)}
}}
\mu_{\widehat{Z}_{\mathcal{D}(j)}|Y_jC_{\mathcal{D}(j)}}
(\widehat{\zz}_{\mathcal{D}(j)}|\boldsymbol{y}_j,\boldsymbol{c}_{\mathcal{D}(j)})
\mu_{Z_{\mathcal{D}(j)}Y_j}(\boldsymbol{z}_{\mathcal{D}(j)},\boldsymbol{y}_j)
}
\notag
\\
&=
\sum_{k=1}^{|\mathfrak{I}|}
E_{(F,G)_{\mathcal{S}_k}}\lrB{
\sum_{\substack{
\boldsymbol{z}_{\mathring{\cS}_k}\in\mathcal{Z}^n_{\mathring{\cS}_k},
\boldsymbol{c}_{\mathcal{S}_k}\in\mathcal{C}_{\mathcal{S}_k},
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{M}_{\mathcal{S}_k}
}}
\mu_{Z_{\mathring{\cS}_k}}(\boldsymbol{z}_{\mathring{\cS}_k})
\lrbar{
\frac{
1
}{
|\mathcal{C}_{\mathcal{S}_k}|
|\mathcal{M}_{\mathcal{S}_k}|
}
-
\mu_{Z_{\mathcal{S}_k}|Z_{\mathring{\cS}_k}}(
\mathfrak{C}_{(F,G)_{\mathcal{S}_k}}(\boldsymbol{c}_{\mathcal{S}_k},\boldsymbol{m}_{\mathcal{S}_k})
|\boldsymbol{z}_{\mathring{\cS}_k})
}
}
\notag
\\*
&\quad
+
\sum_{j\in\mathcal{J}}
E_{F_{\mathcal{D}(j)}}\lrB{
\mu_{Z_{\mathcal{D}(j)}\widehat{Z}_{\mathcal{D}(j)}}\lrsb{\lrb{
(\boldsymbol{z}_{\mathcal{D}(j)},\widehat{\zz}_{\mathcal{D}(j)}):
\widehat{\zz}_{\mathcal{D}(j)}\neq\boldsymbol{z}_{\mathcal{D}(j)}
}
}
}
\notag
\\
&\leq
\sum_{k=1}^{|\mathfrak{I}|}
\sqrt{\textstyle
\alpha_{(F,G)_{\mathcal{S}_k}}-1
+\sum_{
\mathcal{S}'\subset\mathcal{S}_k:
\mathcal{S}'\neq\emptyset
}
\alpha_{(F,G)_{\mathcal{S}_k\setminus\mathcal{S}'}}
[\beta_{(F,G)_{\mathcal{S}'}}+1]
2^{-n\gamma(k,\mathcal{S}')}
}
+
2
\sum_{k=1}^{|\mathfrak{I}|}
\mu_{Z_{\mathring{\cS}_k\cup\mathcal{S}_k}}(\underline{\mathcal{T}}_k^{\complement})
\notag
\\*
&\quad
+2\sum_{j\in\mathcal{J}}
\sum_{\substack{
\mathcal{D}'\subset\mathcal{D}(j):
\mathcal{D}'\neq\emptyset
}}
\alpha_{F_{\mathcal{D}'}}\lrB{\beta_{F_{\mathcal{D}(j)\setminus\mathcal{D}'}}+1}
2^{
-n\gamma(j,\mathcal{D}')
}
+2 \sum_{j\in\mathcal{J}}
\beta_{F_{\mathcal{D}(j)}}
+2 \sum_{j\in\mathcal{J}}
\mu_{Z_{\mathcal{D}(j)}Y_j}(\overline{\mathcal{T}}_j^{\complement}),
\label{eq:proof-channel-error-ave}
\end{align}
where
\begin{align*}
\gamma(k,\mathcal{S}')
&\equiv
\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}_k})-\sum_{s\in\mathcal{S}'}[r_s+R_s]-\varepsilon
\\
\gamma(j,\mathcal{D}')
&\equiv
\sum_{s\in\mathcal{D}'}r_s-\overline{H}(\boldsymbol{Z}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{Z}_{\mathcal{D}(j)\setminus\mathcal{D}'})-\varepsilon
\\
\underline{\mathcal{T}}_k
&\equiv
\lrb{
(\boldsymbol{z}_{\mathcal{S}_k},\boldsymbol{z}_{\mathring{\cS}_k}):
\begin{aligned}
&\frac 1n
\log_2\frac1{
\mu_{Z^n_{\mathcal{S}'}|Z^n_{\mathring{\cS}_k}}(\boldsymbol{z}_{\mathcal{S}'}|\boldsymbol{z}_{\mathring{\cS}_k})
}
\geq
\underline{H}(\boldsymbol{Z}_{\mathcal{S}'}|\boldsymbol{Z}_{\mathring{\cS}_k})-\varepsilon
\\
&\text{for all}\ \mathcal{S}'\subset\mathcal{S}_k
\ \text{satisfying}\ \emptyset\neq\mathcal{S}'\subset\mathcal{S}_k
\end{aligned}
}
\\
\overline{\mathcal{T}}_j
&\equiv
\lrb{
(\boldsymbol{z}_{\mathcal{D}(j)},\boldsymbol{y}):
\begin{aligned}
&
\frac1n\log
\frac1{
\mu_{Z_{\mathcal{D}'}|Z_{\mathcal{D}(j)\setminus\mathcal{D}'}Y_j}
(\boldsymbol{z}_{\mathcal{D}'}|\boldsymbol{z}_{\mathcal{D}(j)\setminus\mathcal{D}'},\boldsymbol{y}_j)
}
\leq
\overline{H}(\boldsymbol{Z}_{\mathcal{D}'}|\boldsymbol{Y}_j,\boldsymbol{Z}_{\mathcal{D}(j)\setminus\mathcal{D}'})+\varepsilon
\\
&\text{for all}\ \mathcal{D}'\ \text{satisfying}\ \emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)
\end{aligned}
}.
\end{align*}
The first inequality comes from
(\ref{eq:channel-error})--(\ref{eq:channel-error-2-1})
and the fact that
\begin{align}
E_{C_{\mathcal{S}^{k-1}}}\lrB{
\lrB{
\prod_{k'=1}^{k-1}
|\mathcal{C}_{\mathcal{S}_{k'}}|
}
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
}
&=
\sum_{
\boldsymbol{c}_{\mathcal{S}^{k-1}}\in\mathcal{C}_{\mathcal{S}^{k-1}},
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\notag
\\
&=
1
\label{eq:proof-channel-ave-CS(1:k-1)}
\end{align}
implies
\begin{align*}
&
E_{F_{\mathcal{S}}G_{\mathcal{S}}C_{\mathcal{S}}}\lrB{
\sum_{k=1}^{|\mathfrak{I}|}
\lrB{
\prod_{k'=1}^{k-1}
|\mathcal{C}_{\mathcal{S}_{k'}}|
}
\sum_{
\boldsymbol{z}_{\mathcal{S}^{k-1}}\in\mathfrak{C}_{f_{\mathcal{S}^{k-1}}}(\boldsymbol{c}_{\mathcal{S}^{k-1}})
}
\mu_{Z_{\mathcal{S}^{k-1}}}(\boldsymbol{z}_{\mathcal{S}^{k-1}})
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((F,G)_{\mathcal{S}_k},C_{\mathcal{S}_k})
}
\frac1{
|\mathcal{M}_{\mathcal{S}_k}|
}
}
\notag
\\*
&=
\sum_{k=1}^{|\mathfrak{I}|}
E_{F_{\mathcal{S}_k}G_{\mathcal{S}_k}C_{\mathcal{S}_k}}\lrB{
\sum_{
\boldsymbol{m}_{\mathcal{S}_k}\in\mathcal{E}((F,G)_{\mathcal{S}_k},C_{\mathcal{S}_k})
}
\frac{
1
}{
|\mathcal{M}_{\mathcal{S}_k}|
}
},
\end{align*}
the second equality comes from (\ref{eq:mu_ZSk|ZoSk}),
and the last inequality comes from
Lemmas~\ref{lem:hash-FG}, \ref{lem:channel-bcp}, \ref{lem:channel-crp}
in Appendixes \ref{sec:hash}--\ref{sec:channel-crp},
and the relations $r_s=\log_2(|\mathcal{C}_s|)=\log_2(|\mathrm{Im}\mathcal{F}_s|)/n$,
$R_s=\log_2(|\mathcal{M}_s|)/n=\log_2(|\mathrm{Im}\mathcal{G}_s|)/n$.
Finally, let us assume that $\{(r_s,R_s)\}_{s\in\mathcal{S}}$ satisfies
(\ref{eq:rate-positive})--(\ref{eq:rate-decoder})
for all $(k,\mathcal{S}')$ satisfying
$k\in\{1,\ldots,|\mathfrak{I}|\}$, $\emptyset\neq\mathcal{S}'\subset\mathcal{S}_k$,
and $(j,\mathcal{D}')$ satisfying $j\in\mathcal{J}$, $\emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)$.
We have $\gamma(k,\mathcal{S}')>0$ and $\gamma(j,\mathcal{D}')>0$ for all $(k,\mathcal{S}',j,\mathcal{D}')$
satisfying $k\in\{1,\ldots,|\mathfrak{I}|\}$, $\emptyset\neq\mathcal{S}'\subset\mathcal{S}_k$,
$j\in\mathcal{J}$, $\emptyset\neq\mathcal{D}'\subset\mathcal{D}(j)$.
Then, by letting
$\alpha_{F_s}\to1$, $\beta_{F_s}\to0$,
$\alpha_{G_s}\to1$, $\beta_{G_s}\to0$,
$\mu_{Z_{\mathring{\cS}_k\cup\mathcal{S}_k}}(\underline{\mathcal{T}}_k^{\complement})\to0$,
$\mu_{Z_{\mathcal{D}(j)}Y_j}(\overline{\mathcal{T}}_j^{\complement})\to0$,
$\varepsilon\to0$,
and using the random coding argument,
we have the fact that
for all $\delta>0$ and sufficiently large $n$
there are $f_{\mathcal{S}}=\{f_s\}_{s\in\mathcal{S}}$, $g_{\mathcal{S}}=\{g_s\}_{s\in\mathcal{S}}$,
and $\boldsymbol{c}_{\mathcal{S}}=\{\boldsymbol{c}_s\}_{s\in\mathcal{S}}$
such that $\mathrm{Error}(f_{\mathcal{S}},g_{\mathcal{S}},\boldsymbol{c}_{\mathcal{S}})\leq\delta$.
\hfill\IEEEQED
|
2,877,628,090,222 | arxiv | \section{Introduction}
\subsection{Settings}
\label{sec-settings}
Let $\Omega$ be a bounded and connected open subset of $\mathbb{R}^d$, $d \geq 2$,
with Lipschitz boundary $\partial \Omega$.
Let $a:=(a_{i,j})_{1 \leq i,j \leq d} \in L^\infty(\Omega;\mathbb{R}^{d^2})\cap H^1(\Omega;\mathbb{R}^{d^2})$
be symmetric, that is
$$
a_{i,j}(x)=a_{j,i}(x),\ x \in \Omega,\ i,j = 1,\ldots,d,
$$
and fulfill the ellipticity condition: there exists a constant
$c>0$ such that
\begin{equation} \label{ell}
\sum_{i,j=1}^d a_{i,j}(x) \xi_i \xi_j \geq c |\xi|^2, \quad
\mbox{for a.e. $x \in \Omega,\ \xi=(\xi_1,\ldots,\xi_d) \in \mathbb{R}^d$}.
\end{equation}
Assume that $q \in L^{\frac{d}{2}}(\Omega)$ is non-negative and define the operator $\mathcal{A}$ by
$$
\mathcal{A} u(x) :=-\sum_{i,j=1}^d \partial_{x_i}
\left( a_{i,j}(x) \partial_{x_j} u(x) \right)+q(x)u(x),\ x\in\Omega.
$$
We set also $\rho \in L^\infty(\Omega)$ obeying
\begin{equation} \label{eq-rho}
0<c_0 \leq\rho(x) \leq C_0 <+\infty,\ x \in \Omega.
\end{equation}
From now on we set $\mathbb{R}_+=(0,+\infty)$ and we introduce the function $K\in L^1_{loc}(\mathbb{R}_+;L^\infty(\Omega))\cap C^\infty(\mathbb{R}_+;L^\infty(\Omega))$ satisfying th following condition
\begin{equation} \label{K} \inf\{\tau>0:\ e^{-\tau t}K\in L^1(\mathbb{R}_+;L^\infty(\Omega))\}=0.\end{equation}
Then, we define the operator $I_K$ by
$$I_Kg(t,x)=\int_0^tK(t-s,x)g(s,x)ds,\quad g\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega)),\ x\in\Omega,\ t\in\mathbb{R}_+.$$
We introduce the Caputo and Riemann-Liouville fractional derivative with kernel $K$ as follows
$$\partial^{K}_tg(t,x)=I_K\partial_tg(t,x),\quad D^{K}_tg(t,x)=\partial_tI_Kg(t,x),\quad g\in W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega)),\ x\in\Omega,\ t\in\mathbb{R}_+.$$
In the present article we consider the following initial boundary value problem (IBVP):
\begin{equation} \label{eq1}
\left\{ \begin{array}{rcll}
(\rho(x) \partial^{K}_t+\mathcal{A} ) u(t,x) & = & F(t,x), & (t,x)\in
\mathbb{R}_+\times\Omega,\\
u(t,x) & = & 0, & (t,x) \in \mathbb{R}_+\times\partial\Omega, \\
u(0,x) & = & u_0(x), & x \in \Omega.
\end{array}
\right.
\end{equation}
Namely, for different values of the kernel $K$, corresponding to variable order, distributed order and multiterm fractional diffusion equations, we prove the existence of weak solutions of \eqref{eq1} in the sense of a definition involving the Riemann-Liouville fractional derivative $D^{K}_t$. In addition, we would like to prove that this unique weak solution is described by a suitable Duhamel type of formula and its Laplace transform in time has the expected properties for such equations. Our goal is to unify the two different main approaches considered so far for defining solutions of \eqref{eq1}. That is the definition of solutions of \eqref{eq1} in a variational sense involving Riemann-Liouville fractional derivative (see e.g. \cite{EK,Z}) and the definition of solutions in term of Laplace transform (see e.g. \cite{KSY,KY1,KY2,LKS}).
\subsection{Definitions of weak and Laplace-weak solutions}
Before stating our results, we give the definition of solutions under consideration in the present article. Inspired by \cite{EK,Z},
we give the definition of weak solutions of this initial boundary value problem (IBVP in short) \eqref{eq1} as follows.
\begin{defn}\label{d1} \emph{(Weak solution)}
Let the coefficients in \eqref{eq1} satisfy
\eqref{ell}--\eqref{eq-rho}. We say that
$u\in L_{loc}^1(\mathbb{R}_+;L^2(\Omega))$ is a weak solution to \eqref{eq1} if it satisfies
the following conditions.
\begin{enumerate}
\item[{\rm(i)}] The following identity
\begin{equation}\label{d1a}
\rho( x)D_t^K[u-u_0](t, x)+\mathcal A u(t, x)=F(t,x),\quad t\in\mathbb{R}_+,\ x\in\Omega
\end{equation}
holds true in the sense of distributions in $\mathbb{R}_+\times\Omega$.
\item[{\rm(ii)}] We have $I_K [u-u_0]\in W_{loc}^{1,1}(\mathbb{R}_+;D'(\Omega))$ and the following initial condition
\begin{equation}\label{d1b}
I_K[u-u_0](0, x)=0,\quad x\in\Omega,
\end{equation}
is fulfilled.
\item[{\rm(iii)}] We have
$$
p_0=\inf\{\tau>0:\ e^{-\tau t}u\in L^1(\mathbb{R}_+;L^2(\Omega))\}<\infty
$$
and there exists $p_1\ge p_0$ such that for all $p\in\mathbb C$ satisfying $\mathfrak R p>p_1$ we have
$$
\widehat u(p,\,\cdot\,):=\int_0^\infty e^{-p t}u(t,\,\cdot\,)d t\in H^1_0(\Omega)
$$
\end{enumerate}
\end{defn}
\begin{rmk} The conditions in Definition \ref{d1} describe the different aspects of the IBVP \eqref{eq1}. Namely, condition (i) is associated with the equation in \eqref{eq1}, condition (ii) describes the link with the initial condition of \eqref{eq1} and condition (iii) gives the boundary condition of \eqref{eq1}. Let us also observe that in the spirit of the works \cite{EK,Z} \emph{(}see also \cite{KRY,KuY,P}\emph{)}, we use in the equation \eqref{d1a} the fact that for $u\in W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega))$ we have $\partial_t^K u=D_t^K[u-u(0,\cdot)]$. In that sense the expression $\partial_t^K u$ in \eqref{eq1} can be defined in a more general context by considering instead the expression $D_t^K[u-u(0,\cdot)]$.
\end{rmk}
In the present article we study the unique existence of a weak solution of the IBVP \eqref{eq1} in the sense of Definition \ref{d1} in three different context:\\
1) Variable order fractional diffusion equations where for $\alpha\in L^\infty(\Omega)$ satisfying
\begin{equation} \label{alpha}
0 < \alpha_0 \leq \alpha(x) \leq \alpha_M<1,\quad \alpha_M<2\alpha_0,\ x \in \Omega,
\end{equation}
we fix
\begin{equation} \label{Kvariable} K(t,x)=\frac{t^{-\alpha(x)}}{\Gamma(1-\alpha(x))},\quad t\in\mathbb{R}_+,\ x\in\Omega.\end{equation}
2) Distributed order fractional diffusion equations where for a non-negative weight function $\mu \in L^\infty(0,1)$, obeying the following condition:
\begin{equation} \label{mu}
\exists \alpha_0 \in(0,1),\ \exists \varepsilon \in (0,\alpha_0),\ \forall \alpha \in (\alpha_0-\varepsilon,\alpha_0),\ \mu(\alpha) \ge \frac{\mu(\alpha_0)}{2}>0,
\end{equation}
we define
\begin{equation} \label{Kdistributed} K(t,x)=\int_0^1 \mu(\alpha)\frac{t^{-\alpha}}{\Gamma(1-\alpha)}d\alpha,\quad t\in\mathbb{R}_+,\ x\in\Omega.\end{equation}
3) Multiple order fractional diffusion equations where, for $N\in\mathbb N$, $1<\alpha_1<\ldots<\alpha_N<1$ and for $\rho_j\in L^\infty(\Omega)$, $j=1,\ldots,N$, satisfying \eqref{eq-rho} with $\rho=\rho_j$, we fix
\begin{equation} \label{Kmultiple} K(t,x)=\sum_{j=1}^N\rho_j(x) \frac{t^{-\alpha_j}}{\Gamma(1-\alpha_j)},\quad t\in\mathbb{R}_+,\ x\in\Omega.\end{equation}
Let us also recall an alternative definition of weak solutions of \eqref{eq1} defined in terms of Laplace transform (see e.g. \cite{KSY,KY1,KY2,LKS}). In order to distinguish these two definitions of solutions, in the remaining part of this article, this class of weak solutions will be called Laplace-weak solutions. From now on and in all the remaining parts of this article, we denote by $\mathcal J$ the set of functions $F\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ for which there exists $J\in\mathbb N$ such that
$t\mapsto(1+t)^{-J}F(t,\cdot)\in L^1(\mathbb{R}_+;L^2(\Omega))$.
Following \cite{KSY,KY1,KY2,LKS}, we give the following definition of Laplace-weak solutions of \eqref{eq1}.
\begin{defn}\label{d2}\emph{(Laplace-weak solution)} Assume that $K$ is given by either of the three expressions \eqref{Kvariable}, \eqref{Kdistributed} and \eqref{Kmultiple}. Let $F\in \mathcal J$ and let the coefficients and the source term in \eqref{eq1} satisfy
\eqref{ell}--\eqref{eq-rho}. We say that
$u\in L_{loc}^1(\mathbb{R}_+;L^2(\Omega))$ is a Laplace-weak solution to \eqref{eq1} if it satisfies
the following conditions.
\begin{enumerate}
\item[{\rm(i)}]
$\inf\{\tau>0:\ e^{-\tau t}u\in L^1(\mathbb{R}_+;L^2(\Omega))\}=0$.
\item[{\rm(ii)}] There exists $p_1\ge 0$ such that for all $p\in\mathbb C$ satisfying $\mathfrak R p>p_1$, the Laplace transform $\widehat u(p,\cdot)$ of
$u(t,\cdot\,)$ with respect to $t$ is lying in $H^1_0(\Omega)$ and it solves the following boundary value
problem
\begin{equation}\label{d2a}
\begin{cases}
(\mathcal A+\rho p\widehat K(p,\cdot))\widehat u(p,\cdot)=\widehat F(p,\cdot)+\rho\widehat K(p,\cdot)u_0 & \mbox{in }\Omega,\\
\widehat u(p,\cdot\,)=0 & \mbox{on }\partial\Omega.
\end{cases}
\end{equation}
Note that here $\widehat K(p,\cdot)$ is well defined for all $p\in\mathbb C$ satisfying $\mathfrak R p>0$ thanks to condition \eqref{K}.
\end{enumerate}
\end{defn}
\begin{rmk} In Definition \ref{d2} all the properties of the IBVP \eqref{eq1} are described by the boundary value problem \eqref{d2a}.
Indeed, for a solution $u$ of \eqref{eq1} satisfying the condition
$$p_2=\inf\{\tau>0:\ e^{-\tau t}u\in W^{1,1}(\mathbb{R}_+;L^2(\Omega))\}<\infty,$$
we have
$$\widehat{\partial^{K}_t u}(p,\cdot\,)=\widehat K(p,\cdot)[p\widehat u(p,\cdot)-u(0,\cdot)]=p\widehat K(p,\cdot))\widehat u(p,\cdot)-\widehat K(p,\cdot)u_0,\quad p\in\mathbb C,\ \mathfrak R p>p_2.$$
Therefore, applying the Laplace transform in time to the equation of \eqref{eq1} we deduce that, for all $p\in\mathbb C$ satisfying $\mathfrak R p>p_1$, $\widehat u(p,\cdot)$ is the unique solution of \eqref{d2a}. Combining this with the uniqueness and the analyticity of the Laplace transform
we can conclude that such solution of \eqref{eq1} coincides with the Laplace-weak solution of \eqref{eq1}. In that sense, this notion of Laplace-weak solutions allows to define solutions of \eqref{eq1} in terms of properties of their Laplace transform in time.
\end{rmk}
The goal of the present article is to unify these two definitions by proving the equivalence between Definition \ref{d1} and Definition \ref{d2} in some general context with the kernel $K$ given by \eqref{Kvariable}, \eqref{Kdistributed}, \eqref{Kmultiple}. For this purpose, assuming that $F\in\mathcal J$ and $u_0\in L^2(\Omega)$, we will show the unique existence of Laplace-weak solutions of \eqref{eq1} in the sense of Definition \ref{d2}. After that we prove that the Laplace-weak solutions of \eqref{eq1} coincides with the unique weak solution of \eqref{eq1} in the sense of Definition \ref{d1}. This property shows in particular the equivalence between these definitions. In addition to this equivalence, we give also a Duhamel type of representation of the weak solutions of \eqref{eq1} taking the form \eqref{sol1}, \eqref{di1} and \eqref{mul}.
\subsection{Motivations}
Recall that anomalous diffusion in complex media have been intensively studied these last decades in different fields
with multiple applications in geophysics, environmental
and biological problems. The diffusion properties of homogeneous media are
currently modeled, see e.g. \cite{AG,CSLG}, by constant order
time-fractional diffusion processes where in \eqref{eq1} the kernel $K$ takes the form $\frac{t^{-\beta}}{\Gamma(1-\beta)}$ with a constant values $\beta\in(0,1)$. However, in some complex media, several physical properties lead to more general model involving variable order, distributed order and multiterm fractional diffusion equations. For instance, it has been proved that the presence of heterogeneous regions displays space inhomogeneous variations and the constant order fractional dynamic models are not robust for long times (see \cite{FS}). In this context the variable order time-fractional model, corresponding to kernel $K$ given by \eqref{Kvariable}, is more relevant for describing the space-dependent anomalous diffusion process (see e.g. \cite{SCC}). In this context, several variable order diffusion models have been successfully applied in numerous applications in sciences and engineering, including Chemistry \cite{CZZ},
Rheology \cite{SdV}, Biology \cite{GN}, Hydrogeology \cite{AON} and Physics \cite{SS, ZLL}. In the same way, some anomalous diffusion process such as ultra-slow diffusion, where the mean squared variance grows only logarithmically with time, are modeled by fractional diffusion equations with distributed order fractional derivatives with applications in polymer physics and kinetics of particles (see e.g. \cite{MMPG,MS}). For these different physical models, the goal of the present article is to prove existence of weak solutions of \eqref{eq1} enjoying several important properties such as resolution of the equation in the sense of distributions, suitable Duhamel representation formula and expected properties of the Laplace transform in time of the solutions stated in Definition \ref{d2}.
Beside these physical motivations, our analysis is also motivated by applications in other class of mathematical problems where the Duhamel representation formula, the properties of the Laplace transform in time of solutions as well as the resolution in the sense of distributions of the equation in \eqref{eq1}, stated in \eqref{d1a}-\eqref{d1b}, play an important role. This is for instance the case for several inverse problems (see e.g. \cite{JLLY,JK,KLLY,KLY,KOSY,KSXY,LIY}) as well as the study of some dynamical properties (see e.g. \cite{KSS,LLY}), the derivation of analyticity properties in time of solutions (see e.g. \cite{LLY1}) and the numerical resolution (see e.g. \cite{B,JLSZ}) of these equations. In this context our goal is to exhibit weak solutions that satisfy simultaneously all the above mentioned properties.
\subsection{Known results}
Recall that the well-posdness of the IBVP \eqref{eq1} has received a lot of attention these last decades among the mathematical community. For constant order fractional diffusion equations, where in \eqref{eq1} the kernel $K$ takes the form $\frac{t^{-\beta}}{\Gamma(1-\beta)}$ with a constant values $\beta\in(0,1)$, several approaches have been considered for defining solutions of \eqref{eq1}. This includes the definition of solutions of \eqref{eq1} in a variational and strong sense considered by \cite{EK,KuY,KRY,KY2,Z}, the definition of solutions in the mild-sense in \cite{SY} and the definition of solutions by mean of their Laplace transform in time given by \cite{KY1,KY2}. Such analysis includes also the study of the IBVP \eqref{eq1} with a time-dependent elliptic operator as stated in \cite{KuY,KRY,Z}. Several authors considered also the well-posedness of more general class of diffusion equations. For instance, the analysis of \cite{KJ,Z} in some abstract framework can be applied to some class of distributed order and multiterm fractional diffusion equation of the form \eqref{eq1} with a kernel $K$ independent of $x$ (for $K$ given by \eqref{Kmultiple} the coefficients $\rho_1,\ldots,\rho_N$ are constants). In the same way, we can mention the work of \cite{KR,LKS} for the study of distributed order fractional diffusion equations and the work of \cite{LHY,LLY2} devoted to the study of well-posedness of multiterm fractional diffusion equations with both constant and variable coefficients $\rho_1,\ldots,\rho_N$ in \eqref{Kmultiple}. To the best of our knowledge, in the article \cite{KSY} one can find the only result available in the mathematical literature devoted to the study of the well-posedness of variable order fractional diffusion equations (the kernel $K$ given by \eqref{Kvariable}) with non-vanishing initial condition and general source term. In this last work, the authors give a definition of solutions in term of Laplace transform comparable to Definition \ref{d2}. As far as we know, there is no result showing existence of weak solutions of \eqref{eq1} satisfying the properties described by Definition \ref{d1} for variable order fractional diffusion equations.
In all the above mentioned results the authors have either considered a variational definition of solutions comparable to Definition \ref{d1}
or a definition of solutions in terms of Laplace transform comparable to the Laplace-weak solutions of Definition \ref{d2}. However, as far as we know, there has been no result so far proving the unification of these two definitions of solutions for variable order, distributed order or multiterm fractional diffusion equations.
\subsection{Main results}
The main results of this article state the unique existence of Laplace-weak solutions of \eqref{eq1} in the sense of Definition \ref{d2} as well as the equivalence between Definition \ref{d1} and \ref{d2} for the weight function $K$ given by \eqref{Kvariable}, \eqref{Kdistributed}, \eqref{Kmultiple} which correspond to variable, distributed order and multiterm fractional diffusion equations.
For variable order fractional diffusion equations, our result can be stated as follows.
\begin{thm}\label{t1}
Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled. Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $\alpha\in L^\infty(\Omega)$ satisfy \eqref{alpha} and let $K$ be given by \eqref{Kvariable}. Then there exists a unique Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} in the sense of Definition \ref{d2}. Moreover, the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} is the unique weak solution of \eqref{eq1} in the sense of Definition \ref{d1}. In addition, $u$ is described by a Duhamel type of formula taking the form \eqref{sol1}.
\end{thm}
For distributed order fractional diffusion equations, our result can be stated as follows.
\begin{thm}\label{t2}
Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled. Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $\mu \in L^\infty(0,1)$ be a non-negative function satisfying \eqref{mu} and let $K$ be given by \eqref{Kdistributed}. Then there exists a unique Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} in the sense of Definition \ref{d2}. Moreover, the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} is the unique weak solution of \eqref{eq1} in the sense of Definition \ref{d1}. In addition, $u$ is described by a Duhamel type of formula taking the form \eqref{di1}.
\end{thm}
For multiterm fractional diffusion equations, our result can be stated as follows.
\begin{thm}\label{t3}
Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled with $\rho\equiv1$. Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $1<\alpha_1<\ldots<\alpha_N<1$ and $\rho_j\in L^\infty(\Omega)$, $j=1,\ldots,N$, satisfying \eqref{eq-rho} with $\rho=\rho_j$, and let $K$ be given by \eqref{Kmultiple}. Then there exists a unique Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} in the sense of Definition \ref{d2}. Moreover, the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} is the unique weak solution of \eqref{eq1} in the sense of Definition \ref{d1}. In addition, $u$ is described by a Duhamel type of formula taking the form \eqref{mul}.
\end{thm}
\subsection{Comments about our results}
To the best of our knowledge, in Theorem \ref{t1} we obtain the first result of unique existence of weak solutions of variable order fractional fractional diffusion equations solving in the sense of distributions the equation in \eqref{eq1}, as stated in \eqref{d1a}, and with explicit connection to the initial condition $u_0$ stated in \eqref{d1a}-\eqref{d1b}. As far as we know, the only other comparable results can be found in \cite{KSY} where the authors proved only existence of Laplace-weak solution of \eqref{eq1} with $K$ given by \eqref{Kvariable}. In that sense, Theorem \ref{t1} gives the first extension of the the analysis of \cite{KSY} by proving that the unique Laplace-weak solution under consideration in \cite{KSY} is also the unique weak solution in the sense of Definition \ref{d1}.
Let us observe that, in Theorem \ref{t2} and \ref{t3} we show, for what seems to be the first time, the unique existence of weak solutions of distributed order and multiterm fractional diffusion equations that enjoy simultaneously the following properties; 1) The weak solution solves in the sense of distributions the equation in \eqref{eq1} as stated in \eqref{d1a}; 2) The weak solution is explicitly connected with the initial condition $u_0$ by \eqref{d1a}-\eqref{d1b}; 3) The weak solution is described by a Duhamel type of formula; 4) The weak solution is also a Laplace-weak solution in the sense of Definition \ref{d1}. Indeed, several authors proved unique existence of solutions of distributed order and multiterm fractional diffusion equations enjoying the above properties 1) and 2) (see e.g. \cite{KJ,KR,LHY}) or the above properties 3) and 4) (see e.g. \cite{KSY}). Nevertheless, we are not aware of any results proving existence of weak solutions of distributed order fractional diffusion equations or multiterm fractional diffusion equations with variable coefficients enjoying simultaneously the above properties 1), 2), 3) and 4). In that sense, Theorem \ref{t2} and \ref{t3} show that these different properties of solutions of distributed order and multiterm fractional diffusion equations can be unified.
In contrast to Definition \ref{d2}, where the solutions are described by mean of the properties of Laplace transform in time of such class of fractional diffusion equations (see e.g. \cite{P} for more details), Definition \ref{d1} gives more explicit properties of solutions of \eqref{eq1}. Namely, the weak solution of \eqref{eq1}, in the sense of Definition \ref{d1}, solves the equation in \eqref{eq1} in the sense of distribution, as stated in \eqref{d1a}. Moreover, this class of weak solutions are also explicitly connected with the initial condition $u_0$ by mean of properties \eqref{d1a}-\eqref{d1b} and condition (iii) gives the boundary condition imposed to weak solutions in the sense of Definition \ref{d1}. By proving the equivalence between Definition \ref{d1} and Definition \ref{d2} of weak solutions, we show that the weak solution of \eqref{eq1} combine the explicit properties of Definition \ref{d1} with the properties of Laplace transform of solutions as stated in Definition \ref{d2}.
Let us observe that the results of Theorem \ref{t1}, \ref{t2} and \ref{t3} can be applied to the unique existence of solutions of the IBVP \eqref{eq1} at finite time (see the IBVP \eqref{eqq1}). This aspect is discussed in Section 5 of the present article with a definition of weak solutions stated in Definition \ref{d3} by mean of a weak solution at infinite time in the sense of Definition \ref{d1}. Our results for these issue are stated in Theorem \ref{t4}, where we show that the unique solution in the sense of Definition \ref{d3} is independent of the choice of the final time.
Let us observe that the boundary condition under consideration in \eqref{eq1} can be replaced, at the price of some minor modifications, by more general homogeneous Neumann or Robin boundary condition. In the spirit of the work \cite{KY2}, it is also possible to consider non-homogeneous boundary conditions. For simplicity we restrict our analysis to homogenous Dirichlet boundary conditions.
\subsection{Outline}
This paper is organized as follows. In Section 2, we prove the existence of a Laplace-weak solutions of the IBVP \eqref{eq1} in the sense of Definition \ref{d2} as well as the equivalence between Definition \ref{d1} and \ref{d2}, when $K$ is given by \eqref{Kvariable}, stated in Theorem \ref{t1}. In the same way, Section 3 and 4 are respectively devoted to the proof of Theorem \ref{t2} and \ref{t3}. Moreover, in Section 5, we study the same problem at finite time (see the IBVP \eqref{eqq1}) and we give a definition of solutions in that context stated in Definition \ref{d3}. We prove also in Theorem \ref{t3} the unique existence of solutions in the sense of Definition \ref{d3} as well as the independence of the unique solution in the sense of Definition \ref{d3} with respect to the final time.
\section{Variable order fractional diffusion equations}
In this section, we prove the unique existence of a weak
solution to the problem \eqref{eq1} as well as the equivalence between Definition \ref{d1} and \ref{d2} of weak and Laplace-weak solutions of \eqref{eq1} for weight $K$ given by \eqref{Kvariable} with $\alpha\in L^\infty(\Omega)$ satisfying \eqref{alpha}. For this purpose, let us first recall that the unique existence of solutions close to the Laplace-weak solutions for \eqref{eq1} has been proved by \cite[Theorem 1.1]{KSY} in the case of source terms $F\in L^\infty(\mathbb{R}_+;L^2(\Omega))$. We will recall here the representation of Laplace-weak solutions of \eqref{eq1} given by \cite{KSY}. For this purpose, we fix
$\theta\in(\frac{\pi}{2},\pi)$, $\delta>0$ and we define the contour in $\mathbb C$,
\begin{equation} \label{cont1}
\gamma(\delta,\theta):=\gamma_-(\delta,\theta)\cup\gamma_0(\delta,\theta)\cup\gamma_+(\delta,\theta),\end{equation}
oriented in the counterclockwise direction with
\begin{equation}\label{g2}
\gamma_0(\delta,\theta):=\{\delta\, e^{i\beta}:\ \beta\in[-\theta,\theta]\},\quad\gamma_\pm(\delta,\theta)
:=\{s\,e^{\pm i\theta}\mid s\in[\delta,\infty)\}.
\end{equation}
We denote also by $A$ the Dirichlet realization of the operator $\mathcal A$ acting on $L^2(\Omega)$ with domain $H^2(\Omega)\cap H^1_0(\Omega)$. Then, following
\cite{KSY}, we define the operators
\begin{equation}\label{S0}
S_0(t)\psi:=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}e^{t p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\rho p^{\alpha(\cdot)-1}\psi d p,
\quad t>0,
\end{equation}
\begin{equation}\label{S1}
S_1(t)\psi:=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}e^{t p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\psi d p,
\quad t>0.
\end{equation}
According to
\cite[Theorem 1.1]{KSY}, the definition of the operator valued functions $S_0,S_1$ are independent of the choice of
$\theta\in\left(\frac{\pi}{2},\pi\right)$, $\delta>0$. In light of \cite[Theorem 1.1]{KSY} and
\cite[Remark 1]{KSY}, for $u_0\in L^2(\Omega)$ and $F\in L^\infty(\mathbb{R}_+;L^2(\Omega))$, the function $u$ defined by
\begin{equation}\label{sol1}
u(t, \cdot )=S_0(t)u_0+
\int_0^t S_1(t-\tau)F(\tau,\cdot) d\tau,\quad t>0,
\end{equation}
is the unique tempered distribution with respect to the time variable $t\in\mathbb{R}_+$ taking values in $L^2(\Omega)$ whose Laplace transform in time solves \eqref{d2a}. This means that the function $u$ given by \eqref{sol1} will be the unique Laplace-weak solution of problem \eqref{eq1} in the sense of Definition \ref{d1} provided that $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$.
We start by proving an extension of this result to the unique existence of a Laplace-weak solution of problem \eqref{eq1} when $u_0\in L^2(\Omega)$ and $F\in\mathcal J$. For this purpose, we need the following intermediate result about the operator valued functions $S_0$ and $S_1$.
\begin{lem}\label{l1}
Let $\theta\in\left(\frac{\pi}{2},\pi\right)$. The maps $t\longmapsto S_j(t)$, $j=0,1$, defined by
\eqref{S0}-\eqref{S1} are lying in $L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$ and there
exists a constant $C>0$ depending only on $\mathcal A,\rho,\alpha,\theta,\Omega$
such that the estimates
\begin{equation}\label{l1a}
\|S_0(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{2(\alpha_M-\alpha_0)},
t^{2(\alpha_0-\alpha_M)},1\right),\quad t>0,
\end{equation}
\begin{equation}\label{ll2a}
\|S_1(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{2\alpha_M-\alpha_0-1},t^{2\alpha_0-\alpha_M-1},1\right),\quad t>0,
\end{equation}
hold true
\end{lem}
\begin{proof}
Throughout this proof, by $C>0$ we denote generic constants
depending only on $\mathcal{A},\rho,\alpha,\theta,\Omega$, which may change from line to line. In this lemma we only consider the proof of this lemma for the operator valued function $S_0$, for $S_1$ one can refer to \cite[Lemma 6.1]{KLY} for the proof of \eqref{ll2a}.
In light of \cite[Proposition 2.1]{KSY}, for all $\beta\in(0,\pi)$, we have
\begin{equation}\label{ll2b}
\left\|\left(A+\rho(r\,e^{i\beta_1})^{\alpha(\cdot)}\right)^{-1}\right\|_{\mathcal B(L^2(\Omega ))}
\le C\max\left(r^{\alpha_0-2\alpha_M},r^{\alpha_M-2\alpha_0}\right),
\quad r>0,\ \beta_1\in(-\beta,\beta).
\end{equation}
Using the fact that the operator $S_0$ is independent of the choice of $\delta>0$, we can
decompose
\[
S_0(t)=H_-(t)+H_0(t)+H_+(t),\quad t>0,
\]
where
\[
H_m(t)=\frac{1}{2i\pi}\int_{\gamma_m(t^{-1},\theta)} e^{t p}(\rho p^{\alpha(\cdot)}+A)^{-1}\rho p^{\alpha-1} dp,
\quad m=0,\mp,\ t>0.
\]
In order to complete the proof of the lemma, it suffices to prove
\begin{equation}\label{ll2c}
\|H_m(t)\|_{\mathcal B(L^2(\Omega))}\leq C\max\left(t^{2(\alpha_M-\alpha_0)},t^{2(\alpha_0-\alpha_M)},1\right),\quad t>0,\ m=0,\mp.
\end{equation}
Indeed, these estimates clearly implies \eqref{l1a}. Moreover, condition \eqref{alpha} implies that $2(\alpha_0-\alpha_M)>-\alpha_M>-1$ and we deduce from \eqref{l1a} that $S_0\in L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$. For $m=0$, using \eqref{ll2b}, we find
\[\begin{aligned}
\|H_0(t)\|_{\mathcal B(L^2(\Omega))}&\leq C\int_{-\theta}^\theta
t^{-1}\left\|\left(A+(t^{-1}e^{i\beta})^{\alpha(\cdot)}\right)^{-1}\right\|_{\mathcal B(L^2(\Omega)}\norm{|t^{-1}e^{i\beta}|^{\alpha(\cdot)-1}}_{L^\infty(\Omega)}
d\beta\\
&\leq C\max\left(t^{2(\alpha_M-\alpha_0)},t^{2(\alpha_0-\alpha_M)},1\right),\end{aligned}
\]
which implies \eqref{ll2c} for $m=0$. For $m=\mp$, again we
employ \eqref{ll2b} to estimate
$$\begin{aligned}
\|H_\mp(t)\|_{\mathcal B(L^2(\Omega))} & \leq C\int_{t^{-1}}^\infty e^{r t\cos\theta}
\left\|\left(A+(r\,e^{i\theta})^{\alpha(\cdot)}\right)^{-1}\right\|_{\mathcal B(L^2(\Omega)}d r\\
& \leq C\int_{t^{-1}}^\infty e^{r t\cos\theta}
\max\left(r^{\alpha_0-2\alpha_M},r^{\alpha_M-2\alpha_0}\right) \norm{r^{\alpha(\cdot)-1}}_{L^\infty(\Omega)}d r\\
& \leq C\int_{t^{-1}}^\infty e^{r t\cos\theta}
\max\left(r^{2(\alpha_0-\alpha_M)-1},r^{2(\alpha_M-\alpha_0)-1}\right) d r
\end{aligned}$$
For $t>1$, we obtain
$$\begin{aligned}
\|H_\mp(t)\|_{\mathcal B(L^2(\Omega))} & \leq C\int_1^\infty e^{r t\cos\theta}
r^{2(\alpha_M-\alpha_0)-1}d r+C\int_{t^{-1}}^1r^{2(\alpha_0-\alpha_M)-1} d r\\
& \leq C\int_0^\infty e^{r t\cos\theta}r^{2(\alpha_M-\alpha_0)-1}\,d r
+C\left(t^{2(\alpha_0-\alpha_M)}+1\right)\\
& \leq C\,t^{-1}\int_0^\infty e^{r\cos\theta}
\left(\frac{ r}{ t}\right)^{2(\alpha_M-\alpha_0)-1}d r
+C\left(t^{2(\alpha_0-\alpha_M)}+1\right)\\
& \leq C\max\left(t^{2(\alpha_0-\alpha_M)},t^{2(\alpha_M-\alpha_0)},1\right).
\end{aligned}$$
In the same way, for $t\in(0,1]$, we get
\[
\|H_\mp(t)\|_{\mathcal B(L^2(\Omega))}\leq C\int_1^\infty e^{r t\cos\theta}
r^{2(\alpha_M-\alpha_0)-1}d r\leq C\,t^{2(\alpha_M-\alpha_0)}.
\]
Combining these two estimates, we obtain
\[
\|H_\mp(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{2(\alpha_0-\alpha_M)},t^{2(\alpha_M-\alpha_0)},1\right),\quad t>0.
\]
This proves that \eqref{ll2c} also holds true for $m=\mp$. Therefore, estimate \eqref{l1a} holds true and we have $S_0\in L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$ which completes the proof of the lemma.
\end{proof}
We are now in position to state the existence of a unique Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1}, given by \eqref{sol1}, for any source term $F\in\mathcal J$.
\begin{prop}\label{p1} Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled.
Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $\alpha\in L^\infty(\Omega)$ satisfy \eqref{alpha} and let $K$ be given by \eqref{Kvariable}. Then there exists a unique Laplace-weak
solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ to \eqref{eq1} given by \eqref{sol1}.
\end{prop}
\begin{proof}
According to \cite[Theorem 1.1]{KSY} and Lemma \ref{l1}, we only need to prove this result for $u_0\equiv0$. Using Lemma \ref{l1}, we will complete the proof of Proposition \ref{p1} by mean of density arguments. Fix $$G(t,x)=(1+t)^{-J}F(t,x),\quad (t,x)\in\mathbb{R}_+\times\Omega$$
and recall that $G\in L^1(\mathbb{R}_+;L^2(\Omega))$. Therefore, we can find a sequence $(G_n)_{n\in\mathbb N}$ lying in $\mathcal C^\infty_0(\mathbb{R}_+\times\Omega)$ such that
$$\lim_{n\to\infty}\|G_n-G\|_{L^1(\mathbb{R}_+;L^2(\Omega))}=0.$$
Fixing $(F_n)_{n\in\mathbb N}$ a sequence of functions defined by
$$F_n(t,x)=(1+t)^{J}G_n(t,x),\quad (t,x)\in\mathbb{R}_+\times\Omega,\ n\in\mathbb N,$$
we deduce that the sequence $(F_n)_{n\in\mathbb N}$ is lying in $\mathcal C^\infty_0(\mathbb{R}_+\times\Omega)$ and we have
\begin{equation}\label{p1a}
\lim_{n\to\infty}\|(1+t)^{-J}(F_n-F)\|_{L^1(\mathbb{R}_+;L^2(\Omega))}=\lim_{n\to\infty}\|G_n-G\|_{L^1(\mathbb{R}_+;L^2(\Omega))}=0.
\end{equation}
In light of Lemma \ref{l1}, for $t> 0$ we can introduce
$$\begin{aligned}
u_n(t,\cdot\,) & :=\int_0^t S_1(t-\tau)F_n(\tau,\cdot)d\tau=\int_0^t S_1(\tau)F_n(t-\tau,\cdot)d\tau,
\quad n\in\mathbb N,\\
u(t,\,\cdot\,) & :=\int_0^t S_1(t-\tau)F(\tau,\cdot)d\tau
\end{aligned}$$
as elements of $L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$. We will prove that for all $p\in\mathbb C_+:=\{z\in\mathbb C:\ \mathfrak R z>0\}$, the
Laplace transform $\widehat u(p)$ of $u$ is well-defined in $L^2(\Omega)$ and we
have
\begin{equation}\label{p1c}
\lim_{n\to\infty}\|\widehat{u_n}(p)-\widehat u(p)\|_{L^2(\Omega)}=0.
\end{equation}
Applying estimate \eqref{ll2a}, we obtain
$$\begin{aligned}
\left\|e^{-p t}u(t,\,\cdot\,)\right\|_{L^2(\Omega)} & \leq\int_0^t
e^{-\mathfrak R p\tau}\|S_1(\tau)\|_{\mathcal B(L^2(\Omega))}\,
e^{-\mathfrak R p(t-\tau)}\|F(t-\tau,\cdot)\|_{L^2(\Omega)}d\tau\\
& \leq C\left(e^{-\mathfrak R p t}\max
\left(t^{2\alpha_0-\alpha_M-1},t^{2\alpha_M-\alpha_0-1},1\right)\right)*
\left(e^{-\mathfrak R p t}\|F(t,\cdot)\|_{L^2(\Omega)}\right)
\end{aligned}$$
for all $t>0$ and $p\in\mathbb C_+$, where $*$ denotes the convolution in
$\mathbb R_+$. Therefore, applying Young's convolution inequality
and condition \eqref{alpha}, we deduce
$$\begin{aligned}
&\|\widehat u(p)\|_{L^2(\Omega)}\\
& \leq\int_0^\infty
\left\| e^{-p t}u(t,\,\cdot\,)\right\|_{L^2(\Omega)}d t\\
& \leq C\left(\int_0^\infty e^{-\mathfrak R p t}
\max\left(t^{2\alpha_0-\alpha_M-1},t^{2\alpha_M-\alpha_0-1},1\right)d t\right)
\left(\int_0^\infty e^{-\mathfrak R p t}\|F(t,\cdot)\|_{L^2(\Omega)}d t\right)\\
&\leq C_p\left(\int_0^\infty e^{-\mathfrak R p t}
\max\left(t^{2\alpha_0-\alpha_M-1},t^{2\alpha_M-\alpha_0-1},1\right)d t\right) \|(1+t)^{-J}F(t,\cdot)\|_{L^1(\mathbb{R}_+;L^2(\Omega))}<\infty.
\end{aligned}$$
for all $p\in\mathbb C_+$. This proves that $\widehat u(p)$ is well defined for
all $p\in\mathbb C_+$ in the sense of $L^2(\Omega)$. In the same way, for all
$t>0$, $p\in\mathbb C_+$ and $n\in\mathbb N$, we get
$$\begin{aligned}
&\left\| e^{-p t}(u_n-u)(t,\,\cdot\,)\right\|_{L^2(\Omega)}\\
& \leq\int_0^te^{-\mathfrak R p(t-\tau)}\|S_1(t-\tau)\|_{\mathcal B(L^2(\Omega))}
e^{-\mathfrak R p\tau}\|F_n(\tau,\cdot)-F(\tau,\cdot)\|_{L^2(\Omega)}d\tau\\
& \le C\left(e^{-\mathfrak R p t}
\max\left(t^{2\alpha_0-\alpha_M-1},t^{2\alpha_M-\alpha_0-1},1\right)\right)*
\left(e^{-\mathfrak R p t}\|F_n(t,\cdot)-F(t,\cdot)\|_{L^2(\Omega)}\right).
\end{aligned}$$
Thus, applying Young's convolution inequality again, we have
$$\begin{aligned}
& \quad\,\|\widehat{u_n}(p)-\widehat u(p)\|_{L^2(\Omega)}
\le\int_0^\infty\left\|e^{-p t}(u_n-u)(t,\,\cdot\,)\right\|_{L^2(\Omega)}d t\\
& \le C\left(\int_0^\infty e^{-\mathfrak R p t}
\max\left(t^{2\alpha_0-\alpha_M-1},t^{2\alpha_M-\alpha_0-1},1\right)d t\right)
\left(\int_0^\infty e^{-\mathfrak R p t}\|F_n(t,\cdot)-F(t,\cdot)\|_{L^2(\Omega)}\,d t\right)\\
& \le C_p\left(\sup_{t\in\mathbb{R}_+}(1+t)^{J}e^{-\mathfrak R p t}\right)\|(1+t)^{-J}(F_n-F)\|_{L^1(\mathbb{R}_+;L^2(\Omega))}
\end{aligned}$$
for all $p\in\mathbb C_+$ and $n\in\mathbb N$, and \eqref{p1a} implies \eqref{p1c}. On the other
hand, in view of \cite[Theorem 1.1 and Remark 1]{KSY}, since
$F_n\in L^\infty(\mathbb{R}_+;L^2(\Omega))$, we have
\[
\left(\rho p^{\alpha(\cdot)}+A\right)\widehat{u_n}(p,\cdot)=\left(\int_0^\infty e^{-p t}F_n(t,\,\cdot\,)\,d t\right)
,\quad p\in\mathbb C_+,\ n\in\N.
\]
In addition, \eqref{p1a} implies that, for all $p\in\mathbb C_+$, we have
\[\begin{aligned}
\limsup_{n\to\infty}\norm{\widehat{F_n}(p)-\widehat{F}(p)}_{L^2(\Omega)}&\leq \limsup_{n\to\infty}\int_0^\infty e^{-\mathfrak R p t}\|F_n(t,\cdot)-F(t,\cdot)\|_{L^2(\Omega)}\,d t\\
\ &\leq \left(\sup_{t\in\mathbb{R}_+}(1+t)^{N}e^{-\mathfrak R p t}\right) \limsup_{n\to\infty}\|(1+t)^{-J}(F_n-F)\|_{L^1(\mathbb{R}_+;L^2(\Omega))}\\
\ &\leq0.\end{aligned}
\]
Therefore, we obtain
\[
\lim_{n\to\infty}
\left\|\widehat{u_n}(p)-\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\widehat F(p,\cdot)\right\|_{L^2(\Omega)}=0
\]
and \eqref{p1c} implies that $\widehat u(p)=\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\widehat F(p,\cdot)$,
$p\in\mathbb C_+$. From the definition of the operator $A$, we deduce that
$\widehat u(p)\in H^1_0(\Omega)$ solves the boundary value problem \eqref{d2a} for
all $p\in\mathbb C_+$. Recalling that the uniqueness of Laplace-weak solutions can be deduced easily from the uniqueness of the solution of \eqref{d2a} and the uniqueness of Laplace transform, we conclude that $u$ is the unique Laplace-weak
solution of \eqref{eq1} and the proof is completed.
\end{proof}
In view of Proposition \ref{p1} the first statement of Theorem \ref{t1} is fulfilled. Let us complete the proof of Theorem \ref{t1}.
\textbf{Proof of Theorem $\ref{t1}$.} Let us first observe that the first statement of Theorem \ref{t1} is a direct consequence of Proposition \ref{p1}. Therefore, in order to complete the proof of Theorem \ref{t1} we need to prove that \eqref{eq1} admits a unique weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ in the sense of Definition \ref{d1} given by \eqref{sol1} which is the unique Laplace-weak solution of \eqref{eq1}. We divide the proof of our result into three steps. We start by proving the uniqueness of the solution of \eqref{eq1} in the sense of Definition \ref{d2}. Then we prove that \eqref{sol1} is a weak solution of \eqref{eq1} in the sense of Definition \ref{d1} for $u_0\equiv0$. Finally, we show that \eqref{sol1} is a weak solution of \eqref{eq1} in the sense of Definition \ref{d1} for $F\equiv0$.
{\bf Step 1. } This step will be devoted to the proof of the uniqueness of weak solutions of \eqref{eq1} in the sense of Definition \ref{d1}.
For this purpose, let $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ be a weak solution of \eqref{eq1} with $F\equiv0$ and $u_0\equiv0$. In view of condition (iii) of Definition \ref{d1}, we can fix
$$
p_0=\inf\{\tau>0:\ e^{-\tau t}u\in L^1(\mathbb{R}_+;L^2(\Omega))\}.
$$
Then, using the fact that $a:=(a_{i,j})_{1 \leq i,j \leq d} \in H^1(\Omega;\mathbb{R}^{d^2})$, for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_0$, we have $e^{-p t}\mathcal A u\in L^1(\mathbb{R}_+;D'(\Omega))$ and condition \eqref{d2a} implies that
$$
e^{-p t}\rho D_t^K u=-e^{-p t}\mathcal A u,\quad t\in\mathbb{R}_+.
$$
It follows that, for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_0$, $e^{-p t}\rho D_t^K u\in L^1(\mathbb{R}_+;D'(\Omega))$. Combining this with condition (ii) of Definition \ref{d1}, we deduce that, for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_0$, $e^{-p t}\rho I_Ku\in W^{1,1}(\mathbb{R}_+;D'(\Omega))$. Therefore, multiplying \eqref{d1a} by $e^{-p t}$ with $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_0$, integrating over $t\in\mathbb{R}_+$ and using condition \eqref{d1b}, we find
\begin{align*}
0 & =\int_0^\infty e^{-p t}\left(\rho D_t^K u(t,\,\cdot\,)+\mathcal A u(t,\,\cdot\,)\right)d t=\int_0^\infty e^{-p t}\left(\partial_t[\rho I_Ku(t,\,\cdot\,)]+\mathcal A u(t,\,\cdot\,)\right)d t\\
& =p\rho\widehat{I_Ku}(p,\,\cdot\,)+\widehat{\mathcal A u}(p,\,\cdot\,)=\rho p^{\alpha(\cdot)}\widehat u(p,\,\cdot\,)+\mathcal A\widehat u(p,\,\cdot\,).
\end{align*}
Combining this with condition (iii) of Definition \ref{d2}, we deduce that, for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_1$, $\widehat u(p,\,\cdot\,)\in H^1_0(\Omega)$ solves the boundary value problem
$$
\begin{cases}
(\mathcal A+p^{\alpha(\cdot)}\rho)\widehat u(p,\,\cdot\,)=0 & \mbox{in }\Omega,\\
\widehat u(p,\,\cdot\,)=0 & \mbox{on }\partial\Omega.
\end{cases}
$$
On the other hand, applying \cite[Proposition 2.1]{KSY} we deduce that for all $p\in\mathbb C_+$ the operator $A+p^{\alpha(\cdot)}\rho$ is invertible as an operator acting on $L^2(\Omega)$. Therefore, we get that , for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_1$,
$\widehat u(p;\,\cdot\,)\equiv0$ and, combining this with the analyticity and the uniqueness of Laplace transform, we deduce that $u\equiv0$. This completes the proof of the uniqueness of weak solution of problem \eqref{eq1}.\medskip
{\bf Step 2.} In this step we will prove that the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1}, given by \eqref{sol1}, is the weak solution of \eqref{eq1} in the sense of Definition \ref{d1} when $u_0\equiv0$. Note that the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} clearly satisfies condition (iii) of Definition \ref{d1}. Thus, we only need to prove that the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} satisfies conditions (i) and (ii) of Definition \ref{d1}. In a similar way to Proposition \ref{p1}, we fix a sequence $(G_n)_{n\in\mathbb N}$ lying in $\mathcal C^\infty_0(\mathbb{R}_+\times\Omega)$ such that
$$\lim_{n\to\infty}\|G_n-G\|_{L^1(\mathbb{R}_+;L^2(\Omega))}=0$$
and $(F_n)_{n\in\mathbb N}$ a sequence of functions of $\mathcal C^\infty_0(\mathbb{R}_+\times\Omega)$ defined by
$$F_n(t,x)=(1+t)^{J}G_n(t,x),\quad (t,x)\in\mathbb{R}_+\times\Omega,\ n\in\mathbb N.$$
Then condition \eqref{p1a} is fulfilled. According to Proposition \ref{p1}, for all $n\in\mathbb N$, the Laplace-weak solution $u_n$ of \eqref{eq1} with $F=F_n$ is given by
$$
u_n(t,\,\cdot\,)=\int_0^t\,S(t-\tau)F_n(\tau,\cdot)d\tau=\int_0^t\,S(\tau)F_n(t-\tau,\cdot)d\tau,\quad t\in \mathbb R_+.
$$
Using the fact that $F_n\in\mathcal C^\infty_0(\mathbb{R}_+\times\Omega)$, $n\in\mathbb N$, and applying estimate \eqref{ll2a}, we deduce that $u_n\in C^1([0,\infty);L^2(\Omega))$ and $u_n(0,x)=0,\ x\in\Omega$.
Moreover, in view of \eqref{ll2a}, applying Young's convolution inequality, for all $p\in\mathbb C_+$, we get
$$\begin{aligned}
& \quad\,\| e^{-p t}u_n\|_{L^1(\mathbb{R}_+;L^2(\Omega))}+\|e^{-p t}\partial_tu_n\|_{L^1(\mathbb{R}_+;L^2(\Omega))}\\
& \leq C\left\|\left(e^{-(\mathfrak R\,p)t}(\|F_n(t)\|_{L^2(\Omega)}+\|\partial_tF_n(t)\|_{L^2(\Omega)})\right)*\left(e^{-(\mathfrak R\,p)t}\|S(t)\|_{\mathcal B(L^2(\Omega))}\right)\right\|_{L^1(\mathbb{R}_+)}\\
& \leq C\|F_n\|_{W^{1,1}(\mathbb{R}_+;L^2(\Omega))}\int_0^\infty e^{-(\mathfrak R\,p)t}\max\left(t^{2\alpha_M-\alpha_0-1},t^{2\alpha_0-\alpha_M-1},1\right)d t<\infty.
\end{aligned}$$
Thus, for all $n\in\mathbb N$, we have $D_t^K u_n=\partial_t^K u_n$ and we deduce that, for all $p\in\mathbb C_+$, we have
$$
\widehat{D_t^K u_n}(p,\,\cdot\,)=p^{\alpha(\cdot)}\widehat{u_n}(p,\,\cdot\,).
$$
Therefore, using the fact that for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_1$, $\widehat{u_n}(p,\,\cdot\,)$ solves \eqref{d2a} with $F=F_n$ and $u_0\equiv0$, we deduce that
$$
\widehat{D_t^K u_n}(p,\,\cdot\,)=p^{\alpha(\cdot)}\widehat{u_n}(p,\,\cdot\,)=-\rho^{-1}\mathcal A\widehat u_n(p,\,\cdot\,)+\rho^{-1}\widehat{F_n}(p,\cdot)=\widehat{w_n}(p,\,\cdot\,),\quad p\in\mathbb C,\ \mathfrak R\,p>p_1,
$$
where $w_n(t, x)=-\mathcal A u_n(t, x)+F_n(t,x)$, $(t, x)\in \mathbb{R}_+\times\Omega$.
Combining this with the uniqueness and the analyticity of the Laplace transform in time of $u_n$, we deduce that the identity
\
\rho(x) D_t^{K}u_n(t, x)+\mathcal A u_k(t, x)=F_n(t,x),\quad (t, x)\in\mathbb{R}_+\times\Omega
\
holds true in the sense of distributions on $\mathbb{R}_+\times\Omega$. In the same way, using the fact that $u_n\in C^1([0,\infty);L^2(\Omega))$ with $u_n(0,\,\cdot\,)\equiv0$, we deduce that $I_Ku_n\in C^1([0,\infty);L^2(\Omega))$ and
\begin{equation}\label{t1c}
I_Ku_n(0, x)=0,\quad x\in\Omega.
\end{equation}
From now on, we will prove that the above properties can be extended by density to $u$. Fix $T_1>0$. Applying this last identity, we will show that $D_t^K u_n$ converges in the sense of $L^1(0,T_1;D'(\Omega))$ to $-\mathcal A u(t, x)+F(t,x)$ as $n\to\infty$, and then we will complete the proof of the theorem.
Define the space $L^2(\Omega;\rho dx)$ corresponding to the space of function $L^2$ on $\Omega$ with a measure of density $\rho$. We define the operator $A_*=\rho^{-1}\mathcal A$ acting on $L^2(\Omega;\rho dx)$ with domain $D(A_*)=\{v\in H^1_0(\Omega):\ \rho^{-1}\mathcal A v\in L^2(\Omega)\}$ and we recall that $A_*$ is a selfadjoint operator with a compact resolvent whose spectrum consists of a non-deacreasing unbounded positive eigenvalues. In view of Proposition \ref{p1}, we have $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ and we deduce that $\rho^{-1}\mathcal A u\in L^1(0,T_1;D(A_*^{-1}))$. Moreover,
there exists $C>0$ depending only on $\mathcal A$, $\rho$ and $\Omega$ such that, for all $n\in\mathbb N$, we have
\begin{equation}\label{t1d}
\|\rho^{-1}\mathcal A (u_n- u)\|_{L^1(0,T_1;D(A_*^{-1}))}\leq C\|u_n-u\|_{L^1(0,T_1;L^2(\Omega))}.
\end{equation}
In addition, we have
$$
u_n(t,\,\cdot\,)-u(t,\,\cdot\,)=\int_0^tS(t-\tau)[F_n(\tau)-F(\tau)]d\tau,\quad t\in\mathbb{R}_+,\ n\in\mathbb N,
$$ and applying Lemma \ref{l1} and Young's convolution inequality, we obtain
$$\begin{aligned}
\|u_n-u\|_{L^1(0,T_1;L^2(\Omega))}&\leq C\|t^{2\alpha_0-\alpha_M-1}\|_{L^1(0,T_1)}\|F_n-F\|_{L^1(0,T_1;L^2(\Omega))}\\
&\leq C\|(1+t)^{-J}F_n-F\|_{L^1(0,T_1;L^2(\Omega))}\leq C\|(1+t)^{-J}F_n-F\|_{L^1(\mathbb{R}_+;L^2(\Omega))}\end{aligned}
$$
and \eqref{p1a} implies that
$$
\lim_{n\to\infty}\|\rho^{-1}\mathcal A u_n-\rho^{-1}\mathcal A u\|_{L^1(0,T_1;D(A_*^{-1}))}=0.
$$
In the same way, we have
$$
\lim_{n\to\infty}\|\rho^{-1}F_n-\rho^{-1}F\|_{L^1(0,T_1;D(A_*^{-1}))}\leq C\lim_{n\to\infty}\|(1+t)^{-J}F_n-F\|_{L^1(\mathbb{R}_+;L^2(\Omega))} =0
$$
and it follows that $(D_t^K u_n)_{n\in\mathbb N}$ converges in the sense of $L^1(0,T_1;D(A_*^{-1}))$ to $-\rho^{-1}\mathcal A u+F$ as $n\to\infty$. On the other hand, for all $\psi\in C^\infty_0(0,T_1)$, we have
\begin{equation} \label{t1e}\begin{aligned}
\langle D_t^K u_n(t,\,\cdot\,),\psi(t)\rangle_{D'(0,T_1),C^\infty_0(0,T_1)} & =\left\langle\partial_t I_Ku_k(t,\,\cdot\,),\psi(t) \right\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}\nonumber\\
& =-\left\langle I_Ku_n(t,\,\cdot\,),\psi'(t)\right\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}.
\end{aligned}\end{equation}
In addition, repeating the above arguments and applying \eqref{p1a}, one can check that the sequence $(u_n)_{n\in\mathbb N}$ converges to $u$ in the sense of $L^1(0,T_1;L^2(\Omega))$ as $n\to\infty$ and, applying again Young's convolution inequality, we deduce that the sequence $I_Ku_n$ converges to $I_Ku$
in the sense of $L^1(0,T_1;L^2(\Omega))$ as $n\to\infty$. Thus, sending $n\to\infty$, we find
$$\begin{aligned}
\lim_{n\to\infty}\langle D_t^K u_n(t,\,\cdot\,),\psi(t)\rangle_{D'(0,T_1),C^\infty_0(0,T_1)} & =-\lim_{n\to\infty}\left\langle I_Ku_n(t,\,\cdot\,),\psi'(t)\right\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}\\
& =-\left\langle I_Ku(t,\,\cdot\,),\psi'(t) \right\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}\\
& =\left\langle \partial_tI_Ku(t,\,\cdot\,),\psi(t)\right\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}\\
& =\langle D_t^K u(t,\,\cdot\,),\psi(t)\rangle_{D'(0,T_1),C^\infty_0(0,T_1)}.
\end{aligned}$$
It follows that $D_t^K u_n$ converges in the sense of $D'(0,T_1;L^2(\Omega))$ to $D_t^K u$ as $n\to\infty$. Therefore, by the uniqueness of the limit in the sense of $D'(0,T_1;D(A_*^{-1}))$, we deduce that
\
D_t^K u(t,x)=-\rho^{-1}\mathcal A u(t, x)+\rho^{-1}(x)F(t,x),\quad t\in(0,T_1),\ x\in\Omega
\
holds true and $D_t^K u\in L^1(0,T_1;D(A_*^{-1}))$. Using the fact that $T_1>0$ is arbitrarily chosen, we deduce that condition (i) of Definition \ref{d1} holds true. In addition, using the fact $D_t^K u=\partial_t I_Ku$, we obtain that $I_Ku\in W^{1,1}_{loc}(\mathbb{R}_+;D(A_*^{-1}))$. Recalling that $\mathcal C^\infty_0(\Omega)\subset D(A_*)$, one can check that $D(A_*^{-1})\subset D'(\Omega)$ which implies that $I_Ku\in W^{1,1}_{loc}(\mathbb{R}_+;D'(\Omega))$. Combining this with \eqref{p1a}, \eqref{t1d} and fixing $T_1=1$, we deduce that
$$\begin{aligned}
& \quad\,\limsup_{n\to\infty}\|I_Ku_n-I_Ku\|_{W^{1,1}(0,1;D(A_*^{-1}))}\\
& \leq\limsup_{n\to\infty}\|I_Ku_n-I_Ku\|_{L^1(0,1;D(A_*^{-1}))}+\|D_t^K u_k-D_t^K u\|_{L^1(0,1;D(A_*^{-1}))}\\
& \leq C\left(\limsup_{n\to\infty}\|u_n-u\|_{L^1(0,1;L^2(\Omega))}+\limsup_{k\to\infty}\|-\rho^{-1}[\mathcal A (u_k-u)+F_n-F] \|_{L^1(0,1;D(A_*^{-1}))}\right)\\
& \leq0.
\end{aligned}$$
Thus, we have that $(I_Ku_n)_{n\in\mathbb N}$ converge to $I_Ku$ in the sense of $W^{1,1}(0,1;D(A_*^{-1}))$ and from
\eqref{t1c} we deduce that \eqref{d1b} is fulfilled. This proves that the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1}, given by \eqref{sol1}, satisfies condition (ii) of Definition \ref{d1} which implies that $u$ satisfies all the conditions of Definition \ref{d1}. Thus \eqref{sol1} is a weak solution of \eqref{eq1} in the sense of Definition \ref{d1}. This completes the proof of the theorem when $u_0\equiv0$.
{\bf Step 3.} In this step we will prove that the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1}, given by \eqref{sol1}, is the weak solution of \eqref{eq1} in the sense of Definition \ref{d1} when $F\equiv0$. Again, since the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} clearly satisfies condition (iii) of Definition \ref{d1}, we only need to check conditions (i) and (ii). For this purpose, let us first consider the following intermediate result.
\begin{lem}\label{l3} Let $\delta>0$ and $\theta\in(\pi/2,\pi)$. Then we have
\begin{equation} \label{l3a}\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p} d p=1,\quad t\in\mathbb{R}_+.\end{equation}
\end{lem}
We postpone the proof this lemma to the end of the present demonstration. Fix $(u_{0,n})_{n\in\mathbb N}$ a sequence of $\mathcal C^\infty_0(\Omega)$ such that
\begin{equation} \label{t1f} \lim_{n\to\infty}\norm{u_{0,n}-u_0}_{L^2(\Omega)}=0\end{equation}
and consider
$$u_n(t,\cdot)=S_0(t)u_{0,n}.$$
Fix $n\in\mathbb N$. In view of \eqref{l3a} and \eqref{S0}, we have
$$\begin{aligned}u_n(t,\cdot)-u_{0,n}&=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p} \left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\rho p^{\alpha(\cdot)} u_{0,n}-\left(\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p} d p\right)u_{0,n}\\
\ &=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p} \left[\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\rho p^{\alpha(\cdot)} u_{0,n}-u_{0,n}\right]dp.\end{aligned}$$
On the other hand, for all $p\in\mathbb C\setminus(-\infty,0]$, we have
$$\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\rho p^{\alpha(\cdot)} u_{0,n}-u_{0,n}=-\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}$$
and it follows that
$$u_n(t,\cdot)-u_{0,n}=-\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp.$$
On the other hand, in view of \cite[Proposition 2.1]{KSY} there exists a constant $C>0$ depending on $\mathcal A$, $\rho$, $\Omega$ and $\alpha$ such that, for all $p\in \gamma(\delta,\theta)$, we have
\begin{equation} \label{t1g}
\left\|\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}\right\|_{\mathcal B(L^2(\Omega ))}
\leq C\max\left(|p|^{\alpha_0-2\alpha_M},|p|^{\alpha_M-2\alpha_0}\right).\end{equation}
Therefore, for all $p\in \gamma(\delta,\theta)$ and all $t\in\mathbb{R}_+$, we have
$$\norm{\frac{e^{t p}}{p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}}_{L^2(\Omega)}\leq e^{t\mathfrak R p}C\max\left(|p|^{\alpha_0-2\alpha_M-1},|p|^{\alpha_M-2\alpha_0-1}\right),$$
$$\norm{\partial_t\frac{e^{t p}}{p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}}_{L^2(\Omega)}\leq e^{t\mathfrak R p}C\max\left(|p|^{\alpha_0-2\alpha_M},|p|^{\alpha_M-2\alpha_0}\right).$$
It follows that
\begin{equation} \label{t1i}\partial_tu_n(t,\cdot)=\partial_t[u_n(t,\cdot)-u_{0,n}]=-\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}e^{t p}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp=-S_1(t)Au_{0,n}.\end{equation}
Combining this with Lemma \ref{l1} and the fact that $Au_{0,n}\in L^2(\Omega)$, we deduce that the map $u_n$ is lying in $W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega))$. Moreover, we have
$$u_n(0,\cdot)-u_{0,n}=-\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp.$$
Let us prove that
\begin{equation} \label{t1h}u_n(0,\cdot)-u_{0,n}=-\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp\equiv0.\end{equation}
For this purpose, we fix $\delta<1$, $R>1$ and we consider the contour
\begin{equation} \label{cont2}\gamma(\delta,R,\theta):=\gamma_-(\delta,R,\theta)\cup\gamma_0(\delta,\theta)\cup\gamma_+(\delta,R,\theta)\end{equation}
oriented in the counterclockwise direction, where $\gamma_0(\delta,\theta)$ is given by \eqref{g2} and where
\[\gamma_\pm(\delta,R,\theta):=\{r e^{\pm i\theta}:\ r\in(\delta,R)\}.
\]
Applying the Cauchy formula, for any $R>1$, we have
$$\frac{1}{2i\pi}\int_{\gamma(\delta,R,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}d p=\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n} d p$$
with $\gamma_0(R,\theta)$ given by \eqref{g2} with $\delta=R$. Sending $R\to+\infty$, we obtain
$$\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp=\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}dp.$$
On the other hand, applying \eqref{t1g}, we deduce that
$$\begin{aligned}&\norm{\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n} d p}_{L^2(\Omega)}\\
&\leq C\int_{-\theta}^{\theta}\norm{\left(A+\rho(Re^{i\beta})^{\alpha(\cdot)}\right)^{-1}}_{\mathcal B(L^2(\Omega)}\norm{Au_{0,n}}_{L^2(\Omega)} d \beta\\
\ &\leq C\max\left(R^{\alpha_0-2\alpha_M},R^{\alpha_M-2\alpha_0}\right)\norm{Au_{0,n}}_{L^2(\Omega)}.\end{aligned}$$
In view of \eqref{alpha}, it follows
$$\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n}d p=\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\rho p^{\alpha(\cdot)}\right)^{-1}Au_{0,n} d p\equiv0.$$
This proves \eqref{t1f} and in a similar way to Step 2, we deduce that $I_K[u_n-u_{0,n}]\in W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega))$ satisfies
\begin{equation}\label{t1h}
I_K[u_n-u_{0,n}](0,x)=0,\quad x\in\Omega.
\end{equation}
Therefore, $u_n$ satisfies condition (ii) of Definition \ref{d1}. Now let us show that $u_n$ satisfies condition (ii) of Definition \ref{d1}. Applying Lemma \ref{l1} and \eqref{t1i}, we deduce that there exists a constant $C>0$ such that, for all $t>0$, we have
$$\begin{aligned}&\norm{u_n(t,\cdot)}_{L^2(\Omega)}+\norm{\partial_tu_n(t,\cdot)}_{L^2(\Omega)}\\
&\leq \norm{S_0(t)u_{0,n}}_{L^2(\Omega)}+\norm{S_1(t)A u_{0,n}}_{L^2(\Omega)}\\
\ &\leq C\max\left(t^{2(\alpha_M-\alpha_0)},
t^{2(\alpha_0-\alpha_M)},t^{2\alpha_M-\alpha_0-1},t^{2\alpha_0-\alpha_M-1},1\right)\norm{u_{0,n}}_{D(A)}.\end{aligned}$$
Combining this with \eqref{alpha}, for all $p\in\mathbb C_+$, we obtain $t\mapsto e^{-pt}I_K[u_n(t,\cdot)-u_{0,n}]\in W^{1,1}(\mathbb{R}_+;L^2(\Omega))$. Thus, for all $n\in\mathbb N$ and all $p\in\mathbb C_+$, fixing $v_n(t,\cdot)=u_n(t,\cdot)-u_{0,n}$, $t>0$, and applying \eqref{t1h}, we get
$$\begin{aligned}
\widehat{D_t^K v_n}(p,\,\cdot\,)&=\int_0^{+\infty}e^{-pt}\partial_tI_K [u_n(t,\cdot)-u_{0,n}]dt\\
&=p\int_0^{+\infty}e^{-pt}I_K [u_n(t,\cdot)-u_{0,n}]dt\\
&=p\left(\int_0^{+\infty}e^{-pt}I_K u_n(t,\cdot)dt-u_{0,n}\int_0^{+\infty}e^{-pt}\frac{t^{1-\alpha(\cdot)}}{\Gamma(2-\alpha(\cdot))}dt\right)\\
&=p^{\alpha(\cdot)}\widehat{u_n}(p,\,\cdot\,)- p^{\alpha(\cdot)-1}u_{0,n}.\end{aligned}
$$
Therefore, using the fact that for all $p\in\mathbb C$ satisfying $\mathfrak R\,p>p_1$, $\widehat{u_n}(p,\,\cdot\,)$ solves \eqref{d2a} with $F\equiv0$ and $u_0=u_{0,n}$, we deduce that
$$
\widehat{D_t^K v_n}(p,\,\cdot\,)=p^{\alpha(\cdot)}\widehat{u_n}(p,\,\cdot\,)- p^{\alpha(\cdot)-1}u_{0,n}=-\rho^{-1}\mathcal A\widehat u_n(p,\,\cdot\,),\quad p\in\mathbb C,\ \mathfrak R\,p>p_1.
$$
Then in a similar way to Step 2, we find that the identity \
\rho(x) D_t^{K}[u_n-u_{0,n}](t, x)+\mathcal A u_n(t, x)=0,\quad (t, x)\in\mathbb{R}_+\times\Omega
\
holds true in the sense of distributions on $\mathbb{R}_+\times\Omega$. We will now extend this result by density to the Laplace-weak solution of \eqref{eq1} which is given by \eqref{sol1} with $F\equiv0$. For this purpose, fix $T_1>0$. Let us first observe that applying Lemma \ref{l1}, we obtain
$$\lim_{n\to+\infty}\|u_n-u\|_{L^1(0,T_1;L^2(\Omega))}\leq C\|t^{2(\alpha_0-\alpha_M)}\|_{L^1(0,T_1)}\lim_{n\to+\infty}\|u_{0,n}-u_0\|_{L^2(\Omega)}=0.$$
Therefore, repeating the arguments of Step 2, we can prove that $D_t^K u_n$ converges in the sense of $D'(0,T_1;L^2(\Omega))$ to $D_t^K u$ and in the sense of $L^1(0,T_1;D'(\Omega))$ to $\rho^{-1}\mathcal A u$ as $n\to\infty$. Then, repeating the arguments used in the last part of Step 2 we deduce that the Laplace-weak solution $u$ of \eqref{eq1} fulfills the condition (i) and (ii) of Definition \ref{d1}. This completes the proof of Theorem \ref{t1}.\qed
Now that we have completed the proof of Theorem \ref{t1}, let us consider the proof of Lemma \ref{l3}.
\textbf{Proof of Lemma $\ref{l3}$.} Let us first recall that for all $t>0$ the map $z\mapsto \frac{e^{t z}}{z}$ is meromorphic on $\mathbb C$ with a simple pole at $z=0$. Therefore, the residue theorem implies that for all $R>\delta$ we have
\begin{equation} \label{l3b}\frac{1}{2i\pi}\int_{\gamma(\delta,R,\theta)}\frac{e^{t p}}{p} d p=1+\frac{1}{2i\pi}\int_{\gamma_1(R,\theta)}\frac{e^{t p}}{p} d p,\quad t\in\mathbb{R}_+,\end{equation}
where we recall that $\gamma$ is given by \eqref{cont2} and $\gamma_1(R,\theta)$ is given by
$$\gamma_1(R,\theta):=\{R\, e^{i\beta}:\ \beta\in[\theta, 2\pi-\theta]\}.$$
Sending $R\to+\infty$, we obtain
$$\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p}dp=\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma(\delta,R,\theta)}\frac{e^{t p}}{p} d p.$$
On the other hand, we have
$$\begin{aligned}\abs{\frac{1}{2i\pi}\int_{\gamma_1(R,\theta)}\frac{e^{t p}}{p} d p}&\leq \frac{1}{2\pi}\int_{\theta}^{2\pi-\theta} e^{tR\cos \beta} d \beta\\
\ &\leq C e^{tR\cos\theta}.\end{aligned}$$
In view of \eqref{l3b} and the fact that $\theta\in(\pi/2,\pi)$, we find
$$\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p}d p=1+\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma_1(R,\theta)}\frac{e^{t p}}{p} d p=1,\quad t\in\mathbb{R}_+.$$ This proves \eqref{l3a} and it completes the proof of Lemma \ref{l3}.\qed
\section{Distributed order fractional diffusion equations}
In this section, we prove the unique existence of a weak
solution to the problem \eqref{eq1} as well as the equivalence between Definition \ref{d1} and \ref{d2} of weak and Laplace-weak solutions of \eqref{eq1} for weight $K$ given by \eqref{Kdistributed} with $\mu\in L^\infty(0,1)$ a non-negative function satisfying \eqref{mu}. For this purpose, let us first recall that the unique existence of Laplace-weak solutions for \eqref{eq1} has been proved by \cite{LKS} in the case of source terms $F\in L^\infty(\mathbb{R}_+;L^2(\Omega))$ and extended to source terms $F\in L^1(\mathbb{R}_+;L^2(\Omega))$ by \cite[Proposition 5.1]{KSXY}. We will recall here the representation of Laplace-weak solutions of \eqref{eq1} given by these works. Like in the previous section we denote by $A_*$ the operator $\rho^{-1}\mathcal A$ acting in the space $L^2(\Omega;\rho dx)$ with Dirichlet boundary condition. Let $(\varphi_n)_{n\geq1}$ be an $L^2(\Omega;\rho d x)$ orthonormal basis of eigenfunctions of the operator $A_*$ associated with the non-decreasing sequence of eigenvalues $(\lambda_n)_{n\geq1}$
of $A_*$ repeated with respect to there multiplicity. According to \cite[Proposition 2.1]{LKS}, the unique Laplace weak solution $u$ of \eqref{eq1} enjoys the following representation formula
\begin{equation} \label{di1}
u(t,\cdot)=S_{0,\mu}(t)u_0+ \int_0^t S_{1,\mu}(t-s) F(s,\cdot) ds,\ t \in \mathbb{R}_+,
\end{equation}
where
\begin{equation} \label{Smu0}
S_{0,\mu}(t) \psi := \sum_{n=1}^\infty \frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} e^{pt}(\lambda_n+\vartheta(p))^{-1} \frac{\vartheta(p)}{p}\left\langle \psi,\varphi_n\right\rangle_{L^2(\Omega;\rho dx)}\varphi_n,\ \psi \in L^2(\Omega),
\end{equation}
\begin{equation} \label{Smu1}
S_{1,\mu}(t) \psi := \sum_{n=1}^\infty \frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} e^{pt}(\lambda_n+\vartheta(p))^{-1} \left\langle \rho^{-1}\psi,\varphi_n\right\rangle_{L^2(\Omega;\rho dx)}\varphi_n,\ \psi \in L^2(\Omega),
\end{equation}
where $\vartheta(p):=\int_0^1 p^{\alpha} \mu(\alpha) d \alpha$, $\theta\in(\pi/2,\pi)$, $\delta>0$ and $\gamma(\delta,\theta)$ corresponds to the contour \eqref{cont1}. According to \cite{LKS}, the map $S_{j,\mu}$ is independent of the choice of $\theta\in(\pi/2,\pi)$, $\delta>0$. We start by proving an extension of this result to source terms $F\in\mathcal J$ and by proving that the Laplace-weak solution $u$ given by
\eqref{di1} is lying in $L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$. For this purpose, like in the previous section, we need the following intermediate result about the operator valued functions $S_{0,\mu}$ and $S_{1,\mu}$.
\begin{lem}\label{l4}
Let $\theta\in\left(\frac{\pi}{2},\pi\right)$. The maps $t\longmapsto S_{\mu,j}(t)$, $j=0,1$, defined by
\eqref{Smu0}-\eqref{Smu1} are lying in $L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$ and there
exists a constant $C>0$ depending only on $\mathcal A,\rho,\theta,\mu,\Omega$
such that the estimates
\begin{equation}\label{l4a}
\|S_{0,\mu}(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{\alpha_0-\varepsilon-1},t^{\alpha_0}\right),\quad t>0,
\end{equation}
\begin{equation}\label{l4b}
\|S_{1,\mu}(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{\alpha_0-\varepsilon-1},t^{\alpha_0-1}\right),\quad t>0,
\end{equation}
hold true.
\end{lem}
\begin{proof} For the proof of these results for $S_{1,\mu}$ one can refer to \cite[Proposition 5.1]{KSXY} and we only show the above properties for $S_{0,\mu}$. For this purpose, we recall the following estimate from \cite[Lemma 2.2]{LKS},
\begin{equation} \label{est3}
\frac{1}{\abs{\vartheta(p)+\lambda_n}}\leq C\max(\abs{p}^{-\alpha_0+\varepsilon},\abs{p}^{-\alpha_0}),\ p \in \C \setminus (-\infty,0],\ n \in \N,
\end{equation}
where the positive constant $C$ depends only on $\mu$. We recall also that
$$
|\vartheta(p)|\leq C\max(\abs{p},1),\ p \in \C \setminus (-\infty,0],
$$
where the positive constant $C$ depends only on $\mu$. Therefore, we have
\begin{equation} \label{l1c}
\frac{|\vartheta(p)|}{|p|\abs{\vartheta(p)+\lambda_n}}\leq C\max(\abs{p}^{-\alpha_0+\varepsilon},\abs{p}^{-1-\alpha_0}),\ p \in \C \setminus (-\infty,0],\ n \in \N,
\end{equation}
where the positive constant $C$ depends only on $\mu$.
For all $t \in (0,+\infty)$ and all $\psi \in L^2(\Omega)$, by taking $\delta=t^{-1}$ in \eqref{cont1}, \eqref{l1c} implies
\begin{equation} \label{l1d}\begin{aligned}
& \norm{\sum_{n=1}^{+\infty} \left( \int_{\gamma_0(t^{-1},\theta)} \frac{\vartheta(p) e^{pt}}{p(\vartheta(p)+\lambda_n)} dp \right) \langle \psi, \varphi_{n}\rangle_{L_\rho^2(\Omega)} \varphi_{n,k}}_{L^2(\Omega)}\\
&\leq C\norm{\sum_{n=1}^{+\infty} \left( \int_{\gamma_0(t^{-1},\theta)} \frac{\vartheta(p)e^{pt}}{p(\vartheta(p)+\lambda_n)} dp \right) \langle \psi, \varphi_{n}\rangle_{L_\rho^2(\Omega;\rho dx)} \varphi_{n,k}}_{L^2(\Omega;\rho dx)}\\
&\leq C \max(t^{\alpha_0-\varepsilon-1},t^{\alpha_0}) \left(\int_{-\theta}^{\theta}e^{\cos \beta}d\beta\right) \norm{\psi}_{L^2(\Omega;\rho dx)}\leq C \max(t^{\alpha_0-\varepsilon-1},t^{\alpha_0})\norm{\psi}_{L^2(\Omega)},\end{aligned}
\end{equation}
\begin{eqnarray*}
& & \norm{\sum_{n=1}^{+\infty}
\left( \int_{\gamma_\pm(t^{-1},\theta)}\frac{\vartheta(p) e^{pt}}{p(\vartheta(p)+\lambda_n)} dp \right)
\langle \psi,\varphi_{n}\rangle_{L^2(\Omega;\rho dx)} \varphi_{n}}_{L^2(\Omega)}\\
&\leq & C \left(\int_{t^{-1}}^{+\infty} \max( r^{-\alpha_0+\varepsilon},r^{-\alpha_0-1} ) e^{t r \cos \theta} dr\right) \norm{\psi}_{L^2(\Omega)}\\
&\leq & C t^{-1} \left(\int_{1}^{+\infty} \max ( (t^{-1} r)^{-\alpha_0+\varepsilon}, (t^{-1} r)^{-\alpha_0-1} ) e ^{r \cos \theta} dr\right) \norm{\psi}_{L^2(\Omega)}\\
&\leq& C \max(t^{\alpha_0-\varepsilon-1},t^{\alpha_0}) \left(\int_{1}^{+\infty} e^{r\cos\theta}dr\right) \norm{\psi}_{L^2(\Omega)}.
\end{eqnarray*}
Putting these two estimates together with \eqref{g2} we deduce \eqref{l4a} and the fact that $S_{0,\mu}\in L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$.
\end{proof}
Combining Lemma \ref{l4} with the arguments used in Proposition \ref{p1}, we obtain the following result about the unique existence of Laplace-weak solution for problem \eqref{eq1}.
\begin{prop}\label{p2} Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled.
Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $\mu\in L^\infty(0,1)$ be a non-negative function satisfying \eqref{mu} and let $K$ be given by \eqref{Kdistributed}. Then there exists a unique Laplace-weak
solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ to \eqref{eq1} given by \eqref{di1}.
\end{prop}
Armed with this result, we can now complete the proof of Theorem \ref{t2}.
\textbf{Proof of Theorem $\ref{t2}$.} Let us first observe that the first statement of Theorem \ref{t2} is a direct consequence of Proposition \ref{p2}. Moreover, the uniqueness of weak solutions in the sense of Definition \ref{d1} as well as the fact that, for $u_0\equiv0$, \eqref{eq1} admits a unique weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ in the sense of Definition \ref{d1} given by \eqref{di1}, can be deduced by mimicking the proof of Theorem \ref{t1}. For this purpose, we only show that $u$ given by \eqref{di1} is a weak solution of \eqref{eq1} in the sense of Definition \ref{d1} when $F\equiv0$. Since the Laplace-weak solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ of \eqref{eq1} clearly satisfies condition (iii) of Definition \ref{d1}, we only need to check condition (i) and (ii). Fix $(u_{0,n})_{n\in\mathbb N}$ a sequence of $\mathcal C^\infty_0(\Omega)$ such that \eqref{t1f} is fulfilled
and consider
$$u_n(t,\cdot)=S_{0,\mu}(t)u_{0,n},\quad t\in\mathbb{R}_+.$$
Fix $n\in\mathbb N$. In view of \eqref{l3a} and \eqref{Smu0}, we have
$$\begin{aligned}&u_n(t,\cdot)-u_{0,n}\\
&=\sum_{k=1}^\infty \left(\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{\vartheta(p)e^{pt}}{p(\lambda_k+\vartheta(p))} dp\right)\left\langle u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k-\left(\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}\frac{e^{t p}}{p} d p\right)u_{0,n}\\
&=\sum_{k=1}^\infty \left(\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{e^{pt}}{p} \left( \frac{\vartheta(p)}{(\lambda_k+\vartheta(p))} -1\right)dp\right)\left\langle u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k\\
&=-\sum_{k=1}^\infty \left(\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{e^{pt}}{p} \frac{\lambda_k}{(\lambda_k+\vartheta(p))} dp\right)\left\langle u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k\\
&=-\sum_{k=1}^\infty \left(\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{e^{pt}}{p(\lambda_k+\vartheta(p))}dp\right) \left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k.\end{aligned}$$
On the other hand, applying \eqref{est3}, for all $k\in\mathbb N$ and all $p\in\mathbb C\setminus(-\infty,0]$, we have
$$\abs{\frac{e^{pt}}{p(\lambda_k+\vartheta(p))} \left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}}\leq C\max\left(|p|^{-\alpha_0+\varepsilon-1},|p|^{-\alpha_0-1}\right)\abs{\left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}},$$
$$\abs{\partial_t\left(\frac{e^{pt}}{p(\lambda_k+\vartheta(p))} \left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\right)}\leq C\max\left(|p|^{-\alpha_0+\varepsilon},|p|^{-\alpha_0}\right)\abs{\left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}}.$$
Combining this with the fact that $u_{0,n}\in D(A_*)$, we deduce that
\begin{equation} \label{t2c}\partial_tu_n(t,\cdot)=-\sum_{k=1}^\infty \frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{e^{pt}}{\lambda_k+\vartheta(p)} \left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k=-S_{1,\mu}(t)Au_{0,n}.\end{equation}
Applying Lemma \ref{l4} and the fact that $Au_{0,n}\in L^2(\Omega)$, we deduce that the map $u_n$ is lying in $W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega))$. Moreover, we have
$$u_n(0,\cdot)-u_{0,n}=-\sum_{k=1}^\infty \left(\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{1}{p(\lambda_k+\vartheta(p))}dp\right) \left\langle A_*u_{0,n},\varphi_k\right\rangle_{L^2(\Omega;\rho dx)}\varphi_k.$$
Applying the arguments used at the end of the proof of \cite[Proposition 2.1]{LKS}, we obtain
$$\frac{1}{2i\pi} \int_{\gamma(\delta,\theta)} \frac{1}{p(\lambda_k+\vartheta(p))}dp=0,\quad k\in\mathbb N$$
and it follows that $I_K[u_n-u_{0,n}]\in W^{1,1}_{loc}(\mathbb{R}_+;L^2(\Omega))$ satisfies
\begin{equation}\label{t2d}
I_K[u_n-u_{0,n}](0,x)=0,\quad x\in\Omega.
\end{equation}
This proves that $u_n$ satisfies condition (ii) of Definition \ref{d1}. Combining Lemma \ref{l4}, Proposition \ref{p1} with the arguments used in the last step of the proof of Theorem \ref{t2}, we can show that $u_n$ satisfies also condition (i). In the same way, using Lemma \ref{l4}, Proposition \ref{p1} and repeating the arguments used in the last step of the proof of Theorem \ref{t2}, we can show that, by density these properties can be extended to $u$. This proves that the Laplace-weak solution $u$ of \eqref{eq1}, given by \eqref{di1}, fulfills the condition (i) and (ii) of Definition \ref{d1} and it completes the proof of Theorem \ref{t2}.\qed
\section{Multiterm fractional diffusion equations}
In this section, we prove the unique existence of a weak
solution to the problem \eqref{eq1} as well as the equivalence between Definition \ref{d1} and \ref{d2} of weak and Laplace-weak solutions of \eqref{eq1} for weight $K$ given by \eqref{Kmultiple} with $1<\alpha_1<\ldots<\alpha_N<1$ and $\rho_j\in L^\infty(\Omega)$, $j=1,\ldots,N$, satisfying \eqref{eq-rho} with $\rho=\rho_j$. In contrast to variable order and distributed order fractional diffusion equations, we have not find any result in the mathematical literature showing the unique existence of Laplace-weak solutions for multiterm fractional diffusion equations. For this purpose, we will consider first the proof of this result.
For all $p\in\mathbb C\setminus(-\infty,0]$, we can consider the following operator
$$\left(A+\sum_{k=1}^N\rho_k(x)p^{\alpha_k}\right)^{-1}\in \mathcal B(L^2(\Omega)).$$
For all $\theta\in(0,\pi)$ we denote by $\mathcal D_\theta$ the following set $ \mathcal D_{\theta}:=\{re^{i\beta}:\ r>0,\ \beta\in (-\theta,\theta)\}$.
Inspired by \cite[Proposition 2.1]{KSY}, we start with the following properties of the above operator.
\begin{lem}\label{l5} Let $\theta\in(0,\pi)$. Then there exists a constant $C>0$ depending only on $\mathcal A$, $\rho_1,\ldots,\rho_N$, $\alpha_1,\ldots,\alpha_N$, $\Omega$ and $\theta$ such that
\begin{equation}\label{l5a} \norm{\left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1}}_{B(L^2(\Omega))}\leq C|z|^{-\alpha_N},\quad z\in \mathcal D_\theta.\end{equation}
\end{lem}
\begin{proof} Let us observe that, since the spectrum of $A$ is discrete and contained into $\mathbb{R}_+$, it is enough to prove \eqref{l5a} with $z\in \mathcal D_\theta$ satisfying $|z|>1$. For this purpose, from now on we fix $z=re^{i\beta}$ with $r\in[1,+\infty)$, $\beta\in(-\theta,\theta)$ and we will show \eqref{l5a}. In all this proof $c_0$ and $C_0$ denote the constants appearing in \eqref{eq-rho}. We divide the proof of this result into two steps.
\textbf{Step 1:} In this step we will prove that for all $\beta\in(-\theta,\theta)\setminus \{0\}$, we have
\begin{equation}\label{l5d}
\left\| \left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1} \right\|_{\mathcal B(L^2(\Omega))} \leq c_0^{-1} \max\left(|\sin(\alpha_1\beta)|^{-1},|\sin(\alpha_N\beta)|^{-1}\right) r^{-\alpha_N}.\end{equation}
For this purpose, we assume that $\beta \in (0,\theta)$, the case of $\beta \in (-\theta,0)$ being treated in a similar fashion. Let $B_\beta$ be the multiplier in $L^2(\Omega)$, by the function
$$ b_\beta(x) := \left( \sum_{k=1}^N\rho_k(x) r^{\alpha_k} \sin(\beta \alpha_k) \right)^{1 \slash 2},\ x \in \Omega, $$
in such a way that $i B_\beta^2$ is the skew-adjoint part of the operator $A+\sum_{k=1}^N\rho_k(x)r^{\alpha_k}e^{i\beta\alpha_k}$.
Applying \eqref{eq-rho}, we obtain
$$ 0 < c_0^{1\slash2} \min\left(\sin(\alpha_1\beta)^{1 \slash 2},\sin(\alpha_N\beta)^{1 \slash 2}\right) r^{\alpha_N \slash 2}\leq b_\beta(x),\quad x\in\Omega,$$
$$b_\beta(x)\leq (NC_0)^{1 \slash 2} \max\left(\sin(\alpha_1\beta)^{1 \slash 2},\sin(\alpha_N\beta)^{1 \slash 2}\right)r^{\alpha_N \slash 2},\ x \in \Omega.$$
Hence, the self-adjoint operator $B_\beta$ is bounded and boundedly invertible in $L^2(\Omega)$, with
\begin{equation}\label{l5b}
\| B_\beta^{-1} \|_{\mathcal B(L^2(\Omega))} \leq c_0^{-1\slash2} \max\left(\sin(\alpha_1\beta)^{-1 \slash 2},\sin(\alpha_N\beta)^{-1 \slash 2}\right) r^{-\alpha_N \slash 2}.
\end{equation}
Moreover, for each $z=r e^{i \beta}$, it holds true that
\begin{equation}\label{l5c}
A + \sum_{k=1}^N\rho_k(x)z^{\alpha_k} = B_\beta \left( B_\beta^{-1} H_{z} B_\beta^{-1} + i \right) B_\beta,,
\end{equation}
where $H_z:= A + \sum_{k=1}^N\rho_k(x) r^{\alpha_k}\cos(\beta \alpha_k)$. It is clear that the operator $H_z$ is self-adjoint in $L^2(\Omega)$ with domain $D(H_z)=D(A)$, by the Kato-Rellich theorem. Thus, $B_\beta^{-1} H_z B_\beta^{-1}$ is self-adjoint in $L^2(\Omega)$ as well, with domain $B_\beta D(A)$.
Therefore, the operator $B_\beta^{-1} H_z B_\beta^{-1}+i$ is invertible in $L^2(\Omega)$ and satisfies the estimate
$$ \| ( B_\beta^{-1} H_z B_\beta^{-1}+i )^{-1} \|_{\mathcal B(L^2(\Omega))} \leq 1. $$
It follows from this and \eqref{l1c} that $A_q+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}$ is invertible in $L^2(\Omega)$,
with
$$ \left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1} = B_\beta^{-1} (U_\beta^{-1} H_z B_\beta^{-1} +i )^{-1} B_\beta^{-1}, $$
showing that $\left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1}$ maps $L^2(\Omega)$ into $B_\beta^{-1} D(B_\beta^{-1} H_z B_\beta^{-1})=D(A)$. As a consequence, we infer from \eqref{l5b} that
$$\begin{aligned}
\left\| \left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1} \right\|_{\mathcal B(L^2(\Omega))} &\leq \| (B_\beta^{-1} H_z B_\beta^{-1}+i )^{-1} \|_{\mathcal B(L^2(\Omega))} \| B_\beta^{-1} \|_{\mathcal B(L^2(\Omega))}^2 \\
&\leq c_0^{-1} \max\left(|\sin(\alpha_1\beta)|^{-1},|\sin(\alpha_N\beta)|^{-1}\right) r^{-\alpha_N}.\end{aligned}$$
From this last estimate we deduce \eqref{l5d}.
\textbf{Step 2:} We fix $\theta_*\in(0,\min(\theta,\pi/2))$ such that
\begin{equation}\label{l5e}\frac{|\sin(\alpha_N\theta_*)|}{\cos(\alpha_N\theta_*)}\leq \frac{c_0}{2C_0N}.\end{equation}
In this step, we will prove that for all $\beta\in(-\theta_*,\theta_*)$ we have
\begin{equation}\label{l5f}
\left\| \left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1} \right\|_{\mathcal B(L^2(\Omega))} \leq 2c_0^{-1} \cos(\alpha_N\theta_*)^{-1} r^{-\alpha_N},\end{equation}
where we recall that since $\theta_*\in(0,\min(\theta,\pi/2))$ we have $\cos(\alpha_N\theta_*)>0$.
Using the fact that the operator $A$ is positive, for all $v\in D(A)$ and $\beta\in(-\theta_*,\theta_*)$, we get
$$\begin{aligned}\norm{H_zv}_{L^2(\Omega)}\norm{v}_{L^2(\Omega)}&\geq\left\langle H_zv,v\right\rangle_{L^2(\Omega)}\\
\ &\geq \left\langle Av,v\right\rangle_{L^2(\Omega)}+\left\langle \sum_{k=1}^N\rho_k \cos(\beta \alpha_k)v,v\right\rangle_{L^2(\Omega)}\\
\ &\geq \int_\Omega \left(\sum_{k=1}^N\rho_kr^{\alpha_k} \cos(\beta \alpha_k)\right)|v|^2dx\\
\ &\geq \int_\Omega \rho_Nr^{\alpha_N} \cos(\beta \alpha_N)|v|^2dx\\
\ &\geq c_0\cos(\theta_*\alpha_N)r^{\alpha_N}\norm{v}_{L^2(\Omega)}^2.\end{aligned}$$
Choosing $v=H_z^{-1}u$ in the above inequality, we obtain
\begin{equation}\label{l5g}\norm{H_z^{-1}u}_{L^2(\Omega)}\leq c_0^{-1}\cos(\alpha_N\theta_*)^{-1}r^{-\alpha_N}\norm{u}_{L^2(\Omega)}\end{equation}
and we deduce that $\norm{H_z^{-1}}_{\mathcal B(L^2(\Omega))}\leq c_0^{-1}\cos(\alpha_N\theta_*)^{-1}r^{-\alpha_N}$.
Therefore, for all $\beta\in(-\theta_*,\theta_*)$, we get
$$\begin{aligned}\norm{-iH_z^{-1}B_\beta^{2}}_{\mathcal B(L^2(\Omega))}&\leq \norm{H_z^{-1}}_{\mathcal B(L^2(\Omega))}\norm{B_\beta}_{\mathcal B(L^2(\Omega))}^{2}\\
\ &\leq \frac{C_0Nr^{\alpha_N}\sin(\alpha_N\beta)}{c_0\cos(\theta_*\alpha_N)r^{\alpha_N}}\leq \frac{C_0N\sin(\alpha_N\theta_*)}{c_0\cos(\theta_*\alpha_N)}\end{aligned}$$
and, combining this with \eqref{l5e}, we find
$$\norm{-iH_z^{-1}B_\beta^{2}}_{\mathcal B(L^2(\Omega))}\leq \frac{1}{2}.$$
Therefore, for all $\beta\in(-\theta_*,\theta_*)$, the operator $(Id+iH_z^{-1}B_\beta^{2})^{-1}$ is invertible and we have
$$\norm{(Id+iH_z^{-1}B_\beta^{2})^{-1}}_{\mathcal B(L^2(\Omega))}\leq \frac{1}{1-\norm{-iH_z^{-1}B_\beta^{2}}_{\mathcal B(L^2(\Omega))}}\leq 2.$$
It follows that
$$\left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1}=(Id+iH_z^{-1}B_\beta^{2})^{-1}H_z^{-1},\quad \beta\in(-\theta_*,\theta_*)$$
and applying \eqref{l5g} we deduce \eqref{l5f}. Combining \eqref{l5d} and \eqref{l5f}, we deduce \eqref{l5a} by choosing $$C=\min(c_0^{-1} \sin(\alpha_N\theta)^{-1}, c_0^{-1} \sin(\alpha_1\theta_*)^{-1},2c_0^{-1} \cos(\alpha_N\theta_*)^{-1}).$$ This completes the proof of the lemma.\end{proof}
We fix $\theta\in(\frac{\pi}{2},\pi)$, $\delta\in\mathbb{R}_+$ and, applying Lemma \ref{l5}, we consider the operator $\mathcal R_j(t)\in\mathcal B(L^2(\Omega))$, $j=0,1$ and $t\in\mathbb{R}_+$, given by
\begin{equation}\label{R0}
\mathcal R_0(t)h=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}e^{t p}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}\left(\sum_{k=1}^N\rho_kp^{\alpha_k-1}\right)h d p,\quad h\in L^2(\Omega),\ t\in\mathbb{R}_+,
\end{equation}
\begin{equation}\label{R1}
\mathcal R_1(t)h=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}e^{t p}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p,\quad h\in L^2(\Omega),\ t\in\mathbb{R}_+
\end{equation}
Note that here since the map $z\longmapsto\left(A+\sum_{k=1}^N\rho_k(x)z^{\alpha_k}\right)^{-1}$ is holomorphic on $\mathbb C\setminus(-\infty,0]$ as a map taking values in $\mathcal B(L^2(\Omega))$, the definition of $\mathcal R_j$, $j=0,1$, will be independent of the choice of $\delta$ and $\theta\in(\frac{\pi}{2},\pi)$. Let us consider
\begin{equation} \label{mul} u(t,\cdot)=\mathcal R_0(t)u_0+\int_0^t\mathcal R_1(t-s)F(s,\cdot)ds.\end{equation}
Combining the arguments used in Lemma \ref{l1} with estimate \eqref{l5a}, we can show the following properties of the maps $\mathbb{R}_+\ni t\mapsto\mathcal R_j(t)$, $j=0,1$.
\begin{lem}\label{l6}
Let $\theta\in\left(\frac{\pi}{2},\pi\right)$. The maps $t\longmapsto S_j(t)$, $j=0,1$, defined by
\eqref{R0}-\eqref{R1} are lying in $L^1_{loc}(\mathbb{R}_+;\mathcal B(L^2(\Omega))$ and there
exists a constant $C>0$ depending only on $\mathcal A,\rho,\theta,\Omega$
such that the estimates
\begin{equation}\label{l6a}
\|\mathcal R_0(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{\alpha_N-\alpha_1},1\right),\quad t>0,
\end{equation}
\begin{equation}\label{6b}
\|\mathcal R_1(t)\|_{\mathcal B(L^2(\Omega))}\leq
C\max\left(t^{\alpha_N-1},1\right),\quad t>0,
\end{equation}
hold true.
\end{lem}
This proves that, for $u_0\in L^2(\Omega)$ and $F\in\mathcal J$, $u$ given by \eqref{mul} is lying in $L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$.
Let us prove that this function $u$ is the unique Laplace weak solution of \eqref{eq1} when $K$ is given by \eqref{Kmultiple}.
For this purpose, we need two intermediate results.
Combining the result of Lemma \ref{l5} with \cite[Theorem 1.1.]{KSY}, we deduce the following.
\begin{lem}\label{l7} Let $u_0\in L^2(\Omega)$ and $F\in L^\infty(\mathbb{R}_+;L^2(\Omega)$. Then, the function
\begin{equation} \label{fin}v(\cdot,t)=\mathcal R_0(t)u_0+\int_0^t\mathcal R_1(t-s)F(s,\cdot)ds+ BF(t,\cdot),\quad t\in\mathbb{R}_+,\end{equation}
is the unique Laplace weak solution of \eqref{eq1}.
Here $B$ is defined by
$$Bh=\frac{1}{2i\pi}\int_{\gamma(\delta,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p.$$
\end{lem}
\begin{proof} From now on, for any Banach space $Y$ we denote by $\mathcal{S}'(\mathbb{R}_+;Y)$ the set of temperate distributions supported in $[0,+\infty)$ taking values in $Y$. Since the proof of this result is rather long and similar to \cite[Theorem 1.1.]{KSY}, we only give the main idea of its proof when $u_0\equiv0$.
In the first step of this proof we introduce the following family of operators acting in
$L^2(\Omega)$,
$$
\widetilde{W}(p):=p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1} ,\ p \in \mathbb \C
\setminus \mathbb{R}_-.
$$
Combining Lemma \ref{l1} with the arguments used in \cite[Lemma 2.3.]{KSY}, we can define the map
\begin{equation} \label{bm}
\mathcal R_2(t):= \frac{1}{2i\pi} \int_{i\infty}^{i\infty} e^{t p} \widetilde{W}(p+1) dp
= \frac{1}{2 \pi} \int_{-\infty}^{+\infty} e^{ i t\eta} \widetilde{W}(1 + i \eta ) d \eta
, \quad t\in\mathbb{R}\end{equation}
and show that $\mathcal R_2\in L^\infty(\mathbb{R}; \mathcal B(L^2(\Omega))) \cap \mathcal{S}'(\mathbb{R}_+;\mathcal{B}(L^2(\Omega)))$. Moreover, combining Lemma \ref{l1} with Theorem 19.2 and the following remark in \cite{Ru}, we deduce that
$\widehat{\mathcal R_2}(p)=\widetilde{W}(p+1)$ for all $p\in\mathbb C_+$.
As a consequence, the operator
$$
\mathcal R_3(t)=e^{t} \mathcal R_2(t) =\frac{1}{2i\pi} \int_{-i\infty}^{i\infty} e^{t (p+1)}
\widetilde{W}(p+1) dp= \frac{1}{2i\pi} \int_{1-i\infty}^{1+i\infty} e^{t p} \widetilde{W}(p) dp,
\ t \in \mathbb{R}
$$
verifies
$\widehat{\mathcal R_3}(p)=\widehat{\mathcal R_2}(p-1)=\widetilde{W}(p)$ for all $p\in\{z\in\mathbb C;\ \mathfrak R z \in (1,+\infty) \}$.
Following \cite[Lemma 2.4.]{KSY}, we can prove that
\begin{equation} \label{t6a}
\mathcal R_3(t)=\frac{1}{2 i \pi} \int_{\gamma(\delta,\theta)} e^{t p} \widetilde{W}(p) dp,\ t
\in \mathbb{R}_+,
\end{equation}
and $\mathcal R_3\in\mathcal{S}'(\mathbb{R}_+;\mathcal{B}(L^2(\Omega)))\cap L^1_{loc}(\mathbb{R}_+;\mathcal{B}(L^2(\Omega)))$. Using the fact that $\widehat{\mathcal R_3}(p)=\widetilde{W}(p)$ for all $p\in\{z\in\mathbb C;\ \mathfrak R z \in (1,+\infty) \}$, we deduce that
\begin{equation} \label{lap11}
\widehat{\mathcal R_3\psi}(p)= \widetilde{W}(p)\psi,\ p \in \C_+,\ \psi\in L^2(\Omega).
\end{equation}
We denote by $\tilde{F}$ the extension of a function $F$ by $0$ on
$( \Omega\times \mathbb{R}) \setminus (\Omega\times\mathbb{R}_+ )$. Consider the convolution in time of $S_2$ with $\tilde{F}$ given by
$$
(\mathcal R_3*\tilde{F})(x,t) =\int_0^t \mathcal R_3(t-s) F(s,x) \mathds{1}_{\mathbb{R}_+}(s) ds.
\ (t,x) \in \mathbb{R}\times\Omega ,
$$
We show that $\mathcal R_3*\tilde{F}\in \mathcal{S}'(\mathbb{R}_+;L^2(\Omega))$ and
$$
\widehat{\mathcal R_3*\tilde{F}}(p)=\widehat{\mathcal R_3}(p) \widehat{\tilde{F}}(p),\
p\in\mathbb C_+,
$$
with $\widehat{\mathcal R_3}(p)=\int_0^{+\infty} \mathcal R_3(t)e^{-pt}dt$ and $\widehat{\tilde{F}}(p)
= \int_0^{+\infty} F(t)e^{-pt}dt$.
Thus, setting $\tilde{v}:=\partial_t(\mathcal R_3*\tilde{F})\in \mathcal{S}'(\mathbb{R}_+;L^2(\Omega))$,
we derive from \eqref{lap11} that
$$
\widehat{\tilde{v}}(p) = p\widehat{\mathcal R_3*\tilde{F}}(p) = p\widehat{\mathcal R_3}(p)\widehat{\tilde{F}}(p)=\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1} \widehat{\tilde{F}}(p),\ p \in \C_+.
$$
Therefore, the proof will be completed if we show that $\tilde{v}=v$ with $v$ given by \eqref{fin}. For this purpose, applying \eqref{l6a}, we deduce that
$$
(\mathcal R_3*\tilde{F})(t)=\frac{1}{2 i \pi} \int_{\gamma(\delta,\theta)} g(t,p)
dp,\ t \in \mathbb{R}_+
$$
with
\begin{equation} \label{a15b}
g(t,p):= \int_0^te^{(t-s)p} p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}
\tilde{F}(s,\cdot)ds,\ p \in \gamma(\delta,\theta).
\end{equation}
Therefore, for a.e. $t \in \mathbb{R}_+$ and all $p \in \gamma(\delta,\theta)$,
we have
\begin{eqnarray*}
\partial_t g(t,p)
& =& \int_0^te^{(t-s)p} \left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}
\tilde{F}(s,\cdot)ds
+ p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}
\tilde{F}(t,\cdot),
\end{eqnarray*}
and consequently
$$
\norm{\partial_t g(t,p)}_{L^2(\Omega)}
\leq \norm{\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}}_{\mathcal{B}(L^2(\Omega))}
\left( \int_0^t e^{s\mathfrak R p} ds +| p |^{-1} \right) \norm{\tilde{F}}
_{L^\infty(\mathbb{R};L^2(\Omega))}.
$$
From this and \eqref{l5a}, it follows that
$$
\norm{\partial_t g(t,p)}_{L^2(\Omega)}
\leq C| p |^{-(1+ \alpha_N)} \norm{\tilde{F}}
_{L^\infty(\mathbb{R};L^2(\Omega))}=C| p |^{-(1+ \alpha_N)} \norm{F}
_{L^\infty(\mathbb{R}_+;L^2(\Omega))}.
$$
As a consequence, the mapping $p \mapsto \partial_t g(t,p) \in
L^1(\gamma(\delta,\theta);L^2(\Omega))$ for any fixed $t \in \mathbb{R}_+$
and we have $\tilde{v}(t)=\partial_t[\mathcal R_3*\tilde{F}](t)=\frac{1}{2 i \pi}
\int_{\gamma(\delta,\theta)} \partial_t g(t,p) dp$, or equivalently
$$
\tilde{v}(\cdot,t)=\frac{1}{2 i \pi} \int_{\gamma(\delta,\theta)}\left(\int_0^t
e^{(t-s)p} \left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}F(s)ds+p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}F(t)\right)dp
$$
in virtue of \eqref{a15b}. Now, applying the Fubini theorem to the right-hand
side of the above identity, we obtain that $\tilde{v}=v$ with $v$ given by \eqref{fin}. Combining this with Lemma \ref{l6}, we deduce that $v\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ and it is the unique Laplace-weak solution of \eqref{eq1} in the sense of Definition \ref{d2}.
\end{proof}
We can extend the result of Lemma \ref{l7} as follows.
\begin{lem}\label{l8} Let $u_0\in L^2(\Omega)$ and $F\in L^\infty(\mathbb{R}_+;L^2(\Omega))$. Then, the function $u$ given by \eqref{mul} is the unique Laplace weak solution of \eqref{eq1}.
\end{lem}
\begin{proof} According to Lemma \ref{l6} and \ref{l7}, we only need to show that here the map $B$ appearing in Lemma \ref{l6} will be equal to zero.
To see this let us observe that, fixing $\delta<1$ and applying the Cauchy formula, for any $R>1$ and $h\in L^2(\Omega)$, we have
$$\frac{1}{2i\pi}\int_{\gamma(\delta,R,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p=\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p$$
with
$$\gamma(\delta,R,\theta):=\gamma_-(\delta,R,\theta)\cup\gamma_0(\delta,\theta)\cup\gamma_+(\delta,R,\theta)$$
oriented in the counterclockwise direction, where
\[\gamma_\pm(\delta,R,\theta):=\{r e^{\pm i\theta}:\ r\in(\delta,R)\}.
\]
Sending $R\to+\infty$, we obtain
$$Bh=\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p.$$
On the other hand applying Lemma \ref{l1}, we deduce that
$$\begin{aligned}&\norm{\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p}_{L^2(\Omega)}\\
&\leq C\int_{-\theta}^{\theta}\norm{\left(A+\sum_{k=1}^N\rho_k(Re^{i\beta})^{\alpha_k}\right)^{-1}}_{\mathcal B(L^2(\Omega))}\norm{h}_{L^2(\Omega)} d \beta\\
\ &\leq CR^{-\alpha_N}\norm{h}_{L^2(\Omega)}.\end{aligned}$$
Therefore, we have
$$Bh=\lim_{R\to+\infty}\frac{1}{2i\pi}\int_{\gamma_0(R,\theta)}p^{-1}\left(A+\sum_{k=1}^N\rho_kp^{\alpha_k}\right)^{-1}h d p\equiv0.$$
\end{proof}
Combining Lemma \ref{l6}, \ref{l8} with the density arguments used in Proposition \ref{p1}, we obtain the following results about the unique existence of Laplace-weak solutions for \eqref{eq1}.
\begin{prop}\label{p3} Assume that the conditions \eqref{ell}-\eqref{eq-rho} are fulfilled.
Let $u_0\in L^2(\Omega)$, $F\in\mathcal J$, $1<\alpha_1<\ldots<\alpha_N<1$, $\rho_j\in L^\infty(\Omega)$, $j=1,\ldots,N$, satisfy \eqref{eq-rho} with $\rho=\rho_j$, and let $K$ be given by \eqref{Kmultiple}. Then there exists a unique Laplace-weak
solution $u\in L^1_{loc}(\mathbb{R}_+;L^2(\Omega))$ to \eqref{eq1} given by \eqref{mul}.
\end{prop}
Combining Lemma \ref{l6}, Proposition \ref{p3} and mimicking the proof of Theorem \ref{t1}, we deduce Theorem \ref{t3}.
\section{Weak solution at finite time}
In a similar way to \cite{KSY,KY1,KY2,LKS}, following Definition \ref{d1} of weak solutions of \eqref{eq1}, we give the definition of weak solutions of the same problem at finite time. Namely, for $T>0$, let us consider the IBVP
\begin{equation} \label{eqq1}
\left\{ \begin{array}{rcll}
(\rho(x) \partial^{K}_t+\mathcal{A} ) v(t,x) & = & G(t,x), & (t,x)\in
(0,T)\times\Omega,\\
v(t,x) & = & 0, & (t,x) \in (0,T)\times\partial\Omega, \\
v(0,x) & = & u_0(x), & x \in \Omega,\ .
\end{array}
\right.
\end{equation}
We give the following definition of weak solutions of \eqref{eqq1}.
\begin{defn}\label{d3} Let $F$ be the extension of the function $G$ by zero to $\mathbb{R}_+\times\Omega$. Then we call weak solution of \eqref{eqq1}
the restriction on $(0,T)\times\Omega$ of the weak solution $u$ of the IBVP \eqref{eq1} in the sense of Definition \ref{d1}.
\end{defn}
Notice that, according to Definition \ref{d1}, any weak solutions $v$ of \eqref{eqq1} satisfies the following properties:\\
1) $v\in L^1(0,T;L^2(\Omega))$ and the identity
\begin{equation}\label{d3a}
\rho( x)D_t^K[v-u_0](t, x)+\mathcal A v(t, x)=G(t,x),\quad x\in\Omega,\ t\in(0,T).
\end{equation}
holds true in the sense of distributions in $\mathbb{R}_+\times\Omega$.\\
2) We have $I_K [v-u_0]\in W^{1,1}(0,T;D'(\Omega))$ and the following initial condition
\begin{equation}\label{d3b}
I_K[v-u_0](0, x)=0,\quad x\in\Omega,
\end{equation}
is fulfilled. Moreover, applying the result of Theorem \ref{t1}, \ref{t2} and \ref{t3} we can show the unique existence of weak solution of \eqref{eqq1}. Let us also observe that the Definition \ref{d3} of weak solutions depends on the final time $T$. Nevertheless, we can show that the unique weak solution of \eqref{eqq1} in the sense of Definition \ref{d3} is independent of $T$ and by the same way of the extension of the source term $G$ under consideration in Definition \ref{d3}. All these properties can be sum up as follows.
\begin{thm}\label{t4} Assume that the condition of Theorem \ref{t1}, \ref{t2} and \ref{t3} be fulfilled and assume that the weight $K$ is given by \eqref{Kvariable} or \eqref{Kdistributed} or \eqref{Kmultiple}. Then for any $G\in L^1(0,T;L^2(\Omega))$ and $u_0\in L^2(\Omega)$, the IBVP \eqref{eqq1} admits a unique solution $v\in L^1(0,T;L^2(\Omega))$ in the sense of Definition \ref{d3}. Moreover, the unique weak solution of \eqref{eqq1} have a Duhamel type of representation given by:
$$1)\quad v(t,\cdot)=S_{0}(t)u_0+ \int_0^t S_{1}(t-s) G(s,\cdot) ds,\quad t\in(0,T)$$
when $K$ is given by \eqref{Kvariable}. Here $S_0$ \emph{(}resp. $S_1$\emph{)} is defined by \eqref{S0} \emph{(}resp. \eqref{S1}\emph{)}.
$$2)\quad v(t,\cdot)=S_{0,\mu}(t)u_0+ \int_0^t S_{1,\mu}(t-s) G(s,\cdot) ds,\quad t\in(0,T)$$
when $K$ is given by \eqref{Kdistributed}. Here $S_{0,\mu}$ \emph{(}resp. $S_{1,\mu}$\emph{)} is defined by \eqref{Smu0} \emph{(}resp. \eqref{Smu1}\emph{)}.
$$3)\quad v(t,\cdot)=\mathcal R_0(t)u_0+ \int_0^t \mathcal R_1(t-s) G(s,\cdot) ds,\quad t\in(0,T)$$
when $K$ is given by \eqref{Kmultiple}. Here $\mathcal R_0$ \emph{(}resp. $\mathcal R_1$\emph{)} is defined by \eqref{R0} \emph{(}resp. \eqref{R1}\emph{)}.\\
Finally, the solution of the IBVP \eqref{eqq1} in the sense of Definition \ref{d3} is independent of the choice of the final time $T$.
\end{thm}
\begin{proof} The proof the first two claims of this theorem are a direct consequence of Theorem \ref{t1}, \ref{t2} and \ref{t3} and the discussion in Section 2, 3, 4 for the representation of solutions. Therefore, we only need to prove that the unique solution of the IBVP \eqref{eqq1} in the sense of Definition \ref{d3} is independent of the choice of the final time $T$. For this purpose, let us consider $T_1<T_2$ and $G\in L^1(0,T_2;L^2(\Omega))$. For $j=1,2$, consider $v_j$ the weak solution of the IBVP \eqref{eqq1} with $T=T_j$ in the sense of Definition \ref{d3}. In order to prove that the solutions of \eqref{eqq1} in the sense of Definition \ref{d3} are independent of $T$, we need to show that the restriction of $v_2$ to $(0,T_1)\times\Omega$ coincides with $v_1$.
In view of the first claims of the theorem, one of the following identities hold true:
$$1)\quad v_j(t,\cdot)=S_{0}(t)u_0+ \int_0^t S_{1}(t-s) G(s,\cdot) ds,\quad t\in(0,T_j),$$
$$2)\quad v_j(t,\cdot)=S_{0,\mu}(t)u_0+ \int_0^t S_{1,\mu}(t-s) G(s,\cdot) ds,\quad t\in(0,T_j),$$
$$3)\quad v_j(t,\cdot)=\mathcal R_0(t)u_0+ \int_0^t \mathcal R_1(t-s) G(s,\cdot) ds,\quad t\in(0,T_j).$$
Thus, in each case we deduce that
$$v_1(t,x)=v_2(t,x),\quad t\in(0,T_1),\ x\in\Omega.$$
This shows that the unique solution of the IBVP \eqref{eqq1} in the sense of Definition \ref{d3} is independent of the choice of the final time $T$.\end{proof}
\section*{Acknowledgments}
This work was supported by the French National Research Agency ANR (project MultiOnde) grant ANR-17-CE40-0029.
|
2,877,628,090,223 | arxiv | \section{Introduction}
The precise determination of the CKM matrix element $V_{ub}$ remains an important goal in flavour physics, instrumental in performing stringent tests of the CKM matrix unitarity, see
\cite{Bevan:2014iga} for a review.
$|V_{ub}|$ can be extracted from $b\to u \ell\nu$ decays, and in
in particular from $B\to\pi \ell\nu$, using lattice QCD \cite{Lattice:2015tia,Flynn:2015mha} or light-cone sum rules \cite{Imsong:2014oqa,Bharucha:2012wy} calculations of the relevant form factors. The other exclusive channels $\Lambda_b\to p \ell\nu$ \cite{Detmold:2015aaa} and $B\to \pi\pi\ell\nu$ \cite{Faller:2013dwa,Kang:2013jaa} are also actively pursued.
The inclusive determination relies instead on a local Operator Product Expansion (OPE) \cite{Chay:1990da,Bigi:1992su, Bigi:1993fe,Blok:1993va,Manohar:1993qn} which has been successfully applied to $B\to X_c \ell \nu$, see \cite{Alberti:2014yda} for the state of the art.
In the charmless case $B\to X_u \ell \nu$ the convergence of the local OPE is hampered by the experimental cuts that are generally applied to suppress the charm background and that introduce a sensitivity to the Fermi motion of the $b$ quark inside the $B$ meson\footnote{Some of the recent experimental analyses employs sufficiently low cuts to capture up to 90\% of the events, justifying the use of the local OPE. However, these analyses heavily depend on the background subtraction and on the theoretical description of the signal in the shape-function region, whose understanding remains central for an accurate determination of $|V_{ub}|$ from semileptonic $B$ decays.}.
The well-known solution \cite{Neubert:1993ch,Bigi:1993ex} is to introduce a distribution function or {\it Shape Function} (SF) whose moments are dictated by the local OPE. The SF is actually the parton distribution function of the $b$ quark in the meson.
Effects formally suppressed by $\Lambda_{\rm QCD}/m_b$ are also important and lead to the emergence of additional, largely unknown, shape functions \cite{Bauer:2001mh,Leibovich:2002ys,Bosch:2004cb}.
The present HFAG inclusive $|V_{ub}|$ average \cite{hfag} is
based on different approaches \cite{Andersen:2005mj,Gambino:2007rp,Lange:2005yw,Aglietti:2007ik}. In the GGOU approach \cite{Gambino:2007rp} it reads
\begin{equation}
|V_{ub}|_{incl}=(4.51\pm 0.16^{+0.12}_{-0.15})\times 10^{-4}
\end{equation}
and very close values are found with the other methods. The high luminosity expected at Belle-II, together with precise lattice determinations of $f_B$,
will also allow for an accurate determination of $|V_{ub}|$ from decay $B\to \tau \bar\nu$.
The inclusive and exclusive determinations of $|V_{ub}|$ have been in conflict for a long time \cite{pdg}. Although the latest lattice calculations \cite{Lattice:2015tia,Flynn:2015mha} imply a somewhat larger $|V_{ub}|$ than in the past, the discrepancy is still at the level of almost 3$\sigma$ and calls for further scrutiny of all aspects of these determinations. In the case of the inclusive one, the major open problems are
$i)$ the limited knowledge of leading and subleading SFs; $ii)$ the
non-perturbative effects in the high-$q^2$ region\footnote{Weak Annihilation contributions are strongly constrained by semileptonic charm decays \cite{WA}.}; $iii)$ the potential role of higher order perturbative effects. The SFs uncertainty, in particular, has been estimated to affect $|V_{ub}|$ only at the level of a few percent \cite{Gambino:2007rp,Lange:2005yw}. However, these analyses were performed assuming a set of two-parameters functional forms, and it is unclear to what extent the chosen set is representative of the available functional space, and whether the estimated uncertainty really reflects the limited knowledge of the SFs. This point was emphasized in \cite{Ligeti:2008ac}, where a different strategy was also proposed, based on the expansion of the leading SF in a basis of orthogonal functions, fitting its coefficients to the $B\to X_s \gamma$ spectrum, and on the modelling of the subleading SFs.
In this paper we introduce a new method based on the Monte Carlo approach,
with neural networks used as unbiased interpolants for the SFs, in a way similar to what the NNPDF Collaboration do in fitting for Parton Distribution Functions \cite{Ball:2008by} and DIS structure functions \cite{Forte:2002fg}.\footnote{The possible use of neural networks to parameterize the SF in semileptonic $B$ decays has been mentioned in Ref.~\cite{Rojo:2006kr}. }
There are of course several differences with PDF fits, most notably that we parameterize functions of two parameters, and that
direct experimental information on the SFs is presently rather scarse: we only have
measurements of the photon spectrum of $B\to X_s \gamma$ above $\sim1.9$ GeV
\cite{bsgamma} and OPE constraints on the first moments of the SFs. However, the photon spectrum in inclusive radiative decays does not provide direct information on the SFs that appear in the semileptonic decay beyond leading order in $1/m_b$, and the moments do not constrain the functional form much, as we will see.
While there are several methods to determine a probability distribution function like the SF from its first moments (the {\it truncated moment problem} or Stieltjes moment problem in mathematical analysis), see {\it e.g}.\ \cite{math}, the high flexibility of neural networks allows for the straightforward inclusion of additional constraints, such as
the kinematic distributions of $B\to X_u \ell \nu$ which will
be measured with good accuracy at the upcoming Belle-II experiment \cite{future}. The measurement of the $M_X$ or $E_\ell$ shapes, for instance, will contribute useful information on the SFs and in turn reduce the SF uncertainty in the $|V_{ub}|$ extraction.
In the following we adopt the GGOU approach, where inclusive semileptonic decays without charm are described in terms of three $q^2$-dependent SFs, whose first moments are known
from the local OPE. This is the minimal set of SFs, and in this approach they are not split into a leading and several subleading SFs. The kinematic distributions accessible at Belle-II will therefore probe some of their combinations. The neural network method presented here provides a simple way to determine $|V_{ub}|$ taking into account all the constraints on the SFs, including all uncertainties and correlations properly. The SFs appearing in $B\to X_s \gamma$ and $B\to X_s \ell^+\ell^-$ can be treated with the same formalism.
The paper is organized as follows. In the next Section we recall the elements of the GGOU approach which are relevant for our topic. In Section III we discuss artificial neural networks and the way we apply them to our problem. Section IV presents our results
on the resulting SF uncertainty for $|V_{ub}|$, a new extraction of $|V_{ub}|$ from present data, and a preliminary discussion of the improvements possible using a measurement of the $M_X$ spectrum at Belle-II.
\section{Distribution Functions in $B \rightarrow X_u \ell \bar{\nu}_{\ell}$}
Our starting point is the triple differential distribution for $B\to X_u \ell\nu$, which in the case of a massless lepton can be written as
\begin{eqnarray}
\frac{d^3 \Gamma}{dq^2 \,dq_0 \,dE_\ell} &=&
\frac{G_F^2 |V_{ub}|^2}{8\pi^3}
\Bigl\{
q^2 W_1- \!\left[ 2E_\ell^2-2q_0 E_\ell + \frac{q^2}{2} \right] \!W_2
+ q^2 (2E_\ell-q_0) W_3 \Bigr\}
\nonumber\\
&& \hspace{2cm}
\times
\, \theta \left(q_0-E_\ell-\frac{q^2}{4E_\ell} \right)
\theta(E_\ell) \ \theta(q^2)\ \theta(q_0-\sqrt{q^2}),\label{eq:aquila_normalization}
\end{eqnarray}
where $q_0$ and $E_\ell$ are the total leptonic and the charged lepton energies
in the $B$ meson rest frame and $q^2$ is the leptonic invariant mass. The three structure functions $W_i(q_0,q^2)$ are in turn given by the convolution \cite{Gambino:2007rp}
\begin{equation} \label{eq:conv2}
W_i(q_0,q^2) = m_b^{n_i}(\mu)
\int F_i(k_+,q^2,\mu) \ W_i^{pert}
\left[ q_0 - \frac{k_+}{2} \left( 1 - \frac{q^2}{m_b M_B} \right), q^2,\mu \right] dk_+
\end{equation}
where $n_{1,2}=-1,n_3=-2$.
The perturbative kernels $W_i^{pert}$ are computed in the kinetic scheme \cite{kin}
with a hard cutoff $\mu$; in the present implementation \cite{Gambino:2007rp}
effects up to $O(\alpha_s^2\beta_0)$ are included.
Eq.~(\ref{eq:conv2}) defines the SFs, $F_i(k_+, q^2, \mu)$, which describe
the Fermi motion as well as other subleading effects. The $k_+$ moments of the $F_i$
are fixed by the local OPE, which provides $W_i^{pert}$, and by Eq.~(\ref{eq:conv2}). As long as perturbative
corrections to the Wilson coefficients of the power suppressed operators are neglected,
they are given by
\begin{equation}\label{momentsOPE}
\int k_+^n \ F_i(k_+, q^2, \mu) \ dk_+= \left( \frac{2 m_b}{1-\frac{q^2}{m_b M_B}}\right)^n \left[
\delta_{n0} + \frac{I_i^{(n),pow}}{I_i^{(0),tree}}\right],
\end{equation}
where $I_i^{(n)}$ represents the $n$-th central $q_0$ moment of $W_i$, reported in Appendix B of Ref.~\cite{Gambino:2007rp}. All moments but the zero-th one vanish in the limit of infinite $m_b$ and are expressed in terms of the OPE parameters. For illustration, the second moment of $F_3$ is given by
\begin{equation}
\int k_+^2 F_3(k_+, q^2, \mu) dk_+=\left( \frac{1}{1-\frac{q^2}{m_b M_B}}\right)^2
\left[ (1-{\hat q^2} )^2 \left(\frac{\mu_\pi^2}3 - \frac{\rho_{LS}^3}{3m_b}\right) -(1-{\hat q^2} ){\hat q^2} \frac{2\rho_D^3}{3m_b} \right],\label{I2f3}
\end{equation}
up to $ O(\alpha_s\Lambda^2/m_b^2, \Lambda^4/m_b^4)$ corrections. Here ${\hat q^2} =q^2/m_b^2$; $\mu_\pi^2,\mu_G^2,\rho_D^3,\rho_{LS}^3$ are the $B$-meson matrix elements
of the local dimension 5 and 6 operators that appear in the local OPE and are known
from fits to the moments of semileptonic decays into charm, see \cite{Alberti:2014yda}
for recent results.
In the kinetic scheme, the cutoff dependence of these matrix elements propagates to
$F_i$ in such a way that (\ref{eq:conv2}) is order by order independent of $\mu$.
In the limit $q^2\to 0, m_b\to \infty$ one recognizes the moment of the leading SF.
As discussed in \cite{Gambino:2007rp}, the formalism applies only to low and moderate
$q^2$. At high $q^2$ there is no hard scale and the contribution of higher dimensional operators is no longer suppressed. We therefore use Eq.~(\ref{eq:conv2}) only for $q^2<q_*^2=11$GeV$^2$. At higher $q^2$ the rate must be modelled and we employ
the second method described in Sec.~5 of Ref.~\cite{Gambino:2007rp}.
It is worth stressing that the SFs moments typically have a 20-30\% uncertainty, due to missing higher orders in the OPE, and to the limited precision with which the OPE parameters are known.
Since most of the available information on the distribution functions $F_i$ concerns their first two moments, one option is to assume for them a two-parameter functional form,
such as the exponential
\begin{equation}\label{basic1}
F(k_+)= N \,(\bar{\Lambda}-k_+)^a \, e^{b \,k_+}\
\theta(\bar\Lambda-k_+),\nonumber
\end{equation}
determining the normalization $N$ and the parameters $a$, and $b$ from the moments. An extensive set of two-parameter functional forms has been considered in \cite{Gambino:2007rp}, with the two parameters and the normalization determined in bins of $q^2$.
Even though the variation in $|V_{ub}|$ due to the choice of functional form within this set appears rather
small (typically 1-2\%), this method has obvious intrinsic limitations
and lacks the flexibility to adapt to new experimental
information which should become available at Belle-II.
In this paper we explore a different path, training neural networks as functions
of $k_+$ and $q^2$ on the moments.
In the future,
the training will involve also experimentally measured distributions.
The training yields neural network replicas which correspond to analytic parameterizations
of the functions $F_i$; they can be employed in Eqs.~(\ref{eq:aquila_normalization},\ref{eq:conv2}) to compute the branching fraction subject to given experimental cuts and, comparing this with its experimental measurement, to extract $|V_{ub}|$.
After the calculation of the complete $O(\alpha_s^2)$ and $O(\alpha_s \Lambda^2/m_b^2)$ corrections to the moments and rate of $B\to X_u \ell\nu$ \cite{Brucherseifer:2013cu,Alberti:2013kxa,Mannel:2015jka}, the implementation of the GGOU approach described in Ref.~\cite{Gambino:2007rp} needs to be updated. While we do not expect large effects on the extraction of $|V_{ub}|$, we leave this task to a future publication.
\section{Artificial Neural Networks}
Artificial Neural Networks (NNs) provide an unbiased parameterization of a continuous function. They consist of a nonlinear map between a space of inputs and a space of outputs, and are {\it universal approximators}, in the sense that they can approximate any continuous function with arbitrary accuracy, provided that sufficiently many nodes are available (for the case of feedforward NN, see \cite{hornik}).
Finite-size networks are limited in accuracy, but unlike the truncated expansion of a function in a complete basis, their nonlinear nature ensures that this is not a source of bias, as can also be checked by increasing the size of the network. For an elementary introduction to NNs, see \cite{book}. NNs have been successfully employed in many applications in high energy physics, {\it e.g}.\ in the parameterization of PDFs by the NNPDF Collaboration \cite{Ball:2008by}, and in countless experimental analyses, from tagging to triggering.
\subsection{Structure of a Feed-Forward Neural Network}
A simple Feed-Forward Artificial NN is a tuneable analytic sequence of operations on an array of input values in an attempt to recreate a desired output. The most basic system is a single-node (neuron) where a pair of inputs are weighted by adjustable multiplicative parameters, and the sum is then fed into an activation function to produce an output. By changing these weights one can adjust their network to mimic a desired operation. By combining multiple layers of nodes more complex outputs and functions can be obtained.
\begin{figure}[t]
\begin{center}
\def2.5cm{2.5cm}
\begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=2.5cm]
\tikzstyle{every pin edge}=[<-,shorten <=1pt]
\tikzstyle{neuron}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt]
\tikzstyle{input neuron}=[neuron, fill=green!50];
\tikzstyle{output neuron}=[neuron, fill=red!50];
\tikzstyle{hidden neuron}=[neuron, fill=blue!50];
\tikzstyle{annot} = [text width=4em, text centered]
\node[input neuron, pin=left:$k_+$] (I-1) at (0,-1) {};
\node[input neuron, pin=left:$q^2$] (I-2) at (0,-2) {};
\foreach \name / {\hat q^2} in {1,...,3}
\path[yshift=0.5cm]
node[hidden neuron] (H-\name) at (2.5cm,-{\hat q^2} + 0) {};
\foreach \name / {\hat q^2} in {1,...,3}
\path[yshift=0.5cm]
node[hidden neuron] (HH-\name) at (2.5cm + 2.5cm , -{\hat q^2} ) {};
\node[output neuron,pin={[pin edge={->}]right:$F_i(k_+,q^2)$}, right of=HH-2] (O) {};
\foreach \source in {1,...,2}
\foreach \dest in {1,...,3}
\path (I-\source) edge (H-\dest);
\foreach \source in {1,...,3}
\foreach \dest in {1,...,3}
\path (H-\source) edge (HH-\dest);
\foreach \source in {1,...,3}
\path (HH-\source) edge (O);
\node[annot,above of=H-1, node distance=1cm] (hl) {Hidden layer 1};
\node[annot,above of=HH-1, node distance=1cm] (hll) {Hidden layer 2};
\node[annot,left of=hl] {Input layer};
\node[annot,right of=hll] {Output layer};
\end{tikzpicture}
\vspace{0.2cm}
\end{center}
\caption{Neural Network with $\{2,3,3,1\}$ architecture.}\label{fig1}
\end{figure}
The notation used to define the initial and subsequent NN structures will be described by their node layout, i.e. $\mathbf{ \{2,3,3,1\} }$, see Fig.~\ref{fig1}. This represents a NN with 2 inputs, 1 output, and two sequential hidden layers with nodes each. The
inputs $\xi_j^{(l-1)}$ to the node $i$ in layer $l$ are combined into
\begin{equation}
\xi_i^{(l)}=g^{(l)}\left( \sum_{j=1}^{n_{l-1}} w_{ij}^{(l-1)}\xi_j^{(l-1)}-\theta^{l}_i\right),
\end{equation}
where $w_{ij}^{(l-1)}$ are the weights of the connections leading to this particular node and $\theta^{l}_i$ a {\it threshold} which is trained along with the weights. While our standard includes in total 7 hidden nodes, it can be advantageous to increase the system size to ensure convergence, as will be noted further. The number of parameters in a network depends on the number of nodes per layer, $n_\ell$, and for one or more hidden layers is
\begin{eqnarray}
N_p &= &(n_0 + 1) n_1 + (n_1 + 1)n_2 + (n_2 + 1) n_3 + \dots \nonumber
\end{eqnarray}
For example a structure $\mathbf{ \{2,3,3,1\} }$ has $25$ parameters. We have tested one and two layers and eventually employed the architecture $\mathbf{ \{2,7,1\} }$.
Various choices are possible for the activation function $g^{(l)}$, including $g(x)=\tanh x$, $g(x)= x/(1+|x|)$, and the sigmoid function
\begin{equation}
g(x) = \frac1{1+e^{-x}}.
\end{equation}
We generally employ $\tanh$ for the hidden nodes and the sigmoid for the final, to ensure positivity of the output. Changing the activation function in principle should only affect the training time and can be catered to each specific problem. For example, when convolutions of the SFs are required at each training step it is beneficial to switch to a network that employs $g(x)=x/(1+|x|)$, as the performance boost is significant. The two inputs of our NN correspond to the arguments $k_{+}$ and $q^2$ of the SFs $F_i$, both re-scaled to the interval $ (0,1)$.
\subsection{Basic Genetic Algorithm}
The network is trained using a basic Genetic Algorithm, whereupon each child generation is created from a single parent through a series of randomly selected variations on the weights of the NN. The top children as determined through an error analysis of the output are kept and they become the new parents. Each parent-child generation will be referred to as an epoch.
We begin training the NN with randomized weights. The number of variations to be made is randomly selected between 1 and 3. Each selected weight $w_{ji}$ is modified by
\begin{equation}
w'_{ji} = w_{ji} + r_i \times \eta_{NN}(g_e),
\end{equation}
where $r_i$ is a random number from $-1$ to $1$ for each weight; $\eta_{NN}(g_e)$ is a learning rate that starts at $5.0$ initially, but adaptively adjusts depending on the activity of the network; and $g_e$ is the current global epoch number. With this method large variations that would not be beneficial occur less frequently as the epochs pass. If there is no activity for a certain amount of time there is a chance that a local minimum has been found, and $\eta_{NN}$ begins increasing to allow for solutions that can escape this minimum. The learning rate and method with which the weights are adjusted should only affect the learning efficiency and should not introduce a bias in the final replica results.
The process above is repeated 20 independent times on the parent network, and the best resulting child becomes the new parent for the following epoch. The learning criterion, or ``Goodness of Fit'' measurement, is defined by the user. For different cases one can use different requirements for training. We choose to use multiple methods, which are detailed in the following section, in order to gauge the validity of this approach.
\begin{figure}[t]
\begin{center}
\includegraphics[width=11.5cm]{PlotSample0103.pdf}
\end{center}
\caption{Selection of NN replicas of $F_2(k_+,0)$ trained on the first three moments only.}
\label{NNsample}
\end{figure}
\subsection{Error Minimization}
The NN training is governed by an error function which in our case is the $\chi^2$
obtained by comparing the calculation of the first
$k_{+}$-moments of a given SF for a selected set of $q^2 \in [0\, \mbox{GeV}^2, 13\, \mbox{GeV}^2]$ with the OPE
constraints. Initially we use 7 evenly spaced
values of $q^2$. We can select an alternate set of $q^2$ values in the same range to test against over-fitting the sampling points in the $q^2$ direction, but we have verified
that this is not an issue and the functions remain smooth in $q^2$. For each value of $q^2$ we compute the normalization and first three moments in $k_{+}$,
\begin{equation}
M_{n,i}(q^2) = \int_{-\infty}^{\bar\Lambda}k_{+}^n F_i(k_{+},q^2) \,dk_{+} ,
\end{equation}
where $M_{n,i}(q^2)$ are given in Eq.~(\ref{momentsOPE}).
There are therefore 28 quantities to be fit.
However, we generally employ the third moment only as a loose constraint,
and the normalization of the second shape function is fixed to 1 at the order we are working. Throughout the learning phase we monitor the evolution of the $\chi^2$, computed in the various cases as detailed below. The scarcity of data makes it impossible to use a control sample, as done by the NNPDF collaboration. The $\chi^2$ first decreases quickly, with training progressively slowing as expected. We stop the learning when a certain condition is met, typically when
the $\chi^2$ of each replica reaches a certain value.
It is worth stressing that
the first two or three moments do not constrain the SFs much. The point is illustrated
in Fig.~\ref{NNsample} by a representative selection of NN for $F_2(k_+, 0)$, which
are normalized to 1 and satisfy the first two moments within a few \% and
and the third moment within 60\%.
A tighter constraint on the third moment would not change this picture significantly.
Of course, not all the shapes shown in this plot are physically acceptable and
only a handful of them can roughly reproduce the photon spectrum in $B\to X_s \gamma$.
However, this plot demonstrates the capability of NN to properly sample the functional space.
One should be aware that the sampling can be biased in several ways,
for instance by selection based on the speed of learning, by improper choice of random initial weights or by the use of an underlying function to speed the training up.
Indeed, in order to decrease the learning time and to ensure the vanishing of the SFs at the endpoint, we scale the network output by a function that provides the proper behavior. We know the SFs must approach zero at $-\infty$, and cut off at $\bar \Lambda$. To ensure this, one option is to define our full SFs as
\begin{equation}
F_i(k_{+},q^2) = (c_{i0}+ c_{i1} q^2) \,e^{(c_{i2}+c_{i3} q^2) k_{+}} \,(\bar\Lambda - k_{+})^{(c_{i4}+c_{i5} q^2)} \,N_i(k_{+},q^2),
\end{equation}
where $N_i$ is the NN function to be trained. The coefficients $c_{ij}$, are trained simultaneously with the NN weights and are unconstrained. In the case of
the $\mathbf{ \{2,7,1\} }$ architecture, which we generally adopt below, we therefore have a total of 35 parameters.
In order to minimize the bias we have used a set of different underlying functions, although there would be no bias if the SFs were sufficiently constrained by experimental data.
\begin{figure}[t]
\begin{center}
\includegraphics[width=17cm]{GRAPHICGRID4.png}
\end{center}
\caption{Left: examples of accepted and rejected (in red) shapes. Right: sample of NN replicas of $F_3(k_+,0)$ trained on the first three moments only after applying the selection criteria.}
\label{NNsample2}
\end{figure}
As already mentioned, additional information on the SFs comes from the photon spectrum measured in inclusive radiative B decays. One could include these data with
an additional $O(10\%)$ theoretical uncertainty to account for power suppressed corrections to the relation between the photon and semileptonic SFs at $q^2=0$. We
postpone a careful study of the photon spectrum to a future publication.
However, in the present pilot study we include the main qualitative features of the experimental photon spectrum, assuming that the SFs are all dominated by a single peak (without excluding multiple peaks) and are never too steep. As we will illustrate in a moment, these minimal assumptions
strongly reduce the variety of functional forms, as would also do a measurement of the $M_X
$ spectrum at Belle-II.
\section{Results and discussion}
{\bf A. } \ As a first step, we train the NN on the moments only and compare with the functional form error found in \cite{Gambino:2007rp}.
At this stage we are only interested in the spread of the replicas in functional space.
To this end we compute the moments with the same (outdated) input parameters used in \cite{Gambino:2007rp}, neglecting all uncertainties and correlations. Each NN replica is required to reproduce the moments at seven equally spaced
$q^2$ points between $q^2=0$ and 13GeV$^2$.
The training is stopped when
$\chi^2= n$, where $n$ is the total number of constraints, and $\chi^2$ is computed using relative errors of 3\% on the normalization, first and second moment, and of 10\% on the third moment, assuming no correlation between different moments and different bins in $q^2$.
The training is rather long and becomes very slow for smaller errors.
After training a sample of
NNs we select those whose derivative never exceeds 50 in absolute value and which have only one dominant peak (in the case of multiple peaks we check that the height of the subdominant ones is less than 20\% of the height of the dominant one, measured wrt the
common trough). A representative sample of accepted and rejected shapes is shown on the
left in Fig.~\ref{NNsample2}, while on the right we display a
sample of about 150 replicas for $F_3(k_+,0)$ after this pruning.
Each triplet of the selected NN replicas of $F_{1-3}(k_+,q^2)$ then allows for a determination of $|V_{ub}|$ when it is confronted with the
experimental results for a given partial BR. In order
to compare with the results given in the GGOU original paper we compute $|V_{ub}|$
from the same four specific experimental results used there,
namely
\begin{enumerate}
\item [A] $M_X$ cut: $M_X < 1.7, E_{\ell} > 1.0 $ GeV, Belle \cite{Bizjak:2005hn};
\item [B] Combined $M_X$ and $q^2$ cuts: $M_X \leq 1.7 {\rm GeV},\ q^2 > 8 {\rm GeV}^2, E_{\ell} > 1.0$ GeV, Babar \& Belle \cite{Bizjak:2005hn,Aubert:2005hb};
\item [C] Lepton endpoint: $E_{\ell} > 2.0$ GeV, Babar \cite{Aubert:2005mg},
\end{enumerate}
and compare the spread in $|V_{ub}|$ with the functional form dependence
given in \cite{Gambino:2007rp}. This is illustrated in Fig.~\ref{VubA}, where
the spread in the value of $|V_{ub}|$ measures the SFs uncertainty.
We have checked that using different NN architectures
leads to very similar results.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{histogram_A.png}
\end{center}
\caption{Comparison between the $|V_{ub}|$ ranges due to functional form variations given in \cite{Gambino:2007rp} and the values obtained using NN trained on moments computed with the same inputs used there.}
\label{VubA}
\end{figure}
In the calculation of the partial rates we use the same high-$q^2$ setting used by \cite{Gambino:2007rp} for the functional form uncertainty, namely the second method described in Sec.~5 of that paper.
We observe that the central values are very close to those obtained in \cite{Gambino:2007rp}. The spread in the
$|V_{ub}|$ values is larger than in 2007, but the standard deviation of the distributions are
roughly comparable with the functional form errors found in that analysis.
\vspace{2mm}
{ \bf B.} \
As a second step, we include in the analysis the complete theoretical and parametric uncertainty on the moments, with all the correlations between moments and different
$q^2$ bins. Here we want to show that the method allows us to include multiple data with
non-trivial correlations and that the errors and correlations in the inputs are reproduced by
the ensemble of trained replicas.
The OPE parameters are taken from \cite{Alberti:2014yda} and the theoretical uncertainties
of the $F_i$ moments are estimated as in that paper. The theoretical correlation between different $q^2$ bins is
estimated with method C in Sec. 3 of \cite{Gambino:2013rza}. After adding the covariance matrices related to the input parameters and to the theoretical uncertainties, a replica of pseudo-data for the moments of
the three SFs is produced assuming
gaussian distributions. The NN for each $F_i$ are then trained on this replica, keeping track of the
input parameters, and in particular of $m_b$, which is used in the calculation of physical quantities from Eq.\,(\ref{eq:aquila_normalization}).
The training is again ruled by the $\chi^2$ function, which now includes all correlations.
Even though the typical total uncertainty of the first three moments is as large as 25-30\%, high correlations between $q^2$ bins do not allow to speed up the training significantly.
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|c|c|}
\hline Experimental cuts (in GeV or GeV$^2$) & $|V_{ub}| \times 10^{3}$ & $|V_{ub}| \times 10^{3}$\cite{hfag}\\ \hline
$M_X < 1.55, E_{\ell} > 1.0 $ Babar \cite{babarnew}& $4.30(20)(^{26}_{27})$ &$4.29(20)(^{21}_{22}) $ \\
$M_X < 1.7, E_{\ell} > 1.0 $ Babar \cite{babarnew}& $4.05(23)(^{19}_{20})$ & $4.09(23)(^{18}_{19}) $ \\
$M_X \leq 1.7 , q^2 > 8, E_{\ell} > 1.0$ Babar\cite{babarnew}& $4.23(23)(^{26}_{28})$ &$4.32(23)(^{27}_{30}) $ \\
$E_{\ell} > 2.0$ Babar \cite{Aubert:2005mg}& $4.47(26)(^{22}_{27})$ &$4.50(26)(^{18}_{25}) $ \\
$E_{\ell} > 1.0$ Belle \cite{Urquijo:2009tp}&$4.58(27)(^{10}_{11})$ &$4.60(27)(^{10}_{11}) $ \\
\hline
\end{tabular}
\end{center}
\caption{$|V_{ub}|$ determinations using different experimental analyses and comparison with HFAG latest results in the GGOU approach \cite{hfag}. }
\label{table1}
\end{table}
As we adopt up-to-date inputs, we can extract $|V_{ub}|$ from
the latest experimental results and compare the results with the most recent HFAG
compilation \cite{hfag}; this is done in Table 1 for the most representative cases, using
the isospin average $\tau_B=1.582(5)$ps, employed in the derivation of the HFAG values.
The first uncertainty represents the total experimental error, while the second is the sum in quadrature of the standard deviation of the values obtained by the replicas (which
in this case accounts for both functional form and parametric uncertainties),
and all the remaining theoretical uncertainties (perturbative, treatment of the high $q^2$ tail, Weak Annihilation), which are estimated in the same way as in \cite{Gambino:2007rp}.
While we refrain from combining the values of $|V_{ub}|$ originating from different experimental analyses,
we observe that the central values are quite close to those obtained by HFAG. A minor shift downwards is to be expected because, following \cite{Alberti:2014yda}, we adopt a slightly
higher $m_b$ than employed by HFAG.
The theoretical errors, which are asymmetric because of the Weak Annihilation error, are generally slightly larger than those reported by HFAG, especially when the cuts make $|V_{ub}|$ more sensitive to the SFs. This is due to $i)$ a larger spread in
the functional space of the $F_i$ compared with the method of Ref.~\cite{Gambino:2007rp}
used by HFAG; $ii)$ the introduction of a non-negligible theoretical error for the SFs moments, which was not considered in \cite{Gambino:2007rp}.
Given the uncertainties, the agreement between the different rows of Table 1 is good,
and shows that the OPE based approach describes the present data on $B\to X_u \ell \nu$ reasonably well.
We also notice that the HFAG average, Eq.~(1), is dominated by the Belle analysis
\cite{Urquijo:2009tp}, reported in the last row of Table 1,
and by a similar Babar analysis with a $p^*>1.3$GeV cut, since they have a significantly
smaller theoretical error and both prefer a high $|V_{ub}|$.
However, as already mentioned in the Introduction, these analyses heavily depend on background subtraction and signal modelling. On the other hand,
the values reported in the first three rows of Table 1 are consistent with
the recent exclusive $|V_{ub}|$ results given in \cite{Lattice:2015tia} and \cite{Flynn:2015mha} within 1.5$\sigma$.
\vspace{2mm}
{ \bf C.} \ Finally, we consider the possible impact of future Belle-II data on the SFs and consequently on the $|V_{ub}|$ determination. We assume that
Belle-II will measure the $M_X$ spectrum in 8 evenly spaced bins below $M_X=2$GeV, with a total 4\% uncertainty in each bin. A detailed estimate of the potential improvement in
the $|V_{ub}|$ determination would involve a lengthy training of the NN on both the moments and these new data and is beyond the scope of this paper.
Here we demonstrate the discriminating power of the $M_X$ spectrum data
using the NN replicas obtained in step A above, all of which reproduce precisely the first moments. We use randomly selected triplets of these NNs to compute the $M_X$-spectrum
and compare the results with a reference $M_X$-spectrum obtained using one of these triplets. The results are shown in the two insets in Fig.\,\ref{Spectra_Pruned}.
The plot on the lhs refers to replicas which survived the pruning described in {\bf A},
while on the rhs the spectra are produced based on replicas that have been trained on the moments, but failed our acceptance criteria. In both cases the shaded band corresponds to the 1$\sigma$ band around the central value. We observe that the $M_X$-spectrum
is very sensitive to the presence of sharp features in the SFs, which are more likely in the
rejected sample, and that a precise measurement of the spectrum can even exclude many of the replicas in the accepted sample. The Belle-II data therefore have the potential to
constrain significantly the SFs and to validate the OPE-based approach to inclusive charmless
semileptonic $B$ decays.
The above considerations can be made more quantitative by defining a $\chi^2_X$ based on the comparison of the yield in each $M_X$-bin computed from a given triplet with the yield
computed from the reference (``simulated'') $M_X$-spectrum data, assuming a total 4\% error.
The main graph in Fig.\,\ref{Spectra_Pruned} shows the distributions of accepted and rejected replicas as a function of $\chi^2_X$. Most of the rejected replicas and many of the accepted ones would be excluded
by a test based on $\chi^2_X$. Indeed, one can {\it reweight} the NN replicas using
their $\chi^2_X$, giving more importance to the replicas whose $M_X$ spectrum is closer to the experimental one and therefore have lower $\chi^2_X$, see \cite{Ball:2010gb}.
Performing such an exercise on our step A shows that reweighting with the $\chi^2_X$ reduces the uncertainty from the functional forms by 30-70\%, depending on the experimental cuts, and induces a $(0.2-0.4)\%$ negative shift in $|V_{ub}|$.
Of course, the $M_X$-spectrum carries information not only on the SFs, but also on the
HQE parameters $m_b$, $\mu_\pi^2$, etc.\ which have been fixed in the exercise we have just discussed. This is related to what one can learn from the moments of the
$M_X$ spectrum, see \cite{Gambino:2005tp} for a discussion of their sensitivity to the HQE parameters. As a consequence, reweighting the replicas of step B based on the same data would have a much more dramatic effect, because their first moments have a much larger spread.
Unfortunately, in step B the number of available replicas is too limited. The main point to be emphasised, however, is that in our framework the $B\to X_u \ell\nu$ kinematic distributions
($M_X, q^2, E_\ell$ spectra) can be considered together
with all the available relevant information ($B\to X_c \ell\nu$ moments, $B\to X_
\gamma$ spectrum, $m_{b,c}$ determinations, etc.), in the context of a NN training where the HQE parameters are fitted together with the NN parameters. Such analysis
will be mandatory with Belle-II data.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{Spectra_A_Pruning.png}
\end{center}
\caption{Sample $\chi^2_X$ comparison of the step-A sample of NNs with simulated $M_X^2$ spectra data. Inset are the 2-peak pruning analysis and the resulting $M_X^2$ spectra for the pass(blue)/fail(red) ensembles, with the reference spectrum in black.}
\label{Spectra_Pruned}
\end{figure}
\section{Summary}
We have introduced a new parameterization of the SFs characterizing inclusive semileptonic $B$ decays without charm based on artificial neural networks.
The new method allows
for alternative, unbiased estimates of the SFs functional form uncertainty, which turn out
to be in reasonable agreement with previous results obtained using
functional form models.
As we have shown explicitly, a clear advantage of the method is that it permits a straightforward implementation of new experimental data, such as those which will become available at Belle-II. These data will
reduce the SFs uncertainty and, most importantly, their comparison with high-precision theoretical predictions will validate the OPE-based approach in a much more stringent way.
\begin{acknowledgments}
We are grateful to Alberto Guffanti and Hannes Zechlin for very useful discussions. We thank the Mainz Institute for Theoretical Physics (MITP) for hospitality and partial support during the workshop {\it Challenges in semileptonic B decays} in April 2015, where part
of the work was done. Work supported in part by MIUR under contract 2010YJ2NYW 006.
\end{acknowledgments}
|
2,877,628,090,224 | arxiv | \section{Introduction}
\label{s1}
\numberwithin{equation}{section}
Virtually every area of quantum physics is, at least partially, concerned with coherent states. This is due to their relevance in relation to the classical limit of quantum theories. Indeed, a minimum requirement for a family of coherent states is that each state is labelled by a point in the phase space of the theory. This is then interpreted as the quantum state that represents the closest approximation to the classical configuration of that phase space point.
\\
In the context of Loop Quantum Gravity (LQG) \cite{Rov04,AL04,Thi07} -- a non-perturbative quantization of General Relativity (GR) in its 3+1 ADM formulation -- coherent states have been constructed based on the concept of weave states \cite{weave} and heat kernel techniques for compact groups \cite{Hall1,Hall2}. The resulting states have been called ``complexifier coherent states'' \cite{Winkler1,Winkler2,Winkler3,SahThiWin,ThiemanComplex} or ``gauge coherent states'' (owing to the fact that they are covariant under the $SU(2)$ gauge group of LQG). Their properties and their relations to other semiclassical states have been thoroughly investigated \cite{SahThi1,SahThi2,BahThi1,BahThi2,MagliaroPerini}. It is important to notice that gauge coherent states are based on a single graph (or a sum over countably many graphs).\footnote
{
If we naively take a linear combination of such fixed-graph coherent states on the label-set of graphs, we discover that there is no damping factor fast enough to make the norm of the state finite. This problem is due to the non-separability of the kinematical Hilbert space of LQG, and might be solved at the physical level. However, a definition of coherent states even at the diffeomorphism-invariant level is still missing, though proposals exist \cite{ALMMT}.
}
This means that they cannot describe continuous geometries. Nevertheless, if one limits their interest to the Hilbert space of a fixed graph (which we will do), each state in the family represents a (discrete) classical 3-geometry. This is perfectly fine for most applications, and a theoretical justification for working on a fixed graph was given in the framework of Algebraic Quantum Gravity (AQG) \cite{AQG1,AQG2,AQG3}. In this formulation of LQG, one works with an abstract graph (that is, not embedded in a 3-manifold, which has the added benefit of simplifying the implementation of diffeomorphism symmetry), and hence only excitations along already existing edges are allowed.\footnote
{
As a consequence, we can only consider non-graph-changing operators. This is particularly important for the Hamiltonian operator, as one must abandon the original regularization \cite{Thi96_1,Thi96_2} and consider instead a non-graph-changing one \cite{AQG2}.
}
\\
In this series of papers, we wish to consider the applications of gauge coherent states to the description of classical cosmological spacetimes. Due to the homogeneity that characterizes such geometries, the choice of the fixed graph is that of a regular lattice, which we take to be cubic (i.e., each vertex is 6-valent) so that it can be oriented along a fiducial system of coordinates of the 3-geometry. This is the same starting point of Loop Quantum Cosmology (LQC) \cite{LQC1,LQC2,LQC3,LQC4}, which is a LQG-inspired quantization of homogeneous spacetimes: the hope is therefore to shed light on the relation between the full theory of LQG and the mini-superspace models described by LQC. Particularly, we shall be interested in the so-called ``effective dynamics'': in LQC it was numerically shown \cite{AshPawSin} that the expectation value of the Hamiltonian $\hat H_{LQC}$ on gaussian states $\psi_{(c,p)}$ plays the role of effective Hamiltonian. By this, we mean that the Hamiltonian flow produced by $H_{\text{eff}}(c,p) := \langle \psi_{(c,p)} | \hat H_{LQC} | \psi_{(c,p)} \rangle$ on the $(c,p)$-phase space of cosmology coincides with the trajectories followed by the peak of gaussian states when they are evolved via $\hat H_{LQC}$:
\begin{align}
\langle \psi_{(c,p)} | e^{i \phi \hat H_{LQC}} \hat F e^{-i \phi \hat H_{LQC}} | \psi_{(c,p)} \rangle = e^{\{\cdot, H_{\text{eff}}(c,p)\}} \langle \psi_{(c,p)} | \hat F | \psi_{(c,p)} \rangle + \mathcal O(s)
\end{align}
for $\hat F$ an operator on the Hilbert space of LQC, $\phi$ some matter degree of freedom playing the role of physical time (e.g., a massless scalar field), and $s$ the spread of state $\psi_{(c,p)}$. The question of whether this feature lifts to full LQG is still open, but we shall nevertheless refer to the expectation value of the full Hamiltonian on our coherent sates as ``effective Hamiltonian'', and we ask whether the dynamics it generates on the phase space coincides with that of LQC. The answer has been found affirmative in toy models such as Quantum Reduced Loop Gravity \cite{QRLG1,QRLG2} but for full LQG it turns out to be in the negative, and the result was discussed in the short paper \cite{DaporKlaus}. The current series of papers is a detailed presentation of the techniques developed and used to derive this result.
\\
In this first paper, we identify the classical degrees of freedom corresponding to a $k=0$ Robertson-Walker spacetime and propose the subfamily of gauge coherent states that describes such classical spacetimes. Then we develop the techniques needed to compute expectation values (and, ultimately, matrix elements) of operators on this subfamily. Applying this technology to the elementary operators (i.e., holonomies and fluxes) and to their dispersions (spreads), we confirm that each of these states is peaked on the classical Robertson-Walker geometry that labels it. We finally apply the techniques to the volume operator, computing its expectation value to next-to-leading order in the semiclassicality parameter that controls the spread of the states: we find that the leading order is in agreement with the classical volume of a cubic cell in flat Robertson-Walker spacetime.
\\
\\
The article is organized as follows.
\\
In section \ref{s2} we briefly review LQG, with particular attention to the quantization of the scalar constraint, as it is related to the Hamiltonian in the full theory. We also discuss the the basics of gauge coherent states.
\\
In section \ref{s3} the subfamily of cosmological coherent states will be presented. We also prove certain relaions satisfied by these states, which will drastically simplify the computations in the next sections.
\\
In section \ref{s4} we compute the expectation values of monomials in the fundamental operators (holonomy and flux), as these are the basic building blocks for the interesting geometric operators.
\\
In section \ref{s5} the tools are put into action, as we compute the expectation values and dispersions of the fundamental operators, showing that the states are indeed peaked on homogeneous isotropic geometries. We also compute the expectation value of what we call the ``Giesel-Thiemann Volume Operator'' which, as it was shown in \cite{AQG3}, coincides with the Ashtekar-Lewandowski Volume Operator up to a desired order in $\hbar$.
\\
In section \ref{s6} we summarize our results and briefly comment on further research.
\\
Finally, a word on the appendices: appendix \ref{recovering-gauge} deals with the question of gauge-invariance as far as our cosmological coherent states are concerned; appendix \ref{basic-properties} collects some properties of $SU(2)$ irreducible representations and recoupling theory (used in computations throughout the text); appendix \ref{hol-int} contains the explicit computation of certain integrals relevant for expectation values involving the holonomy operator.
\section{Review of LQG and gauge coherent states}
\label{s2}
This section is merely a recap of known results in order to clarify the notation used in this article. The experienced reader may jump directly to section \ref{s3}.
\subsection{Ashtekar-Barbero Variables}
We cast general relativity into its hamiltonian formulation, by splitting our four-dimensional manifold $\mathcal{M}=\mathbb{R}\times \sigma$, where $\sigma$ is a smooth 3-manifold which admits Riemannian metrics. On $\Sigma$ we can define triad fields $e^a_I$, as well as co-triads $e^I_a$ where $a,b,c...\in\{1,2,3\}$ denote tensorial indices and $I,J,K..\in\{1,2,3\}$ can be thought of as $\mathfrak{su}(2)$ algebra indices. These triads are subject to the condition that the 3-metric can be derived from them via $q_{ab}=e^I_ae^J_b \delta_{IJ}$ and that $e^a_Ie^I_b=\delta^a_b$, $e^I_ae^a_J=\delta^I_J$. We then introduce the Ashtekar-Barbero variables \cite{AB-variables1,AB-variables2,AB-variables3}, i.e., the lie-algebra-valued $\text{SU}(2)$-connection $A_a=A_a^I\tau_I$ and the densitized triad $E^a_I$:
\begin{align}
A^I_a (x) := \Gamma^I_a(x)+\beta K_{ab}(x)e^b_J\delta^{JI},\hspace{20pt} E^a_I (x) := \mid \det(e)\mid e^a_I
\end{align}
with $\Gamma^I_a$ the complex-valued spin-connection of $e^I_a$, $K_{ab}$ the extrinsic curvature of $\sigma$, $\det(e):=\det(\{e^J_c\}^J_c)$ and $\beta\in \mathbb{R}$ the Immirzi-parameter. By $\tau_I = -i\sigma_I$ we denote the imaginary Pauli matrices, which in cartesian coordinates $(I=1,2,3)$ are given by
\begin{align}
\tau_1=-\left(\begin{array}{ccc}
0 & i\\
i & 0
\end{array}\right), \hspace{15pt}
\tau_2=-\left(\begin{array}{ccc}
0 & 1\\
-1 & 0
\end{array}\right), \hspace{15pt}
\tau_3=-\left(\begin{array}{ccc}
i & 0\\
0 & -i
\end{array}\right), \hspace{15pt}
\end{align}
They are the generators of the lie algebra $\mathfrak{su}(2)$ and fulfill $\text{Tr}(\tau_I\tau_J)=-2\delta_{IJ}$ and $[\tau_I,\tau_J]=2\epsilon_{IJ}^{\hspace{8pt}K}\tau_K$.
\\
The Ashtekar-Barbero variables form a canonical pair $(A^I_a,E^a_I)$, i.e., we find for their Poisson-bracket that $\{A^I_a (x), E^b_J(y)\}=\delta^b_a \delta^I_J\delta^{(3)}(x,y)$. They uniquely identify a physical geometry if one imposes the {\it Gauss constraint}
\begin{align}
G_J:= \partial_a E^a_J + \epsilon_{JKL}\delta^{LM} A^K_a E^a_M
\end{align}
and the standard constraints of GR: the {\it vector constraint} (smeared with an arbitrary shift-function $N^a$)
\begin{align}
\bar{C}_{Diff}[\bar{N}]=\int dx^3 N^a (x) F_{ab}(x)^IE_I^b(x)
\end{align}
and the {\it scalar constraint} (smeared with an arbitrary lapse function $N$)
\begin{align}
C [N]&=\int dx^3 N \left( F^I_{ab}-(\beta^2+1)\epsilon_{I}^{\hspace{2pt}LM}K_{ac}e^c_LK_{bd}e^d_M\right)\frac{\epsilon^{IJK}E^a_JE^b_K}{\sqrt{\det(q)}}
\end{align}
where $F_{ab}^I:= 2\partial_{[a}A^I_{b]}+\epsilon_{IJK}A^J_aA^K_b$ is the curvature of the connection. For the purpose of quantization, one uses the Thiemann identities \cite{Thi96_1,Thi96_2}
\begin{align}
\{V,A^J_a\}=e^J_a=\frac{1}{2}sgn(\det(e))\epsilon^{JKL}\epsilon_{abc}\frac{E^b_KE^c_L}{\sqrt{\det(q)}},\hspace{20pt}\{V,C_E[1]\}=\beta^2\int dx^3 K_{ab}e^b_I\delta^{IJ} E^a_J
\end{align}
to rewrite the scalar constraint as
\begin{align} \label{ScalarConstraint}
C [N]=\int dx^3 N \left(F^I_{ab}\delta_{IJ} -(\beta^2+1)\epsilon_{JMN} \{\{V,C_E[1]\},A^M_a\}\{\{V,C_E[1]\},A^N_b\} \right) \epsilon^{abc}\{V,A^J_c\}
\end{align}
where $C_E[N] := \int d^3x N F^I_{ab}\delta_{IJ}\epsilon^{abc}\{V,A^J_c\}$ is called the {\it euclidean Hamiltonian} and $V :=\int dx^3 \sqrt{\det(q)}$ is the total volume of the manifold.
\subsection{Quantization}
As we are now in the situation of a gauge theory, we can quantize the phase space variables and the constraints following Dirac procedure. We smear the connection $A_a(x)$ along any curve $e$ in the manifold to obtain the {\it holonomy} $h(e)\in\text{SU}(2)$, i.e., the path-ordered exponential of the connection along $e$. We similarly smear the densitized triad $E^a$ against any 2-dimensional surface $S$ to obtain the {\it flux} $E(S)\in\mathfrak{su}(2)$. For a curve $e$ and a surface $S$ we thus define
\begin{align}
h(e):=\mathcal{P}\exp\left(\int_0^1 dt A_a (c(t))\dot{c}^a(t)\right),\hspace{10pt} E^I(S):=\int_S \epsilon_{abc} dx^a\wedge dx^b E^c_J\delta^{IJ}
\end{align}
The set of $(h,E)$ along {\it all} curves and surfaces constitutes the {\it holonomy-flux algebra}, which one uses to define the Hilbert space by GNS construction. If one fixes finitely many oriented curves, the union of them forms a graph, $\gamma=\bigcup_l e_l$. We call $e_l\in E(\gamma)$ an edge (or link) and any intersection $v \in V(\gamma)$ of two edges a vertex (or node). One can then associate a Hilbert space to $\gamma$ by considering the tensor product of square integrable functions on each edge, $\mathcal{H}:=\otimes_{e\in E(\gamma)} \mathcal{H}_e$ with $\mathcal{H}_e=L_2(\text{SU}(2),d\mu_H)$, $d\mu_H$ being the unique Haar measure on $\text{SU}(2)$. The elements $F_{\gamma}\in \mathcal{H}$ are called {\it cylindrical functions}. The holonomies get promoted to bounded, unitary multiplication operators: for $f_e\in\mathcal{H}_e$ it is
\begin{align}
\hat{h}_{mn}(e)f_e(g):=D^{(\frac{1}{2})}_{mn}(g)f_e(g)
\end{align}
where $D^{(\frac{1}{2})}_{mn}(g)$ is the Wigner-matrix of group element $g$ in the defining irreducible representation of $\text{SU}(2)$ corresponding to spin-1/2 \cite{Brink-Satchler}. The Peter-Weyl Theorem ensures that they form an orthogonal basis, hence any function in $\mathcal H_e$ can be written as $f_e(g_e)=\sum_{j}\sum_{-j\leq m,n\leq j} c_{jmn} D^{(j)}_{mn}(g_e)$, where $j\in \mathbb{N}/2$ (sums over magnetic indices $m,n,...$ will be suppressed in the following). The scalar product is given in the $L_2$ sense:
\begin{align}
\langle F_{\gamma},F_{\gamma'}'\rangle=\delta_{\gamma,\gamma'} \prod_{e\in\gamma} \int d\mu_{H}(g_e) \overline{f_e(g_e)} f_e' (g_e) \\
\int d\mu_H(g) \overline{D^{(j)}_{mn}(g)} D^{(j')}_{m'n'}(g)=\frac{1}{d_j}\delta_{jj'}\delta_{mm'}\delta_{nn'}\label{orthogInt}
\end{align}
where the dimension of spin-j $\text{SU}(2)$-irrep is $d_j=2j+1$. Similiarly, the fluxes become essentially self-adjoint derivation operators:
\begin{align} \label{fluxy}
\hat{E}^K(S)f_e(g):= -\frac{i\hbar\kappa\beta}{4} \sigma(e\cap S) f_{e_1}(g_{e_1}) R^K(e_{2})f_{e_2}(g_{e_2})
\end{align}
where $\sigma(e\cap S)\in\{0,\pm 1\}$ (depending on if edge and surface meet non-transversally or under the same/opposite orientation respectively), $e=e_1 \circ e_2$ such that $s_e=e\cap S$ is the starting point of edge $e_2$ and $g=g_{e_1}g_{e_2}$ (which makes the splitting unique). Finally, the {\it right-invariant vector field} $R^K(e)$ is defined together with the {\it left-invariant vector field} $L^K(e)$ as
\begin{align}
R^K(e)f_e(g):=\left.\frac{d}{ds}\right|_{s=0} f_e(e^{s\tau_K}g),\hspace{20pt} L^K(e)f_e(g):=\left.\frac{d}{ds}\right|_{s=0} f_e(ge^{s\tau_K})
\end{align}
In particular, the action of $R^K$ on the basis element is given by
\begin{align} \label{R-and-D-prime}
R^KD^{(j)}_{mn}(g) = {D'}^{(j)}_{m\mu}(\tau_K)D^{(j)}_{\mu n}(g), \ \ \ {D'}^{(j)}_{mn}(\tau_K) = 2i\sqrt{j(j+1)d_j}(-1)^{j+n}\left(\begin{array}{ccc}
j & 1 & j\\
n & K & -m
\end{array}\right)
\end{align}
as is shown in appendix \ref{basic-properties}. This concludes the description on the {\it kinematical Hilbert space} of LQG.
\\
It remains to incorporate the constraints $G_J, \bar{C}_{Diff}[\bar{N}], C[N]$. The Gauss constraint $G_J$ is easily incorporated by the fact that it is the generator of $\text{SU}(2)$-rotations, hence its solutions are states of $\mathcal{H}$ which are invariant under $\text{SU}(2)$. These can be obtained by group averaging: let $U_G[g]$ be the operator that generates a local $g(x)\in\text{SU}(2)$ transformation and $F_{\gamma}(\{g\})=\prod_{e\in\gamma} f_e(g_e)$; then the corresponding gauge-invariant function is
\begin{align}\label{Group-Averaging}
F_{\gamma}^G(g)= \int D[\{h\}] U_G[\{h\}] F_{\gamma}(\{g\}) := \left( \prod_{v\in V(\gamma)} \int d\mu_H (h_v)\right) \prod_{e\in\gamma} f_e(h_{s_e} gh_{t_e}^{-1})
\end{align}
where $v$ runs through all vertices in $\gamma$, and $s_e, t_e$ denote the vertex at the beginning/end of edge $e$ respectively.\\
The vector constraint $\bar{C}_{Diff}[\bar{N}]$ generates diffeormorphisms of the spatial manifold $\sigma$, and therefore cannot be implemented as an infinitesimal operator due to the action of the diffeomorphism group $\text{Diff}(\sigma)$ not being strongly continuous. Nevertheless, diffeomorphism-invariance can still be implemented via finite diffeomorphisms $\varphi\in\text{Diff}(\sigma)$. For this purpose, in this paper we adopt the idea developed in the context of AQG \cite{AQG1}, where one considers {\it abstract graphs}, that is, graphs which ``forget'' about their embedding in $\sigma$. We will talk about what this explicitly means in section \ref{s3}, when we choose the states with respect to which we compute expectation values.
\\
Finally, let us consider the implementation of the scalar constraint in the quantum theory. Following the strategy of \cite{Thi96_1,Thi96_2}, we rewrite (\ref{ScalarConstraint}) in terms of holonomies and then promote every term to an operator. For the volume -- which was pivotal for using the Thiemann-identities -- this leads to the Ashtekar-Lewandowski volume operator \cite{volume1,volume2}:
\begin{align}
\hat{V}(\sigma) F_{\gamma}(\{g\}) = \ & \frac{(\beta\hbar\kappa)^{3/2}}{2^{5}\sqrt{3}}\sum_{v\in V(\gamma)}\hat{V}_v F_{\gamma}(\{g\}) =\frac{(\beta\hbar\kappa)^{3/2}}{2^{5}\sqrt{3}}\sum_{v\in V(\gamma)}\sqrt{\mid \hat{Q}_v \mid}F_{\gamma}(\{g\}), \label{ALvolume}
\\
\hat{Q}_v := \ & i\sum_{e\cap e'\cap e'' =v } \epsilon(e,e',e'')\epsilon_{IJK}R^I(e)R^J(e')R^K(e'') \label{Q-oper}
\end{align}
with $\epsilon(e, e', e'') := sgn(\det(\dot{e},\dot{e}',\dot{e}''))$ and all edges outgoing at the vertex $v$. (In case of an $e$ being ingoing, one simply replaces $R^K(e)\rightarrow L^K(e)$.) Since the square-root is understood in the sense of the spectral theorem, knowledge of the full spectrum of $\hat{Q}_v$ is required before we can say how $\hat{V}_v$ acts on any state. Unfortunately, despite a lot of research has been done \cite{volume3,volume4,volume5} on the spectrum of (\ref{ALvolume}), a general formula for its eigenstates is still unknown.
\\
Among the various choices of regularization proposed for the scalar constraint, we will use the framework first developed in AQG \cite{AQG2}, where one chooses the scalar constraint to act in a {\it non-graph-changing} way, i.e., one regularizes the curvature of the Ashtekar connection by $F_{ab}(x)\dot{e}^a\dot{e}^b=[h(\square_{ee'})-h(\square_{ee'})^{\dagger}]/2\epsilon^2+\mathcal{O}(\epsilon)$ where $\square_{ee'}$ denotes a small loop starting at $x$ along $e$ and returning along $e'$. Then, we choose for the action of the loop-holonomy the operator $\hat h(\square_{ee'})$, which starts at a vertex $v$ of the graph and goes along already existing, excited edges of $F_{\gamma}$. Thus, the total operator in its symmetrized version looks as follows:
\begin{align}
\hat{C}[N] = \frac{1}{2}\left(\hat{C}_E+\hat{C}_E^{\dagger}\right)-\frac{\beta^2+1}{2}\left(\hat{C}_L+\hat{C}_L^{\dagger}\right)
\end{align}
where
\begin{align}
\hat{C}_E[N]:=&\frac{32}{3i\kappa^2\hbar\beta}\sum_{v\in V(\gamma)}\frac{N_v}{20}\sum_{e\cap e' \cap e''=v}\epsilon(e,e',e'')\frac{1}{2}\times\nonumber\\
&\hspace{30pt}\times\text{Tr}\left((\hat{h}(\square_{ee'})-\hat{h}(\square_{ee'})^{\dagger})\hat{h}(e'')\left[\hat{h}(e'')^{\dagger},\hat{V}_v\right]\right)\label{EuclHam}
\\
\hat{C}_L[N]:=&\frac{128}{3i\kappa^4\hbar^5\beta^5}\sum_{v\in V(\gamma)}\frac{N_v}{20}\sum_{e\cap e'\cap e''=v}\epsilon(e,e',e'')\times\nonumber\\
&\hspace{30pt}\times\text{Tr}\left(\hat{h}(e)\left[\hat{h}(e)^{\dagger},[\hat{C}_E[1],\hat{V}_v]\right]
\hat{h}(e')\left[\hat{h}(e')^{\dagger},[\hat{C}_E[1],\hat{V}_v]\right]
\hat{h}(e'')\left[\hat{h}(e'')^{\dagger},\hat{V}_v\right]\label{LorHam}
\right)
\end{align}
and $N_v$ is the value of lapse function $N$ at $v \in \sigma$.
\subsection{Deparametrization with Gaussian Dust}
Instead of dealing with vacuum GR, where one has to solve the scalar constraint $C[N]$, one can construct observables by adding matter to the theory and trying to find local coordinates such that the constraint acquires the form $C = P + H$ in terms of the conjugated momentum $P$ to the matter degree of freedom. If this form is achieved, one speaks of ``relational observables'' and ``deparametrization'' \cite{dep1,dep2,dep3,dep4,dep5,dep6,dep7}: the function $H$ becomes a physical, conserved Hamiltonian density which drives the physical evolution of the observables with respect to the matter degree of freedom (which therefore plays the role of physical clock, $\tau$). While not all types of matter allow for this decomposition, a good choice is Gaussian dust: in the framework of Torre and Kucha$\check{\text{r}}$ \cite{dep1}, the Lagrangian added to the Einstein-Hilbert action describing Gaussian dust is
\begin{align}\label{TorreKuchar}
\mathcal{L}_{GD}=- \sqrt{\mid \det(g)\mid}\left(\frac{\varrho}{2}(g^{\mu\nu}T_{,\mu}T_{,\nu}+1)+g^{\mu\nu}T_{,\mu}(W_jS^j_{,\nu})\right)
\end{align}
with the field $\varrho$ having dimension $[\text{cm}^{-4}]$, the fields $T,S^j$ having dimension $[\text{cm}]$, and $W_j$ being dimensionless. Performing Legendre transformation, one can show that the time-evolution of an observable ${O}_F$ (associaed with phase space function $F$) is encoded as the Schr\"odinger-like equation $d{O}_F(\tau)/d\tau=\{H, {O}_F(\tau)\}$, where
\begin{align}
H = C[1] = \int dx^3 C(x)
\end{align}
is for this reason called the {\it true Hamiltonian}. We see that $C$ it is not longer a constraint whose vanishing must be imposed, but in fact it generates time-translations. Thus, if one takes this viewpoing, the quantum scalar contraint presented above is understood as the quantum operator producing the dynamics of geometric degrees of freedom wrt the classical observer provided by the dust.
\subsection{Gauge Coherent States}
We have now at our disposal a physical Hilbert space on the fixed graph $\gamma$, and have an understanding of what we mean by dynamics. But while any state $F \in \mathcal H$ can be considered, in this work we will focus on a subset of the {\it gauge coherent state} family. Let us therefore briefly review the general definition and properties of this family.
\\
Following Hall \cite{Hall1, Hall2}, one constructs a coherent state $\psi^t_{e,h_e^{\mathbb{C}}}$ for every edge $e$ of the graph, and glues them together in a cylindrically-consistent way obtaining $\Psi^t_{\gamma,\{h^{\mathbb{C}}\}}(\{g\}):=\prod_{e\in E(\gamma)}\psi^t_{e,h^{\mathbb{C}}_e}(g_e)$. To construct $\psi^t_{e,h_e^{\mathbb{C}}}$ one uses a complex polarization of the classical phase space, i.e., a unitary map $(A,E) \mapsto A^{\mathbb{C}}$ that expresses the complex connection as a function of the real phase space. For example, the left-polar decomposition prescribes
\begin{align} \label{polar-decomposition}
h_e^{\mathbb{C}}:=\exp\left(-\frac{it}{\hbar\kappa\beta} \tau_J E^J(S_e)\right) h(e) \in SL(2,\mathbb C)
\end{align}
where $h(e)$ is the classical holonomy along edge $e$ and $E^J(S_e)$ is the classical flux across the open surface $S_e$ manually assigned to each $e\in E(\gamma)$ in such a way that (1) all $S_e$ are mutually non-intersecting, (2) only $e$ intersects $S_e$ and the intersection is transversal and consists of only one point, (3) both $S_e$ and $e$ carry the same orientation. The dimensionless quantity $t:=\hbar\kappa/a^2 > 0$ is called the {\it semiclassicality parameter}, with $a$ being a length scale that the theory should provide.\footnote
{
As we will see later, $t$ controls the spread in holonomy and flux of the coherent state: The smaller $t$, the smaller the relative dispersions of $h$ and $E$. It has been therefore argued \cite{AQG3} that the natural choice for $a^2$ in a vacuum gravity context is the inverse of cosmological constant, $a^2 = 1/\Lambda$. Using $\kappa = 16\pi G/c^3$, one then finds $t \sim 10^{-120}$.
}
\\
To construct the coherent state in $\mathcal H_e$ peaked on $h_e^{\mathbb{C}} \in SL(2,\mathbb{C})$, one first chooses a {\it complexifier} $\hat{C}_t$ and exponentiates it: this gives rise to the coherent state transform, which for the choice of heat kernel complexifier \cite{Winkler1} reads
\begin{align}
\hat{W}_t := e^{-\frac{1}{\hbar}\hat{C}_t} = e^{\frac{t}{8}\delta_{IJ}R^I(e)R^J(e)}
\end{align}
The {\it (gauge-variant) coherent state} is now obtained by applying $\hat{W}_t$ to the delta-function on $\text{SU}(2)$, $\delta_{h'}$, and continuing analytically the result to $h' \rightarrow h_e^{\mathbb{C}}$:
\begin{align}\label{gauge-variant coh}
\psi^t_{e,h^{\mathbb{C}}_e}(g):=\left(\hat{W}_t\delta_{h'}(g)\right)_{h' \rightarrow h^{\mathbb{C}}_e} =\sum_{j} d_j e^{-\frac{t}{2}j(j+1)}\text{Tr}^{(j)}((h_e^{\mathbb{C}})^\dag g)
\end{align}
where $\text{Tr}^{(j)}(.)$ denotes the trace in the spin-$j$ irrep and the explicit expression $\delta_{h}(g)=\sum_{j}d_j\text{Tr}^{(j)}(hg^{-1})$ has been used.
\\
As was shown in \cite{Winkler2}, these coherent states fulfill a number of useful properties:
\begin{itemize}
\item[(1)] {\it Eigenstates of an annihilation operator}. By defining $\hat{a}(e) := e^{-\frac{1}{\hbar}\hat{C}_{t}}\hat{h}(e)e^{\frac{1}{\hbar}\hat{C}_t}=e^{\frac{3t}{8}}e^{-i\tau_I\hat{E}^I(S_e)/2}\hat{h}(e)$ (where the action of the last exponential has to be understood via Nelson's analytic vector theorem), one finds that the coherent states are simultanous eigenstates for each one:
\begin{align}
\hat{a}_{mn}(e)\Psi^t_{\gamma,\{h^{\mathbb{C}}\}}= [h_e]_{mn}\Psi^t_{\gamma,\{h^{\mathbb{C}}\}}
\end{align}
\item[(2)] {\it Overcompleteness relation}. By considering the measure $d\nu_t(e^{i\tau_J p_J /2}h_e):=d\mu_H(h_e)[\frac{2\sqrt{2}e^{-t/4}}{(2\pi t)^{3/2}}\frac{\sinh(\sqrt{p^2})}{\sqrt{p^2}}e^{-p^2/t}dp^3]$ on $\text{SL}(2,\mathbb{C})$, one can show that
\begin{align}
\int_{\text{SL}(2,\mathbb{C})}d\nu_t(h) \ \psi^t_{e,h_e^{\mathbb{C}}} \ \langle \psi^t_{e,h_e^{\mathbb{C}}}, \cdot \rangle = \mathds{1}_{\mathcal{H}_e}
\end{align}
\item[(3)] {\it Peakedness in holonomy and electric flux}. For all $h,h'\in\text{SL}(2,\mathbb{C})$ there exists a positive function $K_t(h,h')$ decaying exponentially fast as $t\rightarrow 0$ for $h\neq h'$ and such that
\begin{align}
\mid\langle \psi^t_h,\psi^t_{h'}\rangle\mid^2\leq K_t(h,h') ||\psi^t_h||^2||\psi^t_{h'}||^2
\end{align}
Moreover, for holonomies and fluxes one finds
\begin{align}
\langle \psi^t_h,\hat{h}_{mn}(e)\psi^t_{h'}\rangle = \ & h_{mn}(e) \langle \psi^t_h,\psi^t_{h'}\rangle + \mathcal{O}(t)
\\
\langle \psi^t_h,\hat{E}^J(S_e)\psi^t_{h'}\rangle = \ & E^J(S_e) \langle \psi^t_h,\psi^t_{h'}\rangle + \mathcal{O}(t)
\end{align}
\end{itemize}
Most importantly for our purposes, the advantage of using these coherent states is that the evaluation of expectation values of operators involving the Ashtekar-Lewandowski volume (\ref{ALvolume}) can be drastically simplified. Indeed, given a ``good'' coherent state (i.e., one which is peaked at each edge on $|h_{mn}(e)| \gg t$, $|E^J(S_e)| \gg t$), it was shown in \cite{AQG3} that for every polynomial operator $P(\hat{V}_v,\hat{h})$ the following relation holds:
\begin{align}\label{Replacement}
\langle \Psi^t_{\gamma, \{h^{\mathbb{C}}\}}, P(\hat{V}_v,\hat{h})\Psi^t_{\gamma, \{h^{\mathbb{C}}\}} \rangle = \langle \Psi^t_{\gamma, \{h^{\mathbb{C}}\}}, P(\hat{V}_{k,v}^{GT},\hat{h})\Psi^t_{\gamma, \{h^{\mathbb{C}}\}} \rangle + \mathcal{O}(t^{k+1})
\end{align}
where we refer to $\hat{V}^{GT}_v$ as the {\it $k$-th Giesel-Thiemann volume operator}. This is explicitly given by
\begin{align}\label{GTvolume}
\hat{V}^{GT}_{k,v}:= \langle \hat{Q}_v \rangle^{1/2}\left[\mathds{1}_{\mathcal{H}} + \sum_{n=1}^{2k+1}\frac{(-1)^{n}}{n!} \left(0-\frac{1}{4}\right) \left(1-\frac{1}{4}\right) ... \left(n-1-\frac{1}{4}\right) \left(\frac{\hat{Q}_v^2}{\langle \hat{Q}_v \rangle^{2}}-\mathds{1}_{\mathcal{H}}\right)^n
\right]
\end{align}
where $\hat{Q}_v$ is as in (\ref{Q-oper}) and we used the shorthand notation $\langle \hat Q_v \rangle := \langle \Psi^t_{\gamma, \{h^{\mathbb{C}}\}}, \hat Q_v \Psi^t_{\gamma, \{h^{\mathbb{C}}\}} \rangle$.\footnote
{
We observe that operator $\hat{V}^{GT}_{k,v}$ depends explicitly on the coherent state $\Psi^t_{\gamma, \{h^{\mathbb{C}}\}}$ which appears in (\ref{Replacement}), and it therefore makes sense only in the context of equation (\ref{Replacement}).
}
This fact enables us to compute the approximated expectation value (on these coherent states) of any polynomial operator involving Ashtekar-Lewandowski volume, retaining control on the error we make in terms of powers of the semiclassicality parameter $t$.
\section{Cosmological gauge coherent states for LQG}
\label{s3}
In this section we will focus on a subfamily of the coherent states $\Psi^t_{\gamma, \{h^{\mathbb{C}}\}}$, which we claim to be suited to describe flat Robertson-Walker geometries (this claim will be substantiated in the next two sections) at a given instant, i.e., on the spatial manifold $\sigma$. The question of whether these states are actually stable under the dynamics is still open, and will not be addressed here.
\subsection{Choice of States}
We introduce an infrared cutoff by restricting the spatial manifold $\sigma$ to a compact submanifold, $\sigma_R$, which we equip with the topology of a 3-Torus, that is, periodic boundary conditions. With respect to a fiducial metric $\eta$ we identify $R$ as the coordinate length of the Torus, which in principle allows us to remove the cutoff by sending $R \rightarrow \infty$. Thus, we are interested in a fixed graph $\gamma$, which is chosen to be a compact subset of the cubic lattice $\mathbb{Z}^3$ embedded in $\sigma_R$. As such, we shall only consider a subalgebra of the holonomy-flux algebra: the holonomies along the edges of $\gamma$ and the fluxes across the surfaces of the dual cell-complex. To be precise the algebra of the resulting operators reads:
\begin{align}\label{OperatorAlgebra}
\begin{array}{c}
[\hat{h}_{ab}(e),\hat{h}_{cd}(e')]=0,\hspace{15pt}\hspace{15pt}[R^K(e),R^L(e')]=\delta_{ee'}\epsilon^{KL}_{\hspace{8pt}M} R^M(e)
\\
\\
{[}R^K(e),\hat{h}_{ab}(e'){]}=\delta_{ee'}{D'}^{(\frac{1}{2})}_{ac}(\tau_K)\hat{h}_{cb}(e)
\end{array}
\end{align}
with $D'^{(\frac{1}{2})}_{ab}(\tau_K)$ as defined in (\ref{R-and-D-prime}).
\\
The three directions of the lattice can be chosen adapted to the fiducial metric $\eta$, so that the coordinate length of a side of the lattice is $R$. On the other hand, due to $\sigma_R$ being compact, $\gamma$ has a finite number of vertices, $\mathcal{N}^3$. Assuming the lattice to be regular wrt to $\eta$, we therefore find that the coordinate distance between two neighbouring vertices is $\mu := R/\mathcal{N}$.
\\
Now, the classical geometry that we want to reproduce is described by a line element that, in these coordinates, reads
\begin{align}
ds^2=-N^2dt^2+a^2(dx^2+dy^2+dz^2)
\end{align}
with $N$ the lapse function and $a$ the scale factor encoding the spatial geometry. In Ashtekar-Barbero variables, we find for the connection and densitized triad respectively
\begin{align}
A^I_a = c \delta^I_a,\hspace{30pt} E^a_I=p \delta_I^a
\end{align}
with $c$ and $p$ being the fundamental variables. One can now compute the holonomy and the flux along each edge $e_I$ in direction $I$:
\begin{align}\label{coshol&flux}
h(e_I)= e^{-c \mu \tau_I}, \hspace{30pt} E^J(S_{e_I})= p \mu^2 n^J_{(I)}
\end{align}
where $\vec n_{(I)}$ is the unit vector normal to $S_{e_I}$ (so its components wrt cartesian coordinates are $n^J_{(I)} = \delta^J_I$). Therefore, according to (\ref{polar-decomposition}), the element $H_{I}\in \text{SL}(2,\mathbb{C})$ that should label the coherent state $\psi^t_{e_I, H_I} \in \mathcal H_{e_I}$ is (no sum over $I$)
\begin{align}
H_{I} & = \exp \left(-\frac{it}{\hbar\kappa\beta} p \mu^2 \tau_I\right) e^{-c \mu\tau_I} = \exp \left[\left(-2c \mu - i \frac{2t}{\hbar\kappa\beta} p \mu^2\right) \vec n_{(I)} \cdot \vec \tau/2\right] = \notag
\\
& = n_I \exp\left[\left(-2\mu c - i \frac{2\mu^2p}{a^2\beta}\right)\tau_3/2\right] n_I^{\dagger}
\end{align}
where in the third step we used the $SU(2)$-covariance of $\tau_I$ to move the rotation from its basis index to its matrix indices. In particular, $n_I$ are the $SU(2)$ elements that rotate the unit vector $\hat z$ into the unit vector $\vec n_{(I)}$, and are explicitly given by
\begin{align} \label{explicit-n}
n_1=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & -1\\
1 & 1
\end{array}\right),\hspace{20pt}n_2=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & i\\
i & 1
\end{array}\right),\hspace{20pt}n_3=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)
\end{align}
In this way, we have expressed $H_I$ in its {\it holomorphic decomposition}, which for a generic $SL(2,\mathbb C)$ element reads $n \exp(\bar{z} \tau_3/2) n'^{\dagger}$ for $z \in \mathbb C$ and $n,n' \in SU(2)$. While in general $z$, $n$ and $n'$ are independent, in this particularly simple case we find that $n = n'$ are fixed (though different for the three possible orientations of the edges) and we read off
\begin{align}\label{z-label}
z = -2\mu c + i \frac{2\mu^2p}{a^2\beta} \equiv \xi + i\eta
\end{align}
The complex number $z$ is therefore the only label of our coherent states, encoding the classical geometry described by the canonical pair $(c,p)$.
\\
Having the labels $\{h^{\mathbb C}\} = \{H\}$, we finally use (\ref{Appendix-formula for diagonal g}) to find our coherent states:
\begin{align}\label{CosCohSta}
\begin{array}{rl}
\Psi_{(c,p)}(\{g\}) & := \prod_{e\in E(\gamma)}\psi^t_{e,h_e^{\mathbb C}}(g_e) = \prod_{I\in\{1,2,3\}}\prod_{k\in\mathbb{Z}^3_{\mathcal{N}}}\psi_{I,H_I}(g_{k,I})
\\
\\
\psi_{I,H_I}(g) & := \dfrac{1}{\sqrt{\langle1\rangle_z}}\sum_{j\in\mathbb{N}/2}d_je^{-j(j+1)t/2}\sum_{m=-j}^je^{izm}D^{(j)}_{mm}(n_I^{\dagger}gn_I)
\end{array}
\end{align}
where $\langle 1 \rangle_z:=||\psi_{I,H_I}||^2$ is the normalization of the state and $\mathbb{Z}_{\mathcal N}=\{0,1...,\mathcal{N}-1\}$.
\\
These are states on the kinematical Hilbert space: we still have to implement the Gauss constraint and the vector constraint. We will not implement the Gauss constraint explicitly, as we are only interested in the expectation value of gauge-invariant observables, i.e., $\hat O_F$ such that $U_G[g]^{\dagger} \hat O_F U_G[g] = \hat O_F$ for any $SU(2)$ transformation $U_G[g]$. Moreover, it is easy to see by (\ref{gauge-variant coh}) that the coherent states are {\it gauge-covariant}: for a gauge transformation of $\tilde{g}$ at $x=s_e$, the starting point of edge $e$, we get $U_G[\tilde{g}]\psi^t_{e,h^{\mathbb{C}}_e}(g)=\psi^t_{e,h^{\mathbb{C}}_e}(g\tilde g)=\psi^t_{e,h^{\mathbb{C}}_e\tilde{g}^{\dagger}}(g)$. Combining both with the fact that the coherent states are sharply peaked, we get
\begin{align}
\langle \Psi^G_{(c,p)}, \hat{O}_F \Psi^G_{(c,p)}\rangle &= \int D[\{\tilde{g}\}] \int D[\{\tilde{g}'\}] \prod_{e\in E(\gamma)} \langle \psi_{e,H_e}, U[\tilde{g}_{s_e}]^{\dagger} U[\tilde{g}_{t_e}]^{\dagger} \hat{O}_F U[\tilde{g}'_{s_e}] U[\tilde{g}'_{t_e}] \psi_{e,H_e\tilde{g}'_{s_e}}\rangle =\nonumber\\
&= \int D[\{\tilde{g}\}] \int D[\{\tilde{g}'\}] \prod_{e\in E(\gamma)} \langle \psi_{e,\tilde{g}_{t_e}H_e{\tilde{g}_{s_e}}^{-1}}, \hat{O}_F \psi_{e,\tilde{g}'_{t_e}H_e{\tilde{g'}_{s_e}}^{-1}}\rangle\nonumber\\
&=\int D[\{\tilde{g}\}] \int D[\{\tilde{g}'\}] \prod_{e\in E(\gamma)} \langle \psi_{e,\tilde{g}_{t_e}H_e{\tilde{g}_{s_e}}^{-1}}, \hat{O}_F \psi_{e,\tilde{g}_{t_e}H_e{\tilde{g}_{s_e}}^{-1}}\rangle \delta(\tilde{g},\tilde{g}') + \mathcal{O}(t)=\nonumber\\
&=\int D[\{\tilde{g}\}] \langle \Psi_{(c,p)}, U_G[\tilde{g}]^{\dagger} \hat{O}_F U_G[\tilde{g}] \Psi_{(c,p)}\rangle +\mathcal{O}(t)= \nonumber\\
&=\langle \Psi_{(c,p)}, \hat{O}_F \Psi_{(c,p)}\rangle +\mathcal{O}(t)
\end{align}
where in the last step we also used the fact that the Haar measure is normalized. This result guarantees that the expectation values have physical significance at leading order in $t$, without having to impose the Gauss constraint. More work is needed in order to incorporate the first order of quantum corrections (see appendix \ref{recovering-gauge} for a proof of principle).
\\
As for the vector constraint, the naive expectation is that working on an abstract graph automatically takes care of the diffeomorphisms. The situation, however, is more subtle. Indeed, in \cite{diffeo-merda} it was shown that states which appear to be orthogonal from the abstract graph perspective, actually are diffeomorphism-equivalent upon embedding, and therefore correspond to the same state in the diffeomoephism-invariant Hilbert space. This means that there remains a trace of the diffeomorphisms at the level of abstract lattices -- what could be called ``residual diffeomorphisms''. However, the argument presented in \cite{diffeo-merda} relies on the possibility that some links of the lattice are turned off (see fig. 3.1). On the other hand, the cosmological coherent states we just defined are ``maximal'', in the sense that they do not allow any link to be turned off: they live on the lattice $\mathbb Z_{\mathcal N}^3$ and {\it could not} live on any of its sublattices. This is due to the fact that the lattice is finite. If we considered infinite lattices, then there would be the possiblity of completely ``turning off'' one or more rows in it. However, the solution is simple: one simply needs to be aware that the state obtained by such a procedure is in fact identical to the original state. By this reasoning, we see that our cosmological coherent states are already invariant under the residual diffeomorphisms of the lattice.
\begin{figure}[H]
\begin{center}
\includegraphics{loops}
\end{center}
\caption{\footnotesize Examples of graphs which are inequivalent on the abstract lattice, but are related by diffeormorphisms when embedded in the spatial manifold.}
\end{figure}
We therefore conclude that the states introduced above can be considered as physical states. The reminder of the section collects some general results about this particular subfamily, which will be used in the following sections to perform computations. Because the extension to many edges is trivial, we can focus on a single edge, and therefore we shall drop the index `$I$'. Moreover, we will sometimes write $H(z)$ to indicate that the $SL(2,\mathbb C)$ label $H$ effectively depends only on $z$ given in (\ref{z-label}).
\subsection{General Properties of Cosmological Coherent States}
Consider $\psi_{e,H(z)}(g)$ as in (\ref{CosCohSta}), with $H(z)=n e^{z\tau_3/2}n^{\dagger}$ and $z=\xi+i\eta$ as in (\ref{z-label}). The first result gives us a way to simplify expectation values of operators involving left-invariant vector fields.
\\
{\bf Lemma 1:} Let $P(L,\hat{h})$ be a polynomial operator, with $L^K$ the left-invariant vector field. Then:
\begin{align}\label{Lemma1}
\langle \psi_{e,H(z)}, P(L(e),\hat{h}(e)) \psi_{e,H(z)}\rangle = \langle \psi_{e,H(-z)},P(-R(e),\hat{h}(e)^{\dagger}) \psi_{e,H(-z)}\rangle
\end{align}
\begin{proof}
Because of linearity, if suffices to consider a single basis element $\hat{h}_{a_1b_1}(e)^{r_1}L^{K_1}..\hat{h}_{a_nb_n}(e)^{r_n}L^{K_n}$ with $r_i\in\mathbb{N}_0$ and for arbitrary $j,j'$ in the defintion of $\psi_{e,H(z)}$. We recall that $\hat{h}$ is a multiplication operator, while for $R$ we find
\begin{align}\label{R-field-daggered}
R^K D^{(j)}_{ab}(g^{\dagger}) & =(-)^{b-a}R^KD^{(j)}_{-b-a}(g)=(-)^{b-a}\left.\frac{d}{ds}\right|_{s=0}D^{(j)}_{-b-a}(e^{s\tau_K}g) = \notag
\\
& = \left.\frac{d}{ds}\right|_{s=0}D^{(j)}_{ab}((e^{s\tau_K}g)^{\dagger}) = \left.\frac{d}{ds}\right|_{s=0}D^{(j)}_{ab}(g^\dag e^{-s\tau_K}) = \notag
\\
& = -\left.\frac{d}{ds}\right|_{s=0}D^{(j)}_{ab}(g^\dag e^{s\tau_K})
\end{align}
obtained using the properties of Wigner matrices, see appendix \ref{basic-properties}. In light of this, we have
\begin{align}
& \int d\mu_H(g) \overline{D^{(j')}_{m'n'}(n^{\dagger}gn)}D^{(\frac{1}{2})}_{a_1b_1}(g)^{r_1}L^{K_1}\ldots D^{(\frac{1}{2})}_{a_nb_n}(g)^{r_n}L^{K_n}D^{(j)}_{mn}(n^{\dagger}gn)\delta_{m'n'}\delta_{mn}e^{izm}e^{-i\bar{z}m'} = \notag
\\
& = \left.\frac{d}{ds_1}\right|_{s_1=0} ... \left.\frac{d}{ds_n}\right|_{s_n=0}\int d\mu_H(g) \overline{D^{(j')}_{m'm'}(n^{\dagger}gn)}D^{(\frac{1}{2})}_{a_1b_1}(g)^{r_1}D^{(\frac{1}{2})}_{a_2b_2}(ge^{s_1\tau_{K_1}})^{r_2} \times \notag
\\
& \hspace{10pt}\times D^{(\frac{1}{2})}_{a_nb_n}(ge^{s_1\tau_{K_1}} ... e^{s_{n-1}\tau_{K_{n-1}}})^{r_n}D^{(j)}_{mm}(n^{\dagger}g^{\dagger}e^{s_1\tau_{K_n}}n)e^{izm}e^{-i\bar{z}m} = \notag
\\
& = (-1)^n \int d\mu_H(g^{\dagger}) \overline{D^{(j')}_{m'm'}(n^{\dagger}g^{\dagger}n)}D^{(\frac{1}{2})}_{a_1b_1}(g^{\dagger})^{r_1} R^{K_1} ... D^{(\frac{1}{2})}_{a_nb_n}(g^{\dagger})^{r_n}R^{K_n}D^{(j)}_{mm}(n^{\dagger}g^{\dagger}n)e^{izm}e^{-i\bar{z}m'} = \notag
\\
& = (-1)^n\int d\mu_H(g) \overline{D^{(j')}_{-m'-m'}(n^{\dagger}gn)}D^{(\frac{1}{2})}_{a_1b_1}(g^{\dagger})^{r_1} ... D^{(\frac{1}{2})}_{a_nb_n}(g^{\dagger})^{r_n}R^{K_n}D^{(j)}_{-m-m}(n^{\dagger}gn)e^{izm}e^{-i\bar{z}m'} = \notag
\\
& = (-1)^n \int d\mu_H(g)\overline{D^{(j')}_{m'm'}(n^{\dagger}gn)}D^{(\frac{1}{2})}_{a_1b_1}(g^{\dagger})R^{K_1} ... D_{a_nb_n}^{(\frac{1}{2})}(g^{\dagger})^{r_n}R^{K_n}D^{(j)}_{mm}(n^{\dagger}gn)e^{i(-z)m}e^{-i(-\bar{z})m'}
\end{align}
where in the second step we renamed the integration variable $g \rightarrow g^\dag$ and made use of (\ref{R-field-daggered}), in the third step we used $d\mu_H(g)=d\mu_H(g^{\dagger})$ and $D^{(j)}_{mn}(g^{\dagger})=\overline{D^{(j)}_{nm}(g)} = (-1)^{n-m} D^{(j)}_{-n-m}(g)$ (see appendix \ref{basic-properties}), and in the last we renamed $-m\rightarrow m, -m'\rightarrow m'$ (recall that sums over such indices are understood). This gives the statement.
\end{proof}
{\bf Lemma 2:} Let $M(R(e),\hat{h}(e))^{K_1,...,K_n}_{a_1b_1,...,a_{n'}b_{n'}}$ be a monomial operator, with index-structure stemming from $R^{K_i}(e)$ and $\hat{h}_{a_ib_i}(e)$ . Then:
\begin{align}\label{Lemma2}
& \langle \psi_{e,H(z)}, P(R(e),\hat{h}(e))^{K_1,...,K_n}_{a_1b_1,...,a_{n'}b_{n'}} \psi_{e,H(z)}\rangle =
D^{(1)}_{-K_1,-S_1}(n)\ldots D^{(1)}_{-K_n,-S_n}(n)\times\\
&\times D^{(\frac{1}{2})}_{a_1a_1'}(n)D^{(\frac{1}{2})}_{b_1'b_1}(n^{\dagger})\ldots D^{(\frac{1}{2})}_{a_{n'}a_{n'}'}(n)D^{(\frac{1}{2})}_{b_{n'}'b_{n'}}(n^{\dagger})
\langle \psi_{e,H(z)|_{n=1}}, P(R(e),\hat{h}(e))^{S_1,...,S_n}_{a_1'b_1',...,a_{n'}'b_{n'}'} \psi_{e,H(z)|_{n=1}}\rangle\nonumber
\end{align}
where we point out that $H(z)|_{n=1}=e^{z\tau_3/2}$.
\\
\begin{proof}
First, consider the action of $R^K$ on $D^{(j)}_{mn}(n^{\dagger}gn)$:
\begin{align} \label{RD-ngn}
R^K D^{(j)}_{mn}(n^{\dagger}gn) & = D^{(j)}_{mm'}(n^{\dagger}) \left(R^K D^{(j)}_{m'n'}(g)\right) D^{(j)}_{n'n}(n) = D^{(j)}_{mm'}(n^{\dagger}) D'^{(j)}_{m'\mu}(\tau_K) D^{(j)}_{\mu n'}(g) D^{(j)}_{n'n}(n) = \notag
\\
& = D^{(j)}_{mm'}(n^{\dagger}) D'^{(j)}_{m'\mu}(\tau_K) D^{(j)}_{\mu \nu}(n) D^{(j)}_{\nu n}(n^\dag gn) = \notag
\\
& = D^{(1)}_{-K-S}(n) D'^{(j)}_{m\nu}(\tau_S) D^{(j)}_{\nu n}(n^\dag gn)
\end{align}
where in the last step we used (\ref{tau-rotation}). Hence, for the generic monomial, we express numerous times the product of two holonomies as a linear combination with fixed coefficients $c$:
\begin{align}
& \int d\mu_H(g) \overline{D^{(j')}_{m'n'}(n^{\dagger}gn)}\hat{h}_{a_1b_1} ... R^{K_n} D^{(j)}_{mn}(n^{\dagger}gn) =
\\
& = D^{(\frac{1}{2})}_{a_1a_1'}(n)D^{(\frac{1}{2})}_{b_1'b_1}(n^{\dagger}) ... D^{(\frac{1}{2})}_{a_na_n'}(n)D^{(\frac{1}{2})}_{b'_nb_n}(n^{\dagger})\int d\mu_H(g)\overline{D^{(j')}_{m'n'}(n^{\dagger}gn)} \times \notag
\\
&\hspace{10pt} \times D^{(\frac{1}{2})}_{a_1'b_1'}(n^{\dagger}gn)R^{K_1} ... D^{(\frac{1}{2})}_{a_n'b_n'}(n^{\dagger}gn) \left( D^1_{-K_n-S_n}(n){D'}^j_{m\mu_n}(\tau_{S_n}) D^{(j)}_{\mu_nn}(n^{\dagger}gn)\right) = \notag
\\
& = D^{(\frac{1}{2})}_{a_1a_1'}(n)D^{(\frac{1}{2})}_{b_1'b_1}(n^{\dagger}) ... D^{(\frac{1}{2})}_{a_na_n'}(n)D^{(\frac{1}{2})}_{b'_nb_n}(n^{\dagger}) D^1_{-K_n-S_n}(n){D'}^j_{m\mu_n}(\tau_{S_n}) \times \notag
\\
& \hspace{10pt}\times \int d\mu_H(g) \overline{D^{(j')}_{m'n'}(n^{\dagger}gn)} D^{(\frac{1}{2})}_{a_1'b_1'}(n^{\dagger}gn)R^{K_1} ... R^{K_{n-1}}\sum_{j_n} c^n_{j_n,\mu_n'\nu_n}(\mu_n)D^{j_n}_{\mu_n'\nu_n'}(n^{\dagger}gn) \notag =
\\
& = D^{(\frac{1}{2})}_{a_1a_1'}(n)D^{(\frac{1}{2})}_{b_1'b_1}(n^{\dagger}) ... D^{(\frac{1}{2})}_{a_na_n'}(n)D^{(\frac{1}{2})}_{b'_nb_n}(n^{\dagger}) D^1_{-K_n-S_n}(n) ... D^1_{-K_1-S_1}(n) \times \notag
\\
& \hspace{10pt}\times \left({D'}^j_{m\mu_n}(\tau_{S_n}) ... {D'}^j_{m\mu_1}(\tau_{S_1}) \times \int d\mu_H(g) \overline{D^{(j')}_{m'n'}(g)}\sum_{j_n...j_1}
c^1_{j_1,\mu_1',\nu_1}(\mu_1)...c^n_{j_n,\mu_n'\nu_n}(\mu_n)D^{j_1}_{\mu_1'\nu_1'}(g)\right)\nonumber
\end{align}
where in the last line we used invariance of the Haar measure to replace $n^{\dagger}gn\rightarrow g$. We now see that the term in brackets is nothing but the expansion of $\int d\mu_H(g) \overline{D^{(j')}_{m'n'}(g)}\hat{h}_{a_1b_1} ... R^{K_n} D^{(j)}_{mn}(g)$, which was the statement.
\end{proof}
{\bf Lemma 3:} Let $P(R,L,\hat{h})$ be a polynomial operator on $\mathcal H_e$. Then:
\begin{align} \label{r&l-Switch}
\begin{array}{c}
\langle \psi_{e,H(z)|_{n=1}}, P(R,L,\hat{h}) L^K \psi_{e,H(z)|_{n=1}} \rangle = e^{-izK}
\langle \psi_{e,H(z)|_{n=1}}, P(R,L,\hat{h}) R^K \psi_{e,H(z)|_{n=1}} \rangle
\\
\\
\langle \psi_{e,H(z)|_{n=1}}, L^K P(R,L,\hat{h}) \psi_{e,H(z)|_{n=1}} \rangle = e^{-i\bar{z}K}
\langle \psi_{e,H(z)|_{n=1}}, R^K P(R,L,\hat{h}) \psi_{e,H(z)|_{n=1}} \rangle
\end{array}
\end{align}
\begin{proof}
Since ${D'}^j_{mn}(\tau_K)$ enforces $n+K-m=0$ (see appendix \ref{basic-properties}), one gets
\begin{align}
L^K D^{(j)}_{mm}e^{izm} & = D^{(j)}_{m\mu}(g){D'}^{(j)}_{\mu m}(\tau_K)e^{izm} = e^{-izK} D^{(j)}_{\mu m}(g){D'}^{(j)}_{m\mu}(\tau_K) e^{izm}= \notag
\\
& = e^{-izK} R^K D^{(j)}_{mm}(g)e^{izm}
\end{align}
where in the second step we exchanged the dummy indices $\mu \leftrightarrow m$. This is the first property. For the second, we expand $P(R,L,\hat{h})\psi=\sum_{j}c_{jmn}D^{(j)}_{mn}(g)$ with some coefficients $c$:
\begin{align}
& \sum_j c_{jmn} \int d\mu_H(g) \overline{D^{(j')}_{m'm'}(g)}e^{-i\bar{z}m'} L^K D^j_{mn}(g) = \frac{1}{d_{j'}} c_{j'm'n} e^{-i\bar{z}m'}{D'}^{j'}_{m'n}(\tau_K) = \notag
\\
& = e^{-i\bar{z}K}\frac{1}{d_{j'}}c_{j'nm'}e^{-i\bar{z}m'}{D'}^{(j')}_{nm'}(\tau_K) = \notag
\\
& = e^{-i\bar{z}K} \sum_j c_{jn\mu}e^{-i\bar{z}m'}{D'}^{j}_{n\nu}(\tau_K)\int d\mu_H(g)\overline{D^{(j')}_{m'm'}(g)}D^j_{\nu \mu}(g) = \notag
\\
& = e^{-i\bar{z}K} R^K P(R,L,\hat{h})\psi
\end{align}
having exchanged the dummy indices $m'\leftrightarrow n$ in the second step.
\end{proof}
Now, if one uses both relation in (\ref{r&l-Switch}), and the fact that $[R^K(e),L^M(e)]=0$, one gets immediately the following result.
\\
{\bf Corollary:} The following {\it cyclic property} holds:
\begin{align}\label{cyclicity}
\langle \psi_{e,H(z)|_{n=1}}, R^{K_1} .. R^{K_n} \psi_{e,H(z)|_{n=1}} \rangle = e^{-2\eta K_n}
\langle \psi_{e,H(z)|_{n=1}}, R^{K_n}R^{K_1} .. R^{K_{n-1}} \psi_{e,H(z)|_{n=1}} \rangle
\end{align}
where $z=\xi+i\eta$.
\\
As we will see in the next section, this property allows to greatly simplify the computations for the expectation value of any product os $R$'s.
\section{Expectation Values of Monomials on a Single Edge}
\label{s4}
In this section we will compute the expectation values of the various monomials which appear in the geometric operators. Thanks to lemma 2, it suffices to express everything on cosmological coherent states with $H(z)|_{n=1}$, and so we will use a shorthand notation for the {\it non-normalized} expectation values:
\begin{align}
\langle P(R(e),\hat{h}(e))\rangle_{z} & := \langle1\rangle_z \langle \psi_{e,H(z)|_{n=1}}, P(R(e),\hat{h}(e)) \psi_{e,H(z)|_{n=1}}\rangle
\end{align}
Moreover, we will change the basis of $\mathfrak{su}(2)$, meaning that instead of $I,J,K\in \{1,2,3\}$ we will consider the {\it spherical basis}, $s\in \{-,0,+\}$, where $\tau_{\pm}:=\mp (\tau_1 \pm i\tau_2)/\sqrt 2$ and $\tau_0:=\tau_3$. The generators are thus
\begin{align}
\tau_+ = i\sqrt{2}\left(\begin{array}{cc}
0 & 1\\
0 & 0
\end{array}\right),\hspace{20pt}
\tau_- = -i\sqrt{2}\left(\begin{array}{cc}
0 & 0\\
1 & 0
\end{array}\right)\hspace{20pt}\tau_0 = -i\left(\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}\right)
\end{align}
subject to the algebra $[\tau_+,\tau_-]=2i\tau_0$, $[\tau_{\pm},\tau_0]=\pm 2i\tau_{\pm}$.\footnote
{
This does not change the action of geometric operators such as volume (\ref{ALvolume}), since they are by construction $SU(2)$-scalars, and hence invariant under any basis transformation.
}
\subsection{Monomials of right-invariant Vector Field}
Consider first $N$ right-invariant vector fields, all with magnetic index $s_1=..=s_n=0$. We have
\begin{align}\label{MonRight}
& \langle R^{s_1} .. R^{s_N}\rangle_z = \notag
\\
& = \sum_{j,j'}d_jd_{j'}e^{-j(j+1)t/2}e^{-j'(j'+1)t/2}e^{-i\bar{z}m'}e^{izm}\int d\mu_H(g)\overline{D^{(j')}_{m'm'}(g)}{D'}^{(j)}_{m\mu_N}(\tau_0) .. {D'}^{(j)}_{\mu_2\mu_1}(\tau_0)D^{(j)}_{\mu_1m}(g)\notag
\\
& = \sum_{j,j'}d_jd_{j'}e^{-j(j+1)t/2-j'(j'+1)t/2}(-2im)^Ne^{-i\bar{z}m'}e^{izm}\int d\mu_H(g) \overline{D^{(j')}_{m'm'}(g)}D^{(j)}_{mm}(g) \notag
\\
& = \sum_j d_j e^{-j(j+1)t}(-2im)^Ne^{-2\eta m}=\sum_j d_j e^{-j(j+1)t}(i\partial_{\eta})^N e^{-2\eta m}=\nonumber\\
&=(i\partial_{\eta})^N\sum_j d_je^{-j(j+1)t}\frac{\sinh(d_j\eta)}{\sinh(\eta)}=(i\partial_{\eta})^N\langle 1\rangle_z
\end{align}
where we used ${D'}_{mn}(\tau_0)=-2im\delta_{mn}$ (see appendix \ref{basic-properties}) in the second step and the geometric sum $\sum_{m=-j}^j e^{-2\eta m}=\sinh(d_j\eta)/\sinh(\eta)$ to go to the last line. It remains to compute $\langle1\rangle_z$, the normalization of the state, for which we follow closely \cite{Winkler1,Winkler2,Winkler3}. As the authors there have pointed out, the elementary Poisson Summation Formula comes in handy.
\\
{\bf Theorem: (Poisson Summation Formula)} Consider $f\in L_1(\mathbb{R},dx)$ such that the series $\sum_{n\in\mathbb{Z}} f(y+ns)$ is absolutely and uniformly convergent for $y\in [0,s]$, $s>0$. Then
\begin{align}
\sum_{n\in\mathbb{Z}}f(ns)=\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}dx \cdot e^{-i2\pi nx}f(sx)
\end{align}
\begin{proof}
See e.g. the book about fourier analysis by Bochner \cite{Bochner}.
\end{proof}
By realizing that for $d_j=2j+1$ the term in the sum is even, we extend the sum to negative values, thus bringing $\langle1\rangle_z$ in the form to apply this theorem:
\begin{align}
\langle1\rangle_z & =\sum_{d_j=1}^{\infty} d_je^{-(d_j^2-1)t/4}\frac{\sinh(d_j\eta)}{\sinh(\eta)}=\frac{1}{2}\sum_{n=-\infty}^{\infty} n e^{-(n^2-1)t/4}\frac{\sinh(n\eta)}{\sinh(\eta)}=\notag
\\
& = \frac{1}{2}\int_{\mathbb{R}}du\sum_{n\in\mathbb{Z}}e^{-i2\pi n u}e^{-tu^2/4}e^{t/4}u\frac{\sinh(u\eta)}{\sinh(\eta)}
\end{align}
Upon completing the square, in the exponential one gets the term $e^{-4\pi^2n^2/t}$ which, for $t \rightarrow 0$, goes to $0$ faster than any polynomial, unless $n=0$. We conclude that, for $1\gg t$, only the $n=0$ term of the sum contributes, up to an error of order $\mathcal{O}(t^{\infty})$. We thus find
\begin{align}
\langle1\rangle_z=\frac{1}{2}e^{t/4}\int_{\mathbb{R}}du\ u e^{-tu^2/4}\frac{e^{2\eta u}}{\sinh(\eta)}=2e^{t/4}\sqrt{\frac{\pi}{t^3}}\frac{\eta e^{\eta^2/t}}{\sinh(\eta)}
\end{align}
Because of the factor $e^{\eta^2/t}$ in $\langle1\rangle_z$, the leading order of (\ref{MonRight}) in $t$ is obtained when all $N$ derivatives $\partial_{\eta}$ hit $e^{\eta^2/t}$, giving $\mathcal O(1/t^N)$.
\\
Let us now consider the case where some indices $s_1,...,s_n$ are not equal to zero. Since ${D'}^{(j)}_{\mu_{i+1}\mu_{i}}(\tau_{s_i})$ implies $\mu_{i+1}=\mu_i+s_i$ and we have $\mu_0=\mu_{N+1}=m$ it follows that $\sum_i s_i =0$. Consequently, a single non-vanishing $s_i$ is impossible: we shall therefore consider a pair $s_1, s_2$ with opposite sign. Moreover, we will neglect all contributions smaller than $\mathcal{O}(1/t^{N-1})$, since we saw that the leading order (for (\ref{MonRight})) is $\sim 1/t^N$. Using the algebra (for $s_1,s_2,s\neq 0$)
\begin{align}\label{SphAlgebra}
\left[ R^{s_1}, R^{s_2} \right] =-i (s_1-s_2) R^0,\hspace{20pt} \left[R^s,R^0\right]=-2isR^s
\end{align}
we find for the expectation value with a spacing $C$ between $s_1$ and $s_2$
\begin{align} \label{MonRspair}
\langle R^{0} ... R^{s_1} \stackrel{C}{\overbrace{R^0...R^0}}R^{s_2} ... R^{0}\rangle_z & = \langle R^{0} ... R^{0} R^{s_1} R^0...R^0R^{s_2}\rangle_z = \notag
\\
& = \langle \stackrel{N-2}{\overbrace{R^0...R^0}}R^{s_1}R^{s_2}\rangle_z-2iCs_2\langle \stackrel{N-3}{\overbrace{R^0...R^0}}R^{s_1}R^{s_2}\rangle_z+\mathcal{O}(1/t^{N-2}) = \notag
\\
& = \left((i\partial_{\eta})^{N-2}-2iCs_2(i\partial_{\eta})^{N-3}\right)\langle R^{s_1}R^{s_2}\rangle_z +\mathcal{O}(1/t^{N-2})
\end{align}
having used (\ref{cyclicity}) in the first step and (\ref{SphAlgebra}) in the second. We reduced the problem to evaluating the expectation value $\langle R^{s_1}R^{s_2}\rangle_z$. But this can be done without effort by combining the cyclicity property and the algebra: it is
\begin{align}
\langle R^{s_1}R^{s_2}\rangle_z & = e^{-2\eta s_2}\langle R^{s_2}R^{s_1}\rangle_z=e^{-2\eta s_2}(\langle R^{s_1}R^{s_2}\rangle_z-\langle[R^{s_1},R^{s_2}]\rangle_z)=\nonumber
\\
&=e^{-2\eta s_2}\langle R^{s_1}R^{s_2}\rangle_z+e^{-2\eta s_2}i(s_1-s_2)\langle R^0\rangle_z=\nonumber
\\
&=e^{-2\eta s_2}\langle R^{s_1}R^{s_2}\rangle_z-e^{-2\eta s_2}(s_1-s_2)\partial_{\eta}\langle 1\rangle_z
\end{align}
which, solved for $\langle R^{s_1}R^{s_2}\rangle_z$, gives
\begin{align}
\langle R^{s_1}R^{s_2}\rangle_z=\frac{e^{-\eta s_2}}{\sinh(\eta)}\partial_{\eta}\langle1\rangle_z
\end{align}
Again, the leading order is obtained when all $\partial_{\eta}$ hit $e^{\eta^2/t}$. It follows that the term proportional to $C$ in (\ref{MonRspair}) is negligible, and the other is already next-to-leading wrt (\ref{MonRight}). Explicitly, we get
\begin{align}
\langle R^{0} ... R^{s_1} \stackrel{C}{\overbrace{R^0...R^0}}R^{s_2} ... R^{0}\rangle_z = -i \frac{e^{-\eta s_2}}{\sinh(\eta)}(i\partial_{\eta})^{N-1} \langle1\rangle_z +\mathcal{O}(1/t^{N-2})
\end{align}
A similar calculation reveals that four and more non-vanishing indices are of order $\mathcal{O}(1/t^{N-2})$, and will thus be neglected.
\\
The final result up to linear quantum corrections thus read:
\begin{align}\label{ExpMonR}
& \langle R^{s_1}\ldots R^{s_N}\rangle_z =
\\
& = \left[\delta^{s_1...s_n}_0(i\partial_{\eta})^N-\frac{i}{\sinh(\eta)}\sum_{A<B=1}^N\delta^{s_1..\cancel s_A..\cancel s_B..s_N}_0 \left(\delta^{s_As_B}_{+1-1} e^{+\eta} + \delta^{s_As_B}_{-1+1} e^{-\eta}\right)(i\partial_{\eta})^{N-1}\right]\langle1\rangle_z \notag
\end{align}
Making use of lemma 1, equation (\ref{Lemma1}), one can straightforwardly generalize this result to a monomial in left-invariant vector fields:
\begin{align}\label{ExpMonL}
\langle L^{s_1}...L^{s_N}\rangle_z & = (-1)^N\langle R^{s_1}...R^{s_N}\rangle_{-z} = (-1)^{2N}\langle R^{s_N}...R^{s_1}\rangle_{z}=\notag
\\
& = \langle R^{s_N}...R^{s_1}\rangle_{z}
\end{align}
where in the second step we used the explicit expression (\ref{ExpMonR}) to find how a change in sign of $z$ (or $\eta$) influences the expectation value.
\subsection{Monomials of Holonomy Operator}
As is well known from recoupling theory (see appendix \ref{basic-properties}), the product of Wigner matrices can be expressed as a linear combination of a single wigner matrix:
\begin{align}\label{recoupling}
D^{(j_1)}_{ab}(g)D^{(j_2)}_{cd}(g)=\sum_{j=|j_1-j_2|}^{j_1+j_2}d_j(-1)^{m-n}\left(\begin{array}{ccc}
j_1 & j_2 & j\\
a & c & m
\end{array}\right)\left(\begin{array}{ccc}
j_1 & j_2 & j\\
b & d & n
\end{array}\right) D^{(j)}_{-m-n}(g)
\end{align}
This property is extremely useful, since it allows to reduce the problem of computing $\langle \hat h_{a_1b_1} ... \hat h_{a_nb_n} \rangle_z$ to computing $\langle \hat h^{(j)}_{mn} \rangle_z$ (for the required values of $j$), by which we mean the operator whose action is to multiply by $D^{(j)}_{mn}(g)$.
\\
From the explicit expression (\ref{CosCohSta}), we obtain (without normalization)
\begin{align}
\langle \hat h^{(k)}_{ab} \rangle_z & = \sum_{j,j'}d_jd_{j'}e^{-[j(j+1) + j'(j'+1)]t/2} e^{i(zm - \bar z m')} \int d\mu_H(g) \overline{D^{(j')}_{m'm'}(g)} D^{(k)}_{ab}(g) D^{(j)}_{mm}(g) = \notag
\\
& = \sum_{j,j'}d_jd_{j'}e^{-[j(j+1) + j'(j'+1)]t/2} e^{i\xi (m - m')} e^{-\eta(m+m')} \left(
\begin{array}{ccc}
j & k & j'
\\
m & a & -m'
\end{array}
\right) \left(
\begin{array}{ccc}
j & k & j'
\\
m & b & -m'
\end{array}
\right) = \notag
\\
& = \delta_{ab} e^{-i\xi a} \gamma^k_a
\end{align}
where in the second line we performed the integral (see (\ref{basic-recoupling-integral1}) and (\ref{basic-recoupling-integral2})), and in the third we used the observation that $a = m' - m = b$ to extract $e^{i\xi (m - m')} = e^{-i\xi a}$ from the sums and defined the quantity
\begin{align}
\gamma^k_{a}:=\sum_{j,j'} d_j d_{j'} e^{-t[j(j+1)+j'(j'+1)]/2}e^{-\eta(m+m')}\left(\begin{array}{ccc}
k & j & j'\\
a & m & -m'
\end{array}\right)^{2}
\end{align}
If we interchange in $\gamma^k_{a}$ the contracted indices $j\leftrightarrow j'$, $m\leftrightarrow m'$ everything is clearly invariant except for the $3j$-symbol:
\begin{align}
\left(\begin{array}{ccc}
k & j & j'\\
a & m & -m'
\end{array}\right)\rightarrow\left(\begin{array}{ccc}
k & j' &j\\
a & m' & -m
\end{array}\right)=\left(\begin{array}{ccc}
k & j & j'\\
-a & m & -m'
\end{array}\right)
\end{align}
As the index $a$ appeared only in the $3j$-symbol this leads to $\gamma^k_a\rightarrow\gamma^k_{-a}$, but since we only interchanged contracted indices $\gamma^k_a$ must stay invariant: we conclude that
\begin{align}
\gamma^k_a =\gamma^k_{-a}
\end{align}
The various values of $\gamma^k_a$ can now be computed with the Poisson Summation Formula. In appendix \ref{hol-int} the explicit computations are presented for $k=1/2$ and $k=1$ (which are relevant for the expectation value and dispersion of the holonomy operator, and sufficient for the Hamiltonian operator). The results are:
\begin{align} \label{gammas-result}
\begin{array}{rl}
\gamma^{1/2}_{1/2}&= \langle1\rangle_z \left[1+ \dfrac{t}{4\eta}\left(\dfrac{3}{4}\eta-\tanh\left(\dfrac{\eta}{2}\right)\right) + \mathcal O(t^2)\right]
\\
\\
\gamma^1_0 &=\langle 1 \rangle_z \left[1 + t \dfrac{2 \sinh(\eta/2)}{\eta \sinh(\eta)} + \mathcal O(t^2)\right]
\\
\\
\gamma^1_1 &=\langle 1 \rangle_z \left[1 - t \left(\dfrac{1}{4} + \dfrac{1}{2\eta} \tanh(\eta/2)\right) + \mathcal O(t^2)\right]
\end{array}
\end{align}
\subsection{Holonomies and right-invariant Vector Fields}
In this section we present the strategy to compute expectation values of monomials involving both holonomy and right-invariant vector field. We consider a couple of explicit examples.
\\
Let us start with the commutator of an holonomy with $N$ right invariant vector fields. Using the algebra (\ref{OperatorAlgebra}) and dropping all terms of order $\mathcal{O}(1/t^{N-3})$ and lower (since the leading order is $\mathcal{O}(1/t^{N-1})$), we find
\begin{align}\label{ExpCommutator}
& \langle \hat{h}_{ac} [\hat{h}^{\dagger}_{cb}, R^{s_1} ... R^{s_N}] \rangle_z = \delta_{ac}\langle R^{s_1} ... R^{s_N}\rangle_z-\langle\hat{h}_{ab} R^{s_1} ... R^{s_N}\hat{h}^{\dagger}_{cb}\rangle_z = \notag
\\
& = \delta_{ab} \langle R^{s_1} ... R^{s_N}\rangle_z - \langle R^{s_1} \hat{h}_{ac} R^{s_2} ... R^{s_N} \hat{h}^{\dagger}_{cb}\rangle_z + D'^{(1/2)}_{ad}(\tau_{s_1}) \langle \hat{h}_{dc} R^{s_2} ... R^{s_N} \hat{h}^{\dagger}_{cb} \rangle_z = \notag
\\
& = \delta_{ab}\langle R^{s_1} ... R^{s_N}\rangle_z - \langle R^{s_1}R^{s_2} \hat{h}_{ac} ... R^{s_N} \hat{h}^{\dagger}_{cb}\rangle_z + D'^{(1/2)}_{ad}(\tau_{s_2}) \langle R^{s_1} \hat{h}_{dc} R^{s_3} ... R^{s_N} \hat{h}^{\dagger}_{cb} \rangle_z +\notag
\\
& + D'^{(1/2)}_{ad}(\tau_{s_1}) \langle R^{s_2} \hat{h}_{dc}R^{s_3} ... R^{s_N} \hat{h}^{\dagger}_{cb}\rangle_z - D'^{(1/2)}_{ae}(\tau_{s_1}) D'^{(1/2)}_{ed}(\tau_{s_2}) \langle\hat{h}_{dc} R^{s_3} ... \hat{h}^{\dagger}_{cb}\rangle_z = ... = \notag
\\
& = \sum_{A=1}^N D'^{(\frac{1}{2})}_{ab}(\tau_{s_A}) \langle R^{s_1} ... \cancel R^{s_A} ... R^{s_N}\rangle_z - \notag
\\
& - \sum_{A<B=1}^N D'^{(\frac{1}{2})}_{ac}(\tau_{s_A}) D'^{(\frac{1}{2})}_{cb}(\tau_{s_B}) \langle R^{s_1} ... \cancel R^{s_A} ... \cancel R^{s_B} ... R^{s_N}\rangle_z+\mathcal{O}(1/t^{N-3})
\end{align}
So such term can be brought back to expectation values of $R$'s only.
\\
The other type of mixed term is of the form $\hat{h}_{ab} R^{s_1} ... R^{s_N}$. From expression (\ref{CosCohSta}), we get (without normalization)
\begin{align}
\langle \hat{h}_{ab} R^{s_1} ... R^{s_N} \rangle_z & = e^{-i\bar{z}b} \sum_{j,j'} d_jd_{j'}e^{-t[j(j+1)+j'(j'+1)]/2} \times \\
& \times {D'}^{(j)}_{m\mu_N}(\tau_{s_N}) ... {D'}^{(j)}_{\mu_2\mu_1}(\tau_{s_1})\left(\begin{array}{ccc}
\frac{1}{2} & j & j'\\
a & \mu_1 & -m'
\end{array}\right)\left(\begin{array}{ccc}
\frac{1}{2} & j & j' \\
b & m & -m'
\end{array}\right)e^{-2\eta m}\nonumber
\end{align}
where we again used (\ref{recoupling}) and performed the group integral. As we did previously for monomials in $R$'s, let us consider the case $s_1 = ... = s_N = 0$ first. Using ${D'}^{(j)}_{mn}(\tau_0) =-2im\delta_{mn}$ (see appendix \ref{basic-properties}), it is easy to see that
\begin{align} \label{hR..R-allzero}
\langle\hat{h}_{ab}R^0\ldots R^0\rangle_z = e^{-\eta b}(i\partial_{\eta})^Ne^{nb}\langle \hat{h}_{ab}\rangle_z
\end{align}
which has leading order $\mathcal{O}(1/t^N)$. Next, we have the possibility of a single index being nonzero, as well as a pair. The order of these is next-to-leading wrt to (\ref{hR..R-allzero}). Indeed, using $[R^0, R^s]=2isR^s$ for $C\leq N$, we get
\begin{align}\label{holRsRs1}
\langle\hat{h}_{ab} \stackrel{C}{\overbrace{R^0...R^0}} R^s\stackrel{N-1-C}{\overbrace{R^0...R^0}}\rangle_z & = \langle\hat{h}_{ab}\stackrel{C-1}{\overbrace{R^0...R^0}}R^s\stackrel{N-C}{\overbrace{R^0...R^0}}\rangle_z+2i\langle\hat{h}_{ab}\stackrel{C-1}{\overbrace{R^0...R^0}}R^s\stackrel{N-1-C}{\overbrace{R^0...R^0}}\rangle_z = \notag
\\
& = \langle\hat{h}_{ab}R^s\stackrel{N-1}{\overbrace{R^0...R^0}}\rangle_z+\mathcal{O}(1/t^{N-2}) = \notag
\\
& = e^{-\eta b}(i\partial_{\eta})^{N-1}e^{\eta b}\langle \hat{h}_{ab}R^s\rangle_z+\mathcal{O}(1/t^{N-2})
\end{align}
and
\begin{align}\label{holRsRs2}
\langle\hat{h}_{ab}R^0...R^0&R^sR^0...R^0R^{s'}R^0...R^0\rangle_z=(i\partial_{\eta})^{N-2}\langle \hat{h}_{ab}R^sR^{s'}\rangle_z+\mathcal{O}(1/t^{N-2})
\end{align}
We thus reduced the problem to the evaluation of $\hat h_{ab} R^{s}$ and $\hat h_{ab} R^s R^{s'}$. Again, these can be computed by cleverly combining the cyclicity of lemma 3 with the algebra:
\begin{align}
\langle \hat{h}_{ab}R^s\rangle_z & = \langle R^s \hat{h}_{ab}\rangle_z-\langle [R^s,\hat{h}_{ab}]\rangle_z=e^{i\bar{z}s}\langle L^s \hat{h}_{ab}\rangle_z - D'^{(1/2)}_{ac}(\tau_s)\langle \hat{h}_{cb}\rangle_z = \notag
\\
& = e^{i\bar{z}s}\left(\langle \hat{h}_{ab}L^s\rangle_z+\langle [L^s,\hat{h}_{ab}]\rangle_z\right)- D'^{(1/2)}_{ac}(\tau_s) \langle \hat{h}_{cb}\rangle_z = \notag
\\
& = e^{i\bar{z}s}\left(^{-izs}\langle \hat{h}_{ab}R^s\rangle_z+D'^{(1/2)}_{cb}(\tau_s)\langle\hat{h}_{ac}\rangle_z\right)-D'^{(1/2)}_{ac}(\tau_s)\langle\hat{h}_{cb}\rangle_z = \notag
\\
& = e^{2\eta s}\langle \hat{h}_{ab} R^s\rangle_z+e^{i\bar{z}s/2}\left(e^{i\bar{z}s/2}D'^{(1/2)}_{cb}(\tau_s)\langle\hat{h}_{ac}\rangle_z-e^{-i\bar{z}s/2}D'^{(1/2)}_{ac}(\tau_s)\langle\hat{h}_{cb}\rangle_z\right)
\end{align}
leading to
\begin{align} \label{holRs}
\langle \hat{h}_{ab}R^s\rangle_z = \frac{se^{izs/2}}{2\sinh(\eta)}\left(e^{-i\bar{z}s/2}D'^{(1/2)}_{ac}(\tau_s)\langle\hat{h}_{cb}\rangle_z-e^{i\bar{z}s/2}\langle\hat{h}_{ac}\rangle_zD'^{(1/2)}_{cb}(\tau_s)\right)
\end{align}
A similar computation gives
\begin{align}
\langle \hat{h}_{ab} R^{s} R^{s'} \rangle_z & = -i \dfrac{e^{\eta s}}{\sinh(\eta)} \langle \hat{h}_{ab} R^0 \rangle_z + \notag
\\
& + \dfrac{s}{2\sinh(\eta)} e^{izs/2} \left(e^{-i\overline z s/2} D'^{(1/2)}_{ac}(\tau_s) \langle \hat{h}_{cb} R^{s'} \rangle_z - e^{i\overline z s/2} \langle \hat{h}_{ac} R^{s'} \rangle_z D'^{(1/2)}_{cb}(\tau_s)\right)
\end{align}
Now, since (\ref{holRsRs1}) involves only $N-1$ derivatives of $\eta$, we can only get an $\mathcal{O}(1/t^{N-1})$ contribution if all derivatives hit $e^{\eta^2/t}$ in the normalization appearing in $\langle\hat{h}_{ab}\rangle_z=\delta_{ab}e^{-i\xi a}\langle 1\rangle_z$ (which is correct at leading order).
\\
Using the same argument for (\ref{holRsRs2}), and putting the results together with the $s_1 = ... = s_N = 0$ case, we finally obtain
\begin{align}\label{ExpHolR}
\langle \hat{h}_{ab}R^{s_1}\ldots R^{s_n}\rangle_z & = \left[\delta^{s_1...s_N}_0 \delta_{a'b'} e^{-\eta b'} (i\partial_\eta)^N e^{\eta b'} \left(1 + \frac{t}{4\eta}\left(\frac{3}{4}\eta-\tanh\left(\frac{\eta}{2}\right)\right)\right)\right. - \notag
\\
& - \dfrac{\sinh(\eta/2)}{\sinh(\eta)} \left.\sum_{A = 1}^N \delta^{s_1...\cancel s_A ... s_N}_0 (\delta^{s_A}_{+1} + \delta^{s_A}_{-1}) e^{s_A\eta/2} D'^{(\frac{1}{2})}_{a'b'}(\tau^{s_A}) (i \partial_\eta)^{N-1}\right. - \notag
\\
& - i \dfrac{\delta_{a'b'}}{\sinh(\eta)} \left.\sum_{A<B = 1}^N \delta^{s_1...\cancel s_A ... \cancel s_B ... s_N}_0 (\delta^{s_As_B}_{+1-1} + \delta^{s_As_B}_{-1+1}) e^{s_A\eta} (i\partial_\eta)^{N-1}\right] \langle 1\rangle_z
\end{align}
\section{Expectation Values of Geometric Operators}
\label{s5}
The tools developed in the previous section shall now be put into action. We start by computing the expectation value and dispersion of the fundamental operators, therefore discussing the physical interpretation of the semiclassicality parameter t. Afterwards, we investigate the geometric observable Volume.
\subsection{Expectation Values and Spread of Holonomy- and Flux-Operators}
Of particular interest is the expectation value and dispersion of the holonomy operator. Using lemma 2 and the results of (\ref{gammas-result}), for the normalized expectation value of $\hat h_{ab} \equiv \hat h^{(\frac{1}{2})}_{ab}$ on endge $e$ oriented along $\vec n_{(I)}$ we find
\begin{align} \label{h-one-half}
\langle \psi_{e,H(z)}, \hat h_{ab} \psi_{e,H(z)} \rangle & = D^{(\frac{1}{2})}_{aa'}(n_I) D^{(\frac{1}{2})}_{b'b}(n_I^\dag) \frac{1}{\langle 1 \rangle_z} \langle \hat h^{(\frac{1}{2})}_{a'b'}\rangle_z = \notag
\\
& = \left[D^{(\frac{1}{2})}_{a\frac{1}{2}}(n_I) e^{-i\xi/2} D^{(\frac{1}{2})}_{\frac{1}{2}b}(n_I^\dag) + D^{(\frac{1}{2})}_{a-\frac{1}{2}}(n_I) e^{i\xi/2} D^{(\frac{1}{2})}_{-\frac{1}{2}b}(n^\dag_I)\right] \dfrac{1}{\langle 1 \rangle_z} \gamma^{1/2}_{1/2} = \notag
\\
& = D^{(\frac{1}{2})}_{ab}(n_I e^{\xi \tau_3/2} n_I^\dag)\left[1 + \frac{t}{4\eta}\left(\frac{3}{4}\eta-\tanh\left(\frac{\eta}{2}\right)\right) + \mathcal O(t^2)\right]
\end{align}
Recalling that $n_I e^{\xi \tau_3/2} n^\dag_I = e^{\xi \vec n_{(I)} \cdot \vec \tau/2} = e^{-\mu c \tau_I}$, we see that the leading order of this expectation value is exactly the classical holonomy along such an edge, $h(e_{I})$, when it is embedded in flat Robertson-Walker spacetime. For the dispersion, we have (no sum over $a, b$)
\begin{align}
& \langle \psi_{e,H(z)}, \hat h^{(1/2)}_{ab} \hat h^{(1/2)}_{ab} \psi_{e,H(z)} \rangle = d_0 \left(\begin{array}{ccc}
1/2 & 1/2 & 0 \\
a & a & -0
\end{array}\right)
\left(\begin{array}{ccc}
1/2 & 1/2 & 0\\
b & b & -0
\end{array}
\right)
\langle \psi_{e,H(z)}, \hat h^{(0)}_{00} \psi_{e,H(z)} \rangle + \notag
\\
& + d_1 (-1)^{n-m} \left(\begin{array}{ccc}
1/2 & 1/2 & 1 \\
a & a & -m
\end{array}\right)
\left(\begin{array}{ccc}
1/2 & 1/2 & 1\\
b & b & -n
\end{array}
\right)
\langle \psi_{e,H(z)}, \hat h^{(1)}_{mn} \psi_{e,H(z)} \rangle = \notag
\\
& = 3 \left(\begin{array}{ccc}
1/2 & 1/2 & 1 \\
a & a & -2a
\end{array}\right)
\left(\begin{array}{ccc}
1/2 & 1/2 & 1\\
b & b & -2b
\end{array}
\right)
\langle \psi_{e,H(z)}, \hat h^{(1)}_{2a,2b} \psi_{e,H(z)} \rangle
\end{align}
where we used the properties of $3j$-symbols to find that $m = 2a$ and $n = 2b$, and so $(-1)^{2(b-a)} = 1$ since $(b-a)$ is always integer. Using the explicit values of 3j-symbols, we obtain that $\langle (\hat h^{(1/2)}_{ab})^2 \rangle =
\langle \hat h^{(1)}_{2a,2b} \rangle$. This can again be computed from lemma 2 and (\ref{gammas-result}):
\begin{align} \label{h-one}
& \langle \psi_{e,H(z)}, \hat h^{1}_{mn} \psi_{e,H(z)} \rangle = \notag
\\
& = \dfrac{1}{\langle 1 \rangle_z} \left[D^{1}_{m0}(n_I) D^{1}_{0n}(n_I^\dag) \gamma^1_{0} + D^{1}_{m,+1}(n_I) D^{1}_{+1,n}(n_I^\dag) e^{-i\xi} \gamma^1_{1} + D^{1}_{m,-1}(n_I) D^{1}_{-1,n}(n_I^\dag) e^{i\xi} \gamma^1_{-1}\right] = \notag
\\
& = D^{1}_{mn}(n_I e^{\xi \tau_3/2} n_I^\dag) \dfrac{1}{\langle 1 \rangle_z} \gamma^1_{1} + D^{1}_{m0}(n_I) D^{1}_{0n}(n_I^\dag) \dfrac{1}{\langle 1 \rangle_z} (\gamma^1_{0} - \gamma^1_{1}) = \notag
\\
& = D^{1}_{mn}(n_I e^{\xi \tau_3/2} n_I^\dag) \left[1 - t \left(\frac{1}{4} + \frac{1}{2\eta} \tanh(\eta/2)\right)\right] + \notag
\\
& + n_{(I)}^m \overline{n_{(I)}^n} t \left[\frac{1}{4} + \frac{1}{2\eta} \left(\tanh(\eta/2) + \frac{4 \sinh(\eta/2)}{\sinh(\eta)}\right)\right] + \mathcal O(t^2)
\end{align}
where we used the fact that $D_{0m}^1(n_{I}) = n_{(I)}^m$, that is, the component $m$ of unit vector $\vec n_{(I)}$ in spherical basis. So the dispersion is finally
\begin{align}
\Delta h_{ab} & := \langle \psi_{e,H(z)}, (\hat h^{(1/2)}_{ab})^2 \psi_{e,H(z)} \rangle - \langle \psi_{e,H(z)}, \hat h^{(1/2)}_{ab} \psi_{e,H(z)} \rangle^2 =
\\
& = t \left\{-D^{1}_{2a,2b}(n_I e^{\xi\tau_3/2} n_I^\dag) \frac{5}{8} + n_{(I)}^{2a} \overline{n_{(I)}^{2b}} \left[\frac{1}{4} + \frac{1}{2\eta} \left(\tanh(\eta/2) + \frac{4 \sinh(\eta/2)}{\sinh(\eta)}\right)\right]\right\} \notag
\end{align}
The dispersion is linear in $t$, and so it goes to $0$ in the classical limit $t \rightarrow 0$.
\\
The other fundamental operator is the flux, that is proportional to the right-invariant vector field (see (\ref{fluxy})): from lemma 2 and (\ref{ExpMonR}), one immediately finds
\begin{align} \label{R-one}
\langle \psi_{e,H(z)}, R^{k} \psi_{e,H(z)} \rangle & = D^1_{-k-s}(n_I) \dfrac{1}{\langle 1 \rangle_z} \langle R^{s}\rangle_z = D^1_{-k-0}(n_I) \dfrac{1}{\langle 1 \rangle_z} i\partial_{\eta}\langle1\rangle = \notag
\\
& = \frac{2i\eta}{t} n_{(I)}^{-k} \left[1 + \frac{t}{2\eta^2} \left(1 - \eta \coth(\eta)\right)\right]
\end{align}
and similarly
\begin{align} \label{R-two}
& \langle \psi_{e,H(z)}, R^{k_1} R^{k_2} \psi_{e,H(z)} \rangle = D^1_{-k_1-s_1}(n_I) D^1_{-k_2-s_2}(n_I) \frac{1}{\langle 1 \rangle_z} \langle R^{s_1} R^{s_2}\rangle_z = \notag
\\
& = - \left(\frac{2\eta}{t}\right)^2 \left[n^{-k_1}_{(I)} n^{-k_2}_{(I)} \left[1 + \frac{t}{2\eta}\left(\frac{3}{\eta} - 2\coth\eta\right)\right]\right. - \notag
\\
& - \left. t \frac{1}{2\eta \sinh(\eta)}\left(D^{(1)}_{-k_1,-}(n_I) D^{(1)}_{-k_2,+}(n_I) e^{+\eta} + D^{(1)}_{-k_1,+}(n_I) D^{(1)}_{-k_2,-}(n_I) e^{-\eta}\right)\right]
\end{align}
where we used again the fact that $D^1_{m0}(n_I) = n_{(I)}^m$. At this point, we recall that these quantities are expressed in spherical basis. To recover the expectation values in cartesian basis, the following relations must be used:
\begin{align}
R^1 = \frac{(R^+ - R^-)}{\sqrt 2}, \ \ \ \ \ \ \ R^2 = -i\frac{(R^+ + R^-)}{\sqrt 2}, \ \ \ \ \ \ \ R^3 = R^0
\end{align}
Hence, one finds ($K \in \{1,2,3\}$)
\begin{align} \label{R-one-cartesian}
\langle \psi_{e,H(z)}, R^{K} \psi_{e,H(z)} \rangle = \dfrac{2i\eta}{t} n_{(I)}^K \left[1 + \dfrac{t}{2\eta^2} \left(1 - \eta \coth(\eta)\right)\right]
\end{align}
where $n^K_{(I)} = \delta^K_I$ are the cartesian components of $\vec n_{(I)}$ (we used the relation between spherical and cartesian components for vectors: $v^1 = (v^{-} - v^{+})/\sqrt{2}$, $v^2 = (-iv^{-} - iv^{+})/\sqrt{2}$ and $v^3 = v^0$). This equation shows that the cosmological state is peaked in the right-invariant vector field on the value $(2i\eta/t) \vec n_{(I)} = 2i\mu^2p/(\hbar \kappa \beta) \vec n_{(I)}$, which corresponds to the classical value of the flux: $E^J(S_{e_I}) = \mu^2 p \ n^J_{(I)}$. As for the dispersions, we first compute
\begin{align}
\langle \psi_{e,H(z)}, (R^{1})^2 \psi_{e,H(z)} \rangle & = \frac{1}{2} \langle \psi_{e,H(z)}, \left[(R^+)^2 + (R^-)^2 - R^+R^- - R^-R^+\right] \psi_{e,H(z)} \rangle = \notag
\\
& = -\left(\frac{2\eta}{t}\right)^2 \left[\left(\frac{n_{(I)}^- - n_{(I)}^+}{\sqrt 2}\right)^2 \left[1 + \frac{t}{2\eta}\left(\frac{3}{\eta} - 2\coth\eta\right)\right]\right. + \notag
\\
& + \left.t \frac{\coth\eta}{2\eta} \left(D^{(1)}_{++}(n_{I}) - D^{(1)}_{-+}(n_{I})\right) \left(D^{(1)}_{--}(n_{I}) - D^{(1)}_{+-}(n_{I})\right)\right] = \notag
\\
& = -\left(\frac{2\eta}{t}\right)^2 \left[\delta_{I}^1 \left[1 + \frac{t}{2\eta}\left(\frac{3}{\eta} - 2\coth\eta\right)\right] + t \frac{\coth\eta}{2\eta} \left(1 - \delta^1_I\right)\right] = \notag
\\
& = -\left(\frac{2\eta}{t}\right)^2 \left[\delta_{I}^1 + \delta_{I}^1 \frac{3t}{2\eta}\left(\frac{1}{\eta} - \coth\eta\right) + t \frac{\coth\eta}{2\eta}\right]
\end{align}
where in the second-to-last line we used the fact that $(D^{(1)}_{++}(n_{I}) - D^{(1)}_{-+}(n_{I}))(D^{(1)}_{--}(n_{I}) - D^{(1)}_{+-}(n_{I})) = |D^{(1)}_{++}(n_{I}) - D^{(1)}_{-+}(n_{I})|^2 = 1 - \delta^1_I$ (the last equality being cheched by explicit computation using (\ref{explicit-n})). A similar relation holds for $R^2$, while $R^3$ is immediately obtained from (\ref{R-two}). One then finds for the dispersions
\begin{align} \label{R-dispersions-cartesian}
\Delta R^K = \dfrac{2}{t} \left[n_{(I)}^K + (1 - n_{(I)}^K) \eta \coth(\eta)\right]
\end{align}
While it may be worrysome that this dispersion grows with $t \rightarrow 0$, this is expected since no quantum state can be infinitely peaked on both fundamental operators. What matters, however, is that the expectation value of $R^K$ also grows with $t \rightarrow 0$, and it does it in such a way that the ratio (i.e., the {\it relative dispersion}) actually tends to zero as $t \rightarrow 0$:
\begin{align}
\delta R^K := \left|\frac{\Delta R^K}{\langle \psi_{e,H(z)}, R^{K} \psi_{e,H(z)} \rangle^2}\right| = \dfrac{t}{2\eta^2} \left(1 + \dfrac{1 - n_{(I)}^K}{n_{(I)}^K} \eta \coth(\eta)\right)
\end{align}
\subsection{Volume Operator}
We finally turn to the volume operator. Thanks to (\ref{Replacement}), the expectation value of Ashtekar-Lewandowski volume coincides with the expectation value of the ($k=1$)-Giesel-Thiemann volume operator (\ref{GTvolume}) up to next-to-leading order in $t$. But to evaluate that, we only need the expectation values of $\hat Q_v^N$ for $N = 1,2,4$ and $6$. Although these are operators on many edges, the expectation value reduces to the product of expectation values on each edge, so the only quantity we need is the expectation value of a string of $N$ right-invariant vector fields. This was derived in (\ref{ExpMonR}), and restoring the dependence on $n \in SU(2)$, it reads
\begin{align}\label{ExpRexpanded}
& \langle\psi_{e,H(z)}, R^{k_1} .. R^{k_N} \psi_{e,H(z)}\rangle = \left(\frac{2\eta i}{t}\right)^{N} D^{(1)}_{-k_1-s_1}(n) .. D^{(1)}_{-k_N-s_N}(n) \ (\delta_0^{s_1\ldots s_N} +
\\
&\hspace{10pt} + \frac{t}{2\eta} [\delta_0^{s_1...s_N}\left(\frac{N(N+1)}{2\eta}-N\coth(\eta)\right)-\frac{1}{\sinh(\eta)}\sum_{A<B=1}^N\delta_0^{s_1..\cancel s_A ..\cancel s_B ...s_N}(\delta^{s_As_B}_{+1-1}+\delta^{s_As_B}_{-1+1})e^{s_A\eta}])\notag
\end{align}
In $\langle \hat{Q}_v^N \rangle$, one has a products of three such expectation values (one per every edge of the triple). The combinatorics is therefore encoded in $\epsilon_{k_ik'_ik''_i}R^{k_i}(e_1)R^{k_i'}(e_2)R^{k_i''}(e_3)$, which motivates us to consider the object
\begin{align}
\epsilon^{(n)}_{s_is'_is''_i} := \epsilon_{k_ik'_ik''_i} D^{(1)}_{-k_i-s_i}(n_1) D^{(1)}_{-k'_i-s'_i}(n_2) D^{(1)}_{-k'_i-s'_i}(n_3)
\end{align}
Since $n_i$ are fixed $SU(2)$ elements, the components of this tensor can be computed explicitly using (\ref{explicit-n}), and one in particular finds
\begin{align}
\epsilon^{(n)}_{00s} = \delta_{s0}
\end{align}
This is enough for our purposes: indeed, we are interested only in corrections linear in $t$, which means that two of the three strings in the product must be comprised only of $R$'s with vanishing index. $\epsilon^{(n)}_{00s}$ then forces the third index to also vanish, so one obtains
\begin{align}
\langle (R^0)^N \rangle_z = \delta_0^{s_1\ldots s_N} \left(\frac{2\eta i}{t}\right)^{N} \left[1 + \frac{t}{2\eta} \left(\frac{N(N+1)}{2\eta}-N\coth(\eta)\right)\right]
\end{align}
that is, only the terms proportional to $\delta^{s_1..s_N}_0$ will contribute.
\\
Now, the diffeomorphism-invariant quantity $\epsilon(e_a,e_b,e_c):=sgn(\det(a,b,c))=sgn(abc)\epsilon_{abc}$ with $a,b,c\in\{1,2,3\}$ tells us that (calling $R^I_a:=R^I(e_a)$)
\begin{align}
\langle\Psi_{(c,p)},\hat{Q}_v^N\Psi_{(c,p)}\rangle & = \langle\Psi_{(c,p)}, i^N\left(6\epsilon_{IJK} (R^I_1+R^I_{-1})(R^J_2+R^J_{-2})(R^K_3+R^K_{-3})\right)^N\Psi_{(c,p)}\rangle = \notag
\\
& = (6i)^N\prod_{i=1}^3\left(\sum_{n=0}^N \binom{N}{n} \langle (R^0_i)^n\rangle_z\langle (R^0_{-i})^{N-n}\rangle_z
\right) = \notag
\\
& = (6i)^N\left(\sum_{n=0}^N \binom{N}{n} \langle (R^0)^n \rangle_z\langle (R^0)^{N-n}\rangle_z
\right)^3 = \notag
\\
& = (6i)^N\left(\sum_{n=0}^N \binom{N}{n} \left(\frac{2\eta i}{t}\right)^N \left[1+\frac{t}{2\eta} \left(\frac{n(n+1)}{2\eta}-n\coth(\eta)\right)\right] \times \right. \notag
\\
&\hspace{10pt} \times \left.\left[1+\frac{t}{2\eta} \left(\frac{(N-n)(N-n+1)}{2\eta}-(N-n)\coth(\eta)\right)\right]\right)^3 = \notag
\\
& = (6i)^N\left(\frac{2\eta i}{t}\right)^{3N} \left[2^N+\frac{t}{2\eta^2}(N^2+3N)2^{N-2}-\frac{t}{2\eta}N2^N\coth(\eta)\right]^3
\end{align}
where we used
\begin{align}
\sum_{n=0}^N \binom{N}{n} = 2^N, \ \ \ \ \ \ \ \sum_{n=0}^N \binom{N}{n} n = 2^{N-1}N, \ \ \ \ \ \ \ \sum_{n=0}^N \binom{N}{n} n^2 = (N+N^2)2^{N-2}
\end{align}
Thus, we get
\begin{align}
\frac{\langle\Psi_{(c,p)},\hat{Q}_v^N\Psi_{(c,p)}\rangle}{\langle \Psi_{(c,p)},\hat{Q}_v\Psi_{(c,p)}\rangle^N} = 1+\frac{3t}{8\eta^2}N (N-1)
\end{align}
with which one can now compute the expectation value of the Giesel-Thiemann volume operator. For $k=1$, it reads
\begin{align}
\hat{V}^{GT}_{1,v} & = \frac{\langle\Psi_{(c,p)},\hat{Q}_v\Psi_{(c,p)}\rangle^{1/2}}{128} \times
\\
& \times \left[77\cdot\mathds{1}+77\frac{\hat{Q}_v}{\langle\Psi_{(c,p)},\hat{Q}_v\Psi_{(c,p)}\rangle^2}-33\frac{\hat{Q}_v^4}{\langle\Psi_{(c,p)},\hat{Q}_v\Psi_{(c,p)}\rangle^4}+7\frac{\hat{Q}^6_v}{\langle\Psi_{(c,p)},\hat{Q}_v\Psi_{(c,p)}\rangle^6}\right] \notag
\end{align}
so one finds (summing over all $\mathcal N^3$ vertices in the lattice)
\begin{align}
\langle \Psi_{(c,p)}, \hat{V}(\sigma) \Psi_{(c,p)} \rangle = \mathcal N^3 \sqrt{48} \left(\frac{2\eta}{t}\right)^{3/2} \left[1+\frac{3t}{4\eta^2} \left(\frac{7}{8} - \eta\coth(\eta)\right)+\mathcal{O}(t^2)\right]
\end{align}
\section{Conclusion}
\label{s6}
In this work we have constructed a family of coherent states in the full theory of LQC (on a cubic graph) based on gauge coherent states, and have shown that they are peaked on (discretized) flat Robertson-Walker cosmologies. These states are labelled by a parameter $\mu$ controlling how densely embedded is the graph in the spatial manifold. In order to approximate all observables which one needs to describe an isotropic universe, one should choose a sufficiently small $\mu$.
\\
We have presented all the necessary tools for computing the expectation values of any observable including corrections of first order in the semiclassicality parameter $t \sim \hbar$. This parameter is proportional to the spread of the coherent states and thus describes their quantum nature. In other words, this article provides the technology needed for the computation of observables including first order quantum corrections!
\\
We have shown that this works well for the example of the Ashtekar-Lewandowski volume, which can be recasted in polynomial form (instead of a square root) thanks to the result (\ref{Replacement}) by Giesel and Thiemann \cite{AQG3}. This replacement will also be central in the next paper of the series, where we shall turn our attention to the Hamiltonian operator. Specifically, we will use the tools presented here to compute the expectation value of the Hamiltonian on cosmological coherent states. In LQC it has been shown that, if one regards this expectation value as the effective Hamiltonian on the $(c,p)$-phase space, the corresponding effective dynamics agrees with the quantum evolution. Conjecturing that the same is true in LQG, it is important to evaluate this expectation value in the full theory and compare it with LQC. As already reported in \cite{DaporKlaus}, we will find that this expectation value does not coincide with the LQC effective Hamiltonian, due to the presence of the Lorentzian part in the Hamiltonian operator of the full theory.
\section*{Acknowledgements}
The authors would like to thank Kristina Giesel, Hans Liegener and Thomas Thiemann for helpful discussions. AD was partially supported by the Polish National Science Centre grant No. 2011/02/A/ST2/00300. KL thanks the German National Merit Foundation for financial support.
|
2,877,628,090,225 | arxiv | \section{Introduction}
\label{sec:intro}
The era of atmospheric characterization of rocky exoplanets is imminent with the advent of new telescopes, such as the James Webb Space Telescope (JWST, successfully launched in December 2021), the European Extremely Large Telescope (E-ELT), the Large Ultraviolet/Optical/Infrared Surveyor \citep[LUVOIR,][]{Roberge18_luvoir} and Atmospheric Remote-sensing Infrared Exoplanet Large-survey \citep[ARIEL,][]{Tinetti18_ariel}.
This study is motivated by the need to understand the atmospheric circulation on tidally locked exoplanets in order to make the best use of observational data by enabling the community to both refine target selection for observational campaigns and improve confidence in the interpretation of the observations.
Recent simulations of possible climates on TRAPPIST-1e, a rocky planet orbiting an ultracool M-dwarf star, allude to a potential bistability of the atmospheric circulation for this planet \citep[e.g.][]{Sergeev20_atmospheric,Eager20_implications,Turbet22_thai,Sergeev22_thai}.
Here we study the emergence and maintenance of two different circulation regimes of TRAPPIST-1e, a primary observational target, using a 3D general circulation model (GCM).
GCMs help us understand the variety of processes driving planetary climates.
They can reconstruct a simulated three-dimensional state of the atmosphere and its evolution, constrained by a set of parameters observed or assumed for a certain planet.
For a given planetary and atmospheric configuration, we may then obtain a long-term set of statistics (i.e. the climate) compatible with the system of equations of the numerical model.
However, multiple statistically steady solutions may be obtained for the same set of external parameters \citep{Lorenz70_climatic}.
Regions in parameter space where multiple solutions can occur are called bifurcations \citep{SuarezDuffy92_terrestrial,Saravanan93_equatorial} or bistability \citep[e.g.][]{Arnold12_abrupt,Herbert20_atmospheric}.
Moreover, there always exists an uncertainty in GCM parameters, and this is acutely felt in theoretical studies of exoplanetary atmospheres due to the extreme paucity of observational data.
This demands exploration of the model behavior over a range of parameters and configurations, alongside model intercomparisons \citep{Politchouk14_intercomparison,Yang19_simulations,Fauchez21_workshop}.
Earlier studies, focused mostly on non-tidally locked planets, discovered circulation bistability in different scenarios and in models of various degrees of complexity.
One example of circulation bistability concerns the transition to equatorial superrotation in idealized two-layer models of the Earth's atmospheric circulation (\citealp{SuarezDuffy92_terrestrial}; \citealp{Saravanan93_equatorial}; see also discussion in \citealp{Held99_equatorial}).
Here, equatorial superrotation describes a phenomenon where the zonal wind has an excess of angular momentum relative to a state of solid body co-rotation with the underlying planet \citep{Read86_superrotation2,Read18_superrotation}, which can only be obtained when non-axisymmetric disturbances (eddies) transport angular momentum up-gradient \citep{Hide69_dynamics,Gierasch75_meridional,Rossow79_large-scale,Mitchell10_transition}.
For a fast-rotating Earth-like planet, \citet{SuarezDuffy92_terrestrial} and \citet{Saravanan93_equatorial} showed using an idealized two-layer climate model that transient eddies are affected by the strength of a heating perturbation localized at the equator.
The behavior of transient eddies changes the momentum flux balance and leads to the regime transition between a ``conventional'' state, similar to the atmospheric circulation observed on Earth, and a ``superrotating'' state, characterized by a strong eastward jet at the equator.
A positive feedback mechanism was proposed by \citet{Arnold12_abrupt} to explain the bifurcation of the circulation into a subrotating or superrotating state: the resonance of equatorial Rossby waves and background mean flow.
This mechanism was further explored by \citet{Herbert20_atmospheric} who used a simple model to prove that the wave-jet resonance is more robust relative to other feedback mechanisms suggested for the regime bistability.
In the context of tidally locked exoplanets, the bistability of atmospheric circulation was explored by \citet{Thrastarson10_effects}, \citet{Liu13_atmospheric} and \citet{Showman15_three-dimensional} for hot Jupiters, and by \citet{Edson11_atmospheric} and \citet{Noda17_circulation} for terrestrial planets.
\citet{Edson11_atmospheric} found that abrupt transitions occur between two different circulation states, with weak and strong superrotation, at a rotation period of 4--5 days for a dry planet and 3--4 days for an aquaplanet orbiting low-mass stars.
Using an idealized model with no clouds and gray radiation, \citet{Noda17_circulation} mapped the dependence of the large-scale dynamics on a range of rotation periods and identified four circulation patterns, each characterized by either thermally direct day-night circulation, wave-jet resonance, north-south asymmetric effects, or a pair of mid-latitude eastward jets.
Further research of different atmospheric regimes for abstract exoplanetary configurations was conducted by \citet{Carone14_connecting,Carone15_connecting,Carone16_connecting} and later \citet{Kopparapu17_habitable} and \citet{Haqq-Misra18_demarcating}.
The key and sometimes the only parameter demarcating the atmospheric regimes in these studies was the rotation rate of the planet.
With the number of confirmed rocky exoplanets growing, it is pertinent to explore using a 3D GCM whether a specific exoplanet can exhibit regime bistability.
For climate simulations of TRAPPIST-1e, the circulation regime can be sensitive to the representation of convection in the model, as was first noted by \citet{Sergeev20_atmospheric}.
Similar differences in circulation regimes was reported in the TRAPPIST-1 Habitable Atmosphere Intercomparison (THAI), where four different GCMs were used to simulate \ce{N2} and \ce{CO2}-dominated atmospheres of TRAPPIST-1e \citep{Turbet22_thai,Sergeev22_thai}.
The THAI project highlights that the simulated circulation regime for this planet is sensitive to the parameterizations of subgrid-scale processes in GCMs, such as boundary layer processes, radiative transfer, and moist physics.
However, the dynamical feedbacks resulting in different circulation regimes for the same planet were not explored in detail by these studies.
Providing an explanation for them will strengthen our confidence in results from current and future GCM studies, and is the main aim of this study.
In this paper, we study the two distinct circulation regimes that emerge in the atmosphere of a moist nitrogen-dominated atmosphere of TRAPPIST-1e as simulated by a 3D GCM (Sec.~\ref{sec:results_sens}).
We argue that the regime bistability originates during the model spin-up due to the different amount of water vapor in the substellar region and thus different radiative forcing, which affects the emergence of superrotation and the overall climate.
We conduct a series of experiments to test the sensitivity of the regimes to such parts of the model configuration as the initial temperature, slab ocean depth, convection scheme and the cloud radiative effect.
While the model setup is relatively idealized, with a uniform ocean surface at the lower boundary; we use the observed values of planetary radius, rotation rate and insolation with the assumption of a 1:1 synchronous rotation \citep{Grimm18_nature,Fauchez20_thai_protocol}.
We diagnose the regime evolution in the early stages of numerical simulations using various metrics of superrotation and accompanied shifts in the global climate, further supporting our arguments by looking at various terms in the angular momentum budget (Sec.~\ref{sec:results_spinup_dynamics}).
We also describe the surface conditions on the night side of the planet, because the regime shift substantially affects the night side temperature and humidity (Sec.~\ref{sec:results_spinup_cold_traps}).
Finally, we present the steady state of both regimes, showing that they are well-defined with respect to various climate diagnostics (Sec.~\ref{sec:results_mean_state}).
This has consequences for their respective imprint in the atmospheric transmission depth, mostly in the water absorption bands and the continuum level (Sec.~\ref{sec:results_synthobs}).
However, the inter-regime differences in the transmission depth are too small to be detected with the current generation of telescopes.
\section{Methodology}
\label{sec:method}
Transitions of the circulation regime for tidally locked rocky exoplanets were reported in several modeling studies, all based on 3D GCMs \citep[e.g.][]{Edson11_atmospheric,Carone14_connecting,Noda17_circulation}.
It was also noted in recent studies of the TRAPPIST-1e climate \citep{Sergeev20_atmospheric,Eager20_implications,Turbet22_thai,Sergeev22_thai}, in which the dominant climatic feature was either a strong superrotating jet at the equator or two eastward jets in mid-latitudes with a weaker equatorial superrotation.
Throughout this paper, we refer to the former as the ``single jet'' (SJ) regime, and the latter as the ``double jet'' (DJ) regime.
The analysis of the full complexity of this regime bistability, requires a 3D GCM and we employ the Met Office Unified Model (UM).
As we report below (Sec.~\ref{sec:results}), we are able to capture both of the circulation regimes.
The control, or \emph{Base}, experiment develops the SJ regime, whilst various sensitivity simulations settle on either of the two regimes, SJ or DJ, with no intermediate states between them.
We thus choose one of the sensitivity experiments that settled on the DJ regime and compare its evolution and mean state to the SJ regime obtained in the \emph{Base} experiment.
This sensitivity experiment of choice is named \emph{T0\_280} and the only difference in its configuration to that of \emph{Base} is the initial temperature, as described in Sec.~\ref{sec:method_sens}.
Focusing on the \emph{T0\_280} case allows us to explore the bifurcation of the early stages of the simulation in a clearer way, eliminating the effect of e.g. the change of a model parameterization.
At the same time, experiments with different setups that also develop the DJ regime have similar early model evolution as well as the resulting climate, thus making our conclusions robust.
Within this section, the overall model configuration, including the planetary parameters and atmospheric composition, is described in Sec.~\ref{sec:method_model}.
Details of the setup for the base and sensitivity experiments are given in Sec.~\ref{sec:method_sens}.
The method of computing synthetic transmission spectra in our 3D simulations is also detailed in Sec.~\ref{sec:method_synthobs}.
\subsection{Model setup}
\label{sec:method_model}
\begin{deluxetable*}{lll}
\tablecaption{Stellar spectrum and planetary parameters used in this study following \citep{Fauchez20_thai_protocol}. \label{tab:planet}}
\tablehead{
\colhead{Parameter} & \colhead{Units} & \colhead{Value}
}
\startdata
Star and spectrum & & \SI{2600}{\K} BT-Settl with Fe/H=0 \\
Semi-major axis & AU & 0.02928 \\
Orbital period & Earth day & 6.1 \\
Rotation period & Earth day & 6.1 \\
Obliquity & & 0 \\
Eccentricity & & 0 \\
Instellation & \si{\watt\per\square\meter} & 900.0 \\
Planet radius & \si{\km} & 5798 \\
Gravity & \si{\meter\per\second\squared} & 9.12 \\
\enddata
\end{deluxetable*}
All simulations in this study are performed with the UM (code version \texttt{vn12.0}) in the GA7.0 science configuration \citep{Walters19_ga7}.
The UM is configured at a horizontal grid spacing of \ang{2.5} in longitude and \ang{2} in latitude, with 38 vertical levels between the surface and the model top, located at a height of \SI{\approx 80}{\km}.
\footnote{We conducted an additional series of experiments with 60 and 70 vertical levels and different model top heights.
Qualitatively, the conclusions of our study are not affected: both circulation regimes emerge at a higher vertical resolution, although their dependence on the sensitivity parameters (see Sec.~\ref{sec:method_sens} below) does not exactly match those obtained in the model with 38 levels or a different model top height.}
The model is run for 3000 Earth days ($\approx$491 TRAPPIST-1e orbits) to ensure that the atmosphere reaches thermal equilibrium, when the net absorbed stellar radiation is approximately equal to the emitted thermal radiation and when the global mean surface temperature does not have a noticeable long-term trend.
Hereafter we use the word ``day'' to refer to an Earth day, i.e. \SI{86400}{\s}.
In the analysis below (Sec.~\ref{sec:results}), we use daily mean output at high temporal resolution (every day) during the spin-up phase (first 500 days) to capture the emergence of circulation patterns.
The mean-climate state is presented as the average over the days 2000--3000 of simulations (i.e. over $\approx$163 orbits).
We employ a nitrogen-dominated atmospheric configuration, used in the TRAPPIST Habitable Atmosphere Intercomparison (THAI) under the label \textit{Hab~1} \citep{Fauchez20_thai_protocol,Sergeev22_thai}, as well as in many previous exoplanet modeling studies \citep[e.g.][]{Turbet16_habitability,Wolf17_assessing,DelGenio19_albedos,Yang19_simulations}.
Namely, an atmosphere with a total mean pressure of \SI{e5}{\pascal} consisting of \ce{N2}, 400 ppm of \ce{CO2}, and \ce{H2O}, the latter being the main condensible species.
Ozone is not included in our simulations for simplicity, though it may affect our results by modifying the vertical temperature profile in the stratosphere or inhibiting deep convection \citep[see e.g.][]{Gomez-Leal19_climate,Chen19_habitability}.
On the other hand, the radiative influence of ozone is likely to be muted compared to that on Earth, because of the weaker stellar flux in the ozone absorption window \citep[][]{Boutle17_exploring}.
Planetary parameters are the same as in the THAI protocol and are also given here in Table~\ref{tab:planet} for convenience.
The planet is also assumed to be in synchronous rotation, which is justified by the likely time scale of tidal locking compared to the age of its host star \citep{Turbet18_modeling,PierrehumbertHammond19_atmospheric}.
The planet's surface is covered by an immobile slab ocean.
Its bolometric albedo changes depending on the surface temperature as a simple representation of the sea-ice albedo feedback.
The albedo is either 0.06 or 0.25, above or below the freezing point of seawater, respectively.
\subsection{Base and sensitivity experiments}
\label{sec:method_sens}
The \emph{Base} setup is started from an isothermal profile of \SI{300}{\K} and a dry, hydrostatically balanced, atmosphere at rest (first row in Table~\ref{tab:runs}).
Water vapor is then allowed to evaporate from the slab ocean surface and condense into clouds.
Compared to the THAI \textit{Hab~1} UM simulation, here we use a more advanced representation of subgrid cloud variability in the radiation transfer scheme, namely the Monte Carlo Independent Column Approximation (MCICA) while assuming exponential-random overlap \citep{Pincus03_fast,Barker08_mcica}.
To parameterize convection, a mass-flux scheme is used \citep{Gregory90_mass,Walters19_ga7}.
The slab ocean heat capacity is \SI{4e6}{\joule\per\kelvin\per\meter\squared}, corresponding to a depth of \SI{\approx 1}{\meter}, which was also used in previous idealized modeling studies \citep[e.g.][]{Wing18_rcemip,Seeley21_episodic}.
In the \emph{Base} experiment, the surface temperature is allowed to evolve dynamically --- driven by air-sea energy fluxes.
We then run a series of simulations to explore the sensitivity of atmospheric evolution to the initial and boundary conditions, as well as to the choice of the convection parameterization and cloud radiative effects.
The sensitivity experiments are also run for 3000 Earth days, which is sufficient for them to reach a steady state.
We confirmed this by running some of them for 10000 Earth days ($\approx$1639 orbits), during which no further regime transitions happened (not shown).
All simulation setups are summarized in Table~\ref{tab:runs}.
In the first group of experiments, we start the simulation from different temperatures for both the atmosphere and the surface, going from \SI{290}{\K} to \SI{250}{\K}, in increments of \SI{10}{\K}, while holding the rest of the configuration the same as in \emph{Base}.
These experiments are labeled \emph{T0\_290}, \emph{T0\_280}, \emph{T0\_270}, \emph{T0\_260} and \emph{T0\_250}.
In an additional experiment, we test the robustness of the circulation regime by restarting the model in the \emph{Base} configuration from a steady-state snapshot of the DJ regime (labeled \emph{DJ\_start}).
Next, we explore the role of the bottom boundary condition.
We first test the sensitivity of the atmospheric regime to the depth, or equivalently, the heat capacity, of the slab ocean.
In the sensitivity experiments, it is increased to the equivalent depth of \SI{\approx 5}{\meter} and \SI{\approx 10}{\meter} (labeled as \emph{SOD\_5} and \emph{SOD\_10}, respectively)
We then run two experiments with a fixed surface temperature, labeled \emph{FixedSST}.
\footnote{Note that in this configuration the top-of-the-atmosphere (TOA) energy balance fluctuates around a constant non-zero value (which is expected for a fixed temperature setup), implying that the simulation has reached a steady state.}
In \emph{FixedSST\_g}, the surface temperature field is set globally to that obtained in the \emph{T0\_280} experiment (i.e. from the DJ regime).
In \emph{FixedSST\_n}, the DJ-regime surface temperature field is set only to the night hemisphere of the planet, while the day hemisphere surface temperature is fixed to that observed in the SJ regime.
Used extensively in Earth climate modeling \citep[see e.g.][]{Blackburn13_ape,Williamson13_ape}, fixed surface temperature experiments represent a key step in model hierarchy with respect to the lower boundary condition \citep{Maher19_model} and are included in our study to test whether the atmospheric circulation is controlled by the surface thermal forcing on the day or night side of the planet.
With regards to physical parameterizations, we conduct two experiments.
In \emph{Adjust}, we swap the mass-flux convection parameterization for a convection adjustment scheme \citep{Lambert20_continuous}, analogous to the experiment discussed in \citet{Sergeev20_atmospheric}.
Even though convection adjustment schemes are too simplistic to represent the complexity of subgrid-scale convective plumes correctly, they are still often used in modeling planetary atmospheres to understand the key properties of convection \citep[e.g.][]{Lora15_gcm,Labonte20_sensitivity,Turbet21_day-night,Paradise22_exoplasim}.
The \emph{Adjust} experiment thus is designed to test the effect of the representation of convection on the atmospheric circulation.
Similarly, to test the role of cloud radiative feedbacks in the emergence of superrotation, in a separate simulation labeled \emph{CRE\_off} we disable the radiative effect of clouds (both shortwave and longwave).
While somewhat comparable to the benchmark simulations of \citet{Turbet22_thai} with respect to clouds, this experiment still has moisture-related climate processes, such as diabatic heating.
\begin{deluxetable*}{ll}
\tablecaption{Simulation setup. For each sensitivity experiment, only the difference relative to the base experiment is mentioned.}
\label{tab:runs}
\tablewidth{0pt}
\tablehead{
\colhead{Experiment} & \colhead{Description}
}
\startdata
\emph{Base} & Convection scheme: mass-flux\\
& Cloud radiative effect: ON\\
& Initial temperature \SI{300}{\K}\\
& Slab ocean\textsuperscript{\textdagger} depth: \SI{1}{\m}\\
& Surface temperature: dynamic\\
& Start: isothermal atmosphere and surface, dry atmosphere, zero wind speed\\
\emph{T0\_250} & Initial temperature: \SI{250}{\K}\\
\emph{T0\_260} & Initial temperature: \SI{260}{\K}\\
\emph{T0\_270} & Initial temperature: \SI{270}{\K}\\
\emph{T0\_280} & Initial temperature: \SI{280}{\K}\\
\emph{T0\_290} & Initial temperature: \SI{290}{\K}\\
\emph{DJ\_start} & Start: from a steady-state DJ regime\textsuperscript{\textdagger\textdagger} snapshot\\
\emph{SOD\_5} & Slab ocean depth: \SI{5}{\m}\\
\emph{SOD\_10} & Slab ocean depth: \SI{10}{\m}\\
\emph{FixedSST\_g} & Surface temperature: fixed; DJ regime\textsuperscript{\textdagger\textdagger} distribution globally\\
\emph{FixedSST\_n} & Surface temperature: fixed; DJ regime\textsuperscript{\textdagger\textdagger} distribution on the night side \\
\emph{Adjust} & Convection scheme: adjustment\\
\emph{CRE\_off} & Cloud radiative effect: OFF\\
\enddata
\tablecomments{
\textsuperscript{\textdagger} The slab ocean albedo changes depending on the surface temperature between 0.06 and 0.25, above or below the freezing point of seawater, respectively.}
\textsuperscript{\textdagger\textdagger}``DJ regime'' refers to the double mid-latitude jet circulation pattern described in Sec.~\ref{sec:results_sens}.
\end{deluxetable*}
\subsection{Synthetic spectra}
\label{sec:method_synthobs}
To explore the implications of different climates states for observations, we compute synthetic transmission spectra following the method described in \citet{Lines18_exonephology} and applied for terrestrial planets in \citet{Boutle20_mineral}.
In short, the transmission spectra are calculated using spherical geometry within the 3D GCM framework, using the same radiation scheme (SOCRATES) that the UM uses to simulate the climate.
These calculations use high-resolution (280 bands) spectral files and are performed via a second, ``diagnostic'', call to the radiation scheme thereby not affecting the model evolution.
We do not extend the model top to lower pressures, as has been tested in e.g. \citet{Fauchez22_thai}, as this is not required for a temperate climate with Earth-like temperatures and a low cloud deck \citep{Suissa20_dim,SongYang21_asymmetry}.
\section{Results}
\label{sec:results}
In this section, we present the results of our simulations.
We first show in Sec.~\ref{sec:results_sens} that the SJ and DJ regimes can emerge due to small changes in the model setup.
This makes it important to examine what dynamical mechanisms play a role in the formation and maintenance of the regimes.
Thus, using two illustrative simulations, in Sec.~\ref{sec:results_spinup_dynamics} we focus on the earliest stages of the simulations and show that the evolution of the two regimes is associated with subtle differences in the mean and eddy angular momentum fluxes.
In Sec.~\ref{sec:results_spinup_cold_traps}, we then explain that the large night-side surface temperature difference appears between the regimes due to the difference in water vapor content in the night-side atmosphere.
In Sec.~\ref{sec:results_mean_state}, we explore the climate of these regimes in a steady state and demonstrate that they are well-defined with respect to multiple atmospheric diagnostics, such as surface temperature, wind patters, and cloud distribution.
Finally, in Sec.~\ref{sec:results_synthobs} we discuss the implications for transmission spectra and show that while the circulation regimes have consistent differences in the water absorption bands as well as the continuum level, they are too small to be detected with current technology.
\subsection{Circulation regimes across the model simulations}
\label{sec:results_sens}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_mean__all_sim__u_max_eq_jet_300hpa_jet_lat_free_trop_ratio_dn_ep_temp_diff_trop_t_sfc_min__u_temp_vcross.png}
\caption{The mean climate diagnostics in all experiments: (a) maximum equatorial zonal wind speed (x-axis, \si{\m\per\s}) and the latitude of the tropospheric jet (y-axis, degrees); (b) the ratio of the day-night to equator-pole temperature difference (x-axis) and the minimum surface temperature (y-axis, \si{\K}).
Experiments that produce the SJ regime are shown in blue, DJ --- in orange.
Different marker shapes correspond to different groups of sensitivity experiments.
Configuration labels are defined in Table~\ref{tab:runs}.
Also shown is the steady state of the (c) SJ and (d) DJ circulation regimes in the indicative simulations (\emph{Base} and \emph{T0\_280}, respectively).
Panels c and d show the vertical cross-section of the zonal mean eastward wind (shading, \si{\m\per\s}) and zonal mean air temperature (contours, \si{\K}).
Horizontal dashed lines in c and d show the corresponding pressure level of the horizontal cross-sections of (red) temperature and (gray) winds and geopotential height shown in Fig.~\ref{fig:mean_circ}.
``Single Jet'' (SJ) and ``Double Jet'' (DJ) are short-hand descriptive terms rather than precise descriptions: the equatorial jet in the SJ regime exhibits a split at $\sigma\approx 0.5$, and the DJ regime still has an equatorial superrotation, albeit weaker than that in the SJ regime.
\label{fig:all_sim}}
\end{figure*}
As described in Sec.~\ref{sec:method_sens}, we include 13 simulations in our study: the \emph{Base} (control) simulation and 12 sensitivity experiments.
In the sensitivity experiments, we change one aspect of the model configuration at a time, keeping the rest of the configuration the same as that in the \emph{Base} setup.
We change the initial conditions (initial temperature), surface boundary conditions (slab ocean depth and temperature), convection parameterization, and cloud radiative effect (CRE).
Fig.~\ref{fig:all_sim} provides a summary of all our experiments in terms of four key diagnostics of the steady-state climate.
These diagnostics are presented in pairs: Fig.~\ref{fig:all_sim}a shows the strength of the eastward wind at the equator and the latitude of the tropospheric jet, while Fig.~\ref{fig:all_sim}b shows the ratio of the day-night to equator-pole temperature difference and the lowest surface temperature.
It is apparent in both panels that the experiments form two distinct clusters, and there is practically no spectrum between the climate regimes.
Note that the clusters of experiments are the same for all four metrics.
The SJ regime has a higher zonal wind in the equatorial upper troposphere, with its maximum reaching values \SI{\approx 80}{\m\per\s} (x-axis in Fig.~\ref{fig:all_sim}a).
The low values of the jet latitude in this regime demonstrate that the zonal wind maximum is in the tropics (y-axis in Fig.~\ref{fig:all_sim}a).
The DJ regime, on the other hand, has substantially lower zonal wind at the equator --- at about \SI{\approx 40}{\m\per\s}.
However, the DJ regime still maintains an equatorial superrotation, albeit a weaker one compared to that in the SJ regime (Fig.~\ref{fig:all_sim}c,d).
The maximum of the zonal wind speed in the DJ regime is at \ang{\approx 60} latitude, demonstrating that the dominant tropospheric jets are extratropical.
The thermal structure of the SJ regime is such that the temperature difference between the day and night side of the planet is largely equal to the equator-to-pole temperature gradient (x-axis in Fig.~\ref{fig:all_sim}b).
This is mostly due to a colder night side, illustrated by the relatively low surface temperatures in night-side ``cold traps'' in this regime (\SI{<180}{\K}, see the y-axis in Fig.~\ref{fig:all_sim}b).
Note that the only outlier is the \emph{FixedSST\_n} simulation because its night-side temperature is fixed to that of the DJ regime, i.e. a higher value.
Indeed, the DJ regime has consistently higher night-side surface temperatures --- between 220 and \SI{230}{\K}, than that found in SJ simulations.
Consequently, the meridional temperature gradient between the equator and poles is more than 5 times larger than the day-night contrast (x-axis in Fig.~\ref{fig:all_sim}b).
In other words, the DJ thermal structure is more zonally symmetric than that of the SJ regime.
The details of the thermodynamic and circulation patterns of both regimes are discussed in more detail in Sec.~\ref{sec:results_mean_state}.
Fig.~\ref{fig:all_sim} also reveals which of the two climate regime the simulation are sensitive to the following factors: initial conditions overall (\emph{DJ\_start}) and temperature in particular (\emph{T0} group), fixed surface temperature distribution (\emph{FixedSST\_g}), the choice of the convection scheme (\emph{Adjust}), and the inclusion or omission of the radiative impact of clouds (\emph{CRE\_off}).
When one of these aspects of the model setup is changed, the resulting simulation is in the DJ regime (opposite to that in \emph{Base}).
On the other hand, the circulation regime is insensitive to changing the slab ocean depth (experiments \emph{SOD\_5} and \emph{SOD\_10}) or the night-side surface temperature (\emph{FixedSST\_n}).
An important sensitivity simulation is \emph{DJ\_start}, i.e. the simulation started not from an isothermal profile and an atmosphere at rest, but from a previously developed DJ regime.
This steady state snapshot is taken from the \emph{Adjust} simulation and includes all the prognostic model fields \citep{Walters19_ga7}, such as the wind components, atmospheric pressure, temperature, water vapor and cloud content.
As Fig.~\ref{fig:all_sim} demonstrates, such initial conditions appear to determine the end state: the already established DJ regime does not spontaneously transition to the SJ regime, even if the convection scheme is not that used in the \emph{Adjust} case.
This hints at the fact that the DJ regime is more robust than its SJ counterpart, but further conclusions require a separate study.
The simulations started from a different initial temperature $T_0$ provide the most interesting outcome of our model sensitivity study, because moderate initial temperatures (260, 270, and \SI{280}{\K}) result in the DJ regime, while the extremes give the SJ regime (e.g. \emph{T0\_290} with $T_0=\SI{290}{\K}$ and \emph{Base} with $T_0=\SI{300}{\K}$).
It is also the most surprising result, because one would not expect large sensitivity to $T_0$ set within reasonable limits: our simulations do not include a dynamical ocean or sophisticated sea ice schemes \citep[see e.g.][]{DelGenio19_habitable,Yang20_transition,Olson22_effect}, and the slab ocean provides an infinite source of moisture (which would be important only on large time scales).
The \SI{300}{\K} isothermal initial state specified by the THAI protocol, which our control setup inherits, was chosen primarily for its simplicity \citep{Fauchez20_thai_protocol}, but no systematic investigation of the model sensitivity was performed.
In the remaining sections, we deliver a detailed comparison of the SJ and DJ regimes, focusing on the \emph{Base} and \emph{T0\_280} simulations, respectively.
We show that even a \SI{20}{\K} initial temperature difference can lead to a regime bistability within the first tens of days of model evolution.
It happens due to the different amount of water vapor lifted in the atmosphere by convection in the substellar region, which results in a different radiative forcing of the atmosphere and further consequences for superrotation and the overall climate.
\subsection{Emergence of the circulation regimes}
\label{sec:results_spinup_dynamics}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_spinup__base_sens-t280k__u_eq_jet_max_wave_crest_lon_300hpa.png}
\caption{Spin-up diagnostics for the (a) \emph{Base} (b) \emph{T0\_280} simulations: (brown) maximum zonal wind at the equator at \SI{300}{\hecto\pascal} and (cyan) the longitude of stationary Rossby wave crest.
For reference, the wave crest longitude is also shown in Fig.~\ref{fig:mean_circ}c,d.
An animation of the stationary wave pattern and zonal mean atmospheric structure during the first 500 days of the simulations is provided as Supplemental Video 1 (see also Fig.~\ref{fig:spinup_ghgt_anom_wind_temp}).
\label{fig:crest_lon_eq_jet}}
\end{figure*}
While many previous studies discuss the maintenance of superrotation on tidally locked exoplanets in a steady state regime \citep[e.g.][]{ShowmanPolvani11_equatorial,Tsai14_three-dimensional,Carone15_connecting,KomacekShowman16_atmospheric,Noda17_circulation}, its initial acceleration received much less attention, especially in a full-complexity atmospheric GCM.
For hot Jupiter atmospheres, it was explored in 3D GCM simulations by e.g. \citet{Liu13_atmospheric} and \citet{Debras20_acceleration}.
\citet{WangYang20_phase} also briefly discussed it in the context of a wave-jet resonance on a hypothetical tidally locked terrestrial planet.
Building on these studies, we scrutinize the initial phase of the two regimes and track the development of the wave-mean-flow interaction.
The regime evolution described here is unlikely to happen in a real atmosphere, because no atmosphere develops from quiescent isothermal conditions.
However, a small change in forcing (for example due to a stellar flare) on TRAPPIST-1e or a similar exoplanet, whose atmosphere resides on the edge of different regimes, may result in an abrupt change in circulation with consequences for the global climate \citep[e.g.][]{SuarezDuffy92_terrestrial,Caballero10_spontaneous,Arnold12_abrupt,Noda17_circulation}.
It is also crucial to understand how the two different circulation regimes develop in a 3D GCM in order to be confident in the robustness of GCM simulations of an exoplanetary climate.
This will allow for more informed decisions in setting up future single-model studies and GCM intercomparisons \citep{Fauchez21_workshop}.
The evolution of the flow is summarized in Fig.~\ref{fig:crest_lon_eq_jet} by the daily-mean time series of the zonal wind at \SI{300}{\hecto\pascal} along with the phase of the stationary Rossby wave.
The latter is diagnosed by the longitude of the maximum of \SI{300}{\hecto\pascal} eddy geopotential height, i.e. the deviation from the zonal mean of the \SI{300}{\hecto\pascal} isobaric surface height.
It takes between 100 to 250 Earth days for equatorial superrotation to settle into either the SJ or DJ regime in the \emph{Base} and \emph{T0\_280} case, respectively.
Note in the sensitivity experiments with a deeper slab ocean (i.e. with higher heat capacity), the flow evolution takes longer to stabilize (not shown), but the final state does not differ from the control (Fig.~\ref{fig:all_sim}).
The wind speed time series in Fig.~\ref{fig:crest_lon_eq_jet} show that within the first $\approx$80 days, the equatorial superrotation developed in the \emph{Base} setup is weaker than that in the \emph{T0\_280} case (see also Supplemental Video 1 and Fig.~\ref{fig:spinup_ghgt_anom_wind_temp}).
During this first acceleration stage, the planetary-scale wave pattern also develops quicker in the \emph{T0\_280} case and is able to transport eastward momentum to the equator, accelerating the jet to higher velocity relative to that in \emph{Base}.
After approximately day 80, in the \emph{T0\_280} simulation the broad equatorial superrotating flow splits into two separate jet cores, which migrate to mid-latitudes and within a few further days reach their steady-state structure --- the DJ regime.
Accordingly, the eastward momentum supplied to these jets is being taken from the equatorial region, causing the zonal wind at the equator to slow down to \SI{\approx 45}{\m\per\s} (Fig.~\ref{fig:crest_lon_eq_jet}b).
The zonal flow thus fails to achieve resonance with the stationary Rossby wave, whose crest keeps oscillating near the western terminator \citep[see e.g.][]{PierrehumbertHammond19_atmospheric,WangYang20_phase}.
Meanwhile, in the \emph{Base} experiment, the equatorial superrotation continues to develop more gradually and reaches its steady-state maximum by approximately day 200.
A wave-jet resonance develops, which is seen in the acceleration of the equatorial jet to \SI{\approx 90}{\m\per\s} and an eastward shift of the planetary wave, whose crest settles at \ang{\approx 45}E (Fig.~\ref{fig:crest_lon_eq_jet}a, see also Supplemental Video 1 and Fig.~\ref{fig:spinup_ghgt_anom_wind_temp}).
The period needed to reach the steady state is comparable to those found in the idealized experiments of \citet{Noda17_circulation} and \citet{HammondPierrehumbert18_wave-mean}.
The manifestation of the wave-jet resonance in the \emph{Base} case is also similar to that shown in \citet{WangYang20_phase}, though happens over a longer period of time, likely because of the uniform initial conditions in our setup.
Notably, the wind speed and wave crest longitude exhibit oscillations around the steady state.
This time variability is more prominent in the \emph{T0\_280} case, because it is associated with a larger role of transient baroclinic eddies, as we discuss further below.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{ch111_spinup__base_sens-t280k__tseries__wvp_d_dt_sw_d_dt_lw_d_dt_cv_d_dt_diab_d_eady_growth__day000-200_mean.png}
\caption{Spin-up diagnostics for the first 200 days of the (blue lines) \emph{Base} and (orange lines) \emph{T0\_280} simulations: (a) water vapor path, (b) shortwave radiative heating rate, (c) longwave radiative heating rate, (d) convective heating rate, (e) diabatic (radiative plus latent) heating rate, (f) the Eady growth rate.
The water vapor path is averaged over the day side, integrated vertically and has units of \si{\kg\per\m\squared}.
The heating rates are averaged spatially over the day side and vertically over the troposphere and have units of \si{\K\per\day}.
The Eady growth rate is averaged within \SIrange{\approx 850}{500}{\hecto\pascal} in mid-latitudes (\ang{30}--\ang{80}) and has units of \si{\per\day}.
\label{fig:var_tseries}}
\end{figure*}
The regime bifurcation within the first tens of days of the two simulations can be explained using the diagnostics in Fig.~\ref{fig:var_tseries}.
In the \emph{Base} case, the high initial temperature of the surface (\SI{300}{\K}), further increased due to stellar irradiation initially unimpeded by clouds, leads to extremely strong surface latent heat flux via evaporation.\footnote{Note that the initial temperature spike beyond \SI{300}{\K} may not be crucial, because the simulation with a fixed surface temperature, \emph{FixedSST\_g}, settles into the SJ regime (Fig.~\ref{fig:all_sim}).}
The strong latent heat flux induces vigorous convection, manifesting at day 10 as a spike of convective heating of up to \SI{0.6}{\K\per\day}, evident in Fig.~\ref{fig:var_tseries}d (blue curve).
Convective plumes at the substellar region lift significant volumes of water vapor into the atmosphere.
The atmosphere's high initial temperature (also \SI{300}{\K}) allows it to hold a large portion of that moisture before it condenses, as described by the Clausius-Clapeyron equation.
The \emph{Base} case thus experiences a marked increase in the total column water vapor (water vapor path) --- up to \SI{60}{\kg\per\m\squared} --- within the first 20 days (Fig.~\ref{fig:var_tseries}a).
Moistening of the substellar atmosphere produces an increase in shortwave absorption to more than \SI{1.8}{\K\per\day} (Fig.~\ref{fig:var_tseries}b).
However, the water vapor also efficiently radiates energy to space, causing the longwave cooling rate to reach \SI{\approx 2.5}{\K\per\day} (Fig.~\ref{fig:var_tseries}c).
The result is the net cooling of the atmosphere by radiation.
The day-side radiative cooling is offset by heating due to deep convection (Fig.~\ref{fig:var_tseries}d), turbulent fluxes in the boundary layer, and condensation of the water vapor.
The contribution from the latter two processes is smaller relative to convection and thus is not shown.
Consequently, the total forcing of the day side atmosphere is weakly negative in the \emph{Base} case until approximately day 40, after which it increases to the steady-state value of \SI{\approx 1}{\K\per\day} (Fig.~\ref{fig:var_tseries}e).
The \emph{T0\_280} simulation, on the other hand, starts from a profile \SI{20}{\K} colder than that in the \emph{Base} case.
As a result, the surface evaporation and deep atmospheric convection is slightly weaker (orange curve in Fig.~\ref{fig:var_tseries}d).
Furthermore, the saturation water vapor pressure is also lower due to the atmosphere being colder, and the resulting increase in the total column water vapor is about half as much as in the \emph{Base} simulation (Fig.~\ref{fig:var_tseries}a).
While the difference in the shortwave heating between the two experiments is small, the difference in the longwave cooling is larger, with the \emph{T0\_280} atmosphere losing \SI{< 2}{\K\per\day}.
As far as the total latent heating is concerned, it is overall similar in both cases, as exemplified by the convective heating rate in Fig.~\ref{fig:var_tseries}d.
It is also smaller than the radiative heating rates, in agreement with \citet{Boutle17_exploring}.
The overall effect is thus mainly due to the differences in the radiative heating rates, making the net diabatic forcing in the \emph{T0\_280} simulation stronger than that in the control one (Fig.~\ref{fig:var_tseries}e).
This is likely the key difference in the initial stages of the two simulations that places them on different branches of regime evolution.
Namely, the overall weaker forcing in the \emph{Base} simulation, relative to that in \emph{T0\_280}, produces a slower development of the stationary wave and a more gradual acceleration of the equatorial eastward jet (Fig.~\ref{fig:crest_lon_eq_jet}a).
On the contrary, the stronger forcing in the \emph{T0\_280} case establishes the wave pattern and accelerates the equatorial jet (initially) more rapidly (Fig.~\ref{fig:crest_lon_eq_jet}b).
These results agree with the earlier studies based on shallow water models as well as idealized GCMs.
For example, \citet{HammondPierrehumbert18_wave-mean} show the GCM output for a dry tidally locked terrestrial planet with a 5 day rotation period (close to that of TRAPPIST-1e, see Table~\ref{tab:planet}).
The authors demonstrate that for a fixed planetary rotation rate, different circulation regimes emerge depending on the strength of stellar forcing \citep[see also][]{HammondPierrehumbert18_wave-mean}.
Qualitatively, their regime at the highest instellation is similar to that emerging in the \emph{T0\_280} simulation, the defining feature of which is a single broad equatorial eastward jet and a high-amplitude planetary-scale wave.
At the lowest instellation, the authors obtain a regime similar to that in the initial phase of the \emph{Base} simulation with a weaker equatorial superrotation.
Note the change of stellar forcing between the regimes in their study is substantially larger than the changes in forcing in the first days of our simulations.
This is merely a qualitative comparison, however, because \citet{HammondPierrehumbert18_wave-mean} analyze the steady-state circulation, not the acceleration phase.
As discussed below, even though the \emph{Base} simulation starts with a weaker equatorial superrotation, it ends up with a stronger superrotation; whilst the \emph{T0\_280} simulation starts with a stronger superrotation, but ends up with a weaker one.
After this initial development phase ($\approx$80 days), the SJ-like circulation pattern in the \emph{T0\_280} case transforms into the DJ circulation pattern by developing a pair of eastward jets at mid-latitudes.
This corresponds to the decrease in the stationary wave amplitude and the deceleration of the equatorial jet (Fig.~\ref{fig:crest_lon_eq_jet}b).
In other words, the \emph{T0\_280} simulation fails to achieve a wave-jet resonance.
Instead, the \emph{T0\_280} case is characterized by an increase in baroclinicity manifested as baroclinic waves traveling in the zonal direction at high latitudes (see Supplemental Video 1).
The increasing role of baroclinic instability, especially after the first 80 days, is demonstrated by the time series of the Eady growth rate which is calculated following \citet{Vallis17_aofd}:
\begin{equation}
\sigma_E = 0.31|f|\frac{|\partial u/\partial z|}{N},
\end{equation}
where $f=2\Omega sin\phi$ is the Coriolis parameter ($\Omega$ is the planetary rotation rate, $\phi$ is the latitude), $\partial u/\partial z$ is the derivative of the zonal wind velocity with height, $N=\sqrt{g/\theta\partial\theta/\partial z}$ is the Brunt-V\"{a}is\"{a}l\"{a} frequency ($g$ is the acceleration due to gravity, $\theta$ is the potential temperature).
Fig.~\ref{fig:var_tseries}f shows that $\sigma_E$ is consistently higher for the \emph{T0\_280} than for the \emph{Base} simulation (orange and blue curves, respectively).
This indicates that the \emph{T0\_280} case develops conditions more favorable for the baroclinic instability, mostly via the increase of the mean horizontal temperature gradient, which via the thermal wind equation is proportional to $\partial u/\partial z$.
The emergence of baroclinic jets at mid-latitudes marks the mature stage of the DJ regime.
In the \emph{Base} experiment, $\sigma_E$ is substantially smaller, indicating a weaker role of baroclinic instability (Fig.~\ref{fig:var_tseries}f).
The SJ regime reaches its equilibrium and does not develop a strong equator-pole temperature gradient (Fig.~\ref{fig:all_sim}).
Broadly the same chain of events leading to one regime or another is identified across the other sensitivity experiments in our study.
One interesting example is \emph{T0\_250}, which eventually develops an SJ regime, despite its colder initial conditions.
Despite the colder start, which favors stronger initial forcing and thus the evolution similar to that in the \emph{T0\_280} case, the \emph{T0\_250} simulation does not develop strong baroclinicity in mid-latitudes and thus eventually transitions back to the SJ regime (not shown).
From the dynamical perspective, the zonal flow acceleration can be analyzed using the zonal component of the axial angular momentum budget.
Hereafter simply referred to as angular momentum, it is defined per unit mass as $m = (u + \Omega r\cos\phi)r\cos\phi$ where $u$ is the zonal wind speed, $\Omega$ is the rotation rate, $r$ is the planetary radius, and $\phi$ is latitude.
The time and zonal mean budget of $m$, without a shallow atmosphere approximation, can be expressed as
\begin{align}
\frac{\Delta[\rho m]}{\Delta T}&=
\underbrace{ - \frac{[\overline{V}]}{r}\frac{\partial[\overline{m}]}{\partial\phi} }_\text{Term MH}
\underbrace{ - [\overline{W}]\frac{\partial[\overline{m}]}{\partial r} }_\text{Term MV}
\nonumber\\
&\phantom{=\ \,}
\underbrace{ -\frac{1}{r\cos\phi}\frac{\partial}{\partial\phi}([\overline{V}^{\ast}\overline{m}^{\ast}]\cos\phi) }_\text{Term SH}
\underbrace{ - \frac{1}{r^{2}}\frac{\partial}{\partial r}([\overline{W}^{\ast}\overline{m}^{\ast}]r^{2}) }_\text{Term SV}
\nonumber\\
&\phantom{=\ \,}
\underbrace{ -\frac{1}{r\cos\phi}\frac{\partial}{\partial\phi}([\overline{V^{\prime}m^{\prime}}]\cos\phi) }_\text{Term TH}
\underbrace{ - \frac{1}{r^{2}}\frac{\partial}{\partial r}([\overline{W^{\prime}m^{\prime}}]r^{2}) }_\text{Term TV}
\nonumber\\
&\phantom{=\ \,} - r\cos\phi[\overline{\rho G_\lambda}],
\label{eq:aam_main}
\end{align}
where square brackets denote zonal mean and overbars denote time mean, while asterisks and primes denote the deviations from the zonal and time mean, respectively.
The term on the left-hand side is $\Delta[\rho m] = ([\rho m]_{t=\Delta T} - [\rho m]_{t=0})$ divided by $\Delta T$, which is the total change in $m$ over the time period $\Delta T$.
The rest of the notations are as follows: $\rho$ is density, $V=\rho v$ and $W=\rho w$, where $v$ and $w$ are the meridional and vertical wind speeds, respectively; $G_\lambda$ represents friction and dissipation forces.
The derivation of Eq.~\eqref{eq:aam_main} is given in Appendix~\ref{app:aam}.
\eqref{eq:aam_main} states that the change in mean angular momentum can be due to three transport components, each of which can be split into horizontal and vertical parts (H and V, respectively).
The first two terms on the right hand side (MH and MV) represent the advection of mean $m$ by the mean flow, the third and the fourth terms (SH and SV) represent the transport by stationary eddies, while the fifth and the sixth terms (TH and TV) represent the transport by transient eddies.
Note that the mean terms (MH and MV) are written in the advective form.
Eq.~\eqref{eq:aam_main} has a form similar to that for the zonal wind $u$ used in many previous studies \citep[e.g.][]{Kraucunas05_equatorial,Hammond20_equatorial,ZengYang21_oceanic}, but has a more concise form by inherently incorporating the Coriolis force terms within $m$.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{ch111_spinup__base_sens-t280k__ang_mom_bud_ang_mom_time_change__0-20_20-80_80-200_250-450__yprof.png}
\caption{Meridional profiles of the angular momentum budget terms (\si{\joule\per\m\cubed}) in Eq.~\ref{eq:aam_main} during the spin-up phase of the (left) \emph{Base} and (right) \emph{T0\_280} simulations: (orange) mean advection terms, (blue) stationary eddy terms, (purple) transient eddy terms.
The dashed black line shows the residual.
The terms are averaged within the troposphere (\SIrange{\approx 1}{20}{\km}) and over the periods of (a, b) 0--20 days, (c, d) 20--80 days, (e, f) 80--200 days, and (g, h) 250--450 days.
Note the jagged lines in the two top panels are due to a very short period of averaging (20 days).}
\label{fig:ang_mom_yprof}
\end{figure*}
Fig.~\ref{fig:ang_mom_yprof} shows the meridional profiles of the angular momentum budget terms calculated according to Eq.~\ref{eq:aam_main} over four periods of the flow evolution.
During the first stage (0--20~days of the simulation), the dominant terms are the mean advection terms and and are maximized in extratropical latitudes (Fig.~\ref{fig:ang_mom_yprof}a,b).
This is mostly due to the horizontal Coriolis acceleration, which is positive in the mid-troposphere due to the strong meridional divergence of the flow.
This term is roughly the same in both \emph{Base} and \emph{T0\_280} simulations.
The eddy angular momentum transport, however, is notably higher in the \emph{T0\_280} case (Fig.~\ref{fig:ang_mom_yprof}b), corresponding to a stronger acceleration of the equatorial eastward jet (Fig.~\ref{fig:crest_lon_eq_jet}b).
Most of the eddy transport is due to the stationary terms, which transport momentum horizontally from the tropics and mid-latitudes toward the low latitudes and upwards to the upper troposphere at the equator (not shown).
A weak stationary eddy contribution in the \emph{Base} case and a strong one in the \emph{T0\_280} case is in agreement with the initial forcing being likewise weaker and stronger in these simulations.
One can notice that the residual is large in Fig.~\ref{fig:ang_mom_yprof}a, b.
This is likely due to the fact that the simulations are started from rest and the mean-eddy separation is not clear during the earliest stages of the model spin-up.
Another possible source of error is sampling rate: we use daily mean output, which likely leads to an underestimation of the eddy terms.
In the days 20--80 of the simulations, the day-side mean diabatic heating is still stronger in the \emph{T0\_280} case than that in the \emph{Base} case, explaining the slightly stronger stationary eddy transport to the equator (Fig.~\ref{fig:ang_mom_yprof}c,d).
Meanwhile, the mean advection terms increase in magnitude compared to those in the \emph{Base} case and form prominent peaks at mid-latitudes (Fig.~\ref{fig:ang_mom_yprof}d).
The role of transients in this period is small relative to the mean and stationary contributions.
During the next period (80--200~days), while the day-side forcing reaches a steady-state (Fig.~\ref{fig:var_tseries}e), the \emph{Base} case has a weak and meridionally asymmetric acceleration of the zonal flow, indicating that the circulation structure is not yet stable (Fig.~\ref{fig:ang_mom_yprof}e, see also Supplemental Video 1).
The $m$ budget in the \emph{T0\_280} case, on the other hand, experiences a doubling of the magnitude of the mean transport terms and a decrease of the stationary term magnitude at the equator.
This reflects the fact that the balance is tipped in favor of the mean transport of the angular momentum to high latitudes instead of its transport by eddies to the equator (Fig.~\ref{fig:ang_mom_yprof}e).
Accordingly, the equatorial jet decelerates, while the pair of mid-latitude jets accelerates.
By the end of this period (at $\approx$200~days), the equatorial jet in the \emph{Base} experiment increases to its steady state level (Fig.~\ref{fig:crest_lon_eq_jet}a), locking in a resonance with the stationary wave pattern.
This happens as the eastward flow approaches the phase velocity of the wavenumber 1 Rossby wave mode (with the opposite sign), which in our simulations is close to \SI{80}{\m\per\s} estimated according to \citet{WangYang20_phase}.
As this flow speed approaches this threshold, the free Rossby mode becomes stationary relative to the heating in the substellar region and amplifies in magnitude.
Dampened by friction, the wave amplification reaches its maximum when the zonal wind is equal to the Rossby wave speed, which can be thought of as a resonance \citep{Arnold12_abrupt}.
Note that the Kelvin wave speed is much higher and directed opposite to the mean flow in our simulations, so a resonance with the Kelvin wave is not relevant \citep{WangYang20_phase}.
Fig.~\ref{fig:ang_mom_yprof}g and h show contributions to the $m$ budget from each of the terms in Eq.~\ref{eq:aam_main} at equilibrium (beyond 250~days).
The total change of the angular momentum is close to zero, as indicated by the solid black curves, and the mean and eddy terms largely cancel each other out.
In the \emph{T0\_280} case, the shape and the magnitude of the $m$ budget terms remains similar to those in the previous time period, only intensifying the angular momentum transport from the equator by the mean circulation (Fig.~\ref{fig:ang_mom_yprof}h).
Meanwhile, the same mean transport term in the \emph{Base} case decreases substantially, approaching zero at mid-latitudes.
This term's negative values at low latitudes are balanced by the positive stationary eddy term.
Evidently, the stationary term continues to transport angular momentum equatorward, drawing it from high latitudes where it is replenished by the transient eddy term (Fig.~\ref{fig:ang_mom_yprof}g).
The horizontal stationary eddy flux of angular momentum converges in the upper troposphere and diverges in the mid-troposphere, resulting in positive and negative contributions to the momentum budget, respectively (not shown).
The redistribution of angular momentum from the upper layers to the deep layers is performed by the vertical component of the stationary eddy term.
Supporting these findings, the same pattern of eddy acceleration was associated with equatorial superrotation in previous studies of tidally locked planets, assuming various atmospheric conditions and various degree of model complexity \citep[e.g.][]{Tsai14_three-dimensional,Showman15_three-dimensional,HammondPierrehumbert18_wave-mean,Debras20_acceleration,Hammond20_equatorial}.
To sum up, the initial evolution of the SJ and DJ regimes is not monotonic and is driven by a combination of mean overturning circulation, concomitant with higher baroclinicity at mid-latitudes, and zonally asymmetric planetary-scale forcing (with a maximum in the substellar region) due to the planet's synchronous rotation.
Both processes compete in our simulations, and their relative strength during the first 100--200 days determines the trajectory leading to one distinct regime or another.
In the \emph{Base} case, which eventually settles on the SJ regime, the acceleration of the equatorial superrotation is slow and steady, because of a weaker day side radiative forcing, which in turn is damped by the relatively strong longwave cooling due to the high concentration of water vapor.
Nevertheless, the SJ regime is eventually realized, as the equatorial eastward jet reaches the Rossby wave speed, indicating a wave-jet resonance.
The resonance-amplified Rossby wave maintains an excess of angular momentum at the equator, i.e. superrotation.
The \emph{T0\_280} case, while initially developing a strong equatorial jet reminiscent of the SJ regime, experiences a transition to the DJ regime after about 80 days.
The initial equatorial jet acceleration may be attributed to the day side forcing being stronger than that in the \emph{Base} case, while the subsequent transition to the DJ regime is driven by enhanced poleward fluxes of angular momentum due to the mean flow.
It is difficult to pinpoint the root cause for the regime bifurcation in a complex GCM such as the UM.
To explain why a more gradual jet acceleration leads to a SJ regime consistent with a Rossby wave resonance (in the \emph{Base} case), while a more rapid jet acceleration leads to a DJ regime (in the \emph{T0\_280} case), one would likely need a more idealized GCM with an option to prescribe forcing and emulate the regime evolution shown here in a more controlled environment.
The difference in the stationary wave pattern between the regimes is also associated with the position of the night-side cyclonic gyres with cold surface temperatures underneath.
The time evolution of the night-side temperature minima is discussed in detail in the next section (Sec.~\ref{sec:results_spinup_cold_traps}).
Further description of the steady state climate in the SJ and DJ regimes, is given in Sec.~\ref{sec:results_mean_state}, and their imprint in the transmission spectrum --- in Sec.~\ref{sec:results_synthobs}.
\subsection{The night side surface temperature evolution in the two regimes}
\label{sec:results_spinup_cold_traps}
\begin{figure*}
\includegraphics[width=\textwidth]{ch111_spinup__base_sens-t280k__wvp_cwp_wvre_lw_sfc_cre_lw_sfc_t_sfc__cold_traps.png}
\caption{Time series of diagnostics for the night-side cold traps, defined as the region bounded by \ang{45} and \ang{55} in the latitude and \ang{160}--\ang{140}W in the longitude. The panels for the (left) SJ and (right) DJ regime show: (purple, solid) water vapor path in \si{\kg\per\m\squared}, (purple, dashed) cloud water path in \SI{10}{\kg\per\m\squared}, (red, solid) water vapor radiative effect WVRE\textsubscript{LW}\textsuperscript{sfc} in \si{\watt\per\m\squared}, (red, dashed) cloud radiative effect CRE\textsubscript{LW}\textsuperscript{sfc} in \si{\watt\per\m\squared}, and (blue) surface temperature in \si{\K}. The WVRE\textsubscript{LW}\textsuperscript{sfc} is defined as the difference between the radiative fluxes at the surface calculated with and without the water vapor opacity. Likewise, the CRE\textsubscript{LW}\textsuperscript{sfc} is defined as the difference between the ``clear-sky'' and ``cloudy'' radiative fluxes at the surface.\label{fig:cold_traps_wvre_cre}}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{ch111_spinup__base_sens-t280k__dt_sfc_dt_sfc_net_down_lw_sfc_shf_sfc_lhf_sfc_down_lw__cold_traps.png}
\caption{Time series of diagnostics for the night-side cold traps, defined as the region bounded by \ang{45} and \ang{55} in the latitude and \ang{160}--\ang{140}W in the longitude. The panels for the (left) SJ and (right) DJ regime show: (orange) downward longwave radiation flux, (green, solid) net longwave radiation flux, (green, dashed) sensible heat flux in \si{\watt\per\m\squared}, (green, dotted) latent heat flux, and (blue) time derivative of the surface temperature in \si{\K\per\day}. \label{fig:cold_traps_fluxes}}
\end{figure*}
The evolution of the atmospheric circulation during the spin-up period causes a substantial decrease of the temperature and humidity on the night side.
Most strikingly, the night-side average surface temperature in the SJ simulation decreases by \SI{40}{\K}, while its minimum temperature drops by almost \SI{60}{\K} (see the blue curve in Fig.~\ref{fig:cold_traps_wvre_cre}).
This change has been noted in our previous work \citep{Sergeev20_atmospheric} and is investigated in more detail in this section, focusing on the initial period of the simulations.
We present the analysis for the night-side cold traps, defined here as the coldest regions of the night side of the planet.
This region is bounded by \ang{45} and \ang{55} in the latitude and \ang{160}--\ang{140}W in the longitude in our simulations.
The time series of the surface temperature in the night-side cold traps aligns well with the time series of global circulation diagnostics such as the wave crest shift and equatorial jet acceleration (cf. Fig.~\ref{fig:crest_lon_eq_jet} and \ref{fig:cold_traps_wvre_cre}).
After the rapid cooling from the initial warm state, the night-side surface reaches \SI{\approx 236}{\K} in both cases.
The temperature in the DJ case further decreases by a few degrees but stays close to this value throughout the simulation (Fig.~\ref{fig:cold_traps_wvre_cre}b).
In the SJ case however, as the circulation regime develops the strong equatorial superrotation and stationary waves (Fig.~\ref{fig:crest_lon_eq_jet}a), the temperature in the cold traps falls by almost \SI{60}{\K} (Fig.~\ref{fig:cold_traps_wvre_cre}a), fluctuating around \SI{\approx 178}{\K} for the remainder of the simulation (see also Fig.~\ref{fig:t_sfc_wvp_cwp}a).
The night-side surface temperature is dictated mostly by the thermal radiation emitted by the atmosphere to the surface, because there is no incident stellar radiation and no dynamic ocean in our setup.
This is confirmed by the time series of energy fluxes shown in Fig.~\ref{fig:cold_traps_fluxes}.
Turbulent heat fluxes are non-zero, but still an order of magnitude smaller than the longwave radiation flux, suppressed by the near-surface temperature inversion \citep{Joshi20_earth}.
Fig.~\ref{fig:cold_traps_fluxes} demonstrates that the downward longwave radiation (orange curve) is a precursor of the surface temperature in the cold traps.
Its substantial decrease (by \SI{> 100}{\watt\per\m\squared}) after about 150 days of the SJ simulation corresponds to the fall in temperature (blue curve in the negative).
The ability of the atmosphere to radiate heat is controlled by its temperature $T_\text{a}$ and emissivity $\epsilon_\text{a}$ \citep[see discussion in e.g.][]{Lewis18_influence}.
The latter is controlled by the amount of water vapor and cloud condensate in the atmosphere, which is shown in Fig.~\ref{fig:cold_traps_wvre_cre} as the water vapor path, i.e. the mass-weighted vertical integral of the water vapor in the atmosphere.
In both SJ and DJ cases, the model simulation starts from a dry state, which is far from the global equilibrium.
This causes an initial spike in the water vapor path (solid purple curves in Fig.~\ref{fig:cold_traps_wvre_cre}).
Subsequently, the water content in the night-side cold traps decreases, dropping in the SJ case to \SI{\approx 0.1}{\kg\per\m\squared}, but remaining an order of magnitude higher in the DJ case, at \SI{\approx 1.1}{\kg\per\m\squared}.
In the SJ case, as the night-side atmosphere becomes drier, it is less able to radiate heat, causing the decrease in the longwave flux received by the surface (orange curve in Fig.~\ref{fig:cold_traps_fluxes}a), which cools as a result.
To show that the effect of water vapor is larger than that of condensed water (clouds), their contributions to the longwave radiative effect near the planet's surface are also plotted in Fig.~\ref{fig:cold_traps_wvre_cre} (red curves).
Following \citet{Eager20_implications}, the radiative effects of water vapor and clouds are isolated using an additional ``diagnostic'' radiative transfer calculation, which does not affect the simulation itself.
On every time step, these additional calculations omit the opacity of water vapor or clouds and are then compared to the ``cloudy'' calculation.
Their difference (``clear-sky'' minus ``cloudy'') is referred to as the cloud radiative effect (CRE).
The more negative the values of the radiative effect in Fig.~\ref{fig:cold_traps_wvre_cre} are, the more important the contribution of the water vapor or clouds is.
Overall, the radiative effect of water vapor is substantially stronger than that of clouds: in the SJ case their time-average values in the second half of the spin-up period are \SI{-28.3}{\watt\per\m\squared} and \SI{-6.7}{\watt\per\m\squared}, respectively; in the DJ case they are \SI{-43.5}{\watt\per\m\squared} and \SI{-30.9}{\watt\per\m\squared}.
The magnitude of the water vapor radiative effect drops substantially in the SJ case, compared to its initial values or those in DJ, which is a direct consequence of the drying of the night-side cold trap regions (and the night side as a whole).
This decrease of the water vapor content on the night side in the SJ simulation is caused by the reduced transport of warm and moist air from the day to the night side.
For the steady state, this was shown previously by \citet{Sergeev20_atmospheric}: the moist static energy flux divergence in the SJ-like regime (their ``\emph{MassFlux}'' case) was smaller than that in the DJ-like regime (their ``\emph{Adjust}'' case).
\subsection{Steady state of the two circulation regimes}
\label{sec:results_mean_state}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_mean__base_sens-t280k__temp_300hpa_winds_ghgt_dev_map_500hpa.png}
\caption{Steady state atmospheric circulation in the (left) \emph{Base} (SJ regime) and (right) \emph{T0\_280} (DJ regime) simulations.
The panels show (a, b) \SI{500}{\hecto\pascal} temperature (shading, \si{\K}) with \SI{300}{\hecto\pascal} wind vectors, (c, d) \SI{300}{\hecto\pascal} eddy geopotential height (shading, \si{\m}) and eddy wind vectors.
The cyan lines in the bottom panels show the longitude of the planetary wave crest, defined as the maximum of the geopotential height anomaly.
Thin cyan lines show the 50-day mean longitude for several time periods of the steady state climate, while the thick cyan line shows the overall time mean longitude.
The geopotential height anomaly is defined as the deviation from the zonal mean of the height of the \SI{300}{\hecto\pascal} isobaric surface.
\label{fig:mean_circ}}
\end{figure*}
Focusing on the mature stage of the \emph{Base} and \emph{T0\_280} simulations, we now describe the steady state of the SJ and DJ regimes, respectively.
Model output averaged over the last 1000 days of the simulations is used in this section.
We confirm that the regimes are well-defined and have distinct features in the spatial distribution of the key climate diagnostics: surface and air temperature, total column water vapor and cloud content.
Namely, the SJ regime is characterized by a larger day-night temperature contrast due to extremely cold and dry cloudless regions on the night side, while the DJ regime is characterized by a more zonally-oriented morphology of the wind circulation and temperature, reducing the day-night dichotomy of the planet's climate.
We show that the SJ and DJ regimes are similar to those found by \citet{Edson11_atmospheric} and \citet{Noda17_circulation} with respect to the mean tropospheric conditions.
The upper layers of the atmosphere also have notable differences in variables such as water vapor and cloud content, which has implications for the transmission spectrum of TRAPPIST-1e, as discussed in detail in Sec.~\ref{sec:results_synthobs}.
As summarized for all our simulations in Fig.~\ref{fig:all_sim}c,d, the dominant feature of the global tropospheric circulation in both regimes is prograde (eastward) wind, similar to many previous studies, for both abstract \citep[e.g.][]{Edson11_atmospheric,Carone14_connecting,Haqq-Misra18_demarcating} and specific \citep[e.g.][]{Turbet16_habitability,Boutle17_exploring,Fauchez19_impact} tidally locked planetary configurations.
This is also demonstrated by the maps of wind velocity in Fig.~\ref{fig:mean_circ}a and b.
At the equator, for both regimes the atmosphere is superrotating (local maximum of mean angular momentum, see Eq.~\ref{eq:s_def}), though in the SJ regime it is a dominant feature of the circulation, while in the DJ regime it is weaker than the two eastward jets in the mid-latitudes.
The key difference between the two circulation patterns in their steady state is apparent in the location and amplitude of geopotential height anomalies shown in Fig.~\ref{fig:mean_circ}c,d.
The average longitude of the geopotential maxima represents a wave crest and is marked by the cyan vertical lines (their corresponding time evolution is tracked by the cyan curves in Fig.~\ref{fig:crest_lon_eq_jet}).
In the SJ regime, the geopotential maximum (anticyclone) is to the east of the substellar point, while a pair of minima occupy the night side and correspond to cyclonic gyres (Fig.~\ref{fig:mean_circ}c).
This pattern corresponds to an equatorial Rossby wave, analogous to those generated in Earth's tropics \citep{Vallis20_trouble}, but on a global scale (wavenumber 1) and stationary due to the planet's synchronous rotation \citep{PierrehumbertHammond19_atmospheric}.
This planetary-scale wave pattern is Doppler-shifted eastward by the zonal flow, as discussed in Sec.~\ref{sec:results_spinup_dynamics} \citep[see also][]{ShowmanPolvani11_equatorial}.
The wave is largely geostrophically balanced, as evidenced by Fig.~\ref{fig:mean_circ}c,d because the stationary eddy wind vectors are aligned with geopotential height isolines.
At the equator, the geopotential height in the SJ regime also has a prominent planetary-scale perturbation, which corresponds to an equatorial Kelvin wave \citep[e.g.][]{Debras20_acceleration,WangYang20_phase}.
The superposition of Rossby and Kelvin waves is identical to that obtained in shallow water models of exoplanetary atmospheres without the background flow \citep[e.g.][]{HammondPierrehumbert18_wave-mean,WangYang20_phase}.
The temperature field for the \emph{Base} simulation at high latitudes has a weak gradient from equator to pole, but a strong gradient between the day and night sides (shading in Fig.~\ref{fig:mean_circ}a).
This confirms that the SJ regime is less affected by the extratropical baroclinic instability than the DJ regime (Fig.~\ref{fig:var_tseries}f).
In the DJ regime (the \emph{T0\_280} case), the geopotential height pattern is not shifted by the strong superrotation and so the Rossby wave crest is at the western terminator (\ang{-90} longitude) while its trough straddles the eastern terminator (+\ang{90} longitude, Fig.~\ref{fig:mean_circ}d).
With no wave-jet resonance, the geopotential anomalies are also weaker and located closer to poles, while the height perturbation at the equator is small \citep[see e.g.][]{WangYang20_phase}.
In contrast to the SJ regime, the temperature map is dominated by the meridional gradient instead of the zonal, or day-night, gradient (Fig.~\ref{fig:mean_circ}b, see also the x-axis in Fig.~\ref{fig:all_sim}b).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_mean__base_sens-t280k__rotdiv_300hpa.png}
\caption{Helmholtz decomposition of the horizontal wind at \SI{300}{\hecto\pascal} in the (left) \emph{Base} (SJ regime) and (right) \emph{T0\_280} (DJ regime) simulations, corresponding to the wind field shown in Fig.~\ref{fig:mean_circ}a,b.
The panels show (a, b) the zonal mean rotational component, (c, d) the eddy rotational component, (e, f) the divergent component. Note the different scaling of the each of the components.
Also shown is the upward wind velocity (shading, \si{\m\per\s}).
\label{fig:rotdiv}}
\end{figure*}
The full 3D structure of the two circulation regimes can be further elucidated by decomposing the wind field into its rotational and divergent components \citep{HammondLewis21_rotational}.
Fig.~\ref{fig:rotdiv} shows this for the \SI{300}{\hecto\pascal} level.
The dominant eastward jets are immediately revealed by taking the zonal average of the rotational flow: a single equatorial jet in the \emph{Base} case and two mid-latitude jets in the \emph{T0\_280} case (Fig.~\ref{fig:rotdiv}a,b).
The eddy component of the rotational flow (Fig.~\ref{fig:rotdiv}c,d) corresponds to the stationary wave pattern (Fig.~\ref{fig:mean_circ}c,d).
The divergent component of the wind flow has a smaller magnitude relative to the rotational wind, but together with the contours of vertical velocity in Fig.~\ref{fig:rotdiv}e,f, it clearly shows the day-night overturning circulation.
The divergent component is notably weaker in the SJ regime (Fig.~\ref{fig:rotdiv}e) and stronger in the DJ regime (Fig.~\ref{fig:rotdiv}f).
The differences in the rotational and divergent components between our simulations are analogous to those found in the THAI results \citep{Turbet22_thai,Sergeev22_thai}, confirming that it is one of the characteristic features of the two regimes.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_mean__base_sens-t280k__tmm_rot_div_flux_mse__tot_rot.png}
\caption{The steady state moist static energy (MSE) budget for the (a) \emph{Base} (SJ regime) and (b) \emph{T0\_280} (DJ regime) simulations.
See text for details.
\label{fig:mse_rotdiv}}
\end{figure*}
These differences also result in different relative contributions of the rotational and divergent components of the circulation to the energy transport from the day side to the night side.
We assess this by calculating the moist static energy (MSE) flux divergence for each of the components and show the results in Fig.~\ref{fig:mse_rotdiv}.
MSE is defined as
\begin{equation}
h = \underbrace{c_p T + gz}_\text{Dry} + \underbrace{L_v q}_\text{Latent},
\end{equation}
where $c_p$ is the heat capacity at constant pressure, $T$ is temperature, $g$ is the acceleration due to gravity, $z$ is height, $L_v$ is the latent heat of vaporization and $q$ is the water vapor content.
Column-integrated, the divergence of the MSE flux is equal to the total local heating, as expressed by
\begin{equation}
\langle\nabla\cdot h\vec{u}\rangle + F^{net}_{TOA} = 0,
\end{equation}
where the angle brackets denote a mass-weighted vertical integral, $F^{\mathrm{net}}_{\mathrm{TOA}}$ is the top of the atmosphere net energy flux and $\vec{u}$ is the horizontal wind vector, which can be taken as a rotational or divergent component of the total wind field.
Fig.~\ref{fig:mse_rotdiv}a shows that in the SJ regime the surplus of energy on the day side is redistributed roughly equally by the divergent and rotational components of the flow.
Qualitatively, the divergent component tends to transport MSE from the eastern hemisphere of the planet (to the east of the substellar longitude) to its western hemisphere.
It is balanced by the rotational (jet plus eddy) part, which takes MSE from the western hemisphere and deposits it to the east.
Note that despite the partial cancellation of the jet and eddy components of the rotational flow, its magnitude is still larger than that presented in \citet{HammondLewis21_rotational} for a terrestrial planet case.
This is likely due to the assumption of a weak temperature gradient regime in \citet{HammondLewis21_rotational}, which appears to be less applicable in the \emph{Base} simulation (Fig.~\ref{fig:mse_rotdiv}a).
Another likely reason for the discrepancy is the inter-GCM differences in the boundary layer scheme between our studies and warrants further investigation.
The MSE budget for the DJ regime (Fig.~\ref{fig:mse_rotdiv}b) is similar to that in \citet{HammondLewis21_rotational}, despite the circulation pattern in that study being closer to our SJ regime.
The MSE flux divergence is predominantly due to the divergent component of the flow, while the individual rotational components largely cancel out and make the total rotational MSE flux divergence close to zero.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ch111_mean__base_sens-t280k__t_sfc_10m_winds_wvp_cwp.png}
\caption{Steady state thermodynamic conditions in the (left) \emph{Base} (SJ regime) and (right) \emph{T0\_280} (DJ regime) simulations.
The panels show (a, b) surface temperature (shading, \si{\K}) with \SI{10}{\m} wind vectors, (c, d) water vapor path (shading, \si{\kg\per\m\squared}), and (e, f) cloud water path (shading, \si{\kg\per\m\squared}).
\label{fig:t_sfc_wvp_cwp}}
\end{figure*}
We finish the description of the mean climate by briefly discussing the thermodynamic conditions in the SJ and DJ regimes.
The surface temperature has a spatial distribution similar to that of the mid-tropospheric temperature, with a larger day-night gradient in the SJ regime than in DJ (cf. Fig.~\ref{fig:t_sfc_wvp_cwp}a,b and Fig.~\ref{fig:mean_circ}a,b).
The near surface wind vectors shown in Fig.~\ref{fig:t_sfc_wvp_cwp}a,b demonstrate the region of convergent flow towards the substellar point, which is the lower branch of the overturning circulation shown in Fig.~\ref{fig:rotdiv}e,f.
In the SJ regime, the day side's surface attains a maximum temperature of \SI{287}{\K}, while the minimum temperature is in the night-side cold traps, which are aligned with the cyclonic gyres (Fig.~\ref{fig:mean_circ}c,d) and discussed in more detail in Sec.~\ref{sec:results_spinup_cold_traps}.
In the DJ regime, the surface temperature maximum is about \SI{5}{\K} lower than that in the SJ regime, but the minimum is \SI{29}{\K} higher (Fig.~\ref{fig:t_sfc_wvp_cwp}b).
As a result, the average surface temperatures for the SJ and DJ regime are \SI{230}{\K} and \SI{237}{\K}, respectively.
The surface isotherms and the wind convergence have distinct shapes in each of the regimes: they are broadly oriented zonally in the SJ case and meridionally in the DJ case.
This is also true for the rest of our sensitivity experiments (not shown).
Despite the substantially lower surface temperature minimum in the SJ regime than in the DJ regime, the SJ climate stays temperate and does not reach the condition for a potential atmospheric collapse.
Namely, the lowest temperature remains higher than the \ce{CO2} condensation point for a 1 bar atmosphere, so this species is expected to stay in the gaseous phase throughout the simulation \citep[e.g.][]{Turbet18_modeling}.
The mean surface conditions are such that the substellar region retains the temperature above the freezing point of seawater (Fig.~\ref{fig:t_sfc_wvp_cwp}a,b) and is able to maintain the water cycle on the planet (with the caveat of the globally uniform ocean surface in our setup).
The total area of the ice-free surface (with a temperature above the freezing point of seawater), a crude metric of planetary habitability, is similar for both regimes ($\approx 21$ and \SI{18}{\percent} in the SJ and DJ cases, respectively).
This is close to the estimates by other GCMs in the TRAPPIST-1e simulations with a nitrogen-dominated 1 bar atmosphere \citep[between 20 and \SI{24}{\percent}, see][for more details]{Sergeev22_thai}.
The total column water vapor (water vapor path) broadly mirrors the surface temperature map.
The driest areas clearly match the coldest areas of the surface in the SJ case (Fig.~\ref{fig:t_sfc_wvp_cwp}c), and the day-night asymmetry is overall more pronounced than that in the DJ case (Fig.~\ref{fig:t_sfc_wvp_cwp}d).
The absolute values of water vapor path reach 26 and \SI{15}{\kg\per\m\squared} in the SJ and DJ cases, respectively.
While the SJ case has overall more water vapor in the atmosphere, its driest regions are an order of magnitude drier than those in the DJ case.
Fig.~\ref{fig:t_sfc_wvp_cwp}e,f show the total column cloud condensate (cloud water path, including ice and liquid water).
Its absolute values are rather similar across the two regimes, which is dictated by the same cloud parameterization used in all our simulations --- unlike the inter-regime discrepancy in the THAI Hab~1 simulations \citep[which was due to different parameterizations in different GCMs, see][]{Sergeev22_thai}.
The spatial distribution of the cloud water path is different between the regimes, especially on the night side and at the terminators, which imprints on the transmission spectra (see Sec.~\ref{sec:results_synthobs}).
Our steady state results thus demonstrate that in 3D GCM simulations of TRAPPIST-1e there can exist two well-defined climates with different spatial distribution of winds, temperature, and moisture.
This further confirms one of the major findings of the THAI intercomparison project \citep[e.g.][]{Turbet22_thai}, proving that even with the same planetary setup and even in the same GCM, the circulation can settle in two distinct regimes, SJ and DJ.
Circulation patterns similar to the SJ regime have been reported in previous studies at various degrees of GCM complexity and for various terrestrial atmospheres \citep[e.g.][]{Edson11_atmospheric,Carone15_connecting,Noda17_circulation,Haqq-Misra18_demarcating}.
For example, it resembles the circulation obtained at an intermediate range of planetary rotation rate in \citet{Edson11_atmospheric}.
Later it was also found in the idealized experiments of \citet{Noda17_circulation}, who labeled this regime as ``Type II''.
In their setup, this regime developed at the rotation period roughly between 5 and 20 Earth days --- comparable to the rotation rate of TRAPPIST-1e (Table~\ref{tab:planet}).
\citet{Noda17_circulation} likewise attribute the emergence of this regime to the resonant excitation of the planetary-scale stationary waves seen in our \emph{Base} simulation.
Our SJ regime also corresponds to the ``Rhines rotator'' circulation regime in \citet{Haqq-Misra18_demarcating}.
The DJ regime was also identified by \citet{Edson11_atmospheric}, \citet{Noda17_circulation}, and \citet{Haqq-Misra18_demarcating}, in the experiments with the planetary rotation period smaller than 1--4 Earth days.
\citet{Noda17_circulation} labeled this circulation pattern as ``Type IV'' and identified the flow features highly similar to those in the \emph{T0\_280} case here.
The similarity extends even to the precipitation field (as a proxy for convective activity), oriented more zonally than that in the SJ regime (Fig.~\ref{fig:t_sfc_wvp_cwp}e,f).
The key difference between the studies mentioned above and our study is that they typically define different circulation regimes by varying planetary or stellar parameters, such as the planet's rotation rate, over a large range of values; while our study focuses on one specific exoplanet.
The regime bistability in our simulations could be further investigated using 3D GCMs for example by running a model ensemble with slightly different initial conditions or with perturbed parameters in sub-grid parameterizations.
It is also pertinent to extend our study to other exoplanetary atmospheres that may be susceptible to regime bistability explored here for TRAPPIST-1e.
This should help to narrow observational constraints of atmospheric dynamics on rocky exoplanets in general.
\subsection{Implications for observations}
\label{sec:results_synthobs}
\subsubsection{Terminator-mean transmission spectra}
\begin{figure*}
\includegraphics[width=\textwidth]{ch111_synthobs__base_sens-t280k__cloudy_clear.png}
\caption{Simulated transmission spectra for the (blue) SJ and (orange) DJ cases. The transmission spectra are calculated using fluxes (a) with and (b) without the effect of clouds included. Panel c shows the difference, SJ minus DJ, for (solid lines) cloudy and (dashed lines) clear-sky calculations. \label{fig:synthobs}}
\end{figure*}
Atmospheric characterization of transiting terrestrial exoplanets is becoming feasible with powerful new observational facilities such as the James Webb Space Telescope (JWST), successfully launched in December 2021.
TRAPPIST-1e is the most promising target known so far for such studies \citep{Fauchez19_impact,Suissa20_dim}, mostly thanks to the short orbital period of the planet and the small size of its host star, an ultra-cool M8V dwarf.
It has been shown that inter-model differences, namely in the amount of clouds at the terminator, affect the number of transits required for a confident detection of atmospheric features \citep{Fauchez22_thai}.
Here, we test if the distinct circulation regimes with their water vapor and cloud differences have a detectable imprint in a synthetic transmission spectrum.
We use the same two simulations as before, \emph{Base} and \emph{T0\_280}, corresponding to the SJ and DJ regimes, respectively.
Synthetic transmission spectra are computed natively within the radiation scheme of the UM (SOCRATES), once a day over 61 Earth days (10 orbits) during the steady state phase of the simulation (Sec.~\ref{sec:method_synthobs}).
The time mean and typical time variability ($\pm$ standard deviation) for both SJ and DJ regimes are shown in Fig.~\ref{fig:synthobs}.
The most prominent peaks correspond to the \ce{CO2} absorption bands at 2.7, 4.3 and \SI{15}{\micro\m}.
These peaks are a robust feature of our simulations and are largely unaffected by the presence of clouds, as the difference curve shows in Fig.~\ref{fig:synthobs}c.
The same result has been obtained in other 3D GCMs \citep{Fauchez22_thai} and is explained by the fact that even above the cloud deck there is enough \ce{CO2} to saturate the absorption lines.
While small relative to absorption peaks, differences between the SJ and DJ regimes are consistent in the continuum level, which is higher in DJ case across most of the spectrum between 0.6 and \SI{20}{\micro\m} (Fig.~\ref{fig:synthobs}c).
To explain this, we plot the time-mean vertical profiles of cloud content at the terminator in Fig.~\ref{fig:term_diff_climate}.
The profiles reveal that for the DJ case clouds tend to occur at lower altitudes than in the SJ case.
However, the mixing ratio of cloud ice (whose content dominates over cloud water) is noticeably larger in DJ case compared to the SJ case (Fig.~\ref{fig:term_diff_climate}d), and this leads to a slightly higher continuum level for the former, by \SI{\approx 1}{ppm}.
The water vapor band at \SI{6.3}{\micro\m}, on the other hand, is stronger by up to \SI{2.5}{ppm} in SJ than in the DJ case.
This can be attributed to a much lower water vapor content in the upper layers in the DJ regime compared to the SJ regime, as demonstrated by the vertical profiles in Fig.~\ref{fig:term_diff_climate}b.
This difference is similar to that between the LMD-G model and three other GCMs in the THAI project \citep{Fauchez22_thai}.
Note however, that the LMD-G model exhibits an SJ-like circulation regime in the THAI Hab~1 simulation --- so its low humidity in the upper atmosphere is likely a consequence of using a convection adjustment scheme \citep{Sergeev22_thai}.
\subsubsection{East-west terminator differences}
\label{sec:term_asymmetry}
\begin{figure*}
\includegraphics[width=\textwidth]{ch111_mean__vprof_term__base_sens-t280k__temp_sh_cld_liq_mf_cld_ice_mf.png}
\caption{Time mean vertical profiles at the (solid lines) western and (dashed lines) eastern terminators in the (blue) SJ and (orange) DJ. The variables shown are (a) air temperature, (b) specific humidity, (c) cloud liquid water mixing ratio, (d) cloud ice mixing ratio. \label{fig:term_diff_climate}}
\end{figure*}
Even though our simulations assume a uniform ocean surface covering the whole planet, conditions at the western and eastern terminators are not the same.
This is due to the zonal asymmetry in the global circulation introduced by a superposition of the stationary eddies and mean overturning circulation (see Sec.~\ref{sec:results_spinup_dynamics}).
It is important to take the asymmetry into account, because averaging the transmission spectrum over the full terminator may cancel out absorption features.
The terminator asymmetry in the transmission spectra was found to be non-negligible in previous studies, both for hotter gas giants \citep[e.g.][]{Line16_influence,Powell19_transit} and colder rocky planets \citep{SongYang21_asymmetry}.
Here we confirm that the circulation regime differences result in slightly different transmission spectra at the eastern and western terminators.
We find that the terminator asymmetry is roughly twice as large for the SJ regime compared to that in the DJ regime.
\begin{figure*}
\includegraphics[width=\textwidth]{ch111_synthobs__east_minus_west__base_sens-t280k__cloudy_clear_dry.png}
\caption{Differences in the transmission depth (ppm) between terminators (eastern minus western) in the (a) SJ and (b) DJ cases. Different lines show spectral differences assuming (pink) cloudy, (cyan) clear-sky, and (olive) dry atmosphere.\label{fig:term_diff_spectra}}
\end{figure*}
To determine what contributes to the asymmetry the most, we obtain differences in the spectra using all-sky (i.e. cloudy), clear-sky (i.e. cloudless), and dry (i.e. excluding the opacity of water vapor) radiation calculations, which are shown separately for the two regimes in Fig.~\ref{fig:term_diff_spectra}a and b, respectively.
The bulk of terminator asymmetry in both cases is due to water vapor at its absorption bands (e.g. at $\approx$1.4, 1.9, 2.7 and \SI{6.3}{\micro\m}).
This is confirmed by the curves corresponding to cloudy and clear-sky calculations being close together in these regions (compare the pink and cyan curves in Fig.~\ref{fig:term_diff_spectra}).
Between the \ce{H2O} absorption bands, terminator asymmetry is of the order of \SI{\approx 0.25}{ppm}, showing that the eastern terminator is cloudier and thus slightly elevates the continuum level.
The eastern terminator has more clouds, mostly in the form of ice crystals, than its western counterpart (compare solid and dashed lines in Fig.~\ref{fig:term_diff_climate}c,d).
Temperature asymmetry between terminators is different in the SJ compared to DJ case (Fig.~\ref{fig:term_diff_climate}a), but it contributes very little to the asymmetry in transmission depth, being only somewhat visible in the \ce{CO2} absorption bands, as well as in the Rayleigh scattering slope at the shorter wavelengths.
In the SJ regime, the terminator differences (eastern minus western) are overall mostly positive, which is similar to the fast-rotating simulation in \citet{SongYang21_asymmetry}, but have a much lower magnitude of about \SIrange{1}{2}{ppm} --- closer to the slow-rotating simulation in the same study.
The east-west terminator difference has the same sign as that found for cloudy simulations of hot Jupiters, but the magnitude is several orders smaller \citep[e.g.][]{Powell19_transit}, because the atmospheres of rocky planets (as assumed in our study) are much thinner than that of gas giants.
In the DJ regime, the asymmetry is less pronounced, because the overall circulation is more zonally symmetric (Fig.~\ref{fig:mean_circ}) and the differences between terminators are muted (Fig.~\ref{fig:term_diff_climate}b).
The most notable inter-regime difference in the transmission spectra is in the water absorption regions, the largest of which is centered at \SI{6.3}{\micro\m}.
This terminator asymmetry at \SI{6.3}{\micro\m} is positive in the SJ regime, meaning there is more water vapor at the eastern than at the western terminator (Fig.~\ref{fig:term_diff_spectra}a), but in DJ regime the opposite is true.
Note this is difficult to see in the mean vertical profiles in Fig.~\ref{fig:term_diff_climate}b, because the absolute and relative values of water vapor are very small in the upper atmosphere.
The differences in transmission spectra between our simulations, as well as the terminator asymmetry, are too small to be observable with instruments aboard the JWST \citep{May21_water}.
For example, a recent study by \citet{Rustamkulov22_analysis} reports 14 and \SI{10}{ppm} noise floors with 3-$\sigma$ and 1.7-$\sigma$ confidence levels, respectively, for JWST's near infrared spectrograph (NIRSpec) instrument.
The transmission depth differences in our study are even smaller than those simulated for a planet like TRAPPIST-1e by \citet{SongYang21_asymmetry}.
This is because ExoCAM used in that study tends to have higher cloud decks at the terminators compared to other commonly used GCMs, while the UM tends to have lower clouds \citep{Fauchez22_thai}.
Furthermore, ExoCAM was shown to produce consistently moister atmospheres than the UM for simulations of TRAPPIST-1e with an Earth-like atmospheric composition \citep{Sergeev22_thai,Wolf22_exocam}, thus amplifying the potential zonal asymmetry in transmission spectra.
Future telescopes may yet be precise enough to reveal the differences between circulation regimes, which could be elucidated by looking at transmission spectra for both individual and averaged terminator data.
More studies are needed for better understanding of the cloud microphysics affecting the atmospheric opacity on exoplanets like TRAPPIST-1e.
\section{Conclusions}
We investigated the bistability of the atmospheric circulation in the climate simulations of TRAPPIST-1e assuming aquaplanet surface conditions and an \ce{N2}-dominated \SI{1}{bar} moist atmosphere.
The key findings of this study are as follows.
\begin{enumerate}
\item The emerging atmospheric circulation can have two distinct regimes, either dominated by a strong equatorial eastward jet (the SJ regime) or by a pair of mid-latitude eastward jets (the DJ regime).
The SJ and DJ regimes correspond to the ``Type II'' and ``Type IV'' regimes in \citet{Noda17_circulation}, or to ``Rhines rotator'' and ``fast rotator'' in the terminology of \citet{Haqq-Misra18_demarcating}.
The states are well defined and there are practically no intermediate regimes, with respect to key climate diagnostics (e.g. the day-night temperature gradient).
\item In our simulations which of the regimes the climate enters is sensitive to several factors: a change in physical parameterization, a different surface boundary condition for the temperature, or different initial conditions.
However, the regime bistability is not merely an artifact of our GCM: similar circulation regimes were recently identified using other GCMs in both dry \citep{Turbet22_thai} and moist \citep{Sergeev22_thai} simulations of TRAPPIST-1e climate; as well as in earlier idealized studies \citep[e.g.][]{Edson11_atmospheric}.
An interesting outcome of our sensitivity simulations is the bistability due the initial conditions and specifically the initial temperature.
Namely, at certain moderate initial temperatures the DJ regime develops instead of the SJ regime as in the control simulation (started from \SI{300}{\K}).
This finding complements the studies by \citet{Thrastarson10_effects} and \citet{Cho15_sensitivity} who used a 3D GCM of a hot Jupiter with a simplified representation of boundary-layer friction and thermal forcing.
They found that the steady-state atmospheric circulation is sensitive to the initial conditions, unless a much stronger momentum and thermal drag is applied at the bottom of the atmosphere, thus damping any small-scale variability as was done in \citet{Liu13_atmospheric}.
We leave the sensitivity of the atmospheric circulation on TRAPPIST-1e to the initial wind profile for future studies.
\item Using two indicative simulations, one started from \SI{300}{\K} and another from \SI{280}{\K}, we analyze the evolution of the SJ and DJ regimes, respectively.
We show that the initial stage of the regime evolution depends on a fine balance between the zonally asymmetric heating due to the planet's synchronous rotation on the one hand and mean overturning circulation on the other.
The SJ regime appears to be weakly radiatively forced at the beginning due to a higher concentration of water vapor and thus stronger longwave cooling of the atmosphere compared to that in the early stages of the DJ regime.
Consequently, the nascent equatorial jet accelerates more gradually in the SJ case and is able to achieve resonance with the stationary Rossby wave, which in turn markedly amplifies, reinforcing the jet.
The DJ regime appears to be relatively strongly forced because of the lower initial temperature and thus lower water vapor concentration in the first tens of days; thus the nascent equatorial jet accelerates at a higher rate.
However, after about 80 days the circulation transitions to the two mid-latitude jets (i.e. the DJ pattern), which are associated with colder polar regions and thus a higher degree of baroclinicity.
As a result, the wave-jet resonance is not achieved.
\item The zonal angular momentum budget further supports these arguments.
In the initial stage of the regime evolution, the surplus of the angular momentum at the equator (i.e. superrotation) is provided by the stationary eddies, which are initially stronger in the DJ case.
At about day 80, the mean advection terms in the DJ case grow, deplete the zonal momentum at the equator and move it poleward, starting to actively accelerate the mid-latitude jets.
Meanwhile, the SJ regime matures via the stationary eddy contribution to the angular momentum budget at the equator, which intensifies as the stationary Rossby wave pattern resonates with the jet.
\item Having fully developed, the two circulation regimes each have a slightly different imprint on the transmission spectrum, though the differences are too small to be observable with the current technology.
The DJ regime has more clouds at the terminators, so its continuum level is higher than that for SJ over the most part of the analyzed wavelength range (\SIrange{0.6}{20}{\micro\m}).
The upper atmosphere water vapor content, on the other hand, is higher in the SJ regime, so the \ce{H2O} absorption, especially near \SI{6.3}{\micro\m} is higher.
Comparable to the inter-regime differences, there is also an asymmetry between eastern and western terminators, which is more pronounced in the SJ regime.
\end{enumerate}
It is clear from both this study that TRAPPIST-1e resides in a particularly sensitive position with respect to the circulation regime, and even small changes in the model setup can tip the circulation into one regime or another.
This exoplanet is one of the key targets for the upcoming JWST observational programs \citep{Gillon20_trappist1}, so understanding its atmospheric structure is imperative for the best use of observations.
We expect that our results could be applicable to other rocky exoplanets residing in a similar ``sweet spot'' of the planetary size and rotation period.
As indicated by earlier modeling studies of hypothetical planets, there are transition regions between well-defined circulation regimes for which a similar regime bistability and sensitivity to GCM setup can exist (\citealp{Edson11_atmospheric}; \citealp{Noda17_circulation}; see also Fig.~1 in \citealp{Carone18_stratosphere}).
Atmospheric circulation on TRAPPIST-1e appears to be particularly sensitive --- not only to the model choice \citep{Turbet22_thai,Sergeev22_thai}, but also to a small change in the initial conditions (this study).
Furthermore, different regimes can emerge not only for a nitrogen-dominated atmosphere, but for a \ce{CO2} atmosphere too, as noted in the THAI project.
With regards to atmospheric pressure, our preliminary experiments with the total pressure below 1~bar favor the SJ regime, while those with the pressure above 1~bar tend to favor the DJ regime, though a separate study is needed for a confident conclusion.
We have also conducted a series of dry simulations starting the UM from different initial temperatures, i.e. repeating the \emph{T0\_250}, \emph{T0\_260}, \emph{T0\_270}, \emph{T0\_280}, \emph{T0\_290}, and \emph{Base} simulations with a dry atmosphere.
The atmospheric circulation in all of these runs evolves into only one regime, namely SJ.
This result indicates that the bistability is driven primarily by moisture effects, at least in our model.
However, we would like to stress that different states can be obtained for the same planet (and the same initial setup) using different GCMs even in dry conditions \citep{Turbet22_thai}, which is important, even if it is not strictly a bifurcation.
Further work on the underlying dynamical mechanisms of the emergence of the two regimes is required, both for dry and moist setups.
While our study explores the emergence of the circulation regimes in depth and across a few sensitivity experiments, a wider modeling study is needed to outline what GCM configurations favor SJ or DJ regimes.
This has been initiated by the THAI project \citep{Fauchez20_thai_protocol}, but would benefit from expanding the parameter sweep wider, e.g. to other configurations with a non-Earth atmospheric composition.
Additionally, our recommendation for modeling atmospheres prone to regime bistability is to use initial condition and/or physical parameterization ensembles.
The surface boundary conditions are among the factors the regime is sensitive to in our simulations.
As shown by \citet{Lewis18_influence} and \citet{Salazar20_effect}, the presence of a continent on the day side of the planet affects the global circulation.
Such a perturbation to the model may favor one of the circulation regimes, depending on the size and thermodynamic properties of the continent.
Including a dynamic ocean will likely influence the regime bistability too by contributing to the heat transport between the day and night sides of the planet \citep[see e.g.][]{Hu14_role,DelGenio19_habitable}.
These avenues of research are left for the future.
Finally, our simulations are performed with time invariant stellar forcing.
The high sensitivity of the circulation regimes even to initial conditions indicates that the circulation may be prone to an abrupt transition if a temporary forcing is provided.
Such a forcing can be an influx of water vapor into the stratosphere in the aftermath of a large volcanic eruption \citep[e.g.][]{Loffler16_impact,Guzewich22_volcanic} or a series of eruptions \citep[e.g.][]{Joshi03_gcm}.
Another example is periodic change in stellar forcing due to flaring of the host star, which as an M-dwarf.
Performing experiments with periodic or transient stellar forcing \citep[as in e.g.][]{Chen21_persistence} may further elucidate the question of bistability of atmospheric circulation on TRAPPIST-1e.
\acknowledgments
The authors would like to thank two anonymous reviewers for their constructive comments, which helped to improve the manuscript.
Material produced using Met Office Software.
We acknowledge use of the Monsoon2 system, a collaborative facility supplied under the Joint Weather and Climate Research Programme, a strategic partnership between the Met Office and the Natural Environment Research Council.
This work was supported by a Science and Technology Facilities Council Consolidated Grant (\texttt{ST/R000395/1}), UKRI Future Leaders Fellowship (\texttt{MR/T040866/1}), and the Leverhulme Trust (\texttt{RPG-2020-82}).
NTL was supported by Science and Technology Facilities Council Grant \texttt{ST/S505638/1}.
IAB and JM acknowledge support of the Met Office Academic Partnership secondment.
\software{
The Met Office Unified Model is available for use under license; see \url{http://www.metoffice.gov.uk/research/modelling-systems/unified-model}.
Scripts to post-process and visualize the model data are available on GitHub: \url{https://github.com/dennissergeev/t1e_bistability_code} and are dependent on the following open-source Python libraries: \texttt{aeolus} \citep{aeolus}, \texttt{cmcrameri} \citep{Crameri20_misuse}, \texttt{iris} \citep{iris}, \texttt{jupyter} \citep{jupyter}, \texttt{matplotlib} \citep{matplotlib}, \texttt{numpy} \citep{numpy}, \texttt{windspharm} \citep{windspharm}.
}
|
2,877,628,090,226 | arxiv | \section{Introduction}
A major part of non-thermal emission from high energy astrophysical objects is almost always characterized by
the radiation from relativistic electrons moving in magnetic fields.
Usually it is interpreted in terms of the synchrotron radiation.
However, synchrotron approximation is not always valid, in particular when the magnetic fields are highly turbulent.
Electrons suffer from random accelerations and do not trace a helical trajectory.
In general, the radiation spectrum is characterized by the strength parameter
\begin{equation}
a= \lambda_{\mathrm{B}} \frac{e|B|}{mc^2},
\end{equation}
where $\lambda_{\mathrm{B}}$ is the typical scale of turbulent fields, $|B|$ is the mean value of the turbulent magnetic fields,
$e$ is the elementary charge,
$m$ is the mass of electron and $c$ is the speed of light (Reville \& Kirk 2010).
When $a\gg \gamma$, where $\gamma$ is the Lorentz factor of radiating electron,
the scale of turbulent fields is much larger than the Larmor radius $r_g\equiv \gamma m c^2/e|B|$,
and electrons move in an approximately uniform field, so that the synchrotron approximation is valid.
In contrast, when $a\ll1$, $\lambda_{\mathrm{B}}$ is much smaller than the scale $r_g/\gamma = \lambda_{\mathrm{B}}/a$ which
corresponds to the emission of the characteristic synchrotron frequency.
In this regime, electrons move approximately straightly, and jitter approximation
or the weak random field regime of diffusive synchrotron radiation (DSR) can be applied (Medvedev 2010, Fleishman and Urtiev 2010).
For $1\lesssim a \lesssim \gamma$, no simple approximation of the radiation spectrum has been known.
The standard model of Gamma Ray Bursts (GRB) is based on the synchrotron radiation
from accelerated electrons at the internal shocks.
The observational spectra of prompt emission of GRB can be well fitted by a broken power law spectrum which is called the Band function.
Around a third of GRBs show a spectrum in the low energy side harder than the synchrotron theory predicts.
To explain this, other radiation mechanisms are needed.
Medvedev examined relativistic collisionless shocks in relevance to internal shocks of GRB,
and noticed the generation of small scale turbulent magnetic fields near the shock front (Medvedev \& Loeb 1999).
Then he calculated analytically radiation spectrum from electrons moving in small scale turbulent magnetic fields,
to make a harder spectrum than the synchrotron radiation (Medvedev 2000).
However, he assumed that the strength parameter $a$ is much smaller than $1$
and that turbulent field is of one-dimensional structure,
which may be over simplified in general (Fleishman 2006).
Medvedev also calculated 3-dimensional structure assuming that
the turbulent field is highly anisotropic (Medvedev 2006).
He conclude that the harder spectrum is achieved in "head on" case,
and that in "edge on" case, the spectrum is softer than synchrotron radiation.
The spectral index depends on the angle $\theta$ between the particle velocity and shock normal with
hard spectrum obtained when $\theta \lesssim 10^{\circ}$ (Medvedev 2009).
Recently several particle-in-cell (PIC) simulations
of relativistic collisionless shocks have been performed to study the nature of turbulent magnetic fields
which are generated near the shock front (e.g., Frederiksen et al. 2004; Kato 2005; Chang et al. 2008;
Haugbolle 2010).
The characteristic scale of the magnetic fields is the order of skin depth
as predicted by the analysis of Weibel instability.
Then, the wavelength of turbulent magnetic field $\lambda_{\mathrm{B}}$ is described by using a coefficient $\kappa$ as
\begin{equation}
\lambda_{\mathrm{B}} = \kappa \frac{c}{\omega_{\mathrm{pe}} \Gamma_{\mathrm{int}}},
\end{equation}
where $\omega_{\mathrm{pe}}$ is the plasma frequency, and $\Gamma_{\mathrm{int}}$ is the relative Lorentz factor between colliding shells.
The energy conversion fraction into the magnetic fields
\begin{equation}
\varepsilon_{\mathrm{B}} = \frac{B^2/8\pi}{\Gamma_{\mathrm{int}} n m_{\mathrm{p}} c^2}
\end{equation}
is $10^{-3}-0.1$, where $B^2/8\pi$ is energy density of magnetic fields, and
$\Gamma_{\mathrm{int}} n m_{\mathrm{p}} c^2$ is the kinetic energy density of the shell.
The Lorentz factor of electrons is similar to $\Gamma_{\mathrm{int}}$,
and that $\kappa$ is typically $10$ from the result of PIC simulations,
the strength parameter $a$ can be estimated as
\begin{equation}
a = \sqrt{2}\kappa{\varepsilon_{\mathrm{B}}}^{1/2}\sqrt{\frac{m_{\mathrm{p}}}{{m_{\mathrm{e}}}}}
\sim O(1-10^2).
\end{equation}
Thus, the assumption $a\ll1$ on which jitter radiation and DSR weak random field regime
are based is not necessarily valid when we consider the radiation from the internal shock region of GRB.
Fleishman \& Urtiev (2010) calculated the radiation spectrum for $a>1$ using a statistical method,
but their treatment of the "small scale component" is somewhat arbitrary.
They introduced the critical wavelength $\lambda_{\mathrm{crit}}$ and called
components with $\lambda \leq \lambda_{\mathrm{crit}}$ the "small scale component",
where $\lambda_{\mathrm{crit}}$ obeys
the inequality
\begin{equation}
r_{\mathrm{g}} \ll \lambda_{\mathrm{crit}} \ll r_{\mathrm{g}}/\gamma
\label{crit}
\end{equation}
(Toptygin \& Fleishman 1987).
The inequality can be transformed to $1\ll \lambda_{\mathrm{crit}}e|B|/mc^2 \ll \gamma$,
so that the division through $\lambda_{\mathrm{crit}}$ may be
ambiguous when we calculate the radiation spectrum for $1<a<\gamma$.
The synthetic spectra from PIC simulations were calculated recently
(e.g., Hededal 2005; Sironi \& Spitkovsky 2009;
Frederiksen 2010; Reville \& Kirk 2010; Nishikawa et al 2011).
Althogh their magnetic fields are realistic and self-consistent,
it is inevitable that the fields are described by discrete cells in PIC simulation.
Reville \& Kirk (2010) developed an alternative method of calculation of radiation spectra that uses
the concept of photon formation length, which costs much shorter time than the first principle method
utilizing the Lienard-Wiechert potential.
In this paper, we rather use the first principle method to obtain the spectrum as exact as possible.
We adopt the field description method developed
by Giacalone \& Jokipii (1999) and used by Reville \& Kirk (2010).
We assume isotropic turbulent magnetic fields which have broader
power spectra $k_{\mathrm{max}}=100 \times k_{\mathrm{min}}$
and calculate the radiation spectra in the regime of $1<a<\gamma$.
In \S 2 we describe calculation method and numerical results.
In \S 3 we give a physical interpretation.
\section{Method of calculation}
Because we focus our attention on calculating radiation spectrum, we assume the static field with required properties of $a$,
and neglect the back reaction of radiating electrons to the magnetic field.
We solve the trajectory of electron accurately in each time step and calculate the radiation spectrum.
\subsection{Setting}
The isotropic turbulent field is generated by using the discrete
Fourier transform description as developed in Giacalone \& Jokipii (1999).
It is described as a superposition of \textit{N} Fourier modes,
each with a random phase, direction and polarization
\begin{equation}
\bm{B}(\bm{x}) = \sum_{n=1}^N A_n \exp\bigl\{i(\bm{k}_n \cdot \bm{x} + \beta_n)\bigr\}\hat{\xi}_n.
\end{equation}
Here, $A_n,\beta_n,\bm{k}_n \: \mathrm{and} \: \hat{\xi}_n$ are the amplitude, phase,
wave vector and polarization vector for the \textit{n}-th mode, respectively.
The polarization vector is determined by a single angle $0<\psi_n<2\pi$
\begin{equation}
\hat{\xi}_n = \cos{\psi_n}\bm{e_x^{\prime}} + i \sin{\psi_n}\bm{e_y^{\prime}},
\end{equation}
where $\bm{e_x^\prime}$ and $\bm{e_x^\prime}$ are unit vectors, orthogonal to $\bm{e_z^{\prime}} = \bm{k_n}/k_n$.
The amplitude of each mode is
\begin{equation}
A_n^2 = \sigma^2 G_n \Biggl[\sum_{n=1}^N G_n \Biggr]^{-1},
\end{equation}
where the variance $\sigma$ represents the amplitude of the turbulent field.
We use the following form for the power spectrum
\begin{equation}
G_n = \frac{4 \pi k_n^2 \Delta k_n}{1 + (k_n L_{\mathrm{c}})^{\alpha}},
\end{equation}
where $L_c$ is the correlation length of the field. Here, $\Delta k_n$ is chosen such that there is an equal spacing in logarithmic
$k-$space, over the finite interval $k_{\mathrm{min}} \leqq k \leqq k_{\rm{max}}$, where $k_{\rm{max}} = 100 \times k_{\rm{min}}$ and $N = 100$.
where $k_{\rm{min}} = 2\pi/L_{\mathrm{c}}$ and $\alpha = 11/3$.
We have no reliable constraint for value of $\alpha$ from GRB observation,
so we adopt the Kolmogorov turbulence $B^2(k) \propto k^{-5/3}$ tentatively,
where the power spectrum has a peak at $k_{\mathrm{min}}$.
Then we define the strength parameter using $\sigma$ and $k_{\rm{min}}$ as
\begin{equation}
a \equiv \frac{2\pi e \sigma}{mc^2k_{\mathrm{min}}}.
\end{equation}
We inject isotropically 32 monoenergetic electrons with $\gamma=10$ in the prescribed magnetic fields,
and solve the equation of motion
\begin {equation}
\gamma m_{\mathrm{e}} \dot{\bm{v}} = -e \bm{\beta}\times\bm{B}
\end{equation}
using 2nd order Runge-Kutta method.
We pursue the orbit of electrons up to the time $300 \times T_{\mathrm{g}}$ ,
where $T_{\mathrm{g}}$ is the gyro time $T_{\mathrm{g}} = 2\pi \gamma m c / e \sigma$.
We calculate radiation spectrum using acceleration $\dot{\bm{v}} = \dot{\bm{\beta}}c$.
The energy $dW$ emitted per unit solid angle $d\Omega$ (around the direction $\bm{n}$) and per unit frequency $d\omega$ to the direction $\bm{n}$ is computed as
\begin{equation}
\frac{dW}{d\omega d\Omega} = \frac{e^2}{4 \pi c^2}
\Bigl| \int^{\infty}_{-\infty} \:dt^{\prime} \frac{ \bm{n} \times \bigl[ (\bm{n} - \bm{\beta}) \times \dot{\bm{\beta}} \bigr] }
{(1 - \bm{\beta} \cdot \bm{n} )^2 }\exp\bigl\{{i\omega ( t^{\prime} - \frac{\bm{n} \cdot \bm{r}(t^{\prime})}{c})}\bigr\} \Bigr|^2,
\end{equation}
where $\bm{r}(t^{\prime})$ is the electron trajectory, $t^{\prime}$ is retarded time (Jackson 1999).
\subsection{Results}
First, we show the radiation spectrum for $a = 3$ in Figure \ref{a3}.
The frequency is normalized by the fundamental frequency $\omega_{\mathrm{g}} = e\sigma/(\gamma mc)$, and
the magnitude is arbitrarily scaled.
The jagged line is the calculated spectrum, while the straight line drawn in the low frequency region
is a line fitted to a power law spectrum.
The fitting is made in the range of $1-350\omega_{\mathrm{g}}$ and the spectral index turns out to be $0.44$.
The straight line drawn in the high frequency region shows a spectrum of $ \propto \omega^{-5/3}$
expected for diffusive synchrotron radiation for reference (Toptygin \& Fleishman 1987).
The spectrum is well described by a broken power law, and the spectral index of the low energy side is
harder than synchrotron theory predicts.
The peak frequency of this spectrum is located at around $ 10^3 \omega_{\mathrm{g}}$.
This frequency corresponds to approximately the typical frequency of synchrotron radiation
$\omega_{\mathrm{syn}} =3\gamma^2 e\sigma/2mc \sim 10^3 \omega_{\mathrm{g}}$, for $\gamma =10$.
Figure \ref{a5} shows the spectrum for $a = 5$.
The spectral shape changes from that of $a=3$ in both sides of the peak.
The spectrum of the low frequency side becomes a broken power law with a break around $10\omega_{\mathrm{g}}$,
above which the spectrum is fitted by a power law with an index of $0.33$,
as expected for synchrotron radiation, while below the break the index is $0.58$.
The high frequency side above the peak indicates an excess above a power law spectrum $dW/d\omega d\Omega \propto \omega^{-5/3}$.
It looks like an exponential cutoff.
The whole spectrum is described by a superposition of a synchrotron spectrum and a DSR broken power law spectrum.
This spectral shape is totally novel and is different from the one by described Fleishman (2010).
He reported that the spectrum is described by broken power law in the same range of $a$ as this work
($1<a<\gamma,\: a\sim 10^2 \:\mathrm{and}\: \gamma\sim 10^3$) (Fleishman 2010).
To confirm our inference we calculate the case of $a=7$, for three different values of $\alpha$, i.e.,
$\alpha = 14/3,\: 11/3 \: \mathrm{and}\: 8/3$ and the results are shown in Figure \ref{a7}.
The curved black line is a theoretical curve of synchrotron radiation, and three straight black lines are
$dW/d\omega d\Omega \propto \omega^{-2/3}, \omega^{-5/3} \: \mathrm{and} \: \omega^{-8/3}$ expected for DSR theory for reference.
The green line corresponding to $\alpha = 14/3$ reveals a clear exponential cutoff, and reveals DSR component in only the highest frequency region.
The power law index of this component in the highest frequency
region coincides with the expected value $-\alpha + 2 = -8/3$.
The red and blue lines correspond to $\alpha = 11/3 \: \mathrm{and}\: 8/3$, respectively.
They indicate a common feature to the green one.
\section{Interpretation}
We give a physical explanation of the spectra obtained in the previous section.
At first, we interpret the broken power law spectrum for $a=3$ by using DSR theory.
Next, we consider physical interpretation of the complex shape of radiation
spectrum for $a=5$ and $a=7$.
Finally, we compare our work with previous studies.
To begin with, we review the spectral feature of DSR based on the non-perturbative approach for $a<1$ (Fleishman 2006).
The typical spectrum takes a following form:
$dW/d\omega d\Omega \propto \omega^{1/2}$ in the low frequency region, $\propto \omega^{0}$ in the middle frequency region,
and $\propto \omega^{-\alpha + 2}$ in the high frequency region.
The low frequency part and the middle frequency part are separated
at $\omega_{\mathrm{LM}} \sim a \omega_{\mathrm{syn}}$.
This spectral break corresponds to the break of the straight orbit
approximation due to the effect of multiple scattering (Fleishman 2006).
On the other hand, the middle frequency part and the high frequency part are separated at
the typical frequency of jitter radiation $\omega_{\mathrm{MH}} = \omega_{\mathrm{jit}}$ which is estimated by using the method of virtual quanta as
$\omega_{\mathrm{jit}} \sim \gamma^2 k_{\mathrm{min}}c \sim a^{-1} \omega_{\mathrm{syn}}$
(Medvedev 2000, Rybicki \& Lightman 1979).
Then, for $a \sim 1$, $\omega_{\mathrm{LM}} \sim \omega_{\mathrm{MH}}$ is achieved and the middle region may vanish.
The spectrum for $a\sim1$ becomes a broken power law with only one break, which is located at roughly the synchrotron frequency $\omega_{\mathrm{syn}}$.
The power law index of low frequency side is $\sim 0.5$ (which is harder than synchrotron radiation), and that of high frequency side is $-\alpha +2= -5/3$.
Thus, the spectral feature for $a=3$ can be explained by an extrapolation of DSR non-perturbative approach for $a<1$,
if we consider that the middle frequency region is not conspicuous.
Although our spectral index $0.44$ slightly differs from $0.5$ for DSR,
this index is still harder than the synchrotron theory.
The situation $a\sim1$ can be achieved at the internal shock region of GRB,
so that this may be responsible for harder spectral index than synchrotron observed for some GRBs.
Next we interpret the spectral features for $a=5$ and 7.
The conceptual diagram of these spectra for $5\leq a<\gamma$ is depicted in Figure \ref{spece},
and a schematic picture of an electron trajectory is depicted in Figure \ref{sasie}.
We explain the appearance of another break in the low frequency range
seen at around $10\omega_{\mathrm{g}}$ in Figure \ref{a5}.
On the scale smaller than $\lambda_{\mathrm{B}}$, the electron motion may be approximated by a helical orbit,
while it is regarded as a randomly fluctuating trajectory when seen on scales larger than $\lambda_{\mathrm{B}}$.
Therefore, for the former scale, we can apply the synchrotron approximation to the emitted radiation.
The beaming cone corresponding to the frequency $\omega$ is given by
\begin{equation}
\theta_{\mathrm{cone}} = \frac{1}{\gamma}(\frac{3\omega_{\mathrm{syn}}}{\omega})^{1/3}
\label{rlcn}
\end{equation}
(Jackson 1999).
The deflection angle $\theta_0$ of the electron orbit during a time $\lambda_{\mathrm{B}}/c$ is estimated
to be $\theta_0 = a/\gamma$ from the condition
\begin{equation}
\frac{\gamma\lambda_{\mathrm{B}}}{a}\theta_0 = \lambda_{\mathrm{B}}
\end{equation}
as seen in Figure \ref{sasie}.
Thus, the synchrotron theory is applicable only for $\theta_{\mathrm{cone}} < \theta_0$, so that
the break frequency is determined by $\theta_0 = \theta_{\mathrm{cone}}$, and we obtain
\begin{equation}
\omega_{\mathrm{br}} \sim a^{-3}\omega_{\mathrm{syn}}.
\label{br}
\end{equation}
This break frequency is the same as obtained by Medvedev (Medvedev 2010).
We understand that as $a$ is larger, break frequency becomes lower, and when $a$ is comparable to $ \gamma$, $\omega_{\mathrm{br}}$
coincides with the fundamental frequency $e\sigma/\gamma mc$.
Next, we discuss on the high frequency radiation, which results from the electron
trajectory on scales smaller than $\lambda_{B}$.
The synchrotron theory applies between $\lambda_{\mathrm{B}}/a=r_{\mathrm{g}}/\gamma$
and $\lambda_{\mathrm{B}}$.
However, we should notice that electron motion suffers from acceleration
by magnetic turbulence on scales smaller than $\lambda_{\mathrm{B}}/a$.
The trajectory down to the smallest scale of $2\pi/k_{\mathrm{max}}$ is jittering,
which is attributed to higher wavenumber modes as seen in the zoom up of Figure \ref{sasie}.
If the field in this regime is relatively weak, i.e., $\alpha$ is relatively large (Figure \ref{a7}, green line: $\alpha = 14/3$),
the trajectory on the scale smaller than $\lambda_{\mathrm{B}}/a$ does not much deviate from a helical orbit.
In this case, radiation spectrum reveals an exponential cutoff, and a power law component appears only in the highest frequency region.
On the contrary, if the smaller scale field is relatively strong, as in the case of $\alpha = 8/3$ depicted in the blue line in Figure \ref{a7},
the power law component becomes predominant in the high frequency region, and the synchrotron exponential cutoff is smeared out.
The intersection frequency of curved black line and straight black lines at around $10^3\omega_{\mathrm{g}}$ in Figure \ref{a7}
corresponds to $\omega_{\mathrm{jit}}$ as seen in Figure \ref{spece}.
Since the intersection frequency is determined by $a$, the frequency where the power law component appears over the synchrotron cutoff is dependent on $\alpha$.
The excess from the theoretical curve in the middle frequency region in Figure \ref{a7}
may be explained by consideration of two effects.
One is the contribution of hidden DSR component, and the other is a range of synchrotron peak frequency
which is caused by a fluctuation of magnetic field intensity.
Fleishman reported that the spectrum for $1<a<\gamma$ and $3<\alpha<4$ becomes a broken power law (Fleishman \& Urtiev 2010).
Medvedev asserted that the high frequency region of the spectrum reveals an exponential cutoff
for $1<a<\gamma$ (Medvedev 2010).
Our result indicates that an exponential cutoff plus an extra power law component appears,
which is different from Fleishman's remark and from Medvedev's remark on the high frequency region.
On the other hand, similar spectra to ours have been reported in Fleishman (2005) and
Reville \& Kirk (2010) when a uniform field is added to turbulent field.
Because the high energy power law component arises from a turbulent spectrum over the wavenumber space,
this component does not exist when the small scale field
is excited only in a narrow range of wavenumber space.
Since the energy cascade of turbulent magnetic fields should exist at least to some degree,
we regard that the higher wavenumber modes naturally exist.
It depends on the set of parameters of $\sigma \: \mathrm{and} \: k_{\mathrm{max}}$
whether this high energy power law component can be seen or not.
If $ 2\pi e\sigma/mc^2k_{\mathrm{max}} > 1 $, this component will not be seen.
If the magnetic turbulence is excited by Weibel instability at the relativistic shocks,
it is not possible for $k_{\mathrm{\max}}$ to be much larger than $k_{\mathrm{\min}}$
because the wavelength of injection ($\lambda_{\mathrm{B}} = 2\pi/k_{\mathrm{min}}$) is only a few
ten times the skin depth at most.
Therefore, the component will not be seen for $a\gg1$ while for $a\sim O(1)$,
this power law component will be seen.
As for the frequency region lower than the break frequency $\omega_{\rm{br}} = a^{-3} \omega_{\rm{syn}}$,
Medvedev remarked that the spectrum is similar to small angle jitter radiation (Medvedev 2010).
However, it remains to be open if it is so for $1<a<\gamma$, because the assumption that
the straight orbit approximation of radiating particle is broken.
To predict the exact radiation spectrum of the frequency region lower than the break frequency,
it is necessary to pursue the particle orbit to follow the long term diffusion which is a formidable task.
\section{Summary}
We calculate the radiation spectrum from relativistic
electrons moving in the small scale turbulent magnetic fields by using the first principle calculation
utilizing the Lienard-Wiechert potential.
We concentrate our calculation on a range of the strength parameter of $1<a<\gamma$.
We confirm that the spectrum for $a\sim3$ is a broken power law with an index of low energy side
$\sim 0.5$, and that some GRBs with low energy spectral index harder than synchrotron
theory predicts may be explained.
Furthermore, we find that the spectrum for $a\sim 7$ takes a novel shape described by a superposition of
a broken power law spectrum and a synchrotron one.
Especially, an extra power law component appears beyond the synchrotron cutoff in the high frequency region
reflecting magnetic field fluctuation spectrum.
This is in contrast with previous works (Fleishman \& Urtiev 2010, Medvedev 2010).
Our spectra for $a=5$ and $a=7$ are different from both of them.
We have given a physical reason for this spectral feature.
This novel spectral shape may be seen in various other scenes in astrophysics.
For example, the spectrum of 3C273 jet at the knot region may be due to this feature (Uchiyama et al. 2006).
We thank the referee for helpful comments.
We are grateful to T. Okada, S. Tanaka, M. Yamaguchi for discussion and suggestions.
This work is partially supported by KAKENHI 20540231 (F.T.).
|
2,877,628,090,227 | arxiv | \chapter[The cavity method]{The cavity method: from exact solutions to algorithms}
\centerline{\large Alfredo Braunstein$^1$, Guilhem Semerjian$^2$}
\bigskip
\noindent
$^1$ Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129, Torino, Italy,
Italian Institute for Genomic Medicine, IRCCS Candiolo, SP-142, I-10060, Candiolo (TO), Italy and INFN, Sezione di Torino, Italy
\medskip
\noindent
$^2$ Laboratoire de Physique de l'\'Ecole Normale Sup\'erieure, ENS, Universit\'e PSL, CNRS, Sorbonne Universit\'e, Universit\'e Paris Cit\'e, F-75005 Paris, France
\section{Introduction}
The quest of an analytic solution for the simplest mean-field spin-glass model (the Sherrington-Kirkpatrick (SK) one~\cite{SK75}) led Giorgio Parisi to the invention of the replica method~\cite{Pa80}. This method is able to describe and handle the complicated structure of the configuration space of the SK model, with a hierarchical division of the configurations into nested pure states, through the analytical parametrization of matrices of size $n \times n$, in the limit where $n\to 0$, which is, to say the least, a questionable mathematical construction (its predictions have been nevertheless confirmed rigorously later on~\cite{GuTo02,Ta06,Pa13}). In the physics literature an alternative method to solve the SK model was proposed in~\cite{MePaVi86}, and subsequently dubbed the cavity method. In a nutshell the idea is to consider the effect of the addition of one spin in a large SK model, or equivalently to create a ``cavity'' by isolating one spin and modeling the influence that the rest of the system has on it in a self-consistent way. The replica and the cavity methods yield the same predictions for the SK model, with complementary insights on its structure, the cavity method bypassing the ``analytic continuation'' from integer values of $n$ to $0$.
Even if the replica and cavity methods have had an impact inside physics, in particular in the context of structural glasses, they have also been very fruitful in fields which at first sight could seem unrelated, and in particular in computer science, information theory and discrete mathematics. Roughly speaking, the reason for their versatility lies in the rather universal character of the structure of the configuration space evoked above, that appears not only in the SK model but in many other problems with a non-physical origin, notably some random constraint satisfaction problems and error correcting codes. It turns out indeed that these problems can be viewed as mean-field spin-glasses, but slightly different from the SK one: the degrees of freedom in these problems interact strongly with a finite number of neighbors, whereas in the SK all degrees of freedom interact with each other weakly, in a ``fully-connected'' manner. The mean-field character of these sparse, or diluted, models arise from the choice of the neighbors, which is done uniformly at random, without the geometrical constraints of an Euclidean space. In physics terms such a network of interaction is called a Bethe lattice, in mathematics a random graph. This type of model appeared in the physics literature relatively shortly after the fully-connected ones~\cite{VB85}, but it became quickly clear that they were much more challenging to solve, some simplifications of the diverging connectivity (of a central limit theorem flavor) being absent in this case. A line of research extended the replica method to this sparse setting, see in particular~\cite{replica_diluted,BiMoWe} and references therein, at the price of a rather complicated order parameter. It turned out that the cavity method is a more convenient framework than the replica one for these problems, the complex configuration space encoded by the replica symmetry breaking being formulated in a more transparent manner through the cavity approach, as first discussed in~\cite{cavity}; in addition the formalism of the cavity method can be used to develop algorithms that provide informations on a single sample of mean-field spin-glasses, not only on average thermodynamic quantities.
The goal of this chapter is to review the main ideas that underlie the cavity method for models defined on random graphs, as well as present some of its outcomes, focusing on the random constraint satisfaction problems for which it provided both a better understanding of the phase transitions they undergo, and suggestions for the development of algorithms to solve them. It is organized as follows; section~\ref{sec_cavity} focuses on the analytic aspects of the method. It contains an introduction to models defined on random graphs (in Sec.~\ref{sec_rg}), then the equations of the cavity method at the so-called replica symmetric (RS) level and one step of replica symmetry breaking (1RSB) are presented in Sec.~\ref{sec_rs} and \ref{sec_rsb}, before reviewing in Sec.~\ref{sec_predictions} their outcomes concerning the phase diagram of random constraint satisfaction problems. Algorithmic consequences of this approach are detailed in Sec.~\ref{sec_algorithms}.
\section{The cavity method for sparse mean-field models}
\label{sec_cavity}
\subsection{Models on random graphs}
\label{sec_rg}
We shall consider systems made of $N$ elementary degrees of freedom (spins) $\sigma_i$, which take values in some finite alphabet $\chi$, and whose global configuration will be denoted $\underline{\sigma}=(\sigma_1,\dots,\sigma_N) \in \chi^N$. They interact through an energy function (also called Hamiltonian, or cost function), that we decompose as
\begin{equation}
E(\underline{\sigma}) = \sum_{a=1}^M \varepsilon_a(\underline{\sigma}_{{\partial a}}) \ ,
\label{eq_energy}
\end{equation}
where the sum runs over the $M$ basic interactions terms $\varepsilon_a$. We denote $\partial a \subset \{1,\dots,N\}$ the set of variables involved in the $a$'th constraint, and for a subset $S$ of the variables $\underline{\sigma}_S$
means $\{\sigma_i | i \in S \}$. In what follows we assume that all interactions involves a subset of $k$ variables, for a given $k \ge 2$. This framework encompasses usual Ising spin-glass models, with $\chi=\{-1,1\}$, $k=2$ and $\varepsilon_a(\underline{\sigma}_{{\partial a}})=-J_a \sigma_{i_a} \sigma_{j_a}$, $J_a$ being the coupling constant between the spins $i_a$ and $j_a$. It also allows to deal with Potts spins when $\chi=\{1,\dots,q\}$ for a number $q\ge 2$ of spin states, also interpreted as colors; in this case a relevant energy function corresponds to pairwise interactions ($k=2$), with $\varepsilon_a(\underline{\sigma}_{{\partial a}})=\delta_{\sigma_{i_a}, \sigma_{j_a}}$. This yields the Hamiltonian of the Potts antiferromagnetic model, corresponding in the perspective of computer science to the $q$-coloring problem, the cost function counting the number of monochromatic edges among the interacting ones. More generically a constraint satisfaction problem (CSP) corresponds to a cost function of the form (\ref{eq_energy}) with $\varepsilon_a$ taking values $0$ or $1$, and being interpreted as the indicator function of the event ``the $a$-th constraint is not satisfied by the configuration of the variables in $\underline{\sigma}_{{\partial a}}$''. In particular the $k$-SAT and $k$-XORSAT problems can be described in this way with Ising spins and $k$-wise interactions. One calls solution of a CSP a configuration $\underline{\sigma}$ satisfying simultaneously all the constraints, i.e. a zero-energy groundstate, and one says that the CSP is satisfiable if and only if it admits at least one solution.
The Gibbs-Boltzmann probability measure associated to this Hamiltonian for an inverse temperature $\beta$ reads
\begin{equation}
\mu(\underline{\sigma})= \frac{1}{Z} \prod_{a=1}^M w_a(\underline{\sigma}_{\partial a}) \ ,
\quad
Z = \sum_{\underline{\sigma} \in {\cal X}^N} \prod_{a=1}^M w_a(\underline{\sigma}_{\partial a})
\ , \quad
\Phi = \frac{1}{N} \ln Z \ .
\label{eq_mu_first}
\end{equation}
where the partition function $Z$ ensures the normalization of the probability
law, and $w_a(\underline{\sigma}_{\partial a})=e^{-\beta \varepsilon_a(\underline{\sigma}_{\partial a})}$. We introduced the thermodynamic potential $\Phi$ which we shall call a free-entropy, as we did not include the constant $-1/\beta$ that would make it a free-energy. This choice allows to handle the uniform measure over the solutions of a CSP (assumed to be satisfiable), that corresponds to $w_a(\underline{\sigma}_{\partial a})=(1-\varepsilon_a(\underline{\sigma}_{\partial a}))$, in which case $Z$ counts the number of solutions and $\Phi$ is the associated entropy rate. It amounts to set formally $\beta=\infty$ in the Gibbs-Boltzmann definition, in other words to work directly at zero temperature.
A convenient representation of a probability measure $\mu$ of the form (\ref{eq_mu_first}) is provided by a factor graph~\cite{factorgraph}, see Fig.~\ref{fig_fg} for an example, which is a bipartite
graph where each of the $N$ variables $\sigma_i$ is represented by a circle
vertex, while the $M$ weight functions $w_a$ are associated to square vertices. An edge
is drawn between a variable $i$ and an interaction $a$ if and only if
$w_a$ actually depends on $\sigma_i$, i.e. $i \in {\partial a}$. In a similar way
we shall denote $\partial i$ the set of interactions in which $\sigma_i$ appears,
i.e. the graphical neighborhood of $i$ in the factor graph, and call $|{\partial i}|$ the degree of the $i$-th variable. One has a
natural notion of graph distance between two variable nodes $i$ and $j$,
defined as the minimal number of interaction nodes on a path linking $i$ and
$j$.
\begin{figure}
\begin{tikzpicture}
\fill[black] (0,0) circle (3pt);
\draw (1,-0.125) rectangle (1.25,0.125);
\draw (.75,0.6) rectangle (1,.85);
\draw (.75,-0.6) rectangle (1.,-.85);
\fill[black] (2,0.3) circle (3pt);
\fill[black] (2,-0.3) circle (3pt);
\fill[black] (1.8,0.8) circle (3pt);
\fill[black] (1.6,1.4) circle (3pt);
\fill[black] (1.8,-0.8) circle (3pt);
\fill[black] (1.6,-1.4) circle (3pt);
\draw (-1,-0.125) rectangle (-1.25,0.125);
\fill[black] (-2,0.3) circle (3pt);
\fill[black] (-2,-0.3) circle (3pt);
\draw (-1,0) -- (1,0);
\draw (0,0) -- (.75,.6);
\draw (0,0) -- (.75,-.6);
\draw (1.25,0.125) -- (2,0.3);
\draw (1.25,-0.125) -- (2,-0.3);
\draw (1,.85) -- (1.6,1.4);
\draw (1,.6) -- (1.8,0.8);
\draw (1,-.85) -- (1.6,-1.4);
\draw (1,-.6) -- (1.8,-0.8);
\draw (-1.25,0.125) -- (-2,0.3);
\draw (-1.25,-0.125) -- (-2,-0.3);
\fill[black] (-3.4,0.) circle (3pt);
\draw (-2.6,0.5) rectangle (-2.85,0.75);
\fill[black] (-2.725,1.4) circle (3pt);
\draw (-2.6,-0.5) rectangle (-2.85,-0.75);
\fill[black] (-2.725,-1.4) circle (3pt);
\draw (-2.725,1.4) -- (-2.725,.75);
\draw (-3.4,0.) -- (-2.85,0.5);
\draw (-2,0.3) -- (-2.6,0.5);
\draw (-2.725,-1.4) -- (-2.725,-.75);
\draw (-3.4,-0.) -- (-2.85,-0.5);
\draw (-2,-0.3) -- (-2.6,-0.5);
\begin{scope}[xshift=6cm,yshift=1.5cm]
\fill[black] (0,0) circle (3pt);
\draw (1.25,-0.125) rectangle (1.5,0.125);
\draw[directed] (0,0) -- (1.25,0);
\draw (-1.25,-0.125) rectangle (-1.5,0.125);
\draw (-1.25,0.7) rectangle (-1.5,0.95);
\draw (-1.25,-0.7) rectangle (-1.5,-0.95);
\draw[directed] (-1.25,0) -- (0,0);
\draw[directed] (-1.25,0.7) -- (0,0);
\draw[directed] (-1.25,-0.7) -- (0,0);
\draw (0.,-.35) node {$i$};
\draw (1.375,-.35) node {$a$};
\draw (.75,.35) node {$\mu_{i \to a}$};
\draw (-1.375,1.2) node {$b$};
\end{scope}
\begin{scope}[xshift=6cm,yshift=-1.5cm]
\draw (-.125,-0.125) rectangle (.125,0.125);
\fill[black] (1.25,0) circle (3pt);
\draw[directed] (0.125,0) -- (1.25,0);
\fill[black] (-1.25,0) circle (3pt);
\fill[black] (-1.25,0.8) circle (3pt);
\fill[black] (-1.25,-0.8) circle (3pt);
\draw[directed] (-1.25,0) -- (-0.125,0);
\draw[directed] (-1.25,0.8) -- (-0.125,0.125);
\draw[directed] (-1.25,-0.8) -- (-0.125,-0.125);
\draw (0.,-.35) node {$a$};
\draw (1.375,-.35) node {$i$};
\draw (.75,.35) node {$\widehat{\mu}_{a \to i}$};
\draw (-1.375,1.2) node {$j$};
\end{scope}
\end{tikzpicture}
\caption{
Left: an example of a factor graph.
Right: illustration of Eqs.~(\ref{eq_BP_mu},\ref{eq_BP_hmu}).}
\label{fig_fg}
\end{figure}
Our interest lies in disordered systems, in which the probability measure $\mu$ is itself a random object. Suppose indeed that the weight functions $w_a$ are built by drawing, independently for each $a$, the $k$-uplet of variables ${\partial a}$ uniformly at random among the $\binom{N}{k}$ possible choices (and also the coupling constants defining the interaction if necessary). We will denote $\mathbb{E}[\bullet]$ the average with respect to this quenched randomness (let us emphasize that there are two distinct level of probabilities in these systems: the spins $\underline{\sigma}$ are random variables with the probability law $\mu$, and $\mu$ is random because of the stochastic choices in the construction of the factor graph). For $k=2$ the resulting factor graph is drawn from nothing but the celebrated Erd{\H{o}}s-R{\'e}nyi $G(N,M)$ random graph ensemble, the case $k>2$ corresponding to its natural hypergraph generalization. The large size (thermodynamic) limit we shall consider corresponds to $N,M\to \infty$, with $\alpha= M/N$ a fixed parameter. Let us recall some elementary properties of these random factor graphs in this limit:
\begin{itemize}
\item the probability that a randomly chosen variable $i$ has degree
$|{\partial i}|=d$ is $q_d=e^{-\alpha k} (\alpha k)^d/d!$, the Poisson law of
mean $\alpha k$.
\item if one chooses randomly an interaction $a$, then a variable $i\in{\partial a}$,
the probability that $i$ appears in $d$ interactions \emph{besides} $a$, i.e.
that $|{\partial i} \setminus a|=d$, is $\widetilde{q}_d =e^{-\alpha k} (\alpha k)^d/d!$.
\item the random factor graphs are locally tree-like: choosing at random a
vertex $i$, the subgraph made of all nodes at graph distance from $i$ smaller
than some threshold $t$ is, with a probability going to 1 in the thermodynamic
limit with $t$ fixed, a tree.
\end{itemize}
More general ensembles of random factor graphs can be constructed, by fixing
a degree distribution $q_d$ and drawing at random from the set of all graphs
of size $N$ with $N q_0$ isolated vertices, $N q_1$ vertices of degree 1, and
so on and so forth. Then the two distributions $q_d$ and $\widetilde{q}_d$ are
different in general, and related through
$\widetilde{q}_d = (d+1) q_{d+1}/\sum_{d'} d' q_{d'}$. An important example in this class
corresponds to random regular graphs, where $q_d$ is supported by a single
integer.
\subsection{The replica symmetric (RS) cavity method}
\label{sec_rs}
The goal of the cavity method is to describe the properties of the random measure $\mu$ constructed above, for typical samples of the random graph ensemble. The free-entropy $\Phi$ is self-averaging in the thermodynamic limit, its typical value concentrates around its average, the quenched free-entropy $\phi$ defined as
\begin{equation}
\phi = \lim_{N \to \infty} \mathbb{E}[\Phi] = \lim_{N \to \infty}\frac{1}{N} \mathbb{E}[ \ln Z] \ .
\label{eq_def_phi}
\end{equation}
The computation of this quantity is thus the objective of the cavity method, along with a local description of the measure $\mu$, in terms of its marginal distributions on a finite number of spins.
The cavity method relies crucially on the local convergence of random factor
graph models to random trees explained at the end of Sec.~\ref{sec_rg}.
Let us assume momentarily that the factor graph representing
the model under study is a finite tree. Then the problem of characterizing the
measure~(\ref{eq_mu_first}) and computing the associated partition function $Z$
can be solved exactly in a simple, recursive way: one can break the tree into independent subtrees, solve the problems on these substructures, and combine them together to get the solution on the larger problem. This is nothing but a generalization of the transfer matrix method used in physics to solve unidimensional problems, a form of what is known as dynamic programming in computer science. More precisely, for each edge between a variable $i$ and an adjacent interaction $a$ one introduces two directed ``messages'', $\mu_{i \to a}$ and $\widehat{\mu}_{a \to i}$, which are probability measures on the alphabet $\chi$, that would be the marginal probability of $\sigma_i$ if, respectively, the interaction $a$ were removed from the graph, or if all interactions around $i$ except $a$ were removed. A moment of thought reveals that these messages obey the following recursive (so-called Belief Propagation (BP)) equations (see the right part of Fig.~\ref{fig_fg} for an illustration),
\begin{align}
\mu_{i \to a}(\sigma_i) &= \frac{1}{z_{i \to a}} \prod_{b\in {\partial i \setminus a}} \widehat{\mu}_{b \to i}(\sigma_i) \ , \label{eq_BP_mu} \\
\widehat{\mu}_{a \to i}(\sigma_i) &= \frac{1}{\widehat{z}_{a \to i}} \sum_{\underline{\sigma}_{\partial a \setminus i}} w_a(\underline{\sigma}_{\partial a})
\prod_{j \in {\partial a \setminus i}} \mu_{j \to a}(\sigma_j) \ ,
\label{eq_BP_hmu}
\end{align}
with $z_{i \to a}$ and $\widehat{z}_{a\to i}$ ensuring the normalization of the laws.
On a tree factor graph there exists a single solution of these equations, which
is easily determined starting from the leaves of the graph (for which the empty
product above is conventionally equal to 1) and sweeping towards the inside of
the graph. Once the messages have been determined all local
averages with respect to $\mu$ can be computed, as well as the partition
function, in terms of the solutions of these BP equations.
The Belief Propagation algorithm consists in looking for a fixed-point solution
of (\ref{eq_BP_mu},\ref{eq_BP_hmu}), iteratively, even if the factor graph is not a tree; in this case the formula giving $\Phi$ in terms of the messages is only an approximation, known as the Bethe formula for the free-entropy (see for instance~\cite{Yedidia2} for more details on the connections between the stationary points of the Bethe free-entropy and the solutions of the Belief Propagation equations). These equations were discovered independently in Statistical Physics as the Bethe-Peierls
approximation, in artificial intelligence as the Belief Propagation algorithm, and in Information Theory as the Sum-Product algorithm~\cite{MM09}.
Of course random graphs are only locally tree-like, they do possess loops,
even if their lengths typically diverge in the thermodynamic limit. The
cavity method amounts thus to a series of prescriptions to handle these long
loops and to describe the boundary condition they impose on the local tree
neighborhoods inside a large random graph. The simplest prescription, that
goes under the name of replica symmetric (RS) and that is valid for weakly interacting models (i.e. small $\alpha$ and/or large temperature), assumes some spatial correlation decay properties of the probability measure $\mu$. When one removes an interaction $a$ from a factor graph the variables around it becomes strictly independent if one starts from a tree, and asymptotically independent provided only long enough loops join them in absence of $a$, and provided the correlation decays fast enough along these loops. To
compute the average thermodynamic potential (\ref{eq_def_phi}) it is enough in
this case to study the statistics with respect to the quenched disorder
of the messages $\mu_{i \to a}$, $\widehat{\mu}_{a \to i}$ on the edges of the random factor graph.
In other words the order parameter of the RS cavity method is the law of the random variables $\eta$, $\widehat{\eta}$, which are equal to the random messages one obtains by drawing at random a sample, solving the BP equations on it, choosing at random an edge $a-i$, and observing the value of $\mu_{i \to a}$ and $\widehat{\mu}_{a \to i}$. With the assumption of independence underlying the RS cavity method the equations (\ref{eq_BP_mu},\ref{eq_BP_hmu}) translate into Recursive Distributional Equations (RDE) of the form:
\begin{equation}
\eta \overset{\rm d}{=}
f(\widehat{\eta}_1,\dots,\widehat{\eta}_d) \ , \qquad \widehat{\eta} \overset{\rm d}{=} \widehat{f}(\eta_1,\dots,\eta_{k-1})
\ .
\label{eq_eta_RSclass}
\end{equation}
In this equation all the $\eta_i$'s and $\widehat{\eta}_i$'s are independent copies of the random variables $\eta$ and $\widehat{\eta}$, $\overset{\rm d}{=}$ denotes the equality in distribution between random variables, $d$ is drawn according to the law $\widetilde{q}_d$, and the functions $f$ and $\widehat{f}$ are defined by the right hand sides of equations (\ref{eq_BP_mu},\ref{eq_BP_hmu}) (with possibly an additional random draw of the weight $w$).
The RS prediction for
$\phi$ can then be expressed as the average over random copies of $\eta$ and $\widehat{\eta}$ of the local free-entropy contributions obtained from the exact computation of the partition function of
a finite tree. Note that the equation (\ref{eq_eta_RSclass}), if it has in
general no analytic solution, lends itself to a very natural numerical
resolution where the law of $\eta$ is approximately represented as an empirical
distribution over a set of representatives $\eta$ (a population
representation)~\cite{abou1973,cavity}.
The exactness of the predictions of the RS cavity method has been proven rigorously for some models which are not too frustrated (e.g. ferromagnetic systems, or matching models), see for instance ~\cite{DM10,BoLe10,BoLeSa_matchings}. But in general the correlation decay assumption fails, in this case one has to turn to a more sophisticated version of the cavity method, which will be introduced in the next section.
\subsection{Handling the replica symmetry breaking (RSB) with the cavity method}
\label{sec_rsb}
As a matter of fact for low enough temperature, and high enough density of interactions $\alpha$, the configuration space of frustrated random models gets fractured in a large number of pure states (or clusters), and the correlation decay hypothesis only holds for the Gibbs measure restricted to one pure state, not for the
complete Gibbs measure. In the replica method this phenomenon shows up as a breaking of the equivalence between different replicas, we will now explain how the cavity method is able to handle this structure of the configuration space.
It amounts to make further self-consistent hypotheses on the correlated boundary
conditions this induces on the tree-like portions of the factor graph.
Inside each pure state the RS computation is assumed to hold true, and the RSB computation
is then a study of the statistics of the pure states. Let us explain
how this is done in practice at the first level of RSB (1RSB cavity method).
The partition function is written as a sum over the pure states $\gamma$,
that form a partition of the configuration space,
$Z=\sum_\gamma Z_\gamma$, where $Z_\gamma$ is the partition function restricted to
the pure state $\gamma$. It can be written in the thermodynamic limit as
$Z_\gamma = e^{N f_\gamma}$, with $f_\gamma$ the internal free-entropy density of a given pure state. One further assumes
that the number of pure states with
a given value of $f$ is, at the leading
exponential order, $e^{N\Sigma(f)}$, with the so-called configuration entropy, or complexity, $\Sigma$
a concave function of $f$,
positive on the interval $[f_{\rm min},f_{\rm max}]$. In order to compute
$\Sigma$ one introduces a parameter $m$ (called Parisi breaking parameter)
conjugated to the internal thermodynamic potential,
and the generating function of the
$Z_\gamma$ as ${\cal Z}(m) = \sum_\gamma Z_\gamma^m$. In the thermodynamic limit its dominant
behavior is captured by the 1RSB potential $\phi_{\rm 1RSB}(m)$,
\begin{equation}
\phi_{\rm 1RSB}(m)= \lim_{N \to \infty} \frac{1}{N}\log {\cal Z}(m) =
\sup_f \left[\Sigma(f)+ m f \right] \ ,
\label{eq_Phi_1RSBclass}
\end{equation}
where the last expression is obtained by a saddle-point evaluation of the
sum over $\gamma$. The complexity function is then accessible via the inverse
Legendre transform of $\phi_{\rm 1RSB}(m)$~\cite{Mo95}, or in a parametric form
\begin{equation}
f(m) = \phi_{\rm 1RSB}'(m) \ , \qquad
\Sigma(f(m)) = \phi_{\rm 1RSB}(m) - m \phi_{\rm 1RSB}'(m) \ ,
\end{equation}
where $f(m)$ denotes the point where the supremum is reached in
Eq.~(\ref{eq_Phi_1RSBclass}). One has $\Sigma'(f(m))=-m$, i.e. the
introduction of the parameter $m$ allows to explore the complexity
curve by tuning the tangent
slope of the selected point.
The actual computation of $\phi_{\rm 1RSB}(m)$
is done as follows~\cite{cavity}. One introduces on each
edge of the factor graph two distributions $P_{i \to a}$ and $\widehat{P}_{i \to a}$ of messages,
which are the probability over the different pure states $\gamma$, weighted
proportionally to $Z_\gamma^m$, to observe a given value of $\mu_{i \to a}^\gamma$ and $\widehat{\mu}_{a \to i}^\gamma$ respectively,
where $\mu_{i\to a}^\gamma$ and $\widehat{\mu}_{a \to i}^\gamma$ are the messages that appear in Eq.~(\ref{eq_BP_mu},\ref{eq_BP_hmu}),
for the measure restricted to the pure state $\gamma$.
Because $P_{i \to a}$ and $\widehat{P}_{a \to i}$ are themselves random objects with respect to the choices in the generation of the instance of the factor graph,
the order parameter of the 1RSB cavity method becomes the distributions of $P_{i \to a}$ and $\widehat{P}_{a \to i}$ with respect to the disorder. The latter is solution of a self-consistent
functional equation written as
\begin{equation}
P \overset{\rm d}{=} F(\widehat{P}_1,\dots,\widehat{P}_d) \ , \qquad
\widehat{P} \overset{\rm d}{=} \widehat{F}(P_1,\dots,P_{k-1}) \ ,
\label{eq_P_1RSBclass}
\end{equation}
that parallels the equation (\ref{eq_eta_RSclass}) of the RS cavity method, with again independent copies of the distributions $P_i$ and $\widehat{P}_i$.
The right hand sides of these distributional equalities stand for:
\begin{align}
P(\eta) &= \frac{1}{Z} \int \prod_{i=1}^d {\rm d} \widehat{P}_i(\widehat{\eta}_i) \ \delta(\eta-f(\{ \widehat{\eta}_i \})) \
z(\{ \widehat{\eta}_i \})^m \ , \label{eq_1RSB_P} \\
\widehat{P}(\eta) &= \frac{1}{\widehat{Z}} \int \prod_{i=1}^{k-1} {\rm d} P_i(\eta_i) \ \delta(\widehat{\eta}-\widehat{f}(\{ \eta_i \})) \
\widehat{z}(\{ \eta_i \})^m \ ,
\label{eq_1RSB_hP}
\end{align}
with the functions $f$ and $\widehat{f}$ corresponding to the recursion functions at the RS level, see Eq.~(\ref{eq_BP_mu},\ref{eq_BP_hmu}), and $z$ and $\widehat{z}$ the associated normalization factors.
From the solution of this equation
(that again can be found numerically with the population dynamics
method~\cite{cavity}) one
computes the 1RSB potential $\phi_{\rm 1RSB}(m)$ via an expression similar to
the one giving the expression of $\phi$ at the RS level, with now averages
over random distributions $P$ and $\widehat{P}$.
There are different justifications for the appearance of the
``reweighting factors'' $z^m$ and $\widehat{z}^m$ in
Eqs.~(\ref{eq_1RSB_P},\ref{eq_1RSB_hP}). The argument in~\cite{cavity} is based on the
exponential distribution of the free-entropies $N f_\gamma$ of the pure states
with respect to some reference value, and on consistency requirements
on the evolution of the pure states when the cavity factor graph is modified.
One can also study the statistics
of the many fixed point solutions of the BP equations (\ref{eq_BP_mu},\ref{eq_BP_hmu}) and
devise a dual factor graph for the counting of these fixed
points~\cite{MM09}, the reweighting factor allowing to select the fixed
points associated to some internal free-entropy. Another interpretation
was proposed in~\cite{KrMoRiSeZd}, associating the pure states of a large
but finite factor graph model to boundary conditions on trees. This interpretation is particularly relevant in the case $m=1$, for which these boundary conditions are actually drawn from the Gibbs measure itself, and reveals a deep connection between the 1RSB cavity method and the reconstruction on tree problem, as first unveiled in~\cite{MM06}, and with the point-to-set correlations of the Gibbs measure~\cite{MoSe2}.
This construction can be generalized to higher levels of replica symmetry breaking~\cite{Pa80}, with a hierarchical partition of the configuration space into nested pure spaces; the resulting equations for models on sparse random graphs involve a recursive tower of probability distributions over probability distributions, whose numerical resolution becomes extremely challenging beyond 1RSB.
\subsection{Some analytic outcomes of the cavity method}
\label{sec_predictions}
As presented above the cavity method is quite versatile, in the sense that it can address a variety of models defined on random graphs, and it has indeed been applied to several different problems. As an illustration of some of its outcomes we shall now present some results it has provided on the phase diagram of random constraint satisfaction problems (see also chapter 31), and sketch the connections between this qualitative understanding and the quantitative formalism we have introduced before.
In the case of a constraint satisfaction problem the cost function defined in Eq.~(\ref{eq_energy}) is made of a sum of indicator functions of events that the $a$-th constraint is unsatisfied, for instance the number of monochromatic edges in the $q$-coloring problem. The natural questions in this context are: does an instance of the problem admit at least one solution? if yes, how are the solutions organized in the configuration space? It turns out that the answers to these questions have drastically different answers depending on the value of the density of constraints $\alpha$, in other words there exist, in the thermodynamic limit, sharp phase transitions for some threshold values of this parameter.
\begin{figure}
\centerline{\includegraphics[width=.9\textwidth]{clusters.eps}}
\caption{Schematic representation of the phase transitions in a random CSP
ensemble.}
\label{fig_clusters}
\end{figure}
The main transitions that occur for generic ensembles of random CSPs are represented in a schematic way on Fig.~\ref{fig_clusters}. The squares represent the full configuration space, for four different values of $\alpha$ (obviously the representation of this $N$-dimensional hypercube on a two-dimensional
drawing is only a cartoon), while the black area stands for the solutions. For $\alpha > \alpha_{\rm s}$, the satisfiability transition, the square is empty, which translates the absence of solution in typical instances for these density of constraints. The satisfiable regime $\alpha < \alpha_{\rm s}$ is further divided in three regions, separated by structural phase transitions at which the organization of the set of solutions changes qualitatively. For $\alpha<\alpha_{\rm d}$, the so-called clustering, or dynamic transition, all solutions
are somehow close to each other, while in the rest of the satisfiable regime they are broken in clusters of nearby solutions, each cluster being separated from the other ones. The number and size of the relevant clusters further change at the condensation threshold $\alpha_{\rm c}$: for $\alpha_{\rm d}<\alpha < \alpha_{\rm c}$ most solutions are contained in an exponential number of clusters which have all roughly the same size, while in the regime $\alpha_{\rm c}<\alpha < \alpha_{\rm s}$ most solutions are found in a sub-exponential number of clusters with strongly fluctuating sizes.
These qualitative predictions, along with quantitative numerical values for some definite random CSPs families, have been obtained by the analysis of the solutions of the 1RSB cavity equations, according to the following criteria~\cite{MeZe,KrMoRiSeZd}:
\begin{itemize}
\item $\alpha_{\rm d}$ is the smallest value of $\alpha$ such that the 1RSB equations at $m=1$ admit a non-trivial solution.
\item in the regime $[\alpha_{\rm d},\alpha_{\rm c}]$ the configurational entropy, or complexity, associated to the $m=1$ solution, is positive, whereas it becomes negative for $\alpha > \alpha_{\rm c}$.
\item the satisfiability transition is marked by the vanishing of the complexity computed at $m=0$, in the so-called energetic version of the 1RSB cavity method ~\cite{cavity_T0}, that counts all clusters irrespectively of their sizes.
\end{itemize}
\section{Some algorithmic outcomes of the cavity method}
\label{sec_algorithms}
\subsection{Algorithmic applications of the cavity method}
As mentioned above the equations \eqref{eq_BP_mu}-\eqref{eq_BP_hmu} can be used on a single instance to compute (approximately) several properties of the distribution \eqref{eq_mu_first}, including single-site marginals,
joint marginals of variables in a common factor, the free energy and
Shannon's entropy. This approach has been applied to Bayesian networks, in the decoding phase of
communication codes (syndrome-based decoding, Turbo Codes \cite{benedetto_soft-output_1996})
and in stereo image reconstruction. More recently, it has found applications
in a large variety of fields that we shall now review.
\paragraph{BP applications in notable models}
In \cite{kabashima_cdma_2003}, a Belief Propagation algorithm for
CDMA decoding has been presented. Interestingly, it shows how BP can
be efficiently applied to dense models (i.e. in which constraints
involve an extensive number of variables) through an application of
the Central Limit Theorem (the basis of a BP derivative called AMP, see Chapter 19), and it is also shown that solutions are
also fixed points of the famous Thouless-Anderson-Palmer (TAP) equations~\cite{TAP77}
while showing superior iterative convergence properties. A similar
approach has been employed in~\cite{braunstein_learning_2006} for
the binary discrete perceptron learning problem.
In \cite{frey_mixture_2005}, the Affinity Propagation (AP) algorithm
was presented. AP is a BP algorithm for variables with an extensive
number of states. The AP algorithm solves approximately a clustering
problem which is similar in spirit to $K$-means, but with the important
difference of only relying on a distance matrix instead of the original,
possibly high-dimensional, data representation. Auxiliary variables
with a large number of states can be employed to locally enforce global
constraints such as connectivity, by representing in the variables
state the discrete time of an underlying dynamics. BP has been applied
to the resulting extended model \cite{bayati_statistical_2008}.
The dynamic cavity method \cite{neri_cavity_2009} is an application of BP to
study a certain class of out-of equilibrium dynamical models. The
method can be understood as an application of BP to an auxiliary model
in which a variable consists in a couple of time-dependent quantities: one is a single spin trajectory,
the other a local field. Subsequent works showed that a slightly
simpler but equivalent representation can be obtained with a pair
of spin trajectories. On certain models such as discrete, microscopically
irreversible ones (i.e. ones in which a variable can never go back
to a visited state, including the Bootstrap percolation model \cite{altarelli_optimizing_2013},
SI or SIR epidemic models \cite{altarelli_bayesian_2014}), single
trajectories can be efficiently represented by the transition times.
In other cases, some approximations must be employed \cite{aurell_dynamic_2012}. A somehow related variant of the cavity method deals with quantum models, the basic degrees of freedom becoming imaginary-time spin trajectories~\cite{our_review_qaa}.
\paragraph{Exactness of BP on single instances}
Some rigorous results have been proven regarding the exactness of BP algorithms. For certain models and sufficiently large temperature,
the BP update equation becomes a contractive mapping, guaranteeing
the existence and uniqueness of its fixed point and the convergence towards
it under iterations thanks to the Banach theorem. Moreover, this condition
guarantees exactness in the thermodynamical limit on graphs with large
girth \cite{bayati_rigorous_2006}.
On the other side of the spectrum, some exactness results exist in
the small temperature limit as well. Equations to analize models explicitely
at zero temperature can be devised by taking the $T\to0$ limit of the BP equations under
an an opportune change of variables, resulting in equations for energy-shifts
instead of probabilities. These had been known in coding theory as Max-Sum algorithms. Existing proofs of exactness (on
some models) rely on a local optimality condition for BP fixed points.
\cite{bayati_maximum_2005,weiss_optimality_2001,gamarnik_belief_2012}.
Gaussian BP (GaBP) \cite{weiss_correctness_2001} is an application
of BP for a continuous model with positive definite quadratic potential,
i.e. a Multivariate Gaussian. It is shown under certain conditions on
the precision matrix that the GaBP equations converge and give the correct
estimation of the means (but wrong estimation of the variances in
general), effectively solving a linear system iteratively, with convergence
properties that make the method competitive. Note that due to the
fact that the mode is equal to the mean in a Gaussian distribution,
this result can be again thought of as the exactness of the computation of
the maximum.
\paragraph{Survey Propagations and the RSB Phase}
Survey propagation (SP) is the algorithmic counterpart of the 1RSB cavity
method. It has seen its first applications to study the $k$-SAT \cite{mezard_random_2002,braunstein_survey_2005}
and $q-$coloring \cite{col1} problems in the replica symmetry broken phase. SP can
be thought as BP for the combinatorial problem of solutions of a lower
order message passing system (typically Max-Sum or some coarsened
version of it). Such a hierarchical approach can also be employed to analyze problems that possess explicitely such a nested structure, such as the ones coming from
(stochastic) control problems (e.g. the Stochastic Matching problem
\cite{altarelli_stochastic_2011}).
It should also be noted that BP can be used in the RSB phase of constraint
satisfaction problems. In \cite{braunstein_learning_2006} BP has
been applied successfully to the perceptron learning problem with
binary synapses, even in the regime in which it shows a RSB phase.
The solution to this conundrum has been clarified in \cite{baldassi_unreasonable_2016},
where it was shown that BP describes an exponentially small portion
of the solution space that is still exponentally large and has a non-clustered
geometry akin to the dominant region of the solution space in the RS phase.
\paragraph{Decimation and reinforcement.}
An algorithm estimating marginal distributions such as BP can be employed
for sampling, and in particular to find solutions to a constraint
satisfaction problem. The main idea is ancestral sampling, i.e. given
an arbitrary permutation of variable indices $\pi$, one can estimate the marginal distribution
$p\left(x_{\pi_{1}}\right)$ and sample $x_{\pi_{1}}^{*}$ from it,
then restrict the solution space to solutions with $x_{\pi_{1}}=x_{\pi_{1}}^{*}$
and reiterate, effectively sampling $x_{\pi_{i}}^{*}\sim p\left(x_{\pi_{i}}|x_{\pi_{1}}^{*},\dots,x_{\pi_{i-1}}^{*}\right)$
for $i=1,\dots,n$. As $p\left(\underline{x}\right)=\prod_{i=1}^{n}p\left(x_{\pi_{i}}|x_{\pi_{1}},\dots,x_{\pi_{i-1}}\right)$,
this solution provides a fair sample $\underline{x}^{*}$ if
the estimation of the marginals is exact. The analysis of ancestral sampling with BP has been
performed in \cite{Allerton,RiSe09,Coja11,Coja12}. When one is merely
interested in finding \textit{any} solution to a contraint satisfaction problem, and
remembering that marginal estimations are only approximate, it is
convenient to iteratively fix the variable that reduces the solution
space the \emph{less, }which corresponds to fixing the most polarized
variable in the direction of the largest probability of its marginal.
This process is called \emph{decimation}. In practice, decimation corresponds to iteratively selecting the variable with the largest local field and applying an infinite external field to it with the same sign (and then making the equations converge again and reiterating). A soft version of decimation, called reinforcement, can also be conceived, in which a field is applied iteratively to all variables with the same sign of their local field and an intensity that is either a constant \cite{chavas_survey-propagation_2005} or proportional to its magnitude \cite{braunstein_learning_2006, bayati_statistical_2008}. This dynamics slowly drives the system to one with sufficiently large external fields that becomes trivially polarized on one solution. As an additional twist, a backtracking procedure can be implemented on top of decimation, in which variables are occasional freed from their external field when that choice enlarges the solution space sufficiently. This has been implemented for SP, with excellent results \cite{marino_backtracking_2016}.
\section{Conclusions}
The Cavity method is a powerful and versatile approach to the description of disordered systems, that has been shown so far to provide the exact asymptotic solution for many models. For given (finite) system instances, its algorithmic counterpart has many practical applications, ranging from a statistical description of the Boltzman-Gibbs distribution to the individuation of single solutions of a CSP. Moreover, at variance with more traditional methods for inference such as MCMC sampling, it can provide an analytical description, given implicitly by the solution(s) of the cavity equations. This fact enables many possibilities, such as its recursive application (SP), and a functional expression of statistical features as a function of the disorder parameters (see for instance chapter 21 for a discussion of inverse problems).
\bibliographystyle{ws-book-har}
|
2,877,628,090,228 | arxiv | \section{Introduction}
Multi-Agent Reinforcement Learning (MARL) is being increasingly used in various domains such as robotics, computer networks, traffic, resource management, robotic teams and distributed control in general \cite{Busoniu2008}. Many of these situations pose complex challenges to multi-agent systems due to the dynamicity of the environment, defined by uncertainty and non-stationary behaviour. Adding to the complexity in such circumstances is the situation where one might not only encounter dynamicity generated by the stochastic interactions between agents, but also stochasticity due to a possibly continuously changing and evolving environment (independently of agent actions). The environment's behavioural patterns are not being repeated, as the environment is continuously evolving. Moreover, sudden unexpected events which weren't encountered previously are contributing to the element of uncertainty.
Reinforcement Learning (RL) \cite{Sutton1998} can provide an optimal solution for static environments, where a single agent perceives the current state of the environment and takes a decision which affects the environment. There is a finite number of changes in the environment in such cases. Despite these, RL does not require any previous knowledge of the environment. Through a reward based system, the agent eventually learns its optimal behaviour by trial-and-error, where the reward system provides feedback for each of the chosen actions (either a reward or a penalty depending on the case). The agent attempts to maximize its overall reward, and as a results it converges in the end to an optimal solution. The problem arises when several such RL based agents interact within the same environment, transforming (even initially) static environments (from a single agent's perspective) in non-stationary environments, as each agent has a particular influence over the environment itself \cite{Chalkiadakis2003}. All agents attempt to learn simultaneously in such situations, and as a result the guarantees of algorithms convergence are lost. Therefore we are faced with a much more complex problem.
We propose to tackle the problem of non-stationary environments by minimizing the uncertainty component of the problem in order to provide a solution to a \emph{close enough} problem. We augment an online MARL algorithm with a prediction and pattern change detection module in order to mitigate uncertainty. While an initial environment estimate is provided by the prediction side of the model, it also closely monitors the real-time behaviour to check whether there are any significant changes from the expected one (these might occur due to increased uncertainty). If such changes happen, the module triggers the reprediction of the environment's future behaviour, based on previously encountered similar patterns.
Our algorithm, Predictive-Marl (P-MARL), is tested in a real world smart-grid scenario, where a set of Electric Vehicles (EVs) are designated to optimally reach a desired battery state of charge before they are to depart from home. The optimality criteria is defined by the price of electricity, which in this case is considered directly proportional with the aggregated power demand. This is a decentralised control problem, where agents' action affect the state of the environment, and where even the underlying environment's initial state is not known ahead of time with certainty. While there are methods for predicting future energy demands, particular events can lead to unexpected demand patterns which increase the level of uncertainty of such a system. Such events can be caused by natural disasters, power network failures, and unexpected weather phenomena. These are not generally foreseeable.
In the following sections of this paper we introduce the background and related work with regard to the problem, we continue with the generic P-MARL proposition, while the next section presents an implementation on a specific example - a real world Smart Grid scenario. In the final sections we evaluate the algorithm's performance and present our conclusions with regard to its results.
\section{Background and Related Work}
The problem faced is essentially a more complex version of the Distributed Constraint Optimization Problem (DCOP) \cite{Nguyen2014,Wu2013}. While optimal solutions exist for DCOP, these are NP-complete and are not suitable for large scale problems. More than that, a DCOP involving uncertainty \emph{does not have an optimal solution}, precisely due to the uncertainty involved in the environment's next state, which does not pertain a fully defined problem. This particular type of uncertain DCOP has also been defined as a Distributed Coordination of Exploration and Exploitation (DCEE) problem by \cite{Taylor2010}, as DCOP under Stochastic Uncertainty (StochDCOP) by \cite{Leaute2011} or simply dubbed DCOP with uncertainty \cite{Stranders2011}. Since the environment is uncertain and non-stationary, agent rewards will continuously change, therefore a trade-off between exploration and exploitation is necessary in order to lead to a sufficiently good solution. As the environment's state is not known ahead of time, and is continuously undergoing change, the \emph{a priori} design of agent behaviour becomes infeasible.
Centralised solutions have been proposed to such problems, but there are issues concerning computational complexity (they are generally NP-complete), scalability of the solution, the privacy and independence of users, communication overhead involved and resilience/reliability of a centralised controller. Therefore decentralised options are more desirable for this kind of problem, and even more the ones that can adapt in real-time to new conditions produced by the uncertainty element of the environment.
In the following section we present a novel algorithm, P-MARL - a decentralised reinforcement learning based approach to DCOP with uncertainty.
\section{System Architecture}
\label{sec:syst_arch}
Our chosen application environment is inherently non-stationary. In mathematical terms this means that the the underlying generating function \emph{f} of the environment changes over time. The environment effectively experiences \emph{concept drift} \cite{Schlimmer1986}- the environment in this case is also known as being dynamic. To simplify the problem one can attempt to model the outcome of the generating function of the environment, \emph{f}, based on recently observed behaviour. At every time step \emph{t} we have historical data available: $X^{H}=(X_{1},...,X_{t})$. We attempt to predict $X_{t+1}$ considering past observations. The core assumption when dealing with a concept drift problem is uncertainty about the future. Here our target $X_{t+1}$ is not known. It can be assumed, estimated or predicted, but there are no certainty guarantees.
To alleviate the concept drift problem we propose a two step solution, depending on the degree of \emph{driftness}, or the speed of drift:
\begin{enumerate}
\item The passive solution: a continuously updating model based on the most recently observed samples, as suggested by \cite{Widmer1996}
\item The active solution: once a pattern change detection mechanism notices an unexpected shift from the model's expected behaviour it triggers the generation of a new model - such as the solution proposed by \cite{Alippi2008}.
\end{enumerate}
The second part of the solution is obviously more delicate. The new model is proposed based on a pattern matching mechanism that should be able to provide a close match, based on similarly previously encountered behaviour. This would in turn provide help in generating a better model.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{alg_architecture}
\caption[c]{MARL Algorithm Architecture}
\label{fig:alg_architecture}
\end{figure}
As a result, P-MARL's architecture comprises three key components:
\begin{enumerate}
\item \textbf{The Prediction Model}, which effectively considers recent historical values and other key variables that can affect the environment in order to provide an estimate of future behaviour.
\item \textbf{The Pattern Change Detection and Matching Component}, which detects when the prediction model fails in providing reasonable estimations of the future state of the environment. This triggers a new model based subject to the latest observations and findings from a database.
\item \textbf{The Multi-Agent System} (MAS) part based on \textbf{reinforcement learning}, that employs the previous components as an input to improve its performance in the dynamic environment. The RL side is implemented as a multi-objective W-learning process \cite{Humphrys1995}.
\end{enumerate}
The relationship between the components is illustrated in Fig. \ref{fig:alg_architecture}. Eventually, from the first two components, an estimate of the environment's future expected behaviour is provided. The MAS agents evaluate the future behaviour and attempt to optimally reach their goals with respect to the imposed constraints (these constraints can be implemented as extra objectives).
In the following section we define the evaluation testbed environment, and we show in detail how P-MARL is implemented for such case.
\section{Smart Grid Case Study}
In the smart-grid environment, the model provides a good estimate of the next day's demand to the agents. The MAS agents in this case are EVs, evaluate the future demand and attempt to reach their charging goal - a sufficient state of charge (SOC) that would allow them to fulfil the next day's desired trip. At the same time they attempt to minimize the charging cost, which is directly proportional to the power demand in a real-time pricing mechanism.
While such problems are relatively simple from a single RL agent's perspective - as eventually it will reach optimal performance - not the same can be said in an environment where agents compete against each other as well; once a lot of agents decide to charge at the same time, the price significantly increases, leading the agents to decide to charge at other points in time. This can lead to an endless loop, where distracted agents take the same charging point decision at each following timestep, and leads to suboptimal behaviour, as shown by \cite{Karfopoulos2013}. This is a significant obstacle faced by the convergence of the MAS to a desired behaviour.
A summary of P-MARL's decision process is presented in Alg. \ref{alg:p-marl}. The following sections present how P-MARL is implemented in the smart grid scenario, with focus on each of the three components.
\begin{algorithm}
\begin{algorithmic}
\State $initVars = gatherEnvData()$
\State $prediction = initialPrediction(initVars)$
\If{$noChangeDetected()$ }
\State $finalPrediction = prediction$
\Else
\State $matchChangeType()$
\State $input = updateInitVars()$
\State $finalPrediction = anomalyReprediction(input)$
\EndIf
\State$learnBestBehaviour(finalPrediction)$
\State{$exploitLearnedInfo()$}
\caption{P-MARL Algorithm}
\label{alg:p-marl}
\end{algorithmic}
\end{algorithm}
\subsection{1. The Prediction Model}
Electric vehicles generally arrive home at 18:00 and depart he next day at 09:00. Ideally the agent would like to know the periods of lowest demand occurring during that time in order to be as price effective as possible. In a dynamic environment though such \emph{a priori} knowledge is not avaialable.
Techniques such as short-term load forecasting (STLF) deal with power demand estimates, which can give good hints on expected demand. The most appropriate such STLF sub-technique is the one focusing on day-ahead demand forecasting, on a 24 hour basis. The best methods for such forecasts rely not only on historical values of previous power demands but also on other data such as weather variables (temperature, humidity), day of the week and public holidays \cite{Gross1987}. While weekdays and weekends differ significantly in terms of demand patterns, even each weekday has it's own particularities. Special cases occur as well. These could be for example public holidays, so all such cases need to be taken into account.
Our prediction model is implemented based on the work done in \cite{Marinescu2014a}. Here the forecasting method is a hybrid solution exploiting the best features of several forecasting techniques: artificial neural networks, neuro-fuzzy networks and auto-regression.
The hybrid solution uses as input previously recorded power demands, past day's temperature and humidity information, temperature and humidity forecasts for the day to be predicted, and day of the week information. The output is the next day's power demand estimate provided as a sequence of 24 data points, one for each hour of the day.
\subsection{2. The Pattern-Change Detection and Matching Component}
When dealing with uncertainty one can never have quality of service guarantees. Despite the fact that the previous model provides rather accurate predictions, there are particular times when forecasts fail to closely match actual demand. Such particular cases are caused by anomalous events such as electrical grid malfunctions,
unexpected climate phenomena or natural disasters. In order to make the system more robust in the face of such events an additional pattern changed detection component has been added.
This component builds upon the work carried out in \cite{Marinescu2014b}. Once the prediction model provides a forecasting estimate for the next day, the actual demand is compared with the estimate along the first few hours of the morning. If significant deflections from the actual demand occur, the pattern change detection mechanism triggers reprediction as the demand estimate is regarded as flawed. The re-prediction part is based on an artificial neural network which adjusts its historical load input part (24 neurons, one for each hour of the day) to a mash-up between the demand observed so far of the anomalous day (7:00-14:30) and the remaining demand from the closest match. The match is chosen based on similarly previously encountered patterns, which are found in a database of historical recordings - this function is carried out by a classifier involving a self-organizing map. The self-organising map first classifies the type of uncertainty detected and afterwards provides the closest previously such encountered match from the fitting class.
\subsection{3. The Multi-Agent Reinforcement Learning System}
The previously proposed components are designed to be linked with a MARL, thus resulting in P-MARL. Each agent is developed individually by following a reinforcement learning scheme - in this case Q-Learning \cite{Watkins1992} combined with W-Learning.
Initially this is a single-objective problem for each agent: to make sure that the desired charge is reached, reward the agent when charging. This results in a greedy behaviour, with the agent charging at every time-step until it's fully charged. Obviously this would lead to an undesirable behaviour, as the EV can easily charge during periods of high demand. The constraint of avoiding charging at periods of high demand is introduced under the form of another objective. This effectively transforms our problem into a multi-objective one. The second objective rewards the agent if it decides to charge at periods of low demand, and penalises it if it decides to charge at times of high demand . The second objective is highly relevant - it can be further modelled to employ information about the future state of the environment, which is provided by the prediction components. Essentially the prediction component provides a form of reward shaping for the agent through the extra-objective, but this is just an estimate, thus with no guarantees of certainty. If the estimate is good, P-MARLs results will be good; if the estimates are bad, P-MARL will perform sub-optimally. While still uncertain, we argue that a (good) estimate is still better than no estimate at all.
\section{Experimental Setup}
In this paper we apply P-MARL to an environment which is inherently dynamic, a real-world scenario occurring in the smart-grid. The state of the environment is characterised by its energy consumption, which involves randomness due to human users. The environment can be represented as a time-series defined by the half-hourly power demand. The time-series experiences a clear concept drift.
By involving a set of controllable loads on top of a baseload demand (where the baseload is the aggregated demand of inflexible appliances such as lights, television sets or computers, which are exclusively controlled by humans) we reach a situation involving a group of agents with certain objectives to be accomplished, under some predefined constrains. In this case we have a neighbourhood of residential users which contains a set of EVs; the task of each EV is to achieve a desired state of charge (SOC) for the next day's trip. Additionally, this charging process might be constrained by periods of high demand, when electricity is expensive, and when charging is to be avoided. Such periods can change in real-time, for example when all EVs charge simultaneously during a period of relatively moderate energy usage, resulting in very high energy usage. The latter of course is not a desired state of the environment, and ideally should be avoided.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{grid_structure}
\caption[c]{Smart Grid Scenario}
\label{fig:grid_struct}
\end{figure}
The smart grid setup can be visualised in Fig. \ref{fig:grid_struct}. The charging algorithms are evaluated in a real world scenario. Power demands from a community of 230 households are employed, as recorded by an Irish smart meter trial \cite{CER2011}. We have assumed a penetration of EVs of 40\% \cite{Nemry2010}, resulting in a total of 90 EVs. Daily trip is considered to be 50\hspace{2pt}km \cite{EPA2008}, while the EV specifications are borrowed from \cite{Nissan2014}. The vehicles can choose charging slots anytime between 18:00-09:00. This smart grid scenario was implemented in GridLAB-D, a power distribution system simulator \cite{GridLABD2014}.
\begin{figure*}[!ht]
\centering
\subfigure[Perfect Prediction]{
\includegraphics[width=0.31\linewidth]{pshape}
\label{fig:perfect_case}
}\quad
\subfigure[Anomaly Prediction]{
\includegraphics[width=0.31\linewidth]{sshape}
\label{fig:apred_case}
}\quad
\subfigure[Simple Prediction]{
\includegraphics[width=0.31\linewidth]{eshape}
\label{fig:spred_case}
}
\caption{Algorihtms Behaviour}
\label{fig:algorithm_behaviour}
\end{figure*}
\subsection{Benchmark - Optimal Centralised Solution}
In order to efficiently evaluate P-MARL we have to define the Pareto optimal performance of the MAS. As outlined in the previous section, a centralised solution is not suitable in such cases. Nevertheless, it's guaranteed to be optimal with respect to a defined set of constraints. Assuming a system of dynamic pricing, the solution should lead to EVs energy usage at the lowest demand times (given that energy cost is directly proportional with the system power load). The resulting constrained optimization function is presented in Eq. \ref{eq:func}:
\begin{equation}
\centering
\min F(x) = \min \sum\limits_{j=1}^m\left[\sum\limits_{i=1}^n\bigg(x_{ij}+C_{j}\bigg)\right]x_{ij}
\label{eq:func}
\end{equation}
where $F(x)$ is the cost function, $n$ the total number of electric vehicles, $m$ the total hours available for charging (assuming the same availability schedules for EVs), $x_{ij}$ the charging decision (0-off/1-on) of vehicle $i$ at time $j$, and $C_{j}$ the initial cost of energy at time $j$ (based on baseload).
While solutions are achievable, the purpose of the benchmark is not to obtain individual solutions for each agent but to define the aggregated optimal charging solution of the overall vehicles. This is essentially a valley-filling problem, such as the one presented in \cite{Gan2011}.
Ideally the MAS solution should converge to the same result as the centralised solution. In order to evaluate the efficiency of the MAS solution we used a formula based on the mean absolute percentile error (MAPE), as shown in Eq. \ref{eq:eff}.
\begin{equation}
M=\frac{1}{m}\sum\limits_{j=1}^m\bigg(1-\frac{|X_{j}-\hat{X_{j}}|}{TotalNoOfEVs}\bigg)
\label{eq:eff}
\end{equation}
where $X_{j}$ total number of EVs charging at time slot $j$, and $\hat{X_{j}}$ optimal amount of EVs that should be charging at time slot $j$.
It is worth noting that one metric is not enough to show the overall performance of the system. The number of high deviations (above a certain threshold, say 25\%) from the desired optimal solution should also be used, as these can have quite an impact on the physical power networks, in particular at times of high energy usage.
\section{Evaluation}
The experiments are performed on the scenario mentioned before. Four different algorithms are evaluated:
\begin{itemize}
\item \emph{Greedy} Solution - which charges the EVs as soon as possible - unconstrained single-objective MARL
\item \emph{Night Tariff-Aware Greedy} solution - which charges the EVs as soon as possible starting from 23:00, by adjusting to a night-saver time tariff
\item \emph{Optimal} Valley-Filling (V-F) Solution - obtained by the centralised algorithm
\item \emph{P-MARL} solution - given by our multi-objective algorithm presented in the System Architecture section \ref{sec:syst_arch}
\end{itemize}
At the beginning of the experiments the state of charge of each vehicle is randomly initialised with values in between [0.17-0.67]. The experiments are split over 3 different sub-cases: assuming the normal simple prediction of the day as an input, the more accurate re-prediction of the day (anomalous case), and finally assuming perfect forecast of the day. The later one is used just for comparison purposes, in order to see the effects of forecasting accuracy over algorithm performance. The algorithm continuously learns over a period of 100 days (same training day repeated 100 times). The experiments are run 10 times and averaged.
\subsubsection{Simple Prediction}
This sub-case is employed in order to show the difference between accurate daily forecasts and ones that are affected by a higher amount of forecasting errors, in particular in the situation of anomalous days. This can be visualised in Fig. \ref{fig:spred_case}.
\subsubsection{Anomaly Prediction Case}
This sub-case tests the algorithm on a relatively accurate forecast of the baseload demand. In non-anomalous days simple re-prediction achieves similar levels of accuracy, so it could be said that for normal days the other two sub-cases are sufficient in order to show effects of forecasting accuracy. We present the results obtained by the four algorithms in such instances in Fig. \ref{fig:apred_case}.
\subsubsection{Perfect Prediction Case}
Here we assume that our estimation of the environment's future baseload demand is 100\% accurate. Obviously this is not the case in real life, but we use the preliminary results to see the effects of forecasting errors on algorithm performance. The results of the four algorithms in this case are presented in Fig. \ref{fig:perfect_case}.
\subsection{Results and Analysis}
From the graphs we can see that the simple Greedy solution performs worst when compared with the other three methods. This is quite obvious once we look at the efficiency graph in Fig. \ref{fig:perfect_eff}. As soon as the vehicles arrive home they start charging, which occurs exactly during the peak time consumptions (which is the expected behaviour when people come home and start using their home appliances such as ovens, kettles, washing machines, lights, etc.). As a result EVs cause almost an increase of 50\% in demand, which has to be accounted for by turning on additional power plants (expensive). The improved greedy solution leads to the vehicles deciding to charge at night time, starting from 23:00. While this doubles the performance of the Greedy solution (as seen in Fig. \ref{fig:perfect_eff}), if we take a closer look on the actual impact over the power demand curve from Fig. \ref{fig:perfect_case}, we can notice that it also has an undesired effect. The demand suddenly peaks at 23:00 creating a spike significantly higher than the baseload demand at peak time. Again this creates additional problems on the generator side of power networks and leads to unnecessary additional costs. A somewhat similar pattern can also be noticed when analysing the behaviour of the MARL algorithm. As soon as demand decreases several EVs choose to start charging. As the aggregated amount of energy used increases, the EV agents realise immediately that their choice is leading to an undesired state and start backing off. The follow-up is a usage pattern that is much closer to the desired optimal valley-filling behaviour achieved by the centralised algorithm. There are still some random variations which happen due to the same simultaneous decision of several agents. Fig. \ref{fig:perfect_eff} points out the evolution of the MARL as it moves from exploration to exploitation stages. It explores various possibilities for 40 learning episodes (days), and afterwards it starts exploiting. This is clearly noticeable in the graph. We have a sudden jump in efficiency, and then algorithm requires only 10 more learning episodes to converge to it's near-optimal solution (a 92\% Pareto optimal solution).
\begin{table}[b]
\small
\caption{Comparison of Performance}
\begin{tabular}{|l|c|c|c|c|}
\hline
\bf{Method} & \bf{Greedy} & \bf{N. Greedy} &\bf{MARL} & \bf{V-F} \\
\hline
\bf{Perfect} & 46.1\% & 83.1\% & 92.4\% & 100\%\\
\hline
\bf{Repredicted} & 44.1\% & 83.5\% & 92.2\% & 97.6\%\\
\hline
\bf{Simple} & 44.6\% & 83.8\% & 89.6\% & 97.9\%\\
\hline
\end{tabular}
\label{tab:perf_anal}
\end{table}
The greedy solutions are obviously inferior in terms of Pareto optimality and effects on the electrical grid, but they also choose to charge the vehicles more than the other algorithms. While this can be useful in unexpected situations, on average the cars will have a higher SOC at the beginning of the day than required for their daily trips. The valley-filling algorithm will always charge precisely the amount required for the daily trip, while the MARL algorithm shows a somewhat more sensible form of reasoning: if the car is generally charged enough (80\%), the EV agents tend to be satisfied and some stop charging even though they would have more available charge slots. This behaviour does not show any bad effects on the EV performance, as after ten experiments each worth 60 learning episodes, the vehicles have never even once run out of charge. Moreover, they all arrive home with a SOC in between 40\%-60\% - which obviously leaves them space for those extra unexpected trips. Another side effect is the fact that battery life is therefore expected to increase if the battery is not fully charged each time \cite{Hoffart2008}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{perfect_efficiency}
\caption[c]{Efficiency Evaluation}
\label{fig:perfect_eff}
\end{figure}
The performance of the algorithms in terms of Pareto optimality is summarised in Table \ref{tab:perf_anal}. As expected the best performance is achieved in the perfect prediction case, assuming perfect anticipation of the future power demand. Even in the case of reprediction the performance of the MARL is only minimally diminished (0.2\%) as it is able to adjust in real-time, unlike the other algorithms which are schedule based. This can be seen in Fig. \ref{fig:apred_case}. The Greedy and Night-Greedy algorithms have very similar performance in all three cases. Together with the decrease in forecasting accuracy comes a decrease in the performance of P-MARL as well. The increase in forecasting errors from 4.66\%\hspace{2pt}MAPE in the anomalous reprediction method to the 7.66\%\hspace{2pt}MAPE of the simple prediction comes with a price in terms of MARL performance, which is brought down from 92.4\% to 89.6\% Pareto optimality. P-MARL still outperforms the greedy solutions, while the centralised solution's performance doesn't seem to be affected at all in this case. This is due to the fact that, despite significant differences in demand (the initial prediction provides an estimate whose peak is underestimated by 5\%) the actual curve's shape maintains the same pattern in this particular case, thus allowing the valley-filling algorithm to perform similarly. P-MARL's second objective is based on statistics computed from the estimated demand (average demand and standard deviation), thus lower estimated peaks lead to EV power usage in times of actual high demand of the day, as can be noticed in Fig. \ref{fig:spred_case} where some EV charge occurs even during peak times. The EVs expect periods of higher demand to occur later, so they try to take advantage of the periods of supposedly lower demand. This leads to an undesired effect. A small percentage of the EVs actually end up charging in the period with the highest demand, thus the least cost-effective one.
\section{Conclusions and Future Work}
This paper presents P-MARL, a solution which improves MARL performance in uncertain environments. The proposed solution reaches near-optimal results, within 92\% Pareto optimality. As noticed in the Evaluation section, the performance of the P-MARL is closely related to the ability of being able to accurately forecast future states of the environment. The loss in accuracy results in diminished performance, therefore the need of very good environment prediction mechanisms is justified. Uncertainty can be dealt with by pattern-change detection elements which are able to trigger the re-evaluation of the environment's future behaviour. The effects of such a mechanism on forecasting accuracy and algorithm's performance has been noticed in the experimental section of the paper. Even though assuming perfect estimation of the environment's future behaviour, the dynamics implied by multi-agent systems lead to stochastic behaviour resulting sometimes in undesired effects. In this scenario we have proposed P-MARL as an online non-collaborative MARL solution, with no possibilities of communication. If we disregard the privacy concerns and communication overhead of collaborative MARLs, we believe that agents' cooperation in collaborative environments should lead to improved solutions, without significant deflections from the optimal performance. This will be the scope of future investigations, where we intend to apply collaborative algorithms such as DWL \cite{Dusparic2010}.
\newpage
\bibliographystyle{aaai}
|
2,877,628,090,229 | arxiv | \section{Introduction}
While there is a plethora of different reduction notions that have been studied in computational complexity (see e.g.~\cite{hemaspaandraO02} for an overview), it has often been observed that in nearly all $\mathsf{NP}$-completeness proofs in the literature logarithmic space many-one reductions suffice.
In contrast to $\mathsf{NP}$-completeness, many $\mathsf{\#P}$-hardness results in counting complexity are not shown with many-one reductions but with the more permissive Turing-reductions and the $\mathsf{\#P}$-hardness under many-one reductions remains an open problem.
It is natural to ask if there is a fundamental difference between both $\mathsf{\#P}$-hardness notions.
Note that the question of the relative power of reduction notions for $\mathsf{NP}$-completeness has been studied and there are known separations under different complexity assumptions, see e.g.~the survey~\cite{Pavan03}.
In this short note, we answer an analogous question for arithmetic circuit complexity, the algebraic sibling of counting complexity. In arithmetic circuit complexity the most usual reduction notion are so-called \textup{p}-projections. Despite being very restricted, \textup{p}-projections have been used to show nearly all of the completeness results in the area since the ground-breaking work of Valiant~\cite{valiant79}. It was only more recently that \textup{c}-reductions, a more permissive notion more similar to Turing- or oracle-reductions, have been defined in \cite{bur00} and used for some results (see e.g.~\cite{BriquelK09,Rugy-Altherre12,DurandMMRS14}).
Again the question comes up if there is a fundamental difference between these two notions of reductions. In fact, it was exactly this uncertainty about the relative power of \textup{p}-projections and \textup{c}-reductions that motivated the recent work Mahajan and Saurabh~\cite{MahajanS16}:
For the first time they prove a natural problem complete for the arithmetic circuit class $\vp$ under \textup{p}-projections,
where before there existed only such result under \textup{c}-reductions.
In this paper we answer the question of the relative strength of of \textup{p}-projections and \textup{c}-reductions:
We show unconditionally that over every field $\mathbb{F}$ there are explicit families of polynomials that are $\vnp$-complete over $\mathbb{F}$ under \textup{c}-reductions that are not $\vnp$-complete over $\mathbb{F}$ under \textup{p}-projections.
We also show that the question which polynomials are complete under which reductions depends on the underlying field in a rather subtle way.
It is a well known phenomenon that the permanent family,
which is $\vnp$-complete under \textup{p}-projections over fields of characteristic different from 2,
is contained in $\vp$ over fields of characteristic 2 and thus likely not $\vnp$-hard there.
We present a more subtle situation:
We give an explicit family of polynomials that is $\vnp$-complete under \textup{c}-reductions over all fields with more than 2 elements
and that is even $\vnp$-complete under \textup{p}-projections over a large class of fields including the complex numbers,
but over the real numbers it is only $\vnp$-complete under \textup{c}-reductions and not under \textup{p}-projections.
\paragraph*{Acknowledgements.} The authors would like to thank Dennis Amelunxen for helful discussions. Some of the research leading to this article was performed while the authors were at the Department of Mathematics at the University of Paderborn and at Texas A\&M University.
\section{Preliminaries}
We only give some very minimal notions of arithmetic circuit complexity. For more details we refer the reader to the very accessible recent survey~\cite{Mahajan14}.
The basic objects to be computed in arithmetic circuit complexity are polynomials. More precisely, one considers so called \textup{p}-families of polynomials, which are sequences $(f_n)$ of multivariate polynomials such that the number of variables in $f_n$ and the degree of $f_n$ are both bounded by a polynomial in $n$.
We assume that each \textup{p}-family computes polynomials over a field $\mathbb{F}$ which will vary in this paper but is fixed for each \textup{p}-family.
A polynomial $f$ in the variables $X_1, \ldots, X_n$ is a projection of a polynomial $g$, in symbols $f\le g$, if $f(X_1, \ldots, X_n) = g(a_1, \ldots, a_m)$ where the $a_i$ are taken from $\{X_1, \ldots, X_n\} \cup \mathbb{F}$. The first reduction notion we consider in this paper are so-called \textup{p}-projections: A \textup{p}-family $(f_n)$ is a \textup{p}-projection of another \textup{p}-family $(g_n)$, symbol $(f_n) \le_\textup{p} (g_n)$ if there is a polynomially bounded function $t$ such that
\[\exists n_0 \forall n \ge n_0 \colon f_n \le g_{t(n)}.\]
Intuitively, \textup{p}-projections appear to be a very weak notion of reductions although surprisingly the bulk of completeness results in arithmetic circuit complexity can be shown with them. For some \textup{p}-families, though, showing hardness with \textup{p}-projections appears to be hard, and consequently, a more permissive reduction notion called \textup{c}-reductions has also been used.
The oracle complexity $L^{g}(f)$ of a polynomial $f$ with oracle $g$ is the minimum number of arithmetic operations $+$, $-$, $\times$, and evaluations of $g$ at previously computed values that are sufficient to compute~$f$ from the variables $X_1, X_2, \ldots$ and constants in $\mathbb{F}$.
Let $(f_n)$ and $(g_n)$ be \textup{p}-families of polynomials. We call $(f_n)$ a \emph{\textup{c}-reduction} of $(g_n)$, symbol $(f_n)\le_\textup{c} (g_n)$, if and only if there is a polynomially bounded function $t:\mathbb{N}\rightarrow \mathbb{N}$ such that the map $n\mapsto L^{g_{t(n)}}(f_n)$ is polynomially bounded.
Intuitively, if $(f_n) \le_\textup{c} (g_n)$, then we can compute the polynomial in $(f_n)$ with a polynomial number of arithmetic operations and oracle calls
to $g_{t(n)}$, where $t(n)$ is polynomially bounded.
Let $C_n$ denote the group of cyclic cyclic permutations on $n$ symbols and define the $n$th Hamiltonian cycle polynomial $\mathrm{HC}_n$ as $\mathrm{HC}_n := \sum_{\pi \in C_n} \prod_{i=1}^n X_{i,\pi(i)}$.
To keep these preliminaries lightweight, we omit the usual definition of $\vnp$ and instead define $\vnp$ to consist of all \textup{p}-families $(g_n)$ with $(g_n) \le_\textup{p} (\mathrm{HC}_n)$.
A \textup{p}-family $(g_n)$ that satisfies $(f_n)\le_\textup{p} (g_n)$ for all $f_n \in \vnp$ is called
$\vnp$-hard under \textup{p}-projections or $\vnp$-\textup{p}-hard for short.
Analogously,
a \textup{p}-family $(g_n)$ that satisfies $(f_n)\le_\textup{c} (g_n)$ for all $f_n \in \vnp$ is called
$\vnp$-hard under \textup{c}-reductions or $\vnp$-\textup{c}-hard for short.
If $(g_n)$ is $\vnp$-\textup{p}-hard and contained in $\vnp$, then $(g_n)$ is call $\vnp$-\textup{p}-complete.
Analogously for $\vnp$-\textup{c}-completeness.
Clearly if a family is $\vnp$-\textup{p}-complete, then it is also $\vnp$-\textup{c}-complete.
Note that a \textup{p}-family $(g_n)$ is $\vnp$-\textup{p}-hard (resp.~$\vnp$-\textup{c}-hard) iff $(\mathrm{HC}_n)\le_\textup{p} (g_n)$ (resp.~$(\mathrm{HC}_n)\le_\textup{c} (g_n)$).
\section{\textup{c}-reductions are strictly stronger than \textup{p}-projections}
In this section, we show that there are polynomials that are $\vnp$-\textup{c}-complete but not $\vnp$-\textup{p}-complete.
Let $X$ denote a new variable, unused by $\mathrm{HC}_n$ for any $n$.
Define
\[
P_n := X \cdot \mathrm{HC}_n + (\mathrm{HC}_n)^2.
\]
Note that $P_n$ is defined for every field.
We remark that $(P_n)$ can easily be shown to be contained in $\vnp$, because $\mathrm{HC}_n \in \vnp$ and the class $\vnp$ is closed under multiplication and addition~\cite{Valiant82} (see also~\cite[Theorem 2.19]{bur00}).
\begin{lemma}\label{lem:ccomplete}
$(P_n)$ is $\vnp$-\textup{c}-complete over every field.
\end{lemma}
\begin{proof}
Fix a field $\mathbb{F}$.
For field elements $\alpha \in \mathbb{F}$
let $P_n(X \leftarrow \alpha)$ denote $P_n$ with variable $X$ set to $\alpha$.
We observe that
\[
P_n(X \leftarrow 1) - P_n(X \leftarrow 0) = \mathrm{HC}_n
\]
and thus
$P_n$ is $\vnp$-\textup{c}-complete.
\end{proof}
\begin{lemma}\label{lem:notcomplete}
$(P_n)$ is not $\vnp$-\textup{p}-complete over any field.
\end{lemma}
\begin{proof}
Let $f$ be any univariate polynomial in some variable $Y$ and let $f$ be of odd degree at least 3. We show that $f$ is not a projection of~$P_n$ for any $n$,
which finishes the proof because then the constant \textup{p}-family $(f)$ is not a \textup{p}-projection of $(P_n)$.
For a multivariate polynomial $h$ let $\deg_Y(h)$ denote the $Y$-degree of $h$,
which is the degree of $h$ interpreted as a univariate polynomial in $Y$ over the polynomial ring with additional variables.
Let $A$ be an $n \times n$ matrix whose entries are variables and constants.
We denote by $P_n(A)$ the linear projection of $P_n$ given by $A$.
We now analyze $\deg_Y(P_n(A))$.
Clearly $\deg_Y(X(A))\leq 1$.
If $\deg_Y(\mathrm{HC}_n(A)) \leq 1$, then $\deg_Y(P_n(A))\leq 2 < 3 \leq \deg_Y(f)$ and thus $P_n(A)\neq f$.
If $\deg_Y(\mathrm{HC}_n(A)) \geq 2$, then $\deg_Y(P_n(A)) = \deg_Y ((\mathrm{HC}_n(A))^2) = 2 \deg_Y (\mathrm{HC}_n(A))$.
But $\deg_Y(f)$ is an odd number, so in this case we also have $P_n(A)\neq f$.
\end{proof}
As a corollary we get that \textup{c}-reductions yield strictly more complete problems that \textup{p}-projections.
\begin{theorem}\label{thm:main}
For every field $\mathbb{F}$, $(P_n)$ is $\vnp$-\textup{c}-complete over $\mathbb{F}$, but not $\vnp$-\textup{p}-complete over $\mathbb{F}$.
\end{theorem}
\section{The dependence on the field}\label{sct:dependence}
In this section we construct a family $(Q_n)$ of polynomials that is
is $\vnp$-\textup{c}-complete over all fields with more than two elements,
but over the real numbers $(Q_n)$ is not $\vnp$-\textup{p}-complete.
This shows that the relative power of different reductions notions depends on the field and is thus likely quite complicated to characterize in general.
We consider the polynomials $Q_n$ defined on the matrix $(X_{ij})_{i,j\in [n]}$ defined by \[Q_n :=\sum_{\pi \in C_n} \prod_{i\in [n]} X_{i,\pi(i)} + \sum_{\pi \in C_n} \prod_{i\in [n]} X_{i,\pi(i)}^2.\] Note that $Q_n$ is similar to the polynomial $P_n$ considered before.
But unlike $P_n$ the homogeneous part of degree $n^2$ of $Q_n$ is \emph{not} $(\mathrm{HC}_n)^2$ but only contains a subset of the monomials.
Using Valiant's criterion, it is easy to see that $(Q_n) \in \vnp$, see for example \cite{bur00}[Proposition 2.20].
Although from its algebraic properties $Q_n$ might look very different from $P_n$,
the following Lemma can be proved exactly as Lemma~\ref{lem:ccomplete}.
\begin{lemma}\label{lem:ccompleteQ}
$(Q_n)$ is $\vnp$-\textup{c}-complete over every field with more than 2 elements.
\end{lemma}
\begin{proof}
Fix a field $\mathbb{F}$ with more than 2 elements.
The proof is a simple interpolation argument.
Choose $a \in \mathbb{F}$ with $a \notin\{0,1\}$.
For a variable matrix
\[
X=\begin{pmatrix}
X_{1,1} & X_{1,2} & \cdots & X_{1,n}\\
X_{2,1} & X_{2,2} & \cdots & X_{2,n}\\
\vdots & \vdots & \ddots & \vdots \\
X_{n,1} & X_{n,2} & \cdots & X_{n,n}
\end{pmatrix}
\]
let $\bar X$ denote $X$ with the first row scaled by $a$:
\[
\bar X=\begin{pmatrix}
a X_{1,1} & a X_{1,2} & \cdots & a X_{1,n}\\
X_{2,1} & X_{2,2} & \cdots & X_{2,n}\\
\vdots & \vdots & \ddots & \vdots \\
X_{n,1} & X_{n,2} & \cdots & X_{n,n}
\end{pmatrix}.
\]
Clearly $\mathrm{HC}_n(\bar X) = a \mathrm{HC}_n(X)$.
Moreover,
\[
Q_n(\bar X) = a \mathrm{HC}_n(X) + a^2 \sum_{\pi \in C_n} \prod_{i\in [n]} X_{i,\pi(i)}^2.
\]
Therefore
\[
(a-a^2)\mathrm{HC}_n(X) = Q_n(\bar X) - a^2 Q_n(X).
\]
But $a-a^2 = a(1-a) \neq 0$ because $a \notin \{0,1\}$.
We conclude
\[
\mathrm{HC}_n(X) = \frac{1}{a-a^2} Q_n(\bar X) - \frac{a^2}{a-a^2}Q_n(X).
\]
It follows that $Q_n$ is even $\vnp$-\textup{c}-complete under linear $p$-projections, a restricted form of $c$-reductions (see \cite[p. 54]{bur00}).
\end{proof}
We now show that over the real numbers Lemma~\ref{lem:ccompleteQ} cannot be improved from \textup{c}-reductions to \textup{p}-projections.
\begin{lemma}\label{lem:notcompleteQ}
$(Q_n)$ is not $\vnp$-\textup{p}-complete over $\mathbb{R}$.
\end{lemma}
\begin{proof}
We show that the polynomial $X$ is not a projection of~$Q_n$ for any $n$. Assume this were not the case.
Then there is an $(n\times n)$-matrix $A=(a_{ij})$ whose entries are variables or constants such that $P_n(A) = X$.
W.l.o.g.\ we assume that no other variables than $X$ appear in $A$, so $a_{ij} \in \{X\}\cup \mathbb{R}$.
Let $\sigma \in C_n$ be an $n$-cycle such that
$\prod_{i=1}^n a_{i\sigma(i)}$ has maximal degree. Obviously this degree is at least $1$. Then the monomial $\prod_{i=1}^n a_{i\sigma(i)}^2$ has at least degree $2$ and it cannot cancel out in $Q_n$ because
\begin{itemize}
\item it cannot cancel with any $\prod_{i=1}^n a_{i\mu(i)}$ for an $n$-cycle $\mu$, because those all have smaller degrees, and
\item it cannot cancel out with any $\prod_{i=1}^n a_{i\mu(i)}^2$, because those all have positive coefficients in $Q_n(A)$.
\end{itemize}
Thus $Q_n(A)$ has degree at least $2$, which implies that $Q_n(A)\neq X$.
\end{proof}
Interestingly, Lemma \ref{lem:notcompleteQ} does not generalize to arbitrary fields.
\begin{lemma}\label{lem:completeQ}
Let $\mathbb{F}$ be a field such that there are elements $a_1, \ldots, a_s$ with $\sum_{i=1}^s a_i \ne 0$ and $\sum_{i=1}^s a_i^2 = 0$. Then $(Q_n)$ is $\vnp$-\textup{p}-complete over $\mathbb{F}$.
\end{lemma}
\begin{proof}
For an $(n\times n)$-matrix $A$ let $\mathrm{HC}(A)$ be the Hamiltonian cycle polynomial evaluated at $A$ and set $\mathrm{HC}(A^{(2)}) := \sum_{\sigma\in C_n} \prod_{i=1}^n a_{i\sigma(i)}^2$. With this notation clearly $Q_n(A) = \mathrm{HC}(A) + \mathrm{HC}(A^{(2)})$.
>From an $(s\times s)$-matrix $A$ and a $(t\times t)$-matrix $B$ we construct the $(s+t+2)\times(s+t+2)$ \emph{Hamiltonian connection matrix} $\mathrm{con}(A,B)$
as follows.
Let $G_A$ be the labeled digraph with adjacency matrix $A$ and let $G_B$ be the labeled digraph with adjacency matrix $B$.
The vertex corresponding to the first row and column in $A$ is called $v_A$, analogously for $v_B$.
The labeled digraph $G_A'$ is defined by replacing $v_A$ in $G_A$ by two vertices $v_A^{\text{in}}$ and $v_A^{\text{out}}$
such that the edges going into $v_A$ now go into $v_A^{\text{in}}$
and the edges coming out of $v_A$ now come out of $v_A^{\text{out}}$.
This operation increases the total number of vertices by one: $|V(G_A)|+1=|V(G_A')|$.
We create a labeled digraph $G_{\mathrm{con}(A,B)}$ as the union of $G_A'$ and $G_B'$ with two additional edges,
one going from $v_A^{\text{in}}$ to $v_B^{\text{out}}$ and the other from
$v_B^{\text{in}}$ to $v_A^{\text{out}}$, both labelled with 1.
Let $\mathrm{con}(A,B)$ denote the $(s+t+2)\times(s+t+2)$ adjacency matrix of $G_{\mathrm{con}(A,B)}$.
By construction we have a bijection between the set of Hamiltonian cycles in $G_{\mathrm{con}(A,B)}$ and
the set of pairs $(c_A,c_B)$ of Hamiltonian cycles $c_A$ in $G_A$ and $c_B$ in $G_B$.
Thus $\mathrm{HC}(\mathrm{con}(A,B))=\mathrm{HC}(A)\mathrm{HC}(B)$ and $\mathrm{HC}(\mathrm{con}(A,B)^{(2)})=\mathrm{HC}(A^{(2)})\mathrm{HC}(B^{(2)})$.
Therefore
\begin{equation}\label{eqn:decompose}
Q_{s+t+2}(\mathrm{con}(A,B)) = \mathrm{HC}(A)\mathrm{HC}(B) + \mathrm{HC}(A^{(2)})\mathrm{HC}(B^{(2)}).
\end{equation}
Let $a:= \sum_{i=1}^s a_i$ and \[A:= \begin{pmatrix} 0 & a^{-1} & a^{-1} & \ldots & a^{-1} & a^{-1}\\
a_1 & 0 & 0 & \ldots & 0 & 1 \\
a_2 & 1 & 0 & \ldots & 0 & 0 \\
a_3 & 0 & 1 & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
a_s & 0 & 0 & \ldots & 1 & 0 \\
\end{pmatrix}
\]
It is easy to verify that $\mathrm{HC}(A) = \sum_{i=1}^s a_i a^{-1} = 1$ and $\mathrm{HC}(A^2) = \sum_{i=1}^s a_i^2 (a^{-1})^2 = \left(\sum_{i=1}^s a_i^2\right) a^{-2} = 0$.
Thus we get with (\ref{eqn:decompose})
\[Q_{s+t+2}(\mathrm{con}(A,B)) = \mathrm{HC}(A)\mathrm{HC}(B) + \mathrm{HC}(A^{(2)})\mathrm{HC}(B^{(2)}) = \mathrm{HC}(B) \]
for every $(t\times t)$-matrix $B$.
Thus the Hamiltonian cycle family $(\mathrm{HC}_n)$ is a \textup{p}-projection of $(Q_n)$ and the claim follows.
\end{proof}
\begin{corollary}\label{cor:complete2Q}
\begin{enumerate
\item[a)] $(Q_n)$ is $\vnp$-\textup{p}-complete over $\mathbb{C}$.
\item[b)] $(Q_n)$ is $\vnp$-\textup{p}-complete over any field of characteristic greater than $2$.
\end{enumerate}
\end{corollary}
\begin{proof}
a) Set $s:=2$ and $a_1=1$ and $a_2 =i$. We have $a_1 + a_2 =1+i \neq 0$ and $a_1^2 + a_2^2 = 0$ and thus the claim follows by Lemma \ref{lem:completeQ}.
b) Let $p>2$ be the characteristic of the field and set $s:=p$. We have \[\sum_{i=1}^{\frac{p-1}{2}} 1 + \sum_{i=1}^{\frac{p+1}{2}} (-1) = -1 \ne 0\] and \[\sum_{i=1}^{\frac{p-1}{2}} 1^2 + \sum_{i=1}^{\frac{p+1}{2}} (-1)^2 = p\cdot 1 = 0.\] With Lemma \ref{lem:completeQ} the claim follows.
\end{proof}
\section{Conclusion}
We have shown that for all fields \textup{c}-reductions and \textup{p}-projections differ in power.
Note that one could show versions of Theorem~\ref{thm:main} for essentially all other complexity classes from arithmetic circuit complexity, as long as they contain complete families of homogeneous polynomials and the polynomial~$X$. Since the proofs are essentially identical, we have not shown these results here.
We have also shown that the question which families are complete under which reductions also depends on the field. This indicates that understanding the exact power of different reduction notions is probably very complicated.
Another question is with respect to the naturalness of our separating examples. They have been specifically designed for our results and apart from that we do not consider them very interesting. Can one show that the more natural polynomials in \cite{BriquelK09,Rugy-Altherre12,DurandMMRS14} which were shown to be complete under \textup{c}-reductions are not complete under \textup{p}-projections?
\newcommand{\etalchar}[1]{$^{#1}$}
|
2,877,628,090,230 | arxiv | \section{Introduction}
Cooling flow groups and clusters contain large amounts of hot
X-ray emitting gas that should be radiatively cooling on time
scales less than a Hubble time (see Fabian, Nulsen \& Canizares
1984 for an early review of cooling flows). The primary uncertainty
in the original cooling flow scenario was the ultimate fate of
the cooling gas. While diffuse H$\alpha$ emission was commonly found within the central
dominant galaxy (CDG) in cooling flows (Heckman 1981; Hu et al. 1985)
and star formation was observed in cluster cooling flows as early as
McNamara \& O'Connell (1989), the observed star formation rates
were orders of magnitudes less than the inferred mass deposition rates
of the hot gas. The cooling flow scenario underwent a major modification after
the launch of the {\it Chandra} and {\it XMM-Newton} X-ray telescopes.
{\it Chandra}, with its high spatial resolution, discovered
AGN-induced cavities and shocks within cooling flows (e.g., McNamara et al. 2000;
Blanton et al. 2003; Fabian et al. 2003; Forman et al. 2007; Randall et al. 2011),
while both {\it XMM-Newton} and {\it Chandra} found little spectroscopic
evidence for large amounts of cooling gas (David et al. 2001; Peterson et al. 2003).
By compiling observations of cooling flow clusters, several groups
(Birzan et al. 2004; Dunn \& Fabian 2006, O'Sullivan et al. 2011) have shown
that the mechanical power of the AGN in the CDG is
sufficient to prevent the bulk of the hot gas from cooling. While AGN feedback
probably prevents most of the hot gas from cooling, star formation
has now been detected in many CDGs residing at
the centers of cooling flows
(e.g., Allen et al. 1995; Rafferty et al. 2006; Donahue et al. 2007; Quillen et al. 2008)
and there are many indications that
the star formation is a product of the cooling flow, including:
1) a correlation between the star formation rate in CDGs and the spectroscopic
mass deposition rate (O'Dea et al. 2008), 2) the existence of a sharp
threshold for star formation that occurs when the cooling time of the
hot gas is less than about 1~Gyr (Rafferty et al. 2008) or the entropy is less
than 30~keV~cm$^{2}$ (Voit et al. 2008) and 3) a correlation between the
star formation rate derived from {\it Spitzer} and {\it Herschel} data and
the radiative cooling time of the hot gas (Egami et al. 2006a; Rawle et al. 2012).
Single dish CO surveys over the past decade have shown that a
substantial fraction of CDGs in
cluster cooling flows harbor molecular gas (Edge 2001; Salome et al. 2003).
Warm molecular gas has also been detected in CDGs by emission from
vibrationally excited molecular hydrogen (e.g., Jaffe \& Bremer 1997;
Donahue et al. 2000; Egami et al. 2006b).
Recent Atacama Large (sub)Millimeter Array (ALMA) observations
of A1664 (Russell et al. 2014) and
A1835 (McNamara et al. 2014) show that both of these systems
contain more than ${10^{10} M_{\odot}}$ of molecular gas.
There is also evidence that A1835 may have a high velocity
molecular outflow driven by the radio jets or buoyant X-ray cavities.
In this paper we present the results of a cycle 0 ALMA observation
of NGC 5044 in the CO(2-1) emission line. The NGC 5044 group of galaxies
is the X-ray brightest group in the
sky and has a very smooth and nearly spherically symmetric large
scale X-ray morphology. However, the central region of NGC 5044
is highly perturbed with many AGN-inflated cavities, cool X-ray
filaments, cold fronts and multiphase gas (Buote et al. 2003; Gastaldello et al. 2009;
David et al. 2009; 2011). NGC 5044 also hosts a system of very bright H$\alpha$
filaments (Caon et al. 2000; Sun et al. 2014).
The presence of three cold fronts in the Chandra
and XMM-Newton data (Gastaldello et al. 2013; O'Sullivan et al. 2014) and a
peculiar velocity of 140~km~s$^{-1}$ relative to the group mean (Cellone \& Buzzoni 2005)
suggest that NGC 5044 is likely undergoing a sloshing motion within the group center.
Large scale asymmetries in the X-ray morphology (Gastaldello et al. 2013) and
regions of enhanced elemental abundances (O'Sullivan et al. 2014) suggest that
the sloshing orbital plane of NGC 5044 is perpendicular to the plane of the sky based on
the simulations in Roediger et al. (2011).
This paper is organized as follows. Section 2 contains a description
of the ALMA data reduction. Our results concerning the mass and
kinematics of the observed molecular structures and correlations
between the molecular gas, dust, H$\alpha$ filaments and hot gas
are presented in $\S 3$. Implications concerning the origin,
dynamics and confinement of the molecular gas are discussed
in $\S 4$ and our main results are summarized in $\S 5$.
\bigskip
\section{Data Reduction}
We observed NGC 5044 with ALMA during Cycle 0 of the scientific observations.
A single pointing was made towards the center of NGC 5044
(RA=13:15:23.97, Dec-16.23.07.5) on 2012 January 13. The primary beam
of ALMA at 1.3~mm is $\sim$~27$^{\prime\prime}$ arcsec at full-width half maximum and
provides a field of view of $\sim$~4.0 kpc at the distance of NGC 5044. The
observation spanned 1.0~hr, for a total on-source integration time of 29~min.
The quasar J1337-129, located $6.38^{\circ}$ from NGC 5044, served as the complex gain
(secondary) calibrator. Scans of J1337-129 were made every 12 min bracketing scans
of NGC 5044. The quasar 3C 279 served as the bandpass calibrator and
Titan as the absolute flux (primary) calibrator.
The amount of precipitable water vapor during our observation was 1.4 mm
The correlator was configured to provide four spectral windows in two linear
polarizations (no measurements of the cross products were made in ALMA cycle 0).
Each spectral
window spanned a total bandwidth of 1875~MHz and was split into 3840 channels
so that each channel had a width of 488.28125~kHz. One of the spectral windows
was centered on the CO(2-1) line (1.3-mm band) at the systemic velocity
of NGC 5044, providing a velocity resolution of 0.64~km~s$^{-1}$ in CO(2-1).
The other three spectral windows (229.336-231.211 GHz, 241.270-243.145 GHz
and 242.853-244.728 GHz) covered line-free regions to measure the
continuum emission of NGC 5044, necessary for subtracting any continuum
emission at the frequency of the CO(2-1) line. As explained below, we detected
relatively strong continuum emission from the central AGN of NGC 5044.
The data were calibrated using the software package CASA by the ALMA observatory.
The calibrated data, along with a set of continuum-subtracted channel maps in CO(2-1),
were delivered to us on 2013 March 4. To make channel maps suitable to our own needs,
we performed our own continuum subtraction in the CO(2-1) line and made channel
maps at our own desired velocity resolutions and weightings of the visibility
data. The synthesized beam attained in our cycle 0 ALMA observation was
1.4$^{\prime\prime}$ by 2.2$^{\prime\prime}$ FWHM (210 by 330~pc).
We adopt a systemic velocity of 2758~km~s$^{-1}$ for NGC 5044 and a luminosity distance
of 31.2~Mpc (Tonry et al. 2001), which gives a physical scale
in the rest frame of NGC 5044 of $1^{\prime\prime}=150$~pc.
\section{Results}
The most significant CO(2-1) feature in the ALMA Cycle 0 spectrum
within the central 1~kpc diameter region is the
redshifted emission between 0 and 120~km~s$^{-1}$ with a sharp peak at
60~km~s$^{-1}$ (see Fig. 1). The blueshifted emission within this region is
broader and less peaked,
with emission spanning a velocity range from -300 to 0~km~s$^{-1}$.
For comparison, we also show in Fig.1 the IRAM 30m CO(2-1) spectrum
and the ALMA CO(2-1) spectrum extracted from within a 11$^{\prime\prime}$
diameter region (i.e., the full-width half maximum of the IRAM 30m primary
beam). While the redshifted portions of the IRAM 30m and ALMA spectra are in reasonably
good agreement, the integrated flux density in the blueshifted portion of
the ALMA spectrum is only 20\% of that in the IRAM 30m data.
Similar results were found by Russell et al. (2014)
for Abell 1664 and McNamara et al. (2014) for Abell 1835 when
comparing IRAM 30m and ALMA data and is presumably due to the
presence of diffuse CO(2-1) emission in these systems that
is resolved out in the ALMA data.
To search for discrete molecular structures in NGC 5044, we binned the ALMA data cube into
velocity slices with bin widths ranging from 10 to 100~km~s$^{-1}$.
We found that a bin width of 50~km~s$^{-1}$ optimized the signal-to-noise
of the emission lines.
Fig. 2 displays channel maps between -350~km~s$^{-1}$ and 150~km~s$^{-1}$ in
50~km~s$^{-1}$ slices. A total of 24 molecular structures are detected with a
CO(2-1) surface brightness exceeding $4 \sigma$ within the central 2.5~kpc
(16.7$^{\prime\prime}$) where the primary beam response is greater than 0.3. More
structures are detected at larger radii, but we limit all further analysis in
this paper to these 24 molecular structures.
Our ALMA data covered a velocity range from -400 to 400~km~s$^{-1}$ centered
\begin{inlinefigure}
\center{\includegraphics[width=1.00\linewidth,bb=18 144 573 701,clip]{f1.pdf}}
\caption{ALMA CO(2-1) spectrum from within the central
1~kpc ($6.7^{\prime\prime}$) diameter region (solid black line). For comparison,
the IRAM 30m CO(2-1) spectrum (solid red line) and the ALMA cycle 0 spectrum
(dashed black line) from within a 11$^{\prime\prime}$ diameter region
(i.e., the full-width half maximum of the IRAM 30m primary beam) are also shown.}
\end{inlinefigure}
\noindent
on the systemic velocity of NGC 5044. No molecular structures were detected in
emission with redshifts between 150-400~km~s$^{-1}$ or
blueshifts between 350-400~km~s$^{-1}$. However, an absorption feature is detected
in the central continuum source at 250~km~s$^{-1}$ and is discussed further in
$\S 3.2$.
The channel maps show that the velocity distribution of the
molecular structures in NGC 5044 is highly asymmetric. There are 16 blueshifted
structures and only 8 redshifted structures within the central 2.5~kpc.
The blueshifted structures span a broad range in velocity, while the redshifted
structures are more localized in velocity space. The blueshifted structures
also are more centrally concentrated (at least in projection) compared to the
redshifted structures.
CO(2-1) spectra were extracted for all 24 molecular structures identified in
Fig. 2 and fit to a Gaussian profile. The best-fit mean velocity, linewidth and integrated
flux density are shown in Table 1 and the spectra are shown in Fig. 3. For some
molecular structures, the signal-to-noise was insufficient to obtain a well constrained fit.
In addition, the spectra of some structures are not well described by a
simple Gaussian. For example, see the spectrum of molecular structure 13 in Fig. 3,
which has a extended blue wing. For these molecular structures, only the mean velocity
and integrated flux density are listed in Table 1. The molecular mass was computed using
the relation in Bolatto et al. (2013),
\begin{equation}
{M_{mol} = 1.05 \times 10^4~ S_{CO} \Delta v ~D_L^2 ~(1+z)^{-1} M_{\odot}}
\end{equation}
\noindent
where ${S_{CO} \Delta v}$ is the integrated CO (1-0) flux density in
units of Jy~km~s$^{-1}$ and $D_L$ is the luminosity distance in units of Mpc.
This relation is based on the galactic CO-to-H$_2$ conversion factor of
$X_{CO}= 2 \times 10^{20}$~cm$^{-2}$~(K km s$^{-1}$)$^{-1}$.
We assume a CO(2-1) to CO (1-0) flux density ratio of 3.2 to estimate the
molecular mass based on the observed CO(2-1) to CO (1-0) brightness
temperature ratio of 0.8 for molecular clouds in spiral galaxies
(e.g., Braine \& Combes 1992) and the factor of two in frequency.
The molecular masses of the structures range from $3 \times 10^5$~M$_{\odot}$
(corresponding to the $4 \sigma$ sensitivity limit)
to $10^7$~M$_{\odot}$ and the linewidths vary from
15 to 65~km~s$^{-1}$. While resolved GMCs in the Milky Way and local
group galaxies can have masses up to $10^7$~M$_{\odot}$, linewidths seldom
exceed 10~km~s$^{-1}$ (Solomon et al. 1987; Blitz et al. 2006;
Fukui et al. 2008; Bolatto et al. 2013) and we therefore refer to the
molecular structures detected in NGC 5044 as giant molecular associations
(GMAs).
The total molecular mass of the GMAs listed in Table 1 is
$5.1 \times 10^7$~M$_{\odot}$, the mean velocity is $-69.7 \pm 6.3$~km~s$^{-1}$
and the velocity dispersion is 122~km~s$^{-1}$, which is
less than the stellar
velocity dispersion of 237~km~s$^{-1}$. While there is a significant difference
in the number of red and blueshifted GMAs, the molecular mass is evenly divided
between the red and blueshifted GMAs. This is due to the massive redshifted
GMA 18 which contains 20\% of the total molecular mass.
\subsection{The central region}
There are several GMAs within the central 1~kpc diameter region that span
more than one channel map. The channel maps suggest the possibility that GMAs 13
and 18 are contiguous across the central AGN, however the
position-velocity diagram in Fig. 4 shows that GMAs 13 and
18 are distinct molecular structures. Fig. 4 also shows that there
is no molecular structure with a smooth transition in velocities
(from redshifted to blueshifted) across
the systemic velocity of the galaxy, which would indicate
the presence of a central disk. To further investigate the
kinematics of the central GMAs we generated velocity maps in two separate
velocity slices (from -100 to 0~km~s$^{-1}$ and from 0 to 130~km~s$^{-1}$)
to prevent projection effects along the line of sight. Fig. 5 shows that
GMAs 11 and 13 have velocities near the
systemic velocity of NGC 5044 at large
radii and monotonically increasing blueshifted velocities close to the central AGN.
The blue shifted emission from GMA 13 is also visible in Fig. 3.
This suggests that these two GMAs are falling into the center of the galaxy
from the far side of the galaxy, similar to the situation observed in Perseus (Lim et al. 2008).
Fig. 6 shows that the gas velocity in GMA 18 increases along
a SW to NE direction
and suggests that GMA 18 is falling into the central region of the galaxy from
the near side of the galaxy. The feature in Fig. 6 toward the NE of GMA 18 at
a velocity of 20~km~s$^{-1}$ shows up as a separate molecular structure in Fig. 4.
The results shown in Table 1 were derived by fitting spectra
binned into 10~km~s$^{-1}$ channels. GMA 18 produces 20\% of the total
flux within the central 2.5~kpc and the signal is strong enough to permit
a higher resolution study. Fig. 7 shows a 1~km~s$^{-1}$ per channel spectrum
of GMA 18 which clearly shows the presence of two separate components.
Fitting a double Gaussian model yields a mean velocity, linewidth and molecular
mass for the narrow-line component of 58.7~km~s$^{-1}$, 5.5~km~s$^{-1}$ and
${M_{mol}=8.7 \times 10^5}$~M$_{\odot}$. While the molecular mass of the narrow-line
component in GMA 18 is comparable with some of the lower mass GMAs detected in NGC 5044,
the linewidth is much smaller and more typical of an individual GMC (Solomon et al. 1987;
Blitz et al. 2006; Fukui et al. 2008; Bolatto et al. 2013).
Using the CASA task {\it imfit}, we fit 2D Gaussians to GMAs 11, 13 and 18
(the best resolved molecular structures in the center of NGC5044 which also
comprise 35\% of the total observed
\newpage
\begin{inlinefigure}
\center{\includegraphics[width=2.1\linewidth]{f2.pdf}}
\caption{Contours of the integrated CO(2-1) intensity over a
velocity width of 50~km~s$^{-1}$ centered on the velocities given in the images.
Contours are shown at 3$\sigma$, 4$\sigma$, 5$\sigma$, $10 \sigma$ and $20 \sigma$,
where $\sigma=40$~mJy~beam$^{-1}$~km~s$^{-1}$. The circle corresponds to a
radius of 2.5~kpc in the rest frame of the galaxy. The centroid of the continuum source
is marked with an "X". The CO(2-1) emission from some GMAs is
contiguous across successive channel maps and these GMAs are labeled with
the same number and different letters on consecutive contour plots.}
\end{inlinefigure}
\newpage
\begin{inlinefigure}
\center{\includegraphics[width=2.0\linewidth]{f3.pdf}}
\caption{ALMA CO(2-1) spectra of the 24 GMAs along with the best-fit Gaussian profiles.}
\end{inlinefigure}
\newpage
\begin{inlinefigure}
\center{\includegraphics[width=1.00\linewidth,bb=30 119 417 759,clip]{f4.pdf}}
\caption{Position-Velocity diagram (where the horizontal axis corresponds to
the off-set in right ascension from the continuum source) for the central region of
NGC 5044. Note that GMAs 13 and 18 are distinct molecular structures. }
\end{inlinefigure}
\noindent
molecular mass).
The task {\it imfit} computes the best-fit FWHM along the major and minor axes
of the 2D Gaussian. We define the cloud radius as
$r_c = \sqrt{\sigma_{maj} \sigma_{min}}$, where $\sigma=$~FWHM/2.35,
and obtain deconvolved cloud radii of $115 \pm 25$, $140 \pm 15$ and $120 \pm 12$~pc
for GMAs 11, 13 and 18, respectively. Using these radii and the molecular
cloud masses listed in Table 1, we obtain surface mass densities of
$100 \pm 30$, $53 \pm 10$ and $220 \pm 35$~ $M_{\odot}$~pc$^{-2}$, and
${n_{H2}}$ volume densities of $13.0 \pm 4.7$, $5.8 \pm 1.2$ and
$28.2 \pm 5.4$~cm$^{-3}$ for GMAs 11, 13 and 18,
respectively. Assuming an
average density of $n_{H2}=100$~cm$^{-3}$ for true GMCs, the average GMC volume
filling factor for the three GMAs is 15\%.
The virial parameter,
\begin{equation}
\alpha =5 \sigma^2 r_c / (G M_{mol})
\end{equation}
\noindent
can be used to determine if a cloud is self-gravitating ($\alpha=2$)
or self-gravitating and in virial equilibrium ($\alpha=1$).
For GMA 11 and the broad-line component of GMA 18, $\alpha=36$ and 24, respectively,
indicating that these GMAs are not self-gravitating. We do not estimate $\alpha$
for GMA 13 since its spectrum does not follow a simple Gaussian profile.
While GMA 18 as a whole
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=26 306 436 599,clip]{f5.pdf}}
\caption{Intensity-weighted CO(2-1) mean velocity map between -100 and 0~km~s$^{-1}$.
The "X" marks the centroid of the continuum source. GMAs 11 and 13 (see Fig. 2) are
identified in the figure.}
\end{inlinefigure}
\noindent
is gravitationally unbound, the presence of the
narrow-line feature seen in Fig. 7 shows that GMA 18 contains at least
one virialized GMC.
\subsection{The Central AGN and Absorption Feature}
The central AGN in NGC 5044 is surprisingly bright in the ALMA data with a
flux density of ${S_{\nu}= 55.3 \pm 3.9}$~mJy and luminosity of
${\nu L_{\nu} = 1.5 \times 10^{40}}$~erg~s$^{-1}$ measured in
a line-free region. We used the CASA task {\it imfit} to confirm that
the central emission is consistent with a point source. For comparison, the
total flux densities (central point source plus extended emission) for 5044 at
235~MHz and 610~MHz from GMRT data are 229~mJy and 39~mJy (Giacintucci et al. 2008).
Thus, the 230 GHz luminosity of the central AGN is at least
500 times greater than its luminosity at 610~MHz and a factor
of a few greater than the
bolometric X-ray luminosity (Giacintucci et al. 2008; David et al. 2009).
As mentioned above, the large number of X-ray cavities within the central
region of NGC 5044 are probably due to multiple AGN outbursts over the
past $10^8$~yrs. The detection of 230~GHz continuum emission from the
AGN shows that it is presently undergoing another outburst, probably
due to a recent accretion event.
It is unlikely that thermal emission from dust makes a substantial
contribution to the 230~GHz continuum emission. {\it Spitzer} data shows
that the 70~$\mu m$ dust emission is extended (Temi et al. 2007).
In addition, even if all of the 70~$\mu m$ emission was assumed to
originate from within the central region, the dust emission models
developed by Temi et al. (2007) predict a flux density
at 230~GHz of less than 1~mJy. The models in Temi et al. assume
a steady-state balance between dust production and ion sputtering
and include heating from starlight and inelastic collision with
thermal electrons. The FIR emissivity was computed by integrating over
the local grain size (and therefore) temperature distribution.
A pronounced absorption feature is seen in the CO(2-1) spectrum
of the central continuum source (see Fig. 8.).
Fitting a Gaussian profile to the absorption feature gives a mean velocity
of $260.3 \pm 0.8$~km~s$^{-1}$ and a linewidth of $5.2 \pm 0.8$~km~s$^{-1}$.
The line-of-sight velocity is a significant fraction of the circular velocity
indicating that the cloud is falling into the central region of the galaxy on a
nearly radial orbit. The optical depth at line center is $\tau = 0.35$,
indicating that either 30\% of the continuum
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=26 306 436 599,clip]{f6.pdf}}
\caption{Intensity-weighted CO(2-1) mean velocity map between 0 and 150~km~s$^{-1}$.
The "X" marks the centroid of the continuum source. The most massive
molecular structure, GMA 18, is identified in the figure.}
\end{inlinefigure}
\noindent
source is covered by an
optically thick cloud or that the entire continuum source is covered by
diffuse molecular gas with $\tau = 0.35$.
Due to the presence of faint line emission near the AGN, we were limited
to a maximum extraction region of 2$^{\prime\prime}$ by 2$^{\prime\prime}$ for
the AGN spectrum shown in Fig. 8 which excludes some of the AGN continuum emission.
The use of a smaller aperture does not affect
the fraction of the continuum flux that is absorbed which is the main
parameter of interest. We confirmed this by extracting spectra in several
smaller regions and found that 30\%
of the continuum flux is
absorbed at line-center in all spectra.
The linewidth of the absorber is typical of an individual GMC.
Assuming the linewidth-size relation in Solomon et al. (1987), we obtain
a size for the absorbing
cloud of 27~pc, implying that the radius of the continuum emission
is less than 50~pc. Alternatively, the absorption could be due
to diffuse molecular gas that fully covers the continuum emission.
In this case, the column density of the diffuse CO must be
${N_{CO}= 3.2 \times 10^{15}}$~cm$^{-2}$. Since the ratio of ${N_{H2}}$
to ${N_{CO}}$ varies significantly in diffuse molecular gas
(Burgh et al. 2007; Liszt 2012) we cannot accurately estimate ${N_{H2}}$.
\subsection{Correlations between the molecular gas, H$\alpha$ filaments,
dust and hot gas}
The distribution of the 24 GMAs listed in Table 1 is shown on an HST
dust extinction map in Fig. 9. This map was created based on
a F791W image obtained with the WFPC2 Planetary Camera.
The two-dimensional surface brightness distribution of the galaxy was
fit with two Sersic components using GALFIT (Peng et al. 2002; 2010).
The original image was divided by the best fitting model and
resulting ratio image was converted to magnitudes.
While the most massive molecular structure, GMA 18 (the red circle close to the
center of the figure), is spatially coincident with the highest extinction
region toward the SE of the AGN, and a few of the GMAs trace the
dust filaments toward the NW, there is not an obvious spatial correlation between
the population of GMAs and the dust. Most GMCs have visual extinctions significantly
greater than 1~mag (Pineda et al. 2010), which would be easily identified
in Fig. 9. On theoretical grounds, it is difficult for CO to form with less shielding
(Wolfire et al. 2010). It is possible that the GMAs not associated with regions of high
extinction are located in dusty regions on the far side of the galaxy which would
not produce much extinction.
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=20 145 574 699,clip]{f7.pdf}}
\caption{ALMA spectrum of GMA 18 along with the best-fit double Gaussian profile (solid
line). The individual Gaussians are show as dashed lines.}
\end{inlinefigure}
NGC 5044 has the brightest system of H$\alpha$+[NII] filaments among
groups of galaxies (Sun et al. 2014). The H$\alpha$+[NII] filaments
primarily extend along a north-south direction (see Fig. 10). Most of the
GMAs are concentrated within the central region of
the H$\alpha$+[NII] emission. Due to the limited ALMA field-of-view,
it is difficult to determine how well the molecular gas traces
the H$\alpha$+[NII] filaments at larger radii.
Caon et al. (2000) measured the kinematics of the H$\alpha$+[NII]
filaments in NGC 5044 using long slit spectroscopy. They found that the velocity
profiles along the slits were fairly irregular, but the bulk of the
H$\alpha$+[NII] emission was blueshifted by 60-100~km~s$^{-1}$
relative to the systemic velocity of NGC 5044. The predominance
of blueshifted H$\alpha$+[NII] emission is consistent with the
presence of mostly blueshifted GMAs in the central region.
The unsharp masked 0.5-2.0~keV {\it Chandra} ACIS image from
David et al. (2009) is shown in Fig. 11 along with the GMAs detected in the
ALMA data. Most of the GMAs
are clustered within the central region which contains the coolest
and lowest entropy X-ray emitting gas. The radiative cooling time
of the hot gas in the central region is about $4 \times 10^7$~yr, which
is well below the observed threshold for triggering star formation
(Rafferty et al. 2008; Voit et al. 2008). The limited ALMA
field-of-view does not permit a detailed comparison between the molecular gas
and the larger scale structure seen in the ACIS image.
Werner et al. (2014) recently presented {\it Herschel} observations of
a sample of eight elliptical galaxies. All six galaxies
in their sample with H$\alpha$ emission were detected in [C II] (which is produced in the
photodissociation region surrounding molecular clouds), including NGC 5044.
The spatial resolution of {\it Herschel} at the [CII] emission line
is $12^{\prime\prime}$ and is insufficient to perform a detailed comparison
between the GMAs and the [CII] emission. Overall, the [CII] emission is
elongated in a north-south direction, similar to the H$\alpha$ emission.
The velocity structure of the [C II] emission is also
similar to the GMAs with predominately blueshifted emission
relative to the systemic velocity of the galaxy near the center of
NGC 5044 and mostly
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=20 145 574 699,clip]{f8.pdf}}
\caption{ALMA spectrum from within a $2^{\prime\prime}$ by $2^{\prime\prime}$
aperture centered on the AGN. Note the blueshifted absorption feature at
260~km~s$^{-1}$ with a linewidth of $5.2 \pm 0.8$~km~s$^{-1}$ which
is probably due to an infalling GMC.}
\end{inlinefigure}
\noindent
redshifted emission at larger radii. However,
the [C II] emission is only detected over a velocity range of
about 120~km~s$^{-1}$ compared to a velocity range of 450~km~s$^{-1}$
for the CO(2-1) emission.
\section{Discussion}
\subsection{Source of the Molecular gas}
We detect 24 GMAs above a surface brightness limit of
0.16~Jy~beam$^{-1}$ with a total mass of ${5.1 \times 10^7 M_{\odot}}$
within the central 2.5~kpc in NGC 5044. The molecular gas in
NGC 5044 has a roughly azimuthally symmetric distribution, no evidence of any
disk-like structures and a velocity dispersion less than the stellar
velocity dispersion. Werner et al. (2014) found that only systems like NGC 5044
with thermally unstable gas, based on the Field criterion, contain [C II] emitting
gas. All these characteristics point to an intrinsic
(i.e., cooling flow) and not extrinsic (i.e., merger with a gas rich system)
origin for the molecular gas. To estimate the mass shed from evolving stars we use
the 2MASS data to compute ${L_K}$, a specific stellar mass loss rate of
$\alpha=\dot M_*/M_*=5.4 \times 10^{-20}$~s$^{-1}$
(Renzini \& Buzzoni 1986, Mathews 1989) and a mass-to-light ratio
of ${0.8 {\rm M}_{\odot}/L_{\odot K}}$ (based on dynamical measurements of
early-type galaxies by Humphrey et al. 2006). Within the central 10~kpc,
the stellar mass loss rate is ${\dot M_*}$= 0.23~M$_{\odot}$~yr$^{-1}$.
There is evidence for multiphase hot gas out to at least 10~kpc in NGC 5044
(Buote et al. 2003; David et al. 2009), indicating that even in the
presence of multiple AGN outbursts, some hot gas is able to cool.
The classical mass deposition rate
(${\dot M_c = 2 \mu m_p L_x / 5 k T }$) within the central 10~kpc is
5.1~M$_{\odot}$~yr$^{-1}$. It is unclear what fraction of
the X-ray emitting gas is able to cool due to AGN feedback.
Fitting a cooling flow model to the ACIS data within the central region
of NGC5044 does not provide useful constraints on the spectroscopic
mass deposition rate since the ambient gas temperature is only 0.7~keV
and the ACIS response declines rapidly below 0.5 keV.
The cavity power derived in David et al. (2009) is about one-half of the X-ray
luminosity within the
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=51 161 503 603,clip]{f9.pdf}}
\caption{Location of the red and blueshifted GMAs on an HST dust extinction image.
The ellipses represent the ALMA beam in our observation. The yellow "X" marks
the centroid of the continuum source.}
\end{inlinefigure}
\noindent
central 10~kpc, but this number depends on the
uncertain volume of the cavities. As long as 5\% of the hot gas is able to cool,
mass deposition from the cooling flow will be the dominant supply mechanism
for the molecular gas.
Neglecting the suppression of gas cooling by AGN feedback, the
molecular mass contained within the 24 GMAs can be produced
by gas cooling within the central 10~kpc in approximately 8~Myr.
Due to the limited sensitivity and coverage in our cycle 0 ALMA data,
the 24 GMAs we detect is probably a lower limit. The H$\alpha$ and [C II] emission,
which traces molecular gas, extend to at least 7~kpc. The integrated CO(2-1) flux
density in the IRAM 30m data is significantly greater than that in the ALMA data
(especially in the blueshifted emission) which is presumably
due to a significant component of diffuse molecular gas that is resolved out in the
ALMA data. Any additional molecular gas or
reduction in the mass deposition rate due to AGN feedback would require
a longer accumulation time for the molecular gas.
\subsection{Cloud Dynamics}
There are several indications of infalling molecular gas in
NGC 5044 (i.e., the absorption feature seen against the central
continuum source and the velocity gradients in GMAs 11 and 13) and
no indications of disk-like structures.
The smooth X-ray morphology of the NGC 5044 group on
large scales indicates that the gas is in nearly hydrostatic
equilibrium, but the observed structure within the central region
(see Fig. 11) suggests that there is some AGN-driven turbulence.
However, the turbulent velocities are probably a small fraction
of the sound speed of the hot gas. Turbulent velocities greater than
20-40~km~s$^{-1}$ would generate a heating rate due to the dissipation of
turbulent kinetic energy that exceeds the radiative cooling rate
of the hot gas (David et al. 2011). The linewidth of an individual
GMA should reflect the turbulent velocity of the hot gas from which it
formed. It is interesting to note that the observed linewidths of
the GMAs in NGC 5044 are comparable to that required to balance radiative cooling
with the dissipation
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=69 175 494 589,clip]{f10.pdf}}
\caption{Location of the red and blueshifted GMAs on an H$\alpha$ image.
The black box shows the field-of-view of the HST image.}
\end{inlinefigure}
\noindent
of turbulent kinetic energy in the hot gas.
The kinematics of the molecular gas in NGC 5044 are very different from
that observed in the more massive clusters A1835
(McNamara et al. 2014) and A1664 (Russell et al. 2014). ALMA
cycle 0 observations show that both of these systems have more
than ${10^{10} M_{\odot}}$ of molecular gas. Abell 1664 has
a massive central disk-like molecular structure and several high
velocity clumps ($\sim 570$~km~s$^{-1}$) which could be either
infalling or outflowing. The molecular gas in A1835 is
distributed between a nearly face-on disk and a bipolar outflow,
possibly driven by radio jets or buoyant X-ray cavities (McNamara et al. 2014).
Both A1664 and A1835 host much more powerful AGNs compared to NGC 5044.
The cavity power in NGC 5044 is ${P_{cav}=6 \times 10^{42}}$~erg~s$^{-1}$
(David et al. 2009), compared to ${P_{cav}=6-8 \times 10^{43}}$~erg~s$^{-1}$
in A1664 (Russell et al. 2014) and ${P_{cav}=10^{45}}$~erg~s$^{-1}$ in
A1835 (McNamara et al. 2014).
For buoyancy to have a significant impact on the molecular gas, the displaced
mass in the X-ray cavities must equal the molecular gas mass. For
NGC 5044 this would require the displacement of all the hot gas
within the central 2~kpc, which obviously has not happened (see Fig. 11).
The total displaced gas mass in the cavities seen in Fig. 11 is approximately
$8 \times 10^6 M_{\odot}$, which is only 15\% of the observed molecular gas mass.
While an X-ray cavity could still affect the dynamics of an
individual GMA, there is no observed correlation between the velocities of
the GMAs and their proximity to X-ray cavities. GMRT observations also show
that NGC 5044 does not have a large scale collimated jet (Giacintucci et al. 2011).
Thus, due to the presence of only a weak AGN in NGC 5044, we conclude that
the molecular gas essentially follows ballistic trajectories
after condensing out of the hot X-ray emitting gas.
\subsection{Are the GMAs confined?}
We showed above that GMAs 11 and 18 would have to be more than
20 times more massive to be gravitationally bound.
While we cannot make a general statement about the entire GMA population in
NGC 5044, it is clear that the largest GMAs
\begin{inlinefigure}
\center{\includegraphics*[width=1.00\linewidth,bb=54 158 497 586]{f11.pdf}}
\caption{Location of the red and blueshifted GMAs on an unsharp masked ACIS image.
The red box shows the field-of-view of the H$\alpha$ image.}
\end{inlinefigure}
\noindent
are not gravitationally bound.
From the {\it Chandra} ACIS data we can compute the thermal pressure
of the X-ray emitting gas and determine if the GMAs are pressure confined.
The thermal pressure of the hot gas in the vicinity of GMA 18 is
${P_{ext}/k= 2.0 \times 10^6}$~K~cm$^{-3}$, assuming GMA 18 is located at
a radial distance equal to its projected distance.
The CO(2-1) linewidth in GMA 18 is approximately 90 times the sound speed of
the gas (assuming a kinetic temperature of 30~K), indicating that the pressure
in GMA 18 is dominated by supersonic turbulence. The turbulent pressure in
GMA 18 is ${P_{turb}/k = \rho \sigma_v^2/k = 9.5 \times 10^6}$~K~cm$^{-3}$,
which is a factor of four greater than the thermal pressure of the
surrounding hot gas, suggesting that GMA is not pressure confined.
However, even if the computed turbulent pressure was less than the thermal
pressure of the hot gas, it is doubtful that GMA 18 could be confined
by the hot gas.
Our estimate for the turbulent pressure assumes that the bulk of GMA 18 is
filled with gas at the volume-averaged density and that all of this gas has a
turbulent velocity equal to the observed CO(2-1) linewidth. Both of these
assumptions are probably not valid. As mentioned above, the volume filling factor
of the few resolved GMAs is fairly small, so the observed CO(2-1) linewidths should
be thought of as the velocity dispersion of the embedded GMCs, each of which has
a smaller internal velocity dispersion and higher density. Due to the factor
of $10^3$ contrast in densities between the embedded GMCs and the X-ray emitting
gas, the thermal pressure of the hot gas will not decelerate and confine the
ballistically moving GMCs within a GMA. Thus, it appears unlikely that the
few resolved GMAs in NGC 5044 are confined by either self-gravity or thermal pressure.
Individual GMCs will only be dynamically affected by the hot gas
after sweeping up a mass equal to their own mass. Typical GMCs have
radii of a few pc and must therefore travel a few kpc due to the factor of $10^3$
contrast in density between the cold and hot gas before sweeping up their own mass.
This distance is more than an order of magnitude greater than the
largest observed GMA. Since the GMAs are not confined by self-gravity or
thermal pressure, they should disperse in the time it takes a GMC to cross
the GMA. Given the observed
radius and linewidth of GMA 18, embedded GMCs should traverse the GMA in
a time of $t_{disp} \sim 2~r_c/\sigma$=12~Myr, which is less than the central
40~Myr cooling time of the hot gas. This implies that the embedded GMCs within
a GMA must condense out of the hot gas at the same time and that the
observed GMAs in NGC 5044 probably arise from local concentrations
of thermally unstable parcels of hot gas. One possible mechanism
for producing a local concentration of cooling parcels of hot
gas is the presence of dust, possibly transported outward
by AGN outbursts as discussed in Temi et al. (2007).
Dust grains accelerate the cooling of the hot gas due to
inelastic collisions between the thermal electrons and the dust
(Mathews and Brighenti et al. 2003). Reducing the cooling time of
the gas also helps the non self-gravitating GMAs maintain their
integrity as they cool.
\subsection{Effects of Star Formation}
The star formation rate in NGC 5044 based on {\it GALEX} and
{\it WISE} data is 0.073~M$_{\odot}$~yr$^{-1}$ (Werner et al. 2014).
NGC 5044 also has unusual
polycyclic aromatic hydrocarbon (PAH) line ratios compared
to other early-type galaxies (Vega et al. 2010) and a highly uncertain
stellar age (Marino et al. 2010), both of which may result from episodes
of star formation over the past few Gyrs. The depletion time of the molecular
gas due to star formation is approximately 700~Myr, which is typical
for molecular gas in brightest cluster galaxies embedded in cooling flows
(O'Dea et al. 2008; Voit \& Donahue 2011), but the IRAM 30m data shows
that there may be a significant amount of diffuse molecular gas
that was undetected in the ALMA data which would substantially increase
the depletion time.
The Kennicutt (1998) relation between star formation rate and gas
surface density predicts a combined star formation rate from GMAs
11, 13 and 18 of 0.03~M$_{\odot}$~yr$^{-1}$. Considering that these
three GMAs comprise about 35\% of the observed molecular gas, this
estimate is in reasonably good agreement with the observed star formation rate.
Assuming that one Type II supernova (SNeII) is produced for every
100~M$_{\odot}$ consumed into stars and that each SNeII produces
$10^{51}$~erg, the SNeII heating rate in GMA 18 should be
${H_{SN}= 6.7 \times 10^{39}}$~erg~s$^{-1}$.
If we assume that GMA 18 has a lifetime of 12~Myr (i.e., the crossing time
of an individual
GMC), then a total of $2.6 \times 10^{54}$~erg of energy will
be released by SNeII during this time, which corresponds
to 60\% of the turbulent kinetic energy in GMA 18. Thus, depending
on the how SNeII energy is partitioned between radiative
and mechanical energy, SNeII could play a role in driving the
internal turbulence and dissipation of the GMAs.
\section{Summary}
Our ALMA cycle 0 observation shows that the cooling flow
in the X-ray bright group NGC 5044 is a breeding ground for molecular gas.
The CO(2-1) emission within the central 2.5~kpc is distributed among 24
molecular structures with a total molecular mass of $5.1 \times 10^7 M_{\odot}$.
Only a few of the molecular structures are spatially resolved and the
observed radii, linewidths and masses of the these structures indicate that
they are not gravitationally bound. Given the large CO(2-1) linewidths of the
structures, they are likely giant molecular associations
and not individual clouds. The average density of the few resolved
GMAs yields a GMC volume filling factor of 15\%.
One narrow line component is detected within the largest GMA with
a linewidth comparable to a virialized GMC.
While the large scale spatial distribution of the GMAs is azimuthally
symmetric, there are some systematic differences in velocity space.
Approximately 20\% of the molecular gas is contained in a single
massive, redshifted GMA near the center of NGC 5044. The remainder of
the molecular gas is divided between many lower mass GMAs with a
greater central concentration among the blueshifted GMAs compared
to the redshifted GMAs. On global scales, the kinematics of the molecular gas is
similar to that observed in the H$\alpha$ filaments (Caon et al. 2000)
and [C II] emitting gas (Werner et al. 2014).
There are no disk-like molecular structures observed in NGC 5044. Two
of the GMAs have velocity maps suggesting that they are falling into
the center of NGC5044 from the far side of the galaxy. There is also
an absorption feature seen in the CO(2-1) spectrum of the central continuum
source with an infalling velocity of 250~km~s$^{-1}$ and a linewidth typical of an
individual GMC. Combining these results with the presence of infalling
molecular gas in the Perseus cluster (Lim et al. 2008) suggest that
infalling molecular gas is common in cooling flows. The observed molecular
mass and distribution and kinematics of the GMAs in NGC 5044 are consistent with
a scenario in which the molecular gas condenses out of the thermally
unstable hot gas and then follows ballistic trajectories.
AGN-inflated buoyant cavities should not have a
major impact on the dynamics of the molecular gas as a whole, except possibly
in the immediate proximity of an X-ray cavity.
The integrated CO(2-1) flux density in the IRAM 30m observation of NGC 5044
is significantly greater than that produced by the molecular structures
detected in the ALMA data. This is presumably due to a significant
component of diffuse molecular gas that has been resolved out in
the ALMA data.
Due to the lack of gravitational or pressure confinement,
the GMAs detected in NGC 5044 should disperse on a timescale
of about 12~Myr, which is less than the 40~Myr central cooling time of
the hot gas, so the embedded GMCs within a GMA must condense out of
the hot gas at the same time. Thus, the observed GMAs probably arise from local
concentrations of over dense, thermally unstable regions in the hot gas.
The wealth of information obtained from our 30 minute cycle 0 ALMA
observation of the near by NGC 5044 group shows that it
is an ideal target for investigating the formation of molecular
gas in cooling flows and the long standing question about the ultimate
fate of the cooling gas. More sensitive and higher resolution ALMA observations
in future cycles promise to illuminate the internal structure of the
GMAs detected in the cycle 0 observation.
This paper makes use of the following ALMA data:
ADS/JAO.ALMA\#2011.0.00735.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC
(Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
The National Radio Astronomy Observatory is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc. This work was supported in part by NASA grant GO2-13146X.
ACE acknowledges support from STFC grant ST/I001573/1.
We would like to thank M. Birkinshaw for assistance in
learning CASA and analyzing the ALMA data and B. McNamara
and P. Nulsen for discussions about their cycle 0 ALMA observations
of clusters of galaxies.
|
2,877,628,090,231 | arxiv | \section{Introduction} \label{Introduction-Section1}
Inclusive insurance (or microinsurance) relates to the provision of insurance services to low-income populations with limited access to mainstream insurance or alternative effective risk management strategies. Many individuals excluded from basic financial services and those microinsurance aims to proctect, live below the minimum level of income required to meet their basic needs. Currently fixed at \$1.90 USD per day,
9.2\% of the population were estimated to live below the international extreme poverty line in 2017 \citep{Online:WorldBank2021}. Increases in the number of new poor and those returning to poverty as a result of the COVID-19 pandemic are expected to reverse the historically declining poverty trend \citep{Book:WorldBank2020}.
Fundamental features of the microinsurance environment such as the nature of low income risks, limited financial literacy and experience, product accessibility and data availability, create barriers to penetration, particularly in relation to the affordability of products. For the proportion of the population living just above the poverty line, premium payments heighten the risk of poverty trapping and induce a balance between profit and loss as a result of insurance coverage, dependent on the entity's level of capital. Here, poverty trapping refers to the inability of the poor to escape poverty without external help \citep{Article:Kovacevic2011}.
Highlighting vulnerability reduction and investment incentive effects of insurance, \cite{Article:Janzen2020} observe a marked reduction in long-term poverty and the social protection costs required to close the poverty gap following introduction of an asset insurance market. Calibrating their model to risk-prone regions in Africa, their study suggests that those in the neighbourhood of the poverty line do not optimally purchase insurance (without subsidies), suppressing their consumption and mitigating the probability of trapping.
\cite{Article:Kovacevic2011} propose negative consequences of insurance uptake for members of low-income populations closest to the poverty line, applying ruin-theoretic approaches to calculation of the trapping probability. \cite{Article:Liao2020} support these findings in their analysis of a multi-equilibrium model with agricultural output risks on data from rural China. Voluntary insurance would enable individuals close to the poverty threshold to opt out of insurance purchase in favour of alternative risk management strategies, in order to mitigate this risk.
In line with the findings of \cite{Article:Singh2020} on the effectiveness of social protection mechanisms for poverty alleviation, \cite{Article:Jensen2017} observe a greater reduction in poverty through implementation of an integrated social protection programme in comparison to pure cash transfers. Government subsidised premiums are the most common form of aid in the context of insurance.
Besides reducing the impact on household capital growth, lowering consumer premium payments has the potential to increase microinsurance take-up, with wealth and product price positively and negatively influencing microinsurance demand, respectively \citep{Article:Eling2014}.
Poverty traps are typically studied in the context of economics, with a large literature focus on why economic stagnation below the poverty line occurs in certain communities. While the poor could readily grow their way out of poverty by adopting profitable strategies such as productive asset accumulation, opportunistic exchange and implementation of cost-effective production technologies, poverty traps are underlined by poverty reinforcing behaviours induced by the state of being poor \citep{Article:Barrett2016}.
A detailed description of the mechanics of the poverty trap state is provided by \cite{Book:Matsuyama2010}. In studying the probability of falling into such a trap, \lq \lq trapping\rq \rq \ describes the event in which a household falls underneath the poverty line and into the area of poverty.
In this paper, we adopt the ruin-theoretic approach to calculating the trapping probability of households in low-income populations presented by \cite{Article:Kovacevic2011}, adapting the piecewise deterministic Markov process such that households are subject to large shocks of random size. In line with the poverty trap ideology, we assume that the area of poverty to be an absorbing state and so consider only the state of events above the poverty threshold. Obtaining explicit solutions for the trapping probability, we compare the influence of three structures of microinsurance on the ability of households to stay above the poverty line. Specifically, we consider a (i) proportional, (ii) subsidised proportional and (iii) subsidised proportional with barrier microinsurance scheme. Aligning with the essential place for governmental support in the provision of social protection which encompasses risk mitigation, we assess for the first time in this context, to the best of our knowledge, the impact of a (government) subsidised insurance scheme with barrier strategy. We optimise the barrier level in the context of the trapping probability and the governmental cost of social protection, identifying the proportion of the population for which such a product would be beneficial. Here, the cost of social protection is defined to account for the provision of government subsidies, in addition to the cost of lifting a household from poverty, should they fall underneath the threshold. The benefit of subsidy schemes for poverty reduction is measured through observation of this governmental cost, in addition to the trapping probability of the households under consideration.
The remainder of the paper will be structured as follows. In Section \ref{TheCapitalModel-Section2}, we introduce the household capital model and its associated infinitesimal generator. The (trapping) time at which a household falls into the area of poverty is defined in Section \ref{TheTrappingTime-Section3}, and subsequently the explicit trapping probability and the expected trapping time are derived for the basic uninsured model. Links between classical ruin theoretic models and the trapping model of this paper are stated in Sections \ref{TheCapitalModel-Section2} and \ref{TheTrappingTime-Section3}. Microinsurance is introduced in Section \ref{IntroducingMicroinsurance-Section4}, where we assume a proportion of household losses are covered by a microinsurance policy. The capital model is redefined and the trapping probability is derived. Sections \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} and \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6} consider the case where households are proportionally insured through a government subsidised microinsurance scheme, with the impact of a subsidy barrier discussed in Section \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}. Optimisation of the subsidy and barrier levels is presented in Sections \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} and \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}, alongside the associated governmental cost of social protection. Concluding remarks are provided in Section \ref{Conclusion-Section6}.
\section{The Capital Model} \label{TheCapitalModel-Section2}
The fundamental dynamics of the model follow those of \cite{Article:Kovacevic2011}, where the growth in accumulated capital $(X_t)$ of an individual household is given by
\vspace{0.3cm}
\begin{align}
\frac{dX_{t}}{dt}=r \cdot\left[X_{t}-x^{*}\right]^{+}, \label{TheCapitalModel-Section2-Equation1}
\end{align}
\vspace{0.3cm}
where $[x]^{+}=\max(x,0)$. The capital growth rate $r$ incorporates household rates of consumption, income generation and investment or savings, while $x^* > 0$ represents the threshold below which a household lives in poverty. Reflecting the ability of a household to produce, accumulated capital $(X_t)$ is composed of land, property, physical and human capital, with health a form of capital in extreme cases where sufficient health services and food accessibility are not guaranteed \citep{Article:Dasgupta1997}. The notion of a household in this model setting may be extended for consideration of poverty trapping within economic units such as community groups, villages and tribes, in addition to the traditional household structure.
The dynamical process in \eqref{TheCapitalModel-Section2-Equation1} is constructed such that consumption is assumed to be an increasing function of wealth (for full details of the model construction see \cite{Article:Kovacevic2011}). The poverty threshold $x^*$ represents the amount of capital required to forever attain a critical level of income, below which a household would not be able to sustain their basic needs, facing elementary problems relating to health and food security. Throughout the paper, we will refer to this threshold as the critical capital or the poverty line. Since \eqref{TheCapitalModel-Section2-Equation1} is positive for all levels of capital greater than the critical capital, points less than or equal to $x^*$ are stationary (capital remains constant if the critical level is not met). In this basic model, stationary points below the critical capital are not attractors of the system if the initial capital exceeds $x^*$, in which case the capital process $(X_t)$ grows exponentially with rate $r$.
Using capital as an indicator of financial stability over other commonly used measures such as income enables a more effective analysis of a household's wealth and well-being. Households with relatively high income, considerable debt and few assets would be highly vulnerable if a loss of income was to occur, while low-income households could live comfortably on assets acquired during more prosperous years for a long-period of time \citep{Book:Gartner2004}.
In line with \cite{Article:Kovacevic2011}, we expand the dynamics of \eqref{TheCapitalModel-Section2-Equation1} under the assumption households are susceptible to the occurrence of large capital losses, including severe illness, the death of a household member or breadwinner and catastrophic events such as floods and earthquakes. We assume occurrence of these events follows a Poisson process with intensity $\lambda$, where the capital process follows the dynamics of \eqref{TheCapitalModel-Section2-Equation1} between events. On the occurrence of a loss, the household's capital at the event time reduces by a random amount $Z_{i}$. The sequence $(Z_{i})$ is independent of the Poisson process and i.i.d. with common distribution function $G$. In contrast to \cite{Article:Kovacevic2011}, we assume reduction by a given amount rather than a random proportion of the capital itself. This adaptation enables analysis of a tractable mathematical model without threatening the core objective of studying the probability that a household falls into the area of poverty.
A household reaches the area of poverty if it suffers a loss large enough that the remaining capital is attracted into the poverty trap. Since a household's capital does not grow below the critical capital $x^{*}$, households that fall into the area of poverty will never escape. Once below the critical capital, households are exposed to the risk of falling deeper into poverty, with a risk of negative capital due to the dynamics of the model. A reduction in a household's capital below zero could represent a scenario where total debt exceeds total assets, resulting in negative capital net worth. The experience of a household below the critical capital is, however, out of the scope of this paper.
We will now formally define the stochastic capital process, where the process for the inter-event household capital \eqref{TheCapitalModel-Section2-Equation2} is derived through solution of the first order ordinary differential equation \eqref{TheCapitalModel-Section2-Equation1}. This model is an adaptation of the model proposed by \cite{Article:Kovacevic2011}.
\vspace{0.3cm}
\begin{definition}
Let $T_{i}$ be the $i^{th}$ event time of a Poisson process $\left(N_{t}\right)$ with parameter $\lambda$, where $T_{0}=0 .$ Let $Z_{i} \ge 0 $ be a sequence of i.i.d. random variables with distribution function $G$, independent of the process $\left(N_{t}\right)$. For $T_{i-1} \leq t<T_{i}$, the stochastic growth process of the accumulated capital $X_{t}$ is defined as
\vspace{0.3cm}
\begin{align}
X_{t}=\begin{cases} \left(X_{T_{i-1}}-x^{*}\right) e^{r \left(t-T_{i-1}\right)}+x^{*} & \text { if } X_{T_{i-1}}>x^{*}, \\ X_{T_{i-1}} & \text{ otherwise}.
\end{cases}
\label{TheCapitalModel-Section2-Equation2}
\end{align}
\vspace{0.3cm}
At the jump times $t = T_{i}$, the process is given by
\vspace{0.3cm}
\begin{align}
X_{T_{i}}=\begin{cases} \left(X_{T_{i-1}}-x^{*}\right) e^{r \left(T_{i}-T_{i-1}\right)}+x^{*} - Z_{i} & \text { if } X_{T_{i-1}}>x^{*}, \\ X_{T_{i-1}} - Z_{i} & \text{ otherwise}.
\end{cases}
\label{TheCapitalModel-Section2-Equation3}
\end{align}
\end{definition}
\vspace{0.3cm}
The stochastic process $(X_t)_{t\geq 0}$ is a piecewise-determinsitic Markov process \citep{Article:Davis1984} and its infinitesimal generator is given by
\vspace{0.3cm}
\begin{align}
(\mathcal{A} f)(x)=r(x-x^{*}) f^{\prime}(x) +\lambda \int_{0}^{\infty} \left[f(x - z) - f(x)\right] \mathrm{d} G(z), \qquad x \ge x^{*}.
\label{TheCapitalModel-Section2-Equation4}
\end{align}
\vspace{0.3cm}
The capital model as defined in \eqref{TheCapitalModel-Section2-Equation2} and \eqref{TheCapitalModel-Section2-Equation3} is actually a well-studied topic in ruin theory since the 1940s. Here, modelling is done from the point of view of an insurance company. Consider the insurer\rq s surplus process $(U_t)_{t\geq 0}$ given by
\vspace{0.3cm}
\begin{align}
U_t =u+pt+ a \int_0^t U_s \, ds-\sum_{i=1}^{N_t} Z_i,
\label{TheCapitalModel-Section2-Equation5}
\end{align}
\vspace{0.3cm}
where $u$ is the insurer\rq s initial capital, $p$ is the constant premium rate, $a$ is the risk-free interest rate, $N_t$ is a Poisson process with parameter $\lambda$ which counts the number of claims in the time interval $[0,t]$, and $(Z_i)_{i=1}^\infty$ is a sequence of i.i.d. claim sizes with distribution function $G$. This model is also called the insurance risk model with deterministic investment, which was first proposed by \cite{Article:Segerdahl1942} and subsequently studied by \cite{Article:Harrison1977} and \cite{Article:Sundt1995}. For a detailed literature review on this model prior to the turn of the century, readers can consult \cite{Article:Paulsen1998}.
Observe that when $p=0$, the insurance model \eqref{TheCapitalModel-Section2-Equation5} for positive surplus is equivalent to the capital model \eqref{TheCapitalModel-Section2-Equation2} and \eqref{TheCapitalModel-Section2-Equation3} above the poverty line $x^*=0$. Subsequently, the capital growth rate $r$ in our model corresponds to the risk-free investment rate $a$ of the insurer\rq s surplus model. More connections between these two models will be made in the next section after the first hitting time is introduced.
\vspace{0.3cm}
\section{The Trapping Time} \label{TheTrappingTime-Section3}
Let
\vspace{0.3cm}
\begin{align}
\tau_{x}:=\inf \left\{t \geq 0: X_{t}<x^{*} \mid X_{0}=x\right\}
\label{TheTrappingTime-Section3-Equation1}
\end{align}
\vspace{0.3cm}
denote the time at which a household with initial capital $x \ge x^{*}$ falls into the area of poverty (the trapping time), where $\psi(x) = \mathbb{P}(\tau_{x} < \infty)$ is the infinite-time trapping probability. To study the distribution of the trapping time, we apply the expected discounted penalty function at ruin concept commonly used in actuarial science \citep{Article:Gerber1998}, such that with a force of interest $\delta \ge 0$ and initial capital $x \ge x^{*}$, we consider
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)= \mathbb{E}\left[w(\mid X_{\tau_{x}}-x^{*}\mid)e^{-\delta \tau_{x}} \mathbbm{1}_{\{\tau_{x} < \infty\}}\right],
\label{TheTrappingTime-Section3-Equation2}
\end{align}
\vspace{0.3cm}
where $\mid X_{\tau_{x}} -x^{*}\mid$ is the deficit at the trapping time and $w(x)$ is an arbitrary non-negative penalty function. For more details on the so called Gerber-Shiu risk theory, the interested reader may wish to consult \cite{Book:Kyprianou2013}. Using standard arguments based on the infinitesimal generator, $m_{\delta}(x)$ can be characterised as the solution of the Integro-Differential Equation (IDE)
\vspace{0.3cm}
\begin{equation}
r(x-x^{*})m_{\delta}'(x)-(\lambda + \delta) m_{\delta}(x)+\lambda \int_{0}^{x-x^{*}}m_{\delta}(x-z)dG(z)=-\lambda A(x), \qquad x \ge x^{*},
\label{TheTrappingTime-Section3-Equation3}
\end{equation}
\vspace{0.3cm}
where
\vspace{0.3cm}
\begin{align}
A(x) := \int_{x-x^{*}}^{\infty}w(z-x)dG(z).
\label{TheTrappingTime-Section3-Equation4}
\end{align}
\vspace{0.3cm}
Due to the lack of memory property, we consider the case in which losses ($Z_i$) are exponentially distributed with parameter $\alpha >0$. Specifying the penalty function such that $w(x)=1$, $m_{\delta}(x)$ becomes the Laplace transform of the trapping time, also interpreted as the expected present value of a unit payment due at the trapping time. Equation \eqref{TheTrappingTime-Section3-Equation3} can then be written such that
\vspace{0.3cm}
\begin{align}
r(x-x^{*})m_{\delta}'(x)-(\lambda + \delta) m_{\delta}(x)+\lambda \int_{0}^{x-x^{*}}m_{\delta}(x-z)\alpha e^{-\alpha z} dz&=-\lambda e^{-\alpha (x-x^{*})} , \hspace{0.2cm} x \ge x^{*}.
\label{TheTrappingTime-Section3-Equation5}
\end{align}
\vspace{0.3cm}
Applying the operator $\left(\frac{d}{dx}+\alpha\right)$ to both sides of \eqref{TheTrappingTime-Section3-Equation5}, together with a number of algebraic manipulations, yields the second order homogeneous differential equation
\vspace{0.3cm}
\begin{align}
-\frac{(x-x^{*})}{\alpha}m_{\delta}''(x)+\Bigg[\frac{(\lambda+\delta - r)}{\alpha r}-(x-x^{*})\Bigg]m_{\delta}'(x)+\frac{\delta}{r}m_{\delta}(x)=0, \qquad x\geq x^{*}. \label{TheTrappingTime-Section3-Equation6}
\end{align}
\vspace{0.3cm}
Letting $f(y):=m_{\delta}(x)$, such that $y$ is associated with the change of variable $y:=y(x)=-\alpha (x-x^{*})$, \eqref{TheTrappingTime-Section3-Equation6} reduces to Kummer\rq s Confluent Hypergeometric Equation \citep{Book:Slater1960}
\vspace{0.3cm}
\begin{align}
y\cdot f''(y) + (c-y) f'(y) - a f(y) =0, \qquad y<0,
\label{TheTrappingTime-Section3-Equation7}
\end{align}
\vspace{0.3cm}
for $a=-\frac{\delta}{r}$ and $c=1-\frac{\lambda + \delta}{r}$, with regular singular point at $y=0$ and irregular singular point at $y=-\infty$ (corresponding to $x=x^{*}$ and $x=\infty$, respectively). A general solution of \eqref{TheTrappingTime-Section3-Equation7} is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=f(y)= \begin{cases} 1 \hspace{9.5cm} x < x^{*},\\
A_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y(x)\right)+A_{2}e^{y(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y(x)\right) \hspace{0.2cm} x \ge x^{*},
\end{cases}
\label{TheTrappingTime-Section3-Equation8}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $A_{1},A_{2} \in \mathbb{R}$. Here,
\vspace{0.3cm}
\begin{align}
M(a, c; z)={ }_{1} F_{1}(a, c; z)=\sum_{n=0}^{\infty} \frac{(a)_{n}}{(c)_{n}} \frac{z^{n}}{n !}
\label{TheTrappingTime-Section3-Equation9}
\end{align}
\vspace{0.3cm}
is Kummer\rq s Confluent Hypergeometric Function \citep{Article:Kummer1837} and $(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(n)}$ denotes the Pochhammer symbol \citep{Book:Seaborn1991}. In a similar manner,
\vspace{0.3cm}
\begin{align}
U(a, c; z)=\left\{\begin{array}{ll}\frac{\Gamma(1-c)}{\Gamma(1+a-c)} M(a, c; z)+\frac{\Gamma(c-1)}{\Gamma(a)} z^{1-c} M(1+a-c, 2-c; z) & c \notin \mathbb{Z}, \\ \lim _{\theta \rightarrow c} U(a, \theta; z) & c \in \mathbb{Z}\end{array}\right.
\label{TheTrappingTime-Section3-Equation10}
\end{align}
\vspace{0.3cm}
is Tricomi\rq s Confluent Hypergeometric Function \citep{Article:Tricomi1947}. This function is generally complex-valued when its argument $z$ is negative, i.e. when $x \ge x^{*}$ in the case of interest. We seek a real-valued solution of $m_{\delta}(x)$ over the entire domain, therefore an alternative independent pair of solutions, here, $M(a,c;z)$ and $e^{z}U(c-a,c;-z)$, to \eqref{TheTrappingTime-Section3-Equation7} are chosen for $x \ge x^{*}$.
To determine the constants $A_1$ and $A_2$, we use the boundary conditions at $x^*$ and at infinity.
Applying equation (13.1.27) of \cite{Book:Abramowitz1964}, also known as Kummer\rq s Transformation $M(a, c; z) = e^{z} M(c-a, c;-z)$, we write \eqref{TheTrappingTime-Section3-Equation8} such that
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=
e^{y(x)}\left[A_{1}M\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};-y(x)\right)+A_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y(x)\right)\right]
\label{TheTrappingTime-Section3-Equation11}
\end{align}
\vspace{0.3cm}
for $x \ge x^{*}$. For $z \rightarrow \infty$, it is well-known that
\vspace{0.3cm}
\begin{align}
M(a, c; z)=\frac{\Gamma(c)}{\Gamma(a)} e^{z} z^{a-c}\left[1+O\left(|z|^{-1}\right)\right]
\label{TheTrappingTime-Section3-Equation12}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
U(a, c; z)= z^{-a}\left[1+O\left(|z|^{-1}\right)\right]
\label{TheTrappingTime-Equation13}
\end{align}
\vspace{0.3cm}
(see for example, equations (13.1.4) and (13.1.8) of \cite{Book:Abramowitz1964}). Asymptotic behaviours of the first and second terms of \eqref{TheTrappingTime-Section3-Equation11} as $y(x) \rightarrow -\infty$ are therefore given by
\vspace{0.3cm}
\begin{align}
\frac{\Gamma\left(1-\frac{\lambda+\delta}{r}\right)}{\Gamma\left(1-\frac{\lambda}{r}\right)}\left(-y(x)\right)^{\frac{\delta}{r}}\left(1+O\left(|-y(x)|^{-1}\right)\right)
\label{TheTrappingTime-Section3-Equation14}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
e^{y(x)}\left(-y(x)\right)^{\frac{\lambda}{r}-1}\left(1+O\left(|-y(x)|^{-1}\right)\right),
\label{TheTrappingTime-Section3-Equation15}
\end{align}
\vspace{0.3cm}
respectively. For $x \rightarrow \infty$, \eqref{TheTrappingTime-Section3-Equation14} is unbounded, while \eqref{TheTrappingTime-Section3-Equation15} tends to zero. The boundary condition $\lim_{x\to\infty} m_{\delta}(x) = 0$, by definition of $m_{\delta}(x)$ in \eqref{TheTrappingTime-Section3-Equation2}, thus implies
that $A_{1}=0$. Letting $x=x^{*}$ in \eqref{TheTrappingTime-Section3-Equation5} and \eqref{TheTrappingTime-Section3-Equation8} yields
\vspace{0.3cm}
\begin{align}
\frac{\lambda}{(\lambda + \delta)}=A_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right).
\label{TheTrappingTime-Section3-Equation16}
\end{align}
\vspace{0.3cm}
Hence, $A_{2}=\frac{\lambda}{(\lambda + \delta)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)}$ and the Laplace transform of the trapping time is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};0\right)}e^{y(x)}U\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};-y(x)\right).
\label{TheTrappingTime-Section3-Equation17}
\end{align}
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item Figure \ref{TheTrappingTime-Section3-Figure1}(a) shows that the Laplace transform of the trapping time approaches the trapping probability as $\delta$ tends to zero, i.e.
\vspace{0.3cm}
\begin{align}
\lim _{\delta \downarrow 0} m_{\delta}(x) =\mathbb{P}(\tau_{x}<\infty)\equiv\psi(x).
\label{TheTrappingTime-Section3-Equation18}
\end{align}
\vspace{0.3cm}
As $\delta\to 0$, \eqref{TheTrappingTime-Section3-Equation17} yields
\vspace{0.3cm}
\begin{align}
\psi(x) = \frac{1}{U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}e^{y(x)}U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};-y(x)\right).
\label{TheTrappingTime-Section3-Equation19}
\end{align}
\vspace{0.3cm}
Figure \ref{TheTrappingTime-Section3-Figure1}(b) displays the trapping probability $\psi(x)$ for the stochastic capital process $X_{t}$. We can further simplify the expression for the trapping probability using the upper incomplete gamma function $\Gamma(a;z)=\int_{z}^{\infty}e^{-t}t^{a-1}dt$. Applying the relation
\vspace{0.3cm}
\begin{align}
\Gamma(a;z)=e^{-z}U(1-a,1-a;z)
\label{TheTrappingTime-Section3-Equation20}
\end{align}
\vspace{0.3cm}
(see equation (6.5.3) of \cite{Book:Abramowitz1964}) and the fact that $\Gamma(a; 0)=\Gamma(a)$ for $\mathbb{R}(a)>0$, we have
\vspace{0.3cm}
\begin{align}
\psi(x)=\frac{\Gamma\left(\frac{\lambda}{r}; -y(x)\right)}{\Gamma\left(\frac{\lambda}{r}\right)}.
\label{TheTrappingTime-Section3-Equation21}
\end{align}
\vspace{0.3cm}
\begin{figure}[H]
\begin{multicols}{2}
\includegraphics[width=8cm, height=8cm]{Figures/TheTrappingTime-Section3-Figure1-a.pdf}
\centering (a)
\includegraphics[width=8cm, height=8cm]{Figures/TheTrappingTime-Section3-Figure1-b.pdf}
\centering (b)
\end{multicols}
\caption{(a) Laplace transform $m_{\delta}(x)$ of the trapping time when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$ for $\delta = 0, \frac{1}{8}, \frac{1}{32}, \frac{1}{128}$ (b) Trapping probability $\psi(x)$ when $Z_{i} \sim Exp(\alpha)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$ for $\alpha = 0.8, 1, 1.5, 2$.}
\label{TheTrappingTime-Section3-Figure1}
\end{figure}
\item As an application of the Laplace transform of the trapping time, one particular quantity of interest is the expected trapping time. This can be obtained by taking the derivative of $m_{\delta}(x)$, where
\vspace{0.3cm}
\begin{align}
\mathbb{E}\left[\tau_{x}\right]=-\left.\frac{d}{d \delta} m_{\delta}(x)\right|_{\delta=0}.
\label{TheTrappingTime-Section3-Equation22}
\end{align}
\vspace{0.3cm}
As such, we differentiate Tricomi\rq s Confluent Hypergeometric Function with respect to its second parameter. Denote
\vspace{0.3cm}
\begin{align}
U^{(c)}(a, c; z)\equiv \frac{d}{d c} U(a, c; z).
\label{TheTrappingTime-Section3-Equation23}
\end{align}
\vspace{0.3cm}
A closed form expression of the aforementioned derivative can be given in terms of series expansions, such that
\vspace{0.3cm}
\begin{align}
\begin{split}
U^{(c)}(a, c; z)&=(\eta(a-c+1)-\pi \cot (c\pi)) U(a, c; z)\\
&-\frac{\Gamma(c-1) z^{1-c} \log (z)}{\Gamma(a)}{ }M(a-c+1 , 2-c ; z)\\&-
\frac{\Gamma(c-1) z^{1-c}}{\Gamma(a)} \sum_{k=0}^{\infty} \frac{(a-c+1)_{k}(\eta(a-c+k+1)-\eta(2-c+k)) z^{k}}{(2-c)_{k} k !}\\
&-\frac{\Gamma(1-c)}{\Gamma(a-c+1)} \sum_{k=0}^{\infty} \frac{\eta(c+k)(a)_{k} z^{k}}{(c)_{k} k !}, \qquad c \notin \mathbb{Z},
\end{split}
\label{TheTrappingTime-Section3-Equation24}
\end{align}
\vspace{0.3cm}
where $\eta(z)=\frac{d \ln\left[\Gamma(z)\right]}{dz}=\frac{\Gamma '(z)}{\Gamma(z)}$ corresponds to equation (6.3.1) of \cite{Book:Abramowitz1964}, also known as the digamma function. Thus, using expression \eqref{TheTrappingTime-Section3-Equation24}, we obtain the expected trapping time
\vspace{0.3cm}
\begin{align}
\begin{split}
\mathbb{E}\left[\tau_{x}\right]&=
\frac{\Gamma\left(\frac{\lambda}{r};-y(x)\right)}{\lambda U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}-\frac{\Gamma\left(\frac{\lambda}{r};-y(x)\right)U^{(c)}\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}{r\left[U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)\right]^{2}}\\
&+e^{y(x)}\frac{U^{(c)}\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};-y(x)\right)}{rU\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}.
\end{split}
\label{TheTrappingTime-Section3-Equation25}
\end{align}
\vspace{0.3cm}
\vspace{0.3cm}
In line with intuition, the expected trapping time is an increasing function of both the capital growth rate $r$ and initial capital $x$. However, since the capital process grows exponentially, large initial capital and capital growth rates significantly reduce the trapping probability and increase the expected trapping time to the point where it becomes non-finite, making the indicator function in the expected discounted penalty function \eqref{TheTrappingTime-Section3-Equation2} tend to zero. A number of expected trapping times for varying values of $r$ are displayed in Figure \ref{TheTrappingTime-Section3-Figure2}.
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8.5cm, height=7cm]{Figures/TheTrappingTime-Section3-Figure2.pdf}
\caption{Expected trapping time when $Z_{i} \sim Exp(1)$, $\lambda = 1$ and $x^{*} = 1$ for $r = 0.02,0.05,0.08$.}
\label{TheTrappingTime-Section3-Figure2}
\end{figure}
\vspace{0.3cm}
\item The ruin probability for the insurance model \eqref{TheCapitalModel-Section2-Equation5} given by
\vspace{0.3cm}
\begin{align}
\xi (u)= P(U_t<0 \text{ for some }t>0 \mid U_0=u)
\label{TheTrappingTime-Section3-Equation26},
\end{align}
\vspace{0.3cm}
is found by \cite{Article:Sundt1995} to satisfy the IDE
\vspace{0.3cm}
\begin{align}
(au+p)\xi'(u)-\lambda \xi (u)+ \lambda \int_0^ {u} \xi(u-z) \, dG(z)+\lambda(1-G(u))=0, \qquad u\geq 0.
\label{TheTrappingTime-Section3-Equation27}
\end{align}
\vspace{0.3cm}
Note that when $p=0$,
\eqref{TheTrappingTime-Section3-Equation27} coincides with the special case of
\eqref{TheTrappingTime-Section3-Equation3} when $x^*=0$, $w(x)=1$, and $\delta=0$. Thus, the household's trapping time can be thought of as the insurer\rq s ruin time. Indeed, the ruin probability in the case of exponential claims when $p=0$ as shown in Section 6 of \cite{Article:Sundt1995} is exactly the same as the trapping probability \eqref{TheTrappingTime-Section3-Equation21} when $x^*=0$.
\end{enumerate}
\section{Introducing Microinsurance}\label{IntroducingMicroinsurance-Section4}
As in \cite{Article:Kovacevic2011}, we assume that households have the option of enrolling in a microinsurance scheme that covers a certain proportion of the capital losses they encounter. The microinsurance policy has proportionality factor $1-\kappa$, where $\kappa \in [0,1]$, such that $100 \cdot (1-\kappa)$ percent of the damage is covered by the microinsurance provider. The premium rate paid by households, calculated according to the expected value principle is given by
\vspace{0.3cm}
\begin{align}
\pi(\kappa, \theta)=(1+\theta) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z_{i}), \label{IntroducingMicroinsurance-Section4-Equation1}
\end{align}
\vspace{0.3cm}
where $\theta$ is some loading factor. The expected value principle is popular due to its simplicity and transparency. When $\theta = 0$, one can consider $\pi(\kappa, \theta)$ to be the pure risk premium \citep{Book:Albrecher2017}. We assume the basic model parameters are unchanged by the introduction of microinsurance coverage.
The stochastic capital process of a household covered by a microinsurance policy is denoted by $X_{t}^{\scaleto{(\kappa)}{5pt}}$. We differentiate between all variables and parameters relating to the original uninsured and insured processes by using the superscript $(\kappa)$ in the latter case.
Since the premium is paid from a household's income, the capital growth rate $r$ is adjusted such that it reflects the lower rate of income generation resulting from the need for premium payment. The premium rate is restricted to prevent certain poverty, which would occur should the premium rate exceed the rate of income generation. The capital growth rate of the insured household $r^{(\kappa)}$ is lower than that of the uninsured household, while the critical capital is higher.
In between jumps, where $T_{i-1} \leq t<T_{i}$, the insured stochastic growth process $X_{t}^{(\kappa)}$ behaves in the same manner as \eqref{TheCapitalModel-Section2-Equation2}, with parameters corresponding to the proportional insurance case of this section, making particular note of the increased critical capital $x^{\scaleto{(\kappa)*}{5pt}}$:
\vspace{0.3cm}
\begin{align}
X_{t}^{\scaleto{(\kappa)}{5pt}}=\begin{cases} \left(X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}-x^{\scaleto{(\kappa)*}{5pt}}\right) e^{r^{\scaleto{(\kappa)}{5pt}} \left(t-T_{i-1}\right)}+x^{\scaleto{(\kappa)*}{5pt}} & \text { if } X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}>x^{\scaleto{(\kappa)*}{5pt}}, \\ X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}} & \text{ otherwise}.
\end{cases}
\label{IntroducingMicroinsurance-Section4-Equation3}
\end{align}
\vspace{0.3cm}
For $t = T_{i}$, the process is given by
\vspace{0.3cm}
\begin{align}
X_{T_{i}}^{\scaleto{(\kappa)}{5pt}}=\begin{cases} \left(X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}-x^{\scaleto{(\kappa)*}{5pt}}\right) e^{r^{\scaleto{(\kappa)}{5pt}} \left(T_{i}-T_{i-1}\right)}+x^{\scaleto{(\kappa)*}{5pt}} - \kappa \cdot Z_{i} & \text { if } X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}>x^{\scaleto{(\kappa)*}{5pt}}, \\ X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}} - \kappa \cdot Z_{i} & \text{ otherwise}.
\end{cases}
\label{IntroducingMicroinsurance-Section4-Equation4}
\end{align}
\vspace{0.3cm}
By enrolling in a microinsurance scheme, a household\rq s capital losses are reduced to $Y_{i} :=\kappa \cdot Z_{i}$. Considering the case in which losses follow an exponential distribution with parameter $\alpha > 0$, the structure of \eqref{TheTrappingTime-Section3-Equation5} remains the same. However, acquisition of a proportional microinsurance policy changes the parameter of the distribution of the random variable of the losses ($Y_i$). Namely, we have that $Y_{i} \sim Exp\left(\alpha^{\scaleto{(\kappa)}{5pt}}\right)$ for $\kappa \in (0,1]$, where $\alpha^{\scaleto{(\kappa)}{5pt}} := \frac{\alpha}{\kappa}$. We can therefore utilise the results obtained in Section \ref{TheTrappingTime-Section3} to obtain the Laplace transform of the trapping time for the insured process, which is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)=\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};0\right)}e^{y^{\scaleto{(\kappa)}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\kappa)}{5pt}}(x)\right),
\label{IntroducingMicroinsurance-Section4-Equation5}
\end{align}
\vspace{0.3cm}
where $y^{\scaleto{(\kappa)}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}\left(x-x^{\scaleto{(\kappa)*}{5pt}}\right)$. Figure \ref{IntroducingMicroinsurance-Section4-Figure1}(a) displays the Laplace transform $m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)$ for varying values of $\delta$.
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item The trapping probability of the insured process $\psi^{\scaleto{(\kappa)}{5pt}}(x)$, displayed in Figure \ref{IntroducingMicroinsurance-Section4-Figure1}(b), is given by
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\kappa)}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}\right)}.
\label{IntroducingMicroinsurance-Section4-Equation6}
\end{align}
\vspace{0.3cm}
\begin{figure}[H]
\begin{multicols}{2}
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure1-a.pdf}
\centering (a)
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure1-b.pdf}
\centering (b)
\end{multicols}
\caption{(a) Laplace transform $m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)$ of the trapping time when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\kappa)*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $\delta = 0, \frac{1}{8}, \frac{1}{32}, \frac{1}{128}$ (b) Trapping probability $\psi^{\scaleto{(\kappa)}{5pt}}(x)$ when $Z_{i} \sim Exp(\alpha)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\kappa)*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $\alpha = 0.8, 1, 1.5, 2$.}
\label{IntroducingMicroinsurance-Section4-Figure1}
\end{figure}
\vspace{0.3cm}
\item When $\kappa = 0$ the household has full microinsurance coverage, the microinsurance provider covers the total capital loss experienced by the household. On the other hand, when $\kappa = 1$, no coverage is provided by the insurer i.e., $X_{t}=X_{t}^{\scaleto{(\kappa)}{5pt}}$.
\item We are interested in studying significant capital losses, since low-income individuals are commonly exposed to this type of shock. Hence, throughout the paper, the parameter $\alpha >0$ should be considered to reflect the desired loss behaviour.
\end{enumerate}
Figure \ref{IntroducingMicroinsurance-Section4-Figure2} presents a comparison between the trapping probabilities of the insured and uninsured processes. As in \cite{Article:Kovacevic2011}, households with initial capital close to the critical capital (here, the critical capital $x^{*}=1$), i.e. the most vulnerable individuals, do not receive a real benefit from enrolling in a microinsurance scheme. Although subscribing to a proportional microinsurance scheme reduces capital losses, premium payments appear to make the most vulnerable households more prone to falling into the area of poverty. In Figure \ref{IntroducingMicroinsurance-Section4-Figure2}, the intersection point of the two probabilities corresponds to the boundary between households that benefit from the uptake of microinsurance and those who are adversely affected.
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure2.pdf}
\caption{Trapping probabilities for the uninsured and insured capital processes, when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $\kappa = 0.5$, $\theta=0.5$ and $x^{*} = 1$.}
\label{IntroducingMicroinsurance-Section4-Figure2}
\end{figure}
\section{Microinsurance with Subsidised Constant Premiums}\label{MicroinsurancewithSubsidisedConstantPremiums-Section5}
\subsection{General Setting}\label{GeneralSetting-Subection51}
Since microinsurance alone is not enough to reduce the likelihood of impoverishment for those close to the poverty line, additional aid is required. In this section, we study the cost-effectiveness of government subsidised premiums, considering the case in which the government subsidises an amount $\beta = \theta - \theta^{*}$, while the microinsurance provider claims a lower loading factor $\theta^{*}$ \citep{Article:Kovacevic2011}. The following relationship between premiums for the non-subsidised and subsidised microinsurance schemes therefore holds
\vspace{0.3cm}
\begin{align}
\pi(\kappa, \theta) = (1+\theta) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z) \ge (1+\theta^{*}) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z) = \pi(\kappa, \theta^{*}).
\label{GeneralSetting-Subection51-Equation1}
\end{align}
\vspace{0.3cm}
Naturally, we assume governments are interested in optimising the subsidy provided to households. Governments should provide subsidies to microinsurance providers such that they enhance households\rq \, benefits of enrolling in microinsurance schemes, however, they also need to gauge the cost-effectiveness of subsidy provision. Households with capital very close to the critical capital will not benefit from enrolling into the scheme even if the entire loading factor $\theta$ is subsidised by the government, however, more privileged households will. One approach to finding the optimal loading factor $\theta^{*}$ for households that could benefit from the government subsidy is to find the solution of the equation
\vspace{0.3cm}
\begin{align}
\psi^{\left(\kappa, \theta^{*}\right)}(x)=\psi(x),
\label{GeneralSetting-Subection51-Equation2}
\end{align}
\vspace{0.3cm}
where $\psi^{\left(\kappa, \theta^{*}\right)}(x)$ and $\psi(x)$ denote the trapping probabilities of the insured subsidised and uninsured processes, respectively, since all loading factors below the optimal loading factor will induce a trapping probability lower than that of the uninsured process through a reduction in premium. This behaviour can be seen in Figure \ref{GeneralSetting-Subection51-Figure1}(a), while the \lq \lq richest\rq \rq \ households do not need help from the government since the non-subsidised insurance lowers their trapping probability below the uninsured case, the poorest individuals require more support. Moreover, as highlighted previously, there are households that do not receive any benefit from enrolling in the microinsurance scheme even when the government subsidises the entire loading factor (when households pay only the pure risk premium, this could occur if the government absorbs all premium administrative expenses). Note that Figure \ref{GeneralSetting-Subection51-Figure1}(b) illustrates the optimal loading factor $\theta^{*}$ for varying initial capital. Initial capitals are plotted from the point at which households begin benefiting from the subsidised microinsurance scheme, i.e. the point at which the dashed $(\theta=0,\beta=0.5)$ line intersects the solid line in Figure \ref{GeneralSetting-Subection51-Figure1}(a). Additionally, Figure \ref{GeneralSetting-Subection51-Figure1}(b) verifies that, from the point at which the dashed-dotted (insured household) line intersects the solid line in Figure \ref{GeneralSetting-Subection51-Figure1}(a), the optimal loading factor remains constant, with $\theta^{*}=0.5$, i.e. the \lq \lq richest\rq \rq \ households can afford to pay the entire premium.
\begin{figure}[H]
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=8cm, height=8cm]{Figures/GeneralSetting-Subection51-Figure1-a.pdf}
\centering (a)
\end{minipage}%
\hfill%
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=7cm, height=7cm]{Figures/GeneralSetting-Subection51-Figure1-b.pdf}
\centering (b)
\end{minipage}
\caption{(a) Trapping probabilities for the uninsured, insured and insured subsidised capital processes when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$ for loading factors $\theta^{*} = 0, 0.25$ (b) Optimal loading factor $\theta^{*}$ for varying initial capitals when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$.}
\label{GeneralSetting-Subection51-Figure1}
\end{figure}
\subsection{Cost of Social Protection}\label{CostofSocialProtection-Subection52}
Next, we assess government cost-effectiveness for the provision of microinsurance premium subsidies to households. Let $\delta \geq 0$ be the force of interest for valuation, and let $S$ denote the present value of all subsidies provided by the government until the trapping time such that
\vspace{0.3cm}
\begin{align}
S=\beta\int^{\tau_{x}}_{0} e^{-\delta t} dt =\beta \ax*{\angl{\tau_{x}}}. \label{CostofSocialProtection-Subection52-Equation1}
\end{align}
\vspace{0.3cm}
We assume a government provides subsidies according to the strategy introduced earlier, i.e. the government subsidises an amount $\beta = \theta - \theta^{*}$, while the microinsurance provider claims a lower loading factor $\theta^{*}$.
For $x\geq x^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}$, where $x^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}$ denotes the critical capital of the insured subsidised process, let $V(x)$ be the expected discounted premium subsidies provided by the government to a household with initial capital $x$ until trapping time, that is,
\vspace{0.3cm}
\begin{align}
V(x)= \mathbb{E}\left[S \mid X^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}_{0}=x\right].
\label{CostofSocialProtection-Subection52-Equation2}
\end{align}
\vspace{0.3cm}
Since $S=\frac{\beta}{\delta} \left[1-e^{-\delta \tau_{x}}\right]$, we can define $m^{\scaleto{(\kappa,\theta^{*})}{5pt}}_\delta(x)$, the Laplace transform of the trapping time with rate $r^{\scaleto{(\kappa,\theta^{*})}{5pt}}$ and critical capital $x^{\scaleto{(\kappa,\theta^{*})*}{5pt}}$, using the Laplace transform for the insured process previously obtained in \eqref{IntroducingMicroinsurance-Section4-Equation5} to compute $V(x)$ when losses are exponentially distributed with parameter $\alpha^{\scaleto{(\kappa)}{5pt}}>0$. This yields
\vspace{0.3cm}
\begin{equation}
\begin{aligned}
V(x)&= \frac{\beta}{\delta}\left[1- m^{\scaleto{(\kappa,\theta^{*})}{5pt}}_\delta(x)\right]\\
&= \frac{\beta}{\delta}\left[1-\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},0\right)}e^{y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}}, -y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)\right)\right],
\end{aligned}
\label{CostofSocialProtection-Subection52-Equation3}
\end{equation}
\vspace{0.3cm}
\vspace{0.3cm}
where $y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}(x-x^{\scaleto{(\kappa,\theta^{*})*}{5pt}})$. We now formally define the government\rq s cost of social protection.
\vspace{0.3cm}
\begin{definition}
Let $\psi^{\left(\kappa, \theta^{*}\right)}(x)$ be the trapping probability of a household enrolled in a subsidised microinsurance scheme with initial capital $x$. Additionally, let $M > 0$ be a constant representing the cost to lift households below the critical capital out of the area of poverty. The government\rq s cost of social protection is given by
\vspace{0.3cm}
\begin{equation}
\textit{Cost of Social Protection}:= V(x) + M \cdot {\psi^{\left(\kappa, \theta^{*}\right)}(x)}. \label{CostofSocialProtection-Subection52-Equation4}
\end{equation}
\vspace{0.3cm}
\end{definition}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item For uninsured households, the government does not provide subsidies, i.e. $V(x) =0$. Furthermore, we consider their trapping probability to be $\psi(x)$.
\item The government manages selection of an appropriate force of interest $\delta \geq 0$ and constant $M > 0$. For lower force of interest the government discounts future subsidies more heavily, while for higher interest future subsidies almost vanish. The constant $M$ could be defined, for example, using the poverty gap index introduced by \cite{Article:Foster1984}, or in such a way that the government ensures with some probability that households will not fall into the area of poverty. Thus, higher values of $M$ will increase the certainty that households will not return to poverty.
\end{enumerate}
Figure \ref{CostofSocialProtection-Subection52-Figure1} displays the government cost of social protection. Observe that in this particular example, we consider high values for both the force of interest $\delta$ and the constant $M$. The choice of $M$ is motivated by Figure \ref{IntroducingMicroinsurance-Section4-Figure2}, which shows that from $x=8$, the trapping probability for uninsured households is very close to zero. Note that a high value of $\delta$ hands a lower weight to future government subsidies whereas a high value of $M$ grants higher certainty that a household will not return to the area of poverty once lifted out.
It is clear that governments do not benefit by entirely subsidising the \lq \lq richest\rq \rq \ households, since they will subsidise premiums indefinitely, almost surely (dashed line for highest values of initial capital). Hence, as illustrated in Figure \ref{GeneralSetting-Subection51-Figure1}(b), it is favourable for governments to remove subsidies for this particular group since their cost of social protection is even higher than when uninsured (solid line for highest values of initial capital). Conversely, governments perceive a lower cost of social protection when fully subsidising the loading factor $\theta$ for households with initial capital lying closer to the critical capital $x^{*}$. The cost of social protection when households pay only the pure risk premium is lower than when paying the premium entirely for values of initial capital in which the dashed line is below the dotted, in which case the government should support premium payments. However, due to the fact that they will almost surely fall into the area of poverty, requiring governments to pay the subsidy in addition to the cost of lifting a household out of poverty, it is not optimal to fully subsidise the loading factor for the most vulnerable, since the cost of social protection is higher than that for uninsured households. Note that, from the point of view of the governmental cost of social protection, Figure \ref{CostofSocialProtection-Subection52-Figure1} confirms earlier statements asserting the inefficiency of providing premium support to the most vulnerable, i.e. neither individual households nor governments receive real benefit under such a scheme. Thus, alternative risk management strategies should be considered for this sector of the low-income population.
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/CostofSocialProtection-Subection52-Figure1.pdf}
\caption{Cost of social protection for the uninsured, insured and insured subsidised capital processes when $Z_{i}\sim Exp(1)$, $r=0.5$, $\lambda=1$, $x^{*}=1$, $\kappa = 0.5$, $\theta = 0.5$, $\delta =0.9$ and $M=8$ for loading factor $\theta^{*}= 0$.}
\label{CostofSocialProtection-Subection52-Figure1}
\end{figure}
\section{Microinsurance with Subsidised Flexible Premiums}\label{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}
\subsection{General Setting}\label{GeneralSetting-Subsection61}
Since premiums are generally paid as soon as microinsurance coverage is purchased, a household\rq s capital growth could be constrained. It is therefore interesting to consider alternative premium payment mechanisms. From the point of view of microinsurance providers, advance premium payments are preferred so that additional income can be generated through investment, naturally leading to lower premium rates. Conversely, consumers may find it difficult to pay premiums up front. This is a common problem in low-income populations, with consumers preferring to pay smaller installments over time \citep{Book:Churchill2006}. Collecting premiums at a time that is inconvenient for households can be futile. Flexible premium payment mechanisms have been highly adopted by informal funeral insurers in South Africa, where policyholders pay premiums whenever they are able, rather than at a specific time during the month \citep{WorkingPaper:Roth2000}. Similar alternative insurance designs in which premium payments are delayed until the insured\rq s income is realised and any indemnities are paid have also been studied. Under such designs, insurance take-up increases, since liquidity constraints are relaxed and concerns regarding insurer default, also prevalent in low-income classes, reduce \citep{Article:Liu2016}.
In this section, we introduce an alternative microinsurance subsidy scheme with flexible premium payments. We denote the capital process of a household enrolled in the alternative microinsurance subsidy scheme by $X_{t}^{\scaleto{(\mathcal{A})}{5pt}}$. Furthermore, as in Section \ref{IntroducingMicroinsurance-Section4}, we differentiate between variables and parameters relating to the original, insured and alternative insured processes using the superscript $(\mathcal{A})$. Under such an alternative microinsurance subsidy scheme, households pay premiums when their capital is above some capital barrier $B \ge x^{\scaleto{(\mathcal{A})*}{5pt}}$, with the premium otherwise paid by the government. In other words, whenever the insured capital process is below the capital level $B$, premiums are entirely subsidised by the government, however, when a household's capital is above $B$, the premium $\pi$ is paid continuously by the household itself. This method of premium collection may motivate households to maintain a level of capital below $B$ in order to avoid premium payments. Consequently, we assume that households always pursue capital growth. Our aim is to study how this alternative microinsurance subsidy scheme can help households reduce their probability of falling into the area of poverty. We also measure the cost-effectiveness of such scheme from the point of view of the government.
The intangibility of microinsurance makes it difficult to attract potential clients. Most clients will never experience a claim and so cannot perceive the real value of microinsurance, paying more to the scheme (in terms of premium payments) than what they actually receive from it. It is only when claims are settled that microinsurance becomes tangible. The alternative microinsurance subsidy scheme described here could increase client value, since, for example, individuals below the barrier $B$ may submit claims, receive a payout and therefore perceive the value of microinsurance when they suffer a loss, regardless of whether they have ever paid a single premium. Other ways of increasing microinsurance client value include bundling microinsurance with other products and introducing Value Added Services (VAS), which represent services such as telephone hotlines for consultation with doctors or remote diagnosis services (for health schemes) offered to clients outside of the microinsurance contract \citep{Article:Madhur2019}.
Under the alternative microinsurance subsidy scheme, the Laplace transform of the trapping time satisfies the following differential equations:
\vspace{0.3cm}
\small
\begin{align}
0 =
\begin{cases} -\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime \prime}(x)+\left[\frac{(\lambda+\delta-r)}{\alpha^{\scaleto{(\kappa)}{5pt}} r}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] m_{\delta}^{\scaleto{(\mathcal{A})}{5pt} \prime}(x)+\frac{\delta}{r} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) \hspace{1cm} \text{for $ x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$}, \\ -\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime \prime}(x)+\left[\frac{(\lambda+\delta-r^{\scaleto{(\kappa)}{5pt}})}{\alpha^{\scaleto{(\kappa)}{5pt}} r^{\scaleto{(\kappa)}{5pt}}}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)+\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) \hspace{0.3cm} \text{for $x \geq B$}.
\end{cases}
\label{GeneralSetting-Subsection61-Equation1}
\end{align}
\normalsize
\vspace{0.3cm}
As in Section \ref{TheTrappingTime-Section3}, use of the change of variable $y^{\scaleto{(\mathcal{A})}{5pt}}:=y^{\scaleto{(\mathcal{A})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}} (x-x^{\scaleto{(\mathcal{A})*}{5pt}})$ leads to Kummer\rq s Confluent Hypergeometric Equation and thus,
\vspace{0.3cm}
\footnotesize
\begin{align}
m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) =
\begin{cases} C_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) + C_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) & \text{for $ x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$}, \\ C_{3}M\left(-\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ C_{4}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) & \text{for $x \geq B$},
\end{cases}
\label{GeneralSetting-Subsection61-Equation2}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $C_{1},C_{2},C_{3},C_{4} \in \mathbb{R}$. Under the boundary condition $\lim_{x\to\infty} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) = 0$ with asymptotic behaviour of the Kummer function $M(a,c;z)$ as presented in Section \ref{TheTrappingTime-Section3}, we deduce that $C_{3}=0$. Also, since $m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x^{\scaleto{(\mathcal{A})*}{5pt}})=\frac{\lambda}{\lambda + \delta}$, we obtain $C_{1}=\frac{\lambda}{\lambda + \delta} - C_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)$.
Due to the continuity of the functions $ m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ and $m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)$ at $x=B$ and the differential properties of the Confluent Hypergeometric Functions
\vspace{0.3cm}
\begin{align}
\frac{d}{dz}M(a,c;z)=\frac{a}{c}M(a+1,c+1;z),
\label{GeneralSetting-Subsection61-Equation3}
\end{align}
\begin{align}
\frac{d}{dz}U(a,c;z)=-aU(a+1,c+1;z),
\label{GeneralSetting-Subsection61-Equation4}
\end{align}
\vspace{0.3cm}
upon simplification,
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
C_{4}=\frac{\left[\frac{\lambda}{\lambda + \delta} - C_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\right]M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)
+C_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\end{split}
\label{GeneralSetting-Subsection61-Equation5}
\end{align}
\normalsize
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
\begin{split}
C_{2}=\frac{\frac{\lambda}{\lambda+\delta}\left[\frac{\delta \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda-\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)+M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}} - D\right)\right]}{K},
\end{split}
\label{GeneralSetting-Subsection61-Equation6}
\end{align}
\vspace{0.3cm}
where
\vspace{0.3cm}
\begin{align}
D:=\frac{\alpha^{\scaleto{(\kappa)}{5pt}}\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}-1\right)U\left(2-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},2-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\label{GeneralSetting-Subsection61-Equation7}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
K: &= M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}}-D\right)
\\
&+D e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) \\
&+\frac{\delta \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda -\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)
\\
&-\alpha^{\scaleto{(\kappa)}{5pt}}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}\left(\frac{\lambda}{r}-1\right)U\left(2-\frac{\lambda}{r}, 2-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right).
\label{GeneralSetting-Subsection61-Equation8}
\end{split}
\end{align}
\normalsize
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item The trapping probability $\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)$ for the alternative microinsurance subsidy scheme is given by
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\begin{cases} 1 - \frac{\Gamma\left(\frac{\lambda}{r}\right)-\Gamma\left(\frac{\lambda}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)}{ (-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)}\Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) +\Gamma\left(\frac{\lambda}{r}\right)-\Gamma \left(\frac{\lambda}{r}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}, & x \leq B\\ \\
\frac{(-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)} \Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)}{ (-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)}\Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) +\Gamma\left(\frac{\lambda}{r}\right)-\Gamma \left(\frac{\lambda}{r}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}, & x \geq B.
\end{cases}
\label{GeneralSetting-Subsection61-Equation9}
\end{align}
\vspace{0.3cm}
Similar to the subsidised case, we can find the optimal barrier $B^{*}$ by determining the solution of the equation
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\mathcal{A}, B^{*})}{5pt}}(x)=\psi(x), \label{GeneralSetting-Subsection61-Equation10}
\end{align}
\vspace{0.3cm}
where $\psi^{\scaleto{(\mathcal{A}, B^{*})}{5pt}}(x)$ and $\psi(x)$ denote the trapping probability of the capital process under the alternative microinsurance subsidy scheme and the uninsured capital process, respectively. Some examples are presented after the remarks.
\item When $B \rightarrow x^{\scaleto{(\mathcal{A})*}{5pt}}$, the trapping probability for the alternative microinsurance subsidy scheme is equal to the trapping probability obtained for the insured case $\psi^{\scaleto{(\kappa)}{5pt}}(x)$, i.e.
\vspace{0.3cm}
\begin{align}
\lim_{B \rightarrow x^{\scaleto{(\mathcal{A})*}{3.5pt}}}\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}\right)}.
\label{GeneralSetting-Subsection61-Equation11}
\end{align}
\vspace{0.3cm}
Moreover, letting $B \rightarrow \infty$, the trapping probability is given by
\vspace{0.3cm}
\begin{align}
\lim_{B \rightarrow \infty}\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r}\right)}.
\label{GeneralSetting-Subsection61-Equation12}
\end{align}
\item Figure \ref{GeneralSetting-Subsection61-Figure1} displays the expected trapping time under the alternative microinsurance subsidy scheme. Not surprisingly, the expected trapping time is an increasing function of both the capital growth rate $r$ and barrier $B$.
\begin{figure}[H]
\centering
\includegraphics[width=8.5cm, height=7cm]{Figures/GeneralSetting-Subsection61-Figure1.pdf}
\caption{Expected trapping time when $Z_{i} \sim Exp(1)$, $\lambda = 1$, $x = 3.5$, $x^{\scaleto{(\mathcal{A})*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $r = 0.08,0.082,0.084$.}
\label{GeneralSetting-Subsection61-Figure1}
\end{figure}
\end{enumerate}
Figure \ref{GeneralSetting-Subsection61-Figure2}(a) shows the trapping probabilities for varying initial capital values under the uninsured, insured, subsidised and alternatively subsidised schemes. As expected, increasing the value of the capital barrier $B$ helps households to reduce their probability of falling into the area of poverty, since support from the government is received when their capital resides in the region between the critical capital $x^{\scaleto{(\mathcal{A})*}{5pt}}$ and the barrier $B$. Furthermore, as in the previous case, households with higher levels of initial capital do not need support from the government, insurance without subsidies decreases their trapping probability to a level below the uninsured (households with initial capital greater than or equal to the point at which the dotted line intersects the solid line).
The optimal barrier for these individuals is in fact the critical capital, i.e., $B^{*}=x^{\scaleto{(\mathcal{A})*}{5pt}}$, households with higher initial capital can therefore afford to pay for microinsurance coverage themselves, as illustrated in Figure \ref{GeneralSetting-Subsection61-Figure2}(b). Figure \ref{GeneralSetting-Subsection61-Figure2}(b) also shows that for the most vulnerable, the government should set up a barrier above their initial capital to remove capital growth constraints associated with premium payments. This level should be selected until the household reaches a capital level that is adequate in ensuring their trapping probability will be equal to that of an uninsured household. Conversely, for the more privileged (those in Figure \ref{GeneralSetting-Subsection61-Figure2}(b), with initial capital approximately greater than or equal to 2), the government should establish barriers below their initial capital, with households paying premiums themselves as soon as they enrol in the microinsurance scheme. This behaviour is mainly due to the fact that their level of capital is distant from the critical capital $x^{\scaleto{(\mathcal{A})*}{5pt}}$. These households are unlikely to fall into the area of poverty after suffering one capital loss, they are instead likely to fall into the region between the critical capital and the barrier $B$ (i.e. the area within which the government pays premiums), before entering the area of poverty. Thus, the aforementioned region acts as a \lq \lq buffer\rq \rq \ for households, since once in this region they will benefit from coverage without paying any premiums. Increasing the initial capital will lead to a decrease in the size of the \lq \lq buffer\rq \rq \ region until it disappears when $B=x^{\scaleto{(\mathcal{A})*}{5pt}}$, as shown in the lower right corner of Figure \ref{GeneralSetting-Subsection61-Figure2}(b), where a straight line is visible.
\vspace{0.3cm}
\begin{figure}[H]
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=8cm, height=8cm]{Figures/GeneralSetting-Subsection61-Figure2-a.pdf}
\centering (a)
\end{minipage}%
\hfill%
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=7cm, height=7cm]{Figures/GeneralSetting-Subsection61-Figure2-b.pdf}
\centering (b)
\end{minipage}
\caption{(a) Trapping probabilities for the uninsured, insured, insured subsidised with $\theta^{*}=0$ and insured alternatively subsidised with $B = 2, 4$ capital processes when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$ (b) Difference between the optimal barrier and the initial capital, i.e. $B^{*}-x$, for varying initial capitals, when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\mathcal{A})*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta = 0.5$.}
\label{GeneralSetting-Subsection61-Figure2}
\end{figure}
\subsection{Cost of Social Protection}\label{CostofSocialProtection-Subsection62}
Similarly to the previous section, it is reasonable to measure the governmental cost-effectiveness of providing microinsurance premium subsidies to households under the alternative microinsurance subsidy scheme. For this reason, we define $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ as the expectation of the present value of all subsidies provided by the government until the trapping time under the alternative microinsurance subsidy scheme, that is
\vspace{0.3cm}
\begin{equation}
V^{\scaleto{(\mathcal{A})}{5pt}}(x):=\mathbb{E}\left[\int_{0}^{\tau_{x}} \pi e^{-\delta t} \mathbbm{1}_{\left\{X_{t}^{\scaleto{(\mathcal{A})}{5pt}} < B\right\}} dt \middle| X^{\scaleto{(\mathcal{A})}{5pt}}_{0}=x\right].
\label{CostofSocialProtection-Subsection62-Equation1}
\end{equation}
\vspace{0.3cm}
If the derivative exists, then using standard infinitesimal generator arguments for $X^{\scaleto{(\mathcal{A})}{5pt}}_{t}$, one gets the following IDE for $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ under the barrier $B$
\vspace{0.3cm}
\begin{equation}
r(x-x^{\scaleto{(\mathcal{A})*}{5pt}})V^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)-(\lambda + \delta) V^{\scaleto{(\mathcal{A})}{5pt}}(x)+\lambda \int_{0}^{x-x^{\scaleto{(\mathcal{A})*}{3.5pt}}}V^{\scaleto{(\mathcal{A})}{5pt}}(x-z)dG(z)+\pi=0, \qquad x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B.
\label{CostofSocialProtection-Subsection62-Equation2}
\end{equation}
\vspace{0.3cm}
Hence, assuming $Z_{i}\sim Exp(\alpha^{(\kappa)})$, the function satisfies the nonhomogeneous differential equation given by
\vspace{0.3cm}
\footnotesize
\begin{align}
-\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} V^{\scaleto{(\mathcal{A})}{5pt}\prime\prime}(x)+\left[\frac{(\lambda+\delta-r)}{\alpha^{\scaleto{(\kappa)}{5pt}} r}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] V^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)+\frac{\delta}{r} V^{\scaleto{(\mathcal{A})}{5pt}}(x)-\frac{\pi}{r}=0, \qquad x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B.
\label{CostofSocialProtection-Subsection62-Equation3}
\end{align}
\normalsize
\vspace{0.3cm}
Letting $V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ be the homogeneous solution of \eqref{CostofSocialProtection-Subsection62-Equation3}, we have
\vspace{0.3cm}
\small
\begin{align}
V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x) =
R_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right), \hspace{0.3cm} x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B,
\label{CostofSocialProtection-Subsection62-Equation4}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $R_{1}, R_{2} \in \mathbb{R}$, where $y^{\scaleto{(\mathcal{A})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}(x-x^{\scaleto{(\mathcal{A})*}{5pt}})$.
Since the general solution of \eqref{CostofSocialProtection-Subsection62-Equation3} can be written as
\vspace{0.3cm}
\begin{align}
V^{\scaleto{(\mathcal{A})}{5pt}}(x)=V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x)+V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x),
\label{MicroinsuranceSubsidiesandtheCostofSocialProtection-Subsection51-Equation5}
\end{align}
\vspace{0.3cm}
where $V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ is a particular solution, one can easily verify that $V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x)= \frac{\pi}{\delta}$ for all $x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$. Then, letting $x=x^{\scaleto{(\mathcal{A})*}{5pt}}$ in \eqref{CostofSocialProtection-Subsection62-Equation2} yields $V^{\scaleto{(\mathcal{A})}{5pt}}(x^{\scaleto{(\mathcal{A})*}{5pt}})=\frac{\pi}{\lambda + \delta}$ and subsequently
\vspace{0.3cm}
\begin{align}
R_{1}=-\left[\frac{\lambda \pi}{(\lambda + \delta) \delta}+R_{2}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};0\right)\right].
\label{CostofSocialProtection-Subsection62-Equation6}
\end{align}
\vspace{0.3cm}
For $x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$, we therefore have
\vspace{0.3cm}
\begin{align}
\begin{split}
V^{\scaleto{(\mathcal{A})}{5pt}}(x)&=-\left[\frac{\lambda \pi}{(\lambda + \delta) \delta}+R_{2}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};0\right)\right]M\left( - \frac{\delta}{r}, 1 - \frac{\lambda + \delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
\\ &+R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) + \frac{\pi}{\delta}.
\end{split}
\label{CostofSocialProtection-Subsection62-Equation7}
\end{align}
\vspace{0.3cm}
Above the barrier $B$, $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ satisfies \eqref{GeneralSetting-Subsection61-Equation1} for $x\geq B$, and so
\vspace{0.3cm}
\small
\begin{align}
V^{\scaleto{(\mathcal{A})}{5pt}}(x) =
R_{3}M\left(-\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ R_{4}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right), \qquad x \geq B,
\label{CostofSocialProtection-Subsection62-Equation8}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $R_{3},R_{4} \in \mathbb{R}$. Since $\lim_{x\to\infty} V^{\scaleto{(\mathcal{A})}{5pt}}(x) = 0$ by definition, we have that $R_{3}=0$. Using the continuity of the functions $ V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ and $V^{\scaleto{(\mathcal{A})\prime}{5pt}}(x)$ at $x=B$ and the differential properties of the Confluent Hypergeometric Functions,
\footnotesize
\begin{align}
\begin{split}
R_{4}&=\frac{-\left[\frac{\lambda \pi}{(\lambda + \delta)\delta} + R_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\right]M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}\\
& +\frac{R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) + \frac{\pi}{\delta}}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\end{split}
\label{CostofSocialProtection-Subsection62-Equation9}
\end{align}
\normalsize
\vspace{0.3cm}
and
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
R_{2}=-\frac{\frac{\lambda \pi}{\lambda+\delta}\left[\frac{ \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda-\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)+\frac{1}{\delta}M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}} - D\right)\right] + \frac{\pi }{\delta} \left(D - \alpha^{\scaleto{(\kappa)}{5pt}}\right)}{K},
\end{split}
\label{CostofSocialProtection-Subsection62-Equation10}
\end{align}
\normalsize
\vspace{0.3cm}
where $D$ and $K$ are \eqref{GeneralSetting-Subsection61-Equation7} and \eqref{GeneralSetting-Subsection61-Equation8}, respectively.
Figure \ref{CostofSocialProtection-Subsection62-Figure1} compares the cost of social protection for the uninsured, insured, insured subsidised and insured alternatively subsidised households. Cost of social protection for the most vulnerable is not reduced with microinsurance coverage (dotted, dashed and dash-dotted lines are all above the solid line for initial capitals close to the critical capital $x^{*}$). As mentioned previously, this corresponds to the high trapping probability of this section of the population, with governments almost surely needing to lift these households out of the area of poverty through payment of a certain amount $M$, in addition to paying subsidies. On the other hand, the cost of social protection could be reduced by providing subsidies to households with greater levels of initial capital (when dashed and dash-dotted lines are below the solid line). Greater initial capitals lead to lower trapping probabilities and a reduction in the likelihood of the government need to pay the value $M$. As a result, the cost of social protection decreases even though subsidies are provided.
Also illustrated in Figure \ref{CostofSocialProtection-Subsection62-Figure1} is the greater cost-effectiveness of conditionally subsidising premiums, in comparison to proportional subsidisation (dashed-dotted line below all other lines for the majority of households in this particular group). Besides lowering the cost of social protection compared to that of uninsured households, it outperforms the traditional insured subsidised case in which some of the loading factor is absorbed by the government. Moreover, the alternative scheme eliminates the disadvantage of paying subsidies indefinitely for the \lq \lq richest\rq\rq \ households almost surely. In consequence, implementing a microinsurance scheme with barrier strategy reduces both the probability of households falling into the area of poverty and the governmental cost of social protection (shown in Figure \ref{GeneralSetting-Subsection61-Figure2}(a) and Figure \ref{CostofSocialProtection-Subsection62-Figure1}, respectively).
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/CostofSocialProtection-Subsection62-Figure1.pdf}
\caption{Cost of social protection for the uninsured, insured, insured subsidised with $\theta^{*}=0$ and insured alternatively subsidised with $B = 3$ capital processes, when $Z_{i}\sim Exp(1)$, $r=0.5$, $\lambda=1$, $x^{*}=1$, $\kappa = 0.5$, $\theta = 0.5$, $\delta =0.9$ and $M=8$.}
\label{CostofSocialProtection-Subsection62-Figure1}
\end{figure}
\section{Conclusion} \label{Conclusion-Section6}
Comparing the impact of three microinsurance mechanisms on the trapping probability of low-income households, we provide evidence for the importance of governmentally supported inclusive insurance in the strive towards poverty alleviation. The results of Sections \ref{IntroducingMicroinsurance-Section4} and \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} support those of \cite{Article:Kovacevic2011}, highlighting a threshold below which insurance increases the probability of trapping. Further to these findings, we have introduced an alternative mechanism with the capacity to reduce this effect, while strengthening government social protection programs by lowering costs.
Analysis of the subsidised microinsurance scheme with premium payment barrier, suggests that in general, the trapping probability of a household is reduced in comparison to basic microinurance and subsidised microinsurance structures, in addition to that of uninsured households. More significant influence is observed in relation to the governmental cost of social protection, with the cease of subsidy payments when household capital is sufficient facilitating government savings and therefore increasing social protection efficiency. Cost of social protection for those closest to the area of poverty remains greater than the corresponding uninsured cost in both subsidised schemes considered. For such households, governments must account for both their support of premium payments and the likely need for household removal from poverty. Government endorsement of further alternative risk mitigation strategies, such as asset accumulation, would be beneficial for those with capital close to the critical level, minimising their risk of falling beneath the poverty line, while reducing social protection costs.
\setcitestyle{numbers}
\bibliographystyle{plainnat}
\section{Introduction} \label{Introduction-Section1}
Inclusive insurance (or microinsurance) relates to the provision of insurance services to low-income populations with limited access to mainstream insurance or alternative effective risk management strategies. Many individuals excluded from basic financial services and those microinsurance aims to proctect, live below the minimum level of income required to meet their basic needs. Currently fixed at \$1.90 USD per day,
9.2\% of the population were estimated to live below the international extreme poverty line in 2017 \citep{Online:WorldBank2021}. Increases in the number of new poor and those returning to poverty as a result of the COVID-19 pandemic are expected to reverse the historically declining poverty trend \citep{Book:WorldBank2020}.
Fundamental features of the microinsurance environment such as the nature of low income risks, limited financial literacy and experience, product accessibility and data availability, create barriers to penetration, particularly in relation to the affordability of products. For the proportion of the population living just above the poverty line, premium payments heighten the risk of poverty trapping and induce a balance between profit and loss as a result of insurance coverage, dependent on the entity's level of capital. Here, poverty trapping refers to the inability of the poor to escape poverty without external help \citep{Article:Kovacevic2011}.
Highlighting vulnerability reduction and investment incentive effects of insurance, \cite{Article:Janzen2020} observe a marked reduction in long-term poverty and the social protection costs required to close the poverty gap following introduction of an asset insurance market. Calibrating their model to risk-prone regions in Africa, their study suggests that those in the neighbourhood of the poverty line do not optimally purchase insurance (without subsidies), suppressing their consumption and mitigating the probability of trapping.
\cite{Article:Kovacevic2011} propose negative consequences of insurance uptake for members of low-income populations closest to the poverty line, applying ruin-theoretic approaches to calculation of the trapping probability. \cite{Article:Liao2020} support these findings in their analysis of a multi-equilibrium model with agricultural output risks on data from rural China. Voluntary insurance would enable individuals close to the poverty threshold to opt out of insurance purchase in favour of alternative risk management strategies, in order to mitigate this risk.
In line with the findings of \cite{Article:Singh2020} on the effectiveness of social protection mechanisms for poverty alleviation, \cite{Article:Jensen2017} observe a greater reduction in poverty through implementation of an integrated social protection programme in comparison to pure cash transfers. Government subsidised premiums are the most common form of aid in the context of insurance.
Besides reducing the impact on household capital growth, lowering consumer premium payments has the potential to increase microinsurance take-up, with wealth and product price positively and negatively influencing microinsurance demand, respectively \citep{Article:Eling2014}.
Poverty traps are typically studied in the context of economics, with a large literature focus on why economic stagnation below the poverty line occurs in certain communities. While the poor could readily grow their way out of poverty by adopting profitable strategies such as productive asset accumulation, opportunistic exchange and implementation of cost-effective production technologies, poverty traps are underlined by poverty reinforcing behaviours induced by the state of being poor \citep{Article:Barrett2016}.
A detailed description of the mechanics of the poverty trap state is provided by \cite{Book:Matsuyama2010}. In studying the probability of falling into such a trap, \lq \lq trapping\rq \rq \ describes the event in which a household falls underneath the poverty line and into the area of poverty.
In this paper, we adopt the ruin-theoretic approach to calculating the trapping probability of households in low-income populations presented by \cite{Article:Kovacevic2011}, adapting the piecewise deterministic Markov process such that households are subject to large shocks of random size. In line with the poverty trap ideology, we assume that the area of poverty to be an absorbing state and so consider only the state of events above the poverty threshold. Obtaining explicit solutions for the trapping probability, we compare the influence of three structures of microinsurance on the ability of households to stay above the poverty line. Specifically, we consider a (i) proportional, (ii) subsidised proportional and (iii) subsidised proportional with barrier microinsurance scheme. Aligning with the essential place for governmental support in the provision of social protection which encompasses risk mitigation, we assess for the first time in this context, to the best of our knowledge, the impact of a (government) subsidised insurance scheme with barrier strategy. We optimise the barrier level in the context of the trapping probability and the governmental cost of social protection, identifying the proportion of the population for which such a product would be beneficial. Here, the cost of social protection is defined to account for the provision of government subsidies, in addition to the cost of lifting a household from poverty, should they fall underneath the threshold. The benefit of subsidy schemes for poverty reduction is measured through observation of this governmental cost, in addition to the trapping probability of the households under consideration.
The remainder of the paper will be structured as follows. In Section \ref{TheCapitalModel-Section2}, we introduce the household capital model and its associated infinitesimal generator. The (trapping) time at which a household falls into the area of poverty is defined in Section \ref{TheTrappingTime-Section3}, and subsequently the explicit trapping probability and the expected trapping time are derived for the basic uninsured model. Links between classical ruin theoretic models and the trapping model of this paper are stated in Sections \ref{TheCapitalModel-Section2} and \ref{TheTrappingTime-Section3}. Microinsurance is introduced in Section \ref{IntroducingMicroinsurance-Section4}, where we assume a proportion of household losses are covered by a microinsurance policy. The capital model is redefined and the trapping probability is derived. Sections \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} and \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6} consider the case where households are proportionally insured through a government subsidised microinsurance scheme, with the impact of a subsidy barrier discussed in Section \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}. Optimisation of the subsidy and barrier levels is presented in Sections \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} and \ref{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}, alongside the associated governmental cost of social protection. Concluding remarks are provided in Section \ref{Conclusion-Section6}.
\section{The Capital Model} \label{TheCapitalModel-Section2}
The fundamental dynamics of the model follow those of \cite{Article:Kovacevic2011}, where the growth in accumulated capital $(X_t)$ of an individual household is given by
\vspace{0.3cm}
\begin{align}
\frac{dX_{t}}{dt}=r \cdot\left[X_{t}-x^{*}\right]^{+}, \label{TheCapitalModel-Section2-Equation1}
\end{align}
\vspace{0.3cm}
where $[x]^{+}=\max(x,0)$. The capital growth rate $r$ incorporates household rates of consumption, income generation and investment or savings, while $x^* > 0$ represents the threshold below which a household lives in poverty. Reflecting the ability of a household to produce, accumulated capital $(X_t)$ is composed of land, property, physical and human capital, with health a form of capital in extreme cases where sufficient health services and food accessibility are not guaranteed \citep{Article:Dasgupta1997}. The notion of a household in this model setting may be extended for consideration of poverty trapping within economic units such as community groups, villages and tribes, in addition to the traditional household structure.
The dynamical process in \eqref{TheCapitalModel-Section2-Equation1} is constructed such that consumption is assumed to be an increasing function of wealth (for full details of the model construction see \cite{Article:Kovacevic2011}). The poverty threshold $x^*$ represents the amount of capital required to forever attain a critical level of income, below which a household would not be able to sustain their basic needs, facing elementary problems relating to health and food security. Throughout the paper, we will refer to this threshold as the critical capital or the poverty line. Since \eqref{TheCapitalModel-Section2-Equation1} is positive for all levels of capital greater than the critical capital, points less than or equal to $x^*$ are stationary (capital remains constant if the critical level is not met). In this basic model, stationary points below the critical capital are not attractors of the system if the initial capital exceeds $x^*$, in which case the capital process $(X_t)$ grows exponentially with rate $r$.
Using capital as an indicator of financial stability over other commonly used measures such as income enables a more effective analysis of a household's wealth and well-being. Households with relatively high income, considerable debt and few assets would be highly vulnerable if a loss of income was to occur, while low-income households could live comfortably on assets acquired during more prosperous years for a long-period of time \citep{Book:Gartner2004}.
In line with \cite{Article:Kovacevic2011}, we expand the dynamics of \eqref{TheCapitalModel-Section2-Equation1} under the assumption households are susceptible to the occurrence of large capital losses, including severe illness, the death of a household member or breadwinner and catastrophic events such as floods and earthquakes. We assume occurrence of these events follows a Poisson process with intensity $\lambda$, where the capital process follows the dynamics of \eqref{TheCapitalModel-Section2-Equation1} between events. On the occurrence of a loss, the household's capital at the event time reduces by a random amount $Z_{i}$. The sequence $(Z_{i})$ is independent of the Poisson process and i.i.d. with common distribution function $G$. In contrast to \cite{Article:Kovacevic2011}, we assume reduction by a given amount rather than a random proportion of the capital itself. This adaptation enables analysis of a tractable mathematical model without threatening the core objective of studying the probability that a household falls into the area of poverty.
A household reaches the area of poverty if it suffers a loss large enough that the remaining capital is attracted into the poverty trap. Since a household's capital does not grow below the critical capital $x^{*}$, households that fall into the area of poverty will never escape. Once below the critical capital, households are exposed to the risk of falling deeper into poverty, with a risk of negative capital due to the dynamics of the model. A reduction in a household's capital below zero could represent a scenario where total debt exceeds total assets, resulting in negative capital net worth. The experience of a household below the critical capital is, however, out of the scope of this paper.
We will now formally define the stochastic capital process, where the process for the inter-event household capital \eqref{TheCapitalModel-Section2-Equation2} is derived through solution of the first order ordinary differential equation \eqref{TheCapitalModel-Section2-Equation1}. This model is an adaptation of the model proposed by \cite{Article:Kovacevic2011}.
\vspace{0.3cm}
\begin{definition}
Let $T_{i}$ be the $i^{th}$ event time of a Poisson process $\left(N_{t}\right)$ with parameter $\lambda$, where $T_{0}=0 .$ Let $Z_{i} \ge 0 $ be a sequence of i.i.d. random variables with distribution function $G$, independent of the process $\left(N_{t}\right)$. For $T_{i-1} \leq t<T_{i}$, the stochastic growth process of the accumulated capital $X_{t}$ is defined as
\vspace{0.3cm}
\begin{align}
X_{t}=\begin{cases} \left(X_{T_{i-1}}-x^{*}\right) e^{r \left(t-T_{i-1}\right)}+x^{*} & \text { if } X_{T_{i-1}}>x^{*}, \\ X_{T_{i-1}} & \text{ otherwise}.
\end{cases}
\label{TheCapitalModel-Section2-Equation2}
\end{align}
\vspace{0.3cm}
At the jump times $t = T_{i}$, the process is given by
\vspace{0.3cm}
\begin{align}
X_{T_{i}}=\begin{cases} \left(X_{T_{i-1}}-x^{*}\right) e^{r \left(T_{i}-T_{i-1}\right)}+x^{*} - Z_{i} & \text { if } X_{T_{i-1}}>x^{*}, \\ X_{T_{i-1}} - Z_{i} & \text{ otherwise}.
\end{cases}
\label{TheCapitalModel-Section2-Equation3}
\end{align}
\end{definition}
\vspace{0.3cm}
The stochastic process $(X_t)_{t\geq 0}$ is a piecewise-determinsitic Markov process \citep{Article:Davis1984} and its infinitesimal generator is given by
\vspace{0.3cm}
\begin{align}
(\mathcal{A} f)(x)=r(x-x^{*}) f^{\prime}(x) +\lambda \int_{0}^{\infty} \left[f(x - z) - f(x)\right] \mathrm{d} G(z), \qquad x \ge x^{*}.
\label{TheCapitalModel-Section2-Equation4}
\end{align}
\vspace{0.3cm}
The capital model as defined in \eqref{TheCapitalModel-Section2-Equation2} and \eqref{TheCapitalModel-Section2-Equation3} is actually a well-studied topic in ruin theory since the 1940s. Here, modelling is done from the point of view of an insurance company. Consider the insurer\rq s surplus process $(U_t)_{t\geq 0}$ given by
\vspace{0.3cm}
\begin{align}
U_t =u+pt+ a \int_0^t U_s \, ds-\sum_{i=1}^{N_t} Z_i,
\label{TheCapitalModel-Section2-Equation5}
\end{align}
\vspace{0.3cm}
where $u$ is the insurer\rq s initial capital, $p$ is the constant premium rate, $a$ is the risk-free interest rate, $N_t$ is a Poisson process with parameter $\lambda$ which counts the number of claims in the time interval $[0,t]$, and $(Z_i)_{i=1}^\infty$ is a sequence of i.i.d. claim sizes with distribution function $G$. This model is also called the insurance risk model with deterministic investment, which was first proposed by \cite{Article:Segerdahl1942} and subsequently studied by \cite{Article:Harrison1977} and \cite{Article:Sundt1995}. For a detailed literature review on this model prior to the turn of the century, readers can consult \cite{Article:Paulsen1998}.
Observe that when $p=0$, the insurance model \eqref{TheCapitalModel-Section2-Equation5} for positive surplus is equivalent to the capital model \eqref{TheCapitalModel-Section2-Equation2} and \eqref{TheCapitalModel-Section2-Equation3} above the poverty line $x^*=0$. Subsequently, the capital growth rate $r$ in our model corresponds to the risk-free investment rate $a$ of the insurer\rq s surplus model. More connections between these two models will be made in the next section after the first hitting time is introduced.
\vspace{0.3cm}
\section{The Trapping Time} \label{TheTrappingTime-Section3}
Let
\vspace{0.3cm}
\begin{align}
\tau_{x}:=\inf \left\{t \geq 0: X_{t}<x^{*} \mid X_{0}=x\right\}
\label{TheTrappingTime-Section3-Equation1}
\end{align}
\vspace{0.3cm}
denote the time at which a household with initial capital $x \ge x^{*}$ falls into the area of poverty (the trapping time), where $\psi(x) = \mathbb{P}(\tau_{x} < \infty)$ is the infinite-time trapping probability. To study the distribution of the trapping time, we apply the expected discounted penalty function at ruin concept commonly used in actuarial science \citep{Article:Gerber1998}, such that with a force of interest $\delta \ge 0$ and initial capital $x \ge x^{*}$, we consider
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)= \mathbb{E}\left[w(\mid X_{\tau_{x}}-x^{*}\mid)e^{-\delta \tau_{x}} \mathbbm{1}_{\{\tau_{x} < \infty\}}\right],
\label{TheTrappingTime-Section3-Equation2}
\end{align}
\vspace{0.3cm}
where $\mid X_{\tau_{x}} -x^{*}\mid$ is the deficit at the trapping time and $w(x)$ is an arbitrary non-negative penalty function. For more details on the so called Gerber-Shiu risk theory, the interested reader may wish to consult \cite{Book:Kyprianou2013}. Using standard arguments based on the infinitesimal generator, $m_{\delta}(x)$ can be characterised as the solution of the Integro-Differential Equation (IDE)
\vspace{0.3cm}
\begin{equation}
r(x-x^{*})m_{\delta}'(x)-(\lambda + \delta) m_{\delta}(x)+\lambda \int_{0}^{x-x^{*}}m_{\delta}(x-z)dG(z)=-\lambda A(x), \qquad x \ge x^{*},
\label{TheTrappingTime-Section3-Equation3}
\end{equation}
\vspace{0.3cm}
where
\vspace{0.3cm}
\begin{align}
A(x) := \int_{x-x^{*}}^{\infty}w(z-x)dG(z).
\label{TheTrappingTime-Section3-Equation4}
\end{align}
\vspace{0.3cm}
Due to the lack of memory property, we consider the case in which losses ($Z_i$) are exponentially distributed with parameter $\alpha >0$. Specifying the penalty function such that $w(x)=1$, $m_{\delta}(x)$ becomes the Laplace transform of the trapping time, also interpreted as the expected present value of a unit payment due at the trapping time. Equation \eqref{TheTrappingTime-Section3-Equation3} can then be written such that
\vspace{0.3cm}
\begin{align}
r(x-x^{*})m_{\delta}'(x)-(\lambda + \delta) m_{\delta}(x)+\lambda \int_{0}^{x-x^{*}}m_{\delta}(x-z)\alpha e^{-\alpha z} dz&=-\lambda e^{-\alpha (x-x^{*})} , \hspace{0.2cm} x \ge x^{*}.
\label{TheTrappingTime-Section3-Equation5}
\end{align}
\vspace{0.3cm}
Applying the operator $\left(\frac{d}{dx}+\alpha\right)$ to both sides of \eqref{TheTrappingTime-Section3-Equation5}, together with a number of algebraic manipulations, yields the second order homogeneous differential equation
\vspace{0.3cm}
\begin{align}
-\frac{(x-x^{*})}{\alpha}m_{\delta}''(x)+\Bigg[\frac{(\lambda+\delta - r)}{\alpha r}-(x-x^{*})\Bigg]m_{\delta}'(x)+\frac{\delta}{r}m_{\delta}(x)=0, \qquad x\geq x^{*}. \label{TheTrappingTime-Section3-Equation6}
\end{align}
\vspace{0.3cm}
Letting $f(y):=m_{\delta}(x)$, such that $y$ is associated with the change of variable $y:=y(x)=-\alpha (x-x^{*})$, \eqref{TheTrappingTime-Section3-Equation6} reduces to Kummer\rq s Confluent Hypergeometric Equation \citep{Book:Slater1960}
\vspace{0.3cm}
\begin{align}
y\cdot f''(y) + (c-y) f'(y) - a f(y) =0, \qquad y<0,
\label{TheTrappingTime-Section3-Equation7}
\end{align}
\vspace{0.3cm}
for $a=-\frac{\delta}{r}$ and $c=1-\frac{\lambda + \delta}{r}$, with regular singular point at $y=0$ and irregular singular point at $y=-\infty$ (corresponding to $x=x^{*}$ and $x=\infty$, respectively). A general solution of \eqref{TheTrappingTime-Section3-Equation7} is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=f(y)= \begin{cases} 1 \hspace{9.5cm} x < x^{*},\\
A_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y(x)\right)+A_{2}e^{y(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y(x)\right) \hspace{0.2cm} x \ge x^{*},
\end{cases}
\label{TheTrappingTime-Section3-Equation8}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $A_{1},A_{2} \in \mathbb{R}$. Here,
\vspace{0.3cm}
\begin{align}
M(a, c; z)={ }_{1} F_{1}(a, c; z)=\sum_{n=0}^{\infty} \frac{(a)_{n}}{(c)_{n}} \frac{z^{n}}{n !}
\label{TheTrappingTime-Section3-Equation9}
\end{align}
\vspace{0.3cm}
is Kummer\rq s Confluent Hypergeometric Function \citep{Article:Kummer1837} and $(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(n)}$ denotes the Pochhammer symbol \citep{Book:Seaborn1991}. In a similar manner,
\vspace{0.3cm}
\begin{align}
U(a, c; z)=\left\{\begin{array}{ll}\frac{\Gamma(1-c)}{\Gamma(1+a-c)} M(a, c; z)+\frac{\Gamma(c-1)}{\Gamma(a)} z^{1-c} M(1+a-c, 2-c; z) & c \notin \mathbb{Z}, \\ \lim _{\theta \rightarrow c} U(a, \theta; z) & c \in \mathbb{Z}\end{array}\right.
\label{TheTrappingTime-Section3-Equation10}
\end{align}
\vspace{0.3cm}
is Tricomi\rq s Confluent Hypergeometric Function \citep{Article:Tricomi1947}. This function is generally complex-valued when its argument $z$ is negative, i.e. when $x \ge x^{*}$ in the case of interest. We seek a real-valued solution of $m_{\delta}(x)$ over the entire domain, therefore an alternative independent pair of solutions, here, $M(a,c;z)$ and $e^{z}U(c-a,c;-z)$, to \eqref{TheTrappingTime-Section3-Equation7} are chosen for $x \ge x^{*}$.
To determine the constants $A_1$ and $A_2$, we use the boundary conditions at $x^*$ and at infinity.
Applying equation (13.1.27) of \cite{Book:Abramowitz1964}, also known as Kummer\rq s Transformation $M(a, c; z) = e^{z} M(c-a, c;-z)$, we write \eqref{TheTrappingTime-Section3-Equation8} such that
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=
e^{y(x)}\left[A_{1}M\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};-y(x)\right)+A_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y(x)\right)\right]
\label{TheTrappingTime-Section3-Equation11}
\end{align}
\vspace{0.3cm}
for $x \ge x^{*}$. For $z \rightarrow \infty$, it is well-known that
\vspace{0.3cm}
\begin{align}
M(a, c; z)=\frac{\Gamma(c)}{\Gamma(a)} e^{z} z^{a-c}\left[1+O\left(|z|^{-1}\right)\right]
\label{TheTrappingTime-Section3-Equation12}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
U(a, c; z)= z^{-a}\left[1+O\left(|z|^{-1}\right)\right]
\label{TheTrappingTime-Equation13}
\end{align}
\vspace{0.3cm}
(see for example, equations (13.1.4) and (13.1.8) of \cite{Book:Abramowitz1964}). Asymptotic behaviours of the first and second terms of \eqref{TheTrappingTime-Section3-Equation11} as $y(x) \rightarrow -\infty$ are therefore given by
\vspace{0.3cm}
\begin{align}
\frac{\Gamma\left(1-\frac{\lambda+\delta}{r}\right)}{\Gamma\left(1-\frac{\lambda}{r}\right)}\left(-y(x)\right)^{\frac{\delta}{r}}\left(1+O\left(|-y(x)|^{-1}\right)\right)
\label{TheTrappingTime-Section3-Equation14}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
e^{y(x)}\left(-y(x)\right)^{\frac{\lambda}{r}-1}\left(1+O\left(|-y(x)|^{-1}\right)\right),
\label{TheTrappingTime-Section3-Equation15}
\end{align}
\vspace{0.3cm}
respectively. For $x \rightarrow \infty$, \eqref{TheTrappingTime-Section3-Equation14} is unbounded, while \eqref{TheTrappingTime-Section3-Equation15} tends to zero. The boundary condition $\lim_{x\to\infty} m_{\delta}(x) = 0$, by definition of $m_{\delta}(x)$ in \eqref{TheTrappingTime-Section3-Equation2}, thus implies
that $A_{1}=0$. Letting $x=x^{*}$ in \eqref{TheTrappingTime-Section3-Equation5} and \eqref{TheTrappingTime-Section3-Equation8} yields
\vspace{0.3cm}
\begin{align}
\frac{\lambda}{(\lambda + \delta)}=A_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right).
\label{TheTrappingTime-Section3-Equation16}
\end{align}
\vspace{0.3cm}
Hence, $A_{2}=\frac{\lambda}{(\lambda + \delta)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)}$ and the Laplace transform of the trapping time is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}(x)=\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};0\right)}e^{y(x)}U\left(1-\frac{\lambda}{r},1-\frac{\lambda+\delta}{r};-y(x)\right).
\label{TheTrappingTime-Section3-Equation17}
\end{align}
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item Figure \ref{TheTrappingTime-Section3-Figure1}(a) shows that the Laplace transform of the trapping time approaches the trapping probability as $\delta$ tends to zero, i.e.
\vspace{0.3cm}
\begin{align}
\lim _{\delta \downarrow 0} m_{\delta}(x) =\mathbb{P}(\tau_{x}<\infty)\equiv\psi(x).
\label{TheTrappingTime-Section3-Equation18}
\end{align}
\vspace{0.3cm}
As $\delta\to 0$, \eqref{TheTrappingTime-Section3-Equation17} yields
\vspace{0.3cm}
\begin{align}
\psi(x) = \frac{1}{U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}e^{y(x)}U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};-y(x)\right).
\label{TheTrappingTime-Section3-Equation19}
\end{align}
\vspace{0.3cm}
Figure \ref{TheTrappingTime-Section3-Figure1}(b) displays the trapping probability $\psi(x)$ for the stochastic capital process $X_{t}$. We can further simplify the expression for the trapping probability using the upper incomplete gamma function $\Gamma(a;z)=\int_{z}^{\infty}e^{-t}t^{a-1}dt$. Applying the relation
\vspace{0.3cm}
\begin{align}
\Gamma(a;z)=e^{-z}U(1-a,1-a;z)
\label{TheTrappingTime-Section3-Equation20}
\end{align}
\vspace{0.3cm}
(see equation (6.5.3) of \cite{Book:Abramowitz1964}) and the fact that $\Gamma(a; 0)=\Gamma(a)$ for $\mathbb{R}(a)>0$, we have
\vspace{0.3cm}
\begin{align}
\psi(x)=\frac{\Gamma\left(\frac{\lambda}{r}; -y(x)\right)}{\Gamma\left(\frac{\lambda}{r}\right)}.
\label{TheTrappingTime-Section3-Equation21}
\end{align}
\vspace{0.3cm}
\begin{figure}[H]
\begin{multicols}{2}
\includegraphics[width=8cm, height=8cm]{Figures/TheTrappingTime-Section3-Figure1-a.pdf}
\centering (a)
\includegraphics[width=8cm, height=8cm]{Figures/TheTrappingTime-Section3-Figure1-b.pdf}
\centering (b)
\end{multicols}
\caption{(a) Laplace transform $m_{\delta}(x)$ of the trapping time when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$ for $\delta = 0, \frac{1}{8}, \frac{1}{32}, \frac{1}{128}$ (b) Trapping probability $\psi(x)$ when $Z_{i} \sim Exp(\alpha)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$ for $\alpha = 0.8, 1, 1.5, 2$.}
\label{TheTrappingTime-Section3-Figure1}
\end{figure}
\item As an application of the Laplace transform of the trapping time, one particular quantity of interest is the expected trapping time. This can be obtained by taking the derivative of $m_{\delta}(x)$, where
\vspace{0.3cm}
\begin{align}
\mathbb{E}\left[\tau_{x}\right]=-\left.\frac{d}{d \delta} m_{\delta}(x)\right|_{\delta=0}.
\label{TheTrappingTime-Section3-Equation22}
\end{align}
\vspace{0.3cm}
As such, we differentiate Tricomi\rq s Confluent Hypergeometric Function with respect to its second parameter. Denote
\vspace{0.3cm}
\begin{align}
U^{(c)}(a, c; z)\equiv \frac{d}{d c} U(a, c; z).
\label{TheTrappingTime-Section3-Equation23}
\end{align}
\vspace{0.3cm}
A closed form expression of the aforementioned derivative can be given in terms of series expansions, such that
\vspace{0.3cm}
\begin{align}
\begin{split}
U^{(c)}(a, c; z)&=(\eta(a-c+1)-\pi \cot (c\pi)) U(a, c; z)\\
&-\frac{\Gamma(c-1) z^{1-c} \log (z)}{\Gamma(a)}{ }M(a-c+1 , 2-c ; z)\\&-
\frac{\Gamma(c-1) z^{1-c}}{\Gamma(a)} \sum_{k=0}^{\infty} \frac{(a-c+1)_{k}(\eta(a-c+k+1)-\eta(2-c+k)) z^{k}}{(2-c)_{k} k !}\\
&-\frac{\Gamma(1-c)}{\Gamma(a-c+1)} \sum_{k=0}^{\infty} \frac{\eta(c+k)(a)_{k} z^{k}}{(c)_{k} k !}, \qquad c \notin \mathbb{Z},
\end{split}
\label{TheTrappingTime-Section3-Equation24}
\end{align}
\vspace{0.3cm}
where $\eta(z)=\frac{d \ln\left[\Gamma(z)\right]}{dz}=\frac{\Gamma '(z)}{\Gamma(z)}$ corresponds to equation (6.3.1) of \cite{Book:Abramowitz1964}, also known as the digamma function. Thus, using expression \eqref{TheTrappingTime-Section3-Equation24}, we obtain the expected trapping time
\vspace{0.3cm}
\begin{align}
\begin{split}
\mathbb{E}\left[\tau_{x}\right]&=
\frac{\Gamma\left(\frac{\lambda}{r};-y(x)\right)}{\lambda U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}-\frac{\Gamma\left(\frac{\lambda}{r};-y(x)\right)U^{(c)}\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}{r\left[U\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)\right]^{2}}\\
&+e^{y(x)}\frac{U^{(c)}\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};-y(x)\right)}{rU\left(1-\frac{\lambda}{r},1-\frac{\lambda}{r};0\right)}.
\end{split}
\label{TheTrappingTime-Section3-Equation25}
\end{align}
\vspace{0.3cm}
\vspace{0.3cm}
In line with intuition, the expected trapping time is an increasing function of both the capital growth rate $r$ and initial capital $x$. However, since the capital process grows exponentially, large initial capital and capital growth rates significantly reduce the trapping probability and increase the expected trapping time to the point where it becomes non-finite, making the indicator function in the expected discounted penalty function \eqref{TheTrappingTime-Section3-Equation2} tend to zero. A number of expected trapping times for varying values of $r$ are displayed in Figure \ref{TheTrappingTime-Section3-Figure2}.
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8.5cm, height=7cm]{Figures/TheTrappingTime-Section3-Figure2.pdf}
\caption{Expected trapping time when $Z_{i} \sim Exp(1)$, $\lambda = 1$ and $x^{*} = 1$ for $r = 0.02,0.05,0.08$.}
\label{TheTrappingTime-Section3-Figure2}
\end{figure}
\vspace{0.3cm}
\item The ruin probability for the insurance model \eqref{TheCapitalModel-Section2-Equation5} given by
\vspace{0.3cm}
\begin{align}
\xi (u)= P(U_t<0 \text{ for some }t>0 \mid U_0=u)
\label{TheTrappingTime-Section3-Equation26},
\end{align}
\vspace{0.3cm}
is found by \cite{Article:Sundt1995} to satisfy the IDE
\vspace{0.3cm}
\begin{align}
(au+p)\xi'(u)-\lambda \xi (u)+ \lambda \int_0^ {u} \xi(u-z) \, dG(z)+\lambda(1-G(u))=0, \qquad u\geq 0.
\label{TheTrappingTime-Section3-Equation27}
\end{align}
\vspace{0.3cm}
Note that when $p=0$,
\eqref{TheTrappingTime-Section3-Equation27} coincides with the special case of
\eqref{TheTrappingTime-Section3-Equation3} when $x^*=0$, $w(x)=1$, and $\delta=0$. Thus, the household's trapping time can be thought of as the insurer\rq s ruin time. Indeed, the ruin probability in the case of exponential claims when $p=0$ as shown in Section 6 of \cite{Article:Sundt1995} is exactly the same as the trapping probability \eqref{TheTrappingTime-Section3-Equation21} when $x^*=0$.
\end{enumerate}
\section{Introducing Microinsurance}\label{IntroducingMicroinsurance-Section4}
As in \cite{Article:Kovacevic2011}, we assume that households have the option of enrolling in a microinsurance scheme that covers a certain proportion of the capital losses they encounter. The microinsurance policy has proportionality factor $1-\kappa$, where $\kappa \in [0,1]$, such that $100 \cdot (1-\kappa)$ percent of the damage is covered by the microinsurance provider. The premium rate paid by households, calculated according to the expected value principle is given by
\vspace{0.3cm}
\begin{align}
\pi(\kappa, \theta)=(1+\theta) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z_{i}), \label{IntroducingMicroinsurance-Section4-Equation1}
\end{align}
\vspace{0.3cm}
where $\theta$ is some loading factor. The expected value principle is popular due to its simplicity and transparency. When $\theta = 0$, one can consider $\pi(\kappa, \theta)$ to be the pure risk premium \citep{Book:Albrecher2017}. We assume the basic model parameters are unchanged by the introduction of microinsurance coverage.
The stochastic capital process of a household covered by a microinsurance policy is denoted by $X_{t}^{\scaleto{(\kappa)}{5pt}}$. We differentiate between all variables and parameters relating to the original uninsured and insured processes by using the superscript $(\kappa)$ in the latter case.
Since the premium is paid from a household's income, the capital growth rate $r$ is adjusted such that it reflects the lower rate of income generation resulting from the need for premium payment. The premium rate is restricted to prevent certain poverty, which would occur should the premium rate exceed the rate of income generation. The capital growth rate of the insured household $r^{(\kappa)}$ is lower than that of the uninsured household, while the critical capital is higher.
In between jumps, where $T_{i-1} \leq t<T_{i}$, the insured stochastic growth process $X_{t}^{(\kappa)}$ behaves in the same manner as \eqref{TheCapitalModel-Section2-Equation2}, with parameters corresponding to the proportional insurance case of this section, making particular note of the increased critical capital $x^{\scaleto{(\kappa)*}{5pt}}$:
\vspace{0.3cm}
\begin{align}
X_{t}^{\scaleto{(\kappa)}{5pt}}=\begin{cases} \left(X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}-x^{\scaleto{(\kappa)*}{5pt}}\right) e^{r^{\scaleto{(\kappa)}{5pt}} \left(t-T_{i-1}\right)}+x^{\scaleto{(\kappa)*}{5pt}} & \text { if } X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}>x^{\scaleto{(\kappa)*}{5pt}}, \\ X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}} & \text{ otherwise}.
\end{cases}
\label{IntroducingMicroinsurance-Section4-Equation3}
\end{align}
\vspace{0.3cm}
For $t = T_{i}$, the process is given by
\vspace{0.3cm}
\begin{align}
X_{T_{i}}^{\scaleto{(\kappa)}{5pt}}=\begin{cases} \left(X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}-x^{\scaleto{(\kappa)*}{5pt}}\right) e^{r^{\scaleto{(\kappa)}{5pt}} \left(T_{i}-T_{i-1}\right)}+x^{\scaleto{(\kappa)*}{5pt}} - \kappa \cdot Z_{i} & \text { if } X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}}>x^{\scaleto{(\kappa)*}{5pt}}, \\ X_{T_{i-1}}^{\scaleto{(\kappa)}{5pt}} - \kappa \cdot Z_{i} & \text{ otherwise}.
\end{cases}
\label{IntroducingMicroinsurance-Section4-Equation4}
\end{align}
\vspace{0.3cm}
By enrolling in a microinsurance scheme, a household\rq s capital losses are reduced to $Y_{i} :=\kappa \cdot Z_{i}$. Considering the case in which losses follow an exponential distribution with parameter $\alpha > 0$, the structure of \eqref{TheTrappingTime-Section3-Equation5} remains the same. However, acquisition of a proportional microinsurance policy changes the parameter of the distribution of the random variable of the losses ($Y_i$). Namely, we have that $Y_{i} \sim Exp\left(\alpha^{\scaleto{(\kappa)}{5pt}}\right)$ for $\kappa \in (0,1]$, where $\alpha^{\scaleto{(\kappa)}{5pt}} := \frac{\alpha}{\kappa}$. We can therefore utilise the results obtained in Section \ref{TheTrappingTime-Section3} to obtain the Laplace transform of the trapping time for the insured process, which is given by
\vspace{0.3cm}
\begin{align}
m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)=\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};0\right)}e^{y^{\scaleto{(\kappa)}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\kappa)}{5pt}}(x)\right),
\label{IntroducingMicroinsurance-Section4-Equation5}
\end{align}
\vspace{0.3cm}
where $y^{\scaleto{(\kappa)}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}\left(x-x^{\scaleto{(\kappa)*}{5pt}}\right)$. Figure \ref{IntroducingMicroinsurance-Section4-Figure1}(a) displays the Laplace transform $m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)$ for varying values of $\delta$.
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item The trapping probability of the insured process $\psi^{\scaleto{(\kappa)}{5pt}}(x)$, displayed in Figure \ref{IntroducingMicroinsurance-Section4-Figure1}(b), is given by
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\kappa)}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}\right)}.
\label{IntroducingMicroinsurance-Section4-Equation6}
\end{align}
\vspace{0.3cm}
\begin{figure}[H]
\begin{multicols}{2}
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure1-a.pdf}
\centering (a)
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure1-b.pdf}
\centering (b)
\end{multicols}
\caption{(a) Laplace transform $m_{\delta}^{\scaleto{(\kappa)}{5pt}}(x)$ of the trapping time when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\kappa)*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $\delta = 0, \frac{1}{8}, \frac{1}{32}, \frac{1}{128}$ (b) Trapping probability $\psi^{\scaleto{(\kappa)}{5pt}}(x)$ when $Z_{i} \sim Exp(\alpha)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\kappa)*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $\alpha = 0.8, 1, 1.5, 2$.}
\label{IntroducingMicroinsurance-Section4-Figure1}
\end{figure}
\vspace{0.3cm}
\item When $\kappa = 0$ the household has full microinsurance coverage, the microinsurance provider covers the total capital loss experienced by the household. On the other hand, when $\kappa = 1$, no coverage is provided by the insurer i.e., $X_{t}=X_{t}^{\scaleto{(\kappa)}{5pt}}$.
\item We are interested in studying significant capital losses, since low-income individuals are commonly exposed to this type of shock. Hence, throughout the paper, the parameter $\alpha >0$ should be considered to reflect the desired loss behaviour.
\end{enumerate}
Figure \ref{IntroducingMicroinsurance-Section4-Figure2} presents a comparison between the trapping probabilities of the insured and uninsured processes. As in \cite{Article:Kovacevic2011}, households with initial capital close to the critical capital (here, the critical capital $x^{*}=1$), i.e. the most vulnerable individuals, do not receive a real benefit from enrolling in a microinsurance scheme. Although subscribing to a proportional microinsurance scheme reduces capital losses, premium payments appear to make the most vulnerable households more prone to falling into the area of poverty. In Figure \ref{IntroducingMicroinsurance-Section4-Figure2}, the intersection point of the two probabilities corresponds to the boundary between households that benefit from the uptake of microinsurance and those who are adversely affected.
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/IntroducingMicroinsurance-Section4-Figure2.pdf}
\caption{Trapping probabilities for the uninsured and insured capital processes, when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $\kappa = 0.5$, $\theta=0.5$ and $x^{*} = 1$.}
\label{IntroducingMicroinsurance-Section4-Figure2}
\end{figure}
\section{Microinsurance with Subsidised Constant Premiums}\label{MicroinsurancewithSubsidisedConstantPremiums-Section5}
\subsection{General Setting}\label{GeneralSetting-Subection51}
Since microinsurance alone is not enough to reduce the likelihood of impoverishment for those close to the poverty line, additional aid is required. In this section, we study the cost-effectiveness of government subsidised premiums, considering the case in which the government subsidises an amount $\beta = \theta - \theta^{*}$, while the microinsurance provider claims a lower loading factor $\theta^{*}$ \citep{Article:Kovacevic2011}. The following relationship between premiums for the non-subsidised and subsidised microinsurance schemes therefore holds
\vspace{0.3cm}
\begin{align}
\pi(\kappa, \theta) = (1+\theta) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z) \ge (1+\theta^{*}) \cdot(1-\kappa) \cdot \lambda \cdot \mathbb{E}(Z) = \pi(\kappa, \theta^{*}).
\label{GeneralSetting-Subection51-Equation1}
\end{align}
\vspace{0.3cm}
Naturally, we assume governments are interested in optimising the subsidy provided to households. Governments should provide subsidies to microinsurance providers such that they enhance households\rq \, benefits of enrolling in microinsurance schemes, however, they also need to gauge the cost-effectiveness of subsidy provision. Households with capital very close to the critical capital will not benefit from enrolling into the scheme even if the entire loading factor $\theta$ is subsidised by the government, however, more privileged households will. One approach to finding the optimal loading factor $\theta^{*}$ for households that could benefit from the government subsidy is to find the solution of the equation
\vspace{0.3cm}
\begin{align}
\psi^{\left(\kappa, \theta^{*}\right)}(x)=\psi(x),
\label{GeneralSetting-Subection51-Equation2}
\end{align}
\vspace{0.3cm}
where $\psi^{\left(\kappa, \theta^{*}\right)}(x)$ and $\psi(x)$ denote the trapping probabilities of the insured subsidised and uninsured processes, respectively, since all loading factors below the optimal loading factor will induce a trapping probability lower than that of the uninsured process through a reduction in premium. This behaviour can be seen in Figure \ref{GeneralSetting-Subection51-Figure1}(a), while the \lq \lq richest\rq \rq \ households do not need help from the government since the non-subsidised insurance lowers their trapping probability below the uninsured case, the poorest individuals require more support. Moreover, as highlighted previously, there are households that do not receive any benefit from enrolling in the microinsurance scheme even when the government subsidises the entire loading factor (when households pay only the pure risk premium, this could occur if the government absorbs all premium administrative expenses). Note that Figure \ref{GeneralSetting-Subection51-Figure1}(b) illustrates the optimal loading factor $\theta^{*}$ for varying initial capital. Initial capitals are plotted from the point at which households begin benefiting from the subsidised microinsurance scheme, i.e. the point at which the dashed $(\theta=0,\beta=0.5)$ line intersects the solid line in Figure \ref{GeneralSetting-Subection51-Figure1}(a). Additionally, Figure \ref{GeneralSetting-Subection51-Figure1}(b) verifies that, from the point at which the dashed-dotted (insured household) line intersects the solid line in Figure \ref{GeneralSetting-Subection51-Figure1}(a), the optimal loading factor remains constant, with $\theta^{*}=0.5$, i.e. the \lq \lq richest\rq \rq \ households can afford to pay the entire premium.
\begin{figure}[H]
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=8cm, height=8cm]{Figures/GeneralSetting-Subection51-Figure1-a.pdf}
\centering (a)
\end{minipage}%
\hfill%
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=7cm, height=7cm]{Figures/GeneralSetting-Subection51-Figure1-b.pdf}
\centering (b)
\end{minipage}
\caption{(a) Trapping probabilities for the uninsured, insured and insured subsidised capital processes when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$ for loading factors $\theta^{*} = 0, 0.25$ (b) Optimal loading factor $\theta^{*}$ for varying initial capitals when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$.}
\label{GeneralSetting-Subection51-Figure1}
\end{figure}
\subsection{Cost of Social Protection}\label{CostofSocialProtection-Subection52}
Next, we assess government cost-effectiveness for the provision of microinsurance premium subsidies to households. Let $\delta \geq 0$ be the force of interest for valuation, and let $S$ denote the present value of all subsidies provided by the government until the trapping time such that
\vspace{0.3cm}
\begin{align}
S=\beta\int^{\tau_{x}}_{0} e^{-\delta t} dt =\beta \ax*{\angl{\tau_{x}}}. \label{CostofSocialProtection-Subection52-Equation1}
\end{align}
\vspace{0.3cm}
We assume a government provides subsidies according to the strategy introduced earlier, i.e. the government subsidises an amount $\beta = \theta - \theta^{*}$, while the microinsurance provider claims a lower loading factor $\theta^{*}$.
For $x\geq x^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}$, where $x^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}$ denotes the critical capital of the insured subsidised process, let $V(x)$ be the expected discounted premium subsidies provided by the government to a household with initial capital $x$ until trapping time, that is,
\vspace{0.3cm}
\begin{align}
V(x)= \mathbb{E}\left[S \mid X^{\scaleto{\left(\kappa, \theta^{*}\right)*}{5pt}}_{0}=x\right].
\label{CostofSocialProtection-Subection52-Equation2}
\end{align}
\vspace{0.3cm}
Since $S=\frac{\beta}{\delta} \left[1-e^{-\delta \tau_{x}}\right]$, we can define $m^{\scaleto{(\kappa,\theta^{*})}{5pt}}_\delta(x)$, the Laplace transform of the trapping time with rate $r^{\scaleto{(\kappa,\theta^{*})}{5pt}}$ and critical capital $x^{\scaleto{(\kappa,\theta^{*})*}{5pt}}$, using the Laplace transform for the insured process previously obtained in \eqref{IntroducingMicroinsurance-Section4-Equation5} to compute $V(x)$ when losses are exponentially distributed with parameter $\alpha^{\scaleto{(\kappa)}{5pt}}>0$. This yields
\vspace{0.3cm}
\begin{equation}
\begin{aligned}
V(x)&= \frac{\beta}{\delta}\left[1- m^{\scaleto{(\kappa,\theta^{*})}{5pt}}_\delta(x)\right]\\
&= \frac{\beta}{\delta}\left[1-\frac{\lambda}{(\lambda + \delta) U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},0\right)}e^{y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa,\theta^{*})}{5pt}}}, -y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)\right)\right],
\end{aligned}
\label{CostofSocialProtection-Subection52-Equation3}
\end{equation}
\vspace{0.3cm}
\vspace{0.3cm}
where $y^{\scaleto{(\kappa,\theta^{*})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}(x-x^{\scaleto{(\kappa,\theta^{*})*}{5pt}})$. We now formally define the government\rq s cost of social protection.
\vspace{0.3cm}
\begin{definition}
Let $\psi^{\left(\kappa, \theta^{*}\right)}(x)$ be the trapping probability of a household enrolled in a subsidised microinsurance scheme with initial capital $x$. Additionally, let $M > 0$ be a constant representing the cost to lift households below the critical capital out of the area of poverty. The government\rq s cost of social protection is given by
\vspace{0.3cm}
\begin{equation}
\textit{Cost of Social Protection}:= V(x) + M \cdot {\psi^{\left(\kappa, \theta^{*}\right)}(x)}. \label{CostofSocialProtection-Subection52-Equation4}
\end{equation}
\vspace{0.3cm}
\end{definition}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item For uninsured households, the government does not provide subsidies, i.e. $V(x) =0$. Furthermore, we consider their trapping probability to be $\psi(x)$.
\item The government manages selection of an appropriate force of interest $\delta \geq 0$ and constant $M > 0$. For lower force of interest the government discounts future subsidies more heavily, while for higher interest future subsidies almost vanish. The constant $M$ could be defined, for example, using the poverty gap index introduced by \cite{Article:Foster1984}, or in such a way that the government ensures with some probability that households will not fall into the area of poverty. Thus, higher values of $M$ will increase the certainty that households will not return to poverty.
\end{enumerate}
Figure \ref{CostofSocialProtection-Subection52-Figure1} displays the government cost of social protection. Observe that in this particular example, we consider high values for both the force of interest $\delta$ and the constant $M$. The choice of $M$ is motivated by Figure \ref{IntroducingMicroinsurance-Section4-Figure2}, which shows that from $x=8$, the trapping probability for uninsured households is very close to zero. Note that a high value of $\delta$ hands a lower weight to future government subsidies whereas a high value of $M$ grants higher certainty that a household will not return to the area of poverty once lifted out.
It is clear that governments do not benefit by entirely subsidising the \lq \lq richest\rq \rq \ households, since they will subsidise premiums indefinitely, almost surely (dashed line for highest values of initial capital). Hence, as illustrated in Figure \ref{GeneralSetting-Subection51-Figure1}(b), it is favourable for governments to remove subsidies for this particular group since their cost of social protection is even higher than when uninsured (solid line for highest values of initial capital). Conversely, governments perceive a lower cost of social protection when fully subsidising the loading factor $\theta$ for households with initial capital lying closer to the critical capital $x^{*}$. The cost of social protection when households pay only the pure risk premium is lower than when paying the premium entirely for values of initial capital in which the dashed line is below the dotted, in which case the government should support premium payments. However, due to the fact that they will almost surely fall into the area of poverty, requiring governments to pay the subsidy in addition to the cost of lifting a household out of poverty, it is not optimal to fully subsidise the loading factor for the most vulnerable, since the cost of social protection is higher than that for uninsured households. Note that, from the point of view of the governmental cost of social protection, Figure \ref{CostofSocialProtection-Subection52-Figure1} confirms earlier statements asserting the inefficiency of providing premium support to the most vulnerable, i.e. neither individual households nor governments receive real benefit under such a scheme. Thus, alternative risk management strategies should be considered for this sector of the low-income population.
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/CostofSocialProtection-Subection52-Figure1.pdf}
\caption{Cost of social protection for the uninsured, insured and insured subsidised capital processes when $Z_{i}\sim Exp(1)$, $r=0.5$, $\lambda=1$, $x^{*}=1$, $\kappa = 0.5$, $\theta = 0.5$, $\delta =0.9$ and $M=8$ for loading factor $\theta^{*}= 0$.}
\label{CostofSocialProtection-Subection52-Figure1}
\end{figure}
\section{Microinsurance with Subsidised Flexible Premiums}\label{MicroinsurancewithSubsidisedFlexiblePremiums-Section6}
\subsection{General Setting}\label{GeneralSetting-Subsection61}
Since premiums are generally paid as soon as microinsurance coverage is purchased, a household\rq s capital growth could be constrained. It is therefore interesting to consider alternative premium payment mechanisms. From the point of view of microinsurance providers, advance premium payments are preferred so that additional income can be generated through investment, naturally leading to lower premium rates. Conversely, consumers may find it difficult to pay premiums up front. This is a common problem in low-income populations, with consumers preferring to pay smaller installments over time \citep{Book:Churchill2006}. Collecting premiums at a time that is inconvenient for households can be futile. Flexible premium payment mechanisms have been highly adopted by informal funeral insurers in South Africa, where policyholders pay premiums whenever they are able, rather than at a specific time during the month \citep{WorkingPaper:Roth2000}. Similar alternative insurance designs in which premium payments are delayed until the insured\rq s income is realised and any indemnities are paid have also been studied. Under such designs, insurance take-up increases, since liquidity constraints are relaxed and concerns regarding insurer default, also prevalent in low-income classes, reduce \citep{Article:Liu2016}.
In this section, we introduce an alternative microinsurance subsidy scheme with flexible premium payments. We denote the capital process of a household enrolled in the alternative microinsurance subsidy scheme by $X_{t}^{\scaleto{(\mathcal{A})}{5pt}}$. Furthermore, as in Section \ref{IntroducingMicroinsurance-Section4}, we differentiate between variables and parameters relating to the original, insured and alternative insured processes using the superscript $(\mathcal{A})$. Under such an alternative microinsurance subsidy scheme, households pay premiums when their capital is above some capital barrier $B \ge x^{\scaleto{(\mathcal{A})*}{5pt}}$, with the premium otherwise paid by the government. In other words, whenever the insured capital process is below the capital level $B$, premiums are entirely subsidised by the government, however, when a household's capital is above $B$, the premium $\pi$ is paid continuously by the household itself. This method of premium collection may motivate households to maintain a level of capital below $B$ in order to avoid premium payments. Consequently, we assume that households always pursue capital growth. Our aim is to study how this alternative microinsurance subsidy scheme can help households reduce their probability of falling into the area of poverty. We also measure the cost-effectiveness of such scheme from the point of view of the government.
The intangibility of microinsurance makes it difficult to attract potential clients. Most clients will never experience a claim and so cannot perceive the real value of microinsurance, paying more to the scheme (in terms of premium payments) than what they actually receive from it. It is only when claims are settled that microinsurance becomes tangible. The alternative microinsurance subsidy scheme described here could increase client value, since, for example, individuals below the barrier $B$ may submit claims, receive a payout and therefore perceive the value of microinsurance when they suffer a loss, regardless of whether they have ever paid a single premium. Other ways of increasing microinsurance client value include bundling microinsurance with other products and introducing Value Added Services (VAS), which represent services such as telephone hotlines for consultation with doctors or remote diagnosis services (for health schemes) offered to clients outside of the microinsurance contract \citep{Article:Madhur2019}.
Under the alternative microinsurance subsidy scheme, the Laplace transform of the trapping time satisfies the following differential equations:
\vspace{0.3cm}
\small
\begin{align}
0 =
\begin{cases} -\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime \prime}(x)+\left[\frac{(\lambda+\delta-r)}{\alpha^{\scaleto{(\kappa)}{5pt}} r}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] m_{\delta}^{\scaleto{(\mathcal{A})}{5pt} \prime}(x)+\frac{\delta}{r} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) \hspace{1cm} \text{for $ x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$}, \\ -\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime \prime}(x)+\left[\frac{(\lambda+\delta-r^{\scaleto{(\kappa)}{5pt}})}{\alpha^{\scaleto{(\kappa)}{5pt}} r^{\scaleto{(\kappa)}{5pt}}}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)+\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) \hspace{0.3cm} \text{for $x \geq B$}.
\end{cases}
\label{GeneralSetting-Subsection61-Equation1}
\end{align}
\normalsize
\vspace{0.3cm}
As in Section \ref{TheTrappingTime-Section3}, use of the change of variable $y^{\scaleto{(\mathcal{A})}{5pt}}:=y^{\scaleto{(\mathcal{A})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}} (x-x^{\scaleto{(\mathcal{A})*}{5pt}})$ leads to Kummer\rq s Confluent Hypergeometric Equation and thus,
\vspace{0.3cm}
\footnotesize
\begin{align}
m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) =
\begin{cases} C_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) + C_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) & \text{for $ x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$}, \\ C_{3}M\left(-\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ C_{4}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) & \text{for $x \geq B$},
\end{cases}
\label{GeneralSetting-Subsection61-Equation2}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $C_{1},C_{2},C_{3},C_{4} \in \mathbb{R}$. Under the boundary condition $\lim_{x\to\infty} m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x) = 0$ with asymptotic behaviour of the Kummer function $M(a,c;z)$ as presented in Section \ref{TheTrappingTime-Section3}, we deduce that $C_{3}=0$. Also, since $m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x^{\scaleto{(\mathcal{A})*}{5pt}})=\frac{\lambda}{\lambda + \delta}$, we obtain $C_{1}=\frac{\lambda}{\lambda + \delta} - C_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)$.
Due to the continuity of the functions $ m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ and $m_{\delta}^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)$ at $x=B$ and the differential properties of the Confluent Hypergeometric Functions
\vspace{0.3cm}
\begin{align}
\frac{d}{dz}M(a,c;z)=\frac{a}{c}M(a+1,c+1;z),
\label{GeneralSetting-Subsection61-Equation3}
\end{align}
\begin{align}
\frac{d}{dz}U(a,c;z)=-aU(a+1,c+1;z),
\label{GeneralSetting-Subsection61-Equation4}
\end{align}
\vspace{0.3cm}
upon simplification,
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
C_{4}=\frac{\left[\frac{\lambda}{\lambda + \delta} - C_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\right]M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)
+C_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\end{split}
\label{GeneralSetting-Subsection61-Equation5}
\end{align}
\normalsize
\vspace{0.3cm}
and
\vspace{0.3cm}
\begin{align}
\begin{split}
C_{2}=\frac{\frac{\lambda}{\lambda+\delta}\left[\frac{\delta \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda-\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)+M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}} - D\right)\right]}{K},
\end{split}
\label{GeneralSetting-Subsection61-Equation6}
\end{align}
\vspace{0.3cm}
where
\vspace{0.3cm}
\begin{align}
D:=\frac{\alpha^{\scaleto{(\kappa)}{5pt}}\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}-1\right)U\left(2-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},2-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\label{GeneralSetting-Subsection61-Equation7}
\end{align}
\vspace{0.3cm}
and
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
K: &= M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}}-D\right)
\\
&+D e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) \\
&+\frac{\delta \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda -\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)
\\
&-\alpha^{\scaleto{(\kappa)}{5pt}}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}\left(\frac{\lambda}{r}-1\right)U\left(2-\frac{\lambda}{r}, 2-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right).
\label{GeneralSetting-Subsection61-Equation8}
\end{split}
\end{align}
\normalsize
\vspace{0.3cm}
\textbf{Remarks.}
\begin{enumerate}[label=(\roman*)]
\item The trapping probability $\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)$ for the alternative microinsurance subsidy scheme is given by
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\begin{cases} 1 - \frac{\Gamma\left(\frac{\lambda}{r}\right)-\Gamma\left(\frac{\lambda}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)}{ (-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)}\Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) +\Gamma\left(\frac{\lambda}{r}\right)-\Gamma \left(\frac{\lambda}{r}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}, & x \leq B\\ \\
\frac{(-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)} \Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)}{ (-y^{\scaleto{(\mathcal{A})}{5pt}}(B))^{\lambda \left(\frac{1}{r}-\frac{1}{r^{\scaleto{(\kappa)}{3.5pt}}}\right)}\Gamma \left(\frac{\lambda}{r^{\scaleto{(\kappa)}{3.5pt}}}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) +\Gamma\left(\frac{\lambda}{r}\right)-\Gamma \left(\frac{\lambda}{r}; - y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}, & x \geq B.
\end{cases}
\label{GeneralSetting-Subsection61-Equation9}
\end{align}
\vspace{0.3cm}
Similar to the subsidised case, we can find the optimal barrier $B^{*}$ by determining the solution of the equation
\vspace{0.3cm}
\begin{align}
\psi^{\scaleto{(\mathcal{A}, B^{*})}{5pt}}(x)=\psi(x), \label{GeneralSetting-Subsection61-Equation10}
\end{align}
\vspace{0.3cm}
where $\psi^{\scaleto{(\mathcal{A}, B^{*})}{5pt}}(x)$ and $\psi(x)$ denote the trapping probability of the capital process under the alternative microinsurance subsidy scheme and the uninsured capital process, respectively. Some examples are presented after the remarks.
\item When $B \rightarrow x^{\scaleto{(\mathcal{A})*}{5pt}}$, the trapping probability for the alternative microinsurance subsidy scheme is equal to the trapping probability obtained for the insured case $\psi^{\scaleto{(\kappa)}{5pt}}(x)$, i.e.
\vspace{0.3cm}
\begin{align}
\lim_{B \rightarrow x^{\scaleto{(\mathcal{A})*}{3.5pt}}}\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}\right)}.
\label{GeneralSetting-Subsection61-Equation11}
\end{align}
\vspace{0.3cm}
Moreover, letting $B \rightarrow \infty$, the trapping probability is given by
\vspace{0.3cm}
\begin{align}
\lim_{B \rightarrow \infty}\psi^{\scaleto{(\mathcal{A})}{5pt}}(x)=\frac{\Gamma\left(\frac{\lambda}{r}; -y^{\scaleto{(\kappa)}{5pt}}(x)\right)}{\Gamma\left(\frac{\lambda}{r}\right)}.
\label{GeneralSetting-Subsection61-Equation12}
\end{align}
\item Figure \ref{GeneralSetting-Subsection61-Figure1} displays the expected trapping time under the alternative microinsurance subsidy scheme. Not surprisingly, the expected trapping time is an increasing function of both the capital growth rate $r$ and barrier $B$.
\begin{figure}[H]
\centering
\includegraphics[width=8.5cm, height=7cm]{Figures/GeneralSetting-Subsection61-Figure1.pdf}
\caption{Expected trapping time when $Z_{i} \sim Exp(1)$, $\lambda = 1$, $x = 3.5$, $x^{\scaleto{(\mathcal{A})*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta=0.5$ for $r = 0.08,0.082,0.084$.}
\label{GeneralSetting-Subsection61-Figure1}
\end{figure}
\end{enumerate}
Figure \ref{GeneralSetting-Subsection61-Figure2}(a) shows the trapping probabilities for varying initial capital values under the uninsured, insured, subsidised and alternatively subsidised schemes. As expected, increasing the value of the capital barrier $B$ helps households to reduce their probability of falling into the area of poverty, since support from the government is received when their capital resides in the region between the critical capital $x^{\scaleto{(\mathcal{A})*}{5pt}}$ and the barrier $B$. Furthermore, as in the previous case, households with higher levels of initial capital do not need support from the government, insurance without subsidies decreases their trapping probability to a level below the uninsured (households with initial capital greater than or equal to the point at which the dotted line intersects the solid line).
The optimal barrier for these individuals is in fact the critical capital, i.e., $B^{*}=x^{\scaleto{(\mathcal{A})*}{5pt}}$, households with higher initial capital can therefore afford to pay for microinsurance coverage themselves, as illustrated in Figure \ref{GeneralSetting-Subsection61-Figure2}(b). Figure \ref{GeneralSetting-Subsection61-Figure2}(b) also shows that for the most vulnerable, the government should set up a barrier above their initial capital to remove capital growth constraints associated with premium payments. This level should be selected until the household reaches a capital level that is adequate in ensuring their trapping probability will be equal to that of an uninsured household. Conversely, for the more privileged (those in Figure \ref{GeneralSetting-Subsection61-Figure2}(b), with initial capital approximately greater than or equal to 2), the government should establish barriers below their initial capital, with households paying premiums themselves as soon as they enrol in the microinsurance scheme. This behaviour is mainly due to the fact that their level of capital is distant from the critical capital $x^{\scaleto{(\mathcal{A})*}{5pt}}$. These households are unlikely to fall into the area of poverty after suffering one capital loss, they are instead likely to fall into the region between the critical capital and the barrier $B$ (i.e. the area within which the government pays premiums), before entering the area of poverty. Thus, the aforementioned region acts as a \lq \lq buffer\rq \rq \ for households, since once in this region they will benefit from coverage without paying any premiums. Increasing the initial capital will lead to a decrease in the size of the \lq \lq buffer\rq \rq \ region until it disappears when $B=x^{\scaleto{(\mathcal{A})*}{5pt}}$, as shown in the lower right corner of Figure \ref{GeneralSetting-Subsection61-Figure2}(b), where a straight line is visible.
\vspace{0.3cm}
\begin{figure}[H]
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=8cm, height=8cm]{Figures/GeneralSetting-Subsection61-Figure2-a.pdf}
\centering (a)
\end{minipage}%
\hfill%
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[width=7cm, height=7cm]{Figures/GeneralSetting-Subsection61-Figure2-b.pdf}
\centering (b)
\end{minipage}
\caption{(a) Trapping probabilities for the uninsured, insured, insured subsidised with $\theta^{*}=0$ and insured alternatively subsidised with $B = 2, 4$ capital processes when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{*} = 1$, $\kappa = 0.5$ and $\theta = 0.5$ (b) Difference between the optimal barrier and the initial capital, i.e. $B^{*}-x$, for varying initial capitals, when $Z_{i} \sim Exp(1)$, $r = 0.5$, $\lambda = 1$, $x^{\scaleto{(\mathcal{A})*}{5pt}} = 1$, $\kappa = 0.5$ and $\theta = 0.5$.}
\label{GeneralSetting-Subsection61-Figure2}
\end{figure}
\subsection{Cost of Social Protection}\label{CostofSocialProtection-Subsection62}
Similarly to the previous section, it is reasonable to measure the governmental cost-effectiveness of providing microinsurance premium subsidies to households under the alternative microinsurance subsidy scheme. For this reason, we define $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ as the expectation of the present value of all subsidies provided by the government until the trapping time under the alternative microinsurance subsidy scheme, that is
\vspace{0.3cm}
\begin{equation}
V^{\scaleto{(\mathcal{A})}{5pt}}(x):=\mathbb{E}\left[\int_{0}^{\tau_{x}} \pi e^{-\delta t} \mathbbm{1}_{\left\{X_{t}^{\scaleto{(\mathcal{A})}{5pt}} < B\right\}} dt \middle| X^{\scaleto{(\mathcal{A})}{5pt}}_{0}=x\right].
\label{CostofSocialProtection-Subsection62-Equation1}
\end{equation}
\vspace{0.3cm}
If the derivative exists, then using standard infinitesimal generator arguments for $X^{\scaleto{(\mathcal{A})}{5pt}}_{t}$, one gets the following IDE for $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ under the barrier $B$
\vspace{0.3cm}
\begin{equation}
r(x-x^{\scaleto{(\mathcal{A})*}{5pt}})V^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)-(\lambda + \delta) V^{\scaleto{(\mathcal{A})}{5pt}}(x)+\lambda \int_{0}^{x-x^{\scaleto{(\mathcal{A})*}{3.5pt}}}V^{\scaleto{(\mathcal{A})}{5pt}}(x-z)dG(z)+\pi=0, \qquad x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B.
\label{CostofSocialProtection-Subsection62-Equation2}
\end{equation}
\vspace{0.3cm}
Hence, assuming $Z_{i}\sim Exp(\alpha^{(\kappa)})$, the function satisfies the nonhomogeneous differential equation given by
\vspace{0.3cm}
\footnotesize
\begin{align}
-\frac{\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)}{\alpha^{\scaleto{(\kappa)}{5pt}}} V^{\scaleto{(\mathcal{A})}{5pt}\prime\prime}(x)+\left[\frac{(\lambda+\delta-r)}{\alpha^{\scaleto{(\kappa)}{5pt}} r}-\left(x-x^{\scaleto{(\mathcal{A})*}{5pt}}\right)\right] V^{\scaleto{(\mathcal{A})}{5pt}\prime}(x)+\frac{\delta}{r} V^{\scaleto{(\mathcal{A})}{5pt}}(x)-\frac{\pi}{r}=0, \qquad x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B.
\label{CostofSocialProtection-Subsection62-Equation3}
\end{align}
\normalsize
\vspace{0.3cm}
Letting $V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ be the homogeneous solution of \eqref{CostofSocialProtection-Subsection62-Equation3}, we have
\vspace{0.3cm}
\small
\begin{align}
V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x) =
R_{1}M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right), \hspace{0.3cm} x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B,
\label{CostofSocialProtection-Subsection62-Equation4}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $R_{1}, R_{2} \in \mathbb{R}$, where $y^{\scaleto{(\mathcal{A})}{5pt}}(x)=-\alpha^{\scaleto{(\kappa)}{5pt}}(x-x^{\scaleto{(\mathcal{A})*}{5pt}})$.
Since the general solution of \eqref{CostofSocialProtection-Subsection62-Equation3} can be written as
\vspace{0.3cm}
\begin{align}
V^{\scaleto{(\mathcal{A})}{5pt}}(x)=V_{h}^{\scaleto{(\mathcal{A})}{5pt}}(x)+V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x),
\label{MicroinsuranceSubsidiesandtheCostofSocialProtection-Subsection51-Equation5}
\end{align}
\vspace{0.3cm}
where $V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x)$ is a particular solution, one can easily verify that $V_{p}^{\scaleto{(\mathcal{A})}{5pt}}(x)= \frac{\pi}{\delta}$ for all $x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$. Then, letting $x=x^{\scaleto{(\mathcal{A})*}{5pt}}$ in \eqref{CostofSocialProtection-Subsection62-Equation2} yields $V^{\scaleto{(\mathcal{A})}{5pt}}(x^{\scaleto{(\mathcal{A})*}{5pt}})=\frac{\pi}{\lambda + \delta}$ and subsequently
\vspace{0.3cm}
\begin{align}
R_{1}=-\left[\frac{\lambda \pi}{(\lambda + \delta) \delta}+R_{2}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};0\right)\right].
\label{CostofSocialProtection-Subsection62-Equation6}
\end{align}
\vspace{0.3cm}
For $x^{\scaleto{(\mathcal{A})*}{5pt}} \leq x \leq B$, we therefore have
\vspace{0.3cm}
\begin{align}
\begin{split}
V^{\scaleto{(\mathcal{A})}{5pt}}(x)&=-\left[\frac{\lambda \pi}{(\lambda + \delta) \delta}+R_{2}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};0\right)\right]M\left( - \frac{\delta}{r}, 1 - \frac{\lambda + \delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
\\ &+R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1 - \frac{\lambda}{r}, 1 - \frac{\lambda + \delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right) + \frac{\pi}{\delta}.
\end{split}
\label{CostofSocialProtection-Subsection62-Equation7}
\end{align}
\vspace{0.3cm}
Above the barrier $B$, $V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ satisfies \eqref{GeneralSetting-Subsection61-Equation1} for $x\geq B$, and so
\vspace{0.3cm}
\small
\begin{align}
V^{\scaleto{(\mathcal{A})}{5pt}}(x) =
R_{3}M\left(-\frac{\delta}{r^{\scaleto{(\kappa)}{5pt}}},1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right)
+ R_{4}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(x)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(x)\right), \qquad x \geq B,
\label{CostofSocialProtection-Subsection62-Equation8}
\end{align}
\normalsize
\vspace{0.3cm}
for arbitrary constants $R_{3},R_{4} \in \mathbb{R}$. Since $\lim_{x\to\infty} V^{\scaleto{(\mathcal{A})}{5pt}}(x) = 0$ by definition, we have that $R_{3}=0$. Using the continuity of the functions $ V^{\scaleto{(\mathcal{A})}{5pt}}(x)$ and $V^{\scaleto{(\mathcal{A})\prime}{5pt}}(x)$ at $x=B$ and the differential properties of the Confluent Hypergeometric Functions,
\footnotesize
\begin{align}
\begin{split}
R_{4}&=\frac{-\left[\frac{\lambda \pi}{(\lambda + \delta)\delta} + R_{2}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};0\right)\right]M\left(-\frac{\delta}{r},1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}\\
& +\frac{R_{2}e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r}, 1-\frac{\lambda+\delta}{r};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right) + \frac{\pi}{\delta}}{e^{y^{\scaleto{(\mathcal{A})}{5pt}}(B)}U\left(1-\frac{\lambda}{r^{\scaleto{(\kappa)}{5pt}}}, 1-\frac{\lambda+\delta}{r^{\scaleto{(\kappa)}{5pt}}};-y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)}
\end{split}
\label{CostofSocialProtection-Subsection62-Equation9}
\end{align}
\normalsize
\vspace{0.3cm}
and
\vspace{0.3cm}
\footnotesize
\begin{align}
\begin{split}
R_{2}=-\frac{\frac{\lambda \pi}{\lambda+\delta}\left[\frac{ \alpha^{\scaleto{(\kappa)}{5pt}}}{(r-\lambda-\delta)}M\left(1-\frac{\delta}{r}, 2-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)+\frac{1}{\delta}M\left(-\frac{\delta}{r}, 1-\frac{\lambda+\delta}{r};y^{\scaleto{(\mathcal{A})}{5pt}}(B)\right)\left(\alpha^{\scaleto{(\kappa)}{5pt}} - D\right)\right] + \frac{\pi }{\delta} \left(D - \alpha^{\scaleto{(\kappa)}{5pt}}\right)}{K},
\end{split}
\label{CostofSocialProtection-Subsection62-Equation10}
\end{align}
\normalsize
\vspace{0.3cm}
where $D$ and $K$ are \eqref{GeneralSetting-Subsection61-Equation7} and \eqref{GeneralSetting-Subsection61-Equation8}, respectively.
Figure \ref{CostofSocialProtection-Subsection62-Figure1} compares the cost of social protection for the uninsured, insured, insured subsidised and insured alternatively subsidised households. Cost of social protection for the most vulnerable is not reduced with microinsurance coverage (dotted, dashed and dash-dotted lines are all above the solid line for initial capitals close to the critical capital $x^{*}$). As mentioned previously, this corresponds to the high trapping probability of this section of the population, with governments almost surely needing to lift these households out of the area of poverty through payment of a certain amount $M$, in addition to paying subsidies. On the other hand, the cost of social protection could be reduced by providing subsidies to households with greater levels of initial capital (when dashed and dash-dotted lines are below the solid line). Greater initial capitals lead to lower trapping probabilities and a reduction in the likelihood of the government need to pay the value $M$. As a result, the cost of social protection decreases even though subsidies are provided.
Also illustrated in Figure \ref{CostofSocialProtection-Subsection62-Figure1} is the greater cost-effectiveness of conditionally subsidising premiums, in comparison to proportional subsidisation (dashed-dotted line below all other lines for the majority of households in this particular group). Besides lowering the cost of social protection compared to that of uninsured households, it outperforms the traditional insured subsidised case in which some of the loading factor is absorbed by the government. Moreover, the alternative scheme eliminates the disadvantage of paying subsidies indefinitely for the \lq \lq richest\rq\rq \ households almost surely. In consequence, implementing a microinsurance scheme with barrier strategy reduces both the probability of households falling into the area of poverty and the governmental cost of social protection (shown in Figure \ref{GeneralSetting-Subsection61-Figure2}(a) and Figure \ref{CostofSocialProtection-Subsection62-Figure1}, respectively).
\vspace{0.3cm}
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=8cm]{Figures/CostofSocialProtection-Subsection62-Figure1.pdf}
\caption{Cost of social protection for the uninsured, insured, insured subsidised with $\theta^{*}=0$ and insured alternatively subsidised with $B = 3$ capital processes, when $Z_{i}\sim Exp(1)$, $r=0.5$, $\lambda=1$, $x^{*}=1$, $\kappa = 0.5$, $\theta = 0.5$, $\delta =0.9$ and $M=8$.}
\label{CostofSocialProtection-Subsection62-Figure1}
\end{figure}
\section{Conclusion} \label{Conclusion-Section6}
Comparing the impact of three microinsurance mechanisms on the trapping probability of low-income households, we provide evidence for the importance of governmentally supported inclusive insurance in the strive towards poverty alleviation. The results of Sections \ref{IntroducingMicroinsurance-Section4} and \ref{MicroinsurancewithSubsidisedConstantPremiums-Section5} support those of \cite{Article:Kovacevic2011}, highlighting a threshold below which insurance increases the probability of trapping. Further to these findings, we have introduced an alternative mechanism with the capacity to reduce this effect, while strengthening government social protection programs by lowering costs.
Analysis of the subsidised microinsurance scheme with premium payment barrier, suggests that in general, the trapping probability of a household is reduced in comparison to basic microinurance and subsidised microinsurance structures, in addition to that of uninsured households. More significant influence is observed in relation to the governmental cost of social protection, with the cease of subsidy payments when household capital is sufficient facilitating government savings and therefore increasing social protection efficiency. Cost of social protection for those closest to the area of poverty remains greater than the corresponding uninsured cost in both subsidised schemes considered. For such households, governments must account for both their support of premium payments and the likely need for household removal from poverty. Government endorsement of further alternative risk mitigation strategies, such as asset accumulation, would be beneficial for those with capital close to the critical level, minimising their risk of falling beneath the poverty line, while reducing social protection costs.
\setcitestyle{numbers}
\bibliographystyle{plainnat}
|
2,877,628,090,232 | arxiv | \section{Introduction}
The description of the dynamics of open
quantum systems has attracted increasing
attention during the last few years \cite{Breuer02a}. The major
reason for this is the identification of
the phenomena of decoherence and dissipation, which
characterize the dynamics of open
quantum systems interacting with their
surroundings \cite{Joos03a},
as a main obstacle to the realization of quantum computers
and other quantum devices \cite{Nielsen00a}. Secondly, recent
experiments on engineering of environments \cite{Myatt00a}
have paved the way to new proposals aimed at
creating entanglement and superpositions
of quantum states exploiting decoherence and
dissipation \cite{Poyatos96a,Plenio02a}.
A common approach to the dynamics of open
quantum systems consists in deriving a master
equation for the reduced density matrix
which describes the temporal behavior
of the open system.
The solution for the master equation can then
be searched by using analytical
or simulation methods, or the combination
of both.
This article concentrates
on the developing of new Monte Carlo simulation methods
for non-Markovian open quantum systems.
The general feature of the Monte Carlo methods
is the generation of an ensemble of stochastic realizations of
the state vector trajectories. The density matrix
and the properties of the system of interest
are then consequently calculated as an
appropriate average of the generated ensemble.
Some common variants of the Monte Carlo
methods for open systems
include the Monte Carlo wave-function (MCWF) method
\cite{Dalibard92a,Molmer96a}, the
quantum state diffusion (QSD) \cite{Gisin92a,Diosi98a,Strunz99a},
and the non-Markovian
wave function (NMWF) formulation unravelling the master equation
in an extended Hilbert space \cite{Breuer02a,Breuer99a,Breuer04a}.
The MCWF method has been
very successfully used to model the laser cooling of atoms.
Actually, 3D laser cooling
has so far been described only by MCWF simulations
\cite{Molmer95a}.
QSD in turn has been found to have
a close connection to the decoherent
histories approach to quantum mechanics
\cite{Diosi95a},
and NMWF method has been recently applied
to study the dynamics of quantum
Brownian particles \cite{Maniscalco04a,Maniscalco04c}.
The various Monte Carlo methods and related topics have been reviewed
e.g. in Refs.~\cite{Molmer96a,Carmichael93a,Gardiner96a,Plenio98a}
In general, simulating open quantum systems is a challenging
task. It has been shown earlier that the methods mentioned above
can solve a wide variety of problems. Nevertheless, sometimes
there
arise situations in which the complexity of the studied system
or the parameter region under study makes the requirement
for the computer resources so large that the solution
may become impossible in practice, though not
in principle.
Thus, it is important to assess the already existing
methods from this point of view, and
develop new variants
to improve their applicability. This is the key
point of this article.
Here, we address the Monte Carlo simulation methods
for the short time-evolution of non-Markovian systems which are weakly
coupled to their environments.
In this case, the dynamics of the system may exhibit rich
features, whereas the weak coupling may
lead to extremely small quantum jump probabilities,
the consequence being unpractically large requirement
for the size of the generated Monte Carlo ensemble.
To overcome this problem, we present below a method
which in general allows to reduce the ensemble
size.
By studying the Hilbert space path integral
for the propagator of a piecewise deterministic
process (PDP) \cite{Breuer02a}, we show that part of the expectation
value of an arbitrary operator $A$ as a function of time $t$,
$\langle A \rangle (t)$, has scaling properties which can be
exploited in Monte Carlo simulations to speed up
the generation of the ensemble,
in the optimal case by several orders of magnitude.
We derive a scaling equation,
from which the result
for $\langle A \rangle (t)$
can be calculated,
all the quantities in the equation
being easily obtainable from the scaled
Monte Carlo simulations.
We concentrate first on the Lindblad-type
non-Markovian case which can be solved by the standard
MCWF method, and then focus
on the non-Lindblad-type case which
requires the use of the NMWF simulations in
the doubled Hilbert space.
The paper is structured as follows.
Section \ref{sec:dyn} introduces the master equation,
the corresponding stochastic Schr\"{o}dinger equation, and
the appropriate simulation schemes
for the Lindblad- and non-Lindblad-type systems.
The Hilbert space path integral method is then used
to calculate the expectation value of an arbitrary
operator setting
the scene for the scaling method which
is presented in Sec.~\ref{sec:scaling}. Section \ref{sec:examples}
shows explicitly how the scaling can be implemented
and demonstrates the usability of the method,
for the example of quantum Brownian motion.
Finally Sec.~\ref{sec:conclusions} presents discussion
and conclusions.
\section{Dynamics of non-Markovian systems}\label{sec:dyn}
We describe first in Sec.~\ref{subsec:lindblad} the master equation for the
Lindblad-type systems and the corresponding
standard MCWF method.
We then continue in Sec.\ref{subsec:nonlindblad} with the description
of the non-Lindblad-type case with the corresponding
stochastic Schr\"{o}dinger equation and NMWF unravelling in
the doubled Hilbert space. The last subsection
\ref{subsec:path}
presents the calculation of the expectation value
of an arbitrary operator $A$ with the Hilbert space path integral
method which paves the way for the scaling procedure.
We begin by considering
master equations obtained from the
time-convolutionless projection operator technique
(TCL)
of the form \cite{Breuer02a,Breuer99a}
\begin{eqnarray}
\frac{\partial}{\partial t} \rho\left( t \right) &=& A \left( t
\right) \rho \left( t \right) + \rho \left( t \right)
B^{\dag}\left( t \right)
\nonumber \\
&& +
\sum_i C_i\left( t \right) \rho \left( t \right) D^{\dag}_i\left( t \right),
\label{eq:genmaster}
\end{eqnarray}
with time-dependent linear operators $A\left( t \right)$, $B\left(
t \right)$, $C_i\left( t \right)$, and $D_i\left( t \right)$.
\subsection{Lindblad-type case: master equation and MCWF method}
\label{subsec:lindblad}
A specific case of the master equation (\ref{eq:genmaster}) is the one of
Lindblad-type \cite{Gorini76a,Lindblad76a,Maniscalco04b}
\begin{eqnarray}
\frac{d}{dt}\rho(t) &=& -i \left[H_S,\rho(t)\right] + \sum_i \gamma_i(t)
\bigg\{ L_i \rho(t) L_i^\dagger -
\nonumber \\
&&
\left.
\frac{1}{2} L_i^\dagger L_i\rho(t)
- \frac{1}{2} \rho(t) L_i^\dagger L_i \right\},
\label{eq:master}
\end{eqnarray}
where $H_S$ is the system Hamiltonian, $\gamma_i(t)$
the time dependent decay rate to channel $i$, and $L_i$ is the corresponding
Lindblad operator.
We define this non-Markovian master equation to be of Lindblad-type when
the time dependent decay coefficients $\gamma_i(t)\geq 0$ for
all times $t$, and non-Lindblad type when $\gamma_i(t)$ acquire
temporarily negative values during the time-evolution \cite{Maniscalco04b}.
The Lindblad-type case can be treated with the standard
MCWF method introduced in this subsection \cite{Dalibard92a},
and the non-Lindblad case
with the NMWF method in the doubled Hilbert space
presented in the following subsection \cite{Breuer99a,Breuer02a}.
The core idea of the standard MCWF method is to generate
an ensemble of realizations for the state vector $\psi(t)$
by solving the
time dependent Schr\"{o}dinger equation
\begin{equation}
i \hbar \frac{\partial \psi(t)}{\partial t}=
H(t)\psi(t),
\label{eq:Schrodinger}
\end{equation}
with the non-Hermitian Hamiltonian $H(t)$
\begin{equation}
H(t)=H_{S}(t)+H_{DEC}(t) \label{eq:H},
\end{equation}
where $H_S(t)$ is the reduced system's Hamiltonian
and the non-Hermitian part $H_{DEC}(t)$ includes the sum over the various allowed decay
channels $i$,
\begin{equation}
H_{DEC}(t)=-\frac{i\hbar}{2}\sum_{i}\gamma_i(t)L_i^{\dagger}L_i, \label{eq:HDec}
\end{equation}
where the jump operator $L_i$ for channel $i$
coincides with the Lindblad operator
appearing in the master equation (\ref{eq:master}).
During a discrete time evolution step
of length $\delta t$ the norm of the state vector
may shrink due
to $H_{DEC}$. The amount of shrinking gives the probability of a
quantum jump to
occur during the short interval $\delta t$. Based on a random number one then
decides whether a quantum jump occurred or not. Before the next time
step is taken,
the state vector of the system is renormalized. If and when a
jump occurs,
one performs a rearrangement of the state vector components
according to the jump
operator $L_i$, before renormalization of
$\psi$.
The jump probability corresponding to the decay
channel $i$ for each of the time-evolution steps $\delta t$ is
\begin{equation}
P_i(t)=\delta t \gamma_i(t)\langle\psi|L_i^{\dagger}L_i|\psi\rangle. \label{eq:Jp}
\end{equation}
The expectation value of an arbitrary operator A is then
the ensemble average over the generated realizations
\begin{equation}
\langle A \rangle (t) = \frac{1}{N}\sum_{i=1}^N \langle \psi_i | A | \psi_i \rangle,
\end{equation}
where $N$ is the number of realizations.
\subsection{Non-Lindblad-type case:
Stochastic Schr\"odinger equation and NMWF method in the doubled Hilbert space}
\label{subsec:nonlindblad}
The solution of the general master equation (\ref{eq:genmaster})
can be obtained
by using the
NMWF unravelling in the doubled Hilbert space
$\widetilde{\cal H}={\cal H}_S\oplus{\cal
H}_S$ where ${\cal H}_S$ is the Hilbert space of the
system \cite{Breuer02a,Breuer99a} .
The state of the system is described by a pair of
stochastic state vectors
\begin{equation}
\theta\left( t \right) = \left(\begin{array}{c}
\phi \left( t \right) \\
\psi\left( t \right)
\end{array} \right),
\end{equation}
such that $\theta(t)$ becomes a stochastic process in the doubled
Hilbert space $\widetilde{\cal H}$.
Denoting the corresponding probability density functional
by $\widetilde{P}[\theta,t]$, we can define the reduced
density matrix as
\begin{equation}
\rho(t)=\int D \theta D \theta^{\ast} |\phi\rangle \langle\psi |
\tilde{P}[\theta,t].
\label{eq:rhobasic}
\end{equation}
The time-evolution of $\theta\left( t \right)$ can be described as
a piecewise deterministic process (PDP)
and the corresponding stochastic Schr\"{o}dinger equation
reads \cite{Breuer02a}
\begin{eqnarray}
&&
d\theta(t) = -i G\left(\theta,t\right)dt +
\nonumber \\
&&
\sum_i \left\{ \frac{\|\theta(t)\|}{\| J_i(t)\theta(t)\|}J_i(t)\theta(t)-
\theta(t)\right\} dN_i(t),
\label{eq:SSE}
\end{eqnarray}
where the Poisson increments satisfy the equations
\begin{eqnarray}
dN_i(t)~dN_j(t) &=& \delta_{ij}dN_i(t), \nonumber \\
E\left[dN_i(t)\right] &=& \frac{\|J_i(t)\theta(t)\|^2}{\|\theta(t)\|^2}dt,
\end{eqnarray}
and the non-linear operator $G(\theta,t)$ is defined as
\begin{equation}
G(\theta,t) = \left[ F\left( t \right)+\frac{1}{2}\sum_i \frac{\| J_i\left( t
\right)\theta\left( t \right) \|^2}{\| \theta\left( t \right)
\|^2} \right]
\theta\left( t \right),
\end{equation}
with the time-dependent operators
\begin{eqnarray}
F\left( t \right) &=& \left(
\begin{array}{cc}
A\left( t \right) & 0 \\
0 & B\left( t \right)
\end{array} \right)
\nonumber \\
J_i\left( t \right) &=& \left( \begin{array}{cc}
C_i\left( t \right) & 0 \\
0 & D_i\left( t \right)
\end{array} \right),
\end{eqnarray}
where $A\left( t \right)$, $B\left( t \right)$, $C_i\left( t
\right)$, and $D_i\left( t \right)$ are the operators appearing in
Eq.~(\ref{eq:genmaster}).
The deterministic part of the PDP is obtained by solving the
following differential equation
\begin{eqnarray}
i\frac{\partial}{\partial t} \theta \left( t \right) =
G(\theta,t),
\end{eqnarray}
and the jumps of the PDP take the form
\begin{equation}
\theta \left( t \right) \rightarrow \frac{\| \theta \left( t
\right) \|} {\| J_i\left( t \right)\theta \left( t \right) \|}
\left(
\begin{array}{c}
C_i\left( t \right) \phi \left( t \right) \\
D_i\left( t \right) \psi \left( t \right)
\end{array} \right).
\end{equation}
Once the ensemble of stochastic realizations
has been generated, one can then calculate the density matrix
of the reduced system from Eq.~(\ref{eq:rhobasic}).
\subsection{The Hilbert space path integral for the propagator of the PDP
and the expectation value of arbitrary operators}
\label{subsec:path}
For simplicity, we present here the Hilbert space path integral for
the Lindblad-type case. The derivation of the non-Lindblad-type case follows
closely the presentation below.
We assume that the initial state of the system
is a pure state $\psi_0$.
In this case the propagator $T$ of the PDP (conditional transition probability)
coincides with the probability density functional $P$ of the stochastic
process \cite{Breuer02a}
\begin{equation}
P\left[\psi, t\right]=
T\left[\psi,t | \psi_0, t_0 \right].
\end{equation}
This quantity describes the probability of the system being
in the state $\psi$ at time $t$ when
it was in the state $\psi_0$ at some earlier time $t_0$.
For short time non-Markovian evolutions and weak couplings, we assume
that the maximum
number of jumps per realization is one. Thus, the expansion of the propagator
$T$ in terms of number of jumps contains two terms:
deterministic evolution without jumps $T^{(0)}$ and paths
with one jump $T^{(1)}$
\begin{eqnarray}
T\left[\psi,t | \psi_0, 0 \right] =
T^{(0)}\left[\psi,t | \psi_0, 0 \right] +
T^{(1)}\left[\psi,t | \psi_0, 0 \right].
\label{eq:T}
\end{eqnarray}
With the assumptions above, the expectation value
of an arbitrary operator $A$ at time $t$ can be calculated as
\cite{Breuer02a}
\begin{eqnarray}
\langle A \rangle \left( t \right) &=&
\int D \psi D \psi^* \langle\psi | A | \psi \rangle
T\left[\psi,t | \psi_0, t_0 \right]
\nonumber \\
&=&
\int D \psi D \psi^* \langle\psi | A | \psi \rangle
\nonumber \\
&&
\left\{T^{(0)}\left[\psi,t | \psi_0, t_0 \right]
+ T^{(1)}\left[\psi,t | \psi_0, t_0 \right]\right\}.
\label{eq:P}
\end{eqnarray}
By calculating $T^{(0)}$ and $T^{(1)}$, see Appendix \ref{app:a}, we obtain
for $\langle A \rangle (t)$
\begin{eqnarray}
\langle A \rangle \left( t \right) &=&
\int D \psi D \psi^* \langle\psi | A | \psi \rangle
\nonumber \\
&&
\times \left\{
\left[ 1-\int_0^t ds \sum_i \gamma_i(s)
\| L_i g_s(\psi_0) \|^2 \right]
\right. \nonumber \\
&&
\times\delta\left( \psi - g_t\left(\psi_0\right) \right)
\nonumber \\
&&
+
\int_0^t ds \int D \psi_1 D \psi_1^*
\int D \psi_2 D \psi_2^*
\nonumber \\
&&
\times\delta\left( \psi-g_t\left( \psi_2 \right) \right)
\sum_i \gamma_i(s) \| L_i \psi_1 \|^2
\nonumber \\
&&
\times\delta\left( \frac{L_i \psi_1}{\| L_i\psi_1 \|} - \psi_2 \right)
\delta\left( \psi_1 - g_s\left( \psi_0 \right) \right)
\Bigg\}.
\label{eq:P2}
\end{eqnarray}
Here, terms of the form $\delta\left( \psi - g_t\left(\psi_0\right) \right)$
are the functional delta-functions and
the deterministic evolution of $\psi_0$
according to the non-Hermitian Hamiltonian $H$ is
given by
\begin{equation}
g_t\left(\psi_0\right) =
\frac{\exp\left( -i\int_0^{t}H(t')dt'\right) \psi_0}
{\|\exp\left( -i\int_0^{t}H(t')dt'\right) \psi_0\|}.
\end{equation}
The physical interpretation of Eq.~(\ref{eq:P2}) is straightforward.
The expectation value of $A$ is calculated with respect
to all possible paths of $\psi$ with appropriate weights.
The first term in the curly brackets is
the no-jump evolution of $\psi$ multiplied with the
corresponding probability of no-jumps.
The second term includes the integration over
all possible jump times and jump routes
with the appropriate transition rates for the one
jump realization.
\section{Scaling}\label{sec:scaling}
Denoting the expectation value of $A$ with respect
to the no-jump evolution
as
\begin{equation}
\langle A \rangle_0 (t) =
\langle \psi = g_t\left( \psi_0 \right) | A |
\psi = g_t\left( \psi_0 \right) \rangle,
\label{eq:a0}
\end{equation}
we obtain from Eq.~(\ref{eq:P2})
\begin{eqnarray}
&&
\langle A \rangle \left( t \right) -
\langle A \rangle_0 (t)
=
\int D \psi D \psi^*
\langle \psi | A | \psi \rangle
\nonumber \\
&&
\times
\left\{ -
\delta \left( \psi - g_t(\psi_0) \right) \int_0^t ds \sum_i \gamma_i(s)
\|L_i g_s(\psi) \|^2
\nonumber \right. \\
&&+
\int_0^t ds \int D \psi_1 D \psi_1^* \int D \psi_2 D \psi_2^*~
\delta \left( \psi - g_t(\psi_2) \right)
\nonumber \\
&&
\times
\sum_i \gamma_i(s) \| L_i \psi_1 \|^2
\delta \left( \frac{L_i \psi_1}{\| L_i\psi_1 \|} -\psi_2\right)
\nonumber \\
&&
\times
\delta \left( \psi_1 - g_s(\psi_0) \right)
\bigg\}.\label{eq:aini}
\end{eqnarray}
This equation leads to the first key observation of the paper.
We notice that $\langle A \rangle \left( t \right) -
\langle A \rangle_0 (t)$ (but not $\langle A \rangle \left( t \right)$
alone) is directly proportional
to the transition rates of the type
\begin{equation}
W \left[ \psi_2 | \psi_1 \right] =
\sum_i \gamma_i(t) \| L_i \psi_1 \|^2
\delta \left[ \frac{L_i \psi_1}{\| L_i\psi_1 \|} -
\psi_2 \right].\label{eq:W}
\end{equation}
In the corresponding Monte Carlo simulations for the case we
are considering, the required size
of the generated ensemble is related to the transition rates $W$
since the rate defines the number of jumps. In more detail, if the
total (cumulative) jump probability for the time evolution
period of interest is $P_c$, we need on average to
generate $1/P_c$ realizations to produce one realization
which has a jump. To achieve good statistical accuracy
we need obviously
a large enough number of jumps and
the minimum condition for the required
ensemble size $N$ becomes $N\gg 1/P_c$.
This leads us to the following observation which can be used
to optimize the ensemble size of the Monte Carlo simulations
(within the approximations we use).
We can artificially increase the number of jumps
by scaling up the transition rate $W$ by a factor of $\beta$.
At the same time we must leave the non-Hermitian
Hamiltonian $H$ unscaled since the ensemble
average contribution given by realizations
with $H$ only (no jumps) appears
on the l.h.s. of the equation (\ref{eq:aini}).
In other words, we are not allowed to
scale the deterministic evolution of the state
vector (which includes also the rotation of
the state vector towards the state with the smallest decay
rate $\gamma_i$) but only increase the number of jumps
by scaling up the transition rates by a factor of $\beta$.
In the simulation this can be done easily
multiplying the jump probabilities for various
decay channels by a same factor $\beta$.
An explicit example how to do this for both
of the cases we are considering, Lindblad- and
non-Lindblad-type, is shown in the next section.
The question is now how we can calculate
from the scaled simulations the result we are looking for,
namely the expectation value for arbitrary
operator $A$ as a function of time $\langle A \rangle (t)$.
It can be shown, see Appendix \ref{app:b},
that the final result for $\langle A \rangle (t)$ starting from
Eq.~(\ref{eq:aini}) can be obtained as
\begin{eqnarray}
\langle A \rangle (t) &=&
\left( 1-\frac{P_{tot}(t)}{\beta} -
\frac{1}{\beta}\frac{N-N_j(t)}{N} \right) \langle A \rangle_0 (t)
\nonumber \\
&&
+\frac{1}{\beta} \bar{\langle A \rangle}_{tot}(t).
\label{eq:fina}
\end{eqnarray}
This equation is the main result of the paper.
It shows that the ensemble average of the scaled simulations
can be used
to calculate the result for the original problem
we are interested in.
In this equation,
$P_{tot}(t)$ is the total transition rate
(see Appendix \ref{app:b}),
$N$ is the size of the ensemble, $N_j(t)$ the number
of jumps in the simulations
as a function of time, $\beta$ the scaling factor,
$\langle A \rangle_0(t)$ the expectation value
with respect the deterministic time evolution
(see Eq.~(\ref{eq:a0})), and $\bar{\langle A \rangle}_{tot}(t)$
the ensemble average from the modified
simulations
where the scaling has been used (see the discussion above).
All of the quantities on the r.h.s. can be easily
calculated in the simulation. Actually,
from a technical point of view,
the only difference
between the scaled and unscaled simulations
is that in the former one we have
to keep track of the number of jumps
as a function of time. A task which
can be easily done in the simulations.
We also note that
at time $t=0$, $P_{tot}(0)=0$, $N_j(0)=0$,
$\bar{\langle A \rangle}_{tot}(0) = \langle A \rangle_0(0)$
and we obtain correctly for time $t=0$:
$\langle A \rangle (0)=\langle A \rangle_0(0)$.
Thus, we can optimize the ensemble size by using
the following procedure in the Monte Carlo simulations:
i) Scale up jump probabilities by suitable factor $\beta$.
ii) Leave decay rates
$\gamma_i(t)$ untouched in the non-Hermitian Hamiltonian $H$
iii) Calculate the result
for $\langle A \rangle (t)$ from Eq.~(\ref{eq:fina}).
It is worth to emphasize here a common feature of
Monte Carlo wave-function simulations.
The deterministic evolution caused by
the non-Hermitian Hamiltonian $H$ changes the relative
weights of the occupied states due to the different
decay rates of the various states. The scaling procedure
incorporates this rotation by adding to the scaled
ensemble average result [the second term on the r.h.s. of Eq.~(\ref{eq:fina})]
contribution from the deterministic
evolution calculated with the appropriate weight (the first term).
Since the reduction in the required
ensemble size is directly proportional
to the used scaling factor $\beta$,
the issue is now how large scaling factor
we can use to optimize the simulations.
The scaling method that we have developed
is valid when
there is maximally
one jump per realization.
This condition has to
hold also for the scaled simulations as well.
As soon
as the scaling factor is so large that
realizations with two or more jumps begin
to occur, additional error (with respect
to the normal statistical error of Monte Carlo
simulations) starts to appear.
In other words, the probability
of having two jumps per realization
has to be much smaller than the one jump probability.
If the total probability for
one jump is $P_c$ (see the discussion above),
the probability for two jumps equals $P_c^2$
and the estimate for additional
error is simply given by $P_c^2/P_c=P_c$.
Thus we can use the scaling
factor which increases the jump
probabilities e.g. to the order of $0.01$
introducing a manageable $1\%$ error
in addition to the normal statistical error
of the Monte Carlo simulations.
For the standard Monte Carlo simulations
there exists a corresponding measurement scheme
interpretation based on the continuous monitoring of the environment
of the system.
The scaling technique modifies the Monte Carlo
simulation method in such a way that
the measurement scheme interpretation
is lost.
The scaled simulations correspond to a stochastic
Schr\"odinger equation where the deterministic
part generated by $G$, see Eq.~(\ref{eq:SSE}), remains the same but the jump
part is scaled with $\beta$, i.e. the expectation value
of the Poisson increment becomes
\begin{equation}
E\left[dN_i(t)\right] = \beta \gamma_i(t)\|L_i\psi\|^2dt.
\end{equation}
Thus the stochastic Schr\"odinger equation does not have
a corresponding
master equation, and actually does not
need to have one for the scaling to work.
This is because we are not looking
for two master equations whose
results are scalable from each other.
Rather the key point is
to modify in a suitable way the
equations for the simulations in order
to make them faster and more efficient.
Summarizing, we have demonstrated above how the scaling works
for the Lindblad-type master equation with
time-dependent but always positive decay coefficients
$\gamma_i(t)$. For this the standard Monte Carlo wave-function
method can be used \cite{Dalibard92a,Plenio98a}.
In a similar way, it can be shown
that the scaling works also for the non-Lindblad-type
case where $\gamma_i(t)$ may acquire temporarily negative
values. In this case one needs to use the doubled Hilbert
space unravelling \cite{Breuer02a}. We show examples
of the scaling for both of these cases in the next Section.
\section{Examples for scaling}\label{sec:examples}
The discussion above shows how it is possible to reduce the size
of the generated ensemble in the Monte Carlo simulations
for non-Markovian systems. It is worth
noting that for the Markovian case the scaling
is not needed because the jump probabilities
can be increased trivially by increasing e.g. the time
step size $\delta t$ in the simulations. For the non-Markovian
case this does not work because the main features
of the open system dynamics may be given by the
time dependence of the decay rates, and
$\delta t$ has to be kept small compared
to the temporal variations of the decay coefficients.
We show below two examples for the scaling. In these examples
we use the scaling factors $\beta=10^4$ and $10^5$ while
the generated ensembles have the sizes of the order of $10^5$.
In other words, without the scaling, the solution of the presented
problems would require at least $10^9$ ensemble members.
To demonstrate the scaling, we perform
the simulations for the short time non-Markovian dynamics
of a quantum Brownian particle (damped
harmonic oscillator) \cite{Maniscalco04a,Maniscalco04c}.
We demonstrate both
the Lindblad-type, and non-Lindblad type cases.
The dynamics of a harmonic oscillator linearly coupled to a
quantized reservoir, modelled as an infinite chain of quantum
harmonic oscillators, is described, in the secular approximation, by
means of the following generalized master equation
\cite{Intravaia03a,Maniscalco04b}
\begin{eqnarray}
&&\frac{ d \rho(t)}{d t}= \frac{\Delta(t) \!+\! \Gamma (t)}{2}
\left[2 a \rho(t) a^{\dag}- a^{\dag} a \rho(t) - \rho(t)
a^{\dag} a \right]
\nonumber \\
&& +\frac{\Delta(t) \!-\! \Gamma (t)}{2} \left[2 a^{\dag} \rho(t)
a - a a^{\dag} \rho(t) - \rho(t) a a^{\dag}
\right]. \nonumber \\
\label{eq:mqbm}
\end{eqnarray}
In the previous equation, $a$ and $a^{\dag}$ are the annihilation
and creation operators, and $ \rho(t)$ the density matrix of the
system harmonic oscillator. The time dependent coefficients
$\Delta(t)$ and $\Gamma(t)$ appearing in the master equation are
known as diffusion and dissipation coefficients, respectively
\cite{Intravaia03a,Maniscalco04a}.
For an Ohmic reservoir spectral
density with Lorentz-Drude cut-off,
the expression for
$\Delta(t)$ is \cite{Caldeira83a,Maniscalco04a}
\begin{eqnarray}
\Delta(t) &=& 2 \alpha^2 k T \frac{r^2}{1+r^2} \left\{ 1
- e^{-\omega_c t} \left[ \cos (\omega_0 t)\right.\right.
\nonumber \\
&&
- (1/r) \left. \sin (\omega_0 t )\right] \big\},
\label{eq:deltaHT}
\end{eqnarray}
where the assumption of the high temperature
reservoir has been used.
The dissipation coefficient $\Gamma(t)$ can be written
\begin{equation}
\Gamma (t)\! \!=\!\! \frac{\alpha^2 \omega_0 r^2}{r^2+1} \Big[1
\!-\! e^{- \omega_c t} \cos(\omega_0 t) \! - r e^{- \omega_c t}
\sin( \omega_0 t ) \Big]. \label{gammasecord}
\end{equation}
Here, $r=\omega_c/\omega_0$ is the ratio between the environment
cut-off frequency $\omega_c$ and the oscillator frequency
$\omega_0$, $\alpha$ is the dimensionless coupling constant, $k$
the Boltzmann constant, and $T$ the temperature.
When $r>1$, the decay coefficients $\Delta(t)\pm\Gamma(t)>0$
for all times,
and the master equation is of Lindblad type.
When $r<1$, the decay coefficients $\Delta(t)\pm\Gamma(t)$
acquire temporarily negative values
and the master equation is of non-Lindblad type \cite{Maniscalco04a}.
For Lindblad-type master equation, one can apply the standard
MCWF method (Sec.~\ref{sec:Lin} ) where as the non-Lindblad-type case requires
the application of the NMWF method
in the doubled Hilbert space (Sec.~\ref{sec:NonLin})
To demonstrate that the scaling works (in addition
to the rigorous proof presented above), we compare below
the results obtained from the simulations to the
exact analytical results \cite{Maniscalco04a,Intravaia03b}.
\subsection{Lindblad-type master equation and MCWF simulations}\label{sec:Lin}
For the Lindblad type case we choose parameters
$2 \alpha^2 k T /\omega_c=1.2\times10^{-6}$, $r=10$,
$\alpha^2 \omega_0 /\omega_c=0.5\times10^{-8}$,
and the scaling
factor $\beta=10^4$.
The initial state of the system is chosen
to be a coherent state
$|\xi=\sqrt{2}\rangle$ such that, at $t=0$,
$\langle n \rangle = |\xi|^2=2$.
We emphasize that the present
paper generalizes the scaling method we have used
in \cite{Maniscalco04a} for
initial Fock states to
arbitrary system
Hamiltonians and arbitrary initial states.
The non-Hermitian part of the Hamiltonian
is now given by [see Eqs.~(\ref{eq:master}), (\ref{eq:H}), and (\ref{eq:mqbm})]
\begin{equation}
H_{DEC}=-\frac{i\hbar}{2}\left\{
[\Delta(t)-\Gamma(t)]aa^{\dagger} +
[\Delta(t)+\Gamma(t)]a^{\dagger}a
\right\}.
\end{equation}
The jump probabilities for each
time step $\delta t$ and decay channel $i$
are now modified so that the jump probability
for channel $1$ (jump up, absorption
of one quantum of energy from the environment) is
\begin{equation}
P_1(t)=\beta~\delta t [\Delta(t)-\Gamma(t)]
\langle\psi|aa^{\dagger}|\psi\rangle,
\end{equation}
and for channel $2$ (jump down,
emission of one quantum of energy into the environment)
\begin{equation}
P_2(t)=\beta~\delta t [\Delta(t)+\Gamma(t)]
\langle\psi|a^{\dagger}a|\psi\rangle.
\end{equation}
The ensemble average is then calculated
in the usual Monte Carlo way, as presented
in Section \ref{subsec:lindblad} and the
simulation results plugged into Eq.~(\ref{eq:fina})
to get the final result.
Figure \ref{fig:eg1} shows the excellent match between
the analytical curve and the simulations using
the scaling. For the discussion of the analytical solution, see
Ref.~\cite{Maniscalco04a}. The results confirm once more the validity of the scaling
procedure and show the short time quadratic non-Markovian behavior of the
average quantum number $\langle n \rangle=\langle a^{\dag}a\rangle$
of the oscillator.
Moreover, for the parameters used here,
the scaling reduces the required ensemble size by a factor of $10^{4}$.
The simulation here contains $6\times 10^{5}$ realizations.
\begin{figure}[tb
\centering
\includegraphics[scale=0.4]{fig1rev.eps}
\caption[f2]{\label{fig:eg1}
Comparison between analytical (solid line) and scaled simulation
results (circles) with the bars of the standard error
for the Lindblad-type case. The figure shows
the behavior of the expectation value of the quantum number
$\langle n\rangle$ as a function of time. The initial state
of the system is a coherent state $|\xi=\sqrt{2}\rangle$.
For the parameters used here,
the scaling reduces the required ensemble size by a factor
on the order of $10^{4}$.
The simulation here contains $6\times 10^{5}$ realizations.
}
\end{figure}
\subsection{Non-Lindblad-type unravelling in the doubled Hilbert space}\label{sec:NonLin}
For the non-Lindblad-type case we choose
the following parameters $2 \alpha^2 k T /\omega_c=2.4\times10^{-6}$, $r=0.1$,
$\alpha^2 \omega_0 /\omega_c=0.5\times10^{-8}$, and
the scaling factor $\beta=10^5$.
As initial state we choose a
superposition of Fock states $\psi=(|0\rangle+|1\rangle)/\sqrt{2}$.
The doubled Hilbert space state vector for the harmonic oscillator
reads
\begin{equation}
\theta(t) = \left(
\begin{array}{c}
\phi \left( t \right) \\
\psi\left( t \right)
\end{array} \right)
= \left(
\begin{array}{c}
\sum_{n=0}^{\infty} \phi_n(t) |n\rangle \\
\sum_{n=0}^{\infty} \psi_n(t) |n\rangle \\
\end{array} \right),
\end{equation}
where $\phi_n(t)$ and $\psi_n(t)$ are the probability amplitudes
in the Fock state basis.
By comparing Eq.~(\ref{eq:mqbm}) with the master equation
(\ref{eq:genmaster}), the operators $A(t)$ and $B(t)$
have to be chosen as
\begin{eqnarray}
A(t) &=& B(t)= -i \omega_0 a^{\dag} a - \frac{1}{2} \left\{
\left[\Delta(t)+\Gamma(t)\right]a^{\dag} a+ \right.
\nonumber \\
&& \left. \left[\Delta(t)-\Gamma(t)\right]aa^{\dag} \right\}.
\end{eqnarray}
Accordingly, the operators $C_i$ and $D_i$ are
\begin{eqnarray}
C_1(t)&=&D_1(t)= \sqrt {|\Delta(t)-\Gamma(t)|} a^{\dag},
\nonumber \\
C_2(t)&=&D_2(t)= \sqrt{|\Delta(t)+\Gamma(t)|} a
\end{eqnarray}
and the corresponding operators $J_i$,
become
\begin{eqnarray}
J_1(t)= \sqrt {|\Delta(t)-\Gamma(t)|} \left(
\begin{array}{cc}
{\rm sgn}\left[\Delta(t)-\Gamma(t)\right] a^{\dag}& 0 \\
0& a^{\dag}\\
\end{array} \right)
\nonumber \\
J_2(t)= \sqrt {|\Delta(t)+\Gamma(t)|} \left(
\begin{array}{cc}
{\rm sgn}\left[\Delta(t)+\Gamma(t)\right] a& 0 \\
0& a\\
\end{array} \right).
\end{eqnarray}
\begin{figure}[tb
\centering
\includegraphics[scale=0.4]{fig2rev.eps}
\caption[f2]{\label{fig:eg2}
Comparison between analytical (solid line) and scaled simulation
results (circles) with the bars of the standard error for the non-Lindblad-type case.
The figure shows
the behavior of the expectation value of the quantum number
$\langle n\rangle$ as a function of time. The initial state
of the system is the superposition of Fock states
$(|0\rangle+|1\rangle)/\sqrt{2}$.
For the parameters used here,
the scaling reduces the required ensemble size atleast by a factor of $10^{4}$.
The simulation contains $6\times10^{5}$ realizations.
The inset shows (in the same scale as the main plot)
the poor match between the analytical result (solid line) and
the simulation result without the scaling (circles) with
$6\times10^{8}$ realizations which is three orders of magnitude larger
than used in the scaling (see text).
}
\end{figure}
The statistics of the quantum jumps
is described by the waiting
time distribution function $F_w(\tau)$ which represents the
probability that the next jump occurs within the time interval
$[t,t+\tau)$. $F_w(\tau)$, derived from the properties of the
stochastic process, reads
\begin{equation}
F_w(\tau)=1-\exp\left[-\int_0^{\tau} \sum_{i=1,2}
P_i\left(s\right)ds\right],\label{eq:fw}
\end{equation}
where for channel 1 (jump up, the system absorbs a quantum of
energy from the environment)
\begin{equation}
P_1(t)=\beta~\frac{|\Delta(t)-\Gamma(t)|}{\| \theta \left( t \right)
\|^2} \sum_{n=0}^{\infty} (n+1)\left[ |\phi_n(t)|^2 +
|\psi_n(t)|^2\right], \label{p1}
\end{equation}
and for channel $2$ (jump down, the system emits a quantum of energy
into the environment)
\begin{equation}
P_2(t)=\beta~\frac{|\Delta(t)+\Gamma(t)|}{\| \theta \left( t \right)
\|^2} \sum_{n=0}^{\infty} n \left[ |\phi_n(t)|^2 +
|\psi_n(t)|^2\right]. \label{p2}
\end{equation}
Here, the probabilities are scaled with a factor of $\beta$
according to the scaling scheme presented above.
When the jump occurs, the choice of the decay channel is made
according to the factors $P_1(t)$ and $P_2(t)$. The times at which
the jumps occur are obtained from Eq. (\ref{eq:fw}) by using the
method of inversion \cite{Breuer02a}.
Figure~\ref{fig:eg2} displays the short time
oscillatory non-Markovian behavior of the average
quantum number $\langle n \rangle$.
This type of behavior is
studied in detail in Ref.~\cite{Maniscalco04a}.
The results show the excellent match between
the exact analytical solution and the simulation results using
the scaling with $6\times10^{5}$ realizations.
Again, the results confirm the validity of the scaling
procedure. Moreover, the inset shows a very poor match
between the non-scaled simulations with $6\times10^{8}$
realizations~\cite{Piilo05a} and justifies the claim that
the reduction in the ensemble size is at least on the order
of $10^4$ when the scaling procedure is used.
The reduction of the ensemble size can
be estimated also by calculating the maximum
jump probability of a single realization.
In the example considered here, the maximum probability
is of the order of $10^{-7}$, in other words on average
an ensemble size of $10^7$ produces one jump event
in the unscaled simulations.
We estimate that one needs several hundreds jumps
in the simulations to produce accurately the rich dynamical
features of the heating function displayed in Fig.~\ref{fig:eg2},
and consequently the requirement for the ensemble size
is at least $10^9$ without the scaling. Thus, the reduction
in the ensemble size by the scaling method is again found to be at least
on the order of $10^4$.
It is interesting to compare the various terms
in the scaling equation (\ref{eq:fina}) in the non-Lindblad
case. Figure \ref{fig:terms} shows the four
terms of the scaling equation (\ref{eq:fina}).
One can notice that two of the terms
practically cancel each other and the final
result is mostly given by the two
terms presented in Figs.~\ref{fig:terms} (a) and (d).
\section{Discussion and conclusions} \label{sec:conclusions}
We have demonstrated
a scaling method for Monte Carlo wave-function
simulations
which can reduce the size of the generated
ensemble
by several orders of magnitude
especially for weakly coupled
non-Markovian systems.
The scaling is based on the notion that
once in the simulations the jump probabilities are scaled,
and the deterministic evolution given by the
non-Hermitian Hamiltonian left untouched,
one can obtain the time evolution
of the observables of interest
from the
scaling equation (\ref{eq:fina}).
The scaling has been used in a restricted form,
for a specific physical system,
in Ref.~\cite{Maniscalco04a}.
In that case the initial state of the system was
a Fock state. Here, we present a generalized
scaling scheme which is able to treat arbitrary
initial states of the system and arbitrary
Hamiltonians.
We emphasize that the scaling method works very well
for solving the short time dynamics of non-Markovian,
systems, which
bear importance e.g. for the decoherence
studies for quantum information processing \cite{Alicki02a}.
\begin{figure}[tb
\centering
\includegraphics[scale=0.4]{fig3rev.eps}
\caption[f2]{\label{fig:terms}
Contribution from the various terms of the scaling equation
(\ref{eq:fina}).
(a) $T_A = \langle A\rangle_0(t)$,
(b) $T_B = -P_{tot}(t)\langle A \rangle_0(t) / \beta$,
(c) $T_C = -[(N-N_j(t))/(\beta N)] \langle A\rangle_0(t)$,
(d) $T_D = \langle {\bar A}\rangle _{tot}(t)/\beta$.
The terms has been shifted to start from the same
initial value for easier comparison.
Here $A$ is the number operator $A = a^{\dag}a$.
The final result presented in Fig.~2. is mostly given as a sum
of the terms
displayed in (a) and (d).
}
\end{figure}
In general, non-Markovian systems, even when they are weakly
coupled to their environments, can posses rich
dynamical features despite of the fact that the quantum jump
probability per stochastic realization is small
during the time evolution period of interest (see the examples above).
This is the key area where the scaling method we have presented
is useful. The small jump probabilities due to the weak coupling
can lead to the situations where the requirement
for the size of the generated ensemble in the Monte Carlo
wave function simulations is unconveniently large.
In these cases, the scaling method can be used to reduce
and optimize the generated ensemble size for
efficient numerical simulation of weakly coupled
non-Markovian systems.
The scaling method presented here can be used
when the master equation of the open quantum system
can be expressed in the general form of Eq.~(\ref{eq:genmaster})
obtained by the time-convolutionless projection operator
techniques (the one-jump restriction still applies,
see below). To compare our method
to the other simulation methods for non-Markovian systems
one should actually compare the validity of the TCL
with respect to the methods presented e.g.~in Refs.~\cite{Imamoglu96a,Yu99a,Gambetta02a}.
Thus, making a rigorous
comparison is an involved
task and is left for future studies.
We initially note here that our method
is not restricted with respect to the temperature
of the environment (while method presented in
Ref.~\cite{Gambetta02a} is valid for the zero-temperature bath)
and is valid, at least in principle,
to the order used in the TCL expansion
of master equation to be unravelled
(while method presented in Ref.~\cite{Yu99a} is post-Markovian,
i.e. first order correction to Markovian dynamics).
However, it is worth mentioning that the validity
of the TCL expansion is crucially related to the
existence of the TCL generator (see e.g.~page $447$
of Ref.~\cite{Breuer02a}).
The scaling method is limited to the cases
where there is maximally one jump per realization
in the generated Monte Carlo ensemble.
Moreover, it is also important to note that the
same restriction applies also for the scaled simulations.
These limits can be easily checked by calculating
the jump probabilities from Eqs.~(\ref{eq:Jp}) and (\ref{eq:fw})
for the time period of interest
or by monitoring the number of jumps in the simulations.
As soon as more than one jump per realization
in the scaled simulations begin to occur, one can
estimate the error by calculating the ratio between
the two-jump and the one-jump probabilities
per realization.
In the examples we have described, we have not
used very aggressive optimization of the ensemble
size (though the ensemble size reduction is on the order of $10^4$),
and no error has been introduced. This has been confirmed
by monitoring the jumps in the simulations: no two-jump
realizations was generated. Thus, the error bars displayed
in the Figs. (\ref{fig:eg1}) and (\ref{fig:eg2}) correspond to the usual
statistical error (standard deviation) of the Monte Carlo ensemble.
In conclusion, the scaling method has limitations
(one jump per realization)
but it is interesting to note that in the region
where the method
can not be applied (more than one jump
per realization), it is not needed. This
is because in this region there already occurs large enough
number of jumps enhancing the statistical accuracy
of the simulations. In other words, the problem which
the scaling solves appears only
within the region of validity of the method.
\acknowledgments
The authors thank H.-P. Breuer for discussions in Freiburg
and acknowledge CSC - the Finnish IT center for science -
for the computer resources.
This work has been financially supported by
the Academy of Finland (JP, project no. 204777), the Magnus
Ehrnrooth Foundation (JP), and the Angelo Della Riccia Foundation
(SM).
|
2,877,628,090,233 | arxiv | \section{Introduction}
The reasons for investigation of
the charge-nonsymmetric muonic molecules like $H\! eH\mu $ are as
follows. First, a direct charge-exchange reaction from the
ground-state muonic hydrogen atom to helium nuclei is suppressed, the
transfer proceeds through the formation of the molecule in the intermediate
state. Hence, the kinetics of muons in media is defined to a large extent
by the probability of this process. Indeed, the role of the formation of
a muonic molecule in a charge-exchange reaction was confirmed in a number
of experiments ~\cite{jones},~\cite{byst}, ~\cite{balin}, ~\cite{jacot}.
The measurement of the yield of $\gamma $-rays due to the decay of the
$H\! ed\mu $ molecules ~\cite{mats}, ~\cite{ishid} revives
interest in the investigation of this system. The experiment gives
evidence of an additional nonradiative decay channel. This possibility
was discussed in papers ~\cite{kino}, ~\cite{ger}, ~\cite{krav2}.
Next, eigenenergies of both usual molecules of media and muonic
molecules, as it follows from the results of ~\cite{krav},
{}~\cite{hara}, are comparable. For this reason one can expect an
active interaction of muonic molecules with media.
As soon as the
charge-nonsymmetric molecules are produced, the possibility of nuclear
transitions arises. The investigation of a nuclear reaction at
typical mesomolecular energies has a fundamental importance due to
absence of any experimental data on the strong interaction of
charged particles in this energy range.
Qualitatively properties of the $H\! eH\mu $ system are defined as
follows. Coulomb interaction is not able to
bind the systems under consideration due to the repulsion in the $H\!
e\mu +H$ channel. Only a $3-$body resonant state can be formed. States
like that are supported by the attractive polarization potential in the
$H\mu +H\! e$ channel and therefore are clustered.
The goal of this paper is to perform
systematical calculations of energy levels and nonradiative decay rates of
the $^{3,4}H\! ed\mu $ systems for all possible values of the total angular
momentum. Only the transition to the channel with the $H\! e\mu $ atom
in the ground state has been considered. Transitions from the molecular
states with $L=1,2$ to the channels with the $H\! e\mu $ atom in the $2s,
2p$ states will be suppressed due to exponentially small overlapping of
the initial and final state wave-functions.
The treatment of this problem met essential difficulties due
to necessity to describe the Coulomb three-body system above the
two-body threshold.
Some approaches to describing
these systems have been applied in ~\cite{hara},
{}~\cite{kino},
{}~\cite{ger}, ~\cite{krav2}.
The approach, using the
hyperspherical "surface" functions method ~\cite{mac},
{}~\cite{lin} has been
applied in this paper. The following advantages of this method
in treating the posed problems can be mentioned.
The method operates with a discrete set of coupled one-dimensional
differential equations. Physical boundary conditions for their
solution can be easily formulated. Moreover, coupling of channels
turns out to be rather small in our case and allows one to use the
decoupled one-level approximation. It is worthwhile to mention the
analogous calculation of the $LiH\mu $ and $BeH\mu $
molecules ~\cite{bel}.
The article is organized in the following way. The description
of the method will be given in the next section, section 3 contains
numerical results, section 4 - discussion and conclusion.
\section{Method}
The Hamiltonian of three charged particles in the Jacobi variables is:
\begin{equation}
H=-\Delta _{\displaystyle{\bf x}_{i}}-\Delta _{\displaystyle{\bf y}_{i}}+
\sum\limits_{s=1}^{3}\frac{q_{s}}{x_{s}},
\label{eq:h1}
\end{equation}
where
\begin{equation}\left.
\begin{array}{l}
{\bf x}_{i}=\sqrt{\displaystyle\frac{m_{k}m_{j}}{m(m_{k}+m_{j})}}({\bf r}_{k}
-{\bf r}_{j}),\vspace{.5cm}\\
{\bf y}_{i}=\sqrt{\displaystyle\frac{m_{i}(m_{k}+m_{j})}{m(m_{i}+m_{k}+m_{j})}}
({\bf r}_{i}-\displaystyle\frac{m_{j}{\bf r}_{j}+m_{k}{\bf
r}_{k}}{m_{j}+m_{k}}),
\label{eq:h2}
\end{array}\right.
\end{equation}
\begin{equation}
q_{i}=2Z_{j}Z_{k}\sqrt{\frac{m_{k}m_{j}}{m(m_{k}+m_{j})}}.
\label{eq:h3}
\end{equation}
${\bf r}_{i}, m_{i}, Z_{i}$ - coordinate, mass and charge of the
$i$-th particle. $\hbar ^2/(me^2),\ me^4/(2\hbar ^2)$ have been used
as length and energy units. Here $m$ is
arbitrary mass and was taken equal to the muonic mass. For
definiteness muon, hydrogen nucleus and
nucleus of the charge Z have been enumerated as particles with number 1, 2
and 3.
Coordinates
$\rho ,\alpha _{i}, \theta _{i}$have been introduced by the relations
\begin{equation}\left.
\begin{array}{l}
x_{i}=\rho cos\displaystyle\frac{\alpha _{i}}{2},\\
y_{i}=\rho sin\displaystyle\frac{\alpha _{i}}{2},\\
cos\theta _{i}=\displaystyle\frac{({\bf x}_{i}\cdot {\bf
y}_{i})}{x_{i}y_{i}},\\
0\ \leq \ \alpha _{i}, \theta _{i}\ \leq \pi .
\label{eq:h4}
\end{array}\right.
\end{equation}
Below the notation $\Omega $ will be used for an arbitrary pair
$\alpha_i, \theta_i$.
Since the systems have two heavy and one light particles, it is reasonable
to assume that the main part of the total angular momentum is carried by
the pair of heavy particles. This is reason why the following form of the
solution of the Schr\" odinger equation has been used:
\begin{equation}
\Psi _{LM}({\bf x},{\bf y})=Y_{LM}({\bf\hat x}_1)\Phi _L(\rho ,\Omega ).
\label{eq:mom}
\end{equation}
Under these assumptions the Schr\" odinger equation for the
states with total angular momentum $L$ takes the form:
\begin{equation}
[-\frac{1}{\rho ^5}\frac{\partial }{\partial \rho} (\rho ^5\frac{\partial
}{\partial \rho} )- \frac{4}{\rho^2}\Delta_{\Omega
}+ \sum\limits_{s=1}^{3}\frac{q_{s}}{x_{s}}-E]\Phi _L(\rho ,\Omega )=0,
\label{eq:e5}
\end{equation}
where
\begin{equation}
\Delta_{\Omega }=\frac{1}{sin^2\alpha _{i}}
[\frac{\partial }{\partial \alpha_{i}} (sin^2\alpha _{i}\frac{\partial
}{\partial \alpha_{i}} )+
\frac{1}{sin\theta _{i}}\frac{\partial }{\partial \theta_{i}} (sin\theta
_{i}\frac{\partial }{\partial \theta_{i}})]
-\displaystyle\frac{L(L+1)}{4cos^{2}\alpha _1}.
\label{eq:e6}
\end{equation}
Following ~\cite{mac} and ~\cite{lin} "surface"
functions $\varphi _{n}(\Omega ;\rho )$
can be introduced as finite solutions of the
equation:
\begin{equation}
[\Delta_{\Omega }-\frac{\rho }{4}
\sum\limits_{s=1}^{3}\frac{q_{s}}{x_{s}}
+\lambda _{n}(\rho )]\varphi_{n}(\Omega ;\rho )=0.
\label{eq:e7}
\end{equation}
Expanding
the solution of the equation (\ref{eq:e5}) onto the set of the
hyperspherical "surface" functions:
\begin{equation}
\Phi _L(\rho ,\Omega )=
\rho ^{-5/2}\sum\limits_{n}u_{n}\varphi _{n}(\Omega ;\rho),
\label{eq:e8}
\end{equation}
one immediately comes to the system of one-dimensional equations
\begin{eqnarray}
[\frac{d^2}{d\rho ^2}-\frac{15}{4\rho ^2}-
\varepsilon _{n}(\rho)&+&E]u_{n}(\rho )+\nonumber\\
\sum\limits_{i}[Q_{ni}(\rho )\frac{d}{d\rho} &+&
\frac{d}{d\rho} Q_{ni}(\rho )-P_{ni}(\rho )]u_{n}(\rho )=0,
\label{eq:e9}
\end{eqnarray}
where
\begin{equation}
Q_{ni}(\rho )=\langle \varphi_{n}|\frac{\partial }{\partial \rho}
\varphi_{i}\rangle ,
\label{eq:e10}
\end{equation}
\begin{equation}
P_{ni}(\rho )=\langle \frac{\partial }{\partial \rho}
\varphi_{n}|\frac{\partial }{\partial \rho} \varphi_{i}\rangle ,
\label{eq:e11}
\end{equation}
\begin{equation}
\varepsilon _{n}(\rho)=\frac{4}{\rho^2}\lambda _{n}(\rho ).
\label{eq:e12}
\end{equation}
$\langle \ \cdot \ |\ \cdot \ \rangle$ means the integration on the
hypersphere over $d\Omega =sin^2\alpha _{i}d\alpha _{i}dcos\theta _{i}$.
One of the most complicated problems of this approach is the
computation of $Q_{ni}(\rho )$ and $P_{ni}(\rho )$, defined in
(\ref{eq:e10}) and (\ref{eq:e11}). By this reason, the following exact
expressions have been used: \begin{equation} Q_{ni}=-\frac{1}{4}(\lambda
_{i}-\lambda _{n})^{-1}\langle
\varphi_{n}|\sum\limits_{s=1}^{3}\frac{q_{s}}{x_{s}} |\varphi_{i}\rangle,
\label{eq:q1}
\end{equation}
\begin{equation}
P_{ni}=-(Q^2)_{ni}.
\label{eq:p1}
\end{equation}
The form (\ref{eq:q1}), (\ref{eq:p1}) allow one to avoid the
calculation of the derivatives of the "surface" functions on the
parameter $\rho $ and use only already known matrix elements $V_{ni}(\rho )$
and eigenvalues $\lambda _{i}(\rho )$ of equation (\ref{eq:e7}).
The variational approach has been applied to solve equation
(\ref{eq:e7}). The "surface" functions have been chosen as a linear
combination of trial functions from the following set:
\begin{equation}\left.
\begin{array}{l}
\phi ^{(\sigma )}_{nl}(\alpha _{\sigma })
P_{l}(cos\theta _{\sigma }),\quad \sigma =2,3,\\
sin^l\alpha _{3}C^{l+1}_{n-l-1}P_{l}(cos\theta _{3}),\\
n>0,\ n>l\geq 0,
\end{array}\right.
\label{eq:s13}
\end{equation}
where
\begin{equation}\left.
\begin{array}{l}
\phi ^{(\sigma )}_{nl}(\alpha )=
R_{nl}(\displaystyle{|q_\sigma |\over n}
\rho cos{\displaystyle\alpha \over 2}),\\
R_{nl}(t)=exp(-t/2)t^{l}L_{n-l-1}^{2l+1}(t)
\end{array}\right.
\label{eq:s14}
\end{equation}
In equations (\ref{eq:s13}) and (\ref{eq:s14}) $P_{l}(x), L_{m}^{k}(x),
C_{n}^{m}(x)$ are the Legendre, Laguerre and Gegenbauer polynomials.
The set of trial functions has been chosen in the
form (\ref{eq:s13}) in order to describe properly the three-body
wave-function at both large and small interparticle distances.
The first line of (\ref{eq:s13}) will describe the system
separated into two clusters. In this case, one of the
clusters is a hydrogen-like atom and hydrogen-like functions
(\ref{eq:s14}) will be proper trial functions. The second line of
(\ref{eq:s13}) will describe the configuration with all three particles
close to each other. In this case, the centrifugal term in (\ref{eq:e5})
dominates and eigenfunctions of the operator (\ref{eq:e6}) are used. The
set of trial functions (\ref{eq:s13})can be easily adjusted to the
different values of the parameter $\rho $. For this purpose numbers of
channel-type functions and hyperspherical harmonics have been changed with
changing $\rho $. It is necessary to emphasize that the dependence of the
numbers of the trial functions on the parameter $\rho $ has not been
exploited in analogous calculations. This dependence gives rise to more
flexibility of the basis and allows one to avoid numerical instabilities
when solving equation (\ref{eq:e7}).
As a result of the solution of equation (\ref{eq:e7}) eigenpotentials
$\varepsilon _{n}(\rho )$, $Q_{12}(\rho )$, $P_{12}(\rho )$ have been
obtained.
The properties of mesomolecules and transition rates
are mostly defined by the specific form of the
effective potentials $\varepsilon _{n}(\rho )$. The lowest
effective potential $\varepsilon _{1}(\rho )$ describes
asymptotically the decay channel $H+He\mu $ and is repulsive at all $\rho
$ values. The next effective potential $\varepsilon
_{2}(\rho )$ describes asymptotically the channel $He+H\mu $.
As it was already mentioned this
potential has an attractive part and supports the resonant state we are
interested. $Q_{12}(\rho )$ and $P_{12}(\rho )$ give rise coupling of
channels. In case of small coupling energy levels of $^{4}\! H\! ed\mu $
and $^{3}\! H\! ed\mu $ will be found as eigenvalues of the equation:
\begin{equation}
[\frac{d^2}{d\rho
^2}-\frac{15}{4\rho ^2}- \varepsilon _{2}(\rho)-P_{22}(\rho
)+E]u_{2}(\rho )=0
\label{eq:s15}
\end{equation}
for zero boundary conditions
\begin{equation}
u_{2}(0)=u_{2}(\infty )=0.
\label{eq:s16}
\end{equation}
Analogously the continuum wave-function has been found in the
one-level approximation as a solution of
the equation:
\begin{equation}
[\frac{d^2}{d\rho
^2}-\frac{15}{4\rho ^2}- \varepsilon _{1}(\rho)-P_{11}(\rho
)+E]u_{1k}(\rho )=0
\label{eq:s17}
\end{equation}
for the following boundary and asymptotic conditions:
\begin{equation}\left.
\begin{array}{l}
u_{1k}(0)=0,\\
{\displaystyle u_{1k}(\rho )\longrightarrow sin(k\rho+\delta),}
\atop{\hspace{-1cm}\rho \to \infty}
\label{eq:s18}
\end{array}\right.
\end{equation}
where $k=(E-\varepsilon _{1}(\infty ))^{1/2}$ and phase $\delta $
is of no interest for our purposes.
The radiationless decay rate is given by
\begin{equation}
\lambda =\frac{1}{k}\left|M_{k}\right|^{2}\cdot
\displaystyle\frac{me^4}{\hbar^3}s^{-1},
\label{eq:s19}
\end{equation}
where the matrix element of the channel coupling operator is
\begin{equation}
M_{k}=\int\limits_{0}^{\infty }d\rho u_{1k}(\rho )[Q_{12}(\rho )\frac{d}{d\rho}
+
\frac{d}{d\rho} Q_{12}(\rho )-P_{12}(\rho )]u_{2}(\rho ).
\label{eq:s20}
\end{equation}
\section{Numerical results}
The following values of the masses were used in calculations:
$m_\mu=206.769m_e$, $m_d=3670.481m_e$, $m_{^{4}\! H\! e}=7294.295m_e$,
$m_{^{3}\! H\! e}=5495.881m_e$. Equation (\ref{eq:e7}) has been
solved for a number of $\rho $ values in the interval $0\leq \rho \leq 45$.
Variations of the upper bound of this interval do not change final
results. Expressions (\ref{eq:e12})-(\ref{eq:p1}) have been used to
calculate $\varepsilon _{n}(\rho ), Q_{ni}(\rho ), P_{ni}(\rho )$
for these $\rho $ values. The set of trial functions (\ref{eq:s13})
has been adjusted in the following way: numbers of channel-type
functions $N_1$ and hyperspherical harmonics $N_2$ were chosen as:
\begin{equation}\left.
\begin{array}{l}
N_1=2,\ N_2=105,\qquad \rho \leq 5;\\
N_1=6,\ N_2=91,\,\ \qquad 5<\rho <7;\\
N_1=6,\ N_2=105,\qquad 7<\rho \leq 15;\\
N_1=12,\ N_2=78,\qquad 15<\rho .
\label{eq:n21}
\end{array}\right.
\end{equation}
The relative accuracy of two lowest eigenpotentials
$\varepsilon _{1}(\rho ), \varepsilon _{2}(\rho )$ calculated in the
above mentioned interval of $\rho $ can be estimated as $10^{-4}$.
Mesomolecular binding energies $E_L$ and radiationless decay rates
$\lambda _L$ for angular momentum values $L=0,1,2$ have been calculated as
described in the previous section.
The integrand in (\ref{eq:s20}) contains the rapidly
oscillating function $u_{1k}(\rho )$, the sharp functions $u_{2}(\rho )$,
$Q_{12}(\rho ), P_{12}(\rho )$ and their derivatives. In consequence of
these facts, special care has been taken of the evaluation of this
integral. For this purpose, $u_2(\rho ), Q_{12}(\rho )$ and $P_{12}(\rho )$
were expressed as a product of sharp functions given in the analytical
form and of smooth functions given numerically. A few
per cent variation of decay rates was found when using different ways for
analytical representation of sharp functions.
The calculated values of the binding energies $E_{BL}=E_{d\mu }-E_L$ and
decay rates for the $^{3,4}\! H\! ed\mu $ systems are presented in Table
1 in comparison with the results of other authors.
{\large {Table 1}
\begin{flushleft}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline system & &
{}~\cite{kino} & ~\cite{hara} & ~\cite{ger} & ~\cite{krav2} & present\\ \hline
$^{4}\! H\! ed\mu $ & $E_{B0}$ & & & 77.96 & 78.7 & 77.49 \\ \cline
{2-7} & $E_{B1}$ & 58.22 & 57.84 & 56.10 & 57.6 & 55.74 \\ \cline {2-7}
& $E_{B2}$ & & & & 20.3 & 17.47\\ \cline {2-7}
& $\lambda _0$ & & & 2.3 & 1.85 & 0.73 \\ \cline {2-7}
& $\lambda _1$ & 1.67 & & 2.4 & 1.38 & 1.20 \\ \cline {2-7}
& $\lambda _2$ & & & & 0.9 & 1.04\\ \hline
$^{3}\! H\! ed\mu $ & $E_{B0}$ & & 70.74 & 69.96 & 70.6 & 69.37 \\
\cline {2-7} & $E_{B1}$ & 48.42 & 47.90 & 46.75 & 48.2 & 46.31 \\ \cline
{2-7} & $E_{B2}$ & & & & 9.6 & 7.11\\ \cline {2-7} & $\lambda _0$ & &
& 8.0 & 3.58 & 2.87 \\ \cline {2-7} & $\lambda _1$ & 5.06 & & 7.0 & 2.77
& 3.22 \\ \cline {2-7} & $\lambda _2$ & & & & 1.54 & 1.74\\ \hline
\end{tabular}
\end{flushleft} }
\noindent
{Table 1. Binding energies $E_{BL}~(eV)$ and decay rates
$\lambda_L~(10^{11}s^{-1})$ of the systems $^{3,4}\! H\! ed\mu $
calculated in Ref.~\cite{kino},~\cite{ger},~\cite{krav2},~\cite{hara}
and in the present paper.}
\section{Discussion}
{}From Table 1 it is clear that binding energies for a given $L$ are close
to each other in all calculations. One can see that energies of the
present paper are higher in comparison with
calculations ~\cite{kino} and~\cite{hara}.
The method of this
work gives an upper bound of eigenenergy if the coupling of channels is
omitted. One can conclude that this fact supports the validity of the
one-level approximation in our approach.
The comparison with the
results obtained in the framework of the Born-Oppenheimer approximation
(~\cite{ger},~\cite{krav2}) cannot be done straightforwardly due to the
following reasons. First, mass values and thresholds are introduced
in these calculations {\it ad hoc} and do not coincide with the physical
ones. The importance of these procedures for the
calculation of the decay rate is not clear. Unlike the eigenenergy
problem, the calculation of the decay rate is very sensitive to the fine
details of wave-functions, as is clear from expression (\ref{eq:s20}).
The quasiclassical approximation used in the calculation of the decay rate
in the paper (~\cite{krav2}) can be an origin of an
additional uncertainty.
Qualitatively, all calculations support the strong isotopic dependence of
the decay rates observed in experiment (\cite{mats},~\cite{ishid})
Nevertheless, the calculated values are quite different and consistency
of theoretical results should be reached.
It is accepted that the formation of $H\! ed\mu $ molecules takes place
in the state with $L=1$. In this connection,
for comparison with experiment, the most important is the ratio
$\lambda_\gamma/(\lambda_\gamma+\lambda_1)$, where $\lambda_\gamma $ is
the radiative decay rate from the molecular state $L=1$. Using
$\lambda_\gamma $ from the paper~\cite{hara} and the present values of
$\lambda_1$, one comes to the ratio
$\lambda_\gamma/(\lambda_\gamma+\lambda_1)=0.585$ for $^{4}\! H\! ed\mu
$ and $\lambda_\gamma/(\lambda_\gamma+\lambda_1)=0.325$ for the $^{3}\! H\!
ed\mu $ systems.
Other processes, which may be important in the experiment, are collisional
transitions to the muonic molecular states with angular momentum $L\neq 1$.
Finally, one would like to emphasize the necessity of the systematic study
in the framework of the same approach of the processes involved in the
formation, rearrangement and decay of systems under consideration.
\section{Acknowledgment}
One of the authors, V.B.Belyaev, would like to thank the Scientific
Division of NATO for the financial support within the Collaborative
Research Grant No.~930102.
|
2,877,628,090,234 | arxiv | \section{Introduction}
\label{sec:intro}
The analysis of network data has achieved increasing interest in the last years. \cite{Golden:10}, \cite{Hunter:12}, \cite{Fien:12} and \cite{Salter:12}, respectively, published survey articles demonstrating the state-of-the-art in the field. We also refer to \cite{Kol:09}, \cite{KolC:14} and \cite{Lush:13} for monographs in the field of statistical network data analysis, see also \cite{Kola:17}. The statistical workhorse model for network data are Exponential Random Graph Models (ERGM) which make use of an exponential family distribution to model the network's adjacency matrix as a random matrix. This model class was proposed by \cite{Frank:86} and is extensively discussed in \cite{Snij:06}.
A different modeling strategy results through comprehending the network adjacency matrix $\boldsymbol{Y} \in \{0,1\}^{N \times N}$ to be generated by a so called graphon. The graphon as data generating model comes into play by assuming that we draw $N$ random variables
\begin{align}
U_1,\ldots, U_N \overset{i.i.d.}{\sim} \mbox{ Uniform}[0,1]
\label{eq:us}
\end{align}
and simulate the network entries $Y_{ij}$ conditional on $U_i$ and $U_j$ and independently through
\begin{align}
Y_{ij}|U_i, U_j \sim \mbox{ Binomial}(1, w(U_i, U_j)).
\label{eq:grm}
\end{align}
The function $w(,)$ is thereby called a graphon (= graph function). In case of symmetric networks we additionally require symmetry so that $Y_{ij} = Y_{ji}$ (and hence in principle we assume $w(u,v) = w(v,u)$). We generally omit so called self-loops and we set $Y_{ii} = 0$ for $i=1,\ldots,N$.
Apparently, additional constraints are necessary to make the graphon function $w(,)$ unique. These are required since any permutation of the indices should yield the same model. This is not guaranteed with (\ref{eq:grm}) unless we impose additional constraints on $w(,)$. The common setting to achieve identifiability is therefore to postulate that
\begin{align}
g(u) = \int w(u,v) \mathop{}\!\mathrm{d} v
\label{eq:g}
\end{align}
is strictly increasing in $u$ which leads to the so called canonical representation of the graphon $w(,)$. Note that $g()$ can be interpreted as (asymptotic) distribution of the degree proportion.
Graphon estimates for modeling network data have recently found attention in the statistical literature. Graphons can be related to ERGMs, at least for simple statistics like two-star or triangles, as shown in \cite{DiacChat:13}. \cite{He:15} make use of this connection and propose to use asymptotic properties of graphons to derive estimates in high dimensional ERGMs. \cite{WolfeOl:13} and \cite{Yangetal:14} discuss non-parametric estimation of graphons including tests on the validity of prespecified graphon shapes, see also \cite{ChAir:14} or \cite{Airoldi:13}. \cite{Gao:15} discuss optimal graphon estimation in stochastic block models, \cite{WolfeOl:14} propose histogram estimates. For a general discussion on graphons we refer to \cite{Bor:08}, \cite{lovasz:12}, \cite{Diac:08} or \cite{Bick:09}. In this paper we propose to use penalized linear B-splines for graphon estimation. This borrows ideas suggested in \cite{Kauermann:13} for copula estimation, since B-splines easily allow to accommodate side constraints such as (\ref{eq:g}) for the resulting estimate. This in contrast is difficult to accommodate in histogram or kernel based estimation. Penalized estimation with B-splines has thereby a long standing tradition in smooth estimation, starting with \cite{Eilers:96} and \citeauthor{Ruppert:09} (\citeyear{Ruppert:03}, \citeyear{Ruppert:09}), see also \cite{Wood:17}. We extend the idea here to graphon estimation.
Smooth estimation is carried out over the unit square considering $U_i$ as given for $i=1,\ldots,N$. However, quantities $U_i$ are latent and the common approach in graphon estimation is to use the ordered network matrix $\boldsymbol{Y}$. Ordering refers to the degree of the nodes so that $degree(i) \leq degree(j)$ of $i < j$, where $degree(i) = \sum_{j\neq i} Y_{ij}$. The underlying reasoning for the ordering is to make the graphon estimate unique reflecting that the (asymptotic) degree distribution $g()$ in (\ref{eq:g}) is monotone. Note that taking (\ref{eq:grm}) with $\boldsymbol{U} = \boldsymbol{u}$ being a sample from (\ref{eq:us}) we have
\begin{align*}
\frac{1}{N} E(degree(i)|\boldsymbol{U} = \boldsymbol{u}) &= \frac{1}{N} \sum_{j\neq i} P(Y_{ij} =1|\boldsymbol{U} = \boldsymbol{u}) \\
&= \frac{1}{N} \sum_{j\neq i} w(u_i, u_j) \stackrel{N \rightarrow \infty}{\longrightarrow} g(u_i)
\end{align*}
which motivates the ordering of $\boldsymbol{Y}$ prior to estimating the graphon. In fact, since $g()$ is strictly increasing the expected average degree asymptotically provides a unique ordering of the nodes of $\boldsymbol{Y}$ such that the ordering of the nodes corresponds with the ordering of the latent variables $U_i$. It should be noted, however, that $degree(i)$ is a random quantity implying that conditional on $U_j < U_i$, we still may observe $degree(j) > degree(i)$. In other words, ordering the nodes based on the observed degree imposes random variability which in fact induces random variability on the graphon estimate just based on the random ordering of the nodes (on top of the randomness based on the random entries $Y_{ij}$). In this paper we aim to explore this variability by taking a Bayesian view. We take model (\ref{eq:grm}) as data generating process and estimate the posterior distribution of $\boldsymbol{U}$ given $\boldsymbol{Y}=\boldsymbol{y}$, i.e.\ $f(\boldsymbol{u}|\boldsymbol{y})$. This is pursued using MCMC simulations which are surprisingly simple. A central requirement for the MCMC to work is however a reasonable estimate of the graphon. We use the graphon estimate based on B-splines proposed above. Combining these two steps in an iterative manner yields in fact the base for the EM algorithm. This will be further explored in the paper.
The paper is organized as follows. Section~\ref{sec:EmpGraph} illustrates what can be said about the univariate posterior distribution of $U_k$ considering only the marginal expected degree proportion of the empirical graphon. In Section~\ref{sec:BayApp} we describe how the multivariate posterior distribution of $\boldsymbol{U}$ can be derived with MCMC sampling if $w(,)$ is known. Graphon estimation will be proposed in Section~\ref{sec:splines} using linear B-splines which is then combined with the Bayesian approach. Section~\ref{sec:EM} extends this idea to an EM algorithm. A discussion concludes the paper.
\section{Empirical Graphon Estimation}
\label{sec:EmpGraph}
\subsection{Graphon Representation}
We assume that $w: [0,1]^2 \rightarrow [0,1]$ is the unique (canonical) representation of a graphon, so that $g(u) = \int w(u,v) \mathop{}\!\mathrm{d} v$ is strictly increasing, see \cite{Bick:09} or \cite{Yangetal:14}. We assume further that $w(,)$ is symmetric and generates a network of size $N$ through the following process. For $N$ independent uniform variables
$$ U_i \sim \text{ Uniform}[0,1], \quad i=1,\ldots,N $$
we obtain the symmetric network through
\begin{align}
P(Y_{ij} = 1| U_i = u_i, U_j = u_j) = w(u_i, u_j)
\label{eq:mod1}
\end{align}
for $1 \leq i < j \leq N$, where $Y_{ji} = Y_{ij}$ and $Y_{ii}=0$. Even though the probability model (\ref{eq:mod1}) used for the construction of networks is simple, it is usually not used for estimation. The reason is that variables $U_i$ are unobservable and hence can not directly be employed to estimate the graphon $w(,)$. Instead, in the recent literature the graphon $w(,)$ is usually estimated from the adjacency matrix $\boldsymbol{Y}$ subject to a rearrangement of the nodes. Let $\sigma: \{1,\ldots,N\} \rightarrow \{1,\ldots,N\}$ be a permutation such that
$$ U_{\sigma (i)} \leq U_{\sigma (j)} $$
with $i < j$. This means $U_{\sigma(i)}= U_{(i)}$, where $U_{(1)} \leq U_{(2)} \leq \ldots \leq U_{(N)}$ define the ordered variables $U_i$. Note that since $U_i$, $i=1,\ldots,N$ are not observable we can also not observe $\sigma()$ which therefore needs to be estimated. This is usually done by making use of the degree. Let therefore $\hat{\sigma} : \{1,\ldots,N\} \rightarrow \{1,\ldots,N\}$ be a permutation such that
\begin{align*}
degree ( \hat{\sigma} ( i )) \leq degree ( \hat{\sigma}( j ))
\end{align*}
for $i < j$. Note that $\hat{\sigma}()$ serves as a direct estimate for $\sigma()$ and we define a resulting initial prediction for $U_j$ based on this simple sorting through
\begin{align}
\hat{u}_j^{emp} = \frac{rank(degree(j))}{N+1},
\label{eq:uinit}
\end{align}
where $rank(degree(j))$ is the rank from smallest to largest of the $j$th element of $\{degree(i), \allowbreak i \in \{1 ,\ldots, N\}\}$. Note that the values $i/(N+1)$, $i=1,\ldots,N$ represent the expected values of $N$ ordered independently $\text{Uniform}[0,1]$ distributed variables. The setting in (\ref{eq:uinit}) is also equivalent to define $\hat{u}_{\hat{\sigma}(j)}^{emp} = j/(N+1)$. \cite{ChAir:14} prove asymptotic convergence rates for $\hat{\sigma}()$. We here aim to explore finite sample properties to investigate numerically what can be said about the difference between $\sigma$ and $\hat{\sigma}$.
To do so we take a Bayesian view by looking at the posterior probability of $\boldsymbol{U} = (U_1,\ldots,U_N)$ given $\boldsymbol{Y}=\boldsymbol{y}$. Note that since $U_1,\ldots,U_N$ are $i.i.d.$ uniform we have
\begin{align*}
f(\boldsymbol{u}|\boldsymbol{y}) \propto \prod_{\substack{i,j \\ j > i}} w(u_i, u_j)^{y_{ij}} (1-w(u_i,u_j))^{1-y_{ij}}.
\end{align*}
If we look at the univariate distribution of a single variable $U_k$ given the entire network $\boldsymbol{Y}=\boldsymbol{y}$, this results through
\begin{align}
f_k (u_k|\boldsymbol{y}) \propto \int \ldots \int \underset{j > i}{\prod_{i,j}} w(u_i, u_j)^{y_{ij}} (1-w(u_i, u_j))^{1-y_{ij}} \mathop{}\!\mathrm{d} u_1 \ldots \mathop{}\!\mathrm{d} u_{k-1} \mathop{}\!\mathrm{d} u_{k+1} \ldots \mathop{}\!\mathrm{d} u_N.
\label{eq:int1}
\end{align}
Apparently, the distribution is too complex to calculate it analytically. In particular if $N$ is large. We will therefore explore (\ref{eq:int1}) in the following sections by pursuing a Bayesian approach. Before doing so we pursue a simple approximation and replace $w(u_i,u_j)$ in (\ref{eq:mod1}) by its empirical version $\hat{w}^{emp}(,)$ which is defined through
\begin{align}
\hat{w}^{emp}(u,v)=y_{\hat{\sigma}(\lceil u N \rceil) \hat{\sigma}(\lceil v N \rceil)},
\label{eq:emp-graphon}
\end{align}
where $\lceil.\rceil$ defines the next largest integer value. Note that $\hat{w}^{emp}(,)$ just mimics the ordered adjacency matrix scaled towards the unit square. If we now look at the conditional distribution of $U_k$ with $U_j$ set to $\hat{u}_j^{emp}$ for $j \neq k$ we get that
\[
\hat{w}^{emp} \left( \hat{u}_i^{emp}, \hat{u}_j^{emp} \right)^{y_{ij}} \left( 1 - \hat{w}^{emp} \left( \hat{u}_i^{emp}, \hat{u}_j^{emp} \right) \right)^{1-y_{ij}} = 1
\]
for $i\neq j$ and $i,j \neq k$. This leads to
\[
\hat{f}_k^{emp} \left( u_k | \hat{\boldsymbol{u}}_{-k}^{emp}, \boldsymbol{y} \right) \propto \prod_{j\neq k} \hat{w}^{emp} \left( u_k , \hat{u}_j^{emp} \right)^{y_{jk}} \left( 1 - \hat{w}^{emp} \left( u_k, \hat{u}_j^{emp} \right) \right)^{1-y_{jk}},
\]
where $\boldsymbol{u}_{-k}$ is $\boldsymbol{u}$ without its $k$th component. Taking this function and integrate over $\boldsymbol{u}_{-k}$ as substitute for $\hat{\boldsymbol{u}}_{-k}^{emp}$ yields an approximation for the marginal distribution of $U_k$ given $\boldsymbol{Y}=\boldsymbol{y}$. This is then defined as
\begin{align}
\begin{split}
\hat{f}^{emp}_k(u_k|\boldsymbol{y}) &\propto \int \ldots \int \prod_{j\neq k} \hat{w}^{emp}(u_k, u_j)^{y_{jk}} (1-\hat{w}^{emp}(u_k, u_j))^{1-y_{jk}} \mathop{}\!\mathrm{d} u_j \\
&\propto \hat{g}^{emp}(u_k)^{degree(k)} (1-\hat{g}^{emp}(u_k))^{N- degree(k)},
\end{split}
\label{eq:margcond}
\end{align}
where $\hat{g}^{emp}(u_k) = \int \hat{w}^{emp}(u_k, v) \mathop{}\!\mathrm{d} v = degree (\hat{\sigma}(\lceil u_k N \rceil))/N$. With (\ref{eq:margcond}) we can approximate the posterior distribution of $U_k|\boldsymbol{Y} = \boldsymbol{y}$ which in turn mirrors how ``close'' $\hat{\sigma}()$ is to $\sigma()$, i.e.\ how reliable is the ordering of the degree of the nodes to represent the ordering of the latent quantities $U_k$. We stress that (\ref{eq:margcond}) gives a univariate statement only, that is we look at the posterior of $U_k$ by canceling out all other $U_j$ with $j \neq k$ by simplified approximations.
\subsection{Facebook Example}
\label{sub:facExp}
We exemplify the estimate (\ref{eq:margcond}) using network data from Facebook, which has been collected by \cite{McAuley:12} and are available on the Stanford Large Network Dataset Collection (\citeauthor{snapnets} \citeyear{snapnets}). We take one of these ego networks with $333$ actors, which is plotted in Figure~\ref{fig:fbNetwork}.
\begin{figure}[tbhp]
\centering
\includegraphics[scale = 0.7]{fb_facebook_net_only}
\caption{Facebook ego network.}
\label{fig:fbNetwork}
\end{figure}
We then order the nodes based on their degree, where the ordered nodes are now labeled with $k=1,\ldots, N$ with respect to an ascending degree, which means $degree(i) \leq degree(j)$ for $i<j$. The emerging empirical graphon estimate $\hat{w}^{emp}(,)$ is depicted in the top left plot in Figure~\ref{fig:fbPostDist1} and the corresponding degree profile is shown in the top right plot.
\begin{figure}[tbhp]
\centering
\begin{minipage}[t]{1\textwidth}
\centering
\includegraphics[width=.99\textwidth]{fb_facebook_multiple_b1}
\end{minipage}
\vfill
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=.99\textwidth]{fb_facebook_multiple_b2}
\end{minipage}
\caption{Empirical graphon estimate $\hat{w}^{emp}(,)$ from (\ref{eq:emp-graphon}) for the Facebook ego network (top left), degree profile (top right) and posterior distribution of $U_k$ based on $\hat{w}^{emp}(,)$ for selected indices (four lower plots). The vertical dashed line (see also number in the box) gives the value $\hat{u}_k^{emp}$.}
\label{fig:fbPostDist1}
\end{figure}
The fitted univariate posterior distribution (\ref{eq:margcond}) for four selected nodes is shown in the four lower plots. For $k=1$, the node with the lowest degree, we see that the posterior of $U_1|\boldsymbol{Y}$ covers roughly a range between zero and $0.4$. For $k=75$ the posterior of $U_{75}|\boldsymbol{Y}$ changes towards the right, which is also seen for $k=235$. For $k=333$, the node with the most edges, we obtain stronger posterior information with a probability mass centered approximately between $0.95$ and $1$. The example demonstrates that based on the empirical graphon we can already draw univariate information on the latent coefficient $U_k$ given the network matrix $\boldsymbol{Y}=\boldsymbol{y}$.
\section{Bayesian Approach}
\label{sec:BayApp}
In the above section we took a univariate view by looking at a single $U_k$. We now extend this and explore the entire posterior distribution of $\boldsymbol{U}$. We pursue this step by constructing an MCMC Gibbs sample from this posterior of $\boldsymbol{U}$. Note that by conditioning on $\boldsymbol{Y}=\boldsymbol{y}$ and all $U_j = u_j$ except of $U_k$ one gets
\begin{align}
f_k(u_k|u_1, \ldots, u_{k-1}, u_{k+1}, \ldots, u_N, \boldsymbol{y}) \propto \prod_{j \neq k} w(u_k, u_j)^{y_{kj}} (1-w(u_k, u_j))^{1-y_{kj}}.
\label{eq:Gibb1}
\end{align}
We pretend in this section that the graphon $w(,)$ is known. This allows to easily sample from (\ref{eq:Gibb1}) using Gibbs sampling and MCMC. To do so, we assume that $\boldsymbol{u}_{(t)} = (u_{(t)1},\ldots,u_{(t)N})$ is the current state of the Markov chain and we aim to update component $k$. Then $u_{(t+1)j} := u_{(t)j}$ for $j \neq k$ and component $u_k$ is updated by drawing from (\ref{eq:Gibb1}). To pursue this step we first need a proposal density. We here make use of a normal proposal using a logit link. To be specific let $z_{(t)k} = \log(u_{(t)k}/(1-u_{(t)k})) = \text{ logit }(u_{(t)k})$. We then propose to draw $z^*_k = z_{(t)k} + N(0, \sigma^2)$ and set $u_k^* = \text{ logit}^{-1} (z_k^*) = \exp(z^*_k) \big/ (1+ \exp(z_k^*))$.
Hence, the proposal density for $U_k$ is proportional to
\begin{align*}
q(u_k^* | u_{(t)k}) &= \frac{\partial u^*_k}{\partial z^*_k} \phi(z^*_k|z_{(t)k}) \\
&\propto \frac{1}{u^*_k (1-u_k^*)} \exp \left(-\frac{1}{2} \frac{(\text{logit } (u_k^*) - \text{ logit }(u_{(t)k}))^2}{\sigma^2}\right),
\end{align*}
where $\phi()$ is the normal density.
Consequently, the ratio of proposals equals
\begin{align*}
\frac{q_k(u_{(t)k} | u^*_k)}{q_k(u_k^*|u_{(t)k})} = \frac{u^*_k(1-u^*_k)}{u_{(t)k} (1- u_{(t)k})}.
\end{align*}
The proposed value $u^*_k$ is accepted (and hence we set $u_{(t+1)k} = u_k^*$) with probability
\begin{align*}
\min \left\{ 1, \quad \prod_{j \neq k} \left[ \left( \frac{w(u_k^*, u_{(t)j})}{w(u_{(t)k}, u_{(t)j})} \right)^{y_{kj}}
\left( \frac{(1-w(u^*_k, u_{(t)j}))}{(1-w(u_{(t)k}, u_{(t)j}))} \right)^{1-y_{kj}} \right]
\frac{u^*_k(1-u^*_k)}{u_{(t)k} (1- u_{(t)k})} \right\}.
\end{align*}
If we do not accept $u_k^*$ we set $u_{(t+1)k} = u_{(t)k}$.
The Gibbs sampling approach is straightforward and simple, but requires the knowledge of the graphon $w(,)$. Apparently, in practice the graphon is not known and we need to replace $w(,)$ in the formula above by an estimate. Working with the empirical graphon $\hat{w}^{emp}(,)$ does not work since the Markov chain will not move appropriately. We therefore need to derive a suitable estimate for $w(,)$ which allows to use the above Bayesian approach. This will be discussed in the next section.
\section{Spline based Graphon Estimation}
\label{sec:splines}
\subsection{Linear B-Splines}
For smooth estimation of the graphon $w(,)$ we first formulate an approximation through spline bases in the form
\begin{align}
w_{\boldsymbol{\theta}}^{approx}(u,v) = \left[ \boldsymbol{B}(u) \otimes \boldsymbol{B}(v) \right] \boldsymbol{\theta},
\label{eq:approx}
\end{align}
where $\boldsymbol{B}(u) \in \mathbb{R}^{1 \times K}$ is a linear B-spline basis on $[0,1]$, normalized to have maximum value 1, see Figure~\ref{fig:splBas}. Parameter vector $\boldsymbol{\theta} \in \mathbb{R}^{K^2 \times 1}$ is indexed through
\[
\boldsymbol{\theta} = \left( \theta_{11},\ldots, \theta_{1K}, \theta_{21}, \ldots, \theta_{K1},\ldots,, \theta_{KK} \right)^\top.
\]
\begin{figure}[tb]
\centering
\includegraphics[scale = 0.7]{splineBasis}
\caption{Normed linear B-spline basis for the approximation of the graphon. The (equidistant) inner knots are denoted by $\tau_j$ with $j=1,\ldots,K$.}
\label{fig:splBas}
\end{figure}
Using (\ref{eq:approx}) we obtain the likelihood
\[
l(\boldsymbol{\theta}) = \sum\limits_{i} \sum\limits_{j \neq i} \left[ y_{ij} \, \log \left( \boldsymbol{B}_{ij} \boldsymbol{\theta} \right) + \left( 1- y_{ij} \right) \, \log \left( 1 - \boldsymbol{B}_{ij} \boldsymbol{\theta} \right) \right],
\]
where $\boldsymbol{B}_{ij} = \boldsymbol{B}(u_i) \otimes \boldsymbol{B}(u_j)$. Taking the derivative leads to the score function
\[
\boldsymbol{s} ( \boldsymbol{\theta}) = \sum\limits_{i} \sum\limits_{j \neq i} \boldsymbol{B}_{ij}^\top \left( \frac{y_{ij}}{w_{\boldsymbol{\theta}}^{approx}(u_i,u_j)} - \frac{1-y_{ij}}{1-w_{\boldsymbol{\theta}}^{approx}(u_i,u_j)} \right).
\]
Moreover, taking the expected second order derivative leads to the Fisher matrix
\[
\boldsymbol{I}(\boldsymbol{\theta}) = \sum\limits_{i} \sum\limits_{j \neq i} \boldsymbol{B}_{ij}^\top \boldsymbol{B}_{ij} \left[ w_{\boldsymbol{\theta}}^{approx} \left( u_i,u_j \right) \cdot \left( 1 - w_{\boldsymbol{\theta}}^{approx} \left( u_i,u_j \right) \right) \right]^{-1}.
\]
Our intention is to maximize $l(\boldsymbol{\theta})$ which could be simply done by Fisher scoring. The resulting maximizer does however not lead to a canonical representation of a graphon since constraint (\ref{eq:g}) is not taken into account. We therefore need to impose additional linear side constraints on $\boldsymbol{\theta}$ to guarantee that (\ref{eq:g}) is fulfilled. Note that with (\ref{eq:approx}) we get $g^{approx}(u)$ through
\begin{align}
g^{approx}(u) = \left[ \boldsymbol{B}(u) \otimes \int\limits_0^1 \boldsymbol{B}(v) \mathop{}\!\mathrm{d} v \right] \boldsymbol{\theta}.
\label{eq:G_1}
\end{align}
For standard B-splines we can easily calculate the integral and for equidistant knots we obtain
\begin{align*}
\int\limits_0^1 \boldsymbol{B}(v) \mathop{}\!\mathrm{d} v &= \left( \int\limits_0^1 B_1(v) \mathop{}\!\mathrm{d} v, \int\limits_0^1 B_2(v) \mathop{}\!\mathrm{d} v, \ldots, \int\limits_0^1 B_K(v) \mathop{}\!\mathrm{d} v \right) \\
&= \underbrace{\frac{1}{K-1} \left( \frac{1}{2}, 1, \ldots, 1, \frac{1}{2} \right)}_{ =: \boldsymbol{A}}.
\end{align*}
This allows to rewrite (\ref{eq:G_1}) to $g^{approx}(u) = \left[ \boldsymbol{B}(u) \otimes \boldsymbol{A} \right] \boldsymbol{\theta}$. Hence, the marginal function $g^{approx}()$ is also expressed as a linear B-spline and a monotonicity constraint is easily accommodated by postulating monotonicity at the knots $\tau_1,\ldots, \tau_K$. That is we need
\begin{align*}
g^{approx} \left( \tau_l \right) - g^{approx} \left( \tau_{l-1} \right) \geq 0 \Leftrightarrow \left[ \left( \boldsymbol{B} \left( \tau_l \right) - \boldsymbol{B} \left( \tau_{l-1} \right) \right) \otimes \boldsymbol{A} \right] \boldsymbol{\theta} \geq 0
\end{align*}
for $l = 2,\ldots,K$, which is a simple linear constraint on the coefficient vector. Besides monotonicity we also impose symmetry on the graphon which is also easily accommodated as linear constraints $\theta_{pq} = \theta_{qp}$ for $p \neq q$. Finally, we need that $0 \leq w_{\boldsymbol{\theta}}^{approx} (,) \leq 1$, which is again a simple linear constraint. All in all we can write the side constraints as $\boldsymbol{C} \boldsymbol{\theta} \geq \boldsymbol{0}$ and $\boldsymbol{D} \boldsymbol{\theta} = \boldsymbol{0}$ for matrices $\boldsymbol{C}$ and $\boldsymbol{D}$ chosen accordingly. With the above linear constraints and the maximization of $l(\boldsymbol{\theta})$ we obtain an (iterated) quadratic programming problem which can be solved using standard software (see e.g.\ \citeauthor{cvxopt} \citeyear{cvxopt} or \citeauthor{quadprog} \citeyear{quadprog}).
\subsection{Penalized Estimation}
Following ideas from the penalized spline estimation (see \citeauthor{Eilers:96} \citeyear{Eilers:96} or \citeauthor{Ruppert:09} \citeyear{Ruppert:09}) we may additionally impose a penalty on the coefficients to achieve smoothness. This is necessary since we intend to choose $K$ large and unpenalized estimation will lead to wiggled estimates. We refer to \cite{Eilers:96} for a motivation of penalized estimation. To do so, we penalize the difference between ``neighbouring'' elements of $\boldsymbol{\theta}$ to achieve smoothness. Let therefore
\[
\boldsymbol{L} = \begin{pmatrix}
1 & -1 & \phantom{-}0 & \multicolumn{2}{c}{\ldots} & \phantom{-}0 \\
0 & \phantom{-}1 & -1 & \multicolumn{2}{c}{\ldots} & \phantom{-}0 \\
\vdots & & \ddots & & & \phantom{-}\vdots \\
0 & \multicolumn{2}{c}{\ldots} & 0 & 1 & -1 \\
\end{pmatrix}
\]
be the first order difference matrix. We then penalize $\left[ \boldsymbol{L} \otimes \boldsymbol{I} \right] \boldsymbol{\theta}$ and $\left[ \boldsymbol{I} \otimes \boldsymbol{L} \right] \boldsymbol{\theta}$, where $\boldsymbol{I}$ is the identity matrix. This is leading to the penalized likelihood
\[
l_{\boldsymbol{P}} (\boldsymbol{\theta}, \lambda) = l (\boldsymbol{\theta}) - \frac{1}{2} \lambda \boldsymbol{\theta}^\top \boldsymbol{P} \boldsymbol{\theta} ,
\]
where $\boldsymbol{P} = \left( \boldsymbol{L} \otimes \boldsymbol{I} \right)^{\top} \left( \boldsymbol{L} \otimes \boldsymbol{I} \right) + \left( \boldsymbol{I} \otimes \boldsymbol{L} \right)^{\top} \left( \boldsymbol{I} \otimes \boldsymbol{L} \right) $ and $\lambda$ serves as smoothing parameter. The corresponding penalization score function is given through
\[
\boldsymbol{s}_{\boldsymbol{P}}(\boldsymbol{\theta}, \lambda) = \boldsymbol{s}(\boldsymbol{\theta}) - \lambda \boldsymbol{P} \boldsymbol{\theta}
\]
and the penalized Fisher matrix in the form
\[
\boldsymbol{I}_{\boldsymbol{P}} (\boldsymbol{\theta},\lambda) = \boldsymbol{I} (\boldsymbol{\theta}) + \lambda \boldsymbol{P}.
\]
We define the resulting estimate with $\hat{\boldsymbol{\theta}}_{\boldsymbol{P}}$. The estimate apparently depends on the penalty parameter $\lambda$ which is suppressed in the notation. Setting $\lambda \rightarrow 0$ gives an unpenalized fit while setting $\lambda \rightarrow \infty$ leads to a constant graphon, i.e.\ an Erd\H{o}s-R\'{e}nyi model. The smoothing parameter $\lambda$ therefore needs to be chosen data driven. We here follow \cite{Kauermann:13} and make use of the Akaike Information Criterion (AIC) (\citeauthor{Hurvich:89} \citeyear{Hurvich:89}, see also \citeauthor{Burnham:10} \citeyear{Burnham:10}). To do so we define the corrected AIC through
\[
AIC_c (\lambda) = - 2 l ( \hat{\boldsymbol{\theta}} ) + 2 \, df (\lambda ) + \frac{2 \, df( \lambda ) \left( df (\lambda) + 1\right) }{ \left( N (N-1) \right) - df( \lambda ) - 1} ,
\]
where $\hat{\boldsymbol{\theta}}$ is the penalized parameter estimate and $df(\lambda)$ is the degree of the model, which we define in the usual way as trace of the product of the inverse penalized Fisher matrix and the unpenalized Fisher matrix. To be specific
\[
df(\lambda ) = tr \left\{ \boldsymbol{I}_{\boldsymbol{P}}^{-1} ( \hat{\boldsymbol{\theta}}, \lambda ) \boldsymbol{I} ( \hat{\boldsymbol{\theta}} ) \right\}.
\]
\subsection{One Step Spline and Bayes Estimation}
Now that we have an estimate for the graphon we can make use of the Bayesian approach proposed in the previous section. That is we employ the Gibbs sampler given in Section~\ref{sec:BayApp} with the graphon estimated by the penalized linear B-spline estimate $\hat{w}^{spline}(,)$. We order the nodes with respect to their degree, that is estimating the graphon using (\ref{eq:uinit}). This now allows to apply the Gibbs sampler from above to obtain the full posterior distribution. More precisely, the MCMC sequence which follows from (\ref{eq:Gibb1}) provides (after appropriate thinning) information about the posterior distribution of $\boldsymbol{U}$ given the network $\boldsymbol{Y}=\boldsymbol{y}$. In principle we could now explore the MCMC samples in the usual way, see e.g.\ \cite{GelSmith:90}. We go a different route here and make use of the structure of the conditional density of $U_k$ given $\boldsymbol{U}_{-k} = \boldsymbol{u}_{-k}$ defined in (\ref{eq:Gibb1}).
Again we therefore replace the graphon by its spline estimate yielding
\[
\hat{f}^{spline}_k(u_k | \boldsymbol{u}_{-k}, \boldsymbol{y}) \propto \prod_{j \neq k} \hat w^{spline}(u_k, u_j)^{y_{kj}} (1-\hat w^{spline}(u_k, u_j))^{1-y_{kj}}.
\]
We then calculate the conditional distribution of $U_k$ using the MCMC samples through
\begin{align}
\hat{f}_k^{spline} (u_k| \boldsymbol{y}) = \frac{1}{n} \sum_{s=1}^n \hat{f}_k^{spline} (u_k|\boldsymbol{u}_{(s\cdot N \cdot r) -k}, \boldsymbol{y}),
\label{eq:postDist}
\end{align}
where $\boldsymbol{u}_{(t) -k} = (u_{(t)1}, \ldots, u_{(t)k-1}, u_{(t)k+1}, \ldots, u_{(t)N})$ is the $t$th state of the Gibbs sampling sequence without the $k$th component, $r \in \mathbb{N}$ describes a thinning factor and $n$ is the number of MCMC states which are taken into account. We demonstrate the idea with two simulation examples.
\subsection{Simulation Examples}
We consider simulated networks for the two graphons given in Table~\ref{tab:graFun}.
\begin{table
\begin{center}
\begin{tabular}{cc}
\toprule
ID & Graphon \\ \midrule
1 & $\begin{aligned} w_1 (u,v) = 1/2 \, (u+v) \end{aligned}$ \\
2 & $\begin{aligned} w_2 (u,v) = 0.8 \left( 1-u \right) \left( 1-v \right) + 0.85 (u \cdot v) \end{aligned}$ \\ \bottomrule
\end{tabular}
\end{center}
\caption{Exemplary graphons considered for simulations.}
\label{tab:graFun}
\end{table}
For both of them we draw a network with dimension $N=500$ using the data generating process (\ref{eq:mod1}). The B-spline based estimation for graphon $w_1 (,)$ is shown in the top right panel in Figure~\ref{fig:sim1GraEst1}. The estimate seems to capture the structure of the real graphon (top left).
\begin{figure}[tbhp]
\centering
\begin{minipage}[t]{1\textwidth}
\centering
\includegraphics[width=.9\textwidth]{4_0_distr_U_given_y_w_hat1}
\end{minipage}
\vfill
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=.85\textwidth]{4_0_distr_U_given_y_w_hat2}
\end{minipage}
\caption{Graphon estimation (top right) based on B-splines and $\hat{\boldsymbol{u}}^{emp}$ for graphon $w_1(,)$ from Table~\ref{tab:graFun} (top left). The two plots in the second row illustrate the observed degree versus the expected degree (left) and $\hat{\boldsymbol{u}}^{emp}$ versus the true simulated $\boldsymbol{u}$ (right). The lower four plots show the posterior distribution of $U_k$ based on the MCMC samples with respect to the degree based estimate $\hat{w}^{spline}(,)$ for some selected indices. The dashed vertical line (see also number in the box) represents the estimation $\hat{u}_k^{emp}$.}
\label{fig:sim1GraEst1}
\end{figure}
Considering the observed degree in comparison to the expected degree (illustrated in the second row, left panel) emphasizes the proximity between those two quantities. The expected degree here is simply given through $(N-1) \cdot g(u_i)$, where $u_i$ are the simulated values. The same consequently applies for the comparison between the estimated $\hat{u}^{emp}_i$ and the simulated $u_i$, which can be seen in the right plot in the second row. The four lower plots show that the estimates $\hat{u}_k^{emp}$ (vertical dashed lines) for four selected indices are adequately represented by the posterior distribution of $U_k$ calculated from the MCMC samples as is designated in (\ref{eq:postDist}). Overall, for graphon $w_1 (,)$ sorting the network by degree is eligible and results in an adequate graphon estimate even in one step. This does not hold for graphon $w_2(,)$ as it is demonstrated in Figure~\ref{fig:sim2GraEst1}. Apparently, in this case the ordering by degree is misleading and the predictions $\hat{u}^{emp}_i$ do not match the simulated values $u_i$ at all, which both is depicted in the second row, respectively. This misplacement leads to an inadequate fit of the graphon as can be seen by comparing the real and the estimated graphon in the first row of Figure~\ref{fig:sim2GraEst1}.
\begin{figure}[tbhp]
\centering
\begin{minipage}[t]{1\textwidth}
\centering
\includegraphics[width=.9\textwidth]{21_0_distr_U_given_y_w_hat1}
\end{minipage}
\vfill
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=.85\textwidth]{21_0_distr_U_given_y_w_hat2}
\end{minipage}
\caption{Graphon estimation (top right) based on B-splines and $\hat{\boldsymbol{u}}^{emp}$ for graphon $w_2(,)$ from Table~\ref{tab:graFun} (top left). The two plots in the second row illustrate the observed degree versus the expected degree (left) and $\hat{\boldsymbol{u}}^{emp}$ versus the true simulated $\boldsymbol{u}$ (right). The lower four plots show the posterior distribution of $U_k$ based on the MCMC samples with respect to the degree based estimate $\hat{w}^{spline}(,)$ for some selected indices. The dashed vertical line (see also number in the box) represents the estimation $\hat{u}_k^{emp}$.}
\label{fig:sim2GraEst1}
\end{figure}
Thus, the ordering by degree is not appropriate in this case. A further indication of the unstructured ordering is depicted in the four lower plots. Here $\hat{u}_k^{emp}$ is for none of the selected indices adequately represented by the corresponding posterior distribution. We show in the next section how the EM approach can correct this issue.
\section{EM based Graphon Estimation}
\label{sec:EM}
\subsection{EM based Spline and Bayes Estimation}
The EM algorithm suggests to replace the missing value of $U_j$ by its mean value $E(U_j|\boldsymbol{Y}=\boldsymbol{y})$ calculated with the parameter estimates of the previous step of the EM algorithm. Given that we are primarily interested in the ordering of the nodes of the network we modify the E-step by looking at the expected ordering only. To do so, we use the MCMC sequence (after applying an appropriate thinning) to estimate the posterior mean in the $m$th step of the EM algorithm through
\begin{align*}
\bar{u}_j^{(m)} = \frac{1}{n} \sum_{s=1}^n u_{(s \cdot N \cdot r)j}.
\end{align*}
We then use the rank of the posterior means to reorder the network matrix accordingly. This corresponds to setting the missing values of $U_j$ to
\begin{align*}
\hat{u}_j^{(m)} = \frac{rank (\bar{u}_j^{(m)}) }{N+1}.
\end{align*}
The principle idea of the EM based algorithm is sketched as follows.
\textbf{\underline{Algorithm:}}
\begin{description}[labelindent=2\parindent]
\item[\textbf{\textit{Step 1:}}] Initial estimation of $\boldsymbol{u}$ by degree $\rightarrow \hat{\boldsymbol{u}}^{(0)}= \hat \boldsymbol{u}^{emp} $
\item[\textbf{\textit{Step 2:}}] Graphon estimation with linear B-splines $\rightarrow \hat{w}^{(m)}(,)$ \hfill \mbox{\textit{(M-step)}}
\item[\textbf{\textit{Step 3:}}] Reordering of $\hat{\boldsymbol{u}}^{(m)}$ based on the MCMC mean $\rightarrow \hat{\boldsymbol{u}}^{(m+1)}$ \hfill \textit{(E-step)}
\item[\textbf{\textit{Step 4:}}] Iteration between \textit{Step 2} and \textit{Step 3} until convergence is reached.
\end{description}
Note that in the beginning the approximation of the posterior mean is allowed to be rather rough as there might be anyway a large gap between its ordering and the ordering of the true $U_j$ due to an incorrect graphon specification. For being more efficient we therefore start with a small number of considered MCMC states $n$ to approximate $E(U_k| \boldsymbol{Y} =\boldsymbol{y})$ which then will be increased successively in each iteration. This is, for example, also supposed by \citeauthor{Tanner:87} (\citeyear{Tanner:87}, sec. 7). We terminate the algorithm if changes on the graphon estimate fall below a threshold, or, of course, if a maximum number of iteration steps is exceeded. The final estimate is defined as $\hat{w}^{EM}(,)$ and the corresponding ordering is denoted as $\hat{\boldsymbol{u}}^{EM}$. To demonstrate the procedure we consider again graphon $w_2(,)$ from Table~\ref{tab:graFun} and the Facebook example from~\ref{sub:facExp}.
\subsection{Simulation and Facebook Example}
\subsubsection{Simulation}
With the iterative approach we achieve for $w_2(,)$ the graphon estimate which is shown in the top right plot in Figure~\ref{fig:sim2GraEst2}.
\begin{figure}[tbhp]
\centering
\begin{minipage}[t]{1\textwidth}
\centering
\includegraphics[width=.9\textwidth]{21_final_distr_U_given_y1}
\end{minipage}
\vfill
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=.85\textwidth]{21_final_distr_U_hat_given_y2}
\end{minipage}
\caption{Graphon estimation (top right) based on the EM algorithm for graphon $w_2(,)$ from Table~\ref{tab:graFun} (top left). The comparison between their marginalizations is illustrated in the second row on the left. The right plot in the second row illustrates the difference between $\hat{\boldsymbol{u}}^{EM}$ and the true simulated $\boldsymbol{u}$. The lower four plots show the posterior distribution of $U_k$ based on the MCMC samples with respect to the EM based estimate $\hat{w}^{EM}(,)$ for some selected indices. The dashed vertical line (see also number in the box) represents the estimation $\hat{u}_k^{EM}$.}
\label{fig:sim2GraEst2}
\end{figure}
The structure of the real graphon (top left) is now well captured by the estimate, in particular compared to the estimation based on $\hat{\boldsymbol{u}}^{emp}$ in Figure~\ref{fig:sim2GraEst1}. Also the proportional degree profile is now represented adequately by $\hat{g}^{EM}(u) = \int \hat{w}^{EM}(u,v) \mathop{}\!\mathrm{d} v $, where the comparison is illustrated in the left plot in the second row. Considering the EM based estimated $\hat{u}_i^{EM}$ a good approximation of the corresponding true simulated $u_i$ can be seen (right plot in the second row). The resulting posterior distribution for selected indices in the four lower plots emphasizes the proximity between $\hat{u}_i^{EM}$ and the posterior mean and hence indicates a good conformity of the components $\hat{\boldsymbol{u}}^{EM}$ and $\hat{w}^{EM}(,)$. All in all, the proposed EM algorithm provides appropriate estimates even if initial ordering of the nodes by their degree is not adequate. A convergence of the algorithm with respect to local graphon values can be seen as of the $11$th iteration, which is illustrated in Figure~\ref{fig:sim1Traj}.
\begin{figure}[tbh]
\centering
\includegraphics[width=.9\textwidth]{21_trajectory}
\caption{Trajectory of $\hat{w}^{(m)} (u,v)$ for graphon $w_2 (,)$ from Table~\ref{tab:graFun} for exemplary pairs of values for the proceeding EM iterations.}
\label{fig:sim1Traj}
\end{figure}
\subsubsection{Facebook Data}
Applying the algorithm to the Facebook network leads to the graphon estimate shown in the top right panel in Figure~\ref{fig:fbPostDist2}.
\begin{figure}[tbhp]
\centering
\begin{minipage}[t]{1\textwidth}
\centering
\includegraphics[width=.99\textwidth]{fb_final_est_graphonFinal_plus_Net}
\end{minipage}
\vfill
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=.99\textwidth]{fb_final_distr_U_hat_given_y2}
\end{minipage}
\caption{Graphon estimation (top right, in log scale) based on the EM algorithm for the Facebook network (top left). The lower four plots show the posterior distribution of $U_k$ based on the MCMC samples with respect to the EM based estimate $\hat{w}^{EM}(,)$ for some selected indices. The dashed vertical line (see also number in the box) represents the estimation $\hat{u}_k^{EM}$.}
\label{fig:fbPostDist2}
\end{figure}
The structure of the network in the top left plot can to some extent be recognized in the estimated graphon, e.g.\ the bundle in the center of the lower section and its behavioral connectivity among themselves and to other nodes, which is located in the graphon approximately between 0.7 and 1. Regarding the posterior distribution in the four lower plots reveal that apparently it is now much more compact compared to the posterior distribution derived using the empirical ordering, which was shown in Figure~\ref{fig:fbPostDist1}. Of course, taking the behavior of connectivity into account instead of merely the overall power of connectivity provides much more information. The corresponding estimates $\hat{u}_k^{EM}$ for some selected indices are again adequately represented by the posterior mean, which demonstrates the well-matching of the graphon estimation $\hat w^{EM} (,)$ and the estimated $\hat{\boldsymbol{u}}^{EM}$.
\section{Discussion}
The paper proposes a novel estimation routine for graphon estimation which explicitly takes the variability of ordering the nodes into account. The proposed semi-parametric estimation based on B-splines allows to incorporate uniqueness restrictions in the estimation. The Bayesian approach relying on Gibbs sampling illuminates the uncertainty about the degree and its distribution. Both steps combined give an EM algorithm which allows for flexible graphon estimation even in large networks. The approach outperforms available routines in two aspects. First, the B-spline estimate has a unique representation guaranteed, that is (\ref{eq:g}) holds for the estimate. Secondly, based on the Bayesian formulation and the EM algorithm one can assess the amount of uncertainty for ordering the nodes based on their degree.
The latter can also be used in more complex models like stochastic block models, where we assume that nodes cluster and form within and between the clusters simple Erd\H{o}s-R\'{e}nyi models. The latter is subject of current research and beyond the scope of the current paper.
\if00
{
\section*{Acknowledgment}
The project was partially supported by the European Cooperation in Science and Technology [COST Action CA15109 (COSTNET)].
} \fi
\clearpage
\bibliographystyle{Chicago}
|
2,877,628,090,235 | arxiv | \section{Introduction}
In magnetic confinement fusion, the stellarator configuration offers viable solutions for what today appear to be serious problems of the more developed tokamak concept. Since the confining field of a stellarator is created by external coils, without the need to drive large amounts of current within the plasma, continuous operation is granted. This improves the plant availability factor, which positively impacts the overall tritium breeding ratio \cite{AbdouNF2021}. Furthermore, since the intrinsic plasma current in a stellarator is typically small, sudden plasma terminations are not expected to limit the reliability or the integrity of a stellarator reactor device. Major disruptions, causing large electromagnetic forces in the structure of the device and generating very energetic beams of run-away electrons, continues to be a co ncern for tokamak reactors.
The confinement laws of both tokamaks and stellarators display a gyro-Bohm dependence \cite{DinklageFST2007}. However, contrary to tokamaks, $H$-mode operation is not considered in baseline stellarator reactor designs. The understanding of $H$-mode transition in stellarators is not any more mature than in tokamaks, but observations suggest that a strong pedestal formation is not typical of $H$-modes in stellarators, {with temperature profiles showing no clear edge pedestal}. On the one hand, this leads to only mild improvements in confinement for those regimes ($\sim20\%$) and, on the other, to a more benign edge localised mode activity \cite{ErckmannPRL1993, HirschPPCF2008, EstradaCPP2010}. Operating tokamak reactors in $H$-mode imposes a minimum power across the separatrix which, in combination with the requirement to radiate a large fraction of the power that enters the scrape-off layer, has been shown to possibly limit the design space of a fusion reactor based on the tokamak concept \cite{ReinkeNF2017, SiccinioNF2017}.
The heat exhaust problem, i.e. the need to handle the vast amount of plasma self-heating as it eventually crosses the magnetic separatrix and travels towards the divertor plates, is shared by all approaches to magnetic confinement fusion. The greater diversity of stellarator configurations results in a more diversified scrape-off layer and divertor concepts and geometries. The helical coils of the heliotron device offer a natural long-legged divertor with a well separated volume \cite{OhyabuNF1994}. The helical axis advanced stellarator (\emph{helias}) relies on the so-called island divertor, which features multiple X-points and long connection lengths. The operation of an island divertor requires to reduce the plasma current by design, and/or actively control it, to position the field resonance in the plasma periphery. Quasi-axisymmetric and helically-symmetric stellarators feature large bootstrap currents and therefore need to resort to `non-resonant' divertors \cite[see e.g.][]{BoozerJPP2015}, that, so far, have not found an experimental realisation\footnote{The helically symmetric experiment (HSX) has been shown to possess a resilient localised pattern of field-line strike points \cite{BaderPoP2017}, but a divertor was not foreseen in its construction.}.
The island divertor concept was first implemented in the Wendestein 7-AS stellarator \cite{SardeiJNM1997} and further engineered in its successor, Wendelstein 7-X (W7-X) \cite{HermannFST2004}. First W7-X divertor operation has demonstrated a potential to handle power \cite{NiemannNF2020} and impurities \cite{WintersPhD2019} favourably. Furthermore, long and stable power detachment has been attained \cite{ZahngPRL2019, SchmitzNF2020}. The characterisation and understanding of the island divertor scrape-off layer physics and plasma wall interaction \cite{MayerPS2020} will occupy the fusion community for years to come, but significant progress is already being made \cite{KillerPPCF2020, FengNF2021, BrezinsekNF2021}.
In this and other respects, the stellarator physics basis is under development and the engineering feasibility of a stellarator power plant needs to be demonstrated. The three-dimensional magnetic configuration poses specific challenges for confining the hot plasma fuel and fusion-generated fast particles. It also complicates engineering aspects, which mainly arise from the three-dimensionality of the electromagnets and the need to have enough of them, close enough to the plasma, to generate an optimised magnetic configuration. Superconducting coil manufacture and support structure, breeding blanket geometry or remote maintenance are all more challenging in stellarator reactors \cite{WarmerFSD2017}. Nevertheless, advances in coil optimization codes are providing new insights and less constraining geometries \cite{LandremanPoP2016, YamaguchiNF2019}.
In the scientific literature, there exist stellarator reactor studies based on several magnetic configurations, including compact quasi-symmetric \cite{NajmabadiFST2008}, force-free heliotron \cite{GotoPFR2011, SagaraNF2017} and helias \cite{GriegerFT1992, BeidlerNF2001, WarmerFST2015}. An early comparison of them was published in \cite{BeidlerToki2001}. \rev{Recently, the fusion reactor systems code PROCESS has been adapted to deal will stellarator configurations \cite{LionNF2021}, which will allow to assess economic and technological feasibility of specific 3D equilibria and coil sets.}
The physics model used in reactor studies is generally rather coarse-grained, such that only the general geometric parameters of the different configurations (e.g. aspect ratio, rotational transform, size and magnetic field) enter the description through scalings and operational limits inferred from experience in one or several devices. Similarly, for devising design points for stellarator reactors, assumptions need to be made on aspects such as the shape of the density and temperature profiles, the concentration of helium ash or the efficiency of the alpha particle heating. This is simply the reflection of an incomplete understanding of the transport processes in magnetic confinement plasmas. Current stellarator research aims at filling those gaps, understanding the relevant equilibrium, stability and transport physics that would allow identifying credible design points for a stellarator reactor.
The present work builds on these studies, and shares their main methods and assumptions, to assess the impact of a coil technology capable of producing strong confining magnetic fields, stronger than those possible with conventional superconductors like niobium-tin, on the physics design point of a stellarator reactor. The demonstration of the fundamental technological principle, namely, the high temperature superconductors (HTS) of the REBCO type, exist since several decades \cite{FiskSSC1987}, but has only recently received a broader attention from the tokamak (see e.g. \cite{SorbomFED2015, WhyteJFE2016, MumgaardAPS2017}) \rev{and stellarator \cite{BrombergFST2011, Paz-SoldanJPP2020} fusion communities (see also the recent review \cite{BruzzoneNF2018})}. As a first step towards the demonstration of the high-field path to commercial fusion energy, researchers have set off to demonstrate the technological feasibility of high-field HTS magnets for its later use in small-scale proof-of-principles experimental devices \cite{CreelyJPP2020}.
Since both the strength of the magnetic field and device size positively affect the energy confinement time, a trade-off between them is possible. Reference \cite{ZohmPTRSA2019} concludes that the ability to produce stronger fields in tokamak configurations (of the order of 10 T on-axis) allows to reduce the size ($\sim$5 m major radius), and potentially also the cost, of a fusion reactor while maintaining sufficient confinement for ignition. In doing so, an obvious problem arises: a smaller wall surface leads to larger neutron wall loads and a faster degradation of in-vessel components. This requires not only alternative neutron stopping and tritium breeding approaches, like those based on molten-salt liquid blankets, but also \rev{developments in maintenance schemes (e.g. demountable coils)} that reduce the reactor down-time \cite{WhyteJFE2016}. {The increase in field and current density on the coils yields larger stresses on the structural components, which can pose important engineering challenges.}
Furthermore, the unmitigated parallel heat flux towards the divertor is also expected to grow as the magnetic field increases (lowering the power flux perpendicular decay length) and the size is reduced (lowering the linear dimension of the strike lines).
But reducing the linear scale of a stellarator reactor carries additional complications besides those related to higher wall fluxes. The radial extent of the tritium breeding blanket cannot be downscaled, so that the magnetic configuration of smaller devices need to be produced by coils that are proportionally further away from the last closed magnetic surface. Flux surface shaping and tailoring of the magnetic field spectrum is consequently more challenging and leads to strongly shaped coil designs, since the higher-order modes of the $B$-field decay quickly when moving away from the coils \cite{KuNF2010, LandremanPoP2016}. One is led to conclude that the exploitation of HTS for fusion applications must be accompanied by important developments in the design and engineering of a stellarator reactor. These developments could certainly build on the above-mentioned proof-of-principle solutions in the tokamak line. \rev{We note that stellarator reactors based on catalysed D-D fusion and low-temperature superconductors have also been studied \cite{SheffieldFST2016}.}
In the present work we address the questions: How would very high field HTS-based electromagnets impact the \emph{physics} design point of a deuterium-tritium stellarator reactor? What stellarator physics research directions could prepare us for making use of such a technological development? We make note that it is a premise of this study that the many engineering challenges associated with the fabrication of strongly shaped electromagnets and their operation at very-high fields are surmounted. Although we discuss some engineering aspects of higher-field, smaller devices, the focus of this study lies on the physics characteristics of the reactor design point.
The rest of the paper is organised as follows. In section \ref{sec:formulas} we present the device geometry, physics assumptions, and formulas that will be used in the one-dimensional study of stellarator reactor design points. A simplified zero-dimensional analysis is first conducted in section \ref{sec:0D}, where we derive the basic field strength / device size relation that stems from the empirical scaling of energy confinement time. We show how that relation leads to the approximate invariance of several important physical parameters for a family of reactor design points. Importantly, the scaling of the density operation point is introduced in this section (and further elaborated in \ref{sec:density}). In section \ref{sec:1D} we complement the findings of the previous section with 1D profile analysis based on prescribed profile shapes. The 0D invariances translate into several archetypal plasma profiles that are shown and discussed. The effect of magnetic field over-engineering on the physics design point is also discussed in this section. Our main conclusions are summarised in section \ref{sec:conclusions}.
\section{Identification of stellarator reactor design points in the $(B, R)$ plane}\label{sec:formulas}
In this section we present the basic parameters, formulas and hypotheses that are used to identify potential reactor design points in the ($B, R$) plane. Here $B$ is the characteristic on-axis magnetic field strength and $R$ is the major radius of the device. We will be working with a fixed device aspect ratio and magnetic geometry, so that $R$ characterises the device size and determines all other dimensions such as the minor radius of the plasma column, $a$, or the confinement volume $V_a = 2\pi^2Ra^2$.
Given certain values for $B$ and $R$, and other confinement-relevant configuration parameters, we would like to find plasma parameters that fulfil the simplified version of the power balance equation,
\begin{equation}\label{eq:PB}
P_h = \frac{W}{\tau_E}~,
\end{equation}
where $P_h$ is the net heating power, $\tau_E$ the energy confinement time and $W$ the total internal plasma energy given by the volume integral
\begin{equation}\label{eq:W}
W = \frac{3}{2}\sum_{s}\int d^3\mathbf{x}\; n_sT_s~,
\end{equation}
where the integration domain is the volume within the last closed magnetic surface.
The index $s$ labels the plasma species with particle density $n_s$ and temperature $T_s$. In the rest of the paper, we will assume that all species have the same temperature and therefore drop the species index and refer to the plasma temperature $T$. Plasma is composed of electrons, with number density $n_e$, deuterium and tritium ions and helium ash from the DT fusion reactions. We assume equal amounts of D and T and a 5\% He concentration, i.e. $n_\mathrm{He}=0.05n_e$, so that $n_\mathrm{D}=n_\mathrm{T}= 0.45n_e$\footnote{We will later show that this concentration of helium implies a ratio of He particle to energy confinement time of about 9 for the family of reactors discussed in section \ref{sec:1D}.}.
The energy confinement time, $\tau_E$ in equation \eqref{eq:PB} is assumed to be given by the ISS04 scaling \cite{YamadaNF2005},
\begin{equation}\label{eq:iss04}
\tau_E = f_c \times 0.134 a^{2.28}R^{0.64}P_h^{-0.61}\overline{n}_e^{0.54}B^{0.84}\iota_{2/3}^{0.41}\; ,
\end{equation}
where $f_c$ is the configuration factor, $\overline{n}_e$ is the line-averaged electron density and $\iota_{2/3}$ is the value of the rotational transform at the $\rho=2/3$ magnetic surface. The normalised radius $\rho$ as well as the radius $r$ will be used to label magnetic surfaces in this work, which relate to each other and to the volume within the flux surface, $V$, as $\rho = r/a = \sqrt{V/V_a}$. In terms of these variables, a volume integral, like the one in \eqref{eq:W}, is given by
\begin{equation*}
\int_0^{V_a} dV\; f(V) = 2V_a\int_0^1d\rho\; \rho f(\rho) = \frac{2V_a}{a^2}\int_0^adr\; r f(r)~,
\end{equation*}
whereas the line-averaged electron density in \eqref{eq:iss04} is given $\bar{n}_e = \int_0^1 n_e(\rho) d\rho$. Note that all physical magnitudes are assumed constant on flux surfaces in this study.
For estimating the net heating power, $P_h$, we consider the plasma self-heating by alpha particles, the power lost by Bremsstrahlung radiation and any auxiliary external heating power. Each of these terms is modelled with a $\rho$-dependent power density, $S_\alpha, S_B$ and $S_\mathrm{aux}$ that are given below:
\begin{equation}\label{eq:Salpha}
S_\alpha = E_\alpha n_D n_T \fsa{\sigma v}_{DT}(T)~,
\end{equation}
with $E_\alpha = 3.5$ MeV. The form of the DT reactivity, $\fsa{\sigma v}_{DT}$, can be seen in figure \ref{fig:DT}. It displays an approximate quadratic temperature dependence in the range of interest.
\begin{figure}
\includegraphics{DTreactivity}
\caption{Reactivity of the D-T fusion reaction as a function of the fuel temperature. The points (+) are calculated by convolving the Maxwellian distributions with the cross section $\sigma$ given by Duane's parametrization \cite{nrl}. The blue curve is the parametric fit to the reactivity given by Bosch and Hale \cite{BoschNF1992}. The red parabola approximates the reaction fairly well below 20 keV (the multiplying coefficient is chosen to match the reactivity at 10 keV).\label{fig:DT}}
\end{figure}
The local Bremsstrahlung radiation density is given by
\begin{equation}\label{eq:SB}
S_B= C_BZ_\mathrm{eff} n_e^2 \sqrt{T}~.
\end{equation}
If temperature and density are expressed in keV and $10^{20}$ m$^{-3}$ respectively, the numerical constant $C_B =5.35\times 10^{-3}$ gives the Bremsstrahlung radiation density in units of MW/m$^3$. The effective ion charge $Z_\mathrm{eff}$ is defined as $Z_\mathrm{eff} = \sum_{s\neq e}Z_s^2n_s/ \sum_{s\neq e}Z_s n_s$ and is equal to 1.1 with our assumed plasma composition. The auxiliary power is modelled by a centred Gaussian
\begin{equation}\label{eq:Saux}
S_\mathrm{aux} = \frac{P_\mathrm{aux}}{V_a w^2} e^{-\frac{\rho^2}{w^2}}~,
\end{equation}
where $P_\mathrm{aux}$ is the total auxiliary power and $w$ is the width of the power deposition profile. The volume integrals of the power densities \eqref{eq:Salpha} and \eqref{eq:SB} are termed $P_\alpha$ and $P_B$ respectively.
The net heating power is then defined here as $P_h = \max(P(\rho))$, where
\begin{equation}\label{eq:P}
P(\rho)= 2V_a\int_0^{\rho} d\rho\; \rho(k_\alpha S_\alpha(\rho) + S_\mathrm{aux}(\rho) - S_B(\rho))~.
\end{equation}
The $k_\alpha$ constant is the efficiency of the alpha heating. We take $k_\alpha = 0.9$, thereby allowing for a 10\% loss alpha particle energy due to fast ion transport\footnote{Lower orbit losses are achievable through optimization in time-independent fields \cite[see e.g. the recent study in][]{BaderJPP2019}, but turbulence and/or Alfvén modes can enhance fast particle transport. The specific choice of $k_\alpha$ is not important for the conclusions of this work, as far as it stays sufficiently close to 1.}.
For the purpose of this study, a reactor design point will be a point in the $(B, R)$ plane for which a device producing a fusion power $P_\mathrm{fus} = 3$ GW with a fusion gain $Q=P_\mathrm{fus}/P_\mathrm{aux} = 40$\footnote{Again, this arbitrary choice does not affect our conclusions. The reactor design point does not depend strongly on the desired $Q$, for large enough $Q$ values (see figure \ref{fig:scan}.j). {Lowering the required auxiliary power would improve the balance of plant but could make active burn control by density necessary.} \rev{Similarly, the lower recirculating power of stellarator power plants could justify targeting lower $P_\mathrm{fus}$ values (see e.g. \cite{MenardNF2011}). We choose to stay with 3 GW to allow a more straightforward comparison with other reactor design points in the stellarator literature.}} can be conceived, on the basis of the presently known scaling of the energy confinement and the density limit. The 3 GW total fusion power is typical of fusion reactor studies and results in an electrical power of about 1 GW, similar to that of present-day fission reactors. A power plant delivering substantially more than 1 GW to the network would have a correspondingly higher capital cost and a more difficult integration in a national power grid. In this work, the total fusion power, $P_\mathrm{fus}$, includes the contribution of the energetic alpha particles and neutrons resulting from the DT reactions, as well as the exothermic neutron-lithium reactions in the breeder \cite{FreidbergPoP2015, RubelJFE2019}. This results in a ratio of fusion to alpha power $P_\mathrm{fus}/P_\alpha = 6.4$.
\begin{table}
\begin{tabular}{p{0.57\columnwidth}cc}
Parameter name & Notation & Value\\
\hline
Aspect ratio & $A$ & 9.0\\
Rotational transform ($\rho =2/3$) & $\iota_{2/3}$ & 0.9\\
Configuration factor & $f_c$ & 1.0\\
Helium concentration & $n_\mathrm{He}/n_e$ & 0.05\\
Fuel concentration & $n_{D,T}/n_e$ & 0.45\\
Effective charge & $Z_\mathrm{eff}$ & 1.1\\
$\alpha$-heating efficiency & $k_\alpha$ & 0.9\\
Fusion-to-alpha power ratio & $P_\mathrm{fus}/P_\alpha$ & 6.4\\
\hline
\end{tabular}
\caption{Reactor design parameters kept fix in this study. Note that the fusion power accounts for alpha and neutron energy, as well as for exothermic breeder reactions \cite{FreidbergPoP2015, RubelJFE2019}. \label{tab:params}}
\end{table}
As mentioned at the beginning of this section, the magnetic geometry will be held fixed in this work. For the formulas presented before, this reduces to fixing the aspect ratio ($A=R/a$), $\iota_{2/3}$ and the configuration factor $f_c$. For the first two magnitudes, we choose values that are representative of the Helias Stellarator Reactor HSR4/18 ($A=9.0$, $\iota_{2/3}=0.9$). The configuration factor, $f_c$, that enters the evaluation of the energy confinement time equation \eqref{eq:iss04} will be set to 1.0 throughout this study. These and other choices are summarised in table \ref{tab:params}.
In the next sections we will be using these equations (or simplified versions of them) to find potential reactor design points and characterise their physical parameters. We note that, since the alpha heating power depends on the plasma temperature, equations \eqref{eq:PB} and \eqref{eq:W} need to be iterated until they converge to a consistent power balance.
\section{Simplified 0D analysis: $B(R)$ scaling of a reactor design point and its main plasma parameters}\label{sec:0D}
Before presenting the analysis of reactor design points based on the formulas introduced in the previous section, we carry out next a simplified 0D analysis that will guide intuition and help understand those results. We will identify the $B(R)$ dependence of a reactor design point and the ensuing dependency of the main physics dimensionless parameters on $B$ (or $R$). In order to do that, we will introduce an \emph{ad hoc} scaling of the plasma density that will also be used later in this article.
For the 0D analysis of this section, we rewrite equation \eqref{eq:W} using $\sum_s n_s \approx 2n_e$ and defining $\fsa{n_eT}_V = \frac{1}{V_a}\int_0^{V_a}dV (n_eT)$, to get
\begin{equation}\label{eq:PB0D}
\fsa{n_eT}_V \approx \frac{P_h\tau_E}{3V_a}~.
\end{equation}
A 0D plasma beta is then defined as
\begin{equation}
\beta_0 = \frac{2\fsa{n_eT}_V}{B^2/2\mu_0}~,
\end{equation}
whereas a characteristic temperature is obtained from the relation
\begin{equation}\label{eq:T0D}
T_{0} = \frac{\fsa{n_eT}_V}{\overline{n}_e}~,
\end{equation}
which is assumed to be close to the volume-averaged temperature. This temperature is used to estimate the alpha and the Bremsstrahlung power. In equation \eqref{eq:PB0D} we include the alpha and auxiliary heating in $P_h$, but otherwise neglect Bremsstrahlung radiation losses\footnote{In a 0D treatment like the one used in this section, the use of an average temperature like \eqref{eq:T0D} leads to an overestimation of $P_B/P_\alpha$.}.
To proceed with the analysis one needs to determine how line-averaged electron density, $\overline{n}_e$, scales with $R$ and $B$. In this work we choose density to scale like
\begin{equation}\label{eq:density}
\overline{n}_e [\mathrm{m}^{-3}] = 1.04\times 10^{19} (B[\mathrm{T}])^2~,
\end{equation}
where the pre-factor is chosen to match the HSR4/18 density ($2.6\times 10^{20}$m$^{-3}$ at 5T). This choice, which fundamentally determines the conclusions of this article, is consistent with the increase of the cut-off densities of the electron-cyclotron resonance heating an it results in a constant ratio of density to the critical density, $\overline{n}_e/\overline{n}_{ec}$, along the lines constant fusion power and gain in the $(B, R)$ plane. Here $\overline{n}_{ec}$ is the line-averaged critical electron density determined by a Sudo-like radiative limit. Further elaboration of this choice is given in \ref{sec:density}.
The result of the 0D $(B, R)$ scan is shown in figure \ref{fig:scan}.
%
\begin{figure*}
\includegraphics{Reactors_3GWplant}
\caption{Magnetic field $B$ and device major radius $R$ scan of several physics and engineering parameters. The line-averaged electron density is initially set to the value given by \eqref{eq:density} and the auxiliary power is determined to give a total fusion power of 3GW\rev{, shown in subfigure (g)}. For small, low-field devices this condition cannot be met \rev{(points marked with 0GW in (g))}. For the $Q=\infty$ part of the plane (where $\ensuremath{P_\mathrm{aux}}{} = 0$) the density is lowered to meet the 3GW target. Marked in red is the $Q=40$ line that can be considered for design points of stellarator reactors. We note that the 1D study conducted later in the paper will lower this curve by about 15\%.\label{fig:scan}}
\end{figure*}
The plots in this figure are made as follows: with the density initially set to \eqref{eq:density} we proceed, for each $(B, R)$ pair, to find the plasma temperature required to meet the $P_\mathrm{fus} =3$GW target (figure \ref{fig:scan}.g) together with the auxiliary power (figure \ref{fig:scan}.h) that is needed to get to that temperature, according to the $\tau_E$ scaling \eqref{eq:iss04}. For those points for which this process leads to $\ensuremath{P_\mathrm{aux}}{}<0$, the density is lowered until $P_\mathrm{fus} = 3$GW and $\ensuremath{P_\mathrm{aux}}{} = 0$ (this corresponds to the $Q=\infty$ part of the plane). The resulting $(n_e, T)$ pairs are shown in the plots \ref{fig:scan}.a and \ref{fig:scan}.b. The rest of the physical and engineering parameters (\ref{fig:scan}.c to \ref{fig:scan}.l) are derived from them and the $(B,R)$ values. The definitions of the normalised tritium gyro-radius and collisionality (figures \ref{fig:scan}.d and \ref{fig:scan}.e) are standard and are presented in \ref{sec:rhoandnu} for convenience.
The consequences of increasing $B$ at a fixed device size while staying with a 3GW fusion target can be seen in figure \ref{fig:scan}: the fusion gain increases rapidly reaching $Q=\infty$. Beyond this point the density operation point needs to be lowered to moderate the increase in the confinement time and reduce the fusion reaction rate. Temperature needs to be increased, which reduces the collisionality. The plasma beta is lowered by the increase in the field strength. While the second of these reductions alleviates the MHD equilibrium and stability requirements on the magnetic configuration, the reduction of density and collisionality makes it more difficult to confine fast and thermal particles in the three dimensional configuration. In consequence we conclude that, unless the reduction of $\beta_0$ design point is a strong requirement, the optimum location of design points for the stellarator reactor in the $B,R$ plane are those just below the $Q=\infty$ line. The red line shown in all plots in figure \ref{fig:scan} are the $Q=40$ $(B, R)$ pairs, which are considered potential reactor design points in this work\footnote{We note that the 1D study conducted later in the paper will shift the $Q=40$ line towards lower fields / smaller device size. In section \ref{sec:Bovereng} we will further discuss the consequences of stepping out of this line.}.
It is apparent in this figure that points along constant-$Q$ lines (or $Q$-lines) feature a very similar temperature $T_0$ (figure \ref{fig:scan}.b). This is a consequence of the $n\sim B^2$ scaling\footnote{Since all densities are related to each other by constant proportionality factors in this work, we will sometimes use $n$, without subindex, when discussing scalings with either electron or ion densities.}, for the 0D temperature \eqref{eq:T0D} scales as $T_0\sim \tau_E/nV_a$ ($P_h$ held constant). According to \eqref{eq:iss04}, this gives a weak
\begin{equation}\label{eq:Tconstant}
T_0\sim (RB)^{-0.08}
\end{equation}
dependence. Together with a constant fusion power, $P_\mathrm{fus} \propto n^2V_a\fsa{\sigma v}_{DT}(T)$, this results in the $B(R)$ relation,
\begin{equation}\label{eq:BRrelation}
B \sim R^{-3/4}~, ~(\textrm{constant } P_\mathrm{fus} \textrm{ and } Q)~,
\end{equation}
where the weak dependence \eqref{eq:Tconstant} has been neglected. Note that the aspect ratio is assumed constant throughout this article, so that $V_a\sim R^3$. The $Q=40$ line in the plots in figure \ref{fig:scan} closely follows this $B(R)$ dependence.
The scaling \eqref{eq:BRrelation} can be understood as the maximal reduction in device size made possible by an increase in the magnetic field strength. That is, given a reactor design point $(R_0, B_0)$, the ability to generate a larger field $B_1$ allows to reduce the relative reactor size as $R_1/R_0 = (B_0/B_1)^{4/3}$. Further reducing the device size would require to increase the $\overline{n}_e/\overline{n}_{ec}$ ratio to recover the $P_\mathrm{fus} = 3$ GW, $Q=40$ target. It can be shown that the scalings \eqref{eq:Tconstant} and \eqref{eq:BRrelation} stem from the approximate gyro-Bohm dependence of the ISS04 energy confinement time \eqref{eq:iss04}, together with the $n\sim B^2$ scaling used here. In fact, it is straightforward to check that balancing a gyro-Bohm diffusive heat flux, $\Gamma_Q \propto n\chi_{\mathrm{gB}}dT/dr$ (see \ref{sec:chi}), against a heating power divided by the flux-surface area $\sim P_h/R^2$ yields a constant temperature, independent of $B$ and $R$, when the heating power and the normalised temperature scale length are kept constant. The $B(R)$ scaling \eqref{eq:BRrelation} then follows from imposing a constant alpha power with a fixed temperature.
Several other important physics parameters are kept almost constant along the $Q$-lines in figure \ref{fig:scan}. The Bremsstrahlung power has a dependence similar to that of the alpha power, $\ensuremath{P_\mathrm{B}}{}\sim n^2V_af(T_0)$, where $f(T_0)\sim \sqrt{T_0}$ , and is similarly constant along the $Q$-lines. Plasma beta, $\beta_0 \sim nT_0/B^2$, is approximately constant along the $Q$-lines, whereas triton collisionality ($\nu_T^*\sim nR/T^2\sim R^{-1/2}$) and gyroradius ($\rho_T
^*\sim 1/RB\sim R^{-1/4}$) vary only slowly with $R$ (or $B$). The combination $(\nu_T^*)^{-1}(\rho_T^*)^2$ is approximately constant along the $Q$-lines. On the contrary, at least two engineering parameters, the neutron wall loading (NWL) and the magnetic energy (figures \ref{fig:scan}.k and \ref{fig:scan}.l), do show dependencies on the device size. The NWL decreases inversely to the first wall surface area $\sim R^2$ when the neutron power is kept constant, as in this scan. The values shown in the first of these figures have been calculated assuming an average plasma-wall distance required for hosting the diverting magnetic structure equal to 20\% of the plasma minor radius $a$. We defer a longer discussion about the implications of these neutron fluxes and other engineering aspects of high-field devices to section \ref{sec:NWL}. The vacuum magnetic energy is estimated by $\mathcal{E}_B = V_c B^2/2\mu_0$, where the volume enclosed by the coil set of average radii $c$ is approximated by $V_c=2\pi^2Rc^2$. $\mathcal{E}_B$ decreases for smaller, higher-field reactors somewhat slower than $\sim R^3B^2\sim R^{3/2}$ along the $Q=40$ line, since the thickness of the neutron shield and breading blanket is kept equal to 1.3 m in the scan\footnote{\rev{It should be noted that the simplistic estimation of the average coil radius, $c = 1.3 \mathrm{m} + 1.2a$, results in a considerably smaller $c$ and total magnetic energy $\mathcal{E}_B$ than those quoted in \cite{BeidlerNF2001} for $a = 2$ m. Missing a specific MHD equilibrium and coil design, the values and tendencies shown in figure \ref{fig:scan}.l should be considered indicative.}}.
\section{1D analysis of reactor design points with prescribed $(n, T)$ profiles}\label{sec:1D}
The 0D analysis presented in the previous section allowed to identify dependencies of the main design parameters that are implied by the ISS04 energy confinement time. We found that a quadratic scaling of the density, $n\sim B^2$ leads to a relation $B\sim R^{-3/4}$ of the size and field strength of stellarator reactors with the same fusion energy and gain. This, in turn, leads to the invariance of several physics parameters for a family of reactor design points. However, the precise identification of a design point requires considering profile effects, for most of the fusion reactions are produced in the hottest central part of the plasma column. In this section we will carry out a 1D study of reactor design points and show how the invariances derived in the 0D analysis lead to archetypal profiles for several plasma parameters. Furthermore, we will show that the reactor operation map, $P_\mathrm{fus}(P_\mathrm{aux}, n/n_\mathrm{DP})$, is also shared by the family of stellarator reactors. Here $n_\mathrm{DP}$ is the density at the design point.
The formulas that will be employed for the 1D analysis were presented in section \ref{sec:formulas}, and involve plasma density and temperature profiles. To date, no validated first-principle transport model has been developed that can be used to predict these profiles in a stellarator reactor confidently. Neoclassical transport theory and codes are well established and have been tested, often positively, in 3D magnetic configurations. Neoclassical heat fluxes provide an irreducible transport, found to be generally smaller than the experimental fluxes \cite{DinklageNF2013, BozhenkovNF2020}. Turbulence is thought to cause the additional heat transport, the computation and validation of which is an active subject of current research \cite{BarnesJCP2019, MauerJCP2020, XanthopoulosPRL2020}. An empirical parametrization of turbulent fluxes based on experimental data was used in \cite{WarmerFST2015}. The analysis presented here will be conducted with prescribed shapes of density and temperature profiles. The line-averaged density is given by \eqref{eq:density}, whereas the temperature profile is scaled to provide consistency with the power balance \eqref{eq:PB} and the ISS04 energy confinement time \eqref{eq:iss04}. Details on the form of these profiles are given in \ref{sec:profiles}. We note that the shape of the temperature profile is chosen to depend on the particle and heating power densities with a constant-$\chi$ ansatz. The effect of varying the density profile shape on the reactor design point will be briefly inspected in section \ref{sec:shapes}.
\begin{figure*}
\includegraphics{Reactor18mQ40}
\caption{\label{fig:HSR4/18}Plasma profiles for an $R=18$m, $P_\mathrm{fus} = 3$GW, $Q=40$ stellarator reactor. The average values obtained are very close to those of the HSR4/18 (c.f. \cite{BeidlerNF2001}).}
\end{figure*}
An example of the $n$ and $T$ profiles is shown in figure \ref{fig:HSR4/18}.a and b, which approximates the design point of the HSR4/18 device with $R=18$ m, $B=5$ T and $P_\mathrm{fus} = 3$ GW, $Q=40$. Other plasma parameters, like the volume-averaged plasma beta $\fsa{\beta}$, the energy confinement time $\tau_E$, or the total power radiated through Bremsstrahlung $P_B$, closely resemble design values reported in \cite{BeidlerNF2001}. Note that the scaling of the density \eqref{eq:density} is indeed chosen to match the HRS4/18 line-averaged density at its design field of 5 T.
The resulting alpha heating and Bremsstrahlung power density profiles are shown in figure \ref{fig:HSR4/18}.c, together with the auxiliary heating density of width $w=0.36$ (see equation \eqref{eq:Saux}). Figure \ref{fig:HSR4/18}.d shows the plasma and fast particle beta, $\beta_\alpha$, the tritium burnup profile and total burnup fraction, $f_\mathrm{burnup}$. The classical slowing-down distribution function of alpha particles is used to obtain an estimate for their pressure and beta (see \ref{sec:betaalpha}), which is important to determine the properties of the Alfvén spectrum and the Alfvén-induced energetic particle transport (see e.g. \cite{HeidbrinkPoP2008, } for a review). Recently, a potential stabilising effect of the fast particle pressure on the ion temperature-gradient driven turbulence has been put forward in \cite{DiSienaPRL2020}.
The tritium burnup fraction is defined as {the number of DT reactions in the plasma per second over the number of injected tritium atoms per second} (see \ref{sec:fburnup}). The inverse of the burnup fraction can be understood as the average number of times that a tritium atom needs to be cycled through the vacuum vessel before it undergoes a DT reaction. Each of the times that a tritium atom is cycled, there is a certain probability that it be lost from the fuel cycle. The burnup fraction is therefore an important factor to determine the overall tritium breeding ratio. It is, nevertheless, subject to considerable uncertainty. Nishikawa sets a $f_\mathrm{burnup}>0.5$\% requirement to enable tritium self-sufficiency \cite{NishikawaFST2010, NishikawaFST2011}, upon consideration of production of tritium in the blanket system and several losses due to trapping, permeation and decay in the vacuum vessel and fuelling and storage circuits. The same references quote values around 3 - 4\% to ease the requirements on the efficiency of tritium recovery and breeding. A recent review \cite{AbdouNF2021} concludes that tritium self-sufficiency can be achieved with confidence with $f_\mathrm{burnup}>2$\% together with a plant availability factor greater than 50\%. In figure \ref{fig:HSR4/18}.d we estimate the burnup fraction using several, partially compensating approximations; namely, zero recycling, 100\% fuelling efficiency and a equal tritium particle and energy confinement times. This results in an estimated $f_\mathrm{burnup} = 1.2\%$, which falls in the right range\footnote{{Since there is a large uncertainty in this estimate, the quantitative burnup fractions presented in this work should be de-emphasised. More important for the discussion is the fact that, under our assumptions, this fraction does not depend on the reactor size (see below).}}.
Finally, fundamental quantities related to the neoclassical and turbulent transport are plotted in figure \ref{fig:HSR4/18}.e (normalised tritium gyro-radius and collisionality profiles) and figure \ref{fig:HSR4/18}.f (temperature and density scale lengths and the thermal diffusivity at mid radius, $\chi_{0.5}$; see \ref{sec:chi}).
The $(B,R)$ position of the design point with profiles shown in figure \ref{fig:HSR4/18}, is plotted in figure \ref{fig:Q40} together with those of smaller, higher-field devices and a larger $R=22$ m device.
\begin{figure}
\includegraphics{Q40lines}
\caption{Design point of stellrator reactors in the $B,R$ plane. The 0D analysis conducted in this article overestimates the required field by using an average temperature. The $B\sim R^{-3/4}$ scaling is plotted alongside the $B,R$ pairs obtained from the 1D analysis for various devices sizes $R=$ 7, 9, 13.5, 18 and 22 m (empty squares). Design parameters for the last four of these radii are listed in table \ref{tab:Q40}. These design points share profiles of several parameters shown in figure \ref{fig:invariant} and the operation map shown in figure \ref{fig:OPmap}. The effect of reducing the configuration factor $f_c$ in \eqref{eq:iss04} on the required magnetic field is illustrated for $R=18$m (full squares; note that the line-averaged density for the design points with $f_c = 0.6$ and $0.8$ is still scaled according to $n\sim B^2$, which leads to an increasing ratio of $\overline{n}_e/\overline{n}_{ec}$ for those points).\label{fig:Q40}}
\end{figure}
These are calculated similarly, keeping the auxiliary power constant at $P_\mathrm{aux} = 75$ MW and varying the magnetic field $B$ until the fusion power target, $P_\mathrm{fus} = 3$ GW, is met. As found in the previous section, the resulting design points fall very close to the $B\sim R^{-3/4}$ curve. Table \ref{tab:Q40} summarises the characteristics of those design points. We note that several global magnitudes not listed in that table are nearly constant for the five design points shown in the table (as well as for the other points along the $Q=40$ line in figure \ref{fig:Q40}) as anticipated in section \ref{sec:0D}. This includes the volume averaged plasma $\beta$, the net heating $P_h$ and Bremsstrahlung power $P_B$, the fusion burnup fraction $f_\mathrm{burnup}$ and the ratio of helium particle confinement time $\tau_\mathrm{He}$ to the energy confinement time (see caption in table \ref{tab:Q40}). Furthermore, we will show next that the constancy holds as well for the profiles of certain physics parameters.
\begin{table*}
\begin{tabular}{r r r r r r r r r r}
$R$(m) & $B$(T) & $\overline{n}_e (10^{20}\mathrm{m}^3)$ & $\tau_E$ (s) & $\chi_{0.5}$ (m$^2$/s) & $\nu_T^*(0)
(10^{-3})$ & W (MJ) & NWL (MW/m$^2$)\\
\hline
7.00 & 10.02 & 10.43 & 0.55 & 0.30 & 2.18 & 223 & 7.32\\
9.00 & 8.32 & 7.20 & 0.80 & 0.34 & 1.95 & 325 & 4.43\\
13.50 & 6.18 & 3.97 & 1.48 & 0.41 & 1.63 & 598 & 1.97\\
18.00 & 5.00 & 2.60 & 2.29 & 0.47 & 1.44 & 922 & 1.11\\
22.00 & 4.31 & 1.93 & 3.10 & 0.52 & 1.32 & 1247 & 0.74\\
\hline
\end{tabular}
\caption{Parameters of the design points marked in fig.\ref{fig:Q40} with red squares. The parameters $\ensuremath{P_\mathrm{fus}}{} =3$GW ($P_\alpha = 469$MW) and $Q=40$ ($\ensuremath{P_\mathrm{aux}}{}$ = 75MW) are imposed by our definition of reactor design point. For all points in the table $P_h = 416.5 \pm 1.5$ MW, $P_B = 119.6 \pm 2.9$ MW, $\fsa{T}_V = 5.27 \pm 0.07$ keV, $\fsa{\beta}_V = 4.28 \pm 0.06$\%, $\beta_\alpha(0)/\beta(0) = 0.25 \pm 0.01$, $f_\mathrm{burnup} = 1.21 \pm 0.01$, $\tau_\mathrm{He}/\tau_E = 9.06\pm0.09$. {The helium particle confinement time is estimated using the fusion reactions as the only source.}\label{tab:Q40}}
\end{table*}
At this point we note that, as already stated in \cite{WarmerFST2015}, the synchrotron radiation is not a relevant loss of power for the usual helias reactor design points. This is also the case for the higher field points considered here. In fact the vacuum synchrotron emission scales as $\sim V_aB^2n_eT_e$, which is constant along the $Q=40$ line in figure \ref{fig:Q40}. The plasma opacity factor depends on the density, field and plasma size as $\sim an_e/B$ (see e.g. \cite{Tamor1988}), which slightly increases with the size of the reactor, $\sim R^{1/4}$, along that line. Also note that, in general, the stellarator design points feature higher densities and lower temperatures compared to the tokamak's, which result in substantially lower synchrotron radiation losses for a similar volume and field strength.
\subsection{Invariant profiles and common optimization requirements}
There are a number of important plasma profiles that are approximately the same for all the reactor design points with $P_\mathrm{fus}= 3$ GW and $Q=40$ (dashed line in figure \ref{fig:Q40}). This was anticipated in the 0D analysis of section \ref{sec:0D} and is recovered here using a 1D analysis (see figure \ref{fig:invariant}). Shown in this figure are profiles for selected plasma parameters computed for the five design points listed in table \ref{tab:Q40}. Line thicknesses show the maximum differences in the profiles obtained from the 1D power balance described in section \ref{sec:1D}.
\begin{figure*}
\includegraphics{InvariantProfiles}
\caption{Invariant profiles for the HSR4/18 family (dashed line in figure \ref{fig:Q40}). Line thickness corresponds to the profile variation for the design points listed in table \ref{tab:Q40}.\label{fig:invariant}}
\end{figure*}
These can be considered archetypal profiles for the reactor family that includes the HSR4/18. It should be noted that details of these profiles do depend on specific choices, like the model density and temperature profiles. Their constancy, however, depends solely on the assumption that the same profile shape can be used to represent density for all reactor sizes and on the ISS04 and $n\sim B^2$ scalings. The temperature profile (figure \ref{fig:invariant}.a) is found to be almost constant, and so is the temperature scale-length $a/L_T = T^{-1}dT/d\rho$ ($a/L_n$ is also constant by assumption). Both the thermal and alpha-particle $\beta$ profiles (figure \ref{fig:invariant}.b) and the diffusivity in gyro-Bohm units (\ref{fig:invariant}.c) are nearly invariant. Figure \ref{fig:invariant}.d shows the combination $(\rho^*_T)^2/\nu^*_T$ of normalised gyro-radius $\rho^*_T$ and collisionality $\nu^*_T$. The fact that these profiles are independent of the reactor size (or field) as indicated above, leads to the conclusion that similar levels of optimization of the neoclassical and turbulent transport, MHD equilibrium and stability and fast ion confinement are to be imposed on the magnetic configuration for these design points. We further elaborate on each of these points next.
\subsubsection{MHD optimization.}
That similar equilibrium currents, $\beta$ limits, and MHD stability properties are required is implied by the invariance of the $\beta(\rho)$ profile in figure \ref{fig:invariant}. Modifications to the vacuum magnetic field produced by the plasma current densities $\mathbf{j}$ are given by the equation
\begin{equation}\label{eq:rotB}
\nabla\times \delta\mathbf{B} = \mu_0\mathbf{j}~.
\end{equation}
The equilibrium plasma currents include the diamagnetic, parallel Pfirsch-Schlüter and parallel bootstrap current, which can be written as
\begin{equation}\label{eq:j}
\mathbf{j} = \frac{dp}{d\rho}\left(\frac{\mathbf{B}\times\nabla\rho}{|\mathbf{B}|^2} + h\mathbf{B}\right) + \frac{\fsa{\mathbf{j}\cdot\mathbf{B}}}{\fsa{|\mathbf{B}|^2}}\mathbf{B}~,
\end{equation}
where $h$ is a function of the space and the magnitude of the magnetic field, defined such that it cancels the total divergence of the term in brackets. This first term, containing the diamagnetic and Pfirsch-Schlüter currents, scales as $\sim (d\beta/d\rho)B/R$ which, plugged into \eqref{eq:rotB}, leads to $\delta B/B\sim(d\beta/d\rho)$.
In the absence of external current drive, the second term in \eqref{eq:j} is given by the bootstrap current. While an explicit expression for the bootstrap current, valid for any collisionality regime, does not exist, its asymptotic form in the $1/\nu$ regime \cite{HelanderJPP2017} displays a scaling $\fsa{\mathbf{j}\cdot\mathbf{B}}\sim nT/R$, which, similarly leads to a relative field modification, $\delta B/B\sim\beta$, independent of the device size. A similar scaling follows from an earlier work by Boozer \cite{BoozerPoFB1990}. Note that the electrostatic potential profile, $\phi(\rho)$, which enters those expressions, is also expected to be nearly invariant, as the neoclassical ambipolarity implies $eT^{-1}(d\phi/d\rho)\sim d\log p/d\rho$, where $e$ is the elementary charge.
\subsubsection{Transport optimization.}\label{sec:transport}
The transport properties of the devices that share the invariant profiles in figure \ref{fig:invariant} should also bear important similarities.
For all of them, the total thermal diffusivity, $\chi$, must be a similarly small fraction ($\sim 10^{-2}-10^{-1}$) of the gyro-Bohm diffusivity, $\chi_{\mathrm{gB}}$, in the central part of the plasma, which will be shown to be consistent with the expected dependencies in the neoclassical and turbulent diffusivities. Note that this order of magnitude is imposed by the need to reach a sufficient core temperature to achieve the target fusion power of 3 GW and does not depend strongly on the profiles chosen in this study.
Neoclassical energy transport is deemed an important, possibly dominant, transport mechanism in high-temperature stellarator reactor cores. In 3D fields, it features a specific regime of high diffusivity that is inversely proportional the collisionality. In this so-called $1/\nu$ regime, the heat diffusivity of a plasma species $s$ is $\chi^s_{1/\nu}\propto (a^2v_{ts}/R)(\epsilon_\mathrm{eff}^{3/2}\rho^{*2}_s/\nu_s^*) = (a/R)(\epsilon_\mathrm{eff}^{3/2}/\nu_s^*)\chi_{\mathrm{gB}}^s$, where the proportionality constant is of order unity (see e.g. \cite{SeiwaldJCP2008}). The effective ripple, $\epsilon_\mathrm{eff}$, is a characteristic of the magnetic field structure, the minimisation of which is targeted by stellarator optimization. An order-of-magnitude figure for how small this coefficient needs to be, can be estimated by imposing $\chi \geq \chi_{1/\nu}$ for the tritium ions. Figure \ref{fig:invariant}.c shows that $\chi/\chi_{\mathrm{gB}}\sim 3\times 10^{-2}$ is necessary in the centre of the family of stellarator reactors. Together with the weakly size-dependent value of the central tritium collisionality in table \ref{tab:Q40} this conditions leads to a central $\epsilon_\mathrm{eff} \lesssim 1\%$. Values around or below this are characteristic of several existing stellarator configurations (see e.g. \cite{BeidlerNF2011} and references therein), including the HSR4/18 \cite{BeidlerNF2001}, new compact quasi-axisymmetric \cite{HennebergNF2019} and several W7-X configurations. In fact, the analysis conducted in \cite{WarmerFST2015} shows that neoclassical transport is compatible with fusion conditions for a scaled version of the high-mirror W7-X configuration. It should also be noted that the radial electric field moderates ion neoclassical losses at the very low central collisionalities, where other regimes like the $\sqrt{\nu}$ become increasingly important. This makes the $1/\nu$ estimates a worst-case scenario for the neoclassical transport channel.
The approximate invariance of scale lengths $a/L_T$, $a/L_n$ and $eT^{-1}(d\phi/d\rho)$, an assumed constant magnetic geometry and the only weak size-dependence of collisionality ($\sim R^{-1/2}$, see the end of section \ref{sec:0D}) and $\rho^*_T\sim R^{-1/4}$ leads to the conclusion that $\chi_\mathrm{neo}/\chi_{\mathrm{gB}}$ would be similar for all reactor design points. Nevertheless, it is well known that microturbulence enhances energy transport above neoclassical levels. In neoclassically optimized stellarators like W7-X, the turbulent component can even dominate the radial energy fluxes over the entire plasma volume \cite{BozhenkovNF2020}. Given a magnetic configuration, $\chi_\mathrm{tb}/\chi_{\rm \mathrm{gB}}$ depends on $a/L_T$, $a/L_n$,and the collisionality. In the family of stellarator reactors discussed in this paper, $a/L_T$, $a/L_n$ do not vary, whereas the collisionality varies only slightly. One is then similarly led to conclude that, $\chi_{\rm tb}/\chi_{\mathrm{gB}}$ would not vary significantly within this family\footnote{\rev{The neoclassical and turbulent diffusivities $\chi_\mathrm{neo}$ and $\chi_{\rm tb}$ are defined as the coefficients that relate the size of the
corresponding energy flux with the typical scale length of the temperature profile. Namely, $Q_\mathrm{neo}\sim \chi_\mathrm{neo} {nT} L_T^{-1}$ and $Q_\mathrm{tb}\sim \chi_\mathrm{tb} {nT} L_T^{-1}$.}}.
To conclude the discussion on transport, we note that the fact that $\chi_\mathrm{neo}/\chi_{\mathrm{gB}}$ and $\chi_\mathrm{tb}/\chi_{\mathrm{gB}}$ do not vary much within the reactor family is consistent with the approximate invariance of the required $\chi/\chi_{\mathrm{gB}}$ (figure \ref{fig:invariant}.c). This does not come as a surprise, for the ISS04 energy confinement time displays an approximate gyro-Bohm dependence that is also characteristic of both neoclassical and turbulent transport mechanisms\footnote{This is not to say that deviations in particular parametric dependencies of the neoclassical and turbulence diffusivities with respect to the gyro-Bohm diffusivity are excluded. The neoclassical 1/$\nu$ limit discussed before is an example of this. Collisional stabilization of turbulent trapped electron modes has been shown to possibly lead to an isotope mass dependence inverse to that of gyro-Bohm \cite{NakataPRL2017}. \rev{What is important from the previous discussion is that the other parameters that define the neoclassical and turbulent transport regimes, and might introduce deviations with respect to the gyro-Bohm diffusivity, are themselves nearly equal across the reactor family.}. }.
\subsubsection{Fast ion optimization.}
Collisionless fast ion losses in stellarator proceed in two time-scales. Trapped particles cause prompt losses, which have a very short characteristic time given by $v_M/a$, where $v_M$ is the characteristic size of the radial magnetic drift. Since this is much shorter than the collisional slowing-down time, the magnetic configuration needs to be designed to reduce prompt losses for any of the reactor sizes considered in figure \ref{fig:Q40}. On a longer time-scale, fast ion losses due to stochastic diffusion set in, which can be moderated by lowering the collisional slowing-down time. A diffusion coefficient for the stochastic losses was derived in \cite{BeidlerPoP2001} using an analytical representation of the magnetic field spectrum. Aside from size-independent geometric factor, the stochastic diffusion scales as $D_\alpha\sim R^2\Omega_\alpha(\rho^*_\alpha)^4$, where $\Omega_\alpha$ is the alpha particle cyclotron frequency. In terms of our size ($R$) and field strength ($B$) variables, one gets $D_\alpha\sim R^{-2}B^{-3}$, which reflects a reduction in the diffusivity of alpha particles for larger devices and stronger fields. To compare the importance of stochastic losses for the different design points, we use an estimate of the diffusive radial excursion in a slowing-down time , $\fsa{\Delta \rho} \sim a^{-1}\sqrt{D_\alpha \tau_s}$. For constant electron temperature, the slowing-down time $\tau_s$ is inversely proportional to the density (see equation \eqref{eq:slowing-down}). Using the density and $B(R)$ scaling (equations \eqref{eq:density} and \eqref{eq:BRrelation}) leads to $\fsa{\Delta \rho}\sim R^{-1/8}$. By this measure, stochastic diffusion of alpha particles is expected to be of very similar importance for the various reactor design points considered in figure \ref{fig:Q40} and table \ref{tab:Q40}.
Alpha particles can excite Alfvén modes which, in turn, can enhance their radial transport. The characteristics of this interaction and enhanced transport would be similar for the different reactor design points in the following sense: first, the population of alpha particles displays an invariant $\beta_\alpha$ profile shown in figure \ref{fig:invariant} \rev{(see also \ref{sec:betaalpha})}. Second, the Alfvén speed $v_A=B/\sqrt{\mu_0\sum_sm_sn_s}$, where $m_s$ is the mass of the species $s$, is constant according to the density scaling \eqref{eq:density}.
\subsection{Invariant operational map}
The family of reactors in figure \ref{fig:Q40} and table \ref{tab:Q40} share another important characteristic; namely, the operation map (i.e. the fusion power dependence on the auxiliary power $P_\mathrm{aux}$ and relative density variation with respect to the design point $n/n_{DP}$), shown in figure \ref{fig:OPmap}.
\begin{figure}
\includegraphics{PnscansAroundDP.pdf}
\caption{Operation map ($P_\mathrm{aux}, n/n_{DP}$) for the HSR4/18 reactor family (dashed line in figure \ref{fig:Q40}). Thermal stability refers to the sign of $P_h-W/\tau_E$ caused by a 1\% constant variation of the equilibrium temperature profile. The steep variation of fusion power in the low $P_\mathrm{aux}$, high $n/n_{DP}$ region is due to the appearance of a higher temperature solution of the $P_h = W/\tau_E$ power balance. Note that the fuel mix is kept constant in this scan.\label{fig:OPmap}}
\end{figure}
That this is indeed the case can be shown from the dependences $P_\mathrm{fus} \sim V_a n^2T^2$ and $nT\sim P_h\tau_E/V_a$, which leads to $P_\mathrm{fus}\sim R^{0.005}(n/n_{DP})^{1.08}P_h^{0.78}$. Since the alpha and Bremsstrahlung power within $P_h$ are approximately constant along the $Q$ lines, the operation map $P_\mathrm{fus}(P_\mathrm{aux}, n/n_{DP})$ (figure \ref{fig:OPmap}) is also approximately independent of the reactor size along the constant $Q$ curve (figure \ref{fig:Q40}).
In figure \ref{fig:OPmap} the thermal stability of the operation points is probed by varying the temperature profile by 1\% and looking at the sign of $P_h - W/\tau_E$, which is, by construction, equal to zero at each operation point for the equilibrium temperature and density profiles. If a positive (negative) increment in the temperature profile leads to a faster increase (decrease) of the heating power, $P_h$, compared to the transported power, $W/\tau_E$, then the operation point is labelled thermally unstable. It is important to note that the shape of the temperature perturbation can affect the sign of the resulting $P_h - W/\tau_E$. Points below the red curve in figure \ref{fig:OPmap} are at least unstable to a temperature perturbation like the one referred above. While operating a stellarator reactor in thermally unstable conditions might be possible, it would presumably require an active control of the burning point for a stable power output. It should also be noted that the way thermal stability is calculated assumes that the energy confinement scales like equation \eqref{eq:iss04} also ``locally'', but deviations from it (including confinement transitions) are observed in present individual devices.
\subsection{Effect of the profile shapes}\label{sec:shapes}
The shape of the density profile has been so far fixed to that shown in figure \ref{fig:invariant}, with a mild peaking given by $k_2=0.5$ in equation \eqref{eq:nemodel}. The shape of the temperature profile \eqref{eq:Tmodel} also depends on this choice. In a reactor, the density profile will be determined by the fuelling method and the particle transport characteristics. Peaked density profiles are thought to be beneficial for confinement in Wendelstein-7X \cite{BozhenkovNF2020}. In the absence of a core particle source, neoclassical thermodiffusion is however expected to lead to core particle depletion \cite{BeidlerPPCF2018}. Hollow density profiles have not been reported in W7-X so far, but are common in the LHD device.
For inspecting the effect of the profile shape on the operation point of a reactor we reproduce the operation map of figure \ref{fig:OPmap} for a hollow ($k_2 = -1$) and a more peaked ($k_2=5$) density profiles. This is shown in figure \ref{fig:peakings}, which shows that relatively small adjustments of the auxiliary power and density allow to recover the $P_\mathrm{fus}=3$ GW, $Q=40$ operation point. However, the operation landscape is substantially changed. The sensitivity of $P_\mathrm{fus}$ is greatly increased for the peaked density profile, such that the operation point lies in a thermally unstable region (in the sense described previously) and sits close to the jump to a higher temperature solution of the power balance equation. It needs to be noted that the operation map is calculated assuming that the D-T fuel mix can be kept constant. In practice, the accumulation of He ash would likely lead to the moderation of the reaction rate for the higher fusion powers.
\begin{SCfigure*}
\includegraphics{PnScans3Peaking}
\caption{Operation maps for density peaking factors $k_2 = 0.5, -1.0$ and $5.0$ (profiles are shown as insets in each figure) for the HSR4/18 reactor family. Colour curves are contours of constant $P_\mathrm{fus}$ with 1GW increments. Gray curves are contours of constant $Q$ with $\times 2$ increments. Circles mark the intersection of the $P_\mathrm{fus} = 3$ GW and $Q=$40 lines.\label{fig:peakings}}
\end{SCfigure*}
\subsection{Discussion on the neutron wall loads and breeding technology}\label{sec:NWL}
As discussed earlier around table \ref{tab:Q40}, smaller reactor devices inevitably suffer more intense neutron bombarding per unit area on the first wall and breeding blanket. To qualify the neutral wall load (NWL) numbers shown in that table we consider that, as a rule of thumb, a maximum NWL of 1.97 MW/m$^2$ (corresponding to a 13.5 m device) would translate into a damage rate of 19.7 dpa/fpy (displacements per atom in a full-power year) in steel and around $4\times10^{18}$ n/m$^2$/fpy neutron flux at the first wall. This could be still acceptable under the damage of the first wall, shielding of the other structures (Vacuum Vessel and Coils) and heat recovery points of view, as preliminary assessed in \cite{PalermoNF2021}, although some improvement on shielding and minor modification on maintenance scheme would be necessary.
Nevertheless, such aspects would be not easily manageable with higher NWL. For example, 4.43 MW/m$^2$ ($R=9$ m in table \ref{tab:Q40}) would lead to damage around 40 dpa/fpy and neutron fluxes of almost $8\times10^{18}$ n/m$^2$/fpy at the first wall. This would complicate the breeding blanket replacement, since the current Eurofer first wall material suggested to be used in stellarators power plants, as extrapolated from DEMO, is qualified up to 20 dpa for the first DEMO phase (1.57 fpy). Besides, the requirements to the shield components that would need to be very exigent in order to get viable values at the different coil structures On the other hand, the NWL of the largest devices considered in table \ref{tab:Q40}, 0.74 MW/m$^2$ would probably lead to low power deposited inside the coolants and accordingly low thermal efficiency. The fourth case with a NWL of 1.11 MW/m$^2$, which indeed corresponds to the HSR4/18 design point, could be the best engineering solution to explore since it seems to offer the best compromise between damage/shielding performances, maintenance schemes and heat recovery/thermal efficiency.
Apart from the considerations on the NWL, a substantial increase of the magnetic field can influence other technological aspects related with the in-vessel components. Firstly, important electromagnetic forces can be developed on the structures which form the breeding blanket \cite{MaioneFED2019}. It has been demonstrated that the mechanical behaviour of the blanket segments can be compromised. The breeding material can also be affected by the magnetic field. Usually, Li compounds are required to breed the tritium in order to maintain the reactor self-sufficiency. One of the most extended breeding materials is the PbLi eutectic alloy, which indeed is an excellent electric conductor. This means that, when moving inside the blankets, magnetohydrodynamic effects can appear \cite{UrgorriNF2018}. The main consequence is the increase on the pressure drop of the liquid metal, which has a critical impact on the plant electric efficiency. Moreover, some MHD effects can cause important reverse flows that can compromise the route of the effluent impacting the tritium permeation through the structures or the formation of He clusters. Another important point is corrosion of structural materials due to the interaction with the liquid metal. It has been demonstrated that the presence of an intense magnetic field enhances the corrosion phenomena on EUROFER samples \cite{CarmonaJNM2015}.
The conclusion of this discussion is that current breeding blanket technology and maintenance schemes can severely limit the viability of small-size, high-field reactors. The full exploitation of high temperature superconductors necessitates parallel technological developments on these fronts, that allow one to cope with the increased neutron fluxes. Continuous molten-salt breeders and demountable coil joints, for easier and faster maintenance, have been proposed in the tokamak high-field path to commercial fusion \cite{WhyteJFE2016}. Some of these ideas have also been considered in stellarator reactor studies \cite{SagaraNF2017}.
\subsection{Magnetic field over-engineering}\label{sec:Bovereng}
So far we have centred our analysis on design points on the $Q=40$ line in figure \ref{fig:Q40}. In the 0D analysis of section \ref{sec:0D} we argued that increasing the field over those design points, while keeping the $P_\mathrm{fus} = 3$ GW target, implied to operate at lower densities and higher temperatures. In a 3D device, operating at lower collisionality and longer alpha particle slowing-down times are disadvantages, as they impose higher standards on the neoclassical and fast particle confinement properties of the magnetic configuration. Since the energy confinement time in its usual form \eqref{eq:iss04} has a positive dependence on the line average density, lowering it is, in a sense, a waste of the confinement potential of the magnetic configurations. However, over-engineering the magnetic field strength in the sense just described brings in two positive consequences of large potential impact: it lowers the plasma $\beta$ and increases the tritium burnup fraction. In reduced-size tokamak reactor studies, high field operation is exploited to move away from operational boundaries \cite{WhyteJFE2016}. The increase of the magnetic field and the decrease in the operation density both act to lower the critical density fraction $\overline{n}_e/\overline{n}_{ec}$. Although it is not in the scope of this article to assess the viability of heat exhaust solutions, we note in passing that those same changes could complicate power handling in the scrape-off layer and divertor regions.
In this subsection we take a look at the consequences of over-engineering the magnetic field with the 1D prescribed-profile analysis utilised in this section. \rev{We inspect the effect of a 25\% increase in $B$ that for the case of a $R=13.5$ m device (see table \ref{tab:Q40}). The results are shown in figure \ref{fig:Bovereng}.
\begin{figure}
\includegraphics{BoverengDT.pdf}
\caption{Magnetic field over-engineering with respect to the $R=13.5$m design point shown in \ref{fig:Q40}. The design point profiles at $B=6.18$T are shown in the left column. \rev{Middle and right column profiles correspond to a 25\% increase of the magnetic field. In all cases $P_\mathrm{fus}=3$ GW and $P_\mathrm{aux}=75$ MW. In the middle column (with a $n_D/n_T$ density ratio of 1), the density needs to be lowered to moderate the fusion power. In the right column, the density is slightly increased, as allowed by the increase in the critical density (see equation \eqref{eq:nc}), and the $n_D/n_T$ ratio is adjusted instead to recover 3 GW fusion power. The labelling of the power densities in the middle row is the same as in figure \ref{fig:HSR4/18}.c.}\label{fig:Bovereng}}
\end{figure}
The profiles for the starting $R=13.5$ m reactor design point are shown in the left column. Those in the centre and right column correspond to the 25\% over-engineered field case. In the centre profiles, the ratio of deuterium to tritium densities, $n_D/n_T$, is kept at 1. In this situation, the increase in the fusion power due to the increase of $B$ can only be moderated by reducing the plasma density.} Temperature needs to change in roughly inverse proportion to keep up with the 3 GW fusion power. The higher temperatures broaden the alpha particle generation and heating profile and increase the burnup fraction. The thermal plasma $\beta$ decreases as $B^{-2}$. However, the alpha particle $\beta_\alpha$ slightly increases and broadens as a result of the longer slowing-down time and the broader profile of alpha particle production. These two effects would demand better fast particle confinement properties (in particular for the longer time-scale stochastic diffusion losses) in a more extended radial range for this over-engineered $B$-field case \rev{with $n_D/n_T = 1$}. It should also be noted that for helias-type configurations, a \emph{sufficient} plasma beta is required to reduce fast ion prompt losses, but the necessary minimum beta is subject to some adjustment at the stage of the optimization of the magnetic configuration. The last row of figure \ref{fig:Bovereng} shows the comparison of the normalised thermal diffusivity $\chi$ (see \ref{sec:chi}) with the proxy for the neoclassical $1/\nu$ diffusivity. The effective ripple is set to $\epsilon_\mathrm{eff} = 0.5\%$. These plots illustrate that further reductions of $\epsilon_\mathrm{eff}$ could be necessary to make the core profiles compatible with the neoclassical heat transport levels in the over-engineered case \rev{with $n_D/n_T = 1$}.
\rev{Instead of lowering the plasma density, in the right column of figure \ref{fig:Bovereng} the ratio of deuterium to tritium has been increased to moderate the fusion power. This is shown to result in a moderation of the negative consecuences refered to above, that stemmed from the necessary decrease of density and increase of temperature. While the reduction of the plasma $\beta$ is not as pronounced, the $\beta_\alpha$ is also reduced and the collisionality increases only slightly (bottom plot). The reduction of the tritium concentration increases the burnup faction three-fold with respect to the normal field case. We conclude that magnetic field over-engineering could be taken advantage of by reducing the tritium concentration, thereby allowing to reduce the plasma and alpha particle normalised pressures and increase the burnup fraction, without strongly reducing plasma collisionality.}
\rev{\subsection{Design points of stellarator reactors with constant neutron wall loading}
As mentioned before, the neutron wall loading is an important technological parameter. The breeding blanket and heat-to-electricity transformation technology determine an optimal range for the NWL. Therefore, one could be interested to search for reactor desing points of constant NWL rather than constant fusion power. Reducing the size of the reactor design point while keeping a constant NWL still gives favorable reduction of the ratio of GW per m$^3$ of reactor material, but the larger electro-mechanical stresses of smaller, higher-field devices could make the support structure bigger and more expensive than that implied by a simple $R^3$ scaling.
Referring to the operation map in figure \ref{fig:OPmap}, valid for all reactor design points in table \ref{tab:Q40}, one sees that reducing the auxiliary heating power and adapting the density, one can go down the $Q=40$ line to the fusion power compatible with a given reduced NWL. However, in doing so, one would require the reactor to operate at larger ratios of density over the critical density $\overline{n}_e/\overline{n}_{ec}$, since $\overline{n}_{ec}\sim P_h^{0.57}B^{0.34}R^{-1.25}$. In consequence, if $\overline{n}_e/\overline{n}_{ec}$ is to be kept constant while \emph{reducing} the fusion power at fixed device size, the magnetic field needs to be \emph{increased}. This is a consequence of the strong power dependence of the critical density characteristic of stellarators \cite{GreenwaldPPCF2002}. The resulting $(B, R)$ design points and some of their physics characteristics are shown in \ref{fig:constantNWL} an table \ref{tab:constantNWL} respectively. Besides the critical density fraction, the fusion gain is kept constant at $Q=40$ in those points. As shown in table \ref{tab:constantNWL}, smaller, higher-field devices with constant NWL (i.e. $P_\mathrm{fus} \sim R^{-2}$) display a rapidly decreasing plasma beta, whereas the central collisionality decreases more slowly. The plasma temperature varies only slightly in the explored range.
}
\begin{table*}
\begin{tabular}{r r r r r r r r r}
$R$(m) & $B$(T) & $\overline{n}_e (10^{20}\mathrm{m}^3)$ & $\langle T\rangle$ (keV) & $P_\mathrm{fus}$ (GW) & $\langle\beta \rangle$ (\%) & $\beta(0)/\beta_\alpha(0)$ & $\nu_T^*(0) (10^{-3})$ & $(\chi/\chi_\mathrm{gB})_{0.5}$\\
\hline
7.00 & 14.58 & 3.89 & 5.54 & 0.45 & 0.79 & 3.68 & 0.77 & 0.05\\
9.00 & 10.86 & 3.58 & 5.34 & 0.75 & 1.27 & 3.91 & 0.96 & 0.06\\
13.50 & 6.89 & 2.98 & 5.25 & 1.69 & 2.58 & 4.04 & 1.23 & 0.07\\
18.00 & 5.00 & 2.60 & 5.21 & 3.00 & 4.24 & 4.10 & 1.44 & 0.07\\
\hline
\end{tabular}
\caption{\rev{
Reactor design points of constant NWL = 1.11 MW/m$^2$, $\overline{n}_e/\overline{n}_{ec}$ and Q = 40 shown in figure \ref{fig:constantNWL}.\label{tab:constantNWL}}}
\end{table*}
\begin{figure}
\includegraphics{ConstantNWL.pdf}
\caption{\rev{Comparison of reactor design points of power (`$\square$') and NWL (`$\times$') similar to those of the HSR4/18. Note that Q and $\overline{n}_e/\overline{n}_{ec}$ are kept constant for all points. The $B\sim R^{-1.15}$scaling is obtained under the assumption of a nearly invariant temperature profile (see also table \ref{tab:constantNWL}).\label{fig:constantNWL}}}
\end{figure}
\rev{It should be emphasised that the present understanding of the Sudo-type density limit in stellarators does not allow to determine an absolute limit as is the case for tokamaks' Greenwald limit \cite{GreenwaldPPCF2002}. In this study we have used the published working point of the HSR4/18 stellarator to define the reference $\overline{n}_e/\overline{n}_{ec}$, but the possibility of operating at substantially higher densities cannot be ruled out. If this were to be the case, the magnetic field of the design points could be lowered with respect to those listed in table \ref{tab:constantNWL}. According to the present understanding (see e.g. \cite{FuchertNF2020} and references therein), the density limit in stellarators is connected to the properties and concentration of the edge radiator impurity. Gaining a more predictive capability for the determination of the critical density in reactor conditions is therefore linked with the validation of models of divertor and SOL/edge impurity transport and with the constraints on the concentration of impurities imposed by power exhaust.}
\section{Summary and conclusions}\label{sec:conclusions}
In this article we have shown that the design points of stellarators of different scale and field, but otherwise similar fusion power and gain, share many similarities under the assumption that the plasma density design point can be scaled as $n\sim B^2$. Archetypal profiles for temperature, plasma $\beta$, gradient scale lengths or thermal diffusivity in gyro-Bohm units, among others, characterise a family of varying-field stellarator reactors with maximally-reduced size, $R\sim B^{-4/3}$ (figure \ref{fig:invariant}). The suitability requirements on the magnetic configuration (e.g. good flux surfaces and MHD stability at high-$\beta$, reduced neoclassical transport in the low-collisionality regime or sufficient confinement of alpha particles) are therefore largely independent of the device size/field strength \rev{across the family of reactor design points}. In this sense, the qualification of optimization criteria in devices like the Wendelstein 7-X is still relevant in a future scenario in which electromagnets based on high temperature superconductors can be applied in stellarator reactors. Furthermore, we have shown that the operation landscape of fusion power and gain as function of the auxiliary power and relative density is also shared among the family of reactors (figure \ref{fig:OPmap}). In consequence, if the technological development of high-field-capable electromagnets progresses to make ten-Tesla-class stellarators accessible, a demonstration device of much reduced size ($<$10 m) is conceivable, that would qualify larger-scale devices in several meaningful ways.
High field and reduced size make conditions particularly harsh for the breeding structure, which calls for alternative approaches for breeder and wall maintenance \cite{WhyteJFE2016}. \rev{An additional complication for devising reduced size stellarator reactor stems from the need to increase the distance from the magnets to the plasma edge (relative to the plasma radius)} to allow for the irreducible space of the breeding blanket and neutron shield. \rev{This results in an increase in the complexity of the coils that would be required to generate an optimised magnetic configuration \cite{LandremanPoP2016}.} It is in this respect that higher-field operation would call for the search of optimised configurations with a larger relative distance between the current and control surfaces. The ability to relax some of the optimization targets, based on possibly over-simplified physics models, would make this line appear more promising.
If the tritium breeding technology or limitations in remote maintenance were to set the minimum size of a stellarator reactor, one could wish to increase the field strength while maintaining a certain device size. Magnetic field over-engineering is accompanied by reduced thermal $\beta$ and higher tritium burnup fraction, but conditions on thermal and alpha particle confinement become more stringent. \rev{However, these drawbacks can be largely mitigated with the adjustment of the D/T density ratio (figure \ref{fig:Bovereng}).}
\rev{As an alternative to the constant-$P_\mathrm{fus}$ scaling of reactor design points adopted throughout this article, the scaling at constant neutron wall loading (NWL) and gain has also been inspected in this article. The stellarator-specific critical density dependence on heating power results in a stronger field scaling ($R\sim B^{-1.15}$) when the reactor linear size is reduced at constant NWL and critical density fraction (figure \ref{fig:constantNWL}). The resulting reactor design points feature lower plasma $\beta$ and slightly lower collisionality compared to same size, higher $P_\mathrm{fus}$ design points.}
The conclusions of this work depend on a number of assumptions that have been presented but cannot be justified on solid physics grounds. The close invariance of plasma profiles and operation map (figures \ref{fig:invariant} and \ref{fig:OPmap}) depends on the use of prescribed shapes for the density and temperature profiles, that are assumed to be the same for all reactors. Other assumptions such as the fuel-mix composition and the alpha heating efficiency add quantitative uncertainty to the design points that have been considered. Our analysis critically relies on the ISS04 scaling of the energy confinement time. The assessment of any limitation to this scaling in high-beta, low-collisionality regimes should be regarded a stellarator research priority. Finally, it should not be left unnoticed that the viability of power exhaust has not been considered in any respect in our study. The development of stellarator scrape-off layer and divertor physics models that allow to incorporate exhaust conditions to reactor studies also appears to be a necessary step in the development of a credible stellarator reactor concept. {Significant departures with respect to the tokamak results \cite{ReinkeNF2017, SiccinioNF2017} could arise from the absence of a well defined threshold power across the separatrix and from differences in the scaling of the scrape-off layer width \cite{EichNF2013, NiemannNF2020}. \rev{This development bears also the importance of allowing to quantify the Sudo-type density limit that is compatible with exhaust conditions in stellarator reactor studies.}}
|
2,877,628,090,236 | arxiv | \section{Introduction}
\label{intro}
Many textbooks on Quantum Mechanics, like Feynman's lectures \cite{Feynman}, describe a thought-experiment based on the Young's double-slit interference setup, and realised with independent particles sent one at a time through the interferometer. The striking feature is that the image we have for interference is a wave passing simultaneously through both slits, incompatible with the image of a particle, which goes through either one or the other slit but not both. While the last statement comes naturally for objects primarily known as particles like electrons \cite{Tonomura}, neutrons \cite{Neutrons}, atoms \cite{Carnal} and molecules \cite{Arndt}, it can be questioned in the case of the ''LichtQuanten" introduced by Einstein \cite{Einstein} in 1905, since light is primarily described as a wave.
\begin{figure}
\includegraphics[width=8.5cm]{fig1.pdf}
\caption{Wavefront-splitting setup based on a Fresnel's biprism (FB). APDs are avalanche silicon photodiodes operating in photon counting regime. An intensified CCD camera (dash line) records interference fringes in the overlapping region of the two deviated wavefronts. When the CCD is removed, it is then possible to demonstrate the single photon behaviour by recording the time coincidences events between the two output channels of the interferometer.}
\label{general}
\end{figure}
One century later, we present a realization of this textbook experiment consisting in single-photon interference. Our experiment, depicted on Fig.\ref{general}, has several new striking features compared to previous works \cite{Grangier,Zeilinger,Benson,Jelezko}: (i) we use a clock-triggered single-photon source from a single emitting dipole, which is both conceptually and practically simple \cite{Kun,Brouri}, (ii) we use a wavefront-splitting interferometer based on a Fresnel's biprism, very close to the basic Young's double-slit scheme, (iii) we register the ''single-photon clicks'' in the interference plane using an intensified CCD camera which provides a real-time movie of the build-up of the single-photon fringes.
In early experiments \cite{Taylor}, the so-called single-photon regime was reached by attenuating light. But Poissonian photon number statistics in faint light pulses leads to the unwanted feature that more than one photon may be present between the source and the interference fringes observation plane. To demonstrate the single-photon behaviour, we also perform the experiment with two detectors sensitive to photons that follow either one or the other interference path. Evidence for single photon behaviour can then be obtained from the absence of time coincidence between detections in these two paths \cite{Grangier,Gerry}. Such a measurement also provides a ''which-path'' information complementary to the interference observation.
The paper is organized as follows. In section $2$, we describe the triggered single-photon source used for the experiment. Section 3 is dedicated to the demonstration of the particle-like behaviour. Finally, single photon interferences are presented in section 4.
\section{Triggered single-photon source using the photoluminescence of a single N-V
colour centre in a diamond nanocrystal }
\label{sec:1}
A lot of efforts have been put in the realization of single-photon-source (SPS) over the recent years. Since first proposal \cite{Imamoglu}, a wide variety of schemes have been worked out, based on the fluorescence from different kinds of emitters, such as molecules, atoms, colour centres and semiconductor structures \cite{General SPS}.
The clock-triggered single-photon source at the heart of our experiment, previously developed for quantum key distribution \cite{Brouri,Alleaume}, is based on the pulsed, optically excited photoluminescence of a single N-V colour centre in a diamond nanocrystal. This system, which consists in a substitionnal nitrogen atom (N) associated to a vacancy (V) in an adjacent lattice site of the diamond crystalline matrix (Fig.2-(a)), has shown an unsurpassed efficiency and photostability at room temperature \cite{Gruber,Kurtsiefer}.
In bulk diamond, the high index of refraction of the material
($n=2.4$) makes difficult to extract N-V colour centre fluorescence efficiently. One way to circumvent this problem is to use diamond nanocrystals, with a size much smaller than the radiated light wavelength, deposited on a microscope glass coverplate~\cite{Beveratos}. For such sub-wavelength size, refraction becomes irrelevant and the colour centre can
simply be assimilated to a point source radiating at the air-glass
interface. Samples were prepared by a procedure described in Ref.\cite{Beveratos}. N-V colour centre are created by irradiation of type Ib diamond powder with high-energy electrons followed by annealing at $800^{\circ}$C. Under well controlled irradiation dose, the N-V colour centre density
is small enough to allow independent
addressing of a single emitter using standard confocal microscopy, as depicted on Fig.\ref{confoc}-(b).
\begin{figure}[h!]
\centerline{\includegraphics[width=8.8cm]{fig2.pdf}}
\caption{(a)-The N-V colour centre consists in a substitionnal nitrogen atom (N), associated to a
vacancy (V) in an adjacent lattice site of the diamond crystalline matrix.
(b)-Confocal microscopy setup: pulsed excitation laser
beam at $\lambda = 532$ nm is tightly focused on the
diamond nanocrystals with a high numerical aperture (NA=$0.95$) microscope
objective. Fluorescence light emitted by the N-V colour centre
is collected by the same objective and then spectrally filtered from the
remaining pump light. Following standard
confocal detection scheme, collected light is focused onto a $100$
microns diameter pinhole.
The reflected part from a $50$/$50$ beamsplitter (BS) goes then into an imaging spectrograph,
while the transmitted part is detected by a silicon avalanche photodiode in photon counting regime and used for fluorescence raster scan as shown in (c). The central peak corresponds to the
photoluminescence of the single N-V colour centre used in the single-photon interference experiment. (d)-Fluorescence spectrum of this single N-V colour centre recorded with a back-illuminated cooled CCD matrix (2 minutes integration duration) placed in the image plane of the spectrograph. The peak fluorescence wavelength is at
670~nm. The two sharp lines (1) and (2) are respectively the two-phonon Raman
scattering line of the diamond matrix associated to the excitation wavelength
and the N-V centre zero phonon line at 637~nm which characterizes the negatively charged N-V colour centre photoluminescence.}
\label{confoc}
\end{figure}
Under pulsed excitation with a pulse duration shorter than the
radiative lifetime, a single dipole emits photons one by one \cite{Brouri_PRA00,De Martini}. As described in Ref.\cite{Kun}, we use a home-built pulsed laser at a wavelength of 532~nm with a 800~ps pulse duration to excite a single N-V colour centre. The 50~pJ energy per pulse is high enough to ensure efficient pumping of the defect centre in its excited level. The
repetition rate, synchronized on a stable external clock, can be adjusted
between 2 to 6 MHz so that successive fluorescent decays are well
separated in time from each other. Single photons are thus emitted by
the N-V colour centre at predetermined times within the accuracy of
its excited state lifetime, which is about 45~ns (See Fig.\ref{fig_biprisme_HBT}) for the centre used in
the experiment.
As a proof of the robustness of this single-photon source, note that all the following experiments have been realised with the same emitting N-V colour centre.
\section{"Which-path" experiment : particle-like behaviour}
Single photons emitted by the N-V colour centre are now sent at normal incidence onto a
Fresnel's biprism. Evidence for a particle-like behaviour can be obtained using the arrangement of Fig.\ref{general}. If light is really made of quanta, a single photon should either be deviated upwards or downwards, but should not be split by the biprism. In that case, no coincidences corresponding to joint photodetections on the two output beams should be observed. On the opposite, for a semi-classical model that describes light as a classical wave, the input wavefront will be split in two equal parts, leading to a non-zero probability of joint detection on the two photodetectors. Observation of zero coincidences, corresponding to an anticorrelation effect, would thus give evidence for a particle-like behaviour.
For a realistic experiment aimed at evidencing this property, we need to establish a criterion which enables us to discriminate between a particle-like behaviour and another one compatible with the semi-classical model for light. For that, we faithfully follow the approach introduced in Ref.\cite{Grangier} for interpreting single photon anticorrelation effect on the two output channels of a beamsplitter. We consider a run corresponding to $N_{T}$ trigger pulses applied to the emitter, with $N_{1}$ (resp. $N_{2}$) counts detected in path 1 (resp. 2) of the interferometer, and $N_{C}$ detected coincidences. It is straightforward to show that any semi-classical theory of light, in which light is treated as a wave and photodetectors are quantized, predicts that these numbers of counts should obey the inequality
\begin{equation}
\alpha = \frac{N_{C}N_{T}}{N_{1}N_{2}} \geq 1.
\end{equation}
Violation of this inequality thus gives a criterion which characterizes nonclassical behaviour. For a single photon wavepacket, perfect anticorrelation is predicted since the photon can only be detected once, leading to $\alpha=0$ in agreement with the intuitive image that a single photon cannot be detected simultaneously in the two paths of the interferometer. On the other hand, inequality (1) cannot be violated even with faint laser pulses. Indeed, in this case the number of photons in the pulse follows a Poisson law predicting $\alpha=1$. This value indicates that coincidences will then be observed preventing the particle-like behaviour to be evidenced. We measured the $\alpha$ correlation parameter for triggered single-photon pulses and for faint laser pulses. To establish a valid comparison between these two cases, all data have been taken for an identical mean number of detected photons per pulse, below $10^{-2}$.
\begin{figure}[t]
\centerline{\includegraphics[width=9cm]{fig3.pdf}}
\caption{Histogram of time intervals between consecutive photodetection events in the two paths of the wavefront-splitting interferometer. The pulsed excitation laser repetition period is 444~ns, and the total counting time was 148.15~s . Lines are exponential fits for each peak, corresponding to a radiative lifetime of $44.6\pm 0.8$~ns. Values written above each peak correspond to their respective area normalized to the corresponding value for poissonian photon number statistics. The strong reduction of coincidences around zero delay gives evidence for single-photon emission by the excited N-V colour centre and for its particle-like behaviour as the wavefront is split in two parts by the Fresnel's biprism.}
\label{fig_biprisme_HBT}
\end{figure}
\begin{table}
\caption{Measurements of the correlation parameter $\alpha$ associated to ten independant sets of $10^5$ photodetections (lasting about 5~s) registered for ({\it a}) faint laser pulses with a mean number of photons per pulse below $10^{-2}$ and ({\it b}) single-photon pulses emitted by the N-V colour centre.
As the laser emits coherent states of light, the number of photons in each pulse is given by Poissonian statistics, leading to a correlation parameter $\alpha = 1.00 \pm 0.06$, the precision being inferred from simple statistical analysis with a 95\% confidence interval. On the other hand, an anticorrelation effect, corresponding to $\alpha = 0.13 \pm 0.01<1$, is clearly observed for single-photon pulses propagating through the Fresnel's biprism.}
\label{tab:1}
\begin{center}
({\it a}) {\it Faint laser pulses}\\
\vspace{0.15cm}
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Counting time (s)
& $N_{1}$
& $N_{2}$
& $N_{C}$
& $\alpha$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
4.780 & 49448 & 50552 & 269 & 1.180\\
4.891 & 49451 & 50449 & 212 & 0.937\\
4.823 & 49204 & 50796 & 211 & 0.934\\
4.869 & 49489 & 50511 & 196 & 0.875\\
4.799 & 49377 & 50623 & 223 & 0.981\\
4.846 & 49211 & 50789 & 221 & 0.982\\
4.797 & 49042 & 50958 & 232 & 1.021\\
4.735 & 49492 & 50508 & 248 & 1.077\\
4.790 & 49505 & 50495 & 248 & 1.090\\
4.826 & 49229 & 50771 & 219 & 0.970\\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\vspace{0.35cm}
\begin{center}
({\it b}) {\it Single-photon pulses}\\
\vspace{0.15cm}
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Counting time (s)
& $N_{1}$
& $N_{2}$
& $N_{C}$
& $\alpha$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
5.138 & 49135 & 50865 & 28 & 0.132\\
5.190 & 49041 & 50959 & 23 & 0.109\\
5.166 & 49097 & 50903 & 23 & 0.109\\
5.173 & 49007 & 50996 & 28 & 0.133\\
5.166 & 48783 & 51217 & 29 & 0.137\\
5.167 & 48951 & 51049 & 31 & 0.147\\
5.169 & 49156 & 50844 & 30 & 0.142\\
5.204 & 49149 & 50851 & 32 & 0.152\\
5.179 & 49023 & 50977 & 26 & 0.124\\
5.170 & 48783 & 51217 & 26 & 0.123\\
\hline
\end{tabular}
\end{center}
\end{table}
Since the single-photon source emits light pulses triggered by a stable external clock and
well separated in time, the value of $\alpha$ can be directly inferred from the record of all photon arrival times. Every photodetection event produced by the two avalanche photodiodes is time-stamped using a time-interval-analyser (TIA) computer board
(GT653, GuideTech). Straightforward processing of these timestamps over a discrete time base allows us to reconstruct the number of detection events on each output channel of the biprism interferometer, and thus gives an access to the "which-path" information. Furthermore, time intervals between two successive photodetections are directly inferred from the timestamps ensemble, so as to estimate if a coincidence has occurred or not for each registered photodetection. However the notion of coincidence is meaningful only accordingly to a temporal gate: there will be a coincidence if two detections happen within the same gate. As the radiative lifetime of the emitting N-V colour centre
used for the experiment is approximately equal to 45~ns (see Fig.\ref{fig_biprisme_HBT}), we set the gate duration to 100~ns. Such a value is much smaller than the 436~ns time interval between two successive excitation pulses. It also ensures that about 90\% of the detected photons are considered for data analysis.
Using the results given on Table~1
and simple statistical analysis associated to a 95\% confidence interval, we infer
$\alpha = 1.00 \pm 0.06$ for faint laser pulses and
$\alpha = 0.13 \pm 0.01$ for single-photon pulses.
As expected, light pulses emitted by the single-photon source lead to a strong violation of the inequality $\alpha \geq 1$ valid for any semi-classical theory of light. The non-ideal value $\alpha \not= 0 $ is due to remaining background fluorescence of the sample and to the two-phonon Raman scattering line, which induces a non-vanishing probability of having more than one photon in the emitted light pulse. Table~1 also shows that the number of coincidences observed with faint laser pulses, within a given integration time, is much higher than in the experiment with single-photon pulses, and corresponds to $\alpha$ equal to unity within one standard deviation. This result is a clear confirmation that it is not possible to demonstrate the particle-like behaviour with attenuated pulses from a classical light source, involving many emitters simultaneously and independently excited.
From the set of photodetection timestamps, one can also build the histogram of delays between two consecutive detections on the two paths of the wavefront-splitting interferometer. In the limit of low collection efficiency and short timescale, the recorded histogram coincides with a measurement of the second-order autocorrelation function $g^{(2)}$ \cite{Reynaud}.
As shown in Fig.\ref{fig_biprisme_HBT}, strong reduction of coincidences around zero delay gives again clear evidence for single-photon behaviour \cite{Brunel}. Note that the area of the zero-delay peak, normalized to its value for coherent pulses with poissonian statistics, is strictly equivalent to the $\alpha$ correlation parameter previously defined.
\section{Single-photon interference setup and data analysis}
\label{sec:3}
\subsection{Experimental setup and observations}
\label{sec:2.1}
Using the same N-V colour centre, we now observe the interference fringes in the
intersection volume of the two separated wavefronts, as shown in Fig.\ref{general}.
Interference patterns are detected in a single-photon recording mode,
using an intensified CCD camera (\emph{iStar} from
Andor Technologies, Ireland) cooled at $-25^\circ{\rm C}$. An eye-piece, equivalent
to a short focal length achromatic lens ($f=22$~mm) is inserted between the biprism and the camera,
in order to obtain a fringe spacing much larger than the $25~\mu$m pixel size of the camera.
Image acquisition parameter are optimized by adjusting the camera gain and the detection threshold, so that more than $\approx 88\%$ of the bright pixels correspond to detections of single photons.
\begin{figure}[ht]
\centerline{\includegraphics[width=9cm]{fig4.pdf}}
\caption{Observation of the interference pattern expanded by the eyepiece and recorded by the intensified CCD camera. Image (a) (resp. (b) and (c)) is made of 272 photocounts (resp. 2240 and 19773) corresponding to an exposure duration of 20~s (resp. 200~s and 2000~s). Graph (d) displays the resulting interference fringes obtained by binning columns of CCD image (c) and fit of this interference pattern using coherent beam propagation in the Fresnel diffraction regime, and taking into account the finite temporal coherence due to the broad spectral emission of the NV colour centre. A visibility of 94\% can be associated to the central fringe.}
\label{interf_setup}
\end{figure}
Snapshots with exposure duration of 1~s are acquired one after each
other, with a mean number of eight photons detected on the CCD
array per snapshot. Accumulated images show how
the interference pattern builds up (Fig.\ref{interf_setup}). Less than 200 accumulated snapshots are required to clearly see the fringes pattern, and the final image contains about $2\times 10^4$ single photodetection events. This is a demonstration of the wave-like behaviour of light, even in the single-photon regime. To display fringe pattern gradual build-up, we generated a movie from the 2000 snapshots (available as Supplementary File). The movie frame rate is set to a value of 30 images per second resulting in a build-up 30 times faster than in real time.
\subsection{Fit of the interference pattern}
\label{sec:2.2}
The interference pattern obtained with a monochromatic plane wave incident on Fresnel's biprism would result in sinusoidal fringes with a visibility equal to unity. In our experiment, the finite spatial extension of the single-photon wavefront yields more complicated interference patterns. In order to fit the fringes shown on Fig.\ref{interf_setup}-(c), we performed a computer simulation of the beam propagation through the Fresnel's biprism using free space propagation theory from classical optics.
Starting at the input of the Fresnel's biprism with a TEM$_{00}$ wavefront associated to a measured 1.25~mm FWHM gaussian intensity distribution, the output field amplitude is calculated in a virtual observation plane located at a given distance $z$ behind the biprism input plane~\cite{Note}. The expected pattern in the CCD plane simply results
from the expansion by the eye-piece of this calculated intensity pattern.
In order to fit more accurately the fringes minima, we also take into account temporal coherence effects due to the N-V colour centre broadband incoherent emission (see Fig.\ref{confoc}-(d)). Finally the calculated intensity pattern is normalized to the total recorded intensity so that the only remaining fitting parameter is the observation distance $z$.
The quality of the fit is illustrated by Fig.\ref{interf_setup}-(d), recorded approximately in the middle of the interference field. All over the interference pattern, the visibility of the fringes is very well described by the beam propagation simulation with the addition of finite temporal coherence of the source.
\begin{figure} [t]
\centerline{\includegraphics[width=8.8cm]{fig5.pdf}}
\caption{Examples of interference patterns observed in the overlapping region of the two transmitted beams through the Fresnel's biprism, for (a) $z = 11 $ mm and (b) $z = 98$ mm. Overlayed blue curves are fits evaluated from a model taking into account Fresnel diffraction and temporal coherence effects.}
\label{fresnel}
\end{figure}
Validity of the model is furthermore confirmed by fits of interference patterns
observed for two others $z$ positions, as shown in Fig.\ref{fresnel}.
The good agreement between the fit and the experimental data is an illustration of the well-known result from Quantum Optics that phenomena like interference, diffraction, propagation, can be computed with the classical theory of light even in the single-photon regime \cite{Gerry}.
\section{Conclusion}
\label{sec:conc}
Summarizing, we have carried out the two complementary experiments of "interference observation vs. which-path detection" in the single-photon regime. While our results may not appear as a big surprise, it is interesting to note that one century after Einstein's paper the intriguing properties of the photon still give rise to sometimes confused debates \cite{Afshar}. We hope that our experiment can contribute not only to clarify such discussions, but also to arouse the interest and astonishment of those who will discover the photon during the century to come.
|
2,877,628,090,237 | arxiv | \section{Introduction}\label{sec:motivation}
Classification tasks on data sets with large feature dimensions are very common in real-world machine learning applications. Typical examples include microarray data for gene selection and text documents for natural language processing. Despite the large number of features present in the data sets, usually only small subsets of the features are relevant to the particular learning tasks, and local correlation among the features is often observed. Hence, feature selection is required for good model interpretability. Popular classification techniques, such as support vector machine (SVM) and logistic regression, are formulated as convex optimization problems. An extensive literature has been devoted to optimization algorithms that solve variants of these classification models with sparsity regularization \cite{pardalos-hansen-2008,sra-nowozin-wright-2011}.
Many of them are based on first-order (gradient-based) methods, mainly because the size of the optimization problem is very large. The advantage of first-order methods is that their computational and memory requirements at each iteration are low and as a result they can handle the large optimization problems occurring in classification problems. Their major disadvantage is their slow convergence, especially when a good approximation of the feature support has been identified. Second-order methods exhibit fast local convergence, but their computational and memory requirements are much more demanding, since they need to store and invert the Newton matrix at every iteration. It is therefore very important to be able to intelligently combine the advantages of both the first and the second order optimization methods in such a way that the resulting algorithm can solve large classification problems efficiently and accurately. As we will demonstrate in this paper such combination is possible by taking advantage of the problem structure and the change in its size during the solution process. In addition, we will also show that our algorithmic framework is flexible enough to incorporate prior knowledge to improve classification performance.
\subsection{Related Work}\label{sec:related_work}
The above requirements demand three features from a learning algorithm: 1. it should be able to automatically select features which are possibly in groups and highly correlated; 2. it has to solve the optimization problem in the training phase efficiently and with high accuracy; and 3. the learning model needs to be flexible enough so that domain knowledge can be easily incorporated. Existing methods are available in the literature that meet some of the above requirements {\em individually}. For enforcing sparsity in the solution, efficient optimization algorithms such as that proposed in \cite{koh2007interior} can solve large-scale sparse logistic regression. On the other hand, the $L_1$-regularization is unstable with the presence of highly correlated features - among a group of such features, essentially one of them is selected in a random manner. To handle local correlation among groups of features, the elastic-net regularization \cite{zou2005regularization} has been successfully applied to SVM \cite{wang2006doubly} and logistic regression \cite{ryali2010sparse}. However, incorporating domain knowledge into the logistic regression formulation is not straightforward. For SVM, including such knowledge in the optimization process has been demonstrated in \cite{fung2003knowledge}. Recently, an Alternating Direction Method of Multipliers (ADMM) has been proposed for the elastic-net SVM (ENSVM) \cite{yeefficient}. ADMM is quick to find an approximate solution to the ENSVM problem, but it is known to converge very slowly to high accuracy optimal solutions \cite{boyd2010distributed}. The interior-point methods (IPM) for SVM are known to be able to achieve high accuracy in their solutions with a polynomial iteration complexity, and the dual SVM formulation is independent of the feature space dimensionality. However, the classical $L_2$-norm SVM is not able to perform automatic feature selection. Although the elastic-net SVM can be formulated as a QP (in the primal form), its problem size grows substantially with the feature dimensionality. Due to the need to solve a Newton system in each iteration, the efficiency of IPM quickly deteriorates as the feature dimension becomes large.
\subsection{Main Contributions}
In this paper we propose a new hybrid algorithmic framework for SVM to address {\em all} of the above challenges and requirements {\em simultaneously}. Our framework combines the advantages of a first-order optimization algorithm (through the use of ADMM) and a second-order method (via IPM) to achieve both superior speed and accuracy. Through a novel algorithmic approach that is able to incorporate expert knowledge, our proposed framework is able to exploit domain knowledge to improve feature selection, and hence, prediction accuracy. Besides efficiency and generalization performance, we demonstrate through experiments on both synthetic and real data that our method is also more robust to inaccuracy in the supplied knowledge than existing approaches.
\section{A Two-phase Hybrid Optimization Algorithm}\label{sec:two_phase_alg}
As previously mentioned, for data sets with many features, the high dimensionality of the feature space still poses a computational challenge for IPM. Fortunately, many data sets of this kind are very sparse, and the resulting classifier $w$ is also expected to be sparse, i.e. only a small subset of the features are expected to carry significant weights in classification. Naturally, it is ideal for IPM to train a classifier on the most important features only.
Inspired by the Hybrid Iterative Shrinkage (HIS) \cite{shi2010fast} algorithm for training large-scale sparse logistic regression classifiers, we propose a two-phase algorithm to shrink the feature space appropriately so as to leverage the high accuracy of IPM while maintaining efficiency. Specifically, we propose to solve an elastic-net SVM (ENSVM) or doubly-regularized SVM (DrSVM) \cite{wang2006doubly} problem during the first phase of the algorithm. The elastic-net regularization performs feature selection with grouping effect and has been shown to be effective on data sets with many but sparse features and high local correlations \cite{zou2005regularization}. This is the case for text classification, microarray gene expression, and fMRI data sets. The support of the weight vector $w$ for ENSVM usually stabilizes well before the algorithm converges to the optimal solution. Taking advantage of that prospect, we can terminate the first phase of the hybrid algorithm early and proceed to solve a classical SVM problem with the reduced feature set in the second phase, using an IPM solver.
\subsection{Solving the Elastic Net SVM using ADMM}\label{sec:admm}
SVM can be written in the regularized regression form as
\begin{equation}\label{eq:unconstr_svm}
\min_{\textbf{w},b}\frac{1}{N}\sum_{i=1}^N(1 - (y_i(\textbf{x}_i^T\textbf{w} + b)))_+ + \frac{\lambda}{2}\|\textbf{w}\|_2^2,
\end{equation}
where the first term is an averaged sum of the hinge losses and the second term is viewed as a ridge regularization on $w$. It is easy to see from this form that the classical SVM does not enforce sparsity in the solution, and $w$ is generally dense. The ENSVM adds an $L_1$ regularization on top of the ridge regularization term, giving
\begin{equation}\label{eq:en_svm}
\min_{\textbf{w},b}\frac{1}{N}\sum_{i=1}^{N}(1-y_i(\textbf{x}_i^T\textbf{w} + b))_+ + \lambda_1\|\textbf{w}\|_1 + \frac{\lambda_2}{2}\|\textbf{w}\|_2^2.
\end{equation}
Compared to the Lasso ($L_1$-regularized regression) \cite{tibshirani1996regression}, the elastic-net has the advantage of selecting highly correlated features in groups (i.e. the grouping effect) while still enforcing sparsity in the solution. This is a particularly attractive feature for text document data, which is common in the hierarchical classification setting. Adopting the elastic-net regularization as in (\ref{eq:en_svm}) brings the same benefit to SVM for training classifiers.
To approximately solve problem (\ref{eq:en_svm}), we adopt the alternating direction method of multipliers (ADMM) for elastic-net SVM recently proposed in \cite{yeefficient}. ADMM has a long history dating back to the 1970s \cite{gabay1976dual}. Recently, it has been successfully applied to problems in machine learning \cite{boyd2010distributed}. ADMM is a special case of the inexact augmented Lagrangian (IAL) method \cite{rockafellar1973multiplier} for the structured unconstrained problem
\begin{equation}\label{eq:struct_unc}
\min_x F(x) \equiv f(x) + g(Ax),
\end{equation}
where both functions $f(\cdot)$ and $g(\cdot)$ are convex. We can decouple the two functions by introducing an auxiliary variable $y$ and convert problem (\ref{eq:struct_unc}) into an equivalent constrained optimization problem
\begin{eqnarray}\label{eq:struct_c}
\min_{x,y} && f(x) + g(y), \quad s.t. \>\> Ax = y.
\end{eqnarray}
This technique is often called variable-splitting \cite{combettes2011proximal}. The IAL method approximately minimizes in each iteration the augmented Lagrangian of (\ref{eq:struct_c}) defined by
$\mathcal{L}(x,y,\gamma) := f(x) + g(y) + \gamma^T(y-Ax) + \frac{\mu}{2}\|Ax-y\|_2^2$
, followed by an update to the Lagrange multiplier $\gamma \gets \gamma - \mu(Ax - y)$.
The IAL method is guaranteed to converge to the optimal solution of (\ref{eq:struct_unc}), as long as the subproblem of approximately minimizing the augmented Lagrangian is solved with an increasing accuracy \cite{rockafellar1973multiplier}. ADMM can be viewed as a practical implementation of IAL, where the subproblem is solved approximately by minimizing $\mathcal{L}(x,y;\gamma)$ with respect to $x$ and $y$ {\em alternatingly once}.
Eckstein and Bertsekas \cite{eckstein1992douglas} established the convergence of ADMM for the case of two-way splitting.
Now applying variable-splitting and ADMM to problem (\ref{eq:en_svm}), \cite{yeefficient} introduced auxiliary variables $(\textbf{a},\textbf{c})$ and linear constraints so that the non-smooth hinge loss and $L_1$-norm in the objective function
are decoupled, making it easy to optimize over each of the variables. Specifically, problem (\ref{eq:en_svm}) is transformed into an equivalent constrained form
\begin{eqnarray}\label{eq:en_svm_constr}
\min_{\textbf{w},b,\textbf{a},\textbf{c}} && \frac{1}{N}\sum_{i=1}^N (a_i)_+ + \lambda_1\|\textbf{c}\|_1 + \frac{\lambda_2}{2}\|\textbf{w}\|_2^2 \\
\nonumber s.t. && \textbf{a} = \textbf{e} - Y(X\textbf{w} + b\textbf{e}) \quad \mbox{and} \quad \textbf{c} = \textbf{w}
\end{eqnarray}
where $\textbf{x}_i^T$ is the $i$-th row of $X$, and $Y = \textrm{diag}(\textbf{y})$.
The augmented Lagrangian
\begin{multline}
\mathcal{L}(\textbf{w},b,\textbf{a},\textbf{c},\gamma_1,\gamma_2) := \frac{1}{N}\sum_{i=1}^N a_i + \lambda_1\|\textbf{c}\|_1 + \frac{\lambda_2}{2}\|\textbf{w}\|_2^2 + \gamma_1^T(\textbf{e} - Y(X\textbf{w} + b\textbf{e})-\textbf{a})\\
+\gamma_2^T(\textbf{w}-\textbf{c}) + \frac{\mu_1}{2}\|\textbf{e} - Y(X\textbf{w} + b\textbf{e})-\textbf{a}\|_2^2 + \frac{\mu_2}{2}\|\textbf{w}-\textbf{c}\|_2^2
\end{multline}
is then minimized with respect to $(\textbf{w},b), \textbf{a},$ and $\textbf{c}$ sequentially in each iteration, followed by an update to the Lagrange multipliers $\gamma_1$ and $\gamma_2$. The original problem is thus decomposed into three subproblems consisting of computing the proximal operator of the hinge loss function (with respect to $\textbf{a}$), solving a special linear system (with respect to $(\textbf{w},b)$ ), and performing a soft-thresholding operation (with respect to $\textbf{c}$), which can all be done in an efficient manner. Due to lack of space in the paper, we have included the detailed solution steps in the Appendix (see Algorithm \ref{alg:admm-ensvm} ADMM-ENSVM), where we define by $\mathcal{S}_\lambda(\cdot)$ the proximal operator associated with the hinge loss
\begin{eqnarray*}
\mathcal{S}_\lambda(\omega) &=& \left\{
\begin{array}{ll}
\omega - \lambda, & \hbox{$\omega > \lambda$;} \\
0, & \hbox{$0 \leq \omega \leq \lambda$;} \\
\omega, & \hbox{$\omega < 0$.}
\end{array}
\right.
\end{eqnarray*}
and $\mathcal{T}_\lambda(\omega) = sgn(\omega)\max\{0,|\omega|-\lambda \}$ is the shrinkage operator.
\subsection{SVM via Interior-Point Method}\label{sec:ipm}
Interior Point Methods enjoy fast convergence rates for a wide class of QP problems. Their theoretical polynomial convergence ($O(n\log \frac{1}{\epsilon})$) was first established by Mizuno \cite{mizuno1994}. In addition, Andersen {\em et al} \cite{Andersen1996implement} showed that the number of iterations needed by IPMs to converge is $O(\log n)$, which demonstrates that their computational effort increases in a slower rate than the size of the problem.
Both the primal and the dual SVM are QP problems. The primal formulation of SVM \cite{vapnik2000nature} is defined as
\begin{eqnarray*}
\textrm{(SVM-P)} \qquad \min_{\textbf{w},b,\xi,\textbf{s}} && \frac{1}{2}\textbf{w}^T\textbf{w} + c\textbf{e}^T\xi \\
\qquad s.t. && y_i(\textbf{w}^T\textbf{x}_i - b) + \xi_i - s_i = 1, i=1,\dots,N, \\
&& \textbf{s} \geq 0, \xi \geq 0.
\end{eqnarray*}
whereas the dual SVM has the form
\begin{eqnarray*}
\textrm{(SVM-D)} \quad \min_{\alpha} && \frac{1}{2}\alpha^TQ\alpha - \textbf{e}^T\alpha \\
s.t. && \textbf{y}^T\alpha = 0, \quad \mbox{and} \quad 0 \leq \alpha_i \leq c, \quad i = 1,\cdots,N,
\end{eqnarray*}
where $Q_{ij} = y_iy_j\textbf{x}_i^T\textbf{x}_j = \bar{X}\bar{X}^T$.
By considering the KKT conditions of (SVM-P) and (SVM-D), the optimal solution is given by
$\textbf{w} = \bar{X}^T\alpha = \sum_{i\in SV}\alpha_i y_i \textbf{x}_i$,
where $SV$ is the set of sample indices corresponding to the support vectors. The optimal bias term $b$ can be computed from the complementary slackness condition $\alpha_i(y_i(\textbf{x}_i^T\textbf{w} + b)-1 + \xi_i) = 0$.
Whether to solve (SVM-P) or (SVM-D) for a given data set depends on its dimensions as well as its sparsity. Even if $X$ is a sparse matrix, $Q$ in (SVM-D) is still likely to be dense, whereas the Hessian matrix in (SVM-P) is the identity. The primal problem (SVM-P), however, has a larger variable dimension and more constraints. It is often argued that one should solve (SVM-P) when the number of features is smaller than the number of samples, whereas (SVM-D) should be solved when the number of features is less than that of the samples. Since in the second phase of Algorithm \ref{alg:hipad} we expect to have identified a small number of promising features, we have decided to solve (SVM-D) by using IPM.
Solving (SVM-D) is realized through the OOQP \cite{gertz2003object} software package that implements a primal-dual IPM for convex QP problems.
\subsection{The Two-phase Algorithm}
Let us keep in mind that the primary objective for the first phase is to appropriately reduce the feature space dimensionality without impacting the final prediction accuracy. As we mentioned above, the suitability of ADMM for the first phase depends on whether the support of the feature vector converges quickly or not. On an illustrative dataset from \cite{yeefficient}, which has 50 samples with 300 features each, ADMM converged in 558 iterations. The output classifier $\textbf{w}$ contained only 13 non-zero features, and the feature support converged in approximately 50 iteration (see Figure \ref{fig:fea_supp_plots} in the Appendix for illustrative plots showing the early convergence of ADMM). Although the remaining more than 500 iterations are needed by ADMM in order to satisfy the optimality criteria, they do not offer any additional information regarding the feature selection process. Hence, it is important to monitor the change in the signs and indices of the support and terminate the first phase promptly. In our implementation, we adopt the criterion used in \cite{shi2010fast} and monitor the relative change in the iterates as a surrogate of the change in the support, i.e.
\begin{equation}\label{eq:hipad_transition}
\frac{\|\textbf{w}^{k+1}-\textbf{w}^k\|}{\max(\|\textbf{w}^k\|,1)} < \epsilon_{tol}.
\end{equation}
We have observed in our experiments that when the change over the iterates is small, the evolution of the support indices stabilizes too.
Upon starting the second phase, it is desirable for IPM to warm-start from the corresponding sub-vector of the solution returned by ADMM. It should also be noted that we apply IPM during the second phase to solve the classical $L_2$-regularized SVM (\ref{eq:unconstr_svm}), instead of the ENSVM (\ref{eq:en_svm}) in the first phase. There are two main reasons for this decision. First, although ENSVM can be reformulated as a QP, the size of the problem is larger than the classical SVM due to the additional linear constraints introduced by the $L_1$-norm. Second, since we have already identified (approximately) the feature support in the first phase of the algorithm, enforcing sparsity in the reduced feature space becomes less critical. The two-phase algorithm is summarized in Algorithm \ref{alg:hipad}.
\begin{algorithm}
\caption{HIPAD (Hybrid Interior Point and Alternating Direction method)}
\begin{algorithmic}[1]\label{alg:hipad}
\STATE Given $\textbf{w}^0, b^0, \textbf{a}^0, \textbf{c}^0, \textbf{u}^0,$ and $\textbf{v}^0$.
\STATE \textbf{PHASE 1: ADMM for ENSVM}
\STATE $(\textbf{w}^{\textrm{ADMM}},b^{\textrm{ADMM}}) \gets \textrm{ADMM-ENSVM}(\textbf{w}^0, b^0, \textbf{a}^0, \textbf{c}^0, \textbf{u}^0,\textbf{v}^0)$
\STATE \textbf{PHASE 2: IPM for SVM}
\STATE $\widetilde{\textbf{w}} \gets $ non-zero components of $\textbf{w}^{\textrm{ADMM}}$
\STATE $(\widetilde{X}, \widetilde{Y}) \gets $ sub-matrices of $(X, Y)$ corresponding to the support of $\textbf{w}^{\textrm{ADMM}}$
\STATE $(\textbf{w},b) \gets \textrm{SVM-IPM}(\widetilde{X}, \widetilde{Y}, \widetilde{\textbf{w}}$), through (SVM-D).
\RETURN $(\textbf{w},b)$
\end{algorithmic}
\end{algorithm}
\section{Domain Knowledge Incorporation}\label{sec:enk-svm}
Very often, we have prior domain knowledge for specific classification tasks. Domain knowledge is most helpful when the training data does not form a comprehensive representation of the underlying unknown population, resulting in poor generalization performance of SVM on the unseen data from the same population. This often arises in situations where labeled training samples are scarce, while there is an abundance of unlabeled data.
For high dimensional data, ENSVM performs feature selection along with training to produce a simpler model and to achieve better prediction accuracy. However, the quality of the feature selection depends entirely on the training data. In pathological cases, it is very likely that the feature support identified by ENSVM does not form a good representation of the population. Hence, when domain knowledge about certain features is available, we should take it into consideration during the training phase and include the relevant features in the feature support should them be deemed important for classification.
In this section, we explore and propose a new approach to achieve this objective. We consider domain knowledge in the form of class-membership information associated with features.
We can incorporate such information (or enforce such rules) in SVM by adding equivalent linear constraints to the SVM QP problem (KSVM) \cite{fung2003knowledge,lauer2008incorporating}. To be specific, we can model the above information with the linear implication
\begin{equation}\label{eq:linear_implication}
B\textbf{x} \leq \textbf{d} \quad \Rightarrow \quad \textbf{w}^T\textbf{x} + b \geq 1,
\end{equation}
where $B \in \mathbb{R}^{k_1\times m}$ and $\textbf{d} \in \mathbb{R}^{k_1}$.
It is shown in \cite{fung2003knowledge} that by utilizing the non-homogeneous Farkas theorem of the alternative,
(\ref{eq:linear_implication}) can be transformed into the following equivalent system of linear inequalities with a solution $\textbf{u}$
\begin{equation}\label{eq:equiv_lin_constr}
B^T\textbf{u} + \textbf{w} = \textbf{0}, \quad \textbf{d}^T\textbf{u} - b + 1 \leq 0, \quad \textbf{u} \geq \textbf{0}.
\end{equation}
Similarly, for the linear implication for the negative class membership we have:
\begin{equation}\label{eq:lin_imp_neg}
D\textbf{x} \leq \textbf{g} \Rightarrow \textbf{w}^T\textbf{x} + b \leq -1, \quad D \in \mathbb{R}^{k_2\times m}, g\in\mathbb{R}^{k_2},
\end{equation}
which can be represented by the set of linear constraints in $\textbf{v}$
\begin{equation}\label{eq:equiv_lin_constr_neg}
D^T\textbf{v} - \textbf{w} = \textbf{0}, \quad \textbf{g}^T\textbf{v} + b + 1 \leq 0, \quad \textbf{v} \geq \textbf{0}.
\end{equation}
Hence, to incorporate the domain knowledge represented by (\ref{eq:linear_implication}) and (\ref{eq:lin_imp_neg}) into SVM, Fung {\em et al} \cite{fung2003knowledge} simply add the linear constraints (\ref{eq:equiv_lin_constr}) and (\ref{eq:equiv_lin_constr_neg}) to (SVM-P).
Their formulation, however, increases both the variable dimension and the number of linear constraints by at least $2m$, where $m$ is the number of features in the classification problem we want to solve. This is clearly undesirable when $m$ is large, which is the scenario that we consider in this paper.
In order to avoid the above increase in the size of the optimization probelm, we choose to penalize the quadratics $\|B^T\textbf{u}+\textbf{w}\|_2^2$ and $\|D^T\textbf{v}-\textbf{w}\|_2^2$ instead of their $L_1$ counterparts considered in \cite{fung2003knowledge}. By doing so the resulting problem is still a convex QP but with a much smaller size. Hence, we consider the following model for domain knowledge incorporation.
\begin{eqnarray*}
\textrm{(KSVM-P)} \qquad
\min_{\textbf{w},b,\xi,\textbf{u},\textbf{v},\eta_u,\eta_v} && \frac{1}{2}\textbf{w}^T\textbf{w} + c\textbf{e}^T\xi + \frac{\rho_1}{2}\|B^T\textbf{u}+\textbf{w}\|_2^2 \\
&& + \rho_2\eta_u + \frac{\rho_3}{2}\|D^T\textbf{v}-\textbf{w}\|_2^2 + \rho_4\eta_v \\
\nonumber s.t. && y_i(\textbf{w}^T\textbf{x}_i + b) \geq 1 - \xi_i, \quad i = 1,\cdots,N, \\
\nonumber && \textbf{d}^T\textbf{u} - b + 1 \leq \eta_u, \\
\nonumber && \textbf{g}^T\textbf{v} + b + 1 \leq \eta_v,\\
\nonumber && (\xi,\textbf{u},\textbf{v},\eta_u,\eta_v) \geq \textbf{0}.
\end{eqnarray*}
We are now ready to propose a novel combination of ENSVM and KSVM, and we will explain in the next section how the combined problem can be solved in our HIPAD framework. The main motivation behind this combination is to exploit domain knowledge to improve the feature selection, and hence, the generalization performance of HIPAD. To the best of our knowledge, this is the first method of this kind.
\subsection{ADMM Phase}
Our strategy for solving the elastic-net SVM with domain knowledge incorporation is still to apply the ADMM method. First, we combine problems (\ref{eq:en_svm}) and (KSVM-P) and write the resulting optimization problem in an equivalent unconstrained form (by penalizing the violation of the inequality constraints through hinge losses in the objective function)
\begin{equation*
\textrm{(ENK-SVM)} \quad \min_{\textbf{w},b,\textbf{u} \geq \textbf{0},\textbf{v} \geq \textbf{0}} F(\textbf{w},b,\textbf{u},\textbf{v}),
\end{equation*}
where $F(\textbf{w},b,\textbf{u},\textbf{v}) \equiv \frac{\lambda_2}{2}\|\textbf{w}\|_2^2 + \lambda_1\|\textbf{w}\|_1 + \frac{1}{N}\sum_{i=1}^{N}(1-y_i(x_i^Tw + b))_+
+ \frac{\rho_1}{2}\|B^T\textbf{u}+\textbf{w}\|_2^2 + \rho_2(\textbf{d}^T\textbf{u} - b + 1)_+ + \frac{\rho_3}{2}\|D^T\textbf{v}-\textbf{w}\|_2^2
+ \rho_4(\textbf{g}^T\textbf{v} + b + 1)_+$.
We then apply variable-splitting to decouple the $L_1$-norms and hinge losses and obtain the following equivalent constrained optimization problem:
\begin{eqnarray}\label{eq:ensvm_ksvm_c}
\min_{\textbf{w},b,\textbf{u},\textbf{v},\textbf{a},\textbf{c},p,q} &&\>\>\> F(\textbf{w},b,\textbf{u},\textbf{v},\textbf{a},\textbf{c},p,q) \\
\nonumber s.t. && \>\>\> \textbf{a} = \textbf{e} - (\bar{X}\textbf{w} + \textbf{y} b), \>\>\> \textbf{c} = \textbf{w}, \\
\nonumber && \>\>\> q = \textbf{d}^T\textbf{u} - b + 1, \>\>\> p = \textbf{g}^T\textbf{v} + b + 1, \\
\nonumber && \>\>\> \textbf{u} \geq \textbf{0}, \>\>\> \textbf{v} \geq \textbf{0}.
\end{eqnarray}
with $F(\textbf{w},b,\textbf{u},\textbf{v},\textbf{a},\textbf{c},p,q)
\equiv \frac{\lambda_2}{2}\|\textbf{w}\|_2^2 + \lambda_1\|\textbf{c}\|_1 + \frac{1}{N}\textbf{e}^T(\textbf{a})_+
+ \frac{\rho_1}{2}\|B^T\textbf{u}+\textbf{w}\|_2^2 + \rho_2(q)_+
+ \frac{\rho_3}{2}\|D^T\textbf{v}-\textbf{w}\|_2^2 + \rho_4(p)_+$.
As usual, we form the augmented Lagrangian $\mathcal{L}$ of problem (\ref{eq:ensvm_ksvm_c}),
\begin{multline*}
\mathcal{L} := F(\textbf{w},b,\textbf{u},\textbf{v},\textbf{a},\textbf{c},p,q)
+ \gamma_1^T(\textbf{e}-(\bar{X}\textbf{w}+\textbf{y} b)-\textbf{a})
+ \frac{\mu_1}{2}\|\textbf{e}-(\bar{X}+\textbf{y} b)-\textbf{a}\|_2^2\\
+ \gamma_2^T(\textbf{w}-\textbf{c}) + \frac{\mu_2}{2}\|\textbf{w}-\textbf{c}\|_2^2
+ \gamma_3(\textbf{d}^T\textbf{u}-b+1-q) + \frac{\mu_3}{2}\|\textbf{d}^T\textbf{u}-b+1-q\|_2^2 \\
+ \gamma_4(\textbf{g}^T\textbf{v}+b+1-p) + \frac{\mu_4}{2}\|\textbf{g}^T\textbf{v}+b+1-p\|_2^2
\end{multline*}
and minimize $\mathcal{L}$ with respect to $\textbf{w},b,\textbf{c},\textbf{a},p,q,\textbf{u},\textbf{v}$ individually and in order. For the sake of readability, we do not penalize the non-negative constraints for $\textbf{u}$ and $\textbf{v}$ in the augmented Lagrangian.
Given $(\textbf{a}^k, \textbf{c}^k, p^k, q^k)$, solving for $(\textbf{w},b)$ again involves solving a linear system
\begin{equation}\label{eq:ksvm_admm_linsys_wb}
\left(
\begin{array}{cc}
\kappa_1 I + \mu_1 X^TX & \mu_1 X^T\textbf{e} \\
\mu_1\textbf{e}^TX & \mu_1 N + \kappa_2 \\
\end{array}
\right)
\left(
\begin{array}{c}
\textbf{w}^{k+1} \\
b^{k+1} \\
\end{array}
\right) =
\left(
\begin{array}{c}
\textbf{r}_{\textbf{w}}\\
\textbf{r}_b\\
\end{array}
\right),
\end{equation}
where $\kappa_1 = \lambda_2 + \mu_2 + \rho_1 + \rho_3, \kappa_2 = \mu_3 + \mu_4, \textbf{r}_{\textbf{w}} = X^TY\gamma_1^k + \mu_1 X^TY(\textbf{e}-\textbf{a}^k) - \gamma_2^k + \mu_2\textbf{c}^k + \rho_3D^T\textbf{v}^k - \rho_1B^T\textbf{u}^k$ and $\textbf{r}_{b} = \textbf{e}^TY\gamma_1^k + \mu_1\textbf{e}^TY(\textbf{e}-\textbf{a}^k) + \gamma_3^k + \mu_3(\textbf{d}^T\textbf{u}^k + 1 - q^k) - \gamma_4^k - \mu_4(\textbf{g}^T\textbf{v}^k+1-p^k)$.
Similar to solving the linear system in Algorithm \ref{alg:admm-ensvm} ADMM-ENSVM, we can compute the solution to the above linear system through a few PCG iterations, taking advantage of the fact that the left-hand-side matrix is of low-rank.
To minimize the augmented Lagrangian with respect to $\textbf{u}$, we need to solve a convex quadratic problem with non-negative constraints
\begin{equation}\label{eq:ksvm_admm_usub}
\min_{\textbf{u} \geq \textbf{0}} \quad \frac{\rho_1}{2}\|B^T\textbf{u} + \textbf{w}^{k+1}\|_2^2 + \gamma_3^k\textbf{d}^T\textbf{u}
+ \frac{\mu_3}{2}\|\textbf{d}^T\textbf{u} - b^{k+1} + 1 - q^{k}\|_2^2.
\end{equation}
Solving problem \eqref{eq:ksvm_admm_usub} efficiently is crucial for the efficiency of the overal algorithm. We describe a novel way to do so.
Introducing a slack variable $\textbf{s}$ and transferring the non-negative constraint on $\textbf{u}$ to $\textbf{s}$, we decompose the problem into two parts which are easy to solve. Specifically, we reformulate (\ref{eq:ksvm_admm_usub}) as
\begin{eqnarray*}
\min_{\textbf{u},\textbf{s} \geq \textbf{0}} && \>\>\> \frac{\rho_1}{2}\|B^T\textbf{u} + \textbf{w}^{k+1}\|_2^2 + \gamma_3^k\textbf{d}^T\textbf{u}
+ \frac{\mu_3}{2}\|\textbf{d}^T\textbf{u} - b^{k+1} + 1 - q^{k}\|_2^2 \\
\nonumber s.t. && \>\>\> \textbf{u} - \textbf{s} = \textbf{0}.
\end{eqnarray*}
Penalizing the linear constraint $\textbf{u} - \textbf{s} = \textbf{0}$ in the new augmented Lagrangian, the new subproblem with respect to $(\textbf{u},\textbf{s})$ is
\begin{multline}\label{eq:ksvm_admm_ussub}
\min_{\textbf{u},\textbf{s} \geq \textbf{0}} \quad \frac{\rho_1}{2}\|B^T\textbf{u} + \textbf{w}^{k+1}\|_2^2 + \gamma_3^k\textbf{d}^T\textbf{u} \\
+ \frac{\mu_3}{2}\|\textbf{d}^T\textbf{u} - b^{k+1} + 1 - q^k\|_2^2 + \gamma_5^T(\textbf{s}-\textbf{u}) + \frac{\mu_5}{2}\|\textbf{u}-\textbf{s}\|_2^2.
\end{multline}
Given an $\textbf{s}^k \geq \textbf{0}$, we can compute $\textbf{u}^{k+1}$ by solving a $k_1 \times k_1$ linear system
\begin{equation}\label{eq:ksvm_admm_linsys_u}
(\rho_1 BB^T + \mu_3 \textbf{d}\dbold^T + \mu_5 I)\textbf{u}^{k+1} = \textbf{r}_\textbf{u},
\end{equation}
where $\textbf{r}_\textbf{u} = -\rho_1B\textbf{w}^{k+1} + \mu_3 \textbf{d} b^{k+1} + \mu_3\textbf{d}(q^k-1) - \textbf{d}\gamma_3^k + \gamma_5 + \mu_5\textbf{s}^k$.
We assume that $B$ has full row-rank. This is a reasonable assumption since otherwise there is at least one redundant domain knowledge constraint and we can simply remove it. The number of domain knowledge constraints ($k_1$ and $k_2$) are usually small, so the system (\ref{eq:ksvm_admm_linsys_u}) can be solved exactly and efficiently by Cholesky factorization.
Solving for $\textbf{s}^{k+1}$ corresponding to $\textbf{u}^{k+1}$ is easy, observing that problem (\ref{eq:ksvm_admm_ussub}) is separable in the elements of $\textbf{s}$. For each element $s_i$, the optimal solution to the one-dimensional quadratic problem with a non-negative constraint on $s_i$ is given by $\max(0,u_i - \frac{(\gamma_5)_i}{\mu_5})$. Writing in the vector form, $\textbf{s}^{k+1} = \max(\textbf{0}, \textbf{u}^{k+1} - \frac{\gamma_5^k}{\mu_5})$.
Similarly, we solve for $\textbf{v}^{k+1}$ by introducing a non-negative slack variable $\textbf{t}$ and solve the linear system
\begin{equation}\label{eq:ksvm_admm_linsys_v}
(\rho_3DD^T + \mu_4 \textbf{g}\gbold^T + \mu_6 I)\textbf{v}^{k+1} = \textbf{r}_\textbf{v},
\end{equation}
where $\textbf{r}_\textbf{v} = \rho_3D\textbf{w}^{k+1} - \mu_4\textbf{g} b^{k+1} - \textbf{g}\gamma_4^k - \mu_4\textbf{g}(1-p^k) + \gamma_6 + \mu_6\textbf{t}^k$, and $\textbf{t}^{k+1} = \max(0,\textbf{v}^{k+1} - \frac{\gamma_6^k}{\mu_6})$.
Now given $(\textbf{w}^{k+1},b^{k+1},\textbf{u}^{k+1},\textbf{v}^{k+1})$, the solutions for $\textbf{a}$ and $\textbf{c}$ are exactly the same as in Lines \ref{line:a_sol} and \ref{line:c_sol} of Algorithm \ref{alg:hipad}, i.e.
\begin{eqnarray*}
\textbf{a}^{k+1} &=& \mathcal{S}_{\frac{1}{N\mu_1}}\left( \textbf{e} + \frac{\gamma_1^k}{\mu_1} - Y(X\textbf{w}^{k+1} + b^{k+1}\textbf{e}) \right), \\
\textbf{c}^{k+1} &=& \mathcal{T}_{\frac{\lambda_1}{\mu_2}}\left( \frac{\gamma_2^k}{\mu_2}+\textbf{w}^{k+1} \right).
\end{eqnarray*}
The subproblem with respect to $q$ is
\begin{eqnarray}\label{eq:ksvm_admm_qsub}
\nonumber \min_q && \quad \rho_2(q)_+ - \gamma_3^k q + \frac{\mu_3}{2}\|\textbf{d}^T\textbf{u}^k - b^{k+1} + 1 - q\|_2^2 \equiv \\
&& \rho_2(q)_+
+ \frac{\mu_3}{2}\|q - (\textbf{d}^T\textbf{u}^k - b^{k+1} + 1 + \frac{\gamma_3^k}{\mu_3})\|_2^2.
\end{eqnarray}
The solution is given by a (one-dimensional) proximal operator associated with the hinge loss
\begin{equation}\label{eq:ksvm_admm_qsol}
q^{k+1} = \mathcal{S}_{\frac{\rho_2}{\mu_3}}\left( \textbf{d}^T\textbf{u}^k - b^{k+1} + 1 + \frac{\gamma_3^k}{\mu_3} \right).
\end{equation}
Similarly, the subproblem with respect to $p$ is
\begin{equation*}
\min_p \quad\quad \rho_4(p)_+ - \gamma_4^k p + \frac{\mu_4}{2}\|\textbf{g}^T\textbf{v}^k + b^{k+1} + 1 - p\|_2^2,
\end{equation*}
and the solution is given by
\begin{equation}\label{eq:ksvm_admm_psol}
p^{k+1} = \mathcal{S}_{\frac{\rho_4}{\mu_4}}\left( \textbf{g}^T\textbf{v}^k + b^{k+1} + 1 + \frac{\gamma_4^k}{\mu_4} \right).
\end{equation}
Due to lack of space in the paper, we summarize the detailed solution steps in the Appendix (see Algorithm \ref{alg:admm_enk} ADMM-ENK)
Although there appears to be ten additional parameters (six $\rho$'s and four $\mu$'s) in the ADMM method for ENK-SVM, we can usually set the $\rho$'s to the same value and do the same for the $\mu$'s. Hence, in practice, there is only one additional parameter to tune, and our computational experience in Section \ref{sec:exp_ksvm} is that the algorithm is fairly insensitive to the $\mu$'s and $\rho$'s.
\subsection{IPM Phase}
The second phase for solving the knowledge-based SVM problem defined by (KSVM-P) follows the same steps as that described in section \ref{sec:ipm}. Note that in the knowledge-based case we have decided to solve the primal problem. This decision was based on extensive numerical experiments with both the primal and dual formulation which revealed that the primal formulation is more efficient.
We found in our experiments that by introducing slack variables and transforming the above problem into a linearly equality-constrained QP, Phase 2 of HIPAD usually requires less time to solve.
\subsection{HIPAD with domain knowledge incorporation}
We formally state the new two-phase algorithm for the elastic-net KSVM in Algorithm \ref{alg:hipad_ksvm}.
\begin{algorithm}
\caption{HIPAD-ENK}
\begin{algorithmic}[1]\label{alg:hipad_ksvm}
\STATE Given $\textbf{w}^0, b^0, \textbf{a}^0, \textbf{c}^0, \textbf{u}^0,\textbf{v}^0,p^0,q^0,\textbf{s}^0\geq\textbf{0},\textbf{t}^0\geq\textbf{0}$.
\STATE \textbf{PHASE 1: ADMM for ENK-SVM}
\STATE $(\textbf{w},b,\textbf{u},\textbf{v}) \gets $ADMM-ENK$(\textbf{w}^0, b^0, \textbf{a}^0, \textbf{c}^0, \textbf{u}^0,\textbf{v}^0,p^0,q^0,\textbf{s}^0,\textbf{t}^0)$
\STATE \textbf{PHASE 2: IPM for KSVM}
\STATE $\widetilde{\textbf{w}} \gets $ non-zero components of $\textbf{w}$
\STATE $(\widetilde{X}, \widetilde{Y}) \gets $ sub-matrices of $(X, Y)$ corresponding to the support of $\textbf{w}$
\STATE $\eta_u^0 \gets \textbf{d}^T\textbf{u}-b+1$
\STATE $\eta_v^0 \gets \textbf{g}^T\textbf{v}+b+1$
\STATE $(\textbf{w},b) \gets \textrm{SVM-IPM}$$(\widetilde{X}, \widetilde{Y}, \widetilde{\textbf{w}},b,\textbf{u},\textbf{v},\eta_u^0,\eta_v^0$)
\RETURN $(\textbf{w},b)$
\end{algorithmic}
\end{algorithm}
\section{Numerical results}
We present our numerical experience with the two main algorithms proposed in this paper: HIPAD and its knowledge-based version HIPAD-ENK. We compare their performance with their non-hybrid counterparts, i.e., ADMM-ENSVM and ADMM-ENK, which use ADMM to solve the original SVM problem.
The transition condition at the end of Phase 1 is specified in (\ref{eq:hipad_transition}), with $\epsilon_{tol} = 10^{-2}$. The stopping criteria for ADMM are as follows: $\frac{|F^{k+1}-F^k|}{\max\{1,|F^k|\}} \leq \epsilon_1, \|\textbf{a}-(\textbf{e}-\bar{X}\textbf{w}-\textbf{y} b)\|_2\leq \epsilon_1, \|\textbf{c}-\textbf{w}\|_2\leq \epsilon_1$ and $ \frac{\|\textbf{w}^{k+1}-\textbf{w}\|_2}{\|\textbf{w}^k\|_2} \leq \epsilon_2$, with $\epsilon_1 = 10^{-5}$, and $\epsilon_2 = 10^{-3}$.
\subsection{HIPAD vs ADMM}
To demonstrate the practical effectiveness of HIPAD, we tested the algorithm on nine real data sets which are publicly available. \textbf{rcv1} \cite{lewis2004rcv1} is a collection of manually categorized news wires from Reuters. The original multiple labels have been consolidated into two classes for binary classification. \textbf{real-sim} contains UseNet articles from four discussion groups, for simulated auto racing, simulated aviation, real autos, and real aviation. Both \textbf{rcv1} and \textbf{real-sim} have large feature dimensions but are highly sparse. The rest of the seven data sets are all dense data. \textbf{rcv1} and \textbf{real-sim} are subsets of the original data sets, where we randomly sampled 500 training instances and 1,000 test instances. \textbf{gisette} is a handwriting digit recognition problem from NIPS 2003 Feature Selection Challenge, and we also sub-sampled 500 instances for training. (For testing, we used the original test set of 1,000 instances.) \textbf{duke}, \textbf{leukemia}, and \textbf{colon-cancer} are data sets of gene expression profiles for breast cancer, leukemia, and colon cancer respectively. \textbf{fMRIa}, \textbf{fMRIb}, and \textbf{fMRIc} are functional MRI (fMRI) data of brain activities when the subjects are presented with pictures and text paragraphs. The data was compiled and made available by Tom Mitchell's neuroinformatics research group \footnote{http://www.cs.cmu.edu/~tom/fmri.html}. Except the three fMRI data sets, all the other data sets and their references are available at the LIBSVM website \footnote{http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets}.
The parameters of HIPAD, ADMM-ENSVM, and LIBSVM were selected through cross validation on the training data.
We summarize the experiment results in Table \ref{tab:real_data_results}.
Clearly, HIPAD produced the best overall predication performance. In order to test the significance of the difference, we used the test statistic in \cite{iman1980approximations} based on Friedman's $\chi^2_F$, and the results are significant at $\alpha=0.1$. In terms of CPU-time, HIPAD consistently outperformed ADMM-ENSVM by several times on dense data.
The feature support sizes selected by HIPAD were also very competitive or even better than the ones selected by ADMM-ENSVM. In most cases, HIPAD was able to shrink the feature space to below 10$\%$ of the original size.
\begin{table*}
\begin{center}
\resizebox{12.25cm}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Data set} & \multicolumn{3}{|c|}{HIPAD} & \multicolumn{3}{|c|}{ADMM-ENSVM} & LIBSVM \\
\cline{2-7}
& Accuracy ($\%$) & Support size & CPU (s) & Accuracy ($\%$) & Support size & CPU (s) & accuracy ($\%$) \\
\hline
\textbf{rcv1} & \textbf{86.9} & 2,037 & 1.18 & 86.8 & 7,002 & 1.10 & 86.1 \\
\textbf{real-sim} & \textbf{94.0} & 2,334 & 0.79 & 93.9 & 2,307 & 0.31 & 93.4 \\
\textbf{gisette} & \textbf{94.7} & 498 & 8.96 & 63.1 & 493 & 45.87 & 93.4 \\
\textbf{duke} & \textbf{90} & 168 & 1.56 & \textbf{90} & 150 & 5.52 & 80 \\
\textbf{leukemia} & \textbf{85.3} & 393 & 1.70 & 82.4 & 717 & 6.35 & 82.4 \\
\textbf{colon-cancer} & \textbf{84.4} & 195 & 0.45 & \textbf{84.4} & 195 & 1.34 & \textbf{84.4} \\
\textbf{fMRIa} & \textbf{90} & 157 & 0.25 & \textbf{90} & 137 & 2.17 & 60 \\
\textbf{fMRIb} & \textbf{90} & 45 & 0.23 & \textbf{90} & 680 & 0.75 & \textbf{90} \\
\textbf{fMRIc} & \textbf{90} & 321 & 0.14 & \textbf{90} & 659 & 1.58 & \textbf{90} \\
\hline
\end{tabular}
}
\end{center}
\caption{Experiment results of HIPAD and ADMM-ENSVM on real data. The best prediction accuracy for each data set is highlighted in bold.}
\label{tab:real_data_results}
\end{table*}
\subsection{Simulation for Knowledge Incorporation}\label{sec:exp_ksvm}
We generated synthetic data to simulate the example presented at the beginning of Section \ref{sec:enk-svm} in the high dimensional feature space. Specifically, four groups of multi-variate Gaussians $K_1,\cdots,K_4$ were sampled from $\mathcal{N}(\mu_1^+, \Sigma_1), \cdots, \mathcal{N}(\mu_4^+, \Sigma_4)$ and $\mathcal{N}(\mu_1^-, \Sigma_1), \cdots, \mathcal{N}(\mu_4^-, \Sigma_4)$ for four disjoint blocks of feature values ($\textbf{x}_{K_1}, \cdots, \textbf{x}_{K_4}$). For positive class samples, $\mu_1^+ = 2, \mu_2^+ = 0.5, \mu_3^+ = -0.2, \mu_4^+ = -1$; for negative class samples, $\mu_1^- = -2, \mu_2^- = -0.5, \mu_3^- = 0.2, \mu_4^- = 1$. All the covariance matrices have 1 on the diagonal and 0.8 everywhere else. The training samples contain blocks $K_2$ and $K_3$, while all four blocks are present in the test samples. A random fraction (5\%-10\%) of the remaining entries in all the samples are generated from the standard Gaussian distribution.
The training samples are apparently hard to separate because the values of blocks $K_2$ and $K_3$ for the two classes are close to each other. However, blocks $K_1$ and $K_4$ in the test samples are well-separated. Hence, if we are given information about these two blocks as general knowledge for the entire population, we could expect the resulting SVM classifier to perform much better on the test data. Since we know the mean values of the distributions from which the entries in $K_1$ and $K_4$ are generated, we can supply the following information about the relationship between the block sample means and class membership to the KSVM:
$\frac{1}{L_1}\sum_{i\in K_1}x_i \geq 4 \Rightarrow \textbf{x} \in A^+$, and
$\frac{1}{L_4}\sum_{i\in K_4}x_i \geq 3 \Rightarrow \textbf{x} \in A^-$
where $L_j$ is the length of the $K_j$, $j=1,\cdots,4$, $A^+$ and $A^-$ represent the positive and negative classes, and the lowercase $x_i$ denotes the $i$-th entry of the sample $\textbf{x}$. Translating into the notation of (KSVM-P), we have
\begin{equation*}
\small
B = \left(
\begin{array}{cccccc}
\textbf{0} & \underbrace{-\frac{\textbf{e}}{L_1}^T } & \textbf{0} & \textbf{0} & \textbf{0} &\textbf{0} \\
& K_1 & & & & \\
\end{array}
\right)
, d = -4, \label{eq:enk_data_knowledge1} \mbox{ and }
D = \left(
\begin{array}{cccccc}
\textbf{0} & \textbf{0} & \textbf{0} & \textbf{0} & \underbrace{-\frac{\textbf{e}}{L_4}^T } & \textbf{0} \\
& & & & K_4 & \\
\end{array}
\right)
, g = -3
\end{equation*}
The information given here is not precise, in that we are confident that a sample should belong to the positive (or negative) class only when the corresponding block sample mean well exceeds the distribution mean. This is consistent with real-world situations, where the domain or expert knowledge tends to be conservative and often does not come in exact form.
We simulated two sets of synthetic data for ENK-SVM as described above, with $(N_{train} = 200, N_{test} = 400, m_{train} = 10,000)$ for \textbf{ksvm-s-10k} and $(N_{train} = 500, N_{test} = 1,000, m_{test} = 50,000)$ for \textbf{ksvm-s-50k}. The number of features in each of the four blocks ($K_1,K_2,K_3,K_4$) is 50 for both data sets.
Clearly, HIPAD-ENK is very effective in terms of speed, feature selection, and prediction accuracy on these two data sets. Even though the features in blocks $K_1$ and $K_4$ are not discriminating in the training data, HIPAD-ENK was still able to identify all the 200 features in the four blocks correctly and exactly.
This is precisely what we want to achieve as we explained at the beginning of Section \ref{sec:enk-svm}. The expert knowledge not only helps rectify the separating hyperplane so that it generalizes better on the entire population, but also makes the training algorithm realize the significance of the features in blocks $K_1$ and $K_4$.
\begin{table*}
\begin{center}
\resizebox{12.25cm}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Data set} & \multicolumn{3}{|c|}{HIPAD-ENK} & \multicolumn{3}{|c|}{ADMM-ENK} & LIBSVM \\
\cline{2-7}
& Accuracy ($\%$) & Support size & CPU (s) & Accuracy ($\%$) & Support size & CPU (s) & accuracy ($\%$) \\
\hline
\textbf{ksvm-s-10k} & \textbf{99} & 200 & 1.99 & 97.25 & 200 & 3.43 & 86.1 \\
\textbf{ksvm-s-50k} & \textbf{98.8} & 200 & 8.37 & 96.4 & 198 & 20.89 & 74.8 \\
\hline
\end{tabular}
}
\end{center}
\caption{Experiment results of HIPAD-ENK and ADMM-ENK on synthetic data. The best prediction accuracy for each data set is highlighted in bold.}
\label{tab:enk_results}
\end{table*}
\section{Conclusion}
We have proposed a two-phase hybrid optimization framework for solving the ENSVM, in which the first phase is solved by ADMM, followed by IPM in the second phase. In addition, we have proposed a knowledge-based extension of the ENSVM which can be solved by the same hybrid framework. Through a set of experiments, we demonstrated that our method has significant advantage over the existing method in terms of computation time and the resulting prediction accuracy. The algorithmic framework introduced in this paper is general enough and potentially applicable to other regularization-based classification or regression problems.
\bibliographystyle{abbrv}
|
2,877,628,090,238 | arxiv | \section*{Appendix: Analysis of $\overline{\text{P}}_{b}(t_w)$}
\addcontentsline{toc}{section}{Appendices}
\renewcommand{\thesubsection}{\Alph{subsection}}
In order to find the optimal value of $t_w$ and correspondingly $t_i$, the derivation of $\overline{\text{P}}_{b}(t_w)$ with respect to $t_w$ is given as follows
{ \begin{equation}
\frac{d \overline{\text{P}}_{b}(t_w)}{d t_w}=\text{PW}_3\frac{Y(t_w)}{\big ({w_1+w_2t_w+w_3e^{-\lambda t_w}}\big )^2},
\label{eq:Awe}
\end{equation}
\noindent where
\begin{align}
\label{eq:uuu}
Y(t_w)& =F_1+(F_2-\lambda F_3t_w)e^{-\lambda t_w} ,\\
\label{eq:f1}
F_1&=(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})(t_s e^{\lambda t_s}+\frac{{1}}{\lambda}) \times \nonumber \\
& \quad \big(\frac{1}{2}(t_{pd}-\phi t_{su})+\frac{{1}}{\lambda}+2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\big)-\nonumber\\
& \quad \big (\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})^2(\overline{\text{D}}_{\max}-\frac{t_s}{2})^2, \\
\label{eq:f2}
F_2&=-F_1+(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})^2 \times \nonumber \\ & \quad (t_s e^{\lambda t_s}+\frac{{1}}{{\lambda}})\big(-2+\lambda(2-\phi) t_{su}+\lambda t_{pd}\big), \\
\label{eq:f3}
F_3&= F_1-2(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})^2(t_s e^{\lambda t_s}+\frac{{1}}{{\lambda}}).
\end{align}}
{Based on typical values of $1\leq\phi< 2$, the condition $0 \leq (2-\phi) t_{su}+t_{pd}$ is met, and hence based on (\ref{eq:f1})-(\ref{eq:f3}), we can conclude that $F_1+F_2> 0, $ $F_2+F_3> 0 $ and $F_1 > F_3$. Furthermore, it can be shown that $F_1$ is a decreasing function of $\lambda$, which has a single root. We refer to its root as $\lambda_t$; where for all $\lambda <\lambda_t$, then $F_1>0$ while, for all $\lambda >\lambda_t$, then $F_1<0$. Root-finding algorithms can be utilized to find $\lambda_t$ as the $\lambda$ value that meets $F_1=0$.}
{To determine whether $\frac{d \overline{\text{P}}_{b}(t_w)}{d t_w}$ is positive or negative, $Y(t_w)$ needs to be analyzed.
\noindent By differentiating $Y(t_w)$ with respect to $t_w$, we obtain
\begin{align}
\frac{{d} Y(t_w)}{d t_w}=-\lambda(F_2+F_3-\lambda F_3 t_w)e^{-\lambda t_w}.
\label{eq:Awer}
\end{align}}
{Additionally, depending on whether $F_3$ and $F_1$ are positive or negative, the $\overline{\text{P}}_{b}(t_w)$ behaves differently. In order to characterize the behaviour of $\overline{\text{P}}_{b}(t_w)$, we define three mutually exclusive cases: Case A ($F_3>0$), Case B ($F_3<0$ and $F_1>0$), and Case C ($F_3<0$ and $F_1<0$). Due to fact that $F_1 > F_3$, in the former case, $F_1$ is always positive.}
Case A ($F_3>0$): {Based on (\ref{eq:Awer}), if $F_3$ is positive, $Y(t_w)$ is a decreasing function for the range of $t_w<\frac{F_2+F_3}{\lambda F_3}$ and an increasing function for $t_w>\frac{F_2+F_3}{\lambda F_3}$. Therefore, $Y(t_w)$ has a minimum point at $t_w=\frac{F_2+F_3}{\lambda F_3}$, where the $Y(t_w)$ at $t_w=\frac{F_2+F_3}{\lambda F_3}$ is positive, i.e., $Y(\frac{F_2+F_3}{\lambda F_3})=F_1-F_3e^{-\lambda t_w}>0$ (due to fact that $F_3<F_1$). As a result, $Y(t_w)$ is always positive and hence $\overline{\text{P}}_{b}(t_w)$ is a monotonous increasing function.}
{Based on (\ref{eq:Awer}), if $F_3$ is negative (Case B or Case C), we can conclude that $Y(t_w)$ is a monotonous decreasing function from $F_1+F_2>0$ to $F_1$.}
Case B ($F_3<0$ and $F_1>0$): {In this case, for all values of wake-up cycle, $Y(t_w)$ is positive, and hence $\overline{\text{P}}_{b}$ is a monotonous increasing function.}
Case C ($F_3<0$ and $F_1<0$): {In this case, $Y(t_w)$ for $t_w<t_{ws}$ is positive, and it is negative for $t_w>t_{ws}$; where $t_{ws}$ is a stationary point, i.e., $Y(t_{ws})=0$ or equivalently, $\frac{\partial \overline{\text{P}}_{b}(t_w)}{\partial t_w}\vert _{t_w=t_{w_s}}=0$. As a result, $\overline{\text{P}}_{b}(t_w)$ is an increasing function within $t_w<t_{ws}$, and a decreasing function for $t_w>t_{ws}$. For the typical range of parameters, we have consistently observed through simulations that $t_{w_s}<t_{w_b}$, so that we can conclude that $\overline{\text{P}}_{b}(t_w)$ is a decreasing function for the feasible range of the wake-up cycle (i.e., $t_{w}>t_{w_b}$).}
{To sum up, for $F_1<0$, or equivalently, ${\lambda}_t<{\lambda}$ (Case C), $\overline{\text{P}}_{b}(t_w)$ is a monotonous decreasing function while, for ${\lambda}<{\lambda}_t$ (Case A or B), $\overline{\text{P}}_{b}(t_w)$ is a monotonous increasing function. Due to the relevance of ${\lambda}_t$, we refer to it as the turnoff packet arrival rate. }
\section{Basic Wake-Up Radio Concept and Assumptions}
\label{sec:Background}
In the considered wake-up radio based scheme, or wake-up scheme (WuS) for short, as presented in \cite{Globecom}, the mobile device is configured to monitor a narrowband wake-up signaling channel in order to enhance its battery lifetime. Specifically, in every wake-up cycle (denoted by $t_w$), the WRx monitors the so-called physical downlink wake-up channel (PDWCH) for a specific on-duration time ($t_{on}$) in order to determine whether data has been scheduled or not. Occasionally, based on the interrupt signal from WRx, the BBU switches on, decodes both PDCCH and physical downlink shared channel (PDSCH), and performs normal connected-mode procedures. The WuS can be adopted for both connected and idle states of radio resource control \cite{Globecom}, and can be configured based on maximum tolerable paging delay that idle users may experience, or alternatively, based on the delay requirements of a specific traffic type at connected state.
The wake-up signaling per each WRx contains a single-bit control information, referred to as the wake-up indicator (WI), where a WI of $1$ indicates the WRx to wake up the BBU, because there is one or multiple packets to be received, while a WI of $0$ signals the opposite. Each WI is code multiplexed with a user-specific signature to selected time-frequency resources, as described in \cite{Globecom}. When a WI of $1$ is sent to WRx, the network expects the target mobile device to decode the PDCCH with a time offset identical to that of start-up time ($t_{su}$). Fig. \ref{fig:drxvswrx} (a) and (b) depict the basic operation and representative power consumption behavior of the conventional DRX-enabled cellular module and that of the cellular module with WRx, respectively, at a conceptual level. As illustrated, the WuS eliminates the unnecessarily wasted energy in the first and second DRX cycles, while also reducing the buffering delay compared to DRX. {Due to the specifically-designed narrowband signal structure of WuS, the WRx power consumption ($\text{PW}_{\text{1}}$) is much lower than that of BBU active, either due to packet decoding ($\text{PW}_{\text{2}}$) or when inactivity timer is running ($\text{PW}_{\text{3}}$) \cite{Globecom}. The lowest power consumption is obtained in sleep state ($\text{PW}_{\text{4}}$). We consider that during the BBU active states, the power consumption due to packet decoding is larger than or equal to that of running inactivity timer (i.e., $\text{PW}_{\text{2}}\geq \text{PW}_{\text{3}})$, but both are fixed. For presentation purposes, we denote the ratio of such power consumption at BBU active states as $\phi=\frac{\text{PW}_2}{\text{PW}_3}$, where $\phi\geq 1$.}
In general, because NR supports wide bandwidth operation, packets can be served in a very short time duration. In addition, in case the user packet sizes are small, packet concatenation in NR is used, so that all buffered packets in a relatively short wake-up cycle can be served in a single transmission time interval (TTI). Accordingly, we assume that radio-link control entity (located at the gNB) concatenates all those packets arriving during the sleep state, and as soon as the BBU is triggered on, the device (UE) can receive and decode the concatenated packets for a service time of $t_s$, which equals to a single TTI. During the serving time, if there was a new packet arrival, the BBU starts serving the corresponding packet by the end of $t_s$. In case that there was no packet arrival by the end of $t_s$, the UE initiates its inactivity timer with a duration of $t_i$. After the inactivity timer is initiated, and if a new PDCCH message is received before the time expiration, the BBU enters the active-decoding state and serves the packet. However, if there is no PDCCH message received before the expiration of the inactivity timer, a sleep period starts, the WRx-enabled cellular module switches to sleep state, and WRx operates according to its {wake-up cycle} \cite{Globecom}. For reference, in case of DRX, the BBU sleeps according to its short and long DRX patterns \cite{pm2,koc}.
The introduction of a PDWCH has two fundamental consequences, namely misdetections and false alarms \cite{Globecom}. In the latter case, WRx wakes up in a predefined time instant, and erroneously decodes a WI of $0$ as $1$, leading to unnecessary BBU power consumption. The former, in turn, corresponds to the case where a WI of $1$ is sent, but WRx decodes it incorrectly as $0$. Such misdetection adds an extra delay and wastes radio resources. {We denote the probability of misdetection and the probability of false alarm as $P_{md}$ and $P_{fa}$, respectively.} The requirements for the probability of misdetection of PDWCH are eventually stricter than those of the probability of false alarm \cite{Globecom}.
\begin{figure}[!t]
\centering
\includegraphics[scale=1.1]{new_wus_drx.pdf}
\caption{{Power consumption profiles of (a) a typical DRX mechanism, and (b) a WRx-enabled cellular module.} }
\label{fig:drxvswrx}
\end{figure}
One of the new features of 5G NR networks to reach their aggressive requirements is latency-optimized frame structure with flexible numerology, providing subcarrier spacings ranging from $15$ kHz up to $240$ kHz with a proportional change in cyclic prefix, symbol length, and slot duration \cite{TS38300}. Regardless of the numerology used, the length of one radio frame is fixed to $10$ ms and the length of one subframe is fixed to $1$ ms~\cite{nr2017}, as in 4G LTE/LTE-Advanced. However, in NR, the number of slots per subframe varies according to the numerology that is configured. Additionally, in order to support further reduced latencies, the concept of mini-slot transmission is introduced in NR, and hence the TTI varies depending on the service type ranging from one symbol, to one slot, and to multiple slots~\cite{nr2017}. In this work, in order to provide consistent and exact timing definitions, different time intervals of the wake-up related procedures are defined as integer multiples of a TTI. {Additionally, according to 3GPP, the packet service time ($t_s$) is one TTI, in which multiple packets can be concatenated.} Furthermore, for the sake of clarity, a TTI duration of $1$ ms is taken as the baseline system assumption for the WuS configurations, which then facilitates applying the proposed concepts also in future evolution of LTE-based systems.
{Finally, it is important to note that from a system-level point of view, the configurable parameters of the WuS are the wake-up cycle ($t_w$) and the inactivity timer ($t_i$), whose values we will optimize in Section~\ref{sec:prob}. The remaining parameters ($t_{on}$, $t_{pd}$, $t_{su}$, $t_{s}$) depend on physical constraints and signal design, and accordingly we assume them to be fixed, i.e., the optimization will be done for fixed (given) values of $t_{on}$, $t_{pd}$, $t_{su}$ and $t_{s}$.}
\section{Conclusions and Future Work}
\label{sec:conc}
In this article, wake-up based downlink access under delay constraints was studied in the context of 5G NR networks, with particular focus on energy-efficiency optimization. It was shown that the performance of the wake-up scheme is governed by a set of two parameters that interact with each other in an intricate manner. To find the optimal wake-up parameters configuration, and thus to take full advantage of the power saving capabilities of the wake-up scheme, a constrained optimization problem was formulated, together with the corresponding closed-form solution. Analytical and simulation results showed that the proposed scheme is an efficient approach to reduce the device energy consumption, while ensuring a predictable and consistent latency. The numerical results also showed that the optimized wake-up system outperforms the corresponding optimized DRX-based reference system in power efficiency. Furthermore, the range of packet arrival rates within which the WuS works efficiently was established, while outside that range other power saving mechanisms, such as DRX or microsleep, can be used.
Future work includes extending the proposed framework to bidirectional communication scenarios with the corresponding downlink and uplink traffic patterns and the associated QoS requirements, as well as to consider other realistic assumptions that impact the energy-delay trade-offs, such as the communication rate and scheduling delays. Additionally, an interesting aspect is to investigate how to configure the wake-up scheme parameters for application-specific traffic scenarios, such as virtual and augmented reality, when both uplink and downlink traffics are considered. Finally, focus can also be given to optimizing the wake-up scheme parameters based on the proposed framework, by utilizing not only traffic statistics but also short-term traffic pattern prediction by means of modern machine learning methods.
\section{Discussions and Final Remarks}
\label{sec:remark}
In this section, three interesting remarks are drawn and discussed.
\vspace{2mm}
\textbf{Remark 1:} The proposed WuS is fully independent of DRX, which means that both methods can co-exist, interact and be used together to reduce energy consumption of the UE even further. Based on the numerical results provided in this paper, our opinion regarding the power saving mechanisms for moderate and generic mobile users is that there is no 'One-Size-Fits-All Solution', unless the UE is well-defined and narrowed to a specific application and QoS requirement. For a broad range of applications and QoS requirements, there is a need for combining and utilizing different power saving mechanisms, and selecting the method that fits best for particular circumstances. For example, the WuS can be utilized for packet arrival rates lower than the turnoff packet arrival rate; for higher packet arrival rates or shorter delay bounds (e.g., smaller than $30$ ms), DRX may eventually be the preferable method of choice; for ultra-low latency requirements, other power saving mechanisms may be needed, building on, e.g., microsleep \cite{Lauridsenthesis} or pre-grant message \cite{pregrant,8616818} concepts. Further, depending on whether an RRC context is established or not, the WuS is agnostic to the RRC states, and can be adopted for idle (delay bound is in range of some hundreds of milliseconds), inactive, and connected modes.
\begin{table*}
\small
\centering
\renewcommand{\arraystretch}{1.4}
\caption{{Minimum power consumption values [mW] as function of TTI size, delay bounds and packet arrival rates}
\label{tti}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$\lambda$ [p/ms]& \multicolumn{3}{c|}{$0.01$} & \multicolumn{3}{c|}{$0.08$} & \multicolumn{3}{c|}{$0.15$} \\ \hline
$\overline{\text{D}}_{\max} $[ms] &$30$&$75$&$500$&$30$&$75$&$500$&$30$&$75$&$500$\\ \hline
$\overline{\text{P}}_c$ @ TTI$~=1$ ms& 54.2 &31.6&6.6 &88.7 & 39& 6.1 &88.9&38.1&5.9 \\ \hline
$\overline{\text{P}}_c$ @ TTI$~=500$ $\mu$s& 50.4&29.4 &5.7 & 83.7 &36.8& 5.8 &85.4 &36.6 &5.7\\ \hline
$\overline{\text{P}}_c$ @ TTI$~=250$ $\mu$s& 48.3& 28.1 &5.5& 81.1 &35.7& 5.7& 83.7&35.9 &5.6 \\ \hline
$\overline{\text{P}}_c$ @ TTI$~=125$ $\mu$s& 47.5&27.7 &5.4& 80.1 &35.3& 5.6& 83.1&35.7 &5.6 \\ \hline
\end{tabular}
\end{table*}
\vspace{2mm}
\textbf{Remark 2:} As mentioned in Section \ref{sec:sysmodel}, a latency-optimized frame structure with flexible numerology is adopted in 5G NR, for which the {slot length scales down when the numerology increases} \cite{nr2017}. In this work, different time intervals within the WuS are defined as multiples of a time unit of a TTI with a duration of $1$ ms. As shown before, the minimum power consumption over the boundary is limited by the minimum feasible value of $t_i$. Therefore, if the TTI can be selected even smaller, i.e., with finer granularity, the optimal power consumption can be further reduced. In this line, Table \ref{tti} presents the power consumption with different TTI sizes (corresponding to NR numerologies 0, 1, 2, and 3~\cite{TS38300}), delay bounds, and packet arrival rates.
Interestingly, Table \ref{tti} shows how the {5G NR numerologies} facilitate the use of WuS and improve the
applicability and energy saving potential of WuS compared to longer TTI sizes. Besides the smaller $t_i$ sizes, with smaller TTIs, the corresponding optimal wake-up cycles are more fine-grained. With shorter TTI sizes down to $125$ $\mu$s, the proposed WuS can provide up to $12 \% $ additional energy savings compared to the baseline $1$ ms TTI. Therefore, on average, the benefit of the flexible NR frame structure is not only for low latency communication but it can also offer energy savings depending on the traffic arrival rates and delay constraints.
\vspace{2mm}
\textbf{Remark 3:} The traffic model assumed in this article is basically well-suited for the periodic nature of DRX. For instance, voice calls and video streaming have such periodic behaviour. However, in other application areas such as MTCs, where sensors can be aperiodically polled by either a user or a machine, the traffic will have more non-periodic patterns. In such case, the DRX may not fit well, while the WuS has more suitable characteristics, being more robust and agnostic to the traffic type.
\section{Introduction}
\label{sec:intro}
In order for the emerging fifth generation (5G) mobile networks to satisfy the ever-growing needs for higher data-rates and network capacities, while simultaneously facilitating other quality of service (QoS) improvements, computationally-intensive physical layer techniques and high bandwidth communication are essential~\cite{nr2017}, \cite{Boccardi2014}. At the same time, however, the device power consumption tends to increase which, in turn, can deplete the mobile device's battery very quickly. {Moreover, it is estimated that feature phones and smartphones consume 2 kWh/year and 7 kWh/year, respectively, based on charging every 60$^{th}$ hour equal to 40\% of battery capacity every day and a standby scenario of 50\% of the remaining time \cite{Fehske}. Also, the carbon footprints of production of feature phones and smartphones are estimated to be 18 kg and 30 kg CO2e per device, respectively, which still is the major contributor of CO2 emission of mobile communication systems \cite{Fehske}. }
In general, battery lifetime is one of the main issues that mobile device consumers consider important from device usability point of view~\cite{Qualcomm2013}. However, since the evolution of battery technologies tends to be slow~\cite{Lauridsenthesis}, the energy efficiency of the mobile device's main functionalities, such as the cellular subsystem, needs to be improved~\cite{Qualcomm2013,Carroll2010,Lauridsen2015}. Furthermore, since the data traffic has been largely downlink-dominated~\cite{ITU2015}, the power saving mechanisms for cellular subsystems in receive mode are of great importance.
The 3rd generation partnership project (3GPP) has specified discontinuous reception (DRX) as one of the \emph{de facto} energy saving mechanism for long-term evolution (LTE), LTE-Advanced and 5G New Radio (NR) networks \cite{TS38300, erik, tr36.213, TS38.30}. DRX allows the mobile device to reduce its energy consumption by switching off some radio modules for long periods of time, activating them only for short intervals. To this end, the modeling and optimization of DRX mechanisms have attracted a large amount of research interest in recent years. The authors in \cite{Liu} proposed an adaptive approach to configure DRX parameters according to users’ activities, aiming to balance power saving and packet delivery latency. Koc \textit{et al.}~formulated the DRX mechanism as a multi-objective optimization problem in \cite{koc}, satisfying the latency requirements of active traffic flows and the corresponding preferences for power saving. In \cite{Mihov}, DRX is modeled as a semi-Markov process with three states (active, light-sleep, and deep-sleep), and the average power consumption as well as the average delay are calculated and optimized. Additionally, the authors in \cite{Ramazanali} utilized exhaustive search over a large parameter set to configure all DRX parameters. Such a method may not be attractive from a computational complexity perspective for real-time/practical applications, however, it provides the optimal DRX-based power consumption and is thus used in this article as a benchmark.
To improve the device's energy-efficiency beyond the capabilities of ordinary DRX, the concept of wake-up radio based access has been discussed, e.g., in \cite{Lauridsenthesis, Demirkol2009, Mazloum2014}. Specifically, in the cellular communications context, the wake-up radio based approaches have been recently discussed and described, e.g., in~\cite{Globecom} and \cite{Lauridsen2016}.
In such concept, the mobile device monitors only a narrowband control channel signaling referred to as wake-up signaling at specific time instants and subcarriers, in an OFDMA-based radio access systems such as LTE or NR, in order to decide whether to process the actual upcoming physical downlink control channel (PDCCH) or discard it. Compared to DRX-based systems, this directly reduces the buffering requirements and processing of empty subframes as well as the corresponding power consumption. Furthermore, in~\cite{Globecom}, the concept of a low-complexity wake-up receiver (WRx) was developed to decode the corresponding wake-up signaling, and to acquire the necessary time and frequency synchronization. {A wake-up scheme that enhances the power consumption of machine type communications (MTC) is introduced in 3GPP LTE Release-15 \cite{TS36.300}, which is based on a narrowband signal, transmitted over the available symbols of configured subframes. It is also considered as the starting point of NR power saving study item in 3GPP NR Release-17~\cite{NR_PS}.}
In general, the existing wake-up concepts and algorithms, such as those described in \cite{Globecom,Lauridsenthesis,Demirkol2009,Mazloum2014, Lauridsen2016,wpwrx, Tang, Wilhelmsson, Kouzayha,Oller,Aoudia, ponna2018saving, eee1, eee5}, build on static operational parameters that are determined by the radio access network at the start of the user’s session, and kept invariant, even if traffic patterns change. Accordingly, methods to optimize such parameters that characterize the employed wake-up scheme are needed, to further reduce energy consumption according to the traffic conditions. This is one of the main objectives of this paper. Specifically, the main contributions of this paper are as follows. Firstly, the wake-up radio based access scheme is modeled by means of a semi-Markov process. {In the model, we consider realistic WRx operation by introducing start-up and power-down periods of the baseband unit (BBU), false alarm and misdetection probabilities of the wake-up signaling, as well as the packet service time.} With such a model, the average power consumption and buffering delay can be accurately quantified and estimated for a given set of wake-up related parameters. Secondly, by utilizing such a mathematical model, the minimization of terminal's power consumption under Poisson traffic model is addressed for a given delay constraint. As a result, a closed-form optimal solution for the operational parameters is obtained. Furthermore, the range of packet arrival rates, for which the wake-up scheme is suitable and energy-efficient, is determined. Finally, simulation-based numerical results are provided in order to validate the proposed model and methods as well as to investigate the power consumption of our proposed solution compared to the optimized DRX-based reference mechanism proposed in \cite{Ramazanali}. The approach described in \cite{Ramazanali} is selected as the benchmark since it provides the optimal power consumption in DRX-based reference systems. Furthermore, to the best of our knowledge, virtually all of the DRX-based literature ignores the start-up and power-down energy consumption, {and \cite{Ramazanali} also neglects the packet service time}. Therefore, we have modified the approach in \cite{Ramazanali} slightly in order to consider such additional energy consumption in the optimization.
The rest of this paper is organized as follows. Section~\ref{sec:Background} presents a brief review of the considered wake-up scheme and its corresponding parameters, and defines basic system assumptions. In
Section~\ref{sec:sysmodel}, we model the wake-up based access scheme by means of a semi-Markov process and derive the power consumption as well as the buffering delay. Then, building on these mathematical models, in Section~\ref{sec:prob}, the optimization problem is formulated and the optimal solution for minimum power consumption is found in closed-form. These are followed by numerical results, remarks, and conclusions in Sections~\ref{sec:eval}, \ref{sec:remark}, and~\ref{sec:conc}, respectively. Some details on the analysis related to the modeling of power consumption are reported in the Appendix. For readers' convenience, the most relevant variables used throughout this paper are listed in Table~\ref {tab:vari}. Terminology-wise, we use gNB to refer to the base-station unit and UE to denote the mobile device, according to 3GPP NR specifications \cite{TS38300}.
\begin{table}[!t]
\scriptsize
\centering
\renewcommand{\arraystretch}{1.3}
\caption{Most important variables used throughout the article }
\label{tab:vari}
\begin{tabular}{cclll}
\cline{1-2}
\multicolumn{1}{|c|} {Variable} & \multicolumn{1}{c|}{Definition } & & & \\ \hhline{==~}
\multicolumn{1}{|c|}{ $\text{PW}_{\text{1}}$ } & \multicolumn{1}{c|}{{power consumption of cellular module while WRx is active}} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\text{PW}_{\text{2}}$ } & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}power consumption of cellular module \\ at active-decoding state (BBU is active) \end{tabular}} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\text{PW}_{\text{3}}$ } & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}power consumption of cellular module at\\ active-inactivity timer state (BBU is active) \end{tabular}}& & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\text{PW}_{\text{4}}$ } & \multicolumn{1}{c|}{{power consumption of cellular module at sleep state}} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\text{S}_{k}$ } & \multicolumn{1}{c|}{UE state in the state machine modeling {($\text{S}_1$, \dots, $\text{S}_4$)}} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $S(\tau_n)$ } & \multicolumn{1}{c|}{{UE state at the $\tau_n$ jump time} } & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\text{P}_{kl}$ } & \multicolumn{1}{c|}{transition probability from state $\text{S}_{k}$ to state $\text{S}_{l}$ } \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\omega_{k}$ } & \multicolumn{1}{c|}{holding time for state $\text{S}_{k}$ } & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$t_{p}$} & \multicolumn{1}{c|}{inter-packet arrival time } & & & \\
\cline{1-2}
\multicolumn{1}{|c|}{$\lambda$} & \multicolumn{1}{c|}{packet arrival rate } & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$t_{su}$} & \multicolumn{1}{c|}{start-up time of cellular module } & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $t_{pd}$ } & \multicolumn{1}{c|}{power-down time of cellular module} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $t_{w}$ } & \multicolumn{1}{c|}{wake-up cycle} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $t_{i}$ } & \multicolumn{1}{c|}{length of inactivity timer} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $t_{s}$ } & \multicolumn{1}{c|}{{
packet service time}} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $t_{on}$ } & \multicolumn{1}{c|}{on-duration time} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\overline{\text{P}}_c$ } & \multicolumn{1}{c|}{average power consumption} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\overline{\text{P}}_b$ } & \multicolumn{1}{c|}{average power consumption over boundary constraint} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\overline{\text{D}}$ } & \multicolumn{1}{c|}{average buffering delay} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{ $\overline{\text{D}}_{\max}$ } & \multicolumn{1}{c|}{maximum tolerable delay or delay bound} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$t_{w_b}$} & \multicolumn{1}{c|}{minimum feasible wake-up cycle over boundary constraint} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$t_w^*$} & \multicolumn{1}{c|}{optimal value of wake-up cycle} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$t_i^*$} & \multicolumn{1}{c|}{optimal value of inactivity timer} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$\lambda_t$} & \multicolumn{1}{c|}{turnoff packet arrival rate} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$\eta$} & \multicolumn{1}{c|}{relative power saving factor} & & & \\ \cline{1-2}
\multicolumn{1}{|c|}{$\phi$} & \multicolumn{1}{c|}{{power consumption ratio of UE at S$_2$ and S$_3$}} & & & \\ \cline{1-2}
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & & &
\end{tabular}
\end{table}
\section*{Acknowledgment}
This work has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk\l{}odowska-Curie grant agreement No. 675891 (SCAVENGE), Tekes TAKE-5 project, {Spanish MINECO grant
TEC2017-88373-R (5G-REFINE), and Generalitat de Catalunya grant 2017 SGR 1195.}
\section{Optimization Problem Formulation and Solution}
\label{sec:prob}
In this section, dual-parameter ($t_w$ and $t_i$)
constrained optimization
problem is formulated with the objective of minimizing the UE power consumption under a buffering delay constraint. Specifically, the average buffering delay is constrained to be less than or equal to a predefined maximum tolerable delay or delay bound, denoted by ${\overline{\text{D}}}_{\max}$, whose value is set based on the service type. To this end, building on the modeling results of the previous section, the optimization problem is now formulated as follows
\begin{eqnarray}
\mathop{\text{minimize}}_{t_w,t_i} && \overline{\text{P}}_c(t_w,t_i) \label{eq:i1} \ \ \\
\text{subject to} && \overline{\text{D}}(t_w,t_i)\leq \overline{\text{D}}_{\max},
\label{eq:c1} \\
&& t_{w}, t_i \in \{1,2,...\},
\label{eq:c6}
\end{eqnarray}
\noindent {where $\overline{\text{P}}_c(t_w,t_i)$ and $\overline{\text{D}}(t_w,t_i)$ are defined in Eq.~\eqref{eq:pc_row_2} and Eq.~\eqref{eq:d1_equation}, respectively.}
The resulting optimization problem in (\ref{eq:i1})-(\ref{eq:c6}) belongs to a class of intractable {mixed-integer non-linear programming (MINLP)} problems \cite{mnlp}. In this work, the corresponding {MINLP} is solved by using the equivalent {non-linear programming problem} with continuous variables, expressed below in (\ref{eq:i2})-(\ref{eq:c7}), which is obtained by means of relaxing the second constraint (\ref{eq:c6}) into a continuous constraint (see Eq. (\ref{eq:c7})), assuming that both parameters are positive real numbers larger than or equal to one (i.e., the minimum TTI unit). The relaxed optimization problem can be expressed as
\begin{eqnarray}
\mathop{\text{minimize}}_{t_w,t_i} && \overline{\text{P}}_c(t_w,t_i) \label{eq:i2} \ \ \\
\text{subject to} && \overline{\text{D}}(t_w,t_i)\leq \overline{\text{D}}_{\max},
\label{eq:c2} \\
&& t_{w} \geq 1 , t_i \geq 1.
\label{eq:c7}
\end{eqnarray}
In general, the optimization problem in \eqref{eq:i2}-\eqref{eq:c7} is not {jointly convex in} $t_w$ and $t_i$. Therefore, finding the global optimum is a challenging task. However, in the next subsections, we exploit the increasing/decreasing properties of the power consumption and delay expressions that we have derived in Section \ref{sec:sysmodel}, in order to derive additional properties of the problem that will allow us to find the optimal solution in closed form.
\subsection{Unbounded Feasible Region}
\label{sec:feasib}
In this section, a schematic approach is used to illustrate the feasible region for the relaxed optimization problem in (\ref{eq:i2})-(\ref{eq:c7}) and then the feasible region is narrowed down to the boundary of the delay constraint, whose points are proved to remain candidate solutions while the other feasible solutions are henceforth excluded.
{Fig. \ref{fig:she_proof} (a) and (b)} show the increasing trend of the power consumption and the decreasing behaviour of the delay constraint as a function of $t_i$, while $t_w$ is fixed at $t_{w_0}$, i.e. $\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_i}>0$ and $\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_i}<0$ (as proved in Section \ref{sec:sysmodel}). Let us consider an arbitrary point $A$ in the interior of the feasible region ($t_{i_A} > t_{i_m}$, where $\overline{\text{D}}(t_{w_0},t_{i_m})=\overline{\text{D}}_{\max}$). As it can be seen from {Fig. \ref{fig:she_proof} (a) and (b)}, there is always a point on the boundary of the delay constraint, referred to as $B$ ($t_{i_B}=t_{i_m}$), where its power consumption $\overline{\text{P}}_{c_B}$ is lower than that of $A$ ($\overline{\text{P}}_{c_B}<\overline{\text{P}}_{c_A}$). Hence, we can conclude that, for any fixed $t_w$, under a given delay constraint, there is a point on the boundary that attains the lowest power consumption.
Similarly, {Fig. \ref{fig:she_proof} (c) and (d)} show the decreasing trend of the power consumption and increasing behaviour of the delay constraint as a function of $t_w$, while $t_i$ is fixed at $t_{i_0}$, i.e. $\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_w}<0$ and $\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_w}>0$ (as proved in Section \ref{sec:sysmodel}). Consider an arbitrary point $C$ in the interior of the feasible region ($t_{w_C} < t_{w_m}$ where $\overline{\text{D}}(t_{w_m},t_{i_0})=\overline{\text{D}}_{\max}$). As it can be seen from {Fig. \ref{fig:she_proof} (c) and (d)}, there is always a point on the boundary of the delay constraint, referred to as $D$ ($t_{w_D}=t_{w_m}$), where its power consumption $\overline{\text{P}}_{c_D}$ is lower than that of $C$ ($\overline{\text{P}}_{c_D}<\overline{\text{P}}_{c_C}$). Then, we can conclude that, for any fixed $t_i$, under a given delay constraint, there is a point on the boundary that attains the lowest power consumption.
Therefore, because for both scenarios (fixed $t_w$ and fixed $t_i$), the lowest power consumption occurs at the boundary of the delay constraint, we can conclude that the optimal point cannot be located in the interior of the feasible region, but rather it lies over the boundary. That is, any arbitrary point $(t_w,t_i)$ in the feasible region of the relaxed optimization problem in (\ref{eq:i2})-(\ref{eq:c7}) cannot be an optimal point, unless it lies on the boundary (rather than the interior) of the delay constraint, i.e., $\overline{\text{D}}(t_w,t_i)= \overline{\text{D}}_{\max}$.
\begin{figure}
\centering
\includegraphics[width=0.50 \textwidth]{newproof.pdf}
\caption{Schematic proof that optimal point lies over the boundary.}
\label{fig:she_proof}
\end{figure}
\subsection{Power Consumption over Boundary}
\label{sec:pcob}
Next, the equation of the boundary curve (expressed through $t_i$ as a function of $t_w$) is derived, and then the power consumption profile of all points on the boundary is calculated as well as formulated as a function of $t_w$ only. In particular, the boundary curve can be obtained by finding all the solutions for which the inequality constraint in (\ref{eq:c2}) is satisfied with equality, while the constraint in (\ref{eq:c7}) is met, i.e.,
\begin{align}
\overline{\text{D}}(t_w,t_i)=\overline{\text{D}}_{\max} \text{~for all~} t_{w}, t_i \geq 1.
\label{eq:c22}
\end{align}
By utilizing Eq. (\ref{eq:d1_equation}), we can isolate $t_i$, and the boundary curve can be formulated as follows
\begin{equation}
\begin{split}
t_i(t_w)=\frac{1}{\lambda}\ln \Big(\frac{{t_w+(t_{su}-\frac{1}{\lambda}){(1-e^{-\lambda t_w})}}-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})}{(\overline{\text{D}}_{\max}-\frac{t_s}{2})(1-e^{-\lambda t_w})(1+e^{\lambda t_s})}\Big) \\\text{for all~} t_{w}\geq t_{w_b},
\label{eq:bn}
\end{split}
\end{equation}
\noindent where $t_{w_b}$ (see Eq. (\ref{eq:twb})) is the minimum feasible value of $t_{w}$ over the boundary.
{By using Eq. (\ref{eq:bn}), one can show that $t_i(t_w)$ is an increasing function with respect to $t_w$ on any feasible $t_w$ point over the boundary of the delay constraint (i.e., $\frac{d t_i(t_w)}{d t_w} \geq 0$), as follows.} Let us use the composite function rule over (\ref{eq:bn}), so that $\frac{d t_i(t_w)}{d t_w}$ can be calculated as follows
\begin{equation}
\frac{d t_i(t_w)}{d t_w} =\frac{1}{\lambda}\frac{d \ln (\text{Arg})}{d \text{Arg}}\frac{d \text{Arg}(t_w)}{d t_w},
\label{eq:part}
\end{equation}
\noindent where $\text{Arg}$ refers to the argument of the logarithm in (\ref{eq:bn}). Since the logarithmic function is monotonically increasing ($\frac{d \ln(\text{Arg})}{d \text{Arg}} \geq 0$), it is sufficient to prove that $\text{Arg}(t_w)$ is increasing with respect to $t_w$,
{ \begin{equation}
\frac{d \text{Arg}(t_w)}{d t_w}=\frac{1-(1-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\lambda +\lambda t_w)e^{-\lambda t_w}}{(\overline{\text{D}}_{\max}-\frac{t_s}{2})(1+e^{\lambda t_s})(1-e^{-\lambda t_w})^2},
\end{equation}
\noindent from which we can write
\begin{equation}
\begin{split}
1-(1+\lambda t_w)e^{-\lambda t_w}>0\Longrightarrow \\
1-(1-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\lambda +\lambda t_w)e^{-\lambda t_w} \geq 0 \Longrightarrow \\
\frac{d \text{Arg}(t_w)}{d t_w} \geq 0.
\end{split}
\label{eq:part1}
\end{equation}}
\noindent Therefore, we can conclude that $\frac{d t_i(t_w)}{d t_w} \geq 0$.
Additionally, one can prove that $t_i=1$ and $t_w=t_{w_b}$ (in which $t_{w_b}$ always exists, and is larger than or equal to one) is located over the boundary, as follows. Based on Eq. (\ref{eq:bn}), and by induction on $t_i=1$, we can write
{ \begin{equation}
\begin{split}
e^{\lambda}=\frac{{t_{w}+(t_{su}-\frac{1}{\lambda}){(1-e^{-\lambda t_{w}})}}-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})}{(\overline{\text{D}}_{\max}-\frac{t_s}{2})(1-e^{-\lambda t_{w}})(1+e^{\lambda t_s})} \Longrightarrow \\
e^{-\lambda t_{w}}=-\frac{-\lambda t_{w} {+\big((e^{\lambda}(1+e^{\lambda t_s})+2)\overline{\text{D}}_{\max} -t_{su}\big)\lambda+1}}{\lambda t_{su}-e^{\lambda}(\overline{\text{D}}_{\max}-\frac{t_s}{2})(1+e^{\lambda t_s})\lambda-1} \Longrightarrow \\
e^{-\lambda t_{w}+F}=-e^{F} \frac{-\lambda t_{w} +F}{H}\Longrightarrow \\
{-\lambda t_{w}+F}=- \mathcal W\Big(\frac{H}{e^{F}}\Big)\Longrightarrow \\
t_{w} = \frac{1}{\lambda} \Big(F+\mathcal W\Big(\frac{H}{e^{F}}\Big)\Big),
\label{eq:sss}
\end{split}
\end{equation}}
\noindent where {$F= \big((e^{\lambda}(1+e^{\lambda t_s})+2)(\overline{\text{D}}_{\max}-\frac{t_s}{2}) -t_{su}\big)\lambda+1$, $H=\lambda t_{su}-e^{\lambda}(\overline{\text{D}}_{\max}-\frac{t_s}{2})(1+e^{\lambda t_s})\lambda-1$} and $\mathcal W(x)$ is the Lambert W function \cite{Corless1996}. For typical $\overline{\text{D}}_{\max}$ and $t_{su}$ values, $1 \ll \frac{F}{\lambda}$ and $H<0$, therefore, the main branch of the Lambert W function ($\mathcal W_0$) can be considered as a solution for (\ref{eq:sss}) that has a value greater than $-1$. Then, we can conclude that
\begin{equation}
\label{eq:twb}
t_{w_b}=\frac{1}{\lambda} \Big(F+\mathcal W_0\Big(\frac{H}{e^{F}}\Big)\Big)\geq 1,
\end{equation}
\noindent and, as a result, all feasible points on the boundary curve can be specified and constrained by $t_i\geq 1$ and $t_w \geq t_{w_b}$.
The $t_{w_b}$ in \eqref{eq:twb} is the smallest feasible $t_w$ over the boundary curve because if we assume that there is a $t_w$ smaller than $t_{w_b}$, based on Eq. (\ref{eq:part}) and (\ref{eq:part1}), its corresponding $t_i$ should become smaller than one, which belongs to the unfeasible region. Therefore, based on proof-by-contradiction, ($t_i=1$, $t_w=t_{w_b}$) lies over the corner part of the boundary. Consequently, ${t_i \geq1}$ and ${t_w \geq t_{w_b}}$ are
equivalent constraints of the boundary of the delay constraint. Therefore, the point ($t_i=1$, $t_w=t_{w_b}$) is an extreme point, and it is located over the boundary curve of the delay constraint, where $t_{w_b}$ is larger than or equal to one.
Finally, by substituting the value of $e^{\lambda t_i}$ over the boundary (argument of logarithm in (\ref{eq:bn})) into (\ref{eq:pc_row_2}), the average power consumption of all the points over the boundary, referred to as $\overline{\text{P}}_b(t_w)$, can be obtained as follows
{ \begin{equation}
\begin{split}
\overline{\text{P}}_{b}(t_w)=\text{PW}_3\frac{u_1+u_2 t_w+u_3e^{-\lambda t_w}}{w_1+w_2t_w+w_3e^{-\lambda t_w}} \text{~~~~for all~} t_{w}\geq t_{w_b},
\label{eq:xc}
\end{split}
\end{equation}}
\noindent where
{ \begin{align}
\begin{split}
u_1&=\big (\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})+\\ & \quad (t_s e^{\lambda t_s}+\frac{1}{\lambda})\big(t_{su}-\frac{1}{\lambda}-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\big),\end{split}\\
\label{eq:Ab}
u_2&=t_s e^{\lambda t_s}+\frac{1}{\lambda},\\
\begin{split}
u_3&= -\big (\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})-\\ & \quad (t_s e^{\lambda t_s}+\frac{1}{\lambda})\big(t_{su}-\frac{1}{\lambda}\big), \end{split} \\ \begin{split}
w_1&=\big (\frac{1}{2}(t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})+\\ & \quad (t_s e^{\lambda t_s}+\frac{1}{\lambda})\big(t_{su}-\frac{1}{\lambda}-2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\big), \end{split} \\
w_2&=t_s e^{\lambda t_s}+\frac{1}{\lambda}+(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2}), \\
\begin{split}
w_3&= -\big (\frac{1}{2}(t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})-\\ & \quad (t_s e^{\lambda t_s}+\frac{1}{\lambda})\big(t_{su}-\frac{1}{\lambda}\big).\end{split}
\end{align}}
In the Appendix, we further analyze the expression in Eq.~(\ref{eq:xc}) in detail.
\subsection{Optimal Parameter Values}
The power consumption over the boundary curve in Eq. (\ref{eq:xc}) depends on the packet arrival rate $\lambda$. Furthermore, as it is shown in the Appendix, $\overline{\text{P}}_{b}(t_w)$ behaves differently for different ranges of $\lambda$. For this purpose, $\frac{d \overline{\text{P}}_{b}(t_w)}{d t_w}$ is calculated (a detailed analysis is provided in the Appendix). Briefly, its sign for different ranges of $\lambda$ within {the feasible region of the wake-up cycle (i.e., $t_{w_b} \leq t_w $)} can be expressed as follows
{ \begin{equation}
\text{sgn}\left(\frac{d \overline{\text{P}}_{b}(t_w)}{d t_w}\right)= \left\{ \,
\begin{IEEEeqnarraybox}[][c]{l?s}
\IEEEstrut
1 & $\text{for} ~ 0< \lambda \leq {\lambda}_t$ , \\
-1 & $\text{for} ~ {\lambda}_t< \lambda < 1$ ,
\IEEEstrut
\end{IEEEeqnarraybox}
\right.
\label{eq:ifbwzz}
\end{equation}}
\noindent where sgn(.) refers to the sign function, and $\lambda_t$ is referred to as the turnoff packet arrival rate. {The turnoff packet arrival rate can be calculated using any typical root-finding algorithm that meets $F_1=0$ (see details in Appendix) where}
{ \begin{align}
F_1&=(1+e^{\lambda t_s})(\overline{\text{D}}_{\max}-\frac{t_s}{2})(t_s e^{\lambda t_s}+\frac{{1}}{\lambda}) \times \\
& \quad \big(\frac{1}{2}(t_{pd}-\phi t_{su})+\frac{{1}}{\lambda}+2(\overline{\text{D}}_{\max}-\frac{t_s}{2})\big)- \\
& \quad \big (\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}\big )(1+e^{\lambda t_s})^2(\overline{\text{D}}_{\max}-\frac{t_s}{2})^2.
\end{align}}
\begin{theorem}
$t_w^*= t_{w_b}~ \text{and} ~ t_i^*=1$ are the optimal parameter values of the optimization problem in (\ref{eq:i2})-(\ref{eq:c7}) for the range $0< \lambda \leq {\lambda}_t$.
\end{theorem}
\begin{proof}
As it can be seen in (\ref{eq:ifbwzz}), for all $0< \lambda \le {\lambda}_t$, the power consumption increases when increasing $t_w$ over the boundary, so that the minimum power consumption is achieved at the minimum feasible $t_w$, i.e., $t_w=t_{w_b}$. Correspondingly, the optimal value of $t_i$ can be calculated by substituting $t_{w_b}$ into Eq. (\ref{eq:bn}), which leads to $t_i=1$. Therefore, for all $0<\lambda\leq{\lambda}_t$, $t_w^*=t_{w_b}$ and $t_i^*=1$ is the optimal solution of the optimization problem in (\ref{eq:i2})-(\ref{eq:c7}).
\end{proof}
{\begin{theorem}
$t_w^*=+\infty~\text{and}~t_i^*=+\infty$ are the optimal parameter values of the optimization problem in (\ref{eq:i2})-(\ref{eq:c7}) for the range ${\lambda}_t< \lambda <1$.
\end{theorem} }
{\begin{proof}
As it can be seen in (\ref{eq:ifbwzz}), for all ${\lambda}_t< \lambda <1$, the power consumption decreases when increasing $t_w$ over the boundary, so that the minimum power consumption is achieved at the maximum feasible $t_w$, i.e., $t_w=+\infty$. Correspondingly, the optimal value of $t_i$ can be calculated by substituting $t_{w}$ into Eq. (\ref{eq:bn}), which is $t_i=+\infty$. Therefore, for all ${\lambda}_t< \lambda <1$, $t_w^*=+\infty$ and $t_i^*=+\infty$ is the optimal solution of the optimization problem in (\ref{eq:i2})-(\ref{eq:c7}).
\end{proof}}
{
\begin{corollary}
For the range ${\lambda}_t< \lambda <1$, the optimal solution is equivalent to not utilizing WuS; the system is always at active-decoding and active-inactivity timer states (P$_{1}+$P$_{4}\approx 0$). Hence, if the energy and delay overhead of switching on/off the BBU are taken into account, the WuS is not effective anymore for high $\lambda$ values. Instead, other power saving mechanisms, such as DRX, microsleep, or pre-grant message could be used in this regime.
\end{corollary}}
{Fig. \ref{fig:lambda_t} (a) and (b)} illustrate how ${\lambda}_t$ changes with $\overline{\text{D}}_{\max} $ and $t_{su}+t_{pd}$, respectively. As it can be observed in {Fig. \ref{fig:lambda_t} (a)}, the turnoff packet arrival rate is independent and insensitive to variations of the value of $\overline{\text{D}}_{\max}$, however, it reduces, when $t_{su}+t_{pd}$ becomes larger (see {Fig. \ref{fig:lambda_t} (b)}). Therefore, in order to decide whether to enable WuS or not, regardless of the QoS requirement of the considered traffic, the network needs to compare the estimated packet arrival rate with pre-calculated and fixed ${\lambda}_t$.
\begin{figure}
\centering
\includegraphics[width=0.5 \textwidth]{lambda_t_new.pdf}
\caption{{${\lambda}_t$ as function of delay bound and sum of transition times.}}
\label{fig:lambda_t}
\end{figure}
Interestingly, for $ {\lambda}_t< \lambda < 1$, the power consumption reduces by increasing $t_w$ {towards infinity (and correspondingly, $t_i$ increases), while the delay constraint is satisfied. This can be interpreted in a way that for packet arrival rates higher than ${\lambda}_t$, the WuS is not effective anymore and only adds overhead energy consumption, thus implying that it is better not to switch off the BBU and to utilize short DRX cycles. As it is shown in {Fig. \ref{fig:lambda_t} (b)}, when $t_{su}+t_{pd}$ becomes larger, turnoff packet arrival rates become smaller, which justifies the fact that for the higher packet arrival rates, the frequent start-up and power-down related energy consumption becomes larger.}
{Additionally, the main reason for such interpretation is due to the fact that for large wake-up cycles, P$_{12}$ approaches to one, which is equivalent to the reduction of number of potential scheduled PDCCHs, and hence there is no gain by using the wake-up scheme over DRX anymore. Furthermore, for higher packet arrival rates, based on the optimal policy, most of the time the BBU is at either S$_2$ or S$_3$ (P$_{1}=$P$_{4}\approx 0$), to avoid wasted energy of start-up and power-down times, and to satisfy the delay constraint (illustrated in Fig. \ref{fig:small_large_lambda} (a)). As it can be seen in Fig. \ref{fig:small_large_lambda} (a), for small packet arrival rates, there is considerable energy consumption for transition between states, however once the packet arrival rate is higher than $\lambda_t$, this trend changes and the UE is mainly at S$_2$ or S$_3$, and it does not waste energy in start-up/power-down stages, due to the need for frequent start-up/power-down of the BBU. Such change in behaviour of the wake-up scheme can be explained by the objective of the system which is to reduce the overall power consumption, as it shown in Fig. \ref{fig:small_large_lambda} (b). Moreover, as it can be seen in Fig. \ref{fig:small_large_lambda} (a), for packet arrival rates higher than the turnoff packet arrival rate, most of the energy is consumed for decoding of the packets, and its energy consumption increases linearly with the packet arrival rate.}
\begin{figure}[!t]
\centering \
\includegraphics[width=0.34 \textwidth]{small_large_lambda1.pdf} \includegraphics[width=0.35 \textwidth]{small_large_lambda2.pdf}
\caption{{a) Normalized average energy consumption of each state, as result of optimal configuration of wake-up scheme parameters; b) average overall power consumption of optimized wake-up scheme.}}
\label{fig:small_large_lambda}
\end{figure}
Finally, it can be shown that for $\lambda$ less than the turnoff packet arrival rate ($0< \lambda \le {\lambda}_t$), the optimal parameter values ($t_w^*$ and $t_i^*$) of the original MINLP (\ref{eq:i1})-(\ref{eq:c6}) can be written based on the optimal values of the equivalent relaxed problem (\ref{eq:i2})-(\ref{eq:c7}) as follows
\begin{align}
t_w^*=\lfloor t_{w_b} \rfloor~ \text{and} ~ t_i^*=1,
\label{eq:A}
\end{align}
\noindent where $\lfloor ~ \rfloor$ refers to the floor function.
\begin{theorem}
$t_w^*=\lfloor t_{w_b} \rfloor~ \text{and} ~ t_i^*=1$ are the optimal parameter values of the MINLP in \eqref{eq:i1}-\eqref{eq:c6} for the range $0< \lambda \leq {\lambda}_t$.
\end{theorem}
\begin{proof}
If $f$ is a function of continuous variables $x$ and $y$, it can easily be shown that
{ \begin{equation}
\begin{split}
\text{if}~ \frac{\partial f}{\partial x}>0 \Longrightarrow \Delta {f}_x > 0, ~\forall x\in \mathbb{R}^+\\
\text{if}~ \frac{\partial f}{\partial x}<0 \Longrightarrow \Delta {f}_x<0, ~\forall x\in \mathbb{R}^+
\label{eq:Arrrr}
\end{split}
\end{equation} }
\noindent where
{ \begin{align}
\Delta {f}_x\triangleq f(\lfloor x \rfloor +1)-f(\lfloor x \rfloor) .
\label{eq:Az}
\end{align}}
Therefore, by assuming $f$ as representative of either $\overline{\text{P}}_{c}(t_w,t_i)$ or $\overline{\text{D}}(t_w,t_i)$ and $x$ as of either $t_i$ or $t_w$, one can prove, similarly to the case with continuous variables (proved in Section \ref{sec:feasib}), that the optimal parameters of MINLP (\ref{eq:i1})-(\ref{eq:c6}) are laid over the boundary. Therefore, the boundary of (\ref{eq:i1})-(\ref{eq:c6}) consists of all combinations of $(\lfloor t_w \rfloor, t_i)$ for which $ t_i \in \{1,2,...\}$ and $ \overline{\text{D}}(t_w,t_i)= \overline{\text{D}}_{\max}$ (as formulated in (\ref{eq:bn})). Similarly, based on (\ref{eq:Arrrr}) and (\ref{eq:part}), as well as the properties of the floor function, we can state that increasing $t_i$ over the boundary may increase $\lfloor t_w \rfloor$ ($\Delta {\lfloor t_w \rfloor}_{t_i}>0$) or $\lfloor t_w \rfloor$ can remain in its previous value ($\Delta {\lfloor t_w \rfloor}_{t_i}=0$). Furthermore, based on (\ref{eq:ifbwzz}), for $0< \lambda < {\lambda}_t$, we can conclude that $\Delta { \overline{\text{P}}_{b}}(t_w,t_i)_{~t_{w}}>0$. Therefore, $t_w^*$ is the smallest feasible value of the wake-up cycle over the boundary, i.e., $t_w^*=\lfloor t_{w_b} \rfloor$. However, $\lfloor t_{w_b} \rfloor$ may correspond to either $t_i=1$ or larger values at the same time. Since the power consumption has the lowest value at the lowest $t_i$, for fixed $t_w$, we can conclude that $t_i^*=1$.
\end{proof}
\begin{corollary}
It turns out that the solution of the relaxed problem in \eqref{eq:i2}-\eqref{eq:c7}, after finding the integer part of one of the two optimization parameters' values, are actual optimal values for the original MINLP optimization problem in (\ref{eq:i1})-(\ref{eq:c6}). Therefore, our relaxation approach yielded an equivalent reformulation.
\end{corollary}
\section{Numerical Results}
\label{sec:eval}
In this section, a set of numerical results are provided in order to validate our concept and the analytical results, as well as to show and compare the average power consumption of the optimized WuS over DRX for packet arrival rates less than the turnoff packet arrival rate. Power consumption of the mobile device in different operating states is highly dependent on the implementation, and also its operational configurations. Therefore, for the numerical results, the power consumption model used in \cite{Globecom, pm2}, \cite{pm1}, \cite{Lauridsen2014} is employed. Its parameters for DRX and WuS are shown in Table \ref{drxpower} and Table \ref{tab:powr}, respectively. LTE-based power consumption values, shown in Table \ref{drxpower}, are considered as a practical example since those of the emerging NR modems are not publicly available yet. {For simulations, we use $\phi=1.1$ as an example numerical value, while methodology wise, numerical results can also be generated for any other value as well.}
\begin{table}[!t]
\scriptsize
\renewcommand{\arraystretch}{1.3}
\caption{{ Average transitional time and representative power consumption values of LTE-based cellular module during short and long DRX when carrier bandwidth is $20$ MHz, and $\phi=1.1$}}
\label{drxpower}
\centering
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
DRX Cycle &$\text{PW}_{\text{sleep}}$& $\text{PW}_{\text{active}}$ &$\text{PW}_{\text{decode}}$ & $t_{su}$& $t_{pd}$ \\
\hline
\hline
short& $395$ mW& $850$ mW & $935$ mW& $1$ ms & $1$ ms\\
\hline
long& $\approx0$ mW& $850$ mW&$935$ mW& $15$ ms & $10$ ms\\
\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\scriptsize
\renewcommand{\arraystretch}{1.3}
\caption{{{Assumed power consumption parameters of the wake-up scheme with assumption of $\phi=1.1$ }}}
\label{tab:powr}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|}
\hline
PW$_{\text{1}}$ & PW$_{\text{2}}$ &PW$_{\text{3}}$ & PW$_{\text{4}}$ &$t_{su}$ & $t_{pd}$& $t_{on}$ \\
\hline
\hline
57mW &935 mW & 850 mW & $\approx0$ mW& 15 ms & 10 ms & 1/14 ms \\
\hline
\end{tabular}
\end{table}
Two different sets of performance results, in terms of power consumption and delay, are presented based on the optimal configuration of the wake-up parameters (\ref{eq:A}). Namely, a) with simplified assumptions of zero false alarm/misdetection rates, and $t_{on}\approx 0$ ms, equivalent to analytical results (ana.), and b) with the realistic assumptions of $P_{fa}=10 \%$, $P_{md}=1 \%$, $t_{on}=1/14$ ms obtained by simulations and \cite{Globecom}, referred to as simulation results (sim.).
Table \ref{optimal_tw} shows the optimal resulting values of $t_w^*$ in (\ref{eq:A}) for different values of $\lambda$ and $\overline{\text{D}}_{\max}$. As it can be observed, for tight delay requirements ($\overline{\text{D}}_{\max}=30$ ms), $t_w^*$ tends to be small, enabling the UE to reduce the duration of packet buffering. Interestingly, for mid range of packet arrival rates ($\lambda=0.1$ p/ms), optimal wake-up cycle for a given delay bound is shorter than for both lower and higher packet arrival rates. The justification is as follows. For higher packet arrival rates, $t_w^*$ becomes larger, the reason being that the inactivity timer is ON most of the time. Therefore, the need for smaller wake-up cycles decreases and correspondingly higher energy overhead is induced. For lower packet arrival rates, in turn, the value of $t_w^*$ is higher due to the infrequent packet arrivals, hence achieving a smaller delay.
\begin{table}[!t]
\scriptsize
\centering
\renewcommand{\arraystretch}{1.3}
\caption{{Optimal values of wake-up cycle under different delay requirements and packet arrival rates ($t_i^*=1$ ms)}}
\label{optimal_tw}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$\lambda$ [p/ms]& \multicolumn{3}{c|}{$0.01$} & \multicolumn{3}{c|}{$0.08$} & \multicolumn{3}{c|}{$0.15$} \\ \hline
$\overline{\text{D}}_{\max} $[ms] &$30$&$75$&$500$&$30$&$75$&$500$&$30$&$75$&$500$\\ \hline
$t_w^*$ [ms] & 180 & 380 &2099 & 124 & 315& 2124 & 125 &328 &2246 \\ \hline
\end{tabular}
\end{table}
{Fig.} \ref {fig:PC_lambda} illustrates the power consumption of the proposed WuS under ideal and realistic assumptions as a function of the packet arrival rate ($\lambda< \lambda_t$), for different maximum tolerable delays.
As it can be observed, for both analytical and simulation results, and for all delay bounds, the average power consumption initially increases, while then remains almost constant (especially for large delay bounds) as $\lambda$ increases, due to the configuration of shorter wake-up cycles for mid packet arrival rates (see Table \ref{optimal_tw}). Moreover, the UE consumes higher power in order to satisfy tighter delay requirements, which, as shown in Table \ref{optimal_tw}, can be translated into shorter $t_w^*$. Furthermore, {Fig.} \ref {fig:PC_lambda} shows that the simulation results closely follow the analytical results, the non-zero gap being due to the non-zero false alarm and misdetection rates. The relative gap between simulation-based and analytical results is somewhat larger for shorter delay bounds, which stems from the correspondingly higher number of wake-up instances.
\begin{figure}[!t]
\centering
\includegraphics[width=0.39 \textwidth]{newPC_lambda_new.pdf}
\caption{{Average power consumption of the optimized wake-up scheme, under ideal and realistic assumptions as a function of packet arrival rate and delay bound.}}
\label{fig:PC_lambda}
\end{figure}
Moreover, {Fig.} \ref{fig:delay_lambda} depicts the average packet delay experienced by the WRx-enabled UE under ideal and realistic assumptions when packet arrival rates vary. As it can be observed, the analytical delay based on the optimal parameter configuration (\ref{eq:A}) is slightly shorter than the maximum tolerable delay. This is because of selecting the greatest integer less than or equal to the optimal wake-up cycle of the relaxed optimization problem. However, the actual average delay is slightly higher than the analytical average delay, especially for high delay bounds. The main reason for such negligible excess delay is the unavoidable misdetections, whose impact is more clear for large wake-up cycles corresponding to high delay bounds. In practice, to compensate for such small excess delay, the delay bound can be set slightly smaller than the actual average delay requirement.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4 \textwidth]{new_delay_new.pdf}
\caption{Average buffering delay of the optimized wake-up scheme, under ideal and realistic assumptions as a function of packet arrival rate and delay bound. For better visualization, the y-axis is deliberately not in linear scale.}
\label{fig:delay_lambda}
\end{figure}
Finally, for comparison purposes, the relative power saving of WuS over DRX representing the amount of power that can be saved with WuS as compared to the DRX-based reference system is utilized, assuming the same delay constraints in both methods. The value of the relative power saving ranges from $0$ to $100\%$, and a large value
indicates that the WuS conserves energy better than the
DRX. Formally, we express the relative power saving ($\eta$) as
\begin{equation}\label{eq:etapgm}
\eta=\frac{\overline{\text{P}}_{\text{DRX}}-\overline{\text{P}}_c}{\overline{\text{P}}_{\text{DRX}}}\times 100,
\end{equation}
\noindent where $\overline{\text{P}}_{\text{DRX}}$ refers to average power consumption of DRX. Furthermore, for a fair comparison, we consider an exhaustive search over a large parameter set of DRX configuration, developed by authors in \cite{Ramazanali}. However, in order to take start-up and power-down power consumption into account, the solution in \cite{Ramazanali} is slightly modified to account for the transitory states.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4 \textwidth]{new_relative_pc_new.pdf}
\caption{{Achieved relative power saving values ($\eta$) of the proposed optimized wake-up scheme as a function of packet arrival rate and delay bound.}}
\vspace{-3mm}
\label{fig:relative_pc}
\end{figure}
{Fig. \ref{fig:relative_pc}} shows the power saving results. It is observed that the proposed WuS, under realistic assumptions, outperforms DRX within the range $\lambda< \lambda_t$, especially for low packet arrival rates with tight delay requirements.
The main reason is that in such scenarios, DRX-based device needs to decode the control channel very often, residing mainly in short DRX cycles, which causes extra power consumption. The WRx, in turn, needs to decode the wake-up signaling frequently, but with lower power overhead. Additionally, as expected, regardless of the delay requirements, for higher packet arrival rates, DRX infers relatively similar power consumption to the WuS. The reason is that in such cases, the DRX parameters can be configured in such a way that there is a low amount of unscheduled DRX cycles, either by utilizing short DRX cycles for very tight delay bounds or by employing long DRX cycles for large delay requirements. Overall, the results in Fig. \ref{fig:relative_pc} clearly demonstrate that WuS can provide substantial energy-efficiency improvements compared to DRX, with the maximum energy-savings being in the order of 40\%.
\section{State Machine based Wake-up System Model}
\label{sec:sysmodel}
For mathematical convenience, the performance of the wake-up based system is studied and analyzed in the context of a Poisson arrival process with a packet arrival rate of $\lambda$ packets per TTI. In the Poisson traffic model, each packet service session consists of a sequence of packets with exponentially distributed inter-packet arrival time ($t_p$) \cite{pm2}.
{The power states of WuS are modeled as a semi-Markov process with four different states that correspond to WRx-ON (state $\text{S}_1$), active-decoding (state $\text{S}_2$), active-inactivity timer (state $\text{S}_3$), and sleep (state $\text{S}_4$), as shown in Fig. \ref{fig:markov}. At S$_1$, WRx monitors PDWCH, and if WRx receives WI=$0$, UE transfers to S$_4$, otherwise (WI=$1$) it transfers to S$_2$. At S$_2$, UE decodes the packets for a fixed duration of $t_s$; if the device is scheduled before the end of $t_s$, it starts decoding the new packet, and remains at S$_2$, otherwise the device transfers to S$_3$. At S$_3$, $t_i$ is running, and if the device is scheduled before the expiry of $t_i$, it enters in S$_2$, otherwise the device transfers to S$_4$. At S$_4$, the device is in sleep state, and cannot receive any signal, as opposed to being fully-functional at S$_2$ and S$_3$. Moreover, at the end of a wake-up cycle in sleep period, the UE moves to S$_1$.} {As noted already in the previous Section~\ref{sec:Background}, each state is associated with a different power consumption level, PW$_k$, $k \in\{1,2,3, 4\}$.}
\begin{figure}[t!]
\centering
\includegraphics[scale=1.6]{new_markof_schematic.pdf}
\caption{{Semi-Markov process for the state transitions of the wake-up scheme, with states $\text{S}_1$, $\text{S}_2$, $\text{S}_3$ and $\text{S}_4$.}}
\label{fig:markov}
\end{figure}
\textbf{Transition probabilities}: The transition probability from UE state ${\text{S}_{k}}$ to ${\text{S}_{l}}$ (${\text{P}_{kl}}$) is defined as
\begin{equation}
{\text{P}_{kl}}=\lim_{n\to\infty} \Pr(S(\tau_n)={\text{S}_{l}} | S(\tau_{n-1})={\text{S}_{k}}),
\end{equation}
\noindent where $S(\tau_n)$ is the UE state at the $\tau_n$ jump time\footnote{{In a semi-Markov process, $S(t)$ is a stochastic process with a finite set of states (${\text{S}_1}, \dots, \text{S}_4$ in our case), having step-wise trajectories with jumps at times $0<\tau_1<\tau_2...$, and its values at the jump times ($S(\tau_n)$) form a Markov chain.
}}.
{When the UE is at ${\text{S}_{1}}$, it moves to ${\text{S}_{2}}$ either because of false alarm or correct detection; otherwise, it moves to ${\text{S}_{4}}$. Accordingly, ${\text{P}_{12}}$ and ${\text{P}_{14}}$ can be expressed as
\begin{equation}
\begin{split}
{\text{P}_{12}}=\Pr[t_{p}>t_{w}|\text{S}_1]P_{fa}+
\Pr[t_{p}\leq t_{w}|\text{S}_1](1-P_{md}) \\
=e^{-\lambda t_w}P_{fa}+(1-e^{-\lambda t_w})(1-P_{md}),~~~~~\,
\end{split}
\end{equation}
\noindent and
\begin{equation}
{\text{P}_{14}}=1-{\text{P}_{12}}.
\end{equation}}
{When the UE is at ${\text{S}_{2}}$, it decodes the packet for a duration of $t_s$, and if the next packet is received before the end of the current service time, the UE starts decoding the new packet at the end of service time; otherwise, it moves to ${\text{S}_{3}}$. Therefore, ${\text{P}_{22}}$ and ${\text{P}_{23}}$ can be obtained as
\begin{equation}
{\text{P}_{22}}=\Pr[t_{p}\leq t_s| \text{S}_2]=1-e^{-\lambda t_s},
\end{equation}
\noindent and
\begin{equation}
{\text{P}_{23}}=1-{\text{P}_{22}}.
\end{equation}}
{When the UE is at ${\text{S}_{3}}$, it moves to ${\text{S}_{2}}$ if the next packet is received before the expiry of $t_i$; otherwise, it moves to ${\text{S}_{4}}$. Therefore, ${\text{P}_{32}}$ and ${\text{P}_{34}}$ can be expressed as
\begin{equation}
{\text{P}_{32}}=\Pr[t_{p}\leq t_i| \text{S}_3]=1-e^{-\lambda t_i},
\end{equation}
\noindent and
\begin{equation}
{\text{P}_{34}}=1-{\text{P}_{32}}.
\end{equation}}
{Finally, at the end of every sleep cycle, the UE decodes PDWCH, and therefore,
\begin{equation}
{\text{P}_{41}}=1.
\end{equation}}
{\textbf{Steady state probabilities}: The steady state probability that the UE is at state ${\text{S}_{k}}$ ($\text{P}_k$) is defined as
\begin{equation}
{\text{P}_{k}}=\lim_{n\to\infty} \Pr( S(\tau_n) = \text{S}_k ).
\end{equation}}
{By utilizing the set of balance equations ($\text{P}_{k}=\sum_{l=1}^{4}\text{P}_{l}\text{P}_{lk}$) and the basic sum of probabilities ($\sum_{k=1}^{4}\text{P}_{k}=1$), the $\text{P}_k$'s can be obtained as follows }
{\begin{equation}
\text{P}_{1}=\text{P}_{4}=\frac{\text{P}_{34}\text{P}_{23}}{ 2\text{P}_{34}\text{P}_{23} +\text{P}_{12}(1+\text{P}_{23}) } ,
\end{equation}
\begin{equation}
\text{P}_{2}=\text{P}_{1}\frac{\text{P}_{12}}{\text{P}_{23}\text{P}_{34}},
\end{equation}
\begin{equation}
\text{P}_{3}=\text{P}_{1}\frac{\text{P}_{12}{}}{\text{P}_{34}}.
\end{equation} }
{\textbf{Holding times}: The corresponding holding time for state ${\text{S}_{k}}$ is denoted by $ \omega_k$, $k \in\{1,2,3, 4\}$. The holding times for $\omega_1$, $\omega_2$ and $\omega_4$ are constant and given by: $\mathbb{E}[\omega_1]= t_{on} $, $\mathbb{E}[\omega_2]= t_{s} $ and $\mathbb{E}[\omega_4]=t_{w}-t_{on}$. However, $\omega_3$ is dependent on the inter-packet arrival time ($t_p$). If a packet arrives before $t_i$, $\omega_3$ is equal to the inter-packet arrival time, otherwise $\omega_3$ equals to $t_i$. Therefore, $\omega_3$ can be calculated as a function of $t_p$ as }
\begin{equation}
\omega_3(t_p)= \left\{ \,
\begin{IEEEeqnarraybox}[][c]{l?s}
\IEEEstrut
t_p,&$\text{for}~~ t_p\leq t_i$ , \\
t_i,&$\text{for}~~ t_p>t_i $ .
\IEEEstrut
\end{IEEEeqnarraybox}
\right.
\end{equation}
\noindent Hence, $\mathbb{E}[\omega_3]$ can be expressed as
\begin{equation}
\mathbb{E}[\omega_3]=\int_{0}^{\infty}\omega_3(t) f_{p}(t)dt=\frac{1-e^{-\lambda t_i }}{\lambda},
\end{equation}
\noindent where $f_{p}(t)=\lambda e^{-\lambda t}$ is the probability density function of the exponentially distributed packet arrival time.
\begin{figure*}[!t]
\normalsize
\setcounter{equation}{14}
\begin{equation}
\overline{\text{P}}_c=\frac{0.5\text{P}_{1}\text{P}_{12}t_{su}(\text{PW}_2-\text{PW}_4)+0.5\text{P}_{3}\text{P}_{34}t_{pd}(\text{PW}_3-\text{PW}_4)+\sum_{n=1}^{4}\text{P}_{n}\mathbb{E}[\omega_n]\text{PW}_{n}}{\text{P}_{1}\text{P}_{12}t_{su}+\text{P}_{3}\text{P}_{34}t_{pd}+\sum_{n=1}^{4}\text{P}_{n}\mathbb{E}[\omega_n]}, \label{eq:pc_row}
\end{equation}
\hrulefill
\end{figure*}
\subsection{Average Power Consumption}
The average power consumption of the UE, denoted by $\overline{\text{P}}_c$, can be calculated as the ratio of the average energy consumption and the corresponding overall observation period. It is given by Eq.~\eqref{eq:pc_row} at the top of the next page,
where $t_{su}$ and $t_{pd}$ correspond to the length of the start-up and power-down stages (transition times), respectively. The corresponding average energy consumption of transitions between states are calculated as the areas under the power profiles of start-up and power-down stages, see Fig.~\ref{fig:drxvswrx}, whose contribution to the average energy consumption is multiplied by its probability of occurrence ($\text{P}_{1}\text{P}_{12}$ and $\text{P}_{3}\text{P}_{34}$, respectively), thus leading to $0.5\text{P}_{1}\text{P}_{12}t_{su}(\text{PW}_2-\text{PW}_4)$ and $0.5\text{P}_{3}\text{P}_{34}t_{pd}(\text{PW}_3-\text{PW}_4)$, respectively.
\setcounter{equation}{15}
For modeling simplicity, we assume that $t_{on}\approx0$, $\text{PW}_{4}\approx0$, $P_{fa}\approx0$, and $P_{md}\approx0$. Therefore, Eq. (\ref{eq:pc_row}) can be expanded as a multivariate function of $t_w$ and $t_i$, {denoted by $\overline{\text{P}}_c(t_w,t_i)$,} as follows
{\begin{equation}
\label{eq:pc_row_2}
\begin{split}
\overline{\text{P}}_c(t_w,t_i)=\\ \text{PW}_3 \frac{e^{\lambda t_i}(\phi t_s e^{\lambda t_s}+\frac{1}{\lambda})+\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}}{ e^{\lambda t_i}(t_s e^{\lambda t_s}+\frac{1}{\lambda})+\frac{t_w}{1-e^{-\lambda t_w}}+t_{su}+t_{pd}-\frac{{1}}{\lambda}}.
\end{split}
\end{equation}}
In order to provide more insight into $\overline{\text{P}}_c(t_w,t_i)$, the instantaneous rate of change of the power consumption with respect to both $t_w$ and $t_i$ is calculated next. Assuming continuous variables, the partial derivatives of $\overline{\text{P}}_c(t_w,t_i)$ with respect to $t_w$ and $t_i$ are
given by
{\begin{equation}
\label{eq:pc_der_t_i}
\begin{split}
\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_i}=\text{PW}_3e^{\lambda t_i}(\lambda\phi t_s e^{\lambda t_s}+1)\times\\\frac{\frac{1}{2}((2-\phi)t_{su}+t_{pd})+\frac{t_w}{1-e^{-\lambda t_w}}}{\big(e^{\lambda t_i}(t_s e^{\lambda t_s}+\frac{1}{\lambda})+\frac{t_w}{1-e^{-\lambda t_w}}+t_{su}+t_{pd}-\frac{{1}}{\lambda}\big)^2},
\end{split}
\end{equation}}
\noindent and
{\begin{equation}
\label{eq:pc_der_t_w}
\begin{split}
\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_w}=\text{PW}_3\big((1+\lambda t_w)e^{-\lambda t_w}-1\big) \times \\
\frac{\big(e^{\lambda t_i}(\phi t_s e^{\lambda t_s}+\frac{1}{\lambda})+\frac{1}{2}(\phi t_{su}+t_{pd})-\frac{{1}}{\lambda}\big)}{\big((1-e^{-\lambda t_w})(e^{\lambda t_i}(t_s e^{\lambda t_s}+\frac{1}{\lambda})+t_{su}+t_{pd}-\frac{{1}}{\lambda})+t_w\big)^2}.
\end{split}
\end{equation}}
It can be seen from Eq. \eqref{eq:pc_der_t_i} that $\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_i}>0$ for all feasible values of $t_w$ and $t_i$. From Eq. (\ref{eq:pc_der_t_w}), we can conclude that $\frac{\partial\overline{\text{P}}_c(t_w,t_i)}{\partial t_w}<0$ due to fact that $(1+\lambda t_w)e^{-\lambda t_w}<1$.
Therefore, the average power consumption
$\overline{\text{P}}_c(t_w,t_i)$ is a strictly increasing function with respect to $t_i$ at $t_i\geq 0$, and it is a strictly decreasing function with respect to $t_w$ at $t_w\geq 0$. As expected, increasing the wake-up cycle $t_w$ for a fixed $t_i$ can reduce the power consumption. However, by increasing $t_i$ for a fixed $t_w$, the power consumption increases.
\subsection{Average Buffering Delay}
We next assume that packets arriving during ${\text{S}_{4}}$ are buffered at the gNB until the UE enters ${\text{S}_{2}}$, thus causing buffering delay. Without loss of generality, we assume that the radio access network experiences unsaturated traffic conditions. {Therefore, all packets that arrive are served without any further scheduling delay. }Furthermore, to simplify the delay modeling, we omit the buffering delay caused by packets arriving on ${\text{S}_{1}}$ or at the start-up state of the modem. This is because the additional buffering delay of such packet arrivals is anyway very small ($t_{on}+t_{su}$). Additionally, thanks to the adoption of the WuS seeking to reduce unnecessary start-ups, the number of occurrences of such scenarios is low.
{Due to slot-based frame structure of NR, where PDCCH is sent at the beginning of the TTI, inherently, all packet arrivals (regardless WuS is utilized or not) suffer from small buffering delay. Since Poisson arrivals are independently and uniformly distributed on any short interval, we assume that the arrival instant of the packet is uniformly distributed within the TTI, and hence an average extra delay of $t_s/2$ will be introduced.}
\begin{figure}[!t]
\centering
\includegraphics[scale=1.1]{delay_shematic.pdf}
\caption{Buffering delay caused by wake-up scheme when (a) there is no misdetection, (b) there is a single misdetection, and (c) there are two consecutive misdetections. }
\label{fig:delay_md}
\end{figure}
Now, as already briefly mentioned in Section \ref{sec:Background}, misdetections can in general increase the buffering delay. For this purpose, Fig. \ref{fig:delay_md} illustrates the buffering delay experienced by the UE with no misdetections, with a single misdetection, and with two consecutive misdetections. The number of consecutive misdetections and the corresponding buffering delay are referred to as $i$ and $d_i$, respectively, and their dependence on $t$ can be written as $d_i(t)=(i+1)t_w+t_{su}+t_{on}-t$, for $i\in \{0,1,...\}$. Therefore, the average buffering delay, denoted by $\overline{\text{D}}$, can be expressed as
\begin{eqnarray}
\overline{\text{D}}= \text{P}_4 \sum_{i=0}^{\infty}(1-P_{md}) { (P_{md} )}^{ i}\int_{0}^{{t_{w}}}f_p(t)d_i(t)dt+\frac{t_s}{2}.
\label{eq:d_equation}
\end{eqnarray}
\begin{figure*}[!t]
\normalsize
\setcounter{equation}{20}
\begin{equation}
\label{eq:delay_der_t_w}
\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_w}=\frac{e^{\lambda t_i}(1+e^{\lambda t_s})\big(1-(1+\lambda t_w)e^{-\lambda t_w}\big)+2(1-e^{-\lambda t_w})+2\lambda t_{su}e^{-\lambda t_w}}{\Big({2+(1-e^{-\lambda t_w})(1+e^{\lambda t_s})e^{\lambda t_i}}\Big)^2} ,
\end{equation}
\begin{equation}
\label{eq:delay_der_t_i}
\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_i}=\frac{-\lambda e^{\lambda t_i}(1+e^{\lambda t_s})(1-e^{-\lambda t_w})\big(t_w+(t_{su}-\frac{1}{\lambda}){(1-e^{-\lambda t_w})}\big)}{\Big({2+(1-e^{-\lambda t_w})(1+e^{\lambda t_s})e^{\lambda t_i}}\Big)^2} .
\end{equation}
\hrulefill
\end{figure*}
\setcounter{equation}{19}
Furthermore, due to the small value of misdetection probability ($P_{md}\approx 0$), the contribution of multiple consecutive misdetections to average buffering delay is small. Thus, the average buffering delay for $P_{md}\approx 0$ can be expanded and solved as a multivariate function of $t_w$ and $t_i$, $\overline{\text{D}}(t_w,t_i)$, as follows
\begin{equation}
\begin{split}
\overline{\text{D}}(t_w,t_i)= \text{P}_4 \int_{0}^{{t_{w}}}f_p(t)d_0(t)dt+\frac{t_s}{2}=\\\frac{t_w+(t_{su}-\frac{1}{\lambda}){(1-e^{-\lambda t_w})}}{2+(1-e^{-\lambda t_w})(1+e^{\lambda t_s})e^{\lambda t_i}}+\frac{t_s}{2}.
\end{split}
\label{eq:d1_equation}
\end{equation}
{We note that $\overline{\text{D}}(t_w,t_i)$ in Eq. \eqref{eq:d1_equation} is strictly-speaking a lower bound of the delay expression in \eqref{eq:d_equation}}.
Similarly to $\overline{\text{P}}_c(t_w,t_i)$, the partial derivatives of $\overline{\text{D}}(t_w,t_i)$ with respect to continuous variables $t_w$ and $t_i$ are given by Eq.~\eqref{eq:delay_der_t_w} and Eq.~\eqref{eq:delay_der_t_i}, respectively, at the top of the page.
\setcounter{equation}{22}
From Eq. (\ref{eq:delay_der_t_w}), it can be easily concluded that $\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_w}>0$, due to fact that $\big(1-(1+\lambda t_w)e^{-\lambda t_w}\big)>0$. Moreover, it can be shown that $\frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_i}<0$ as follows
\begin{equation}
\begin{split}
\lambda t_w >0\Longrightarrow \lambda t_w >1-e^{-\lambda t_w}\Longrightarrow \\
\frac{\lambda t_w}{1-e^{-\lambda t_w}} >1\Longrightarrow\frac{\lambda t_w}{1-e^{-\lambda t_w}} >1-\lambda t_{su}\Longrightarrow \\
t_w+(t_{su}-\frac{1}{\lambda}){(1-e^{-\lambda t_w})} >0\Longrightarrow \frac{\partial\overline{\text{D}}(t_w,t_i)}{\partial t_i}<0.
\end{split}
\end{equation}
Therefore, the average buffering delay
$\overline{\text{D}}(t_w,t_i)$ is a strictly increasing function with respect to $t_w$ at $t_w\geq 0$, and it is a strictly decreasing function with respect to $t_i$ at $t_i\geq 0$. As expected, contrary to the behavior of $\overline{\text{P}}_c(t_w,t_i)$, increasing the wake-up cycle $t_w$ for a fixed $t_i$ increases the buffering delay. On the other hand, by increasing $t_i$ for a fixed $t_w$, the buffering delay can be reduced.
{The findings related to the impact of $t_w$ and $t_i$ on the average delay and power consumption are intuitive while are rigorously confirmed and quantified by the presented expressions.} |
2,877,628,090,239 | arxiv | \section{Introduction}
\indent{} The topology-dependent electronic properties of solids are the subject of considerable current research. Although theoretical and experimental exploration of the electronic implications of topology fall clearly in the realm of materials physics \cite{RevModPhys.82.3045}, the compounds of interest for displaying those properties have a well-defined set of chemical characteristics, including constituent elements with similar electronegativities and strong spin orbit coupling; crystal symmetry also plays a critical role \cite{C3TC30186A}. The fact that such properties are in many cases predictable by theory has generated a wide interest in finding compounds that display them. Although the edges and surfaces of crystals and extremely thin crystals such as graphene might reasonably be expected to display electronic properties dependent on their topology \cite{König02112007,doi:10.1146/annurev-conmatphys-062910-140432,novoselov2005two}, there are also cases where topological properties have been predicted for electrons within bulk three dimensional crystals \cite{PhysRevLett.108.140405,PhysRevLett.107.186806}. Such is the case for the recently emergent compounds Na$_3$Bi and Cd$_3$As$_2$, where early characterization of the real materials indicates that this may indeed be the case \cite{borisenko2013experimental,neupane2013observation,Liu16012014}. One important prediction within this category is that in some cases the electrons will behave like they obey the ``Weyl Hamiltonian", a previously unobserved electronic state \cite{PhysRevB.83.205101,Physics.4.36}. In this instance the presence or absence of a center of symmetry in the host crystal is a critical structural characteristic for the stability of the electronic phase.
Cd$_3$As$_2$ has been well studied in the past \cite{lin1969energy,rosenberg1959cd3as2,wallace1979electronic} but not in this context. Of particular interest have been its semimetallic character and very high electron mobility \cite{rosenberg1959cd3as2}. It is also of interest for solar cells and as an analog for graphene in exploratory device applications \cite{4154290,stoumpos2013semiconducting}. Theoretical work on the electronic structure of Cd$_3$As$_2$ has been done, but the calculations were performed using either the primitive tetragonal (P4$_2$/nmc) intermediate temperature structure (475 $^{\circ}{\rm C}$ - 600 $^{\circ}{\rm C}$)\cite{OrigCd3As2Paper} or the noncentrosymmetric low temperature (Below 475 $^{\circ}{\rm C}$) structure\cite{steigmann1968crystal}(I4$_1$cd) proposed by Steigmann and Goodyear \cite{PhysRevB.88.125427}. The primitive tetragonal structure (1935) (\textit{a} = 8.95 \AA, \textit{c} = 12.65 \AA) has As in a FCC array, with 6 Cd in fluorite-like positions - 2 of the 8 vertices of the distorted cube are fully empty. The empty vertices lie diagonally opposite each other in one face of the cube, and are ordered in a two-dimensional array. The low temperature structure (1968) has a larger unit cell (\textit{a} = 12.67 \AA, \textit{c} = 25.48 \AA) where the empty cube vertices order in a three-dimensional rather than a two-dimensional array. Recent theoretical calculations performed with the reported non-centrosymmetric low temperature I4$_1$cd structure \cite{PhysRevB.88.125427} indicate that Cd$_3$As$_2$ may be a new type of 3D-Dirac semi-metal in part due to the lack of inversion symmetry, which causes the lifting of the spin degeneracy of certain bands in the vicinity of the Dirac point, raising the possibility that Cd$_3$As$_2$ may be an example of a Weyl semimetal. Thus the presence or absence of inversion symmetry has important implications for the electronic properties of Cd$_3$As$_2$.
\newline
\indent{}Here we re-examine the crystal structure of Cd$_3$As$_2$ using current single crystal X-ray diffraction (SXRD) and analysis methods. We also identify the growth direction of needle crystals and the cleavage plane as the [110] and (112), respectively. Through the use of Scanning Tunneling Microscopy (STM) experiments, we identify a 2x2 surface reconstruction of the (112) plane cleavage surface. We find that earlier researchers failed to appreciate the near centrosymmetricity of their reported structure, and that Cd$_3$As$_2$ in its low temperature phase is in fact centrosymmetric, with the space group I4$_1$/acd. The higher symmetry forbids any spin splitting and keeps all bands at least two fold degenerate; this has significant implications for the behavior of the electrons near the Fermi Energy. The inversion symmetry constrains Cd$_3$As$_2$ to be a non-spin-polarized 3D-Dirac semi-metal, and therefore implies that it is a 3D analog to graphene, where the electronic states are also non-spin-polarized.
\section{Experimental}
\indent{}Silvery, metallic colored single crystals of Cd$_3$As$_2$ were grown out of a Cd-rich melt with the stoichiometry (Cd$_5$)Cd$_3$As$_2$. The elements were sealed in an evacuated quartz ampoule, heated to 825 $^{\circ}{\rm C}$, and kept there for 48 hours. The sample was then cooled at a rate of 6 degrees per hour to 425 $^{\circ}{\rm C}$ and was subsequently centrifuged. Only Cd$_3$As$_2$ crystals were formed, with a crystal to flux ratio of 1:5, consistent with the equilibrium phase diagram \cite{ADMA:ADMA19910031215}. Chemical analysis of the crystals was performed in an FEI Quanta 200 FEG Environmental-SEM by energy dispersive x-ray analysis (EDX), which found them to have a Cd:As ratio of 1.500(1):1. Powder X-ray diffraction (PXRD) patterns were collected using Cu K$\alpha$ radiation on a Bruker D8 Focus diffractometer with a graphite monochromator on ground single crystals to confirm the identity of the compound as being Cd$_3$As$_2$ in the low temperature structure. The single crystal X-ray diffraction study was performed on a 0.04 mm x 0.04 mm x 0.4 mm crystal on a Bruker APEX II diffractometer using Mo K$\alpha$ radiation ($\lambda$ = 0.71073 \AA) at 100 K. Exposure time was 35 seconds with a detector distance of 60 mm. Unit cell refinement and data integration were performed with Bruker APEX2 software. A total of 1464 frames were collected over a total exposure time of 14.5 hours. 21702 diffracted peak observations were made, yielding 1264 unique observed reflections (centrosymmetric scaling) collected over a full sphere. No diffuse scattering was observed. The crystal structure was refined using the full-matrix least-squares method on F$^2$, using SHELXL2013 implemented through WinGX. An absorption correction was applied using the analytical method of De Meulenaer and Tompa \cite{deMeulenaera04926} implemented through the Bruker APEX II software. The crystal surfaces were studied with in a home-built cryogenic scanning tunneling microscope at 2 K. No twinning was observed in either the STM or single crystal diffraction measurements. Electronic structure calculations were performed in the framework of density functional theory using the Wien2k code\cite{blaha1990full} with a full-potential linearized augmented plane-wave and local orbitals basis together with the Perdew-Burke-Ernzerhof parameterization of the generalized gradient approximation\cite{perdew1996generalized}. The plane wave cutoff parameter RMTKmax was set to 7 and the Brillouin zone (BZ) was sampled by 250 k-points. The experimentally determined structure was used and spin orbit coupling (SOC) was included.
\section{Results and Discussion}
\subsection{X-ray Diffraction}
\indent{} From SXRD, a body-centered tetragonal unit cell of \textit{a} = 12.633(3) and \textit{c} = 25.427(7) was found, matching PXRD measurements. The systematic absences of the reflections in the SXRD reciprocal lattice are consistent with the possible space groups I4$_1$cd and I4$_1$/acd as previously described \cite{steigmann1968crystal}. It has previously been theorized that Cd$_3$As$_2$ has the ideal Mn$_7$SiO$_{12}$ structure type in the I4$_1$/acd space group, \cite{Zdanowicz1964} but no experimentally determined atomic positions were reported. Here, initial refinements in setting 2 of the centrosymmetric space group I4$_1$/acd were performed with all atoms on the idealized positions of the Mn$_7$SiO$_{12}$ structure type. This idealized model gave very poor fits to the data, with R1 only falling to 60$\%$. It was clear from analysis of the electron density maps that the atoms are in fact not at the ideal positions. In the final refinement, atomic positions as well as anisotropic thermal parameters for all atoms were allowed to vary freely. The refinement results are summarized in Table 1. Table 2 lists final positions for all atoms and Table 3 lists the Cd-As bond lengths. Refined atomic displacement parameters may be found in the .cif file.
\begin{table}[t]
\caption{\small{Refinement Parameters for Cd$_3$As$_2$}}
\scalebox{1}{
\begin{tabular}{lcccc}
\hline
\hline
Phase & Cd$_3$As$_2$ \\
Symmetry & Tetragonal, I4$_1$/acd (No. 142) \\
Cell Parameters (\AA) & \textit{a} = \textit{b} = 12.633(3), \textit{c} = 25.427(7) \\
& $\alpha$ = $\beta$ = $\gamma$ = 90$^{\circ}$ \\
Wavelength (\AA) & Mo K$\alpha$ - 0.71073 \\
Temperature (K) & 100 \\
V (\AA$^3$) & 4058.0(2) \\
Z & 32 \\
Calculated Density (g/cm$^3$)& 6.38 \\
Formula Weight (g/mol) & 487.1 \\
Absorption Coefficient (mm$^{-1}$)& 25.07 \\
Observations & 21702 \\
F000 & 6720 \\
Data/Restraints/Parameters & 1264/0/48 \\
R1 (all reflections)& 0.0480 \\
R1 Fo $>$ 4$\sigma$(Fo) & 0.0220 (893) \\
wR2 (all) & 0.040 (1264) \\
R$_{int}$/R($\sigma$) & 0.0617/0.0238 \\
Difference e$^-$ density (e/\AA$^3$) & 1.24/-1.25 \\
GooF & 1.035 \\
\hline
\hline
\end{tabular}}
\label{}
\end{table}
\indent{}Resolving the ambiguity between a centrosymmetric crystal structure and a noncentrosymmetric crystal structure that is nearly centrosymmetric can sometimes be difficult, and is a well-known problem in crystallography\cite{MarshCentro,Ermera07176}. Unless a clear choice can be made in favor of the noncentrosymmetric model, structures must be described centrosymmetrically\cite{MarshCentro}. The authors of the previous non-centrosymmetric structure report did not appreciate the fact that an alternative origin choice would allow for a centrosymmetric structure and thus did not check a centrosymmetric model against their observed intensities, which were estimated from exposed film densities \cite{steigmann1968crystal}. In order to compare a noncentrosymmetric model for Cd$_3$As$_2$ to the centrosymmetric model, we carried out a refinement in the noncentrosymmetric I4$_1$cd space group, using the published model as a template. The final refinements (see supplementary information) with atomic positions and anisotropic displacement parameters allowed to vary yielded a wR2 of 0.0489 for 2481 data (with noncentrosymmetric scaling) with 93 parameters and an R1 = 0.0265 for 1639 Fo $>$ 4F$\sigma$. The centrosymmetric model displays significantly better R-values. The R1 value for example for all data for the I4$_1$/acd solution is 0.0481 while for the I4$_1$cd solution it is 0.0611. This is an improvement by a factor of 1.22 for the centrosymmetric structure. In addition, we used PLATON\cite{PLATON} to check for and detect missed symmetry\cite{flack2006centrosymmetric}. The ADDSYM analysis (Le Page algorithm for missing symmetry) was used on the previously reported I4$_1$cd noncentrosymmetric structure. For that noncentrosymmetric model, PLATON detected a missing inversion center at (0, $\frac{1}{4}, \frac{1}{4}$) in the unit cell and recommended a 180$^{\circ}$ rotation around the \textit{c}-axis followed by an origin shift to (0, -$\frac{1}{4}$, $\frac{1}{4}$), and a suggested space group of I4$_1$/acd. Further, the deviations of the atoms in the noncentrosymmetric model from their equivalents in the centrosymmetric model are calculated and found to be very small ($\approx$.03 \AA, supplementary info). Finally the Flack parameter (supplementary info) indicated that the structure was not noncentrosymmetric \cite{Flack:a22047}. This analysis thus further confirmed that the centrosymmetric crystal structure is the correct one.
\subsection{Crystal Structure}
\indent{}Cd$_3$As$_2$ has different but related crystal structures as a function of temperature, which all can be considered defect antifluorite types; the Cd is distributed in the cube-shaped array occupied by F in CaF$_2$, while the As is in the FCC positions that are occupied by Ca. Thus in Cd$_3$As$_2$, the formally Cd$^{2+}$ ions are four-coordinated by As and the formally As$^{3-}$ ions are eight-coordinated by Cd. With As$^{3-}$ in VI fold coordination expected to have a radius of about 2.22 \AA~and Cd$^{2+}$ in IV coordination expected to have a radius of about 0.92 \AA, the $\frac{r_{Cd^{2+}}}{r_{As^{3-}}} = 0.41$, which is near the ideal 0.414 cutoff for tetrahedral coordination of the metal in the CaF$_2$-type structure ($\frac{r_M}{r_X}$ = 0.15 - 0.414).\cite{muller1993inorganic}. Further, as evidenced by the stoichiometry, Cd$_3$As$_2$ is Cd deficient of the ideal Cd$_4$As$_2$ antifluorite formula, missing $\frac{1}{4}$ of the Cd atoms needed to form a simple cube around the As. Thus the (distorted) cube can be written as Cd$_6$$\Box$$_2$, where $\Box$ = an empty vertex. The As and Cd coordinations in our Cd$_3$As$_2$ structure are shown in Figure 1. In the ordered, lower temperature variants, the Cd atoms shift from the ideal antifluorite positions toward the empty vertices of the cube (shown in Figure 2a). This distortion makes occupancy of the empty vertices highly energetically unfavored. At high temperatures (T $>$ 600 $^{\circ}{\rm C}$) the Cd ions are disordered\cite{alphaCd3As2} and so Cd$_3$As$_2$ adopts the ideal antifluorite space group Fm-3m with \textit{a} = 6.24 \AA~(The HT structure). Between 600 $^{\circ}{\rm C}$ and 475 $^{\circ}{\rm C}$, the intermediate temperature (IT) P4$_2$/nmc structure is found, where the Cd ions order so that the empty vertices are located on diagonally opposite corners of one face of the Cd$_6$$\Box$$_2$ cubes (see below). These cubes then stack so that the empty vertices form channels parallel to the \textit{a} and \textit{b} axes at different levels along the \textit{c} axis\cite{alphaCd3As2}. On further cooling, another ordering scheme is found. Below 475 $^{\circ}{\rm C}$, the Cd atoms further order in a three-dimensional fashion, such that each distorted Cd$_6$$\Box$$_2$ cube stacks on top of the previous one after a 90$^{\circ}$ rotation (either clockwise or counter-clockwise depending on the particular chain) about the stacking axis (parallel to the \textit{c} axis). This is the low temperature (LT) centrosymmetric I4$_1$/acd structure of the crystals whose structure is determined here and whose physical properties are of current interest. This also appears to be a new structure type. Figures 2b and 2c show how the three structures are related: the P4$_2$/nmc structure is a supercell of the disordered Fm-3m structure, and the I4$_1$/acd structure is a supercell of the P4$_2$/nmc structure.
\begin{table}[t]
\caption{\small{Refined atomic positions for Cd$_3$As$_2$ in space group I4$_1$/acd}}
\scalebox{1}{
\begin{tabular}{ccccc}
\hline
\hline
Atom & Wyckoff & \textit{x} & \textit{y} & \textit{z} \\[1ex]
\hline
Cd1 & 32g & 0.13955(3) & 0.36951(3) & 0.05246(2) \\
Cd2 & 32g & 0.11162(3) & 0.64230(3) & 0.07243(2) \\
Cd3 & 32g & 0.11863(3) & 0.10598(4) & 0.06247(2) \\
As1 & 32g & 0.24602(4) & 0.25779(5) & 0.12315(2) \\
As2 & 16d & 0 & $\frac{1}{4}$ & 0.99931(2) \\
As3 & 16e & $\frac{1}{4}$ & 0.51070(7) & 0 \\
\hline
\hline
\end{tabular}}
\label{}
\end{table}
\begin{table}[h]
\caption{\small{Bond lengths in Cd$_3$As$_2$}}
\scalebox{1}{
\begin{tabular}{ccc|cccc}
\hline
\hline
Atom1 & Atom2 & Distance \AA & & Atom1 & Atom2 & Distance \AA \\[1ex]
\hline
As1 & Cd3 & 2.6250(8) & & Cd1 & As3 & 2.6282(8) \\
& Cd2 & 2.6507(8) & & & As1 & 2.6518(8) \\
& Cd1 & 2.6518(8) & & & As2 & 2.6858(7) \\
& Cd2 & 2.6552(8) & & & As1 & 2.9838(8) \\
& Cd3 & 2.9408(8) \\
& Cd1 & 2.9837(8) \\
& & \\
As2 & Cd2 & 2.6771(8) & & Cd2 & As1 & 2.6507(8) \\
& Cd2 & 2.6771(8) & & & As1 & 2.6552(8) \\
& Cd1 & 2.6858(7) & & & As2 & 2.6771(8) \\
& Cd1 & 2.6858(7) & & & As3 & 3.0351(8) \\
& Cd3 & 2.8522(7) \\
& Cd3 & 2.8522(7) \\
& & \\
As3 & Cd3 & 2.5935(7) & & Cd3 & As3 & 2.5935(7) \\
& Cd3 & 2.5935(7) & & & As1 & 2.6250(8) \\
& Cd1 & 2.6282(8) & & & As2 & 2.8522(7) \\
& Cd1 & 2.6282(8) & & & As1 & 2.9408(8) \\
& Cd2 & 3.0351(8) \\
& Cd2 & 3.0351(8) \\
\hline
\hline
\end{tabular}}
\label{}
\end{table}
\indent{}The I4$_1$/acd structure has 3 unique Cd atoms and 3 unique As atoms and is schematically shown in Figure 3. Since the empty vertices of the Cd$_6$$\Box$$_2$ cubes sit diagonally opposite each other in a face of the distorted cube, the incomplete Cd cube can be thought of as having only one ``closed face". This ``closed face" changes position in either a clockwise or counter-clockwise fashion as the cubes stack along the \textit{c} direction, resulting in a screwing chain of cubes. These chains then align so that each chain is next to a chain of an opposite handedness; a right-handed chain is surrounded by left-handed chains. This results in an inversion center being present between each chain.
\indent{}For physical property measurements, it is important to identify characteristic planes and directions in the as-grown crystals. Cd$_3$As$_2$ grown as described here forms both irregular and rod-like crystals, all of which appear to have planar pseudo-hexagonal surfaces, often in steps, perpendicular to the long crystal axis. While the detailed SXRD measurements for structural refinement were carried out on small crystals, which gave the cleanest diffraction spots, larger crystals are employed for property measurements. Several of these larger crystals (e.g. with typical dimensions of 0.15 mm by 0.05 mm by 1.2 mm) were used in order to ascertain the growth direction of the needles. The crystals were mounted onto flat kapton holders and the Bruker APEX II software\cite{APEXII} was used to indicate the face normals of the crystal after the unit cell and orientation matrix were determined. The largest crystals were also placed onto PXRD slides and oriented diffraction experiments were conducted. The long axis of the needles was consistently found to be the [110] direction. The planar pseudo-hexagonal crystal surfaces with normals perpendicular to the growth direction are the (112) planes; these correspond to the close packed planes in the cubic antifluorite phase transformed into the LT structure supercell Miller indices. STM studies find that the cleavage plane of Cd$_3$As$_2$ is the pseudo-hexagonal (112) plane found here as a well-developed face in the bulk crystals. Figure 4a shows a projection of the structure with the (112) plane shaded in orange. Figure 4b is a topographic STM image (V$_{bias} =$ -250 mV, I = 50 pA, Temp = 2 K) of the (112) cleavage plane. The nearest neighbor spacing is found to 4.4$\pm$0.1 \AA, which is consistent with the As-As spacing or Cd-Cd spacing on a (112) type plane. Also visible in the inset of Figure 4b is the appearance of a 2x2 surface reconstruction, likely due to dangling bonds from the termination of the (112) plane during cleavage. Figure 4c is a projection slightly off of the [110] direction, with the corresponding lattice plane shaded in green. Figure 4d shows the unit cell axes and the growth directions of a large needle crystal of Cd$_3$As$_2$ that is suitable for physical property measurements.
\indent{}A [110] needle direction for a tetragonal symmetry crystal is relatively uncommon \cite{buerger1956elementary}. The crystal growth conditions employed in this study were such that the majority of the slow cooling took place through the stable temperature region of the IT P4$_2$/nmc structure. We hypothesize that the fastest growing direction in this phase is along the chain axis of the empty cube vertices, which is a principal axis of the structure (either the a or b axis). Thus the needle crystals have already grown before the last 50 degrees of the cooling, during which the low temperature annealing results in the ordering of the Cd atoms that yields the I4$_1$/acd structure. As can be seen from Figure 2, the principal axes of the P4$_2$/nmc structure become the set of [110] directions in the I4$_1$/acd structure. Variation of the growth temperature and conditions in future crystal growth studies may result in different types of needle axes, but the (112) close packed plane for crystal face development and cleavage is likely to be strongly preferred.
\subsection{Electronic Structure}
\indent{}The electronic structure calculated from the experimentally determined centrosymmetric structure found here (Table 2) is shown in Figure 5. This band structure is similar to that reported previously for the incorrect noncentrosymmetric crystal structure\cite{PhysRevB.88.125427}. Crucially, however, due to the inversion symmetry present in the I4$_1$/acd structure, there is no spin splitting around the Dirac point (where the bands cross between $\Gamma$ and Z at E$_F$ and all bands are at least two-fold degenerate. Thus, the centrosymmetric structure indicates that Cd$_3$As$_2$ is a 3D-Dirac semi-metal with two fold degenerate bands that come together to a four-fold degenerate Dirac point at the Fermi level. It is therefore the 3D analog to graphene, where there is also no spin-splitting at the Dirac point. Furthermore, the states at the $\Gamma$ point can now be characterized by their full symmetry (including their inversion eigenvalue) thus allowing parity counting to demonstrate the nontrivial topology in this ground state.
\section{Conclusion}
\indent{}In conclusion, we report the correct crystal structure of Cd$_3$As$_2$ in the low temperature, three-dimensionally ordered phase, as well as the corresponding electronic structure. We identify that for crystals grown as reported here, the needle growth direction is the [110] and the cleavage plane and most developed face in crystals is the pseudo-hexagonal (112) plane. Also present is a 2x2 surface reconstruction of the (112) plane cleavage surface. We found that Cd$_3$As$_2$ crystallizes in the centrosymmetric, I4$_1$/acd space group, and as such appears to be a new structure type. The Cd$_6$$\Box$$_2$ cubes order in a spiral, corkscrew fashion along an axis parallel to \textit{c}. Each corkscrew is surrounded by corkscrews of the opposite handedness, which results in the overall structure having inversion symmetry. The previously reported model in the I4$_1$cd space group (\# 110), where an inversion center was omitted, can be related to this structure by rotating by 180$^{\circ}$ about the c-axis and then shifting the origin to 0, -$\frac{1}{4}$, $\frac{1}{4}$. Since the correct centrosymmetric structural model uses only 6 unique atoms to describe the structure, electronic structure calculations become much more facile, which will help with the theoretical analysis of the electronic structure of this phase. In electronic structure calculations based on the centrosymmetric crystal structure, we find that the previously reported bulk band crossing along $\Gamma$-Z at the Fermi level is maintained, however due to the inversion symmetry, no spin splitting is allowed. Therefore Cd$_3$As$_2$ is expected to be a non-spin-split 3D-Dirac semi-metal, and a three-dimensional analog to graphene.
\bigskip
\begin{acknowledgement}
This research was supported by the Army Research Office, grant W911NF-12-1-0461.
\end{acknowledgement}
|
2,877,628,090,240 | arxiv | \section{Introduction
\label{sec:intro}
The past few years have been marked by a slow, but steady, shift from the era of the detections of exoplanets to the new age of the characterization of their atmospheres. Exoplanets transiting in front of their host stars allow for atmospheric features to be imprinted onto the total system light \citep{se00,br01,ch02}. Secondary eclipses allow for photons from the exoplanetary atmosphere to be directly measured \citep{ch05,deming05}. Extracting the spectroscopic signatures of these exoplanetary atmospheres is a challenging task, because they are typically many orders of magnitude fainter than the light from their host stars. Interpreting these signatures requires a profound understanding of radiative transfer and atmospheric chemistry, in order to infer the thermal structure and atomic/molecular abundances of the atmosphere from the data.
Hot Jupiters are particularly accessible to atmospheric characterization via transits and eclipses. They are hardly one-dimensional (1D) objects, but a reasonable first approach is to study them using 1D, plane-parallel model atmospheres \citep{su03,ba05,fo05,fo06,fo08,fo10,bu06,bu07,bu08}, which may be used to mimic the dayside- or nightside-integrated emission. The simplest model one may construct of a dayside emission spectrum (besides a Planck function) is a 1D model with an atmosphere in radiative and chemical equilibrium, if one neglects the effects of atmospheric dynamics and photochemistry. Despite these simplifications, there are several non-trivial demands associated with such a model: it should be able to consider a rich variety of chemistries, metallicities, irradiation fluxes from the star and internal heat fluxes from the interior of the exoplanet. It should be able to take, as an input, arbitrary combinations of molecules and their opacities. The synthetic spectrum computed should be highly customizable, such that it may be readily compared to both photometric and spectroscopic data, often combined in a heterogeneous way across wavelength. To explore such a broad range of parameter space, the numerical implementation of a model (short: ``code") needs to solve for radiative equilibrium very efficiently and also allow for numerical convergence to be checked in several different ways: number of model layers, spectral resolution of opacity function, number of wavelength bins used, etc. Such a code forms the basis of a flexible radiation package that one may couple to a chemical kinetics code or a three-dimensional general circulation model. The challenges of constructing a 1D radiative-convective model are also discussed in the review article by \cite{marley15}, where the ``convective" part stands for the additional consideration of convective stability, which marks the next step in sophistication of an atmospheric model.
In the current work, we present a customizable and built-from-scratch computer code named \texttt{HELIOS}\footnote{Named after the Greek god of the Sun.}, which has or uses the following components.
\begin{itemize}
\item In this initial version, we use the analytical solutions of the radiative transfer equation in the two-stream approximation, as derived by \cite{he14}. These solutions enable us to iteratively and self-consistently solve for the temperature-pressure profile of the atmosphere via iteration with its opacity function, which generally depends on temperature, pressure and wavelength. The synthetic spectrum is obtained as a natural by-product of this self-consistent calculation.
\item For the opacity function of the atmosphere, we use our open-source and custom opacity calculator, \texttt{HELIOS-K}, which was previously published by \cite{gr15}. The finest resolution we have used is $10^{-5}$ cm$^{-1}$ across the entire wavenumber range considered. We then compute $k$-distribution tables from this finely-spaced grid of opacities across temperature, pressure and molecular species.
\item Throughout this work, we assume chemical equilibrium, which effectively means that the chemistry is described by only two parameters: the elemental abundances of carbon and oxygen. Given the input values of these elemental abundances, we then use the validated analytical formulae of \cite{he16b} and \cite{he16c} to calculate the mixing ratios (abundances normalized to that of molecular hydrogen) of the various molecules. We consider water (H$_2$O), carbon monoxide (CO), carbon dioxide (CO$_2$) and methane (CH$_4$).
\item We have built \texttt{HELIOS} to run on graphics processing units (GPUs) to maximize the computational throughput. A \texttt{HELIOS} calculation with 101 model layers and 300 wavelength bins takes only a few minutes to complete on a personal computer with a NVIDIA GeForce 750M GPU.\footnote{Note that these are fully converged and self-consistent models, which require iteration to solve for radiative equilibrium.} This level of efficiency allows us to effectively perform parameter studies.
\end{itemize}
In Section \ref{sec:meth}, we provide a detailed description of our methodology, including the equations and boundary conditions used, the numerical methods, the structure of our grid, the opacity calculations, the chemistry model, and the stellar models used. In Section \ref{sec:res}, we subject \texttt{HELIOS} to various tests, use it to address several lingering ambiguities\footnote{We describe these issues as ``lingering", because studies in the published literature typically omit the details involved, which prevents us from directly comparing our results to them.} in the literature and also to examine 6 case studies of hot Jupiters. In Section \ref{sec:sum}, we summarize our results, compare them to previous work and discuss opportunities for future work.
\section{Methodology
\label{sec:meth}
\subsection{Radiative Transfer Scheme}
\label{sec:rad}
\subsubsection{Preamble}
Any scheme to represent the propagation of radiation through an atmosphere has to solve the radiative transfer equation \citep{ch60,mi70},
\begin{equation}
\mu \frac{\partial I_\lambda}{\partial \tau_\lambda} = I_{\lambda} - S_\lambda,
\label{eq:general}
\end{equation}
where $I_{\lambda}$ is the monochromatic and wavelength-dependent intensity, $\mu \equiv \cos\theta$ is the cosine of the incident angle ($\theta$) relative to the normal and $\tau_\lambda$ is the optical depth measured from the top of the atmosphere downwards. We denote the wavelength by $\lambda$. The crucial ``length" to adopt in radiative transfer is the optical depth. (Only a non-vanishing $\Delta \tau_\lambda$ leads to a change in intensity $\Delta I_\lambda$.) The source function $S_\lambda$ accounts for both radiation scattered into the line of sight and the thermal emission associated with each location in the medium. Equation (\ref{eq:general}) is generally difficult to solve, because it is a partial differential equation in $\tau_\lambda$ and $\mu$.
A commonly used simplification is to reduce equation (\ref{eq:general}) to an ordinary differential equation in $\tau_\lambda$ by integrating over the incoming ($-\pi/2 \le \theta \le 0$ or $-1 \le \mu \le 0$) and outgoing ($0 \le \theta \le \pi/2$ or $0 \le \mu \le 1$) hemispheres and assuming that the ratios of various moments of the intensity are constant and take on specific values. This is known as the ``two-stream approximation" \citep{me80}. One may then solve the ordinary differential equation analytically to obtain solutions for \textit{pairs} of model atmospheric layers \citep{he14}. The moments of the intensity are related by the so-called ``Eddington coefficients". Of particular interest to us is the first Eddington coefficient \citep{he14},
\begin{equation}
\epsilon = \frac{1}{{\cal D}},
\end{equation}
which is related to the ``diffusivity factor" ${\cal D}$. In the next subsection, we show that ${\cal D}$ should take on a value between 1 and 2 depending on the thickness of the atmospheric layers.
In the current study, we use the two-stream solutions previously derived by \cite{he14}. We note that these solutions allow for the inclusion of non-isotropic scattering via two functions: the single-scattering albedo ($\omega_0$) and the scattering asymmetry factor ($g_0$) \citep{go89,pi10}. Pure absorption and scattering correspond to $\omega_0=0$ and $\omega_0=1$, respectively. Forward, backward and isotropic scattering correspond to $g_0=1$, -1 and 0, respectively. Our formulation allows for $\omega_0$ and $g_0$ to be specified as functions of wavelength/frequency/wavenumber, temperature and pressure.
Hereafter, the term ``flux" describes a wavelength-dependent quantity.\footnote{Accordingly, the units of the flux $F$ are $[F]=$ erg s$^{-1}$ cm$^{-3}$.} Integrating the flux over all wavelengths, one obtains the ``bolometric flux". We also neglect for readability the subscript $\lambda$ for $\tau$ and $B$.
\subsubsection{Exact solution of the radiative transfer equation in the pure absorption limit}
As previously shown by \cite{he14} (and references therein), the radiative transfer equation has an exact solution in the limit of pure absorption ($\omega_0=0$). We use a staggered grid (see Section \ref{subsect:grid}), such that the two-stream solutions are applied to the \textit{interfaces} of a model layer. We label the interfaces by ``1" and ``2" and our convention is to locate interace 2 above interface 1 in altitude. If the layer has only one temperature throughout (i.e., it is isothermal), then the fluxes at the interfaces are given by
\begin{equation}
\label{eq:isodirect}
\begin{split}
F_{2,\uparrow} &= \mathcal{T} F_{1,\uparrow} + \pi B_1 (1-\mathcal{T}) , \\
F_{1,\downarrow} &= \mathcal{T} F_{2,\downarrow} + \pi B_1 (1-\mathcal{T}).
\end{split}
\end{equation}
The $\uparrow$ and $\downarrow$ subscripts refer to the outgoing and incoming fluxes, respectively. The blackbody intensity within this layer is given by $B_1$.
We can improve upon the isothermal-layer treatment by considering a (linear) temperature gradient within the layer \citep{toon89}. If we instead Taylor-expand the Planck function in $\tau$ and retain only the constant and linear terms, we obtain
\begin{equation}
\label{eq:nonisodirect}
\begin{split}
F_{2,\uparrow} =& \mathcal{T} F_{1,\uparrow} + \pi B_1 (1 - \mathcal{T}) \\
+& \pi B^\prime \left\{ \frac{2}{3}\left[1-e^{-\Delta \tau} \right] - \Delta \tau \left(1-\frac{\mathcal{T}}{3}\right) \right\} , \\
F_{1,\downarrow} =& \mathcal{T} F_{2,\downarrow} + \pi B_2 (1 - \mathcal{T}) , \\
+& \pi B^\prime \left\{-\frac{2}{3}\left[1-e^{-\Delta \tau} \right] + \Delta \tau \left(1-\frac{\mathcal{T}}{3}\right) \right\}.
\end{split}
\end{equation}
following the derivation in \cite{he14}. The difference in optical depth between the layers is given by $\Delta \tau \equiv \tau_2 - \tau_1$. The gradient of the Planck function is approximated by
\begin{equation}
\label{eq:planck}
B^\prime \approx \frac{B_2 - B_1}{\tau_2 - \tau_1},
\end{equation}
where $B_1$ and $B_2$ are now the Planck functions for the temperatures at the interfaces 1 and 2, respectively.
In both the isothermal and non-isothermal cases, the transmission function or transmissivity is
\begin{equation}
\begin{split}
\label{eq:direct_trans}
\mathcal{T} &= 2 \int^1_0 \mu \exp\left(-\frac{\Delta\tau}{\mu}\right) d\mu , \\
&= (1-\Delta \tau)\exp{\left(-\Delta \tau\right)}+\Delta \tau^2 \mathcal{E}_1(\Delta \tau) ,
\end{split}
\end{equation}
where $\mathcal{E}_1$ is the exponential integral of the first order. Unlike for the two-stream solutions, there is no need to specify $\mathcal{D}$ as an input, because it has an exact solution,
\begin{equation}
\label{eq:diff}
\mathcal{D} = - \frac{1}{\Delta \tau}\ln{\left[\left(1-\Delta \tau \right)\exp{\left(-\Delta \tau\right)}+\Delta \tau^2 \mathcal{E}_1\left(\Delta \tau \right) \right]} .
\end{equation}
For very thin layers ($\Delta \tau \ll 1$), ${\cal D}=2$ is an accurate approximation, but as the layer becomes optically thick the value of $\cal{D}$ approaches unity (Figure \ref{fig:diff}). Operationally, since we pick our model grid to be equally spaced in the logarithm of pressure, it means that the value of $\Delta \tau$ is small near the top of the model atmosphere and gradually becomes large (and exceeds unity) at high pressures. Within the context of the two-stream approximation, assuming ${\cal D}$ to be constant is equivalent to picking a representative or mean value, over the entire atmosphere, of the diffusivity factor.
As already pointed out by \cite{he14}, the analytical expression for ${\cal D}$ when scattering is present (equivalent to eq. \ref{eq:diff}) is unknown.
It is worth emphasizing that equations (\ref{eq:isodirect}) and (\ref{eq:nonisodirect}) are exact solutions and that the two-stream approximation is \textit{not} taken. In Section \ref{subsect:diffuse}, we compare these exact solutions to the two-stream solutions to derive the value of $\mathcal{D}$.
\subsubsection{Different flavors of two-stream solutions}
We now rederive the two-stream solutions of \cite{he14} without setting ${\cal D}=2$, so as to facilitate comparisons with the exact solutions. For all of the solutions presented in this subsection, the transmission function is
\begin{equation}
\label{eq:trans}
\mathcal{T} \equiv \exp{\left[- \mathcal{D} \sqrt{(1 - \omega_0 g_0)(1 - \omega_0)} \Delta \tau \right]}.
\end{equation}
The simplest two-stream solutions are derived in the limit of pure absorption and isothermal atmospheric layers,
\begin{equation}
\label{eq:isonoscat}
\begin{split}
F_{2,\uparrow} &= \mathcal{T} F_{1,\uparrow} + 2\pi\epsilon B_1 (1-\mathcal{T}) , \\
F_{1,\downarrow} &= \mathcal{T} F_{2,\downarrow} + 2\pi\epsilon B_1 (1-\mathcal{T}).
\end{split}
\end{equation}
Without scattering ($\omega_0 = 0$), the coupling coefficients are $\zeta_+ = 1$ and $\zeta_- = 0$, and the transmission function simply becomes $ \mathcal{T} = \exp(-\mathcal{D}\Delta \tau)$. If we increase the sophistication of the model by considering non-isothermal layers and pure absorption, we obtain
\begin{equation}
\begin{split}
F_{2,\uparrow} =& \mathcal{T} F_{1,\uparrow} + 2\pi\epsilon \left[ B_1 - \mathcal{T} B_2 + \epsilon B' (1 - \mathcal{T}) \right] , \\
F_{1,\downarrow} =& \mathcal{T} F_{2,\downarrow} + 2\pi\epsilon \left[ B_2 - \mathcal{T} B_1 - \epsilon B' (1 - \mathcal{T}) \right] .
\end{split}
\end{equation}
For isothermal atmospheric layers with non-isotropic scattering being included, the two-stream solutions for the fluxes read
\begin{equation}
\label{eq:isoscat}
\begin{split}
F_{2,\uparrow} &= \frac{1}{\alpha} \left[\xi F_{1,\uparrow} - \beta F_{2,\downarrow} + 2\pi \epsilon B_1 (\beta - \upsilon) \right] , \\
F_{1,\downarrow} &= \frac{1}{\alpha} \left[\xi F_{2,\downarrow} - \beta F_{1,\uparrow} + 2\pi \epsilon B_1 (\beta - \upsilon) \right].
\end{split}
\end{equation}
The coefficients $\alpha$, $\beta$, $\xi$, $\upsilon$ are defined as
\begin{equation}
\label{eq:cap}
\begin{split}
\alpha &\equiv \zeta^2_- \mathcal{T}^2 - \zeta^2_+ ,\\
\beta &\equiv \zeta_+ \zeta_- (1-\mathcal{T}^2) ,\\
\xi &\equiv (\zeta^2_- - \zeta^2_+) \mathcal{T} ,\\
\upsilon &\equiv (\zeta^2_- \mathcal{T} + \zeta^2_+)(1-\mathcal{T}) ,
\end{split}
\end{equation}
with the coupling coefficients being
\begin{equation}
\mathcal{\zeta_\pm} \equiv \frac{1}{2} \left[ 1 \pm \left(\frac{1 - \omega_0}{1 - \omega_0 g_0}\right)^{1/2} \right].
\end{equation}
In the limit of $\omega_0 = 1$, the equations in (\ref{eq:isoscat}) are replaced by
\begin{equation}
\label{eq:purescat}
\begin{split}
F_{2,\uparrow} &= F_{1,\uparrow} - \frac{\mathcal{D} (1 - g_0) \tau_0 (F_{1,\uparrow} - F_{2,\downarrow})}{2 + \mathcal{D}(1 - g_0) \tau_0} , \\
F_{1,\downarrow} &= F_{2,\downarrow} + \frac{\mathcal{D} (1 - g_0) \tau_0 (F_{1,\uparrow} - F_{2,\downarrow})}{2 + \mathcal{D}(1 - g_0) \tau_0} .
\end{split}
\end{equation}
These solutions give the correct limits of a transparent or opaque atmosphere when $\omega_0=1$ \citep{he14}. The general solutions stated before in equation (\ref{eq:isoscat}) do not reproduce this limit.
Our most sophisticated two-stream solutions include non-isotropic scattering and non-isothermal model atmospheric layers,
\begin{equation}
\label{eq:nonisoscat}
\begin{split}
F_{2,\uparrow} =& \frac{1}{\alpha} \left\{\xi F_{1,\uparrow} - \beta F_{2,\downarrow} + 2\pi \epsilon \left[ B_1 (\alpha + \beta) - B_2 \xi \right.{} \right.{} \\
&+ \left.{} \left.{} \frac{\epsilon}{1+\omega_0 g_0} B' (\alpha - \xi - \beta) \right] \right\}, \\
F_{1,\downarrow} =& \frac{1}{\alpha} \left\{\xi F_{2,\downarrow} - \beta F_{1,\uparrow} + 2\pi \epsilon \left[ B_2 (\alpha + \beta) - B_1 \xi \right.{} \right.{} \\
&+ \left.{} \left.{} \frac{\epsilon}{1+\omega_0 g_0} B' (\xi - \alpha + \beta) \right] \right\}.
\end{split}
\end{equation}
Note that in the non-isothermal approach a single constant gradient of $B'$ is assumed within a layer. Thus $B_1$ and $B_2$ are placed at the interfaces.\footnote{In practice, in the numerical implementation of the equations one layer has to be divided into two sublayers (see Sect. \ref{subsect:grid}).} The coefficients $\alpha$, $\beta$ and $\xi$, as well as the coupling coefficients $\zeta_\pm$, retain the same functional forms as in the case of having isothermal layers.
Generally, we find that the non-isothermal solutions attain more rapid numerical convergence (to radiative equilibrium). In principle, if a large enough number of isothermal layers is used, the isothermal and non-isothermal calculations should agree.
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f1.pdf}
\end{minipage}
\vspace{-0.3cm}
\caption{Diffusivity factor $\mathcal{D}$, as a function of the difference in optical depth $\Delta \tau$ across a layer, in the limit of pure absorption.}
\label{fig:diff}
\end{center}
\end{figure}
\subsubsection{Rayleigh scattering}
To include the effects of Rayleigh scattering by molecules, we use the cross section \citep{sn05},
\begin{equation}
\sigma_{\rm scat, \lambda} = \frac{24 \pi^3}{n_{\rm ref}^2 \lambda^4} \left(\frac{n_\lambda^2 - 1}{n_\lambda^2 + 2}\right)^2 K_\lambda,
\end{equation}
where $n_{\rm ref}$ is the number density at a reference temperature and pressure $n_\lambda$ is the wavelength-dependent refractive index and $K_\lambda$ is the King factor, which is a correction factor for polarization.
In the current study, we focus on Rayleigh scattering by hydrogen molecules, but our approach may be straightforwardly generalized to other molecules. We ignore the contribution due to helium, which is less than 1\% compared to that of molecular hydrogen. For H$_2$, we use $n_{\rm ref} = 2.68678\times10^{19}$cm$^{-3}$, $K = 1$ and
\begin{equation}
n_\lambda = 13.58\times10^{-5}\left(1+7.52\times10^{-11} \mbox{ cm}^2 ~\lambda^{-2} \right)+1.
\end{equation}
The influence of Rayleigh scattering enters via its inclusion, as $\sigma_{\rm scat,\lambda}/\bar{m}$, to the opacity of each model layer, where $\bar{m}$ is the mean molecular mass, and also via the single-scattering albedo $\omega_0$. The dashed line in Figure \ref{fig:opac} shows the opacity of Rayleigh scattering by H$_2$, which dominates in the optical but becomes subdominant, compared to molecular absorption, in the infrared due to its dropoff with $\lambda^{-4}$.
If the scattering dominates and $(1- \omega_0) < 10^{-6}$ in this layer and waveband, then we switch to the pure scattering solutions (eq. \ref{eq:purescat}).
\subsection{Numerical Method}
\label{sec:numerical}
\subsubsection{Model grid}
\label{subsect:grid}
For the isothermal treatment, a staggered grid is used with the layers being separated by interfaces. There are $n$ layers and $n+1$ interfaces. The grid is evenly spaced in height or the logarithm of pressure, which serves as the vertical coordinate. The thickness of the $i$-th layer is given by
\begin{equation}
\Delta z_i = \frac{k_{\rm B} T_i}{\bar{m}g}\ln \left(\frac{P_{i,{\rm inter}}}{P_{i+1,{\rm inter}}}\right),
\end{equation}
with $k_{\rm B}$ being the Boltzmann constant, $g$ the surface gravity. For hydrogen-dominated atmospheres, we set $\bar{m}=2.4 m_p$ with $m_p$ being the mass of the proton. The pressures at the interfaces are represented by $P_{i,{\rm inter}}$ and $P_{i+1,{\rm inter}}$. The preceding expression is obtained from integrating the equation of hydrostatic balance over a model layer and assuming isothermality and the equation of state for an ideal gas.
The contribution to the optical depth\footnote{To be pedantic, the optical depth is a coordinate. It is the difference in optical depth that is needed for radiative transfer. The analogy is to distance versus displacement.} by the $i$-th layer is
\begin{equation}
\Delta\tau_{i} = \Delta m_{{\rm col},i} \kappa_i = \frac{P_{i,{\rm inter}} - P_{i+1,{\rm inter}}}{g} \kappa_i,
\label{eq:layertau}
\end{equation}
where $\kappa_i$ is the opacity and $\Delta m_{{\rm col},i}$ is the difference in column mass, which can be further written in terms of pressure and surface gravity.
For the non-isothermal grid, we require a more sophisticated grid layout, which is shown in Figure \ref{fig:grid}. Each layer has a temperature and pressure, located at its center. To compute the fluxes, we need to interpolate across the temperature and pressure grids to obtain their values at the interfaces. A key quantity to compute is the Planck function $B$, which relates the temperature to the thermal emission of a layer. If one constructs the grid using a single gradient $B{^\prime}$ of the Planck function over the whole layer, one is essentially decoupling the radiative transfer process from the temperature at the center of the layer. We solve this problem by splitting each layer into two sublayers, leading to two $B{^\prime}$ values within a layer. The fluxes are propagated first from the lower interface to the layer center, then from the layer center to the upper interface (and vice versa), similar to the approach taken in e.g. \cite{mendonca2015}. In this manner, both the layer centers and interfaces are involved in the iteration for radiative equilibrium.
Finally, in the non-isothermal grid, a numerical caveat arises in the upper atmosphere. There, the optical depth difference $\Delta\tau_{i}$ of a layer $i$ is tiny (due to the very small pressure) and thus the denominator of eq. (\ref{eq:planck}) vanishes, which in turn leads to numerical issues for $B^\prime$ in eq. (\ref{eq:nonisoscat}). To prevent this, we keep the sub-layered grid of the non-isothermal approach, but switch in each sublayer from the non-isothermal (eq. \ref{eq:nonisoscat}) to the isothermal prescription (eq. \ref{eq:isoscat}) whenever $\Delta\tau_{i} < 10^{-4}$ occurs.
\subsubsection{Boundary conditions}
\label{sec:bound}
At the top of the atmosphere (TOA), which is also the $n$-th interface of the model atmosphere, the flux is given by
\begin{equation}
F_{n, \downarrow} = f\left(\frac{R_\star}{a}\right)^2 \pi B_\star ,
\end{equation}
where $R_\star$ is the stellar radius, $a$ is the orbital distance of the planet and $B_\star$ is the stellar blackbody function. This represents the heating from the incident stellar flux. Most of the quantities in the preceding expression are astronomical observables (or quantities that may be inferred from the observations). It is possible to replace $B_\star$ by a more sophisticated model of the stellar spectrum (see Section \ref{sec:star}).
The quantity $f$ is a parameter that describes the redistribution of heat from the dayside to the nightside of a tidally-locked hot Jupiter, which is dictated by an interplay between atmospheric dynamics and radiative cooling. In principle, its value may be inferred from infrared phase curves. Theoretically, it is bounded between $f=1/4$ (full redistribution) and $f=1$ (no redistribution). Since we are using our 1D, plane-parallel model to describe the dayside emission spectra of hot Jupiters, the value of $f$ is a proxy for the dayside integrated absorption and re-emission of radiation. In the current study, we adopt $f=2/3$ following the arguments in e.g. \cite{bu08} and \cite{sb10}.
At the bottom of the model atmosphere (BOA), we have included the option to specify an internal radiative heat flux ($\pi B_{\rm intern}$), such that
\begin{equation}
\int \pi B_{\rm intern} ~d\lambda= \sigma_{\rm SB} T_{\rm intern}^4,
\end{equation}
where $\lambda$ is the wavelength, $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant, $T_{\rm intern}$ is the internal temperature, $B_{\rm intern} \equiv B(T_{\rm intern})$ and $B$ is the Planck function. The internal heat flux reflects the thermal heating due to gravitational contraction. The BOA is also the 0-th interface. It is important to note that any form of atmospheric heating is associated with the \textit{net flux} (the difference between the outgoing and incoming fluxes) \citep{he14},
\begin{equation}
\pi B_{\rm intern} = F_{0, \uparrow} - F_{0, \downarrow}.
\end{equation}
In our current study, we set $T_{\rm intern}=0$ K in the absence of such constraints on hot Jupiters.
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{f2.pdf}
\end{minipage}
\vspace{-1.3cm}
\caption{Staggered grid used for models with non-isothermal layers. The boundary conditions are applied at the top (stellar irradiation) and bottom (internal heat flux) of the model atmosphere, which are also the $n$-th and 0th interfaces, respectively. The pressure and temperature are located at the center of each layer, while the fluxes tranversing a layer are computed at the layer interfaces. We further divide each layer into two sublayers during the iteration for radiative equilibrium (see text for details). In the schematic, we have focused on the $k$-th layer and the various quantities associated with its center and interfaces. Quantities marked with an asterisk are temporarily used in the computation, but not stored as the final output. The layers are evenly spaced in the logarithm of pressure.}
\label{fig:grid}
\end{center}
\end{figure}
\subsubsection{Iterating for radiative equilibrium}
\label{sec:iter}
Within each model layer of the atmosphere, its temperature and pressure determine its absorption and scattering properties, given by molecular abundances and opacities, which in turn determines the transmission function and fluxes. However, as flux enters and exits the layer, the temperature changes, which in turn changes the opacity. Clearly, this is an iterative process. It turns out that one is iterating for radiative equilibrium, which is a statement of \textit{local} energy conservation \citep{he14}. Local energy conservation implies global energy conservation, but not vice versa \citep{he16b}.
We integrate the fluxes ($F_\uparrow$ and $F_\downarrow$) over the entire spectral range to obtain the bolometric fluxes ($\mathcal{F}_\uparrow$ and $\mathcal{F}_\downarrow$), which in turn allows us to construct the bolometric net flux ($\mathcal{F}_- \equiv \mathcal{F}_\uparrow - \mathcal{F}_\downarrow$). For the $i$-th layer, the divergence\footnote{In 1D, the divergence is simply the vertical gradient.} of the bolometric net flux becomes
\begin{equation}
\frac{ \Delta \mathcal{F}_{i,-}}{\Delta z_i} = \frac{(\mathcal{F}_{i+1,\uparrow} - \mathcal{F}_{i+1,\downarrow}) - (\mathcal{F}_{i,\uparrow} - \mathcal{F}_{i,\downarrow})}{\Delta z_i} .
\end{equation}
Between successive timesteps, the change in temperature of the $i$-th layer then becomes \citep{he14}
\begin{equation}
\label{eq:step}
\Delta T_i = \frac{1}{\rho_i c_{\rm p}}\frac{ \Delta \mathcal{F}_{i,-}}{\Delta z_i} \Delta t_i ,
\end{equation}
where $\rho_i$ is the local density and $\Delta t_i$ is the numerical timestep. The specific heat capacity of an ideal gas at constant pressure is \citep{pi10}
\begin{equation}
c_{\rm p} = \frac{2+n_{\rm dof}}{2\bar{m}}k_{\rm B} ,
\end{equation}
where we set the number of degrees of freedom of the gas to be $n_{\rm dof} = 5$, as is valid for a diatomic molecule (ignoring the vibrational modes) like molecular hydrogen, the main component of gas planet atmospheres. This simplification does not hold should the atmospheric composition change, e.g. by dissociation of molecular hydrogen at very high temperatures. Since in our model the only occurence of $c_{\rm p}$ is in the timestepping algorithm this flaw is for our cause of only minor concern, but would render e.g. the calculation of the entropy inaccurate.
In practice, we start with an arbitrary temperature profile with $\Delta \mathcal{F}_{i,-} \neq 0$. We perform the iteration described until $\Delta \mathcal{F}_{i,-}$ vanishes for each layer, which is our numerical condition for radiative equilibrium (c.f. eq. \ref{eq:condition}). Physically, each atmospheric layer emits the same amount of energy which it receives.
When scattering is present, the flux solutions become coupled. Each array of outgoing or incoming fluxes cannot be populated independently of the other. This is solved iteratively by starting with the coupled dependencies as zero and populating the flux arrays multiple times in each temperature step. We include 4 additional scattering iterations in the full radiative transfer calculation as the flux values are still known from the previous timestep and 80 scattering iterations for pure post-processing (propagating only once through the atmosphere) purposes.
\subsubsection{Numerical timestepping}
For the numerical timestepping, there are two options in \texttt{HELIOS}. The first option uses a fixed and uniform timestep ($\Delta t$) for every model layer. Typically, we choose $10^2 \lesssim \Delta t \lesssim 10^4$ s. The challenge is that $\Delta T \propto \rho^{-1}$ and $\rho$ may vary by several orders of magnitude across our model atmosphere. With a uniform timestep, the upper layers of the atmosphere attain convergence much more rapidly than the lower atmosphere. Thus, this approach is plausible and technically correct, but infeasible.
A more efficient approach is to implement an adaptive timestepping scheme that uses a different timestep for each model layer. Specifically, the timestep in the $i$-th layer is related to the radiative timescale ($t_{i,\rm rad}$),
\begin{equation}
\Delta t_i = f_{i,\rm pre} ~t_{i,\rm rad} ,
\end{equation}
where $f_{i,\rm pre}$ is a pre-factor to adjust to the optimal value of $\Delta t_i$. The radiative timescale is approximated by
\begin{equation}
t_{i,\rm rad} \approx \frac{c_{\rm p} P_i}{\sigma_{\rm SB} g T_i^3},
\end{equation}
where the temperature and pressure of the $i$-th layer is given by $T_i$ and $P_i$, respectively. With this improved timestepping scheme, the timestep becomes larger as one goes deeper into the model atmosphere. The evolution of the model does not strictly correspond to a physical evolution, but is rather a convenient way of reaching a numerical steady state.
To further optimize the efficiency of \texttt{HELIOS}, we also allow the timestep to vary in time as the model approaches radiative equilibrium. Specifically, the algorithm checks in each layer whether the temperature has oscillated for the most recent 6 successive timesteps. We find oscillations in temperature to be a robust and practical indicator of having adopted too large a timestep. If oscillations are detected, the timestep is reduced by 33\%. By contrast, if no oscillations are detected (i.e., the change in temperature is monotonic), then the timestep is increased by 10\%.
We note that the purpose of the pre-factor ($f_{i, {\rm pre}}$) is to dampen sudden spikes in $\Delta \mathcal{F}_{i,-}$. For practical purposes, it takes the form of
\begin{equation}
f_{i, {\rm pre}} = \frac{10^5}{\left[|\Delta \mathcal{F}_{i,-}|/\left({\rm erg \; s}^{-1} {\rm cm}^{-2}\right)\right]^{0.9} },
\end{equation}
which leads the temperature iteration step $\Delta T_i$ to depend only on $\Delta \mathcal{F}_{i,-}^{0.1}$, which guarantees the correct direction of the evolution but substantially smoothes irregularities, making the iteration process significantly more stable.
Finally, we need a condition to judge if radiative equilibrium has been established. Usually, one would assume a criterion demanding the rate of temperature change to be below a certain threshold, $\Delta T/\Delta t < \delta_{\rm limit}$, and evaluate whether this is satisfied in every layer. However, if $\Delta t$ is variable and not representing a physical time, then the utility of this approach becomes suspect. Instead of setting a threshold on the consequence of radiative equilibrium (changes in temperature), we set one on its cause (a vanishing bolometric flux divergence). We use the dimensionless convergence criterion,
\begin{equation}
\label{eq:condition}
\frac{\Delta \mathcal{F_-}}{\sigma_{\rm SB} T^4} < 10^{-7},
\end{equation}
where the change in bolometric net flux is normalized by the thermal emission associated with each layer. In practice, this criterion results in changes in temperature of less than 4 K at the BOA and less than 1 K in the photospheric regions, which impacts the emission spectrum by less than 0.5\%.
\subsection{Calculating Opacities and Transmission Functions}
\label{sec:opac}
Our method for computing the opacities (cross sections per unit mass) of molecules has previously been elucidated in \cite{gr15}, who published an opacity calculator named \texttt{HELIOS-K} that is part of the \texttt{HELIOS} radiation package. As such, we do not repeat the detailed explanations of \cite{gr15} and instead highlight only the salient points. We include the opacities associated with the four main infrared absorbers: H$_2$O, CO$_2$, CO and CH$_4$. We also include the opacities associated with the collision-induced absorption (CIA) of H$_2$-H$_2$ and H$_2$-He pairs. Table \ref{tab:opac} states the spectroscopic line lists used to compute our opacities, while Figure \ref{fig:opac} displays the final weighted opacities used in the code at one temperature and pressure.\footnote{The reader should be aware that, in this first version of \texttt{HELIOS}, we omit greenhouse gases like NH$_3$, HCN, C$_2$H$_2$ and the alkali metals Na and K, which may have an impact on the atmospheric structure. H and H$^-$ absorption may also be important at high temperatures. Nevertheless, our starting set of four molecules is sufficient for us to build up the first version of a radiative transfer code, and we intend to augment this set in the future.}
The first step involves calculating the opacity function (cross section per unit mass as a function of wavelength, temperature and pressure), which includes all of the molecules previously mentioned, at a given spectral resolution. If the spectral resolution is too coarse, then spectral lines may be missed or omitted, which leads to an under-estimation of the true opacity. To avoid this pitfall, we use a resolution of $10^{-5}$ cm$^{-1}$. Since the wavenumber range goes up to $\sim 10^4$ cm$^{-1}$, this means that we are sampling the opacity function at $\sim 10^9$ points, which approaches a true line-by-line calculation.
The shape of each spectral line is described by a Voigt profile. A major uncertainty associated with this approach, which remains an unsolved physics problem, is that the far line wings of the Voigt profile over-estimate or under-estimate the true opacity contribution depending on the molecule (see \citealt{gr15} for a discussion). The common practice is to truncate each Voigt profile at some fixed spectral width. For example, \cite{sh07} use a line-wing cutoff of $\mbox{min}(25P/1\mbox{ atm}, 100)$ cm$^{-1}$. We use a cutoff of 100 cm$^{-1}$ except for water, where we instead use 25 cm$^{-1}$. We emphasize that the correct functional form of these far line wings is unknown.
To speed up our calculations, we wish to avoid having to deal with integrating over $\sim 10^9$ points in the opacity function to obtain the transmissivities. Instead, we employ the $k$-distribution method to calculate the transmission function within each wavelength bin,
\begin{equation}
\mathcal{T} = \int^1_0 \psi ~dy,
\label{eq:trans_gen}
\end{equation}
where the integrand, which is given by $\psi \equiv \exp{(-\mathcal{D} \Delta \tau)}$, is a function of a new variable ($y$) that is bounded between 0 and 1. We refer the reader to \cite{gr15} for a detailed explanation of the $k$-distribution method and instead focus on our method for numerically evaluating the preceding integral, which we solve by applying the Gauss-Legendre quadrature rule,
\begin{equation}
\int^1_0 \psi ~dy = \frac{1}{2} \sum^{20}_{g=1} w_g ~\psi\left(\frac{1 + y_g}{2} \right),
\end{equation}
where $y_g$ is $g$-th root of the 20-th order Legendre polynomial $P_{20}$. The corresponding Gaussian weight $w_g$ is \citep{abramowitz72}
\begin{equation}
w_g = \frac{2}{[1-y_g^2]P_{20}^\prime[y_g]^2},
\end{equation}
with $P_{20}^\prime$ being the derivative of $P_{20}$. We find that using a 20th order Gaussian quadrature rule is sufficient by comparing our calculations to direct integration using Simpson's rule (not shown).
The obvious advantage of using Gaussian quadrature over direct integration is the enhanced computational efficiency. In \texttt{HELIOS}, we propagate the fluxes through the model atmosphere for each of the 20 Gaussian points and perform the Gaussian quadrature sum at the end of the propagation to obtain the flux associated with a wavelength bin. Since the fluxes follow inhomogeneous paths across pairs of layers (i.e., the temperatures and pressures are not constant along these paths) and we also add the $k$-distribution functions of the various molecules, we have to invoke the correlated-$k$ approximation \textit{twice} \citep{gr15}.
Computing the flux through each Gaussian point is equivalent to expressing the transmission function through layer $i$ and waveband $l$ by
\begin{equation}
\mathcal{T}_{i,l} = \sum^{20}_{g = 1} w_g e^{-\kappa_{i,l,g} \Delta m_{{\rm col},i}} ,
\label{eq:trans_sum}
\end{equation}
which is nothing else than a discrete form of equation (\ref{eq:trans_gen}) applied to our model. The $g$-th $k$-coefficient in waveband $l$ is written as
\begin{equation}
\kappa_{i,l,g} = \sum^6_{j=1} \mathcal{X}_j\left(T_i, P_i \right) ~\kappa_{j,l,g}\left(T_i, P_i \right),
\label{eq:kappa_gauss}
\end{equation}
where $T_i$ and $P_i$ are the temperature and pressure at the center of the $i$-th layer in the isothermal layer grid and also at the interfaces in the non-isothermal layer grid where we have sublayers. In the latter case, we calculate the opacity in the center and at the interface and take their average to obtain the value in the connecting sublayer. The mixing ratios and opacities are generally functions of temperature and pressure. At this point, we have to distinguish between the mixing ratios by volume ($X_j$) versus the mixing ratios by mass ($\mathcal{X}_j$). The chemistry formulae (see Sect. \ref{sec:chem}) are constructed to compute $X_j$. However, to construct $\kappa_i$ we need
\begin{equation}
\label{eq:mixing}
\mathcal{X}_j = \frac{X_j m_j}{\bar{m}},
\end{equation}
where $m_j$ is the mass of the $j$-th molecule.
In equations (\ref{eq:kappa_gauss}) and (\ref{eq:mixing}), the indices $j=1,2,3,4$ refer to the 4 molecules being included in the current study: CO, CO$_2$, H$_2$O and CH$_4$. For these molecules, $X_j$ is computed using the chemistry model. The indices $j=5$ and $j=6$ refer to the CIA opacities associated with H$_2$-H$_2$ and H$_2$-He, respectively. For these, we use $X_5=1$ and $X_6=0.1$ to approximately reflect cosmic abundance. We use $m_5 = 2 m_p$ and $m_6 = 4 m_p$.
By using equation (\ref{eq:kappa_gauss}), we inherently assume the spectral lines of the various molecules to be {\it perfectly correlated}. In general, there are three limits: perfectly correlated, randomly overlapping (perfectly uncorrelated) and disjoint lines (see \citealt{pi10} for a review). Real spectral lines behave in a way that is intermediate between these limits. \cite{lacis91} and, more recently, \cite{amundsen16} have implemented a randomly overlapping method for combining the opacities of the different molecules, which is computationally more expensive as it involves multiple summations. As the spectral resolution increases (and the bin size decreases), these approaches should converge to the same answer. The true accuracy of these approaches remains unquantified in the hot atmosphere regime and needs to be tested by a true line-by-line calculation, where each of the $\gtrsim 10^9$ line shapes is numerically resolved. This is the subject of future work and is beyond the scope of the present paper.
In \texttt{HELIOS}, the $k$-coefficients are read in from a four-dimensional, pre-computed table in temperature ($100 \le T \le 2900$ K, $\Delta T = 200$ K), pressure ($10^{-6} \le P \le 10^3$ bar, $\Delta \log_{10} P$ = 0.5)\footnote{If the layer pressure or temperature exceeds the range of the values in the table, the opacity is simply taken to correspond to the closest pre-tabulated value.} and wavelength ($0.33 \le \lambda \le 10^5$ $\mu$m), with the bins subdivided by 20 Gaussian points. The opacities are used at the constructed wavelength (and Gaussian point) values, but are linearly interpolated across $T$ and $\log{P}$.
Finally, we note that we use 300 wavelength bins (equally spaced in wavenumber) when running \texttt{HELIOS} to solve for radiative equilibrium. Upon obtaining the converged temperature-pressure profile, we then use it to compute synthetic spectra in 3000 wavelength bins as a post-processing step. We find that this approach produces essentially identical results to performing the entire calculation using 3000 wavelength bins (not shown).
\begin{table*}
\caption{Opacity Sources used in this work.}
\label{tab:opac}
\vspace{-0.4cm}
\begin{center}
\bgroup
\def1.5{1.5}
\begin{tabular}{| l | l |}
\hline
Name & Source \\ \hline
H$_2$O & HITEMP database\footnote{\label{f:hitran}hitran.org/hitemp/} \citep{ro10} \\
CO$_2$ & HITEMP database\textsuperscript{\ref{f:hitran}} \\
CO & HITEMP database\textsuperscript{\ref{f:hitran}} \\
CH$_4$ & HITRAN database\footnote{www.cfa.harvard.edu/hitran/} \citep{ro13} \\
CIA & HITRAN CIA database \citep{ri12} \\
Rayleigh scattering & \citealt{sn05} \\
\hline
\end{tabular}
\egroup
\end{center}
\end{table*}
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f3.pdf}
\end{minipage}
\vspace{-0.3cm}
\caption{Opacities, as functions of wavelength, for all of the opacity sources used in the current study, computed using \texttt{HELIOS-K} \citep{gr15}. For illustration, we set $T=1500$ K and $P=1$ bar. Each opacity is weighted by its mass mixing ratio. We include only Rayleigh scattering by molecular hydrogen, but CIA associated with both H$_2$-H$_2$ and H$_2$-He pairs.}
\label{fig:opac}
\end{center}
\end{figure}
\subsection{Chemistry Model}
\label{sec:chem}
Given the elemental abundances of carbon ($n_{\rm C}$) and oxygen ($n_{\rm O}$), we would like to compute the mixing ratios (number densities normalized by that of molecular hydrogen) of the 4 molecules used in our model as functions of temperature and pressure. This requires a chemistry model. To this end, we use the analytical calculations of \cite{he16b}. Specifically, \cite{he16a} laid out the theoretical formalism, which led to the formulae in equations (12), (20) and (21) in \cite{he16b} that we are using. \cite{he16c} demonstrated that these formulae are accurate compared to a Gibbs free energy minimization code, even when nitrogen is added to the system. We explicitly demonstrate the agreement between equations (12), (20) and (21) of \cite{he16b} and the calculations from the \texttt{TEA} code of \cite{bl15} in Figure \ref{fig:tea}. Since we do not study atmospheres with C/O $>1$, we ignore C$_2$H$_2$.
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f4.pdf}
\end{minipage}
\vspace{-0.3cm}
\caption{Validation of our analytical chemistry model (\citealt{he16b}; circles) by calculations using the Gibbs free energy minimization code, \texttt{TEA} (\citealt{bl15}; solid curves). For illustration, we have computed the volume mixing ratios as functions of C/O and examined $P=1$ bar and $T=800$ and 3000 K.}
\label{fig:tea}
\end{center}
\end{figure}
We define ``solar element abundance" to be $n_{\rm C} = 2.5 \times 10^{-4}$ and $n_{\rm O} =5 \times 10^{-4}$, such that ${\rm C/O} \equiv n_{\rm C}/n_{\rm O} = 0.5$. In this study, we keep the value of $n_{\rm O}$ fixed and vary $n_{\rm C}$ when we vary C/O. For example, a model with ${\rm C/O} = 0.1$ has $n_{\rm C} = 5 \times 10^{-5}$ and $n_{\rm O} = 5 \times 10^{-4}$.
Following the convention of the astronomers, we refer to the ``metallicity" as the set of values of the elemental abundances that have atomic numbers larger than that of helium. In our model, these are $n_{\rm C}$ and $n_{\rm O}$. These numbers are simply decreased or increased by a constant factor when the metallicity is varied. For example, a model with $3\times$ solar metallicity has $n_{\rm C} = 7.5 \times 10^{-4}$ and $n_{\rm O} = 1.5 \times 10^{-3}$, but still retains ${\rm C/O} =0.5$.
Figure \ref{fig:abundances} shows examples of our calculations of the molecular mixing ratios as functions of temperature, C/O and metallicity. To develop some intuition for the relative abundances of molecules present in our model atmospheres, we have included shaded columns indicating the dayside-averaged temperatures of 5 of the 7 exoplanets being studied in the current paper.\footnote{Two planets are hotter than 3000 K and not visible in Fig. \ref{fig:abundances}. As we have tabulated Gibbs free energies only up to 3000 K, we assume the chemistry to be that at 3000 K if the temperatures exceed 3000 K.}
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f5a.pdf}
\end{minipage}
\vspace{-0.1cm}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f5b.pdf}
\end{minipage}
\vspace{-0.1cm}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f5c.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Elucidating the temperature dependence of the volume mixing ratios of the molecules used in the current study. For illustration, we set $P=1$ bar and explore C/O$=0.1$ (top panel), C/O$=0.5$ (middle panel) and C/O$=1$ (bottom panel). Within each panel, we explore the effects of varying the metallicity by $1/3\times$ and $3\times$ the solar value.}
\label{fig:abundances}
\end{center}
\end{figure}
\subsection{Stellar Models}
\label{sec:star}
For any atmosphere of the exoplanet irradiated by the host star, one needs a description of the incident stellar flux. The simplest approach is to adopt a Planck function, where the only input is the effective temperature of the stellar photosphere ($T_\star$). The next level of sophistication requires the use of models such as \texttt{MARCS}, \texttt{PHOENIX} or Kurucz (\texttt{ATLAS}) that predict the photospheric emission from a star. Specifically in this work, we use the latter two: \texttt{PHOENIX} \citep{al95,hu13} and Kurucz models \citep{ku79,mu04,mu05}.\footnote{The \texttt{PHOENIX} spectra are downloaded directly from their online library at ftp://phoenix.astro.physik.uni-goettingen.de/HiResFITS/ and interpolated in stellar temperature $T_{\star}$, surface gravity $g_\star$, and metallicity to fit the stellar parameters shown in Table \ref{tab:para}. The Kurucz spectra are interpolated in $T_{\star}$ and $g_\star$.} For completeness, Figure \ref{fig:star} shows the stellar spectra we used to model our sample of 6 hot Jupiters in Section \ref{sec:bench}. The choice of stellar model has two primary effects. First, since the secondary emission spectrum is the ratio of the exoplanet's to the star's flux, features in the stellar spectrum are imprinted onto it. Second, differences in the stellar spectrum cause changes in the way the model atmosphere is being heated, which ultimately affects the temperature-pressure profile and synthetic spectrum. As both the \texttt{PHOENIX} and Kurucz stellar models do not extend across the entire wavelength range included in our calculations (0.33 $\mu$m to 10 cm), we patch them using a Planck function.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6b.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6c.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6d.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6e.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f6f.pdf}
\end{minipage}
\vspace{0.0cm}
\caption{Comparison of the \texttt{PHOENIX} and Kurucz stellar models with the stellar blackbody function for the 6 hot Jupiters examined in the current study. Each stellar model was customized according to the specified stellar effective temperature, surface gravity and metallicity, as stated in Table \ref{tab:para}.}
\label{fig:star}
\end{center}
\end{figure*}
\subsection{Numerical Implementation}
\label{sec:num}
The computationally intensive parts of \texttt{HELIOS} are written in \texttt{CUDA C++} \citep{cuda}, a proprietary language extension of \texttt{C++} for general purpose computations on suitable NVIDIA GPUs. Due to the GPU's main purpose of providing a fast 2D graphical image where each pixel needs to be updated simultaneously, their architecture is designed to maximize the throughput of parallel calculations and memory bandwidth. A radiative transfer problem is naturally amenable to parallelization as the flux propagation through the atmosphere can be computed for each wavelength separately if we assume coherent scattering (i.e., no change in the wavelength of the radiation). We also parallelize the interpolation of the pre-computed $k$-distribution tables to determine the correct layer values. For further speed-up, the code offers the possibility to tabulate the Planck and the transmission functions at the model's wavelength values and a grid in $T$ ($\Delta T = 10$ K) and opacity ($ \Delta \log_{10} \kappa$ = 0.1), respectively. These grid resolutions are found to be sufficient for a converged behaviour of the model (not shown here).
With this implementation the temperature iteration, the procedure needs typically the following time: with a NVIDIA Geforce 750M, the atmospheric temperatures converge within 2 to 15 minutes; with a NVIDIA K20 GPU, this takes between 0.5 to 4 minutes. These times have been found for a typical atmospheric set-up with 101 layers and 300 wavelength bins, including a separate iteration for scattering during each numerical timestepping. Without scattering, the convergence times are usually a factor of 2 smaller. Once the converged temperature-pressure profile is found, the calculation of the emission spectrum with very high spectral resolution (3000 wavelength bins), as a post-processing step, takes less than 30 seconds. In our experience, we have found it to be sufficient to run \texttt{HELIOS} on a personal computer with a NVIDIA GPU (i.e., a GPU cluster is unnecessary).
\section{Results
\label{sec:res}
We first subject \texttt{HELIOS} to a battery of tests. We then use it to address several lingering issues in the literature concerning the radiative transfer of exoplanetary atmospheres. Finally, we present 1D, benchmark calculations for the emission spectra of 6 hot Jupiters (HD 189733b, WASP-8b, WASP-12b, WASP-14b, WASP-33b and WASP-43b) that serve as ``null hypothesis" models.
By default, we use 300 wavelength bins and 101 non-isothermal layers in our calculations to solve for radiative equilibrium. The emission spectra are computed using 3000 bins and isothermal layers as a post-processing step. These bins are evenly distributed in wavenumber and cover a range of $0.1 \le \lambda^{-1} \le 30000$ cm$^{-1}$, which corresponds to 0.33 $\mu$m $\le \lambda \le 10$ cm. The layer pressures at the TOA and BOA are set at 1 $\mu$bar and 1 kbar, respectively. Stellar heating is represented by a Planck function. The diffusivity factor is set to $\mathcal{D}=2$ and the redistribution efficiency factor is set to $f=2/3$. Isotropic scattering ($\omega_0 \ne 0$, $g_0=0$) and equilibrium chemistry with solar abundances are assumed. Unless otherwise stated, our fiducial model adopts these default parameter values.
\subsection{Tests}
To check \texttt{HELIOS} for consistency of the implementation, we focus on the case study of the super Earth GJ 1214b. The parameter values used are listed in Table \ref{tab:para}.
\begin{table*}
\caption{Planetary and stellar parameters used in this study.}
\label{tab:para}
\vspace{-0.4cm}
\begin{center}
\bgroup
\def1.5{1.5}
\begin{tabular}{| l | l | l | l | l | l | l | l |}
\hline
Object & GJ 1214b\footnote{\cite{bouchy05}, \cite{an13}, \cite{ha13}} & HD 189733b\footnote{\cite{so10}, \cite{de13}, \cite{bo15}} & WASP-8b\footnote{\cite{qu10}} & WASP-12b\footnote{\cite{he09}, \cite{ch11}} & WASP-14b\footnote{\cite{jo09}} & WASP-33b\footnote{\cite{co10}, \cite{ko13}, \cite{le15}} & WASP-43b\footnote{\cite{gi12}} \\ \hline
mean molecular mass ${\bar m}$ [$m_{\rm p}$] & \multicolumn{7}{c|}{2.4\footnote{Our choice value for a hydrogen dominated atmosphere.}} \\ \hline
surface gravity $g$ [cm s$^{-2}$] & 768 & 1950 & 5510\footnote{This value has been obtained from Newton's law of gravity assuming a spherical shape of the planet and neglecting rotation.} & 1164 & 10233 & 2884 & 4699 \\
orbital separation $a$ [AU] & 0.01411 & 0.03142 & 0.0801 & 0.02293 & 0.036 & 0.0259 & 0.0152 \\
effective temp. $T_{\rm eff}$\footnote{Assuming day-side heat redistribution using a factor $f = 2/3$.} [K]& 775 (660\footnote{\label{f:x}{This value is used for the model comparison with \cite{mi10}.}}) & 1575 & 1185 & 3241 & 2403 & 3494 & 1845 \\
planet. radius $R_{\rm pl}$ [$R_{\rm Jup}$] & 0.2479 & 1.216 & 1.038 & 1.776 & 1.281 & 1.679 & 1.036 \\
stell. temp. $T_{\star}$ [K] & 3252 (3026\textsuperscript{\ref{f:x}}) & 5050 & 5600 & 6300 & 6475 & 7430 & 4520 \\
stell. radius $R_{\star}$ [$R_{\odot}$]& 0.211 & 0.805 & 0.945 & 1.595 & 1.306 & 1.509 & 0.667 \\
stell. s. grav., $\log$ $g_{\star}$ [cm s$^{-2}$]& 5.04 & 4.53 & 4.5 & 4.16 & 4.07 & 4.3 & 4.645 \\
stell. metallicity [F/H] & 0.13 & 0.0 & 0.2 & 0.2 & 0.0 & 0.0 & 0.0 \\
\hline
\end{tabular}
\egroup
\end{center}
\vspace{-0.1cm}
\end{table*}
\subsubsection{Comparison to GJ 1214b model of Miller-Ricci \& Fortney}
We test \texttt{HELIOS} against the results of \cite{mi10} for the planet GJ 1214b, who used the code originally developed by \cite{mc89} and \cite{ma99} for the atmospheres of Solar System planets. It was later adapted to exoplanetary atmospheres by \cite{fo05}. They utilize a radiative transfer technique based on \cite{toon89}, which is a multi-stream approach with a simplified two-stream solution for the scattering, further explained in \cite{cahoy10}, and add a convection model for unstable atmospheric layers. Furthermore, \cite{mi10} use the opacities associated with H$_2$O, CO$_2$, CO, CH$_4$ and NH$_3$ \citep{fr08}, as well as the CIA opacities associated with H$_2$-H$_2$, H$_2$-He, H$_2$-CH$_4$ and CO$_2$-CO$_2$. Their chemistry model is taken from \cite{lo02,lo06} and they include a treatment of Rayleigh scattering by molecular hydrogen. Still, we choose to compare \texttt{HELIOS} with the results of \cite{mi10}, because the employed radiative transfer technique and also the list of absorbers, together with the Rayleigh scattering, are similar to ours.
As a reference, we take their solar-abundance model that has a dayside-averaged temperature of 660 K (see the red, dashed curve in their Figure 1). To permit any reasonable comparison, we use the same astronomical parameters as \cite{mi10}. For instance, we set the stellar temperature to 3026 K and tune the redistribution parameter $f$ (in this test only) so that the dayside-effective temperature attains 660 K like in their set-up. Furthermore, to mimic their use of a stellar spectrum for GJ 1214 from \cite{hauschildt99} we also employ a PHOENIX stellar spectrum (from the updated online database) for the same stellar parameters, extrapolated by a blackbody fit to cover the whole wavelength range.
In Figure \ref{fig:mrcomp}, left panel, we show the temperature-pressure profiles for GJ 1214b, by \cite{mi10}, and as computed with \texttt{HELIOS}. There is excellent agreement around $P=10^{-2} - 1$ bar---essentially, the calculations produce infrared photospheric temperatures that coincide. At $P > 1$ bar, the \texttt{HELIOS} temperature-pressure profile is about 200 K hotter. We suspect that this discrepancy is due to our simpler treatment of the opacities, as we only consider 4 molecules. This leads to greater transparency particularly in the visible wavelengths of our model atmosphere, which in turn produces more heating in the deep atmosphere. To support this hypothesis, we have successfully reproduced the deep atmospheric structure of \cite{mi10} by artificially introducing an opacity of $6 \times 10^{-4}$ cm$^2$ g$^{-1}$ to the shortwave below 1 $\mu$m (see green dashed curve in Fig. \ref{fig:mrcomp}). Since our model does not have any convective treatment, we cannot reproduce the adiabat in their model at the bottom boundary. However, by introducing an internal heat flux, we can somewhat mimic their deep temperatures (shown for $T_{\rm intern}=60$ K).
In the right panel of Figure \ref{fig:mrcomp} we show the ratio of the planetary and the stellar emission for \cite{mi10}'s model and ours. The spectra are of the same magnitude and show similar trends. Their results show a larger variation in intensity across wavelength, particularly enabling emission from deeper, and thus hotter, atmospheric regions. This could be a consequence of several factors: differences in employed molecular line lists, combination of the opacities, chemistry model or the stellar spectrum. Considering all those components it is not surprising that the individual spectral features do not match perfectly and we conclude that \texttt{HELIOS} is still rather consistent with the results of \cite{mi10}.
For completion, we show both the spectra of our fiducial set-up and the one with an added artificial shortwave opacity. As expected, those are very similar because around the emitting photosphere the models only differ slightly.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f7a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f7b.pdf}
\end{minipage}
\vspace{-0.0cm}
\caption{Comparison with the atmospheric model of GJ 1214b from \cite{mi10}. The left panel shows the day-side temperature-pressure profile at $T_{\rm eff} = 660$ K. The temperatures in the infrared photosphere ($\sim 10^{-2} - 1$ bar) match very well. We also reproduce the deep atmosphere temperatures when an artificial opacity of $6 \times 10^{-4}$ cm$^2$g$^{-1}$ is inserted into the visible wavelengths (dashed curve). We can mimic the convective tail by adding internal heat flux; here shown for $T_{\rm intern} = 60$ K (cyan). The right panel depicts the corresponding planetary emission for three models of the left panel, together with a blackbody emission at the same effective temperature for comparison. The spectrum of \cite{mi10} shows more pronounced features, but overall has the same magnitude. The \texttt{HELIOS} runs are similar as the main temperature difference lies below the emitting photosphere.}
\label{fig:mrcomp}
\end{center}
\end{figure*}
\subsubsection{Trends associated with scattering}
As a further consistency check of \texttt{HELIOS}, we examine calculations with idealized descriptions of scattering and check if the trends match our physical intuition.
For illustration, we set $\omega_0=0.5$ across all wavelengths. We then examine models with $g_0=-0.5, 0$ and 0.5, which are also constant across all wavelengths. We emphasize that the two-stream solutions used in \texttt{HELIOS}, which are taken from \cite{he14}, are generally able to take $\omega_0$ and $g_0$ as input \textit{functions} (rather than just scalars/numbers).
Figure \ref{fig:scat} shows the fiducial pure absorption model compared against the 3 models with idealized descriptions of isotropic, backward and forward scattering. For $g_0=-0.5$ and 0, scattering generally shifts the absorption profile of starlight upwards (towards lower pressures), which cools the model atmosphere. As the scattering shifts from being isotropic to being backward, the deep atmosphere becomes cooler. We also observe a trend of the reflected light at $\lesssim 1$ $\mu$m being the strongest for backward scattering, but of the thermal emission at $\gtrsim 1$ $\mu$m being the strongest for forward scattering, which is expected.
Scattering also has the general effect of muting the spectral features in the synthetic spectra. It effectively raises the level of the infrared continuum. This effect is stronger as the scattering becomes more backward-dominated (Figure \ref{fig:scat}). Such an effect mimicks the presence of aerosols or condensates. Overall, these expected trends provide a ``proof-of-concept" validation of \texttt{HELIOS}.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f8a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f8b.pdf}
\end{minipage}
\vspace{0.0cm}
\caption{Consistency check of \texttt{HELIOS} by examining the temperature-pressure profiles (left panel) and synthetic spectra (right panel) in the idealized limits of scattering: $g_0=0$ (isotropic scattering), $g_0=0.5$ (forward scattering) and $g_0=-0.5$ (backward scattering). For illustration, we set $\omega_0=0.5$. Both the single-scattering albedo ($\omega_0$) and scattering asymmetry factor ($g_0$) are assumed to be constant across wavelength, but we note that the two-stream solutions we implemented allow for them to generally be specified as functions of wavelength, temperature and pressure. The qualitative trends associated with the temperature-pressure profiles and synthetic spectra are consistent with physical expectations (see text).}
\label{fig:scat}
\end{center}
\end{figure*}
\subsubsection{Isothermal versus non-isothermal layers}
\label{sec:reslayer}
An essential ingredient of 1D models of atmospheres in radiative equilibrium is the number of layers used in the computation. We perform a series of convergence tests by considering different numbers of layers and employing isothermal versus non-isothermal layer models. We again use the parameters of GJ 1214b as an illustration.
Figure \ref{fig:restest} shows the temperature-pressure profiles associated with models having 51, 201 and 1001 isothermal layers, and also those with 21, 101, 501 non-isothermal layers. First, we note that the temperature-pressure profiles of the models with non-isothermal layers coincide (with differences of less than 3 K), implying that 21 non-isothermal layers is sufficient to attain convergence. By contrast, even with 1001 layers, no convergence is seen for the models with isothermal layers. These results illustrate the superiority of using non-isothermal layers. We recover the same behavior even when a different case study (e.g., WASP-12b) is considered (not shown).
Next, we compute the synthetic spectrum of the model with 501 non-isothermal layers and use it as a reference. We then consider models with 51 and 101 non-isothermal layers, as well as models with 51, 101, 201 and 501 isothermal layers. For each model, we compute the deviation in the synthetic spectrum, from the reference model, as a function of wavelength. Figure \ref{fig:restest} shows that, as expected, the deviation decreases as the resolution increases. Only the model with 51 isothermal layers produces deviations that exceed 1\% in the flux. The model with 101 isothermal layers produces deviations that are typically less than 1\%. Since models with isothermal layers are faster to compute, this motivates us to adopt a model with 101 isothermal layers for our post-processing step. In other words, we use non-isothermal layers to iterate for radiative equilibrium. Upon attaining radiative equilibrium, we post-process the converged temperature-pressure profile, using a model with 101 isothermal layers, to produce synthetic spectra.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f9a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f9b.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Resolution tests to determine the minimum number of isothermal versus non-isothermal layers needed for numerical convergence. The left panel shows various temperature-pressure profiles computed using 51, 201 and 1001 isothermal layers versus 21, 101 and 501 non-isothermal layers, demonstrating that the use of isothermal layers is not an efficient approach. The right panel shows the deviation or error in the synthetic spectrum, as a function of wavelength, using the model with 501 non-isothermal layers as a reference.}
\label{fig:restest}
\end{center}
\end{figure*}
\subsubsection{Obtaining convergence for the $k$-distribution tables}
Another essential ingredient of 1D models of atmospheres in radiative equilibrium is the spectral resolution used in constructing the opacity function, which is then used to construct the $k$-distribution tables. We wish to investigate the errors associated with using different spectral resolutions. The reference case is taken to be a model with a spectral resolution of 10$^{-5}$ cm$^{-1}$. We examine models with resolutions of $10^{-1}$, $10^{-2}$, $10^{-3}$ and $10^{-4}$ cm$^{-1}$ and compare the errors in the synthetic spectra, after we have iterated for radiative equilibrium, as a function of wavelength, relative to the reference. As we are using 3000 wavelength bins, these sampling resolutions correspond to $10^2$, $10^3$, $10^4$ and $10^5$ points per bin, respectively.
Figure \ref{fig:resopac} shows our results for the case studies of GJ 1214b and WASP-12b, which were illustrated to span the range of temperatures for the currently characterizable exoplanetary atmospheres. We find the expected trend that the error decreases as the resolution increases from $10^{-1}$ cm$^{-1}$ to $10^{-4}$ cm$^{-1}$. Using a spectral resolution of only $10^{-1}$ cm$^{-1}$ ($10^{-2}$ cm$^{-1}$) results in errors that are $> 10\%$ ($\sim 1\%-10\%$) in the near-infrared flux. To reduce the error to $\sim 1$\%, we find a minimum resolution of $10^{-3}$ cm$^{-1}$ to be required in our model. This value might change if one is using opacity sampling. We also show the error in the spectra produced by purely post-processing the temperature profile of the reference case, which demonstrates that the errors are not merely associated with iterating for radiative equilibrium.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f10a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f10b.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Elucidating the errors, in the synthetic spectra, associated with using different spectral resolutions to construct the $k$-distribution tables. The reference case uses a spectral resolution of $10^{-5}$ cm$^{-1}$. The label ``ppb" refers to the number of points per bin. All of the synthetic spectra were computed for model atmospheres in chemical and radiative equilibrium using the correlated-$k$ approximation. We either run the whole radiative transfer iterative process (solid) or solely post-process the $T$-$P$ profile of the reference case (dotted). For illustration, we examine models of cool (GJ 1214b; left panel) and hot (WASP-12b; right panel) exoplanetary atmospheres.}
\label{fig:resopac}
\end{center}
\end{figure*}
\subsubsection{Using the correct value of the diffusivity factor}
\label{subsect:diffuse}
As discussed previously, one may obtain an exact solution of the radiative transfer equation, without invoking the two-stream approximation, only in the limit of pure absorption. This solution may be compared to two-stream calculations with different assumed values of the diffusivity factor.
\cite{amundsen14} have recently advocated for the use of ${\cal D}=1.66$ from comparing their two-stream calculations to a different set of calculations computed using the discrete-ordinates radiative transfer method. \cite{armstrong69} also advocate for ${\cal D}=1.66$ based on radiative transfer calculations of water in the atmosphere of Earth. However, the correct value for ${\cal D}$ should depend on the vertical resolution of the model (c.f. Fig. \ref{fig:diff}), which motivates us to perform our own comparisons.
Figure \ref{fig:difffit} displays the computed temperature-pressure profiles and the error in the resulting synthetic spectrum for GJ 1214b for ${\cal D}=1.66, 1.8, 1.9$ and 2 compared to the exact solution. Regarding temperature, the ${\cal D}=1.9$ and 2 models produce the best match to the exact solution. However, ${\cal D}=2$ leads, on average, to the smallest error in the spectrum. We also consider the same set of calculations for a hotter exoplanet, WASP-12b. In this case, ${\cal D}=2$ clearly produces the best match to the exact solution in terms of the temperature as well as the spectrum. In general, the error in the spectrum is smaller than for the cooler planet. It is not unsurprising, that ${\cal D}=2$ provides the most accurate results, because we expect the diffusivity factor to approach a value of 2 when the vertical resolution of the model is sufficient (see Figure \ref{fig:diff}), i.e. the difference in optical depth between the layers is small, at least in the photospheric regions of the atmosphere.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f11a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f11b.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f11c.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f11d.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Determination of the diffusivity factor ($\mathcal{D}$) by comparing two-stream and exact solution in the limit of pure absorption. For illustration, we study the warm super Earth GJ 1214b (top panels) and the hot Jupiter WASP-12b (bottom panels). We show the temperature profiles on the left and the error in the resulting synthetic spectrum compared to the exact solution on the right. A diffusivity factor of $\mathcal{D}=2$ appears to produce the best match to the exact solutions, following closely the temperature-pressure profile of the exact solution and leading on average to the smallest error in the spectrum.}
\label{fig:difffit}
\end{center}
\end{figure*}
\subsection{Testing the Null Hypothesis and Variations on a Theme: Benchmark 1D Models for Hot Jupiters}
\label{sec:bench}
Despite heroic efforts to obtain data for exoplanetary atmospheres, exoplanets are spatially unresolved point sources---and will probably remain so for the foreseeable future---although phase curves and eclipse maps provide some spatial information. As a first approach, theorists have resorted to interpreting the spectra of exoplanetary atmospheres using simple models: 1D, plane-parallel, just as we have constructed. There is a precedent of using 1D models to interpret spectra (see Introduction for references). As a null hypothesis, we make the following assumptions: chemical equilibrium, radiative equilibrium (which we solve for using \texttt{HELIOS}) and solar abundances. This would be the second simplest model after a blackbody emission spectrum \citep{hansen14}. Upon constructing the null hypothesis, we then examine variations in the metallicity and C/O.
\begin{table}
\caption{Spectral data sources}
\label{tab:data}
\vspace{-0.4cm}
\begin{center}
\bgroup
\def1.5{1.5}
\begin{tabular}{| l | l |}
\hline
Planet & Source \\ \hline
HD 189733b & \cite{cr14}, \cite{to14} \\
WASP-8b & \cite{cu13} \\
WASP-12b & \cite{st14} \\
WASP-14b & \cite{bl13} \\
WASP-33b & \cite{de12}, \cite{ha15} \\
WASP-43b & \cite{bl14}, \cite{kr14} \\
\hline
\end{tabular}
\egroup
\end{center}
\end{table}
We have chosen the sample of hot Jupiters to include in this analysis based on a literature search for planets with non-blackbody emission spectra. We have started from \cite{hansen14}, which catalogs all planets with secondary eclipse measurements in at least two bandpasses as of 2014. They found 7 planets that are poorly fit by a blackbody model. We also searched for any more recent non-blackbody results. To select the most precise, reliable measurements from our search, we consider space-based data only. We have also stipulated that the data were reduced with state-of-the-art techniques. Specifically, we only consider Spitzer results that used sophisticated models of the intrapixel sensitivity such as BLISS mapping or pixel-level decorrelation \citep{stevenson12, deming15}. This approach has been demonstrated to be the best practice in Spitzer data analysis \citep{ingalls16}. This search has resulted in the selection of six planets: HD 189733b, WASP-8b, WASP-12b, WASP-14b, WASP-33b, and WASP-43b. Their model parameter values and spectral data sources are shown in Tables \ref{tab:para} and \ref{tab:data}, respectively.
Figure \ref{fig:bench_star} shows the null-hypothesis models for all 6 studies. We have computed synthetic spectra and temperature-pressure profiles using a stellar blackbody, a Kurucz stellar model and a \texttt{PHOENIX} stellar model. All of the stellar models were customized for each case study by specifying, as input parameters, the stellar effective temperature, surface gravity and metallicity. The synthetic spectra in all three cases are qualitatively similar. The largest difference occurs between 3 and 10 $\mu$m. These differences appear to be more pronounced for the hottest hot Jupiters (i.e., WASP-12b and WASP-33b). Interestingly, the choice of stellar model affects the strength of the water-band features between 1.5 and 2.5 $\mu$m, which are partially probed by the WFC3 instrument on the Hubble Space Telescope. This discrepancy between the models is somewhat apparent for HD 189733b and WASP-43b. The shapes of the temperature-pressure profiles, in all 6 cases, are very similar with the largest discrepancies in either the very high optically thin or deep optically thick layers, which are less important for the planetary emission.
Overall, HD 189733b appears to be consistent with a null hypothesis and its dayside emission spectrum is reasonably described by a 1D, plane-parallel model in chemical and radiative equilibrium with solar metallicity. WASP-43b is fairly well described by the null hypothesis. However, our models for WASP-8b, WASP-12b, WASP-14b and WASP-33b consistently under-predict the infrared fluxes. These discrepancies could be either due to an insufficient opacity implementation (lacking partial molecular absorption, aerosol extinction or inaccurate line profiles) or due to a limited methodological framework, lacking chemical disequilibrium (which requires a self-consistent calculation coupled to a chemical kinetics solver), radiative disequilibrium (which requires another self-consistent calculation coupled to atmospheric dynamics) or non-1D effects (which a 1D model prescription with $f$ cannot characterize and which would ideally require coupling to a 3D spatially resolved general circulation model). We will defer this investigation to future work.
For further variations on the theme, we retain the \texttt{PHOENIX} stellar models as they offer higher spectral resolution and more updated atomic/molecular line lists than the Kurucz stellar models. In Figure \ref{fig:bench_metal}, we repeat our calculations with 1/3$\times$, 1$\times$ and 3$\times$ solar metallicity. We find the expected trend that a higher metallicity leads to generally hotter model atmospheres, which has the effect of strengthening the near-infrared water-band features. However, compared to the null hypothesis, decreasing or increasing the metallicity by a factor of 3 appears to have a minimal effect on the synthetic spectra, which is consistent with the retrieval analysis conducted for WASP-43b in \cite{kr14}, where they obtain similar metallicity uncertainties based on data constraints. Our conclusions are qualitatively identical to those visible in Figure \ref{fig:bench_star}.
Varying the C/O has a more marked effect, as we show in Figure \ref{fig:bench_coratio}. Specifically, we examine water-rich (C/O$=0.1$), solar-abundance (C/O$=0.5$) and C/O$=1$ scenarios. Generally, we find that the C/O$=1$ models have consistently colder temperature-pressure profiles, due to the lower abundance of H$_2$O as the oxygen atom is preferentially sequestered by CO, at high temperatures, compared to the water-rich and solar-abundance models. The increasing abundance of CO also leads to stronger absorption features at 2.3, 4.5 and 4.8 $\mu$m, which render the model atmospheres darker (i.e., they have less flux in these bands). This transition to the stronger CO features is more pronounced in the hotter objects (WASP-12b and WASP-33b). Our qualitative conclusions appear to be unchanged: our models for WASP-8b, WASP-12b, WASP-14b and WASP-33b still under-predict the infrared fluxes. It is somewhat difficult to judge if the data favours the water-rich or C/O$=1$ models, for HD 189733b and WASP-43b, without running a detailed atmospheric retrieval model, which we again defer to future work.
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12b.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12c.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12d.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12e.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f12f.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Null-hypothesis models for the 6 hot Jupiters in our current study: 1D, plane-parallel model atmospheres in chemical and radiative equilibrium, with solar metallicity/abundances. The predicted dayside emission spectra were compared to published data (see text for details). For each case study, we computed three models using the \texttt{PHOENIX} and Kurucz stellar models as well as a stellar blackbody. For each assumption of the stellar irradiation flux, we iterated the model atmosphere to attain radiative equilibrium (see text for details).}
\label{fig:bench_star}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13b.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13c.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13d.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13e.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f13f.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Same as Figure \ref{fig:bench_star}, but using only the \texttt{PHOENIX} stellar model and examining the effects of varying the metallicity of the model atmospheres.}
\label{fig:bench_metal}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14a.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14b.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14c.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14d.pdf}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14e.pdf}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{f14f.pdf}
\end{minipage}
\vspace{-0.1cm}
\caption{Same as Figure \ref{fig:bench_star}, but using only the \texttt{PHOENIX} stellar model and examining the effects of varying the C/O (0.1, 0.5 and 1) of the model atmospheres.}
\label{fig:bench_coratio}
\end{center}
\end{figure*}
\section{Summary, Discussion \& Conclusions
\label{sec:sum}
\subsection{Summary}
We have presented the new, extensible code, \texttt{HELIOS}, which solves the equation of radiative transfer for a 1D, plane-parallel atmosphere that allows for non-isotropic scattering via the specification of the functional forms of the single-scattering albedo and the scattering asymmetry factor. It uses a staggered spatial grid with the options of specifying isothermal or non-isothermal layers. We have used \texttt{HELIOS-K} \citep{gr15} to compute the opacities of the four molecules, which are active in the infrared, and combined those by weighing them with the validated analytical formulae of \cite{he16b} and \cite{he16c} for equilibrium chemistry. In order to combine the various gaseous absorbers we have employed a correlated-$k$ approximation, which assumes perfect correlation between the molecular bands. The boundary conditions are the stellar irradiation flux at the top of the model atmosphere and the internal heat flux at the bottom. \texttt{HELIOS} further allows for the stellar irradiation flux to be specified as a simple Planck function or from a stellar model (e.g. Kurucz, \texttt{PHOENIX}). We have constructed and optimized \texttt{HELIOS} to run on GPUs, which allows for fast computation on a single machine. We have exploited this efficiency to explore the parameter space of stellar type, metallicity and C/O ratio.
\subsection{Comparison to previous work}
Several groups have made contributions to a rich body of literature on self-consistent radiative transfer models in exoplanetary atmospheres. The work of \cite{bu06,bu07,bu08} uses the accelerated lambda iteration method, originally developed for stellar atmospheres \citep{hubeny95}. The work of \cite{fo05,fo06,fo08,fo10} and \cite{mo13,mo15} use an atmosphere modeling code and radiative transfer methods with a heritage from brown dwarf and Solar System models \citep{mc89,toon89,burrows97,ma99}. \cite{amundsen14} recently implemented a radiative transfer code using the two-stream approximation in the limit of pure absorption. \cite{molli15} constructed a pure-absorption code using the ``variable Eddington factor" method, which has a heritage from the study of stellar atmospheres (e.g. \citealt{auer70}) and protoplanetary disks \citep{dullemond02}. Our approach and assembly of the various components (see above) and their collective implementation is a novel endeavor and we hope it will contribute to the advancement of this field.
\subsection{Discussion and Opportunities for Future Work}
In the current work, we have considered a small set of the four main infrared absorbers (H$_2$O, CO$_2$, CO, CH$_4$), and included the opacity associated with CIA from H$_2$-H$_2$ and H$_2$-He pairs. Future work should include more opacity sources, especially that associated with C$_2$H$_2$ and HCN, if one is interested in C/O$>1$ models, and Na and K as these are major absorbers in the visible for very hot planets. Also, important at the higher-temperature end of exoplanets is continuum absorption by electrons moving freely in the field or being decoupled from the shell of neutral atoms (e.g. H, He), molecules (e.g. H$_2$) or ions (e.g. H$^-$) \citep{sh07}. Furthermore, it is important to conduct a study examining the accuracy of the employed correlated-$k$ approximation for different combinations of molecular absorbers, since this could be a potential source of error---a study similar to \cite{amundsen16} for the random-overlap scheme. Another opportunity for future work is the inclusion of aerosols and clouds, whose proper implementation remains a subject of debate. Additionally, we will implement convective adjustment as the next step in sophistication and we plan to investigate the effect of disequilibrium chemistry (induced by both atmospheric motion and photochemistry) and radiative disequilibrium by coupling \texttt{HELIOS} to a chemical kinetics code and a general circulation model. Hot Jupiters are complex, three-dimensional entities (e.g., \citealt{burrows10}) and interpreting them, on a detailed case-by-case basis, requires a three-dimensional model (e.g., \citealt{kataria15}). The exact interpretation of the molecular abundances associated with the 6 hot Jupiters may be performed using an atmospheric retrieval code. \texttt{HELIOS} is a key component of the open-source Exoclimes Simulation Platform (ESP; \url{exoclime.net}), which includes a chemical kinetics code \citep{tsai16}, retrieval code \citep{lavie16} and general circulation models (\citealt{mendonca16}; Grosheintz et al., in preparation). The up-to-date version of \texttt{HELIOS} may be downloaded from its main repository \url{github.com/exoclime/HELIOS} and the version used to produce the results in this work is archived under the DOI: 10.5281/zenodo.164176.
\section*{Acknowledgements}
M.M., L.G., S.G., J.M., B.L., D.K., S.T. and K.H. thank the Swiss National Science Foundation (SNF), the Center for Space and Habitability (CSH), the PlanetS National Center of Competence in Research (NCCR) and the MERAC Foundation for partial financial support.
\software{\\
\texttt{HELIOS-K} (\citealt{gr15}; \url{github.com/exoclime/HELIOS-K}), \\
\texttt{CUDA} \citep{cuda}, \\
\texttt{PyCUDA} \citep{pycuda}, \\
\texttt{python} \citep{python}, \\
\texttt{scipy} \citep{scipy}, \\
\texttt{numpy} \citep{numpy}, \\
\texttt{matplotlib} \citep{matplotlib}.
}
|
2,877,628,090,241 | arxiv | \section{Introduction}
It goes without saying that symmetry is a central theme in theoretical physics as it provides strong constraints on the dynamics of physical systems. A textbook example is Noether's theorem in classical physics, which states that every continuous symmetry is associated with a conservation law.
At the quantum level, 't Hooft anomalies of internal or spacetime symmetries give non-trivial constraints on the infrared (IR) fixed points of renormalization group (RG) flows, especially when the IR theories are strongly-coupled.
Recently, the understanding of symmetry has significantly evolved. In the modern point of view, the presence of symmetries in a quantum field theory (QFT) is equivalent to the existence of topological defects \cite{Gaiotto:2014kfa}.
More precisely, given a global symmetry $G$ in a $D$-dimensional QFT, the action of a group element $g\in G$ on a local operator $\cal O$ is implemented by shrinking a topological defect $U_g$ inserted on a $(D-1)$-sphere with the operator $\mathcal O$ placed at a point lying inside the sphere, as shown in the figure below\footnote{We consider zero-form symmetries throughout this paper.}
\vspace{-.8cm}
\ie\label{eqn:TDaction}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,-0.75) circle (1.325) ;
\draw (0,-2.3) node {$U_g$};
\draw (0,1.2) node {};
\draw (-0.5,-0.75)\dotsol {right}{$\mathcal O$};
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw (0,0)\dotsol {right}{$ \widehat{U}_{g}(\mathcal O)$};
\draw (0,.45) node {};
\end{tikzpicture}
\end{gathered}
\vspace{-.5cm}
\fe
\vspace{-.8cm}
\noindent The defect $U_g$ is called a topological defect because physical observables are invariant under continuous deformations of the manifold where $U_g$ is inserted on. The group multiplication of two elements $g_1$, $g_2\in G$ is realized by fusing the corresponding topological defects $U_{g_1}$ and $U_{g_2}$.
With such a conceptual breakthrough, the notion of symmetry has been extended in various directions including higher-form symmetry \cite{Gaiotto:2014kfa}, higher-group symmetry \cite{Gukov:2013zka,Kapustin:2013qsa,Kapustin:2013uxa,Cordova:2018cvg,Benini:2018reh},
and category symmetry (non-invertible symmetry) \cite{Bhardwaj:2017xup,Chang:2018iay}, where the last one is the main subject of this paper. A category symmetry generalizes the group structure of an ordinary symmetry
by including topological defects that do not have an inverse under the fusion.
The prototype example is the symmetry generated by the topological defect lines (TDLs) in the two-dimensional Ising conformal field theory (CFT), where the TDL implementing the Kramers-Wannier duality is known to be non-invertible \cite{Chang:2018iay,Frohlich:2004ef,Frohlich:2006ch,Frohlich:2009gb}.
The generalization from ordinary to category symmetry is expected to provide us with
new tools to understand and constrain various physical systems.
For instance, the generalization of 't Hooft anomaly matching condition for category symmetry had been used to constrain the RG flows between CFTs or gapped phases \cite{Chang:2018iay,Thorngren:2019iar,Komargodski:2020mxz,Thorngren:2021yso,Kikuchi:2021qxz, Kikuchi:2022jbl,Kikuchi:2022gfi}. Non-invertible defects also play important roles in the study of quantum gravity \cite{Rudelius:2020orz,Heidenreich:2021xpr,McNamara:2021cuo,Cordova:2022rer,Arias-Tamargo:2022nlf,Benini:2022hzx}.
In two dimensions, category symmetries (topological defect lines) are ubiquitous, and there is a comprehensive study of them
\cite{Bhardwaj:2017xup,Chang:2018iay,Frohlich:2004ef,Frohlich:2006ch,
Frohlich:2009gb,Thorngren:2019iar,Thorngren:2021yso,Verlinde:1988sn,Petkova:2000ip,Fuchs:2002cm,Fuchs:2003id,Fuchs:2004dz,Fuchs:2004xi,Quella:2006de,Fuchs:2007vk,Fuchs:2007tx,Bachas:2007td,Kong:2009inh,Petkova:2009pe,
Kitaev:2011dxc,Carqueville:2012dk,
Brunner:2013xna,Davydov:2013lma,Kong:2013gca,Petkova:2013yoa,
Bischoff:2014xea,Aasen:2016dop,
Makabe:2017ygy,
Thorngren:2018bhj,
Ji:2019ugf,Lin:2019hks,
Gaiotto:2020iye,Aasen:2020jwb,Chang:2020imq,
Huang:2021zvu,Burbano:2021loy
}.
Viewing symmetries as TDLs allows one to discuss junctions and endpoints of the TDLs, which are described in the mathematical language of the fusion category.
In certain cases, this 2d picture can be lifted to the framework of 3d TQFTs. More precisely, in the boundary-bulk correspondence between 2d RCFTs (rational CFTs) and 3d TQFTs \cite{Moore:1989yh,Elitzur:1989nr}, the Verlinde lines (TDLs preserving the maximal chiral algebra) can be naturally interpreted as anyons in three dimensions.
Recent developments further extend the study of category symmetry to higher dimensional QFTs \cite{Kaidi:2021gbs,Koide:2021zxj,Choi:2021kmx,Kaidi:2021xfk,Roumpedakis:2022aik,Choi:2022zal,Bhardwaj:2022yxj,Bashmakov:2022jtl,Kaidi:2022uux,Choi:2022jqy,Cordova:2022ruw}.
In this work, we extend the study of category symmetry in a different direction to two-dimensional fermionic CFTs. In general, fermionic CFTs have a richer structure than bosonic CFTs due to the existence of fermionic local operators, which are Grassmann-valued operators with half-integer spins. On Riemann surfaces, the fermionic local operators obey anti-periodic (Neveu-Schwarz) or periodic (Ramond) boundary conditions along non-contractible cycles. These structures carry over to the defect operators living at the junctions or endpoints of the TDLs, except that fermionic defect operators could in general have non-half-integer spins.
A universal example in all fermionic CFTs is the TDL associated with the fermion parity $\bZ_2$ symmetry, which assigns $+1$ (or $-1$) charge to bosonic (or fermionic) local operators. The defect operators living at the endpoints of it are nothing but the Ramond sector operators.\footnote{In the example of fermionic minimal models\cite{Runkel:2020zgg,Hsieh:2020uwb}, one can easily check that the spin and statistics of the Ramond sector operators are uncorrelated.}
The existence of fermionic defect operators leads to several modifications and additions to the properties of TDLs.
In particular, they can live at trivalent junctions and be involved in the
F-moves (the crossings) of the TDLs. Due to their fermion statistics, one needs to introduce an ordering to the junctions to keep track of the ordering of the fermionic defect operators in the correlation functions.
Exchanging the order would give extra signs, schematically as
\ie
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,-<-=.56] (-1,0) -- (-.8,0) -- (-.5,0) -- (0,0);
\draw (0,0)\dotsol {right}{$\psi'$};
\draw (-1,0)\dotsol {left}{$\psi$};
\draw (-1,0) node [below]{\tiny\textcolor{red}{1}};
\draw (0,0) node [below]{\tiny\textcolor{red}{2}};
\draw [line,->-=1] (-1,0) -- (-1.5,.87);
\draw [line,->-=1] (-1,0) -- (-1.5,-.87);
\draw [line,->-=1] (0,0) -- (.5,-.87);
\draw [line,->-=1] (0,0) -- (.1,.175) -- (.5,.87);
\end{tikzpicture}
\end{gathered}=\la\cdots \psi \psi'\cdots\ra=-\la\cdots \psi'\psi\cdots\ra=-\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,-<-=.56] (-1,0) -- (-.8,0) -- (-.5,0) -- (0,0);
\draw (0,0)\dotsol {right}{$\psi'$};
\draw (-1,0)\dotsol {left}{$\psi$};
\draw (-1,0) node [below]{\tiny\textcolor{red}{2}};
\draw (0,0) node [below]{\tiny\textcolor{red}{1}};
\draw [line,->-=1] (-1,0) -- (-1.5,.87);
\draw [line,->-=1] (-1,0) -- (-1.5,-.87);
\draw [line,->-=1] (0,0) -- (.5,-.87);
\draw [line,->-=1] (0,0) -- (.1,.175) -- (.5,.87);
\end{tikzpicture}
\end{gathered}\,.
\fe
As a result, the F-moves now satisfy a different consistency condition, called the super pentagon identity, instead of the pentagon identity for the TDLs in bosonic CFTs.
Besides the super pentagon identity, the F-moves are also constrained by a new projection condition if the internal lines of the H-junctions belong to a new type of TDLs, called q-type.\footnote{The q-type object in super fusion category was introduced in \cite{Gu:2010na,Gaiotto:2015zta} and further studied in \cite{Aasen:2017ubm} in the study of 2+1d fermionic topological phases.} Q-type TDLs can host a one-dimensional Majorana fermion on their worldlines, which can be pair-created or pair-annihilated as
\ie
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) -- (0,1);
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{1}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,.
\fe
The aforementioned projection condition follows from the pair-creation on the q-type internal TDLs. The existence of q-type TDLs also leads to several other consequences including a modified Cardy condition for the invertible TDLs, which are used in finding TDLs in many fermionic CFTs \cite{Kikuchi:2022jbl,Kikuchi:2022gfi}.
The mathematical structure of the fermionic defect operators and the q-type TDLs are captured in the language of the super fusion category \cite{brundan2017monoidal,usher2018fermionic}. It is also instructive to relate TDLs in 2d fermionic RCFTs to anyons in 3d spin TQFTs, which can be obtained by gauging a mixture of a $\mathbb Z_2$ one-form symmetry and the spin structure in certain bosonic TQFTs, or in the language of condensed matter, a fermionic anyon condensation \cite{Gu:2010na}. In this picture, the q-type objects are fixed points under the action of the $\mathbb Z_2$ one-form symmetry.
This paper is organized as follows. Section \ref{sec:review2d} briefly reviews the defining properties of TDLs in 2d bosonic CFTs. Section \ref{sec:defining_properties} introduces the defining properties of TDLs in 2d fermionic CFTs including the fermionic defect operator and the q-type TDL. Section \ref{sec:corollaries} derives several consequences from the defining properties including the F-moves, the super pentagon identity, the universal sector of the F-moves, and the modified Cardy condition for q-type invertible TDLs. Section \ref{sec:super_fusion_category} relates the TDLs in 2d fermionic CFTs to the super fusion categories. Section \ref{sec:super_fusion_rank2} solves the super pentagon identity for the rank-2 super fusion categories up to multiplicity 2, and conjectures a classification of the rank-2 super fusion categories.
Section \ref{sec:spin_selection_rules} derives the spin selection rules for the spectrum of defect operators in all the non-trivial rank-2 super fusion categories. Section \ref{sec:summary} ends with a summary and comments on the constraints of the fermionic RG flows from the TDLs. Appendix \ref{sec:projection_condition} gives a solution to the projection condition. Appendix \ref{sec:universal_oriented_TDL} presents the solution to the universal sector of oriented q-type TDLs. Appendix \ref{sec:fusion_coef} gives a detailed derivation of the formula of the fusion coefficients in Section \ref{sec:fusion_coef_main}. Appendix \ref{sec:F-symbols_C2q} gives the detailed data of the TDLs in the super fusion category ${\mathcal C}^2_{\rm q}$. Appendix \ref{sec:SFC_m=2f} gives the F-moves of the super fusion category that shows up in the $m=4$ fermionic minimal CFT, which is of rank-$4$ and can be regarded as the ``tensor product" of two rank-2 super fusion categories discussed in Section \ref{sec:super_fusion_rank2}.
\subsection{Review of TDLs in 2d bosonic CFTs}
\label{sec:review2d}
In a two-dimensional CFT, a global symmetry $G$ can be implemented by topological defect lines (TDLs) \cite{Bhardwaj:2017xup,Chang:2018iay,Frohlich:2004ef,Frohlich:2006ch,Petkova:2000ip,Fuchs:2002cm,Fuchs:2003id,Fuchs:2004dz,Fuchs:2004xi,Quella:2006de,Fuchs:2007vk,Fuchs:2007tx,Bachas:2007td,Kong:2009inh,Petkova:2009pe,Kitaev:2011dxc,Carqueville:2012dk,Brunner:2013xna,Davydov:2013lma,Kong:2013gca,Petkova:2013yoa,Bischoff:2014xea}, supporting on oriented paths. When a charged operator $\mathcal O$ passes through a TDL $\cL_g$ associated with a group element $g \in G$, it gets acted by a $g$-transformation, denoted by $\widehat{\cL}_g (\mathcal O)$. Graphically, we have
\ie\label{eqn:TDLaction}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) node [below=3pt]{} -- (0,1) node [above=-3pt] {$\cL_g$};
\draw (1,0)\dotsol {below}{$\mathcal O$};
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0.3,-1) node [below=3pt]{} -- (0.3,1) node [above=-3pt] {$\cL_g$};
\draw (-0.7,0)\dotsol {below}{$\widehat\cL_g(\mathcal O)$};
\end{tikzpicture}
\end{gathered}
\fe
The group multiplication of $G$ is realized by the fusion of the TDLs,
\ie
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) node [below=3pt]{} -- (0,1) node [above=-3pt] {$\cL_g$};
\draw [line,->-=.55] (1,-1) -- (1,1) node [above=-3pt] {$\cL_{g'}$};
\end{tikzpicture}
\end{gathered}
\quad
=
\quad\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) node [below=3pt]{} -- (0,1) node [above=-3pt] {$\cL_{gg'}$};
\end{tikzpicture}
\end{gathered}
\fe
where we simply bring the two TDLs on top of each other. The fusion satisfies all the group theory axioms: it is associative, the identity element is realized by the trivial TDL $I$, and finally the TDL $\cL_{g^{-1}}$ associated with the inverse of $g$ is the orientation reversal $\overline{\cL}_g$ of the TDL $\cL_g$.
In the language of TDLs, it is natural to generalise the group structure of $G$ to a (semi)ring structure, which comprises a direct sum $+$ in addition to the fusion product. A general fusion takes the form
\ie
\cL_a \cL_b =\sum_c N^c_{ab} \cL_c\,,
\fe
where the RHS is a sum of simple TDLs $\cL_c$, which cannot be decomposed further, and the fusion coefficients $N^c_{ab}$ are nonnegative integers. In general, a TDL $\cL_a$ could be non-invertible under the fusion
with, for example, the fusion rule
\ie\label{eqn:non-invertible_fision}
\cL_a \overline{\cL}_a= I+ \cL_b+\cdots\,.
\fe
When a charged operator $\cO$ passes through the non-invertible TDL $\cL_a$, by applying the fusion rule \eqref{eqn:non-invertible_fision} piece-wisely, one finds
\ie\label{eqn:TDLaction_new}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) node [below=3pt]{} -- (0,1) node [above=-3pt] {$\cL_a$};
\draw (0.5,0)\dotsol {right}{$\mathcal O$};
\end{tikzpicture}
\end{gathered}
\quad
\propto
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw (1.75,0)\dotsol {below}{$\widehat{\cL}_a(\mathcal O)$};
\draw [line,->-=.55] (2.5,-1) node [below=3pt]{} -- (2.5,1) node [above=-3pt] {$\cL_a$};
\end{tikzpicture}
\end{gathered}
\quad
+
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw (1.2,0)\dotsol {below}{$\widehat{\cL}^{v}_a(\mathcal O)$};
\draw [line,->-=0.65] (1.2,0) -- (2.5,0) node[right]{$v$} ;
\draw (1.875,.25) node {$\cL_b$};
\draw [line,->-=.85,->-=.35] (2.5,-1) node [below=3pt]{} -- (2.5,1) node [above=-3pt] {$\cL_a$};
\end{tikzpicture}
\end{gathered}
\quad
+\quad\cdots\,.
\fe
This makes us inevitably discuss junctions and endpoints of TDLs, whose properties will be reviewed in the latter part of this section. Here, let us briefly explain the various ingredients of \eqref{eqn:TDLaction_new}. The trivalent junction of the outgoing TDLs $\cL_a$, $\overline{\cL}_a$, $\overline{\cL}_b$ and the endpoint (or one-way junction) of the TDL $\cL_b$ host defect Hilbert spaces $\cH_{\cL_a,\overline{\cL}_a, \overline{\cL}_b}$ and $\cH_{\cL_b}$, respectively. $v$ is a topological (zero conformal weight scalar) defect operator in the defect Hilbert space $\cH_{\cL_a,\overline{\cL}_a, \overline{\cL}_b}$. The TDL $\cL_a$ induces a linear map $\widehat{\cL}^{v}_a$ from the Hilbert space $\cH$ of local operators to the defect Hilbert space $\cH_{\cL_b}$, i.e. $\widehat{\cL}^{v}_a(\cO)\in\cH_{\cL_b}$.
The sequence of maps $(\widehat \cL_a,\, \widehat \cL^v_a,\,\cdots)$ characterize the category symmetry action of the non-invertible TDL $\cL_a$ on local operators.
To be self-contained, we give an overview of the defining properties of TDLs in \cite{Chang:2018iay} that would be relevant for our discussions in the following sections.
\begin{itemize}
\item {\bf Isotopy invariance:} The deformations of TDLs by the ambient isotopies of the graph embedding on the flat spacetime do not change physical observables such as correlation functions involving these TDLs.\footnote{Note that on curved spacetime, the deformations of TDLs might lead to phases, called isotopy anomalies \cite{Chang:2018iay}.} It leads to the fact that TDLs commute with the stress-tensor.
\item {\bf Defect operators/states:}
TDLs can join at a point-like junction, which is equipped with a defect Hilbert space, e.g. $\mathcal{H}_{\cL_1,\cL_2,\cdots,\cL_k}$ for the $k$-way junction of the TDLs $\cL_1, \cL_2,\cdots, \cL_k$.\footnote{The cyclic permutation of $\cL_1, \cL_2,\cdots, \cL_k$ defines an isomorphic between the defect Hilbert space $\mathcal{H}_{\cL_1,\cL_2,\cdots,\cL_k}$.} The defect Hilbert space inherits the addition from the associated TDLs, for example $\cH_{\mathcal{L}_1+\mathcal{L}_2}=\cH_{\mathcal {L}_1}\oplus \cH_{\mathcal {L}_2}$.
Since TDLs commute with the stress-tensor, the defect Hilbert space forms a representation of the left and right Virasoro algebras. By the state/operator map, the states in the defect Hilbert space correspond to point-like defect operators living at the junction of the TDLs. The junction vector space $V_{\cL_1,\cL_2,\cdots,\cL_k}$ is defined to be the conformal weights $(h,\tilde h)=(0,0)$ subspace of $\mathcal{H}_{\cL_1,\cL_2,\cdots,\cL_k}$. In particular, the junction vector $m\in V_{\cL_1,\cL_2}$ at the two-way junction of the TDLs $\cL_1$ and $\cL_2$ gives a linear map\footnote{In terms of fusion category, these operators define morphisms in $\text{Hom}(\cL_1, \cL_2)$.}
\ie
m: H_{\cL_1} \rightarrow H_{\cL_2}\,.
\fe
Note that each junction comes with an ordering of the TDLs attached to the junction. Following \cite{Chang:2018iay}, we mark the last TDL entering each junction by a cross $\times$.
\item {\bf Locality:} A TDL configuration on a Riemann surface is invariant under the cutting and gluing. One can cut the Riemann surface along a circle that intersects the TDLs transversely, and put on the circle a complete orthonormal basis of states in the Hilbert space associated with that circle.
The states on the circle can be constructed by taking a disc with (defect) operators inserted, which could be further glued back to the Riemann surface with a circle boundary. For example, consider a TDL configuration that contains an H-junction on a Riemann sphere ${\rm S}^2$, as shown in \eqref{eqn:H_in_S^2}. Two junction vectors $v_1\in V_{\cL_1,\cL_2,\overline\cL_5}$ and $v_2\in V_{\cL_3,\cL_4,\overline\cL_5}$ are inserted at the two trivalent junctions of the H-junction. One can cut off a disc containing the H-junction and glue back a disc with a four-way junction inserted inside, graphically duplicated as
\ie\label{eqn:H_in_S^2}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\shade[ball color = gray!40, opacity = 0.4] (-.5,0) circle (2);
\draw [line,dashed] (-.5,0) circle (1);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (-.2,0) node {$\times$} -- (0,0);
\draw [line,->-=.75] (-1,0) .. controls (-1.5,.5) and (-1.207107,0.707107) .. (-1.91421,1.41421) ;
\draw [line,->-=.75] (0,0) .. controls (.5,.5) and (.207107,0.707107) .. (.91421,1.41421) ;
\draw [line,->-=.75] (-1,0) .. controls (-1.5,-.5) and (-1.207107,-0.707107) .. (-1.91421,-1.41421) ;
\draw [line,->-=.75] (0,0) .. controls (.5,-.5) and (.207107,-0.707107) .. (.91421,-1.41421) ;
\draw (-1,0) \dotsol {below}{$v_1$};
\draw (0,0) \dotsol {below}{$v_2$};
\draw (-1.8,.8) node {$\cL_1$};
\draw (-1.8,-.8) node {$\cL_2$};
\draw (.8,.8) node {$\cL_4$};
\draw (.8,-.8) node {$\cL_3$};
\end{tikzpicture}
\end{gathered}
\quad =\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\shade[ball color = gray!40, opacity = 0.4] (-.5,0) circle (2);
\draw [line,dashed] (-.5,0) circle (1);
\draw [line,->-=.75] (-.5,0) to (-1.91421,1.41421);
\draw [line,->-=.75] (-.5,0) to (-1.91421,-1.41421);
\draw [line,->-=.75] (-.5,0) to (.91421,1.41421);
\draw [line,->-=.75] (-.5,0) to (.91421,-1.41421);
\draw (-1.8,.8) node {$\cL_1$};
\draw (-1.8,-.8) node {$\cL_2$};
\draw (.8,.8) node {$\cL_4$};
\draw (.8,-.8) node {$\cL_3$};
\draw (-.5,0) \dotsol {below}{$v_3$};
\draw (-.25,.25) node {\rotatebox[origin=c]{45}{$\times$}};
\end{tikzpicture}
\end{gathered}\,.
\fe
The junction vector $v_3\in V_{\cL_1,\cL_2,\cL_3,\cL_4}$ is determined by the junction vectors $v_1$ and $v_2$ via a bilinear map $H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5):V_{\cL_1,\cL_2,\overline\cL_5}\otimes V_{\cL_3,\cL_4,\cL_5}\to V_{\cL_1,\cL_2,\cL_3,\cL_4}$, which is given by the disc ``correlation functional":
\ie
H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)\equiv\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.55);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (-.2,0) node {$\times$} -- (0,0);
\draw [line,->-=.75] (-1,0) -- (-1.5,1.2) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-1.2) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=.75] (0,0) -- (.5,-1.2) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=.75] (0,0) -- (.5,1.2) node [above right=-3pt] {${\cal L}_4$};
\end{tikzpicture}
\end{gathered}
\,.
\fe
It takes the input of two junction vectors in $V_{\cL_1,\cL_2,\overline\cL_5}$ and $ V_{\cL_3,\cL_4,\cL_5}$ and outputs a state in the Hilbert space associated with the boundary gray circle. The output state should have conformal weights $(h,\tilde h)=(0,0)$ and hence live in the junction vector space $V_{\cL_1,\cL_2,\cL_3,\cL_4}$.
\item {\bf Modular covariance:} The modular covariance of the correlation functions of local operators on Riemann surfaces carries over to those with TDLs and defect operators.
Let us focus on the partition function on a torus $T^2$.
When a TDL $\cL$ is inserted along the time direction of $T^2$, it modifies the quantization by a twisted boundary condition on the space circle, which defines the defect Hilbert space $\mathcal{H}_{\cL}$. The torus partition function is then evaluated by a trace over $\mathcal{H}_{\cL}$,
\ie
Z_{\cL}(
\tau,\bar\tau)=\Tr_{\mathcal{H}_{\cL}}\left( q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{c}{24}}\right).
\fe
After a modular S-transformation,
the TDL $\cL$ becomes inserting along the space direction and implements the $\widehat{\cL}$ action on the Hilbert space $\cH$ of local operators. This gives the twisted partition function $Z^{\cL}(\tau,\bar\tau)= \Tr_{\mathcal{H}}(\widehat{\cL} q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{c}{24}})$. In summary, the modular invariance requires that
\ie
\Tr_{\mathcal{H}_{\cL}}\left( q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{c}{24}}\right)=Z_{\cL}(\tau,\bar\tau)=
Z^{\cL}\left(
-\frac{1}{\tau},-\frac{1}{\bar\tau}\right)=\text{Tr}_{\mathcal{H}}\left(\widehat{\cL} \tilde q^{L_0-\frac{c}{24}}\bar{\tilde q}^{\widetilde L_0-\frac{c}{24}}\right)\,,
\fe
where $\tilde q=e^{-\frac{2\pi i}{\tau}}$.
\end{itemize}
\section{TDLs in fermionic CFTs}
\subsection{Defining properties}
\label{sec:defining_properties}
The TDLs in fermionic CFTs obey all the defining properties stated in Section 2 of \cite{Chang:2018iay} and reviewed in Section \ref{sec:review2d}, except the following modifications and additions.
\paragraph{1. (Fermionic defect operator)}
The defect Hilbert space at the junctions of TDLs admits an ${\mathbb Z}_2$ grading $\sigma$, which assigns $0$ to bosonic and $1$ to fermionic states or defect operators.
The fermionic defect operators are Grassmann-valued, i.e. exchanging the order of two fermionic defect operators inside a correlation function acquires a minus sign. Hence, we add an extra integer label (in red color) to the junctions to specify their ordering inside correlation functions. The junction vector space $V_{\cL_1,\cdots,\cL_n}$ inherits the ${\mathbb Z}_2$ grading of the defect Hilbert space ${\cal H}_{\cL_1,\cdots,\cL_n}$. We denote the bosonic subspace by $V^{\rm b}_{\cL_1,\cdots,\cL_n}$ and the fermionic subspace by $V^{\rm f}_{\cL_1,\cdots,\cL_n}$. For example, the correlation function involving the fermionic junction vectors $\psi\in V^{\rm f}_{\cL_1,\cL_2,\overline\cL
_5}$ and $\psi'\in V^{\rm f}_{\cL_5,\cL_3,\cL_4}$ obeys
\ie\label{eqn:H_odd_junction}
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (0,0);
\draw (0,0)\dotsol {right}{$\psi'$};
\draw (-1,0)\dotsol {left}{$\psi$};
\draw (-1,0) node [below]{\tiny\textcolor{red}{1}};
\draw (0,0) node [below]{\tiny\textcolor{red}{2}};
\draw [line,->-=1] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=1] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=1] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=1] (0,0) -- (.1,.175) node {\rotatebox[origin=c]{60}{$\times$}} -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\end{tikzpicture}
\end{gathered}=\la\cdots \psi \psi'\cdots\ra=-\la\cdots \psi'\psi\cdots\ra=-\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (0,0);
\draw (0,0)\dotsol {right}{$\psi'$};
\draw (-1,0)\dotsol {left}{$\psi$};
\draw (-1,0) node [below]{\tiny\textcolor{red}{2}};
\draw (0,0) node [below]{\tiny\textcolor{red}{1}};
\draw [line,->-=1] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=1] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=1] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=1] (0,0) -- (.1,.175) node {\rotatebox[origin=c]{60}{$\times$}} -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\end{tikzpicture}
\end{gathered}\,.
\fe
On Riemann surfaces, defect (or bulk local) operators satisfy the Neveu-Schwarz (NS) or Ramond (R) boundary condition. More precisely, an operator is identified with $(-1)^{\sigma\nu}$ times its image under a $2\pi$ shift along a non-contractible cycle on a Riemann surface, where $\sigma$ is the $\bZ_2$ grading of the operator and $\nu=1$ ($\nu=0$) for the NS (R) boundary condition. Let us focus on the NS boundary condition.
For example, consider a pair of defect operators $\cO$'s connected by a TDL $\cL$ on a cylinder, we have
\ie
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,-.5) ellipse (1.75 and .5);
\draw [line,lightgray] (1.75,-.5) -- (1.75,3.5) ;
\draw [line,lightgray] (-1.75,-.5) -- (-1.75,3.5) ;
\draw [line,lightgray] (0,3.5) ellipse (1.75 and .5);
\draw [line] (0,.8) -- (0,2.2) ;
\draw (0,0.8)\dotsol {below}{$\cO(x,y_1)$};
\draw (0,2.2)\dotsol {above}{$\cO(x+2\pi,y_2)$};
\draw (0,0.8) node [left]{\tiny\textcolor{red}{2}};
\draw (0,2.2) node [left]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\quad\quad=\quad (-1)^\sigma \quad
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,-.5) ellipse (1.75 and .5);
\draw [line,lightgray] (1.75,-.5) -- (1.75,3.5) ;
\draw [line,lightgray] (-1.75,-.5) -- (-1.75,3.5) ;
\draw [line,lightgray] (0,3.5) ellipse (1.75 and .5);
\draw [line] (0,.8) .. controls (0,1.8) and (1.75,0.7) .. (1.75,1.2);
\draw [line,dashed] (1.75,1.2) .. controls (1.75,1.7) and (-1.75,2.3) .. (-1.75,1.8) ;
\draw [line] (-1.75,1.8) .. controls (-1.75,1.5) and (0,1.2) .. (0,2.2) ;
\draw (0,0.8)\dotsol {below}{$\cO(x,y_1)$};
\draw (0,2.2)\dotsol {above}{$\cO(x,y_2)$};
\draw (0,0.8) node [left]{\tiny\textcolor{red}{2}};
\draw (0,2.2) node [left]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\fe
More explicitly, on the double cover of the cylinder,
\ie
&\begin{gathered}
\begin{tikzpicture}[scale=2]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line,lightgray] (2,0) -- (4,0) -- (4,2) -- (2,2) ;
\draw [line] (1,0.5) -- (3,1.5) ;
\draw [line] (0,1) -- (1,1.5) ;
\draw [line] (3,0.5) -- (4,1) ;
\draw (1,0.5)\dotsol {below}{$\cO(x,y_1)$};
\draw (1,1.5)\dotsol {above}{$\cO(x,y_2)$};
\draw (3,0.5)\dotsol {below}{$\cO(x+2\pi,y_1)$};
\draw (3,1.5)\dotsol {above}{$\cO(x+2\pi,y_2)$};
\draw (1,0.5) node [left]{\tiny\textcolor{red}{2}};
\draw (1,1.5) node [right]{\tiny\textcolor{red}{1}};
\draw (3,0.5) node [left]{\tiny\textcolor{red}{2}};
\draw (3,1.5) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\fe
we identify the operators as
\ie
\cO(x+2\pi,y_i)=(-1)^\sigma \cO(x,y_i)\,.
\fe
By the state/operator correspondence, this rule implies that a $2\pi$ rotation of a defect operator produces a phase $e^{2\pi i(s+\frac{1}{2}\sigma)}$ that depends on both the spin $s$ and the $\bZ_2$ grading $\sigma$ of the defect operator.
We will work in the NS sector throughout this paper. The correlation functions in the R sector can be interpreted as those in the NS sector with extra fermion parity TDL $(-1)^F$ inserted.
\paragraph{2. (Simple TDL)} A TDL ${\cal L}$ is simple if the junction vector space $V_{{\cal L},\overline{\cal L}}$ is isomorphic to either ${\mathbb C}^{1|0}$ or ${\mathbb C}^{1|1}$. The former TDL is called {\it m-type}, and the later is called {\it q-type}. For a q-type TDL $\cL_{\rm q}$, the fermionic state in the junction vector space $V_{{\cal L}_{\rm q},\overline{\cal L}_{\rm q}}\cong \bC^{1|1}$ corresponds to a topological fermionic defect operator $\psi$ (with conformal weight $(h,\tilde h)=(0,0)$) living on the worldline of $\cL_{\rm q}$. We will refer to $\psi$ as the 1d Majorana fermion.\footnote{
The possibility of $V_{{\cal L},\overline{\cal L}}\cong {\mathbb C}^{0|1}$ cannot be consistent with the super pentagon identity \eqref{eqn:super_pentagon} and the isomorphisms $V_{{\cal L},\overline{\cal L}}\cong V_{{\cal L},\overline{\cal L},I}\cong V_{\overline{\cal L},I,{\cal L}}\cong V_{I,{\cal L},\overline{\cal L}}$.}
On an oriented q-type TDL $\cL$, the 1d Majorana fermion $\psi$ can be pair-created, and by a suitable choice of the normalization of $\psi$ we have
\ie\label{eqn:oriented_fermion_pair_creation}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) -- (0,1);
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{1}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
-\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=.55] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{2}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}\,.
\fe
On an unoriented q-type TDL $\cL$, there is no canonical way to completely fix the pair-creation of $\psi$, and we have\footnote{Our convention of the pair-creations \eqref{eqn:oriented_fermion_pair_creation} and \eqref{eqn:unoriented_fermion_pair_creation} is related to the one in \cite{Aasen:2017ubm} by a field redefinition $\psi\to e^{\pm\frac{\pi i}{4}}\psi$.}
\ie\label{eqn:unoriented_fermion_pair_creation}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line] (0,-1) -- (0,1);
\end{tikzpicture}
\end{gathered}
\quad
=
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{1}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad
\,\quad{\rm or}\quad
\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line] (0,-1) -- (0,1);
\end{tikzpicture}
\end{gathered}
\quad
=
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{2}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
=
-\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line] (0,-1) -- (0,1);
\draw (0, 0.5)\dotsol {right}{$\psi$};
\draw (0, -0.5)\dotsol {right}{$\psi$};
\draw (0,0.5) node [left]{\tiny\textcolor{red}{1}};
\draw (0,-.5) node [left]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,.
\fe
For a junction involving a q-type TDL ${\cal L}$, the 1d Majorana fermion on ${\cal L}$ induces an $\bZ_2$-odd map $\Psi$ on the defect Hilbert space of the junction, which preserves the Virasoro algebra and squares to the identity map due to \eqref{eqn:oriented_fermion_pair_creation} or \eqref{eqn:unoriented_fermion_pair_creation}. It follows that the spectrum of the defect Hilbert space is doubly degenerate.
For example, consider a trivalent junction of a q-type TDL ${\cal L}_3$ and two other TDLs ${\cal L}_1$ and ${\cal L}_2$, the 1d Majorana fermion $\psi$ on ${\cal L}_3$ can act on the junction vector space $V_{\cL_1,\cL_2,\cL_3}$ as
\ie\label{eqn:fermion_action}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=1] (-1,0) -- (-.8,0) node {$\times$} -- (0,0) node [right] {$\cL_3$};
\draw [line,->-=1] (-1,0) -- (-1.5,.87) node [above left=-3pt] {$\cL_1$};
\draw [line,->-=1] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {$\cL_2$};
\draw (-1,0) node [left]{\tiny\textcolor{red}{2}};
\draw (-.3,0)\dotsol {below}{$\psi$};
\draw (-.3,0) node [above]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
~=~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,->-=1] (-1,0) -- (-.8,0) node {$\times$} -- (0,0) node [right] {$\cL_3$};
\draw [line,->-=1] (-1,0) -- (-1.5,.87) node [above left=-3pt] {$\cL_1$};
\draw [line,->-=1] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {$\cL_2$};
\end{tikzpicture}
\end{gathered}
\quad\circ~ \Psi_{\cL_3}\,,
\fe
where $\Psi_{\cL_3}:V_{\cL_1,\cL_2,\cL_3}\to V_{\cL_1,\cL_2,\cL_3}$ is a $\bZ_2$-odd linear map, that squares to the identity map, i.e. $\Psi_{\cL_3}^2=1$.
\paragraph{3. (H-junction and partial fusion)} By the locality property, an H-junction involving four external simple TDLs $\cL_1,\cdots,\cL_4$ and an internal simple TDL $\cL_5$ is a bilinear map
\ie\label{eqn:H-junction}
H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)\equiv\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (-.2,0) node {$\times$} -- (0,0);
\draw [line,->-=.75] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=.75] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=.75] (0,0) -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\draw (-1,0) node [left]{\tiny\textcolor{red}{1}};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}~:\quad V_{\cL_1,\cL_2,\overline\cL_5}\otimes V_{\cL_3,\cL_4,\cL_5}\to V_{\cL_1,\cL_2,\cL_3,\cL_4}\,,
\fe
which takes two junction vectors in the junction vector spaces $V_{\cL_1,\cL_2,\overline\cL_5}$ and $V_{\cL_3,\cL_4,\cL_5}$, and produces a state on the gray circle in the junction vector space $V_{\cL_1,\cL_2,\cL_3,\cL_4}$. The ordering of the junction vector spaces in the tensor product corresponds to the ordering of the junction vectors in correlation functions, and is determined by the red integer labels in the graph \eqref{eqn:H-junction}.
If the internal TDL $\cL_5$ is a q-type TDL, a pair of 1d Majorana fermions can be pair-created and act on the junction vector spaces $V_{\cL_1,\cL_2,\overline\cL_5}$ and $V_{\cL_3,\cL_4,\cL_5}$, explicitly as
\ie\label{eqn:pair-creation_on_H}
\hspace{-.5cm}\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (-.2,0) node {$\times$} -- (0,0);
\draw [line,->-=.75] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=.75] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=.75] (0,0) -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\draw (-1,0) node [left]{\tiny\textcolor{red}{1}};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1,0) -- (-.9,0) node {$\times$} -- (-.1,0) node {$\times$} -- (0,0);
\draw (-.5,.2) node [above] {${\cal L}_5$};
\draw [line,->-=.75] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=.75] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=.75] (0,0) -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\draw (-1,0) node [left]{\tiny\textcolor{red}{1}};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\draw (-.75,0) node [above]{\tiny\textcolor{red}{3}};
\draw (-.25,0) node [above]{\tiny\textcolor{red}{4}};
\draw (-.75,0)\dotsol {below}{$\psi$};
\draw (-.25,0)\dotsol {below}{$\psi$};
\end{tikzpicture}
\end{gathered}\quad=\quad\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_5$} -- (-.2,0) node {$\times$} -- (0,0);
\draw [line,->-=.75] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,->-=.75] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_3$};
\draw [line,->-=.75] (0,0) -- (.5,.87) node [above right=-3pt] {${\cal L}_4$};
\draw (-1,0) node [left]{\tiny\textcolor{red}{1}};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\circ (\Psi_{\overline\cL_5}\otimes \Psi_{\cL_5})\,,
\fe
which can be equivalently written as
\ie\label{eqn:H_projection}
H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)=H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)\circ P_{\cL_5}\,,
\fe
where $P_\cL$ is a projection map
\ie\label{eqn:projection_map}
P_\cL\equiv \begin{cases}
1&\text{for m-type $\cL$,}
\\
\frac{1}{2}(1+\Psi_{\overline\cL}\otimes \Psi_{\cL})&\text{for q-type $\cL$,}
\end{cases}
\fe
and the ``1" stands for the identity map.\footnote{When $\cL_5$ is unoriented, the two different choices of the pair-creation in \eqref{eqn:unoriented_fermion_pair_creation} give projection operators onto different linearly independent subspaces.}
On a local patch, a pair of TDLs can be partially fused to a TDL $\cL_1\cL_2$, with a set of junction vectors $v_i\in V_{\cL_1,\cL_2,\overline{\cL_1,\cL_2}}$ and $\tilde v_i\in V_{\ocL_2,\ocL_1,\cL_1\cL_2}$ inserted at the trivalent junctions,
\ie\label{eqn:partial_fusion}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1.069,.1) -- (-.5,.1) node [above] {${\cal L}_1$} -- (.069,.1);
\draw [line,-<-=.56] (-1.069,-.1) -- (-.5,-.1) node [below] {${\cal L}_2$} -- (.069,-.1);
\draw [line] (-1.058,.1) -- (-1.5,.87) ;
\draw [line] (-1.058,-.1) -- (-1.5,-.87) ;
\draw [line] (.058,-.1) -- (.5,-.87) ;
\draw [line] (.058,.1) -- (.5,.87) ;
\end{tikzpicture}
\end{gathered}\quad=\quad\sum_i\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (-.5,0) circle (1.325);
\draw [line,-<-=.56] (-1,0) -- (-.8,0) node {$\times$} -- (-.5,0) node [above] {${\cal L}_1\cL_2$} -- (-.2,0) node {$\times$} -- (0,0);
\draw (0,0)\dotsol {right}{$\tilde v_i$};
\draw (-1,0)\dotsol {left}{$v_i$};
\draw [line,->-=.75] (-1,0) -- (-1.5,.87) node [above left=-3pt] {${\cal L}_1$};
\draw [line,->-=.75] (-1,0) -- (-1.5,-.87) node [below left=-3pt] {${\cal L}_2$};
\draw [line,-<-=.75] (0,0) -- (.5,-.87) node [below right=-3pt] {${\cal L}_2$};
\draw [line,-<-=.75] (0,0) -- (.5,.87) node [above right=-3pt] {${\cal L}_1$};
\draw (-1,0) node [below]{\tiny\textcolor{red}{1}};
\draw (0,0) node [below]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,.
\fe
The combination $\sum_i v_i\otimes \tilde v_i$ is unique in the projection
\ie
\bigoplus_{\cL_i\subset \cL_1\cL_2}P_{\cL_i}\left(V_{\cL_1,\cL_2,\ocL_i}\otimes V_{\ocL_2,\ocL_1,\cL_i}\right)\,.
\fe
\subsection{Corollaries}
\label{sec:corollaries}
Most of the corollaries in Section 2 of \cite{Chang:2018iay} hold for TDLs in fermionic CFTs. We discuss some modifications and additional corollaries.
\subsubsection{F-move}
\label{sec:F-move}
The direct sum of all possible H-junctions with external simple TDLs $\cL_1,\cdots,\cL_4$ gives a map $H^{\cL_1,\cL_4}_{\cL_2,\cL_3}\equiv \bigoplus_{\text{simple}\,\cL_5}H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)$. By \eqref{eqn:H_projection}, $H^{\cL_1,\cL_4}_{\cL_2,\cL_3}$ satisfies
\ie
H^{\cL_1,\cL_4}_{\cL_2,\cL_3}=H^{\cL_1,\cL_4}_{\cL_2,\cL_3}\circ \prod_{\text{simple $\cL_5$}}P_{\cL_5}\,.
\fe
After restricting the domain of $H^{\cL_1,\cL_4}_{\cL_2,\cL_3}$ to the projection
\ie\label{eqn:projected_VxV}
\bigoplus_{\text{simple}~\cL_5} P_{\cL_5}\left(V_{\cL_1,\cL_2,\overline\cL_5}\otimes V_{\cL_3,\cL_4,\cL_5}\right)\,,
\fe
one can find the inverse map $\overline H^{\cL_1,\cL_4}_{\cL_2,\cL_3}$ following the same manipulation as in \cite{Chang:2018iay} using the partial fusion \eqref{eqn:partial_fusion}, i.e. the composition $\overline H^{\cL_1,\cL_4}_{\cL_2,\cL_3}\circ H^{\cL_1,\cL_4}_{\cL_2,\cL_3}$ equals to the identity map on \eqref{eqn:projected_VxV}. One also define $\overline H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_6)$ as the projection of $\overline H^{\cL_1,\cL_4}_{\cL_2,\cL_3}$ onto the subspace $P_{\cL_6}(V_{\cL_1,\cL_2,\overline\cL_6}\otimes V_{\cL_3,\cL_4,\cL_6})$.
The H-junction crossing kernel $\widetilde K$ is define as the composition\footnote{For notation simplicity, here we abuse the notation a bit by denoting the conjugation of the projection map $P_{\cL}$ (by cyclic permutation maps) also as $P_{\cL}$.}$^,$\footnote{We restrict the domain of the crossing kernel $\widetilde K_{\cL_2,\cL_3}^{\cL_1,\cL_4}(\cL_5,\cL_6)$ to be the projection $P_{\cL_5}(V_{\cL_1,\cL_2,\overline\cL_5}\otimes V_{\cL_5,\cL_3,\cL_4})$, because in the following we would like to consider the inverse of the crossing kernel $\widetilde K_{\cL_2,\cL_3}^{\cL_1,\cL_4}$.}
\ie\label{eqn:tK}
\widetilde K_{\cL_2,\cL_3}^{\cL_1,\cL_4}(\cL_5,\cL_6)&\equiv C_{ \cL_4,\cL_1,\cL_6}\circ\overline H^{\cL_2,\cL_1}_{\cL_3,\cL_4}(\cL_6)\circ C_{\cL_1,\cL_2,\cL_3,\cL_4} \circ H^{\cL_1,\cL_4}_{\cL_2,\cL_3}(\cL_5)\circ C_{\cL_5,\cL_3,\cL_4}
\\
&\,:\,P_{\cL_5}(V_{\cL_1,\cL_2,\overline\cL_5}\otimes V_{\cL_5,\cL_3,\cL_4}) \to P_{\cL_6}(V_{\cL_2,\cL_3,\overline\cL_6}\otimes V_{\cL_1,\cL_6,\cL_4})\,.
\fe
The F-symbol $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}$ is defined as the inverse of the direct sum of the crossing kernel as
\ie
F_{\cL_4}^{\cL_1,\cL_2,\cL_3}\equiv \left(\bigoplus_{\text{simple $\cL_5$, $\cL_6$}} \widetilde K_{\cL_2,\cL_1}^{\cL_3,\overline\cL_4}(\cL_6,\cL_5)\right)^{-1}\,.
\fe
Further projecting $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}$ to the subspace ${\rm Hom}(P_{\cL_5}(V_{\cL_2,\cL_1,\overline\cL_5}\otimes V_{\cL_3,\cL_5,\overline\cL_4}), P_{\cL_6}(V_{\cL_3,\cL_2,\overline\cL_6}\otimes V_{\cL_6,\cL_1,\overline\cL_4}))$ gives $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)$. Graphically, the F-symbol $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)$ gives the F-move
\ie\label{eqn:F-move}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,-<-=.55] (0,0) -- (0,-1) node [below=-3pt] {$\cL_4$};
\draw [line,->-=.55] (-1,1) -- (-2,2) node [above=-3pt] {$\cL_1$};
\draw [line,->-=.55] (0,0) -- (-1,1) ;
\draw [line,->-=.55] (0,0) --(2,2) node [above=-3pt] {$\cL_3$};
\draw [line,->-=.55] (-1,1) --(0,2) node [above=-3pt] {$\cL_2$};
\draw (-.5,.5) node [above right=-3pt] {$\cL_5$};
\draw (0,0) node [left]{\tiny\textcolor{red}{2}};
\draw (-1,1) node [below]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}~=~\sum_{\text{simple $\cL_6$}}\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,-<-=.55] (0,0) -- (0,-1) node [below=-3pt] {$\cL_4$};
\draw [line,->-=.55] (0,0) -- (-2,2) node [above=-3pt] {$\cL_1$};
\draw [line,->-=.55] (0,0) -- (1,1) ;
\draw [line,->-=.55] (1,1) --(2,2) node [above=-3pt] {$\cL_3$};
\draw [line,->-=.55] (1,1) --(0,2) node [above=-3pt] {$\cL_2$};
\draw (.5,.5) node [above left=-3pt] {$\cL_6$};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\draw (1,1) node [below]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}~\circ~ F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)\,.
\fe
Here, in the remaining of this section, and in Section \ref{sec:super_fusion_rank2}, we adopt the no $\times$-mark convention such that
the lines pointing down (up) from a junction correspond to the lines with (without) the $\times$-marks.
It would be useful to regard the F-symbol $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)$ as a map between $V_{\cL_2,\cL_1,\overline\cL_5}\otimes V_{\cL_3,\cL_5,\overline\cL_4}$ and $V_{\cL_3,\cL_2,\overline\cL_6}\otimes V_{\cL_6,\cL_1,\overline\cL_4}$ that satisfies the projection condition\footnote{Our treatment of the F-moves differs from \cite{Zhou:2021ulc} that we impose the projection condition \eqref{eqn:projection_F} instead of the projective unitary condition (26) in \cite{Zhou:2021ulc}.}
\ie\label{eqn:projection_F}
F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)=P_{\cL_6}\circ F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)\circ P_{\cL_5}\,.
\fe
We give the solution to the projection condition in the matrix representation in Appendix \ref{sec:projection_condition}.
\subsubsection{Super pentagon identity}
As illustrated in the following commutative diagram
\ie\label{eqn:super_pentagon_diagram}
\begin{tikzcd}
& V_{2,1,\overline i}\otimes V_{3,i,\overline j}\otimes V_{4,j,\overline 5 }
\arrow[rd, "{F^{i,3,4}_5(j,k)}"]
\arrow[ld,"{F^{1,2,3}_j(i,m)}" ]
& & &
\\
V_{3,2, \overline m}\otimes V_{m,1,\overline j}\otimes V_{4,j,\overline 5 }
\arrow[dd,"{F^{1,m,4}_5(j,l)}"]
& &V_{2,1,\overline i}\otimes V_{4,3, \overline k}\otimes V_{k,i ,\overline 5 }
\arrow[dd,"{S^{4,3, k}_{2,1, i}}"]
\\
\\
V_{3,2,\overline m}\otimes V_{4,m,\overline l}\otimes V_{l,1,\overline 5}
\arrow[rd,"{F^{2,3,4}_l(m,k)}"]
&& V_{4,3, \overline k}\otimes V_{2,1,\overline i}\otimes V_{k,i ,\overline 5 }
\arrow[ld,"{F^{1,2,k}_5(i,l)}"]
\\
&V_{4,3, \overline k}\otimes V_{k,2,\overline l}\otimes V_{l,1,\overline 5} &
\end{tikzcd}
\fe
the F-move satisfies the super pentagon identity
\ie\label{eqn:super_pentagon}
F^{1,2,k}_5(i,l)\circ S_{2,1,i}^{4,3,k} \circ F^{i,3,4}_5(j,k)=\sum_{m}F^{2,3,4}_l(m,k)\circ F^{1,m,4}_5(j,l)\circ F^{1,2,3}_j(i,m)\,,
\fe
where we abbreviated $F_{\cL_4}^{\cL_1,\cL_2,\cL_3}(\cL_5,\cL_6)$ by $F^{1,2,3}_4(5,6)$. The map $S_{2,1,i}^{4,3,k}$ exchanges the ordering of a pair of junction vectors. More precisely, we have $S_{1,2,3}^{4,5,6}~:~ V_{\cL_1,\cL_2,\ocL_3}\otimes V_{\cL_4,\cL_5,\ocL_6}\to V_{\cL_4,\cL_5,\ocL_6}\otimes V_{\cL_1,\cL_2,\ocL_3}$ defined by
\ie\label{eqn:ordering_s}
S_{1,2,3}^{4,5,6}(v\otimes v')=(-1)^{\sigma(v)\sigma(v')} v'\otimes v\,,
\fe
where $\sigma$ is the $\bZ_2$-grading of the junction vector spaces.
We emphasize that when any of the internal simple TDLs is q-type, the both hand sides of the super pentagon identity are maps between the projections of the tensor products of the junction vector spaces, $P_{\cL_i} P_{\cL_j}(V_{\cL_2,\cL_1,\cL_i}\otimes V_{\cL_3,\cL_i,\cL_j}\otimes V_{\cL_4,\cL_j,\cL_5})$ and $P_{\cL_k}P_{\cL_l}(V_{\cL_3,\cL_4,\cL_k}\otimes V_{\cL_k,\cL_2,\cL_l}\otimes V_{\cL_l,\cL_1,\cL_5})$.
\subsubsection{Universal sector}
\label{sec:universal_sector}
The F-moves of the H-junctions with no more than two non-trivial external TDLs form a simple set of data that is universal for all the TDLs, and can be solved by the super pentagon equations with also no more than two non-trivial external TDLs. We will call it the universal sector.
For the m-type TDLs, the analysis of the universal sector is identical to the TDLs in bosonic CFTs \cite{Chang:2018iay}, and we review it here. For simplicity, we consider unoriented m-type TDLs. First, let us choose some convenient gauge to fix some F-moves. Note that for the following trivalent junctions, we can perform the following ``gauge transformations",
\ie
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad x\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,& \begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
y\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,
\fe
where the dotted lines denote the trivial TDL $I$.
By using these, we can always normalize the F-moves of the following graphs,
\ie
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line,dotted] (-1,1) -- (-2,2);
\draw [line,dotted] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\quad=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line,dotted] (-2,2) -- (0,0);
\draw [line,dotted] (0,2) --(1,1);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (1,1);
\draw [line] (0,0) -- (0,-1) ;
\end{tikzpicture}
\end{gathered}\,,
\quad\quad\quad
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line] (-1,1) -- (-2,2);
\draw [line,dotted] (-1,1) --(0,2);
\draw [line,dotted] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\quad=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line] (-2,2) -- (0,0);
\draw [line,dotted] (0,2) --(1,1);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (1,1);
\draw [line] (0,0) -- (0,-1) ;
\end{tikzpicture}
\end{gathered}.
\label{graph:m_type_normalization}
\fe
With the above normalization, one can further focus on all super pentagon equations with the F-moves of H-junctions containing only two non-trivial external 1m-type TDLs. The solution is unique, and all these F-symbols are represented by $1\times 1$ identity matrices.
Next, we focus on the universal sector of an unoriented q-type TDL $\cL$. The result for the universal sector of oriented q-type TDLs will be summarized in Appendix \ref{sec:universal_oriented_TDL}. The junction vector space of a trivalent junction involving two $\cL$'s and a trivial TDL $I$ is isomorphic to $\mathbb C^{1|1}$. More precisely, we have three isomorphic junction vector spaces
\ie
V_{I,\cL,\cL}\cong V_{\cL, I ,\cL }\cong V_{\cL,\cL ,I}\cong \mathbb C^{1|1}\,.
\fe
The 1d Majorana fermion on $\cL$ acts on these junction vector spaces as
\ie\label{eqn:1d_mf_action}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (-0.17,.1)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad\alpha_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}\,,\qquad& \begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,-.2)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\alpha_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0.17,.1)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\beta_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}\,,& \begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,-.2)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\beta_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0.17,.1)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\gamma_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\,,&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (-0.17,.1)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\gamma_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {right}{};
\end{tikzpicture}
\end{gathered}
\,,
\fe
where the dotted lines denote the trivial TDL $I$, and the junctions with (or without) a black dot are associated with the fermionic (or bosonic) junction vector subspaces.
Changing the basis of the junction vector spaces gives the ``gauge transformations"
\ie\label{eqn:universal_gauge}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad x_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,~& \begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
x_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
y_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,~&\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
y_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
z_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\,,~&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
z_2\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\,.
\fe
The gauge symmetry can be partially fixed by the gauge conditions
\ie\label{eqn:gf_abc}
&\alpha_1=\frac{1}{\alpha_2}\equiv\alpha\,,\quad\beta_1=\frac{1}{\beta_2}\equiv\beta\,,\quad\gamma_1=\frac{1}{\gamma_2}\equiv\gamma\,.
\fe
The above gauge conditions are invariant under the residual gauge transformations given by \eqref{eqn:universal_gauge} with the constraints
\ie\label{eqn:residual_gauge}
&x_1=\pm x_2\,,\quad y_1=\pm y_2\,,\quad z_1=\pm z_2\,.
\fe
Let us focus on the F-symbols with only two non-trivial external TDLs
\ie\label{eqn:universal_F}
&F^{\cL,\cL,I}_I(I,\cL)\,,~F^{\cL,I,\cL}_I(\cL,\cL)\,,~ F^{\cL,I,I}_\cL(\cL,I)\,,~ F^{I,\cL,\cL}_I(\cL,I)\,,~ F^{I,\cL,I}_\cL(\cL,\cL)\,,~ F^{I,I,\cL}_\cL(I,\cL)\,.
\fe
The F-symbols will be regarded as maps between the unprojected junction vector spaces that satisfy the projection condition \eqref{eqn:projection_F}, and represented by $4\times 4$ matrices ${\cal F}$'s. The projection matrices can be computed using the 1d Majorana fermion actions \eqref{eqn:1d_mf_action}. For example, the projection matrix involved in the projection condition of the F-matrix ${\cal F}^{\cL,\cL,I}_I(I,\cL)$ are given by the relations
\ie\label{eqn:ex_pc_third}
&\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
~=~
\epsilon_1\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (.2,.2) \dotsol {right}{\tiny\textcolor{red}{2}};
\draw (.8,.8) \dotsol {right}{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
~=~
\frac{\epsilon_1\gamma}{\alpha}~\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (0,0) \dotsol {right}{\tiny\textcolor{red}{2}};
\draw (1,1) \dotsol {right}{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (1,1) \dotsol {right}{};
\end{tikzpicture}
\end{gathered}
~=~
\alpha\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (.5,.5) \dotsol {right}{};
\end{tikzpicture}
\end{gathered}
~=~
\alpha\gamma\begin{gathered}
\begin{tikzpicture}[scale=.5]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line,dotted] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (0,0) \dotsol {right}{};
\end{tikzpicture}
\end{gathered}\,,
\fe
where $\epsilon_1=\pm 1$ is the sign ambiguity of the pair-creation on unoriented q-type TDLs \eqref{eqn:unoriented_fermion_pair_creation}. Similarly, we could compute the projection matrices for the other F-matrices. There is a sign ambiguity for each pair-creation, and we need to introduce seven other more undetermined signs $\epsilon_2$, $\epsilon_2'$, $\epsilon_3$, $\epsilon_4$, $\epsilon_5$, $\epsilon_5'$, $\epsilon_6=\pm1$.
Solving the projection conditions as in Appendix \ref{sec:projection_condition}, we find the following form of the F-matrices
\ie\label{eqn:universal_F-matrices}
&{\cal F}^{\cL,\cL,I}_I(I,\cL)=\begin{pmatrix}
f_{1b} & 0
\\
0 & f_{1f}
\end{pmatrix}\begin{pmatrix}
1 & \frac{\epsilon_1\gamma}{\alpha} & 0 & 0
\\
0 & 0 & 1 & \alpha\gamma
\end{pmatrix}\,,
\quad
{\cal F}^{\cL,I,\cL}_I(\cL,\cL)=
\begin{pmatrix}
1 & 0
\\
\epsilon_2\alpha \gamma & 0
\\
0 & 1
\\
0 & \frac{\gamma}{\alpha}
\end{pmatrix}
\begin{pmatrix}
f_{2b} & 0
\\
0 & f_{2f}
\end{pmatrix}
\begin{pmatrix}
1 & \frac{\epsilon_2'\gamma}{\beta} & 0 & 0
\\
0 & 0 & 1 & \beta\gamma
\end{pmatrix}\,,
\\
&{\cal F}^{\cL,I,I}_\cL(\cL,I)=\begin{pmatrix}
1 & 0
\\
\epsilon_3 & 0
\\
0 & 1
\\
0 & \frac{1}{\alpha^2}
\end{pmatrix}
\begin{pmatrix}
f_{3b} & 0
\\
0 & f_{3f}
\end{pmatrix}\,,
\quad
{\cal F}^{I,\cL, \cL}_I(\cL,I)=\begin{pmatrix}
1 & 0
\\
\epsilon_4\beta\gamma & 0
\\
0 & 1
\\
0 & \frac{\gamma}{\beta}
\end{pmatrix}
\begin{pmatrix}
f_{4b} & 0
\\
0 & f_{4f}
\end{pmatrix}\,,
\\
&{\cal F}^{I,\cL,I}_\cL(\cL,\cL)=\begin{pmatrix}
1 & 0
\\
\frac{\epsilon_5\beta}{\alpha} & 0
\\
0 & 1
\\
0 & \frac{1}{\beta\alpha}
\end{pmatrix}
\begin{pmatrix}
f_{5b} & 0
\\
0 & f_{5f}
\end{pmatrix}
\begin{pmatrix}
1 & \frac{\epsilon_5'\beta}{\alpha} & 0 & 0
\\
0 & 0 & 1 & \alpha\beta
\end{pmatrix}
\,,
\quad
{\cal F}^{I,I, \cL}_\cL(I,\cL)=\begin{pmatrix}
f_{6b} & 0
\\
0 & f_{6f}
\end{pmatrix}
\begin{pmatrix}
1 & \epsilon_6 & 0 & 0
\\
0 & 0 & 1 & \beta^2
\end{pmatrix}\,.
\fe
The residual gauge symmetry (\eqref{eqn:universal_gauge} with \eqref{eqn:residual_gauge}) can be used to fix
\ie\label{eqn:gf_f12}
f_{1b}=f_{2b}=\frac{1}{2}\,.
\fe
There are ten super pentagon equations with only two non-trivial external TDLs. Solving them, we find two gauge inequivalent solutions of the F-moves, \ie\label{eqn:universal_unoriented_sols}
&\alpha=\beta=1\,,\quad f_{1f}=\frac{1}{2}\gamma^{-1}\,,\quad f_{2f}=-\frac{\gamma^2}{2}\,,
\\
& f_{3b}=f_{3f}=f_{4b}=1\,,\quad f_{4f}=\gamma^{-1}\,,\quad f_{5b}=f_{5f}=f_{6b}=f_{6f}=\frac{1}{2}\,,
\\
& \epsilon_1=\epsilon_2=\epsilon_2'=\epsilon_3=\epsilon_4=\epsilon_5=\epsilon_5'=\epsilon_6=1\,,
\fe
for $\gamma=e^{\frac{i\pi}{4}},\,e^{\frac{3i\pi}{4}}$. Note that the ambiguities $\epsilon_1,\cdots, \epsilon_6$ are completely fixed. This effectively gives the unoriented TDL $\cL$ an orientation such that all the lines in \eqref{eqn:1d_mf_action} (and also \eqref{eqn:ex_pc_third}) have an upward pointing orientation.
\paragraph{Constraints on the projection matrices}
Since the 1d Majorana fermions can freely move along the q-type TDLs, there are additional constraints on the actions \eqref{eqn:1d_mf_action}, as shown in the following graphs,
\ie\label{eqn:universal_T_relations}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (-.68,.4) \dotsol {left}{\tiny\textcolor{red}{1}};
\draw (-.17,.1) \dotsol {left}{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad \alpha^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,-.2)\dotsol {left}{\tiny\textcolor{red}{1}};
\draw (0,-.7)\dotsol {left}{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad \alpha^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line,dotted] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0.17,.1)\dotsol {right}{\tiny\textcolor{red}{2}};
\draw (0.68,.4)\dotsol {right}{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\quad=\quad \beta^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\draw (0,-.2)\dotsol {right}{\tiny\textcolor{red}{1}};
\draw (0,-.7)\dotsol {right}{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad \beta^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line,dotted] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,
\\
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0.17,.1)\dotsol {right}{\tiny\textcolor{red}{2}};
\draw (0.68,.4)\dotsol {right}{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\quad=\quad -
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0.17,.1)\dotsol {right}{\tiny\textcolor{red}{1}};
\draw (0.68,.4)\dotsol {right}{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad
-\gamma^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (-.68,.4) \dotsol {left}{\tiny\textcolor{red}{1}};
\draw (-.17,.1) \dotsol {left}{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad=\quad -\gamma^4
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\,,
\fe
where we have used eq.\,(\ref{eqn:unoriented_fermion_pair_creation}) and (\ref{eqn:1d_mf_action}) for the unoriented q-type line. Therefore we have the constraints on the $\alpha$, $\beta$ and $\gamma$ as
\ie
\alpha^4=1\,,\ \ \ \beta^4=1\,,\ \ \ {\rm and}\ \ \ \gamma^4=-1\,.
\fe
One can check that our universal sector solutions (\ref{eqn:universal_unoriented_sols}) indeed satisfy the above constraints.\footnote{There are actually sign ambiguities in the relations \eqref{eqn:universal_T_relations}, but the universal sector solutions (\ref{eqn:universal_unoriented_sols}) fix those sign ambiguities.}
\subsubsection{Rotation on defect operators}
As discussed in the previous section, there are phases and signs producing from rotating the defect operators, switching the order of the defect operators in correlation functions, and moving the defect operators around non-contractible cycles. They are consistent with each other as we now see. Consider a defect operator $\cO$ in the defect Hilbert space ${\cal H}_{\cL}$ of an unoriented TDL $\cL$, which can be normalized such that the two-point function of a pair of $\cO$'s connected by $\cL$ is
\ie
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line] (0,0) .. controls (-1,0) and (-1,.3) .. (0,.3)
(0,.3) .. controls (1,.3) and (1,0) .. (2,0);
\draw (0,0)\dotsol {below}{$\cO(z_1,\bar z_1)$};
\draw (2,0)\dotsol {below}{$\cO(z_2,\bar z_2)$};
\draw (0,0) node [right]{\tiny\textcolor{red}{1}};
\draw (2,0) node [right]{\tiny\textcolor{red}{2}};
\draw (0,.675) node {};
\end{tikzpicture}
\end{gathered}
~=~ z_{12}^{-2h}\bar z_{12}^{-2\tilde h}\,.
\fe
Bringing $z_2$ to $z_1$ and $z_1$ to $z_2$ counterclockwisly gives
\ie
\begin{gathered}\begin{tikzpicture}[scale=1.25]
\draw [line] (0,0) .. controls (-1,0) and (-1,-.3) .. (0,-.3)
(0,-.3) .. controls (1,-.3) and (1,0) .. (2,0);
\draw (0,0)\dotsol {above}{$\cO(z_1,\bar z_1)$};
\draw (2,0)\dotsol {above}{$\cO(z_2,\bar z_2)$};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\draw (2,0) node [right]{\tiny\textcolor{red}{1}};
\draw (0,-.375) node {};
\end{tikzpicture}
\end{gathered}
~=~ e^{-2\pi i s}z_{12}^{-2h}\bar z_{12}^{-2\tilde h}\,,
\fe
where the phase on the RHS is given by taking $z_{12}\to e^{i\pi}z_{12}$. Rotating the left operator by degree $2\pi$ gives a phase $e^{2\pi i (s+\frac{\sigma}{2})}$, and we have
\ie
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line] (0,0) .. controls (-1,0) and (-1,.3) .. (0,.3)
(0,.3) .. controls (1,.3) and (1,0) .. (2,0);
\draw (0,0)\dotsol {below}{$\cO(z_1,\bar z_1)$};
\draw (2,0)\dotsol {below}{$\cO(z_2,\bar z_2)$};
\draw (0,0) node [right]{\tiny\textcolor{red}{2}};
\draw (2,0) node [right]{\tiny\textcolor{red}{1}};
\draw (0,.675) node {};
\end{tikzpicture}
\end{gathered}
~=~ e^{\pi i\sigma }z_{12}^{-2h}\bar z_{12}^{-2\tilde h}\,.
\fe
We see that the resulting phase $e^{\pi i\sigma }$ is consistent with the sign given by exchanging the order of the two defect operators in the two-point function.
To see the consistency between the rotation phase and the boundary condition of the defect operators, let us consider a pair of 1d Majorana fermions $\psi$ on antipodal points of a q-type TDL $\cL$ inserted on a unit circle centered at the origin of a complex plane with complex coordinate $z$. Under the exponential map $z=e^{iw}$, the complex plane is mapped to an infinite cylinder with the NS boundary condition for the non-contractible cycle,
\ie
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,4) -- (4,4) -- (4,0) -- (0,0);
\draw [line,->-=.75] (2,2) circle (1);
\draw (1,2)\dotsol {left}{$\psi$};
\draw (3,2)\dotsol {right}{$\psi$};
\draw (1,2) node [right]{\tiny\textcolor{red}{1}};
\draw (3,2) node [left]{\tiny\textcolor{red}{2}};
\draw (2,.75) node {$\cL$};
\end{tikzpicture}
\end{gathered}
\quad\quad\longrightarrow\quad\quad
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) ellipse (1.75 and .5);
\draw [line,lightgray] (1.75,0) -- (1.75,3) ;
\draw [line,lightgray] (-1.75,0) -- (-1.75,3) ;
\draw [line,lightgray] (0,3) ellipse (1.75 and .5);
\draw [line,-<-=.5] (1.75,1.5) arc(0:-180:1.75 and .5);
\draw [line,dashed] (1.75,1.5) arc(0:180:1.75 and .5);
\draw (-1,1.09)\dotsol {below}{$\psi$};
\draw (1,1.91)\dotsol {below}{$\psi$};
\draw (-1,1.09) node [above]{\tiny\textcolor{red}{1}};
\draw (1,1.91) node [above]{\tiny\textcolor{red}{2}};
\draw (0,1.3) node {$\cL$};
\end{tikzpicture}
\end{gathered}
\fe
Now, moving both of the two $\psi$'s on the cylinder counterclockwisly by half of the circle gives a minus sign due to the NS-boundary condition. On the plane, it corresponds to first moving the $\psi$'s to the middle configuration in \eqref{eqn:Lpsipsimove}
then rotating the two $\psi$'s by degree $\pi$.
\ie\label{eqn:Lpsipsimove}
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,4) -- (4,4) -- (4,0) -- (0,0);
\draw [line,->-=.75] (2,2) circle (1);
\draw (1,2)\dotsol {left}{$\psi$};
\draw (3,2)\dotsol {right}{$\psi$};
\draw (1,2) node [right]{\tiny\textcolor{red}{1}};
\draw (3,2) node [left]{\tiny\textcolor{red}{2}};
\draw (2,.75) node {$\cL$};
\end{tikzpicture}
\end{gathered}
~=~
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,4) -- (4,4) -- (4,0) -- (0,0);
\draw [line, ->-=.3, ->-=.8] (1,2) .. controls (1,2.5) and (.25,2) .. (.25,1.5)
(.25,1.5) .. controls (.25,0) and (3.25,3) .. (3.25,2.5)
(3.25,2.5) .. controls (3.25,2.25) and (3,2.25) .. (3,2)
(3,2) .. controls (3,1.5) and (3.75,2) .. (3.75,2.5)
(3.75,2.5) .. controls (3.75,4) and (.75,1) .. (.75,1.5)
(.75,1.5) .. controls (.75,1.75) and (1,1.75) .. (1,2) ;
\draw (1,2)\dotsol {above}{$\psi$};
\draw (3,2)\dotsol {below}{$\psi$};
\draw (1,2) node [right]{\tiny\textcolor{red}{2}};
\draw (3,2) node [left]{\tiny\textcolor{red}{1}};
\draw (2,1.5) node {$\cL$};
\end{tikzpicture}
\end{gathered}
~=
\left(e^{\frac{i\pi}{2}}\right)^2~\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,4) -- (4,4) -- (4,0) -- (0,0);
\draw [line,->-=.75] (2,2) circle (1);
\draw (1,2)\dotsol {left}{$\psi$};
\draw (3,2)\dotsol {right}{$\psi$};
\draw (1,2) node [right]{\tiny\textcolor{red}{2}};
\draw (3,2) node [left]{\tiny\textcolor{red}{1}};
\draw (2,.75) node {$\cL$};
\end{tikzpicture}
\end{gathered}
\fe
The rotation produces the phase $(e^{\frac{i\pi}{2}})^2$ which matches with the previous minus sign in the cylinder case. By a similar manipulation, we also find that
\ie
\begin{gathered}\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,4) -- (4,4) -- (4,0) -- (0,0);
\draw [line,->-=.75] (2,2) circle (1);
\draw (1,2)\dotsol {left}{$\psi$};
\draw (2,.75) node {$\cL$};
\end{tikzpicture}
\end{gathered}~=0\,.
\fe
\subsubsection{Fusion coefficients}
\label{sec:fusion_coef_main}
The fusion coefficients are determined by the dimensions of the junction vector spaces as
\ie\label{eqn:general_fusion_rule}
{\cal L}_1 {\cal L}_2 = \sum_{\text{m-type ${\cal L}_{i_m}$}}\dim_{\mathbb C} (V_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_m}}) {\cal L}_{i_m}+\sum_{\text{q-type ${\cal L}_{i_q}$}}\dim_{{\mathbb C}^{1|1}} (V_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_q}}) {\cal L}_{i_q}\,.
\fe
The derivation of this formula is given in Appendix \ref{sec:fusion_coef}. Due to this formula, the dimension of the junction vector space will be also referred as the multiplicity.
Since the TDL ${\cal L}_{i_q}$ is q-type, the dimensions of the bosonic and fermionic subspaces of $V_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_q}}$ coincide, i.e.
\ie
\dim_{{\mathbb C}^{1|1}} (V_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_q}}) = \dim_{{\mathbb C}} (V^{\rm f}_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_q}})= \dim_{{\mathbb C}} (V^{\rm f}_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_q}})\equiv n_{12{i_q}}\,.
\fe
On the other hand, the dimensions of the bosonic and fermionic subspaces of $V_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_m}}$ are different in general, and we define
\ie
n_{{\rm b},12i_m}\equiv \dim_{\mathbb C} (V^{\rm b}_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_m}})\,,\quad n_{{\rm f},12i_m}\equiv\dim_{\mathbb C} (V^{\rm f}_{{\cal L}_1,{\cal L}_2,{\cal L}_{i_m}})\,.
\fe
For example, the fusion of a q-type TDL ${\cal L}$ with its orientation reversal gives
\ie
{\cal L}\overline{\cal L}=\dim_{\mathbb C}(V_{{\cal L},\overline{\cal L},I})I + \cdots=(1_{\rm b}+1_{\rm f}) I + \cdots\,,
\fe
where we use $1_{\rm b}$ and $1_{\rm f}$ to emphasize the fact that $V^{\rm b}_{\cL,\overline\cL,I}$ and $V^{\rm f}_{\cL,\overline\cL,I}$ are both one-dimensional.
\subsubsection{q-type invertible TDL and modified Cardy condition}
Let us consider a set of $N_{\rm m}$ m-type invertible TDLs $\cL_{i_{\rm m}}$ for $i_{\rm m}=1,\cdots,N_{\rm m}$ and $N_{\rm q}$ q-type TDLs $\cL_{i_{\rm q}}$ for $i_{\rm q}=1,\cdots,N_{\rm q}$, that closes under the fusion. Invertibility of the m-type TDLs implies
\ie\label{eqn:inv_m-typ}
\cL_{i_{\rm m}} \overline \cL_{i_{\rm m}}=I\,.
\fe
We further impose the following fusion rules for the q-type TDLs
\ie\label{eqn:inv_q-typ}
\cL_{i_{\rm q}} \overline \cL_{i_{\rm q}}=(1_{\rm b}+1_{\rm f})I\,.
\fe
A q-type TDL that satisfies the above fusion rule is called a {\it q-type invertible TDL}, by the reason we will see momentarily.
The above fusion rules \eqref{eqn:inv_m-typ} and \eqref{eqn:inv_q-typ} imply that absolute values of the loop expectation values (quantum dimensions) are
\ie
|\la \cL_{i_m}\ra_{\bR^2}|
=1\,,
\quad
|\la \cL_{i_q}\ra_{\bR^2}|
=\sqrt{2}\,.
\fe
This further implies that the products $\cL_{i_{\rm m}}\cL_{j_{\rm q}}$ and $\cL_{j_{\rm q}}\cL_{i_{\rm m}}$ are q-type TDLs, the product $\cL_{i_{\rm m}}\cL_{j_{\rm m}}$ is a m-type TDL, and the product $\cL_{i_{\rm q}}\cL_{j_{\rm q}}$ is $1_{\rm b}+1_{\rm f}$ times a m-type TDL, i.e. the fusion rules should take the form
\ie\label{eqn:invertible_fusions}
\cL_{i_{\rm m}}\cL_{j_{\rm q}}&=\cL_{k_{\rm q}}\,,& \cL_{j_{\rm q}}\cL_{i_{\rm m}}&=\cL_{l_{\rm q}}\,,
\\
\cL_{i_{\rm m}}\cL_{j_{\rm m}}&=\cL_{k_{\rm m}}\,,&\cL_{i_{\rm q}}\cL_{j_{\rm q}}&=(1_{\rm b}+1_{\rm f})\cL_{k_{\rm m}}\,.
\fe
Note that the above fusion rules admit an extra $\bZ_2$ grading, where the m-type TDLs are $\bZ_2$ even and the q-type TDLS are $\bZ_2$ odd, but one should not confused this with the $\bZ_2$ grading $\sigma$ of the defect Hilbert vector space.
The above fusions are almost group multiplications, except that the coefficient on the RHS of the last equation in \eqref{eqn:invertible_fusions} is not 1. If we define
\ie\label{eqn:def_widetilde_L}
\widetilde \cL_{i_q}=\frac{1}{\sqrt{2}}\cL_{i_q}
\fe
and ignore the differences between $1_{\rm b}$ and $1_{\rm f}$, then the fusion forms a group with the elements $\{\cL_{i_m},\widetilde\cL_{i_q}\}$.\footnote{Dividing a factor of $\sqrt{2}$ in \eqref{eqn:def_widetilde_L} could be interpreted as factoring out the 1d Majorana fermion living on the worldline of the TDLs $\cL_{i_{\rm q}}$.}
The TDLs $\widetilde\cL_{i_q}$ together with the m-type invertible TDLs $\cL_{i_m}$ generate an (invertible) symmetry by their actions on local operators and states in the fermionic CFT.\footnote{We adopt the strict sense of symmetry here, as opposed to the non-invertible or category symmetry.}
Note importantly that some parts of the symmetry are generated by $\widetilde\cL_{i_q}$ that do not have a well-defined defect Hilbert space. Hence, the usual Cardy condition should be modified. Consider the twisted partition function
\ie
&Z^{\widetilde \cL_{i_{\rm q}}}(\tau,\bar\tau)=\Tr_{\cal H}\left(\widehat{\widetilde\cL\,}_{i_{\rm q}}q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{c}{24}}\right)\,.
\fe
Its S-transformation equals to $\frac{1}{\sqrt{2}}$ times the partition function for the defect Hilbert space ${\cal H}_{\cL_{i_{\rm q}}}$
\ie
&Z^{\widetilde \cL_{i_{\rm q}}}\left(-\frac{1}{\tau},-\frac{1}{\bar\tau}\right)=\frac{1}{\sqrt{2}}\Tr_{{\cal H}_{\cL_{i_{\rm q}}}}\left(q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{c}{24}}\right)\,.
\fe
The spectrum of the defect Hilbert space ${\cal H}_{\cL_{i_{\rm q}}}$ is doubly degenerate because ${\cal H}_{\cL_{i_{\rm q}}}$ forms a representation of a one-dimensional Clifford algebra generated by the action of the 1d Majorana fermion $\psi$ living on the q-type TDL $\cL_{i_{\rm q}}$. Therefore, $Z^{\widetilde \cL_{i_{\rm q}}}\left(-\frac{1}{\tau},-\frac{1}{\bar\tau}\right)$ should admit a $q$, $\bar q$-expansion with coefficients in $\sqrt{2}\bZ_{\ge 0}$. We will refer to this as the {\it modified Cardy condition}.
\subsection{Relation to super fusion categories}
\label{sec:super_fusion_category}
The relation between TDLs in fermionic CFTs and super fusion categories is analogous to the relation between TDLs in bosonic CFTs and fusion categories discussed in \cite{Chang:2018iay}. Here, we highlight some modifications and additions, following the treatment of super fusion category in \cite{Aasen:2017ubm}.
The defect Hilbert space ${\cal H}_{\cL}$ of a TDL $\cL$ is an object in the super fusion category. The morphisms between objects in the super fusion category are given by junction vectors in the two-way junction vector spaces. More explicitly, a weight $(h,\tilde h)=(0,0)$ operator $m$ in the defect Hilbert space ${\cal H}_{\cL_1,\ocL_2}$ (i.e. $m\in V_{\cL_1,\ocL_2}$) gives an isomorphism between the defect Hilbert spaces,
\ie
m:{\cal H}_{\cL_1}\to {\cal H}_{\cL_2}\,.
\fe
The endomorphism spaces for m-type and q-type TDLs are
\ie\label{eqn:end_space}
{\rm End}({\cal H}_{\cL_{\rm m}})=V_{\cL_{\rm m},\ocL_{\rm m}}\cong\bC\,,\quad {\rm End}({\cal H}_{\cL_{\rm q}})=V_{\cL_{\rm q},\ocL_{\rm q}}\cong\bC^{1|1}\,.
\fe
The junction vector space $V_{\cL_{i_1},\cdots, \cL_{i_n}}$ (the subspace of the weight-$(0,0)$ operators in ${\cal H}_{\cL_{i_1},\cdots, \cL_{i_n}}$) corresponds to the fusion space $V^{i_{n-1},\cdots, i_1}_{i_n}$ in the super fusion category \cite{Aasen:2017ubm}. In the discussion of the F-moves, the tensor products between the fusion spaces are usually taken to have coefficients valued in the endomorphism spaces \eqref{eqn:end_space} (as in (351) of \cite{Aasen:2017ubm}). It is equivalent to considering the projected tensor products \eqref{eqn:projected_VxV} with $\bC$-number coefficients. An important new ingredient of the pentagon equation in the super fusion category is the Koszul sign from exchanging the order of the fusion spaces. Such a sign corresponds to the sign \eqref{eqn:ordering_s} from exchanging the ordering of the junction vectors.
\section{Super fusion categories of rank-2}
\label{sec:super_fusion_rank2}
Let us consider a system of two simple TDLs, the trivial TDL $I$ and a non-trivial TDL $W$. The system is described by super fusion categories of rank-2. When $W$ is m-type, the most general fusion rule is
\ie\label{eqn:rank2_m}
W^2= I+ (n_{\rm b}+n_{\rm f})W\,.
\fe
When $W$ is q-type, the most general fusion rule is
\ie
\label{sec3:qtypefusion}
W^2=(1_{\rm b}+1_{\rm f}) I+ nW\,.
\fe
The corresponding (super-)fusion categories are labeled as $\mathcal C_{\rm m}^{(n_{\rm b}, n_{\rm f})}$ and $\mathcal C_{\rm q}^{n}$.
We solve the super pentagon identity \eqref{eqn:super_pentagon} for $(n_{\rm b},n_{\rm f})\in\{(0,0),(1,0),(0,1),(2,0),(1,1),$ $(0,2),(3,0),(2,1),(1,2),(0,3),(3,1),(2,2),(1,3)\}$, and $n\le 2$, in the following procedure.
We first use \texttt{Mathematica} to spell out the super pentagon identity as polynomial equations whose variables are the F-symbols. We then use \texttt{Magma} to find the Gr\"{o}bner basis of the polynomial equations. Finally, we input the Gr\"{o}bner basis back to \texttt{Mathematica} to solve for the solutions. We find that the solutions only exist for $n_{\rm b},\,n_{\rm f}\in \{0, 1\}$ and $n\in\{0,2\}$. We conjecture that these are all the rank-2 super fusion categories.
We also list the CFT realizations of the above rank-2 super fusion categories in the table below.
\begin{table}[H]
\begin{center}
\begin{tabular}{c|c|c}
& \# categories & CFT realization\\\hline
${\mathcal C}_{\rm m}^{(0, 0)}$ & 2 & $(-1)^{F}$ in any fermionic CFT
\\\hline
${\mathcal C}_{\rm m}^{(1,0)}$ & 2 & $m=4$ fermionic minimal model
\\\hline
${\mathcal C}_{\rm m}^{(0,1)}$& 2 & ?
\\\hline
${\mathcal C}_{\rm m}^{(1,1)}$ & 2 & $m=7$ fermionic minimal model
\\\hline
${\mathcal C}_{\rm q}^{0}$ & 4 & $m=3$ (and $m=4$) fermionic minimal model
\\\hline
\multirow{2}{*}{${\mathcal C}_{\rm q}^{2}$} & \multirow{2}{*}{4} & $m=11$ fermionic minimal model
\\\cline{3-3}
& & $m=12$ fermionic minimal model
\end{tabular}.
\end{center}
\caption{The rank-2 super fusion categories and the CFT realizations, of those with a positive loop expectation value $\vev{W}_{\bR^2}>0$, in the fermionic minimal models \cite{Runkel:2020zgg,Hsieh:2020uwb,Kulp:2020iet}. We leave a question mark for the ${\mathcal C}_{\rm m}^{(0,1)}$ categories, as we do not know their CFT realization.}
\end{table}
\subsection{Gauge fixings}
Before proceeding to solve the super pentagon identity, we first briefly discuss some generalities on the gauge fixings. As we have seen in Section \ref{sec:defining_properties}, for every trivalent junction, one is free to change the basis of the junction vector space. When the junction vector space is one-dimensional, this corresponds to a simple rescaling of the graph by a complex number,
\begin{align}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\node (l3) at (0, -1.3) {$c$};
\draw [line] (0,0) -- (.87,.5) ;
\node (l2) at (1.07,.7) {$b$};
\draw [line] (0,0) -- (-.87,.5) ;
\node (l1) at (-1.07,.7) {$a$};
\draw (0,0)\dotsol {}{};
\node at (0.3,-0.2) {$\alpha$};
\end{tikzpicture}
\end{gathered}
\quad\to\quad g^{a,b,c}_{\alpha}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\node (l3) at (0, -1.3) {$c$};
\draw [line] (0,0) -- (.87,.5) ;
\node (l2) at (1.07,.7) {$b$};
\draw [line] (0,0) -- (-.87,.5) ;
\node (l1) at (-1.07,.7) {$a$};
\draw (0,0)\dotsol {}{};
\node at (0.3,-0.2) {$\alpha$};
\end{tikzpicture}
\end{gathered}
\end{align}
where $a,\, b,\, c,\,$ label the type of TDLs and $\alpha=0$ or $1$ for the bosonic or fermionic junction. If the junction vector space is multi-dimensional, say
\ie
V_{a,b,c}=\mathbb C^{m|n}\,,
\fe
the above scaling factor will be promoted to a $m\times m$ or $n\times n$ matrices for $\alpha=0$ or $1$, i.e.
\begin{align}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\node (l3) at (0, -1.3) {$c$};
\draw [line] (0,0) -- (.87,.5) ;
\node (l2) at (1.07,.7) {$b$};
\draw [line] (0,0) -- (-.87,.5) ;
\node (l1) at (-1.07,.7) {$a$};
\draw (0,0)\dotsol {}{};
\node at (0.6,-0.2) {\footnotesize$\alpha\,, i$};
\end{tikzpicture}
\end{gathered}
\quad\to\quad \left(g^{a,b,c}_{\alpha}\right)^{i}_j
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\node (l3) at (0, -1.3) {$c$};
\draw [line] (0,0) -- (.87,.5) ;
\node (l2) at (1.07,.7) {$b$};
\draw [line] (0,0) -- (-.87,.5) ;
\node (l1) at (-1.07,.7) {$a$};
\draw (0,0)\dotsol {}{};
\node at (0.6,-0.2) {\footnotesize$\alpha\,, j$};
\end{tikzpicture}
\end{gathered}
\end{align}
where the addition indexes $i,\, j$ label the multiplicities. Correspondingly, the F-matrices for the following graph
\begin{align}
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line] (-1,1) -- (-2,2);
\node at (-2.3, 2.3) {$a$};
\draw [line] (-1,1) --(0,2);
\node at (0.3, 2.3) {$b$};
\draw [line] (0,0) --(2,2);
\node at (2.3, 2.3) {$c$};
\draw [line] (0,0) -- (0,-1.5);
\node at (0.3, -1.8) {$d$};
\draw [line] (0,0) -- (-1,1) ;
\node at (-0.3, 0.7) {$e$};
\draw (0,0)\dotsol {}{};
\draw (-1,1)\dotsol {}{};
\node at (-1.8,0.6) {\footnotesize$\alpha,\, i$};
\node at (-0.8,-0.4) {\footnotesize$\beta,\, j$};
\end{tikzpicture}
\end{gathered}
\quad=\quad
\sum_{f,{\delta,\gamma}}\sum_{k,l}
\begin{gathered}
\begin{tikzpicture}[scale=.50]
\draw [line] (-2,2) -- (0,0);
\node at (-2.3, 2.3) {$a$};
\draw [line] (0,2) --(1,1);
\node at (0.3, 2.3) {$b$};
\draw [line] (1,1) --(2,2);
\node at (2.3, 2.3) {$c$};
\draw [line] (0,0) -- (1,1);
\node at (0.3, 0.7) {$f$};
\draw [line] (0,0) -- (0,-1.5) ;
\node at (0.3, -1.8) {$d$};
\draw (0,0)\dotsol {}{};
\draw (1,1)\dotsol {}{};
\node at (1.7,0.6) {\footnotesize$\gamma,\,k$};
\node at (0.7,-0.4) {\footnotesize$\delta,\,l$};
\end{tikzpicture}
\end{gathered}
\mathcal F^{a,b,c;\, \alpha, \beta}_{d;\, \delta,\gamma}(e, f)^{i,j}_{k,l}\,,
\end{align}
will transform as
\begin{align}
\mathcal F^{a,b,c;\, \alpha, \beta}_{d;\, \delta,\gamma}(e, f)^{i,j}_{k,l}\longrightarrow\,
\sum_{i^\prime,j^\prime,k^\prime,l^\prime}
\left(g^{b,c,f}_\gamma\right)^{k^\prime}_k\cdot
\left(g^{a,f,d}_\delta\right)^{l^\prime}_l\cdot
\left(g^{a,b,e}_\alpha\right)^{-1\,i}_{i^\prime}\cdot
\left(g^{e,c,d}_\beta\right)^{-1\,j}_{j^\prime}
\,\mathcal F^{a,b,c;\, \alpha, \beta}_{d;\, \delta,\gamma}(e, f)^{i^\prime,j^\prime}_{k^\prime,l^\prime}\,.
\label{eqn:F_gauge}
\end{align}
In section \ref{sec:Cq2F}, we will solve the super pentagon identities of the $\cC^2_{\rm q}$ categories, which have multiplicity two for both the bosonic and fermionic fusion channels. For their gauge fixings, we will present a detailed discussion in App.\ref{sec:F-symbols_C2q}. In the discussion below, for simplicity, we focus on the case of multiplicity one. In this case, the F-matrices transform as
\begin{align}
\mathcal F^{a,b,c;\, \alpha, \beta}_{d;\, \delta,\gamma}(e, f)^{i,j}_{k,l}\longrightarrow\, \frac{g^{b,c,f}_\gamma\cdot g^{a,f,d}_\delta}{g^{a,b,e}_\alpha\cdot g^{e,c,d}_\beta}\,\mathcal F^{a,b,c;\, \alpha, \beta}_{d;\, \delta,\gamma}(e, f)\,.
\label{eqn:F_scaling}
\end{align}
It is easy to verify that the super pentagon equations (\ref{eqn:super_pentagon}) are homogeneous under the scaling \eqref{eqn:F_scaling}. Therefore, we need to first choose some gauge conditions to remove the scaling redundancies, which in turn helps us to fix some components of the F-matrices to certain values.
Clearly, if two components of the F-matrices share the same scaling factor, one can only use the factor to fix one of the components, while the other one would be later determined by the super pentagon identities. Given a set of trivalent junctions and the F-matrices associated with them, we need to determine a maximal set of components of the F-matrices that can be gauge fixed. The gauge transformation \eqref{eqn:F_scaling} can be schematically written as
\begin{align}
\mathcal F_\mu\longrightarrow \left(\prod_{i}^n g_i^{\mu_i}\right) \mathcal F_\mu\,,
\end{align}
where $i$ denotes the collection of the indices $({a,b,c;\, \alpha})$, and $\mu$ collectively denotes the indices of the F-matrices. The exponent $\mu_i$ can only take values in $\{-2,\,-1,\,0,\,1,\,2\}$.
Therefore, for each F-matrix $\cF_\mu$, we can assign a ``gauge vector" $\vec v$ to it,
\begin{align}
\vec v_\mu=(\mu_1,\,\mu_2,\,\dots,\, \mu_n)\,.
\end{align}
The gauge vectors for all the F-matrices form a matrix $\mathcal G=\{\vec v_\mu\}$.
The maximal number of gauges we can choose is given by the rank of $\mathcal G$.
In practice, we can choose rank($\mathcal G$) number of components of the F-matrices with linearly independent gauge vectors,
and fix them to some convenient values. In the categories $\mathcal C^{(1,0)}_{\rm m}$ and $\mathcal C^{(0,1)}_{\rm m}$, there are three linearly independent gauge vectors, and thus we fix three components in the F-matrices. In $\mathcal C^{(1,1)}_{\rm m}$, there are four to fix. At last, in $\mathcal C^{0}_{\rm q}$, there are five to fix, which has been discussed throughout in Section \ref{sec:universal_sector}, see \eqref{eqn:gf_abc} and \eqref{eqn:gf_f12}. The overall scaling ($x_1=x_2=y_1=y_2=z_1=z_2$ in \eqref{eqn:universal_gauge}) actually does not change any F-matrix.
\subsection{m-type:}
Let us present the F-symbols of the super fusion categories $\mathcal C_{\rm m}^{(n_{\rm b}, n_{\rm f})}$.
\subsubsection{${\mathcal C}_{\rm m}^{(0, 0)}$ and ${\mathcal C}_{\rm m}^{(1, 0)}$}
In these two cases, they are ordinary fusion categories with two objects, which have been studied in \cite{Chang:2018iay}. The fusion category ${\mathcal C}_{\rm m}^{(0, 0)}$ has two solutions corresponding to the (non-)anomalous $\mathbb Z_2$ symmetry.
To compare with the fermionic $\mathcal C_{\rm m}^{(0, 1)}$ later, we here highlight the non-trivial F-moves of the H-junctions in the bosonic $\mathcal C_{\rm m}^{(1, 0)}$,
\ie\label{eqn:bosonic_Fibonacci}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\quad=\quad
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-2,2) -- (0,0);
\draw [line] (0,2) --(1,1);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (1,1);
\draw [line,dotted] (0,0) -- (0,-1) ;
\end{tikzpicture}
\end{gathered}\,,
\qquad\qquad
&\hspace{-.5cm}\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\end{pmatrix}
\quad=\quad
\begin{pmatrix}
\zeta^{-1} & 1 \\
\zeta^{-1} & -\zeta^{-1}
\end{pmatrix}
\cdot
\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\fe
where $\zeta=\frac{1\pm \sqrt{5}}{2}$.
\subsubsection{${\cal C}_{\rm m}^{(0, 1)}$}
Our first non-trivial super-fusion category with a m-type TDL has the fusion rule,
\ie
W^2=I+1_f\,W\,.
\fe
Besides the universal sector, we solved the rest of the super-pentagon equations induced by the above fusion rule. We found two consistent solutions and determines the F-moves in the following H-junctions,
\ie\label{eqn:fermionic_Fibonacci}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad=\quad -
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-2,2) -- (0,0);
\draw [line] (0,2) --(1,1);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (1,1);
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw (1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
\quad\quad\quad\quad
\hspace{-.5cm}\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (-1,1)\dotsol {}{};
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}
\quad=\quad
\begin{pmatrix}
\zeta^{-1} & 1 \\
-\zeta^{-1} & \zeta^{-1}
\end{pmatrix}
\cdot
\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\fe
where the dots at the vertices denote the fermionic junction vector in the junction vector space. In addition, we also used the gauge freedom,
\ie
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad z\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,dotted] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,
\fe
to normalize the upper right entry $\mathcal F^{W, W, W}_W(I, W)$ to $1$.
\subsubsection{$\mathcal C_{\rm m}^{(1, 1)}$}
In a similar fashion, for the super-category $\mathcal C_{\rm m}^{(1, 1)}$ with fusion rule,
\ie
W^2=I+(1_{\rm b}+1_{\rm f})\,W\,,
\fe
we find two consistent solutions to the super pentagon equations. Besides the universal sector, we spell out these F-moves as follows:
\ie
&\hspace{-.5cm}\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\cdot
\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line,dotted] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,\\ \\
&\hspace{-.5cm}\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (0,0)\dotsol {}{};
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (-1,1) ;
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}
=
\begin{pmatrix}
\frac{1}{w} & -\frac{w}{w+1} & 1 & 0 & 0 \\
\frac{1-w}{w} & \frac{1}{w+1} & 1 & 0 & 0 \\
-\frac{1}{w} & -\frac{1}{2} & \frac{1-w}{2 w} & 0 & 0 \\
0 & 0 & 0 & -\frac{w}{w+1} & -\frac{w}{w+1} \\
0 & 0 & 0 & -\frac{w}{w+1} & \frac{w}{w+1} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line,dotted] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,
\label{Cm11Fmoves},
\fe
where $w=1\pm \sqrt{2}$ and we have once again used up the gauge freedoms,
\ie
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}
\quad&\to\quad g_0\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\end{tikzpicture}
\end{gathered}\,,&
\quad\quad
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\quad&\to\quad
g_1\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line] (0,0) -- (0,-1) ;
\draw [line] (0,0) -- (.87,.5) ;
\draw [line] (0,0) -- (-.87,.5) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
\fe
to normalize the 13 and 23 entries of $\mathcal F^{W, W, W}_W$ to $1$. Using the above F-moves, we compute the loop expectation value
\ie\label{eqn:Cm11_vevW}
\vev{W}_{\bR^2}=w\,.
\fe
We would like to further remark that the first solution is gauge equivalent to the F-moves in the super fusion category obtained by the fermion condensation of the $R_\bC( \widehat{ so(3)}_6)$ fusion category given in \cite{Zhou:2021ulc}.
\subsection{q-type:}
We present the F-symbols of the super fusion categories $\mathcal C_{\rm q}^n$.
\subsubsection{$\mathcal C_{\rm q}^0$}
\label{sec:Cq0F}
$\mathcal C_{\rm q}^0$ is the first non-trivial super-fusion category of rank-2 containing a q-type TDL, with the following fusion rule,
\begin{align}
W^2=(1_{\rm b}+1_{\rm f})I\,.
\end{align}
The q-type TDL is always unoriented for the category of rank-2. In Section \ref{sec:universal_sector}, we have solved the F-matrices $\mathcal F^{W, W, I}_I$, $\mathcal F^{W, I, W}_I$, $\mathcal F^{W, I, I}_W$, $\mathcal F^{I, W, W}_I$, $\mathcal F^{I, W, I}_W$ and $\mathcal F^{I, I, W}_W$. There are two gauge inequivalent solutions given in \eqref{eqn:universal_unoriented_sols}. With respect to each of the above solutions, one can further find two inequivalent solutions to the F-matrix $\mathcal F^{W,
W, W}_W$. So overall there are \emph{four} inequivalent solutions. We list them as follows:
\ie
&\hspace{-.5cm}\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (-1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (-1,1) ;
\draw (0,0)\dotsol {}{};
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (-1,1) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (-1,1) -- (-2,2);
\draw [line] (-1,1) --(0,2);
\draw [line] (0,0) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (-1,1) ;
\draw (-1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}
=
\begin{large}
\frac{\kappa}{\sqrt{2}}\begin{pmatrix}
1 & \gamma & 0 & 0 &\\
\gamma & -\gamma^2 & 0 & 0 & \\
0 & 0 & 1 & \gamma & \\
0 & 0 & \gamma & -\gamma^2 & \\
\end{pmatrix}
\end{large}
\cdot
\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (1,1) ;
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (1,1) ;
\draw (0,0)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\\
\begin{gathered}
\begin{tikzpicture}[scale=.3]
\draw [line] (0,0) -- (-2,2);
\draw [line] (1,1) --(0,2);
\draw [line] (1,1) --(2,2);
\draw [line] (0,0) -- (0,-1);
\draw [line, dotted] (0,0) -- (1,1) ;
\draw (1,1)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\fe
for $\kappa=\pm 1$ and $\gamma=e^{\frac{i \pi}{4}},\,e^{\frac{3i \pi}{4}}$.\footnote{The F-moves of the $\cC^0_{\rm q}$ categories are also obtained in \cite{Wang:thesis}. We thank Yanzhen Wang for sharing his undergraduate thesis with us.} Using the above F-moves, we compute the loop expectation value
\ie
\vev{W}_{\bR^2}= \sqrt{2}\kappa\,.
\fe
The solution with $\kappa=1$ and $\gamma=e^{\frac{3\pi i}{4}}$ is gauge equivalent to the F-moves in the super fusion category obtained by the fermion condensation of the $\bZ_2$ $\rm TY$ fusion category given in \cite{Zhou:2021ulc}. Actually, all of the four solutions can be obtained via the fermionic anyon condensation of the 3d Ising TQFT. In the condensation picture, there are two ways \cite{Kitaev_2006} to braid the anyons $\sigma$ and $\psi$, corresponding to the duality $\mathcal N$-line and the $\mathbb Z_2$ $\eta$-line in the 2d Ising CFT. The two braidings are precisely encoded in the values of $\gamma$, while the Frobenius-Schur indicator $\kappa$ is inherited from its bosonic parent. We thus have overall four ways of fermionic condensations.\footnote{We thank Qing-Rui Wang for a discussion on this result from the perspective of fermionic condensation.}
\subsubsection{$\mathcal C_{\rm q}^2$}
\label{sec:Cq2F}
Our last example is the $\cC^2_{\rm q}$ super fusion categories, whose fusion rule is given by
\ie
W^2=({\rm 1_b+1_f})I+2W\,,
\fe
where $W$ is a q-type TDL. We separate the non-trivial F-symbols in the following four sectors classified by the number of their external trivial TDL:
\ie
&\mathcal S_4:\ \ \mathcal F^{I,I,I}_I\,;\notag\\
&\mathcal S_2:\ \ \mathcal F^{W, W, I}_I\,,\ \ \mathcal F^{W, I, W}_I\,,\ \ \mathcal F^{W, I, I}_W\,,\ \ \mathcal F^{I, W, W}_I\,,\ \ \mathcal F^{I, W, I}_W\,,\ \ {\rm and}\ \ \ \mathcal F^{I, I, W}_W\,;\notag\\
&\mathcal S_1:\ \ \mathcal F^{I, W, W}_W\,,\ \ \mathcal F^{W, I, W}_W\,,\ \ \mathcal F^{W, W, I}_W\,,\ \ {\rm and}\ \ \ \mathcal F^{W, W, W}_I\,;\notag\\
&\mathcal S_0:\ \ \mathcal F^{W, W, W}_W\,.
\fe
Overall there are $361$ variables in the F-symbols constrained by 9601 super-pentagons as well as the projection conditions. We find four solutions to these super-pentagons, where one of them is gauge equivalent to the solution obtained by fermionic condensation of the (bosonic) $\frac{1}{2}E_6$ fusion category as discussed in \cite{Zhou:2021ulc}. Now we spell them out in detail. One first needs to fix the gauge freedoms of these F-symbols and solve the projection conditions. We present the discussion of those in Appendix \ref{sec:F-symbols_C2q}. The trivial F-symbol $\mathcal F^{I,I,I}_I=1$ in $\mathcal S_4$ as usual; The $\mathcal S_2$ sector is the universal sector for q-type TDLs that has been discussed before. We thus start from the sector $\mathcal S_1$. The F-matrices take the form \eqref{eqn:C^2_Q_single_1} and \eqref{eqn:C^2_Q_single_234}. Using the solution (\ref{eq:Cq2_gauge1}),
\ie
\sigma=
\begin{pmatrix}
1\ \ & 0\, \\
0\ \ & 1\,
\end{pmatrix}\,,\ \
\rho=
\begin{pmatrix}
0\ \ & 1\,\\
-1\ \ & 0\,
\end{pmatrix}
\,,\ \ {\rm and}\ \
\tau=
\begin{pmatrix}
1\ \ & 0\,\\
0\ \ & -1\,
\end{pmatrix}\,,
\fe
the super pentagon equations give
\ie
&f_{7b}=\frac{1}{2}
\begin{pmatrix}
1\ \ & 0\, \\
0\ \ & 1\,
\end{pmatrix}\,,\ \
{\rm and}\ \ \
f_{7f}=\frac{1}{2}
\begin{pmatrix}
0\ \ & 1\,\\
-1\ \ & 0\,
\end{pmatrix}\,,\notag\\
&f_{8b}=\frac{1}{2}
\begin{pmatrix}
1\ \ & 0\, \\
0\ \ & 1\,
\end{pmatrix}\,,\ \
{\rm and}\ \ \
f_{8f}=\frac{1}{2}
\begin{pmatrix}
0\ \ & 1\,\\
-1\ \ & 0\,
\end{pmatrix}\,,\notag\\
&f_{9b}=f_{9f}=\frac{1}{2}
\begin{pmatrix}
1\ \ & 0\, \\
0\ \ & 1\,
\end{pmatrix}\,,\notag\\
&f_{10b}=\frac{1}{8}
\begin{pmatrix}
\xi \ \ & -\frac{2}{\xi} \,\\
-\xi \ \ & -\frac{2}{\xi} \,
\end{pmatrix}+\frac{\gamma^2}{8}
\begin{pmatrix}
\frac{2}{\xi}\ \ & \xi\,\\
-\frac{2}{\xi} \ \ & \xi\,
\end{pmatrix}\,,\ \
{\rm and}\ \ \
f_{10f}=\frac{1}{8}
\begin{pmatrix}
\frac{2}{\xi} \ \ & \xi\,\\
\frac{2}{\xi} \ \ & -\xi \,
\end{pmatrix}+\frac{\gamma^2}{8}
\begin{pmatrix}
-\xi\ \ & \frac{2}{\xi} \,\\
-\xi \ \ & -\frac{2}{\xi} \,
\end{pmatrix}\,.
\fe
And finally, for the most non-trivial sector $\mathcal S_0$, the F-matrices take the form \eqref{eqn:1/2E6_q_all}. Solving the super pentagon equations, we find the matrices $\mathcal F_b$ and $\mathcal F_f$ as,
\ie
&\mathcal F_b=\frac{1}{2}
\begin{pmatrix}
\frac{\xi}{2}\ \ & 0\ \ & 0\ \ & -\frac{1+\gamma^2}{2} \ \ & 2\ \ & 2\gamma\ \,\\
0\ \ & \frac{1+\gamma^2}{2}\ \ & \frac{\xi}{2} \ \ & 0\ \ & -2\gamma^2\ \ & -2\gamma^{-1}\ \,\\
-\frac{1+\gamma^2}{2}\ \ & 0\ \ & 0 \ \ & -\frac{\xi}{2}\ \ & -2\gamma^2 \ \ & 2\gamma^{-1}\ \,\\
0\ \ & -\frac{\xi}{2}\ \ & \frac{1+\gamma^2}{2}\ \ & 0 \ \ & 2\ \ & -2\gamma\ \,\\
-\frac{1-\gamma^2}{4}-\frac{\xi}{4}\ \ & 0\ \ & 0\ \ & \frac{1+\gamma^2}{4}+\frac{\gamma^2\,\xi}{4}\ \ & -\xi \ \ & -\gamma\,\xi\ \,\\
0\ \ & \frac{\gamma+\gamma^{-1}}{4}+\frac{\gamma\,\xi}{4}\ \ & \frac{-\gamma+\gamma^{-1}}{4}+\frac{\gamma^{-1}\xi}{4}\ \ & 0\ \ & -\gamma\,\xi\ \ & \gamma^2\xi\ \,
\end{pmatrix}\,,\notag
\\\notag\\
&\mathcal F_f=\frac{1}{2}
\begin{pmatrix}
0\ \ & \frac{1+\gamma^2}{2}\ \ & \frac{\xi}{2}\ \ & 0 \ \ & 2\gamma^{-1}\ \ & 2\gamma^2\ \,\\
-\frac{\xi}{2}\ \ & 0\ \ & 0 \ \ & \frac{1+\gamma^2}{2} \ \ & -2\gamma\ \ & -2\ \,\\
0\ \ & \frac{\xi}{2}\ \ & -\frac{1+\gamma^2}{2} \ \ & 0\ \ & -2\gamma \ \ & 2\ \,\\
-\frac{1+\gamma^2}{2}\ \ & 0\ \ & 0\ \ & -\frac{\xi}{2} \ \ & 2\gamma^{-1}\ \ & -2\gamma^2\ \,\\
0\ \ & -\frac{\gamma+\gamma^{-1}}{4}-\frac{\gamma\,\xi}{4}\ \ & \frac{\gamma-\gamma^{-1}}{4}-\frac{\gamma^{-1}\xi}{4}\ \ & 0\ \ & \gamma^2\xi \ \ & -\gamma\,\xi\ \,\\
-\frac{1-\gamma^2}{4}-\frac{\xi}{4}\ \ & 0\ \ & 0\ \ & \frac{1+\gamma^2}{4}+\frac{\gamma^2\xi}{4}\ \ & -\gamma\,\xi\ \ & -\xi\ \,
\end{pmatrix}\,,
\fe
where we have defined $\xi=1\pm\sqrt{3}$, and $\gamma=e^{\frac{\pi i}{4}}$ or $e^{\frac{3\pi i}{4}}$ as before.
Overall we have two choices of $\xi$'s and $\gamma$'s, and thus four solutions.
From the F-moves, we compute the loop expectation value
\ie
\vev{W}_{\bR^2}=-\frac{2}{\xi}\,.
\fe
Finally, the solution with $\xi=1-\sqrt{3}$ and $\gamma=e^{\frac{3\pi i}{4}}$ is gauge equivalent to the F-moves in the super fusion category obtained by the fermion condensation of the $\frac{1}{2}E_6$ fusion category given in \cite{Zhou:2021ulc}.
\section{Spin selection rules}
\label{sec:spin_selection_rules}
Following the analysis in \cite{Chang:2018iay}, the fractional part of the spins of the states in the bosonic or fermionic defect Hilbert space ${\cal H}^{\rm b}_\cL$ or ${\cal H}^{\rm f}_\cL$ can be determined by a sequence of modular T transformations and the F-moves, schematically as
\ie\label{eqn:FTchain}
\begin{tikzcd}
&
\arrow[dl, "T^{-2}"]
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (1.25, 0.55) -- (1, 1);
\draw (1.2,0.575) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\draw (.915,0.875) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(2,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line] (1.3, .9) -- (.8, 0.54);
\draw (1.2,0.755) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\draw (.7,0.51) node{\rotatebox[origin=c]{60}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
\arrow[l, "F^{-1}"]
&
\arrow[l, ""]
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.5) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.5) ;
\draw (0.875,1.1) node{\rotatebox[origin=c]{40}{\footnotesize $\times$}} ;
\draw (1,.8) node{\rotatebox[origin=c]{0}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
\arrow[dd, "F"]
\\
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\\
&
\arrow[ul, "T^2"]
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(2,.7) .. controls (1,.7) and (1,1.3) .. (0,1.3)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.75, 0.55) -- (1, 1);
\draw (0.65,0.53) node{\rotatebox[origin=c]{70}{\footnotesize $\times$}} ;
\draw (0.95,0.835) node{\rotatebox[origin=c]{60}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
&
\arrow[l, "F"]
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (0,1.3)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line] (0.7, .9) -- (1.2, 0.54);
\draw (0.6,0.74) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\draw (1.125,0.525) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
&
\arrow[l, ""]
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.5) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.5) ;
\draw (0.875,0.765)
node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\draw (1,1.05) node{\rotatebox[origin=c]{0}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
\end{tikzcd}
\fe
In the following subsections, we derive the spin selection rules for the super fusion categories ${\cal C}^{(m,n)}_{\rm m}$ and ${\cal C}^n_{\rm q}$. For simplicity, we will suppress the $\times$-marks and the trivial lines in the following TDL graphs, which could be recovered by comparing with the graphs in \eqref{eqn:FTchain}.
\subsection{${\cal C}_{\rm m}^{(1,0)}$}
Let us first focus on the bosonic Fibonacci fusion category ${\cal C}_{\rm m}^{(1,0)}$ with the fusion rule
\ie
W^2=I+W\,.
\fe
For any pair of states $\ket{\psi}, \ket{\psi'}\in {\cal H}^{\rm b}_\cL$ or ${\cal H}^{\rm f}_\cL$ with equal conformal weight, we consider the matrix element of the cylinder propagator
\ie
\bra{\psi'}q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{\tilde c}{24}}\ket{\psi}
\quad=\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}\,.
\fe
Performing modular $T^2$ transformation, and applying the F-move in \eqref{eqn:bosonic_Fibonacci}, we find
\ie\label{eqn:T^2_fibo_spin_selec}
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad =\quad
\zeta^{-1}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad +\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}\,.
\fe
Similarly, considering a modular $T^{-2}$ transformation on the matrix element $\bra{\psi'}q^{L_0-\frac{c}{24}}\bar q^{\widetilde L_0-\frac{\tilde c}{24}}\ket{\psi}$ gives
\ie\label{eqn:T^2_fibo_spin_selec_hp_r}
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(0,1.3) .. controls (1,1.3) and (1,.7) .. (2,.7)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad =\quad
\zeta^{-1}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad +\quad
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}\,.
\fe
Applying the F-move again on the last graph in above, we obtain
\ie\label{eqn:T^2_fibo_spin_selec_hp}
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}
\quad =\quad
\zeta^{-1}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,0.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
\quad -\quad
\zeta^{-1}~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
\,.
\fe
In summary, we find the following three equations
\ie\label{eqn:bosonic_fibo_spin_selec_eqns}
e^{4\pi i (s+\frac{1}{2}\sigma)} = \zeta^{-1} + \widehat W_+\,,
\quad
e^{-4\pi i (s+\frac{1}{2}\sigma)}=\zeta^{-1}+\widehat W_-\,,
\quad
\widehat W_- = \zeta^{-1} e^{2 \pi i (s+\frac{1}{2}\sigma)}-\zeta^{-1}\widehat W_+\,.
\fe
where $s$ and $\sigma$ are the spin and the $\bZ_2$-grading of the states $\ket\psi$ and $\ket{\psi'}$, and $\widehat W_\pm$ are the lassoing linear operators acting on ${\cal H}_W$ defined by an $W$ line wrapping the spatial circle splitting over the temporal $W$ line as in the last graph in \eqref{eqn:T^2_fibo_spin_selec} and \eqref{eqn:T^2_fibo_spin_selec}, respectively.
The solution to \eqref{eqn:bosonic_fibo_spin_selec_eqns} gives the spin selection rules: the states in the bosonic defect Hilbert space ${\cal H}^{\rm b}_W$ should have spins in
\ie\label{fibo_Hb}
s\in \bZ + \begin{cases}
0,\,\pm \frac{2}{5}\quad {\rm for}\quad \zeta=\frac{1+\sqrt{5}}{2}\,,
\\
0,\,\pm \frac{1}{5}\quad {\rm for}\quad \zeta={1-\sqrt{5}\over 2}\,,
\end{cases}
\fe
and the states in the fermionic defect Hilbert space ${\cal H}^{\rm f}_W$ should have spins in
\ie\label{fibo_Hf}
s\in \bZ + \begin{cases}
\frac{1}{2},\,\pm \frac{1}{10}\quad {\rm for}\quad \zeta=\frac{1+\sqrt{5}}{2}\,,
\\
\frac{1}{2},\,\pm \frac{3}{10}\quad {\rm for}\quad \zeta=\frac{1-\sqrt{5}}{2}\,.
\end{cases}
\fe
The super fusion category ${\cal C}_{\rm m}^{(1,0)}$ of $\zeta=\frac{1+\sqrt{5}}{2}$ is realized by the TDLs $\{I,W\}$ in the $m=4$ fermionic minimal model \cite{Kikuchi:2022jbl}, and we find that the spin selection rules \eqref{fibo_Hb} and \eqref{fibo_Hf} are indeed satisfied by the defect Hilbert space ${\cal H}_W$.
\subsection{${\cal C}_{\rm m}^{(0,1)}$}
Let us carry out the same manipulation in the fermionic Fibonacci fusion category $\cC_{\rm m}^{(0,1)}$.
We first perform modular $T^2$ transformation, upon which we apply for the F-move in \eqref{eqn:fermionic_Fibonacci}, and we find
\ie\label{eqn:T2_fFf}
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad =\quad
\zeta^{-1}~&&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad +\quad
&&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1,1.3)\dotsol {}{};
\draw (1,.7)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(0,1.3) .. controls (1,1.3) and (1,.7) .. (2,.7)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad =\quad
\zeta^{-1}~&&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad -\quad
&&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1,1.3)\dotsol {}{};
\draw (1,.7)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1,1.3)\dotsol {}{};
\draw (1,.7)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\quad =\quad
-\zeta^{-1}~&&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
\quad +\quad
\zeta^{-1}~&&
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1,1.3)\dotsol {}{};
\draw (1,.7)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\,,
\fe
where the sign differences between \eqref{eqn:T2_fFf} and \eqref{eqn:T^2_fibo_spin_selec}, \eqref{eqn:T^2_fibo_spin_selec_hp}, \eqref{eqn:T^2_fibo_spin_selec_hp_r} are due to the sign differences in the F-moves \eqref{eqn:fermionic_Fibonacci} and \eqref{eqn:bosonic_Fibonacci} and also minus signs coming from moving the fermionic junction vectors around the spatial circle. We find the equations
\ie\label{eqn:fermionic_fibo_spin_selec_eqns}
e^{4\pi i (s+\frac{1}{2}\sigma)} = \zeta^{-1} + \widehat W_+\,,
\quad
e^{-4\pi i (s+\frac{1}{2}\sigma)}=\zeta^{-1}+\widehat W_-\,,
\quad
\widehat W_- = -\zeta^{-1} e^{2 \pi i (s+\frac{1}{2}\sigma)}-\zeta^{-1}\widehat W_+\,,
\fe
which give the spin selection rules
\ie\label{eqn:Cm01_SSR}
s+\frac{\sigma}{2}\in \bZ + \begin{cases}
\frac{1}{2},\,\pm \frac{1}{10}\quad {\rm for}\quad \zeta=\frac{1+\sqrt{5}}{2}\,,
\\
\frac{1}{2},\,\pm \frac{3}{10}\quad {\rm for}\quad \zeta=\frac{1-\sqrt{5}}{2}\,.
\end{cases}
\fe
Note interesting that the spin selection rules \eqref{eqn:Cm01_SSR} are related to \eqref{fibo_Hb} and \eqref{fibo_Hf} with the fermion parity $(-1)^\sigma$ flipped. We do not know any CFT that realizes the $\cC^{(0,1)}_{\rm m}$ super fusion categories.
\subsection{${\cal C}_{\rm m}^{(1,1)}$}
Let us apply the manipulations in \eqref{eqn:FTchain} to the super fusion category ${\cal C}_{\rm m}^{(1,1)}$:
\ie
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (1.2,.6) .. controls (1,.9) and (1,.9) .. (1,1) ;
\end{tikzpicture}
\end{gathered}
~=
\frac{1}{w}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
~-\frac{w}{w+1}~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
~+~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(0,1.3) .. controls (1,1.3) and (1,.7) .. (2,.7)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.7,.6) -- (0.7,1.1) ;
\end{tikzpicture}
\end{gathered}
~=
\frac{1}{w}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
~ -\frac{w}{w+1}
~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}
~ -
~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (.5,1) .. (0,1)
(1,2) .. controls (1,1.5) and (1.5,1) .. (2,1) ;
\end{tikzpicture}
\end{gathered}
~=
\frac{1}{w}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
~-\frac{w}{w+1}
~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
~ +
~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;-
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}
~=
\frac{1-w}{w}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
~+\frac{1}{w+1}
~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
~ +
~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;-
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,,
\\
&
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
~=
-\frac{1}{w}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
~ -\frac{1}{2}
~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
~+\frac{1-w}{2w}~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\,,
\fe
which gives the equations
\ie\label{eqn:bosonic_fermi_fibo_spin_selec_eqns}
e^{4\pi i (s+\frac{1}{2}\sigma)}& =\frac{1}{w} - \frac{w}{w+1} \widehat W_+ -\widehat W_+^f\,,
\\
e^{-4\pi i (s+\frac{1}{2}\sigma)}&=\frac{1}{w}-\frac{w}{w+1}\widehat W_- +\widehat W_-^f\,,
\\
e^{-2\pi i (s+\frac{1}{2}\sigma)}&=\frac{1}{w}-\frac{w}{w+1}\widehat W_+ +\widehat W_+^f\,,
\\
\widehat W_- &= \frac{1-w}{w} e^{2 \pi i (s+\frac{1}{2}\sigma)}+\frac{1}{w+1}\widehat W_++\widehat W_+^f\,,
\\
\widehat W_-^f &= -\frac{1}{w} e^{2 \pi i (s+\frac{1}{2}\sigma)}-\frac{1}{2}\widehat W_++\frac{1-w}{2w}\widehat W_+^f\,.
\fe
Solving the above equations, we find
\ie
s+\frac{\sigma}{2}\in \bZ + \left\{
0,\, \frac{1}{4},\,\frac{1}{2},\,\frac{3}{4}\right\}\quad {\rm for}\quad w=1\pm\sqrt{2}\,.
\fe
The spin selection rule is satisfied by the defect Hilbert space of the TDL $\cL_{\frac{3}{4},\frac{3}{4}}$ in the $m=7$ fermionic minimal model \cite{Kikuchi:2022gfi}. Furthermore, since the $m=7$ fermionic minimal model is unitary, it should realize the $\cC_{\rm m}^{(1,1)}$ category of $w=1+\sqrt{2}$ which has a positive loop expectation value \eqref{eqn:Cm11_vevW}.
\subsection{${\cal C}_{\rm q}^{0}$}
Let us consider the super fusion category ${\cal C}_{\rm q}^{0}$ with the F-moves given in Section \ref{sec:Cq0F}. Following the bottom sequence of the modular T transformations and the F-moves in \eqref{eqn:FTchain}, we find
\ie
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(2,.7) .. controls (1,.7) and (1,1.3) .. (0,1.3)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.75, 0.55) -- (1, 1);
\draw (0.6,0.625) node{\rotatebox[origin=c]{70}{\footnotesize $\times$}} ;
\draw (0.915,0.85) node{\rotatebox[origin=c]{60}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
\quad &= \frac{\kappa}{\sqrt{2}}\quad
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (0,1.3)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.7, .9) -- (1.2, 0.54);
\draw (0.6,0.815) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\draw (1.05,0.64) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\end{tikzpicture}
\end{gathered}
\quad+\frac{\kappa\gamma}{\sqrt{2}}\quad
\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (0,1.3)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.7, .9) -- (1.2, 0.54);
\draw (0.6,0.815) node{\rotatebox[origin=c]{30}{\footnotesize $\times$}} ;
\draw (1.05,0.64) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\draw (0.7, .9)\dotsol {}{};
\draw (1.2, 0.54) \dotsol {}{} ;
\draw (0.7, .9) node [right]{\tiny\textcolor{red}{2}};
\draw (1.2, 0.54) node [below]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\\
&= \frac{\kappa}{\sqrt{2}}\quad\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
\quad - \frac{\kappa\gamma}{\sqrt{2}} \quad \begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,1.5) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.5) ;
\draw (0.85,0.85) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\draw (1,1.125) node{\rotatebox[origin=c]{0}{\footnotesize $\times$}} ;
\draw (1, .5)\dotsol {}{};
\draw (1, 1.5) \dotsol {}{};
\draw (1, .5) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.5) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}\,,
\fe
where again the black dots denote the 1d Majorana fermion on the q-type TDL. Next, we apply the rules in \eqref{eqn:1d_mf_action}, and find
\ie
&\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,1.5) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.5) ;
\draw (0.85,0.85) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\draw (1,1.125) node{\rotatebox[origin=c]{0}{\footnotesize $\times$}} ;
\draw (1, .5)\dotsol {}{};
\draw (1, 1.5) \dotsol {}{};
\draw (1, .5) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.5) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad =\gamma \quad
&\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,1.5) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.5) ;
\draw (0.85,0.85) node{\rotatebox[origin=c]{50}{\footnotesize $\times$}} ;
\draw (1,1.125) node{\rotatebox[origin=c]{0}{\footnotesize $\times$}} ;
\draw (1, .85)\dotsol {}{};
\draw (1, 1) \dotsol {}{};
\draw (1, .85) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
\quad =
-\gamma\quad\begin{gathered}
\begin{tikzpicture}[scale=1.25]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}\,.
\fe
The spin in the defect Hilbert space ${\cal H}_W$ should satisfy the spin selection rules
\ie
s=\frac{1}{2}{\mathbb Z}+\begin{cases}
\frac{7}{16}&{\rm for}~ \kappa=1,\,\gamma=e^{\frac{i\pi}{4}}\,,
\\
\frac{3}{16}&{\rm for}~ \kappa=-1,\,\gamma=e^{\frac{i\pi}{4}}\,,
\\
\frac{1}{16}&{\rm for}~ \kappa=1,\,\gamma=e^{\frac{3i\pi}{4}}\,,
\\
\frac{5}{16}&{\rm for}~ \kappa=-1,\,\gamma=e^{\frac{3i\pi}{4}}\,.
\end{cases}
\fe
The third (first) spin selection rule is satisfied by the $\bZ_2$ holomorphic (anti-holomorphic) fermion parity TDL $(-1)^{F_L}$ ($(-1)^{F_R}$) in the fermionic $m=3$ minimal model \cite{Ji:2019ugf,Kikuchi:2022jbl} and the anti-holomorphic (holomorphic) $\bZ_2$ R-symmetry TDL in the fermionic $m=4$ minimal model \cite{Kikuchi:2022jbl}.
\subsection{${\cal C}_{\rm q}^{2}$}
To compute the spin selection rule for ${\cal C}_{\rm q}^{2}$, we need to use the bosonic part of the F-move \eqref{eqn:1/2E6_q_all}, which is represented by the $10\times 10$ matrix
\ie
{\bf F}\equiv\begin{pmatrix}
1\otimes 1 & 0
\\
\tau^{-1}\otimes \rho^{-1} & 0
\\
0 & 1
\\
\end{pmatrix}
{\cal F}_b
\begin{pmatrix}
1\otimes 1 & \tau\otimes\sigma & 0
\\
0 & 0 & 1
\end{pmatrix}\,.
\fe
The inverse of this F-move is represented by the pseudoinverse of ${\bf F}$,
\ie
{\bf F}^+=\begin{pmatrix}
\frac{1}{2}(1\otimes 1) & 0
\\
\frac{1}{2}(\tau^{-1}\otimes \sigma^{-1}) & 0
\\
0 & 1
\\
\end{pmatrix}
{\cal F}_b^{-1}
\begin{pmatrix}
\frac{1}{2}(1\otimes 1) & \frac{1}{2}(\tau\otimes\rho) & 0
\\
0 & 0 & 1
\end{pmatrix}\,.
\fe
Let us define the following vectors
\ie
{\bf v}_1&=\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\\
{\bf v}_2&=\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\\
{\bf v}_3&=\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\draw[line, dotted] (.68,1.32) -- (1.32,.68) ;
\draw (.68,1.32)\dotsol {}{};
\draw (1.32,.68)\dotsol {}{};
\draw (.68,1.32) node [right]{\tiny\textcolor{red}{2}};
\draw (1.32,.68) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,,
\\
{\bf v}_4&=\begin{pmatrix}
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (.5,1) .. (0,1)
(1,2) .. controls (1,1.5) and (1.5,1) .. (2,1) ;
\end{tikzpicture}
\end{gathered}
&
\begin{gathered}
\begin{tikzpicture}[scale=.75]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (.5,1) .. (0,1)
(1,2) .. controls (1,1.5) and (1.5,1) .. (2,1) ;
\draw[line, dotted] (.68,.68) -- (1.32,1.32) ;
\draw (.68,.68)\dotsol {}{};
\draw (1.32,1.32)\dotsol {}{};
\draw (.68,.68) node [right]{\tiny\textcolor{red}{2}};
\draw (1.32,1.32) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
\end{pmatrix}\,.
\fe
Applying the F-moves, we find the equations
\ie\label{eqn:SSR_C2q_eq1}
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (2,.7)
(0,.7) .. controls (1,.7) and (1,1.3) .. (2,1.3)
(0,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (1.2,.6) .. controls (1,.9) and (1,.9) .. (1,1) ;
\end{tikzpicture}
\end{gathered}
~=
~({\bf F}^+ {\bf v}_1^T)_9\,,\quad \begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.3) and (1,.7) .. (0,.7)
(0,1.3) .. controls (1,1.3) and (1,.7) .. (2,.7)
(2,1.3) .. controls (1,1.3) and (1,1.7) .. (1,2) ;
\draw [line,dotted] (0.7,.6) -- (0.7,1.1) ;
\end{tikzpicture}
\end{gathered}~=~({\bf F} {\bf v}_2^T)_9\,,\quad {\bf v}_4^T= {\bf F}{\bf v}_3^T\,.
\fe
Using the equations \eqref{eqn:1d_mf_action} and \eqref{eqn:dot_action_unoriented2}, we also find
\ie\label{eqn:SSR_C2q_eq2}
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
~=~ \gamma~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}\,,
&&
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line,dotted] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}
~=~ \gamma^{-1}~\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\draw[line, dotted] (.68,1.32) -- (1.32,.68) ;
\draw (.68,1.32)\dotsol {}{};
\draw (1.32,.68)\dotsol {}{};
\draw (.68,1.32) node [right]{\tiny\textcolor{red}{2}};
\draw (1.32,.68) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}~=~\gamma~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (1.5,1) .. (2,1)
(1,2) .. controls (1,1.5) and (.5,1) .. (0,1) ;
\end{tikzpicture}
\end{gathered}\,,
&&
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (.5,1) .. (0,1)
(1,2) .. controls (1,1.5) and (1.5,1) .. (2,1) ;
\draw[line, dotted] (.68,.68) -- (1.32,1.32) ;
\draw (.68,.68)\dotsol {}{};
\draw (1.32,1.32)\dotsol {}{};
\draw (.68,.68) node [right]{\tiny\textcolor{red}{2}};
\draw (1.32,1.32) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}~=~\gamma^{-1}~
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) .. controls (1,.5) and (.5,1) .. (0,1)
(1,2) .. controls (1,1.5) and (1.5,1) .. (2,1) ;
\end{tikzpicture}
\end{gathered}\,,
\\
&\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\end{tikzpicture}
\end{gathered}
~=~(\tau\otimes \sigma)~ \begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,.7) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,1.3) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{1}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{2}};
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\end{tikzpicture}
\end{gathered}\,,
~&&
\begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}
~=~(\tau\otimes \rho)~ \begin{gathered}
\begin{tikzpicture}[scale=1]
\draw [line,lightgray] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) ;
\draw [line] (1,0) -- (1,2) ;
\draw [line] (1,1.3) .. controls (1,1) and (1.5,1) .. (2,1)
(0,1) .. controls (.5,1) and (1,1) .. (1,.7) ;
\draw (1,0.7)\dotsol {}{};
\draw (1,1.2)\dotsol {}{};
\draw (1, .7) node [right]{\tiny\textcolor{red}{2}};
\draw (1, 1.2) node [right]{\tiny\textcolor{red}{1}};
\end{tikzpicture}
\end{gathered}\,.
\fe
Solving the equations \eqref{eqn:SSR_C2q_eq1} and \eqref{eqn:SSR_C2q_eq2}, we find the spin selection rules
\ie
s+\frac{\sigma}{2}\in \bZ + \begin{cases}
0,\, \frac{1}{6},\, \frac{1}{2},\,\frac{2}{3}&{\rm for}~\xi=1+\sqrt{3},~\gamma=e^\frac{i\pi}{4}~{\rm or}~\xi=1-\sqrt{3},~\gamma=e^\frac{3i\pi}{4}\,,
\\
0,\, \frac{1}{3},\, \frac{1}{2},\,\frac{5}{6}&{\rm for}~\xi=1+\sqrt{3},~\gamma=e^\frac{3i\pi}{4}~{\rm or}~\xi=1-\sqrt{3},~\gamma=e^\frac{i\pi}{4}\,.
\end{cases}
\fe
The first (second) spin selection rule is satisfied by the TDLs in the exceptional $m=12$ ($m=11$) fermionic minimal model \cite{Kikuchi:wip}. Furthermore, since the $m=11$ and $12$ fermionic minimal models are unitary, they should realize the $\cC^2_{\rm q}$ super fusion categories of $\vev{W}_{\bR^2}=-\frac{2}{\xi}=1+\sqrt{3}$.
\section{Summary and discussions}
\label{sec:summary}
This paper is devoted to developing the theory of TDLs in fermionic CFTs.
\begin{enumerate}
\item We formulated the defining properties of TDLs in 2d fermionic CFTs, which are mostly carried over from those for 2d bosonic CFTs, except that the Hilbert spaces associated with junctions and endpoints of TDLs can now also host fermionic operators. In addition, there is a new type of simple TDLs (q-type TDLs), whose two-way junction vector spaces host fermionic operator of conformal weights $(h,\tilde h)=(0,0)$.
\item We derived several consequences from the defining properties and discussed their relation to super fusion categories. In particular, the F-moves (crossings) \eqref{eqn:F-move} of the TDLs are constrained by the projection condition \eqref{eqn:projection_F} and the super pentagon identity \eqref{eqn:super_pentagon}. The later differs from the pentagon identity in the bosonic CFT by an extra edge (the map $S^{4,3,k}_{2,1,i}$) in the commutative diagram \eqref{eqn:super_pentagon_diagram}, that exchanges the order of two junction vectors.
\item We gave a conjectural classification of the rank-2 super fusion categories. The nontrivial ones are $\cC_{\rm m}^{(0,0)}$, $\cC_{\rm m}^{(1,0)}$, $\cC_{\rm m}^{(0,1)}$, $\cC_{\rm m}^{(1,1)}$, $\cC_{\rm q}^0$, $\cC_{\rm q}^2$. We gave the F-moves explicitly for all of them.
\item We derived the spin selection rules for the defect Hilbert spaces of the TDLs in the aforementioned nontrivial super fusion categories, and discussed their realizations in the fermionic minimal models.
\end{enumerate}
Finally, we comment on the constraints from the TDLs on the RG flows between fermionic fixed points. A TDL is preserved along an RG flow if it commutes with the conformal primary that triggers the RG flow.
If a unitary RG flow preserves a TDL with non-integer quantum dimension, then the IR phase cannot be a non-degenerate gapped state \cite{Chang:2018iay}. Following the argument in \cite{Chang:2018iay}, we derive a refinement of this statement for the preserving TDL being q-type. Let $\cL_{\rm q}$ be a q-type TDL preserved along an RG flow. Suppose the IR fixed point is a TQFT with a unique vacuum, it follows that
\ie
\la \cL_{\rm q} \ra_{{\rm S}^1\times \bR} = \Tr \widehat \cL_{\rm q} = \Tr_{{\cal H}_{\cL_{\rm q}}}1 = \dim {\cal H}_{\cL_{\rm q}}\,,
\fe
where on the second equality we use the modular S transformation. The defect Hilbert space ${\cal H}_{\cL_{\rm q}}$ is even dimensional. Hence, we find
\ie
\la \cL_{\rm q} \ra_{{\rm S}^1\times \bR}\in 2{\mathbb Z}_{\ge 0}\,.
\fe
In other words, an RG flow cannot be trivially gapped if it preserves a q-type TDL whose quantum dimension is not a non-negative even integer.
\section*{Acknowledgements}
We thank Yongchao L\"u and Qing-Rui Wang for insightful discussions. CC is partly supported by National Key R\&D Program of China (NO. 2020YFA0713000). The work of J.C. is supported by the National Thousand-Young-Talents Program of China. F. Xu is partly supported by the Research Fund for International Young Scientists, NSFC grant No. 11950410500.
|
2,877,628,090,242 | arxiv | \section{Introduction and Main Result}
A $2$-$(v,k,\lambda )$ \emph{design} $\mathcal{D}$ is a pair $(\mathcal{P},%
\mathcal{B})$ with a set $\mathcal{P}$ of $v$ points and a set $\mathcal{B}$
of blocks such that each block is a $k$-subset of $\mathcal{P}$ and each two
distinct points are contained in $\lambda $ blocks. We say $\mathcal{D}$ is
\emph{non-trivial} if $2<k<v$, and symmetric if $v=b$. All $2$-$(v,k,\lambda
)$ designs in this paper are assumed to be non-trivial. An automorphism of $%
\mathcal{D}$ is a permutation of the point set which preserves the block
set. The set of all automorphisms of $\mathcal{D}$ with the composition of
permutations forms a group, denoted by $\mathrm{Aut(\mathcal{D})}$. For a
subgroup $G$ of $\mathrm{Aut(\mathcal{D})}$, $G$ is said to be \emph{%
point-primitive} if $G$ acts primitively on $\mathcal{P}$, and said to be
\emph{point-imprimitive} otherwise. In this setting we also say that $%
\mathcal{D}$ is either \emph{point-primitive} or \emph{point-imprimitive}
respectively. A \emph{flag} of $\mathcal{D}$ is a pair $(x,B)$ where $x$ is
a point and $B$ is a block containing $x$. If $G\leq \mathrm{Aut(\mathcal{D})%
}$ acts transitively on the set of flags of $\mathcal{D}$, then we say that $%
G$ is \emph{flag-transitive} and that $\mathcal{D}$ is a \emph{%
flag-transitive design}.\
In 1987, Davies \cite{Da} proved that in a flag-transitive and
point-imprimitive $2$-$(v,k,\lambda )$ design, the block size is bounded
for a given value of the parameter $\lambda $, where $\lambda \geq 2$ by a result of Higman-McLaughlin \cite{HM} dating back to 1961. In 2005, O'Reilly
Regueiro \cite{ORR} obtained an explicit upper bound. Later that year,
Praeger and Zhou \cite{PZ} improved that upper bound and gave a complete
list of feasible parameters. In 2020, Mandi\'{c} and \v{S}ubasi\'{c} \cite{MS} classified the
flag-transitive point-imprimitive symmetric $2$-designs with $\lambda \leq
10 $ except for two possible numerical cases. Recently, Montinaro \cite{Mo2} has classified those with $k>\lambda
\left(\lambda -3 \right)/2$ and such that a block of the $2$-design
intersects a block of imprimitivity in at least $3$ points. In this paper we
complete the work started in \cite{Mo2} by classifying $\mathcal{D}$ with $%
k>\lambda \left(\lambda -3 \right)/2$ regardless the intersection size of a block of $\mathcal{D}$ with a block of
imprimitivity. More precisely, our result is the following.
\bigskip
\begin{theorem}
\label{main} Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a
symmetric $2$-$(v,k,\lambda )$ design admitting a flag-transitive,
point-imprimitive automorphism group. If $k> \lambda (\lambda-3)/2$ then
one of the following holds:
\begin{enumerate}
\item $\mathcal{D}$ is isomorphic to one of the two $2$-$(16,6,2)$ designs.
\item $\mathcal{D}$ is isomorphic to the $2$-$(45,12,3)$ design.
\item $\mathcal{D}$ is isomorphic to the $2$-$(15,8,4)$ design.
\item $\mathcal{D}$ is isomorphic to one of the four $2$-$(96,20,4)$ designs.
\end{enumerate}
\end{theorem}
\bigskip
In 1945, Hussain \cite{Hu} and in 1946 Nandi \cite{Na} independently proved the existence of three symmetric $2$-$(16,6,2)$ designs. In 2005, O’Reilly Regueiro \cite{ORR} proved that exactly two of them are flag-transitive and point-imprimitive. In the same paper O’Reilly Regueiro constructed a $2$-$(15,8,4)$ design. Such $2$-design was proved to be unique by Praeger and Zhou \cite{PZ} in 2006. One year later, Praeger \cite{P2} constructed and proved there is exactly one flag-transitive and point-imprimitive $2$-$(45,12,3)$ design. Finally, in 2009, Law, Praeger and Reichard \cite{LPR} proved there are four flag-transitive and point-imprimitive $2$-$(96,20,4)$ designs.
\bigskip
The outline of the proof is as follows. We start by strengthening the classification result obtained in \cite{MS}. Indeed, in \cite{MS} it is proven that, if $\lambda \leq 10$ then $\mathcal{D}$ is known except for two numerical values for the parameters for $\mathcal{D}$. The two exceptions are ruled out here in Theorem \ref{MMS}. Subsequently, we focus on the case $\lambda>10$. In Proposition \ref{DivanDan} it is
shown that $G^{\Sigma }$ acts primitively on $\Sigma $ by using the results
contained in \cite{Mo2}. Then, it is proven in Theorem \ref{AQP} that, either
$G$ acts point-quasiprimitively on $\mathcal{D}$ or $G(\Sigma )\neq 1$, $%
G^{\Sigma }$ is almost simple and $\mathcal{D}$ has parameters $%
(2^{d+2}(2^{d-1}-1)^{2}, 2\left( 2^{d}-1\right) \left(
2^{d-1}-1\right) ,2(2^{d-1}-1))$ where $d\geq 4$. Afterwards, by combining
the O'Nan-Scott theorem for quasiprimitive groups achieved in \cite{P1}
with an adaptation of the techniques developed by \cite{ZZ}, in Theorem \ref{ASPQP} we
show that $G^{\Sigma }$ is almost simple also in the quasiprimitive case.
Moreover, if $L$ is the preimage in $G$ of $\mathrm{Soc}(G^{\Sigma })$ and $%
\Delta \in \Sigma $, in Proposition \ref{unapred}, Corollary \ref{QuotL} and Theorem \ref{Large}
it is proven that, either $G(\Sigma )=1$, $L_{\Delta
}$ is contained in a semilinear $1$-dimensional group and $\left\vert L \right \vert \leq 2 \left \vert L_{\Delta
}^{\Delta } \right \vert ^{2}\left\vert \mathrm{Out}(L)\right \vert^{2}$, or $G(\Sigma )=1$, $L_{\Delta
}$ is a non-solvable $2$-transitive permutation group of degree $\left\vert \Delta
\right\vert $ and $\left\vert L\right\vert \leq \left\vert L_{\Delta }\right\vert ^{2}$, or $G(\Sigma )\neq 1$ and a quotient of $L_{\Delta }^{\Sigma }
$ is isomorphic either to $SL_{d}(2)$, or to $A_{7}$ for $d=4$. In particular, in each case $L_{\Delta }^{\Sigma }$ is a large subgroup of $L^{\Sigma }$. Finally, we use all the above mentioned constraints on $ L^{\Sigma }$ and on $ L_{\Delta }^{\Sigma
}$ together with the results contained in \cite{AB} and \cite{LS} to precisely determine the admissible pairs $( L^{\Sigma },L_{\Delta}^{\Sigma })$ and from these to prove that there are no examples of $\mathcal{D}$ for $\lambda >10$. At
this point, our classification result follows from Theorem \ref{MMS}.
\bigskip
\section{Preliminary reductions}
\medskip It is well know that, if $\mathcal{D}$ is a symmetric $2$-$%
(v,k,\lambda )$ design, then $r=k$, $b=v$ and $k(k-1)=(v-1)\lambda $ (for instance, see \cite{Demb}). Moreover, the following fact holds:
\begin{lemma}
\label{PP}If $\mathcal{D}$ admits a flag-transitive automorphism group $G$
and $x$ is any point of $\mathcal{D}$, then $\left\vert y^{G_{x}}\right\vert
\lambda =k\left\vert B\cap y^{G_{x}}\right\vert $ for any point $y$ of $%
\mathcal{D}$, with $y\neq x$, and for any block $B$ of $\mathcal{D}$
incident with $x$.
\end{lemma}
\begin{proof}
Let $x$, $y$ be points of $\mathcal{D}$, $y\neq x$, and $B$ be any block of $%
\mathcal{D}$ incident with $x$. Since $(y^{G_{x}},B^{G_{x}})$ is a tactical
configuration by \cite[1.2.6]{Demb}, it follows that $\left\vert
y^{G_{x}}\right\vert \lambda =k\left\vert B\cap y^{G_{x}}\right\vert $.
\end{proof}
\bigskip
The following theorem, which is a summary of \cite{Mo2} and some of the results contained
in \cite{PZ}, is our starting point.
\bigskip
\begin{theorem}
\label{PZM}Let $\mathcal{D}=(\mathcal{P} ,\mathcal{B})$ be a symmetric $2$%
-design admitting a flag-transitive, point-imprimitive automorphism group $G$
that leaves invariant a non-trivial partition $\Sigma =\left\lbrace
\Delta_{1},...,\Delta_{d} \right\rbrace$ of $\mathcal{P} $ such that $%
\left\vert \Delta_{i} \right \vert =c$ for each $i=1,...,d$. Then the
following hold:
\begin{enumerate}
\item[I.] There is a constant $\ell$ such that, for each $B\in \mathcal{B}$
and $\Delta _{i}\in \Sigma $, the size $\left\vert B\cap \Delta
_{i}\right\vert $ is either $0$ or $\ell$.
\item[II.] There is a constant $\theta$ such that, for each $B\in \mathcal{B}
$ and $\Delta _{i}\in \Sigma $ with $\left\vert B\cap \Delta _{i}\right\vert
>0$, the number of blocks of $\mathcal{D}$ whose intersection set with $%
\Delta _{i}$ coincides with $B\cap \Delta _{i}$ is $\theta$.
\item[III.] If $\ell=2$ then $G_{\Delta _{i}}^{\Delta _{i}}$ acts $2$%
-transitively on $\Delta_{i}$ for each $i=1,...,d$.
\item[IV.] If $\ell\geq 3$ then $\mathcal{D}_{i}=\left( \Delta_{i}, \left(B
\cap \Delta_{i} \right)^{G_{\Delta _{i}}^{\Delta _{i}}}\right)$ is a
flag-transitive non-trivial $2$-$\left(c,\ell,\lambda/\theta\right)$ design
for each $i=1,...,d$.
\end{enumerate}
Moreover, if $k>\lambda (\lambda -3)/2$ then one of the following holds:
\begin{enumerate}
\item[V.] $\ell=2$ and one of the following holds:
\begin{enumerate}
\item[1.] $\mathcal{D}$ is a symmetric $2$-$(\lambda ^{2}(\lambda
+2),\lambda (\lambda +1),\lambda )$ design and $\left(c,d
\right)=\left(\lambda+2,\lambda^{2}\right)$.
\item[2.] $\mathcal{D}$ is a symmetric $2$-$\left( \left(\frac{\lambda+2}{2}%
\right) \left(\frac{\lambda^2-2\lambda+2}{2}\right),\frac{\lambda^2}{2}%
,\lambda \right)$ design, $\left(c,d \right)=\left(\frac{\lambda+2}{2},\frac{%
\lambda^2-2\lambda+2}{2}\right)$, and either $\lambda \equiv 0 \pmod{4}$, or
$\lambda=2w^{2}$, where $w$ is odd, $w \geq 3$, and $2(w^2-1)$ is a square.
\end{enumerate}
\item[VI.] $\ell\geq 3$ and one of the following holds:
\begin{enumerate}
\item[1.] $\mathcal{D}$ is isomorphic to the $2$-$(45,12,3)$ design of \cite[%
Construction 4.2]{P2}.
\item[2.] $\mathcal{D}$ is isomorphic to one of the four $2$-$(96,20,4)$
designs constructed in \cite{LPR}.
\end{enumerate}
\end{enumerate}
\end{theorem}
\bigskip
We are going to focus on case (V) of the previous theorem. \emph{Throughout the paper, a $2$%
-design as in case (V.1) or (V.2) of Theorem \ref{PZM} will be simply
called a $2$-design of type 1 or 2 respectively.}
\bigskip
Let $\Delta \in \Sigma $ and $x \in \Delta $. Since $G(\Sigma) \trianglelefteq G_{\Delta}$ and $G(\Delta) \trianglelefteq G_{x}$, it is immediate to verify that $(G^{\Sigma})_{\Delta}=(G_{\Delta})^{\Sigma}$ and that $\left( G_{\Delta}^{\Delta} \right)_{x}=(G_{x})^{\Delta}$. Hence, in the sequel $(G^{\Sigma})_{\Delta}$ and $\left( G_{\Delta}^{\Delta} \right)_{x}$ will simply be denoted by $G^{\Sigma}_{\Delta}$ and $G_{x}^{\Delta}$ respectively. Moreover, the following holds:
\begin{equation}\label{salvaMAOL}
\frac{G^{\Sigma}_{\Delta}}{G(\Delta)^{\Sigma}} \cong \frac{G_{\Delta}}{G(\Delta)G(\Sigma)} \cong \frac{G^{\Delta}_{\Delta}}{G(\Sigma)^{\Delta}}.
\end{equation}
\bigskip
A further reduction is the following theorem.
\bigskip
\begin{theorem}
\label{MMS}Let $\mathcal{D}=\left( \mathcal{P},\mathcal{B}\right) $ be a
symmetric $2$-$(v,k,\lambda )$ design admitting a flag-transitive,
point-imprimitive automorphism group $G$. If $\lambda \leq 10$ then
\begin{enumerate}
\item $\mathcal{D}$ is isomorphic to one of the two $2$-$(16,6,2)$ designs
\item $\mathcal{D}$ is isomorphic to the $2$-$(45,12,3)$ design.
\item $\mathcal{D}$ is isomorphic to the $2$-$(15,8,4)$ design.
\item $\mathcal{D}$ is isomorphic to one of the four $2$-$(96,20,4)$ designs.
\end{enumerate}
\end{theorem}
\begin{proof}
This results is proven in \cite[Theorem 1]{MS} with the following possible
exceptions of $(v,k,\lambda ,c,d)=(288,42,6,8,36)$ or $(891,90,9,81,11)$.
Note that $k>\lambda (\lambda -3)/2$ in both exceptional cases. Actually,
the latter does not correspond to any case of Theorem \ref{PZM}(V--VI),
and hence it cannot occur. The former corresponds to Theorem \ref{PZM}%
(V.1) for $\lambda =6$. Also, if $\Delta \in \Sigma$ then $G_{\Delta }^{\Delta }\cong AGL_{1}(8)$, $ A\Gamma L_{1}(8)$, $A_{8}$, $S_{8}$,
$PSL_{2}(7)$ or $PGL_{2}(7)$ by \cite[Lists (A) and (B)]{Ka} since $G_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta$.
Assume that $u$ divides the order of $G(\Sigma )$, where $u$ is an odd prime, and let $\psi $ be a $u$-element of $G(\Sigma )$. Then $\psi $
fixes at least a point on each $\Delta \in \Sigma $ since $\left\vert \Delta \right\vert =8$. Thus $\psi $ preserves
at least two distinct blocks of $\mathcal{D}$ by \cite[Theorem 3.1]{La}, say $B_{1}$ and $B_{2}$. Actually, $\psi $ fixes $B_{1}$ and $B_{2}$ pointwise since any of these intersect each element of $\Sigma$ in $0$ or $2$ points and $%
\psi \in G(\Sigma )$. Hence, $\psi $ fixes at least $2\cdot 42-6$ points of $\mathcal{D}$, but this contradicts \cite[Corollary 3.7]{La}. Thus, the
order of $\left\vert G(\Sigma )\right\vert =2^{i}$ with $i \geq 0$.
Assume that $w$ divides the order of $G(\Delta )$, where $w$ is an odd prime, $w \geq 7$, and let $\phi $ be a $w$-element of $G(\Delta )$. Then $\phi
$ fixes the $6$ blocks incident with any pair of distinct
points of $\Delta $. Therefore, $\phi $ fixes at least $6\cdot
\binom{8}{2}$ points of $\mathcal{D}$ by \cite[Theorem 3.1]{La}, and we again reach a contradiction by \cite[Corollary 3.7]{La}%
. Thus, $G(\Delta )$ is a $\{2,3,5\}$-group.
Any Sylow $7$-subgroup of $G$ is of order $7$, since $\left\vert \Sigma \right\vert =36$, $G_{\Delta }^{\Delta }$ is one of the groups listed above and the order of $G(\Delta )$ is coprime to $7$. Then, by \cite[Table B.4]{DM} one of the following holds:
\begin{enumerate}
\item $A_{9} \trianglelefteq G^{\Sigma} \leq S_{9}$ and $S_{7}\trianglelefteq G^{\Sigma}_{\Delta} \leq S_{7} \times Z_{2}$;
\item $PSL_{2}(8) \trianglelefteq G^{\Sigma} \leq P \Gamma L_{2}(8)$ and $ D_{14} \trianglelefteq G^{\Sigma}_{\Delta}\leq F_{42}$;
\item $PSU_{3}(3)\trianglelefteq G^{\Sigma} \leq P \Gamma U_{3}(3) $ and $PSL_{2}(7) \trianglelefteq G^{\Sigma}_{\Delta}\leq PGL_{2}(7)$;
\item $ G^{\Sigma} \cong Sp_{6}(2)$ and $G^{\Sigma}_{\Delta} \cong S_{8}$.
\end{enumerate}
Since $G_{\Delta }^{\Delta }$ is one of the $2$-transitive-groups listed above, (1) is immediately ruled out by (\ref{salvaMAOL}). In (2) the group $G_{\Delta}$ is solvable since $G^{\Sigma}_{\Delta}$ is solvable and $G(\Sigma )$ is a $2$-group. Thus $G_{\Delta }^{\Delta }$ is solvable and hence it is isomorphic to $AGL_{1}(8)$ or $A\Gamma L_{1}(8)$. Moreover, a quotient group of $G_{\Delta }^{\Delta }$ must contain a subgroup isomorphic to $D_{14}$ by (\ref{salvaMAOL}) since $G(\Delta )$ is a $\{2,3,5\}$-group, but this is clearly impossible. It follows that only (3) and (4) are admissible. Also, $G(\Sigma)=G(\Delta)=1$ in both cases by comparing (\ref{salvaMAOL}) with the possibilities for $G_{\Delta}^{\Delta}$ provided above.
Assume that (4) occurs. Then $G$ acts $2$-transitively on $\Sigma$. Let $x \in \Delta$ and $\Delta^{\prime} \in \Sigma \setminus \{ \Delta \}$. Then $G_{x} \cong S_{7}$ and $G_{x,\Delta^{\prime}} \leq G_{\Delta,\Delta^{\prime}} \cong (S_{4} \times S_{4}):Z_{2}$ by \cite{At}, hence $G_{x}$ acts transitively on $\Sigma \setminus \{ \Delta \}$. On the other hand, $G_{x}=G_{B}$ for some block $B$ of $\mathcal{D}$ since $G$ has a unique conjugate class of subgroups isomorphic to $S_{7}$ by \cite{At}. Therefore, $G_{x}$ permutes transitively the $21$ elements of $\Sigma$ intersecting $B$, whereas $G_{x}$ acts transitively on the $35$ elements $\Sigma \setminus \{ \Delta \}$. So this case is excluded. Finally, (3) is ruled out with the aid of \textsf{GAP} \cite{GAP}, and the proof is thus completed.
\end{proof}
\bigskip
On the basis of Theorem \ref{MMS}, in the sequel we may assume that $\lambda
>10$.
\bigskip
\begin{lemma}
\label{Mpomoc}If $G$ preserves a further partition $\Sigma ^{\prime }$ of
the point set of $\mathcal{D}$ in $d^{\prime}$ blocks of imprimitivity of size $c^{\prime}$. Then $d^{\prime}=d$ and $c^{\prime}=c$, and one of the following holds:
\begin{enumerate}
\item $\Sigma =\Sigma ^{\prime }$;
\item $\left\vert \Delta \cap \Delta ^{\prime }\right\vert \leq 1$ for for each $\Delta \in \Sigma $, $\Delta ^{\prime }\in
\Sigma ^{\prime }$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose there is a $G$-invariant partition $\Sigma ^{\prime }$ of the point
set of $\mathcal{D}$. Let $B$ any block of $\mathcal{D}$ and let $\Delta
^{\prime }\in \Sigma ^{\prime }$. If $\left\vert B\cap \Delta ^{\prime }\right\vert \geq 3$ then
either $(v,k,\lambda )=(45,12,3)$ or $(96,20,4)$ by \cite[Theorem 1.1]{Mo2},
whereas $\lambda >10$ by our assumptions. Thus, $\left\vert B\cap \Delta ^{\prime }\right\vert =2$.
If $\mathcal{%
D}$ is of different type with respect to $\Sigma
$ and to $\Sigma ^{\prime }$, then $%
k=\lambda +2=\lambda ^{2}/2$, which has no integer solutions. Therefore $%
\mathcal{D}$ is of the same type with respect to $\Sigma $ and to $\Sigma ^{\prime }$, and hence $d^{\prime}=d$ and $c^{\prime}=c$.
Let $\Delta
\in \Sigma $ such that $\Delta \cap \Delta ^{\prime }\neq \varnothing $. If $%
\left\vert \Delta \cap \Delta ^{\prime }\right\vert >1$ then $\Delta =\Delta
^{\prime }$ since $G$ induces a $2$-transitive group on $\Delta $ and $%
\left\vert \Delta \right\vert =\left\vert \Delta ^{\prime }\right\vert $.
Therefore $\Sigma =\Sigma ^{\prime }$ which is (2).
\end{proof}
\bigskip
\begin{proposition}
\label{DivanDan}$G^{\Sigma }$ acts primitively on $\Sigma $. Moreover, and
one of the following holds:
\begin{enumerate}
\item $G(\Sigma )\neq 1$ and $\mathrm{Soc}(G_{\Delta }^{\Delta })\trianglelefteq
G(\Sigma )^{\Delta }$;
\item $G(\Sigma )=1$ and $G$ acts point-quasiprimitively on $\mathcal{D}$.
\end{enumerate}
\end{proposition}
\begin{proof}
It follows from Lemma \ref{Mpomoc} and \cite[Theorem 1.5A]{DM}
that $G^{\Sigma }$ acts primitively on $\Sigma $.
Assume that $G(\Sigma )\neq 1$. If there is $\Delta ^{\prime }\in \Sigma $
such that $G(\Sigma )\leq G(\Delta ^{\prime })$. Then $G(\Sigma )\leq
G(\Delta ^{\prime \prime })$ for each $\Delta ^{\prime \prime }\in \Sigma $,
and hence $G(\Sigma )=1$, since $G(\Sigma )\vartriangleleft G$ and $G$ acts
transitively on $\Sigma $. However, it contradicts the assumption $G(\Sigma )\neq 1$.
Thus $G(\Sigma )\nleq G(\Delta )$ for each $\Delta \in \Sigma $. Then $1\neq
G(\Sigma )^{\Delta }\trianglelefteq G_{\Delta }^{\Delta }$, and hence $%
\mathrm{Soc}(G_{\Delta }^{\Delta })\trianglelefteq G(\Sigma )^{\Delta }$ by \cite[Theorem 4.3B]{DM} since $%
G_{\Delta }^{\Delta }$ is $2$-transitive on $\Delta $.
Assume that $G(\Sigma )=1$. Let $N$ be any normal subgroup of $G$. Then $N=G$
since $G(\Sigma )=1$ and since $G^{\Sigma }$ acts primitively on $\Sigma $.
Therefore, $G$ acts point-quasiprimitively on $\mathcal{D}$.
\end{proof}
\begin{lemma}
\label{Fixpoints}Let $\Delta \in \Sigma $ and $\gamma \in G(\Delta )$, $%
\gamma \neq 1$. Then one of the following holds:
\begin{enumerate}
\item $\mathcal{D}$ is of type 1 and $\left\vert \mathrm{Fix}(\gamma
)\right\vert \leq (\lambda +2)\lambda $;
\item $\mathcal{D}$ is of type 2 and $\left\vert \mathrm{Fix}(\gamma
)\right\vert \leq \lambda ^{2}/2+\sqrt{\lambda ^{2}/2-\lambda }$
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\Delta \in \Sigma $ and $\gamma \in G(\Delta )$, $\gamma \neq 1$. It
follows from by \cite[Theorem 3.1]{La} that%
\begin{equation}
\left\vert \mathrm{Fix}(\gamma )\right\vert \leq \frac{\lambda }{k-\sqrt{%
k-\lambda }}\cdot \left\vert \Delta \right\vert \cdot \left\vert \Sigma
\right\vert \label{Fixgamma}
\end{equation}%
where $\frac{\lambda }{k-\sqrt{k-\lambda }}\cdot \left\vert \Delta
\right\vert \cdot \left\vert \Sigma \right\vert $ is either $(\lambda
+1)\lambda $ or $\lambda ^{2}/2+\sqrt{\lambda ^{2}/2-\lambda }$ according to
whether $\mathcal{D}$ is of type 1 or 2 respectively.
\end{proof}
\bigskip
\begin{corollary}
\label{CFixT}The following hold:
\begin{enumerate}
\item If $\mathcal{D}$ is of type 1, each prime divisor of $%
\left\vert G(\Delta )\right\vert $ divides $\lambda (\lambda -1)$.
\item If $\mathcal{D}$ is of type 2, each prime divisor of $%
\left\vert G(\Delta )\right\vert $ divides $\lambda (\lambda -1)(\lambda-2)(\lambda-3)$.
\end{enumerate}
\end{corollary}
\begin{proof}
Let $\gamma $ be $w$-element of $G(\Delta )$, where $w$ is a prime such that
$w\nmid \lambda $. Since $\gamma $ fixes $\Delta $ pointwise, $\gamma $
fixes at least $\mu $ of the $\lambda $ blocks of $\mathcal{D}$ incident
with any two distinct points of $\Delta $, where $\mu =\lambda \pmod{w}$.
Clearly, these fixed blocks are pairwise distinct.
If $\mathcal{D}$ is of type 1 then $\gamma $ fixes $\mu \frac{%
(\lambda +2)(\lambda +1)}{2}$ blocks of $\mathcal{D}$, and hence $\left\vert \mathrm{Fix}(\gamma )\right\vert \geq \mu
\frac{(\lambda +2)(\lambda +1)}{2}$. Therefore $\mu =1$ by Lemma \ref%
{Fixpoints}(1) and the assertion (1) follows.
If $\mathcal{D}$ is of type 2 then $\gamma $ fixes $\mu \frac{\lambda}{4}\left(\frac{\lambda}{2}+1\right)$ blocks of $\mathcal{D}$, and hence $\left\vert \mathrm{Fix}(\gamma )\right\vert \geq
\mu \frac{\lambda}{4}\left(\frac{\lambda}{2}+1\right)$. Therefore $\mu \leq 3$ by Lemma \ref%
{Fixpoints}(2) and the assertion (2) follows.
\end{proof}
\bigskip
\begin{lemma}
\label{2S}Let $x$ be any point of $\mathcal{D}$. Then $G(\Sigma )_{x}$ lies in
a Sylow $2$-subgroup of $G(\Sigma)$.
\end{lemma}
\begin{proof}
Let $x$ be any point of $\mathcal{D}$ and let $\varphi $ be any $w$-element of $G(\Sigma )_{x}$%
, where $w$ is an odd prime. Then $\varphi $ fixes at least a blocks $B$ of $%
\mathcal{D}$ by \cite[Theorem 3.1]{La}. Since $B$ intersects each element of
$\Sigma $ in $0$ or $2$ points, and since $\varphi $ is a $w$-element of $%
G(\Sigma )$, it follows that $\varphi $ fixes $B$ pointwise. Therefore, $%
\varphi $ fixes at least $k$ points of $\mathcal{D}$. Then $\varphi $ fixes
at least $k$ blocks of $\mathcal{D}$ again by \cite[Theorem 3.1]{La}. Let $%
B^{\prime }$ be further block fixed by $\varphi $. We may repeat the
previous argument with $B^{\prime }$ in the role of $B$ thus obtaining $%
\varphi $ fixing $B^{\prime }$ pointwise. Then $\varphi $ fixes at least $%
2k-\lambda $ points of $\mathcal{D}$, as $\left\vert B\cap B^{\prime
}\right\vert =\lambda $. Thus $\left\vert \mathrm{Fix}(\alpha )\right\vert
\geq 2k-\lambda $ and hence $\left\vert \mathrm{Fix}(\alpha )\right\vert $
is greater than or equal to $\lambda (2\lambda +1)$ or $\lambda ^{2}-\lambda
$ according to whether $\mathcal{D}$ is of type 1 or 2 respectively.
However, this contradicts Lemma \ref{Fixpoints}, as $\lambda >2$.$%
\allowbreak $
\end{proof}
\bigskip
\begin{lemma}
\label{strike}If $G(\Sigma )\neq 1$ and $v$ is odd, then the following hold:
\begin{enumerate}
\item $G(\Sigma )= Soc(G_{\Delta })$ is an elementary abelian $p$-group, $p$ an odd prime, acting regularly on $\Delta $;
\item $G^{\Sigma }\leq GL_{d}(p)$, with $p^{d}=\left\vert \Delta \right\vert
$ and $\Delta \in \Sigma $.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $G(\Sigma )_{x}\neq 1$, where $x$ is any point of $\mathcal{D}$.
Then $G(\Sigma )_{x}$ is a Sylow $2$-subgroup of $G(\Sigma )$ by Lemma \ref%
{2S}. Denote $G(\Sigma )_{x}$ simply by $S$. Then $S$ fixes the same number $%
t$ of points in each $\Delta \in \Sigma $, where $t \geq 1$ since $v$ is odd. Therefore, $\left\vert \mathrm{Fix%
}(S)\right\vert =t\left\vert \Sigma \right\vert $. Now, if $\alpha $ is any
non-trivial element of $S$ then $\left\vert \mathrm{Fix}(\alpha )\right\vert
\geq t\left\vert \Sigma \right\vert $, and hence $t=1$ by Lemma \ref%
{Fixpoints}. Thus $S$ fixes a unique point on each $\Delta $.
Suppose that $\left\vert S\right\vert \geq 4$ and let $B$ be any block preserved by $S$
and $\Delta ^{\prime }$ be such that $\left\vert B\cap \Delta ^{\prime
}\right\vert =2$. Then there is a subgroup $S_{0}$ of $S$ of index at most $%
2 $ fixing $y$ and $y^{\prime }$ where $\left\{ y,y^{\prime }\right\} =B\cap
\Delta ^{\prime }$. Then $S_{0}\leq G(\Sigma )_{y}\cap G(\Sigma )_{y^{\prime
}}$, where $G(\Sigma )_{y}$ and $G(\Sigma )_{y^{\prime }}$ are two distinct
Sylow $2$-subgroups of $G(\Sigma )$, each of these fixing a unique point in $\Delta^{\prime} $. Suppose there is a point $z$ of $\mathcal{D}$ such that $%
z\in \mathrm{Fix}(G(\Sigma )_{y})\cap \mathrm{Fix}(G(\Sigma )_{y^{\prime }})$%
. The $\left\langle G(\Sigma )_{y},G(\Sigma )_{y^{\prime }}\right\rangle
\leq G(\Sigma )_{z}$ and hence $G(\Sigma )_{y}=G(\Sigma )_{y^{\prime
}}=G(\Sigma )_{z}$ since $G(\Sigma )_{z}$ is a Sylow $2$-subgroup of $%
G(\Sigma )$ by Lemma \ref{2S}. Then $G(\Sigma )_{y}$ fixes also $y^{\prime }$
in $\Delta ^{\prime }$ with $y^{\prime }\neq y$, and we reach a
contradiction. Thus $\mathrm{Fix}(G(\Sigma )_{y})\cap \mathrm{Fix}(G(\Sigma
)_{y^{\prime }})=\varnothing $, and hence $\left\vert \mathrm{Fix}%
(S_{0})\right\vert \geq 2\left\vert \Sigma \right\vert $ since $S_{0}\leq
G(\Sigma )_{y}\cap G(\Sigma )_{y^{\prime }}$. Now, if we use the above
argument this time with $\alpha \in S_{0}$, $\alpha \neq 1$, we reach a contradiction.
Therefore $\left\vert S\right\vert =2$. Also, $G(\Sigma )=O(G(\Sigma )).S$ by Proposition \ref{DivanDan}(1), and $%
\left\vert \mathrm{Fix}(S)\right\vert =\left\vert \Sigma \right\vert $.
Let $\Lambda =\left\{ Fix(S)^{\gamma }:\gamma \in G(\Sigma )\right\} $.
Since $S=G(\Sigma )_{x}$ is a Sylow $2$-subgroup of $G(\Sigma )$, $G(\Sigma
)\vartriangleleft G$ and $G$ acts point-transitively on $\mathcal{D}$, it
follows that $\Lambda $ is a $G$-invariant partition of the point set of $\mathcal{D}$ in $\left\vert \Delta
\right\vert $ blocks each of size $\left\vert \Sigma \right\vert $. Then $%
\left\vert \Sigma \right\vert =\left\vert \Delta \right\vert $ by
Lemma \ref{Mpomoc}(2) and hence $\lambda =2$, but this contradicts our
assumptions.
Assume that $G(\Sigma )_{x}=1$. Then $G(\Sigma )=O(G(\Sigma ))$. Moreover, $%
Soc(G_{\Delta }^{\Delta })\trianglelefteq G(\Sigma )^{\Delta }\cong G(\Sigma
)$ by Proposition \ref{DivanDan}(1). Then $G(\Sigma )= Soc(G_{\Delta
}^{\Delta })$ is an elementary abelian $p$-group for some odd prime $p$,
since $G(\Sigma )$ acts regularly on $\Delta $ and $\left\vert \Delta
\right\vert $ is odd. Hence $G(\Sigma )$ is abelian, and so $G(\Sigma
)\trianglelefteq C_{G}(G(\Sigma ))\trianglelefteq G$. If $C_{G}(G(\Sigma
))\neq G(\Sigma )$, then $G=C_{G}(G(\Sigma ))$ since $G^{\Sigma }$ is
primitive on $\Sigma $ by Proposition \ref{DivanDan}. This implies $%
G(\Sigma ) \leq Z(G)$ and hence $G_{x}\leq G(\Delta )$
for any $x\in \Delta $. This is a contradiction since $G_{\Delta }^{\Delta }$
is $2$-transitive on $\Delta $. Therefore $C_{G}(G(\Sigma ))=G(\Sigma
) = Soc(G_{\Delta }^{\Delta })$. Therefore $G^{\Sigma }\leq \mathrm{Aut}%
(G(\Sigma ))\cong GL_{d}(p)$.
\end{proof}
\bigskip
\section{Further Reductions}
The aim of this section is to prove the following reduction result:
\begin{theorem}
\label{AQP}One of the following holds:
\begin{enumerate}
\item $G(\Sigma )=1$ and $G$ acts point-quasiprimitively on $\mathcal{D}$.
\item $G(\Sigma )$ is a non-trivial self-centralizing elementary abelian $2$%
-subgroup of $G$ and the following hold:
\begin{enumerate}
\item $\mathcal{D}$ is a symmetric $2$-$(2^{d+2}(2^{d-1}-1)^{2},\allowbreak
2\left( 2^{d}-1\right) \left( 2^{d-1}-1\right) ,2(2^{d-1}-1))$ design with $%
d\geq 4$;
\item Either $G_{x}^{\Delta }\cong SL_{d}(2)$, or $G_{x}^{\Delta }\cong
A_{7} $ and $d=4$.
\item $G^{\Sigma }$ is almost simple.
\end{enumerate}
\end{enumerate}
\end{theorem}
Its proof is structured as follows. Case (1) is an immediate consequence of Proposition \ref{DivanDan}(2), whereas the proof of (2) relies mainly on Corollary \ref{CFixT} and on O'Nan Scott theorem applied to $G^{\Sigma}$ for $v$ even, and on Corollary \ref{CFixT} and on \cite[Theorem 3.1]{BP} for $v$ odd.
\bigskip
\subsection{Designs of type 1 and quasiprimitivity}
\begin{lemma}
\label{T1even}Let $\mathcal{D}$ be of type 1. If $G(\Sigma )\neq 1$ and $v$ is even, then the following hold:
\begin{enumerate}
\item $\mathcal{D}$ is a symmetric $2$-$(2^{d+2}(2^{d-1}-1)^{2},\allowbreak
2\left( 2^{d}-1\right) \left( 2^{d-1}-1\right) ,2(2^{d-1}-1))$ design, $d \geq 4$;
\item $G(\Sigma )$ is a non-trivial self-centralizing elementary abelian $2$%
-subgroup of $G$. Also $G_{\Delta }^{\Sigma }/G(\Delta )^{\Sigma }\cong
G_{x}^{\Delta }$ and one of the following holds:
\begin{enumerate}
\item $G_{x}^{\Delta }\cong SL_{d}(2)$;
\item $G_{x}^{\Delta }\cong A_{7}$ and $d=4$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $\mathcal{D}$ is of type 1, $G(\Sigma )\neq 1$ and $v$ is even. Since $Soc(G_{\Delta
}^{\Delta })\trianglelefteq G(\Sigma )^{\Delta }$ by Proposition \ref%
{DivanDan}(1), it follows that $Soc(G_{\Delta }^{\Delta })_{x}\trianglelefteq
G(\Sigma )_{x}^{\Delta }$, where $x\in \Delta $. Hence, $Soc(G_{\Delta
}^{\Delta })_{x}$ is either trivial or a $2$-group by Lemma \ref{2S}. If $G_{\Delta }^{\Delta
} $ is almost simple, then $Soc(G_{\Delta }^{\Delta })$ acts $2$%
-transitively on $\Delta $, with $Soc(G_{\Delta }^{\Delta })_{x}$ a $2$%
-group. However, this is impossible by \cite[List (A)]{Ka}. Therefore, $%
G_{\Delta }^{\Delta }$ is of affine type, and hence $\left\vert \Delta
\right\vert =2^{d}$ since $v=\lambda^{2}(\lambda+2)$ is even. Then $\lambda =2(2^{d-1}-1)$ and so $%
\left\vert \Sigma \right\vert =2^{2}(2^{d-1}-1)^{2}$, where $d \geq 4$ since $\lambda >10$. In particular, $%
\mathcal{D}$ is a symmetric $2$-design with parameters as in (1). Moreover,
by \cite[List (B)]{Ka} one of the following holds:
\begin{enumerate}
\item[(i)] $G_{x}^{\Delta }\leq \Gamma L_{1}(2^{d})$;
\item[(ii)] $SL_{d/h}(2^{h})\trianglelefteq G_{x}^{\Delta }\leq \Gamma
L_{d/h}(2^{h})$ with $d/h>1$;
\item[(iii)] $Sp_{d/h}(2^{h})\trianglelefteq G_{x}^{\Delta }\leq \Gamma
Sp_{d/h}(2^{h})$ with $d/h>1$ and $d/h$ even,
\item[(iv)] $G_{2}(2^{d/6})\trianglelefteq G_{x}^{\Delta }\leq
G_{2}(2^{d/6}):Z_{d/6}$ with $d\equiv 0\pmod{6}$.
\item[(v)] $G_{x}^{\Delta }\cong A_{6}$ or $A_{7}$ and $d=4$.
\end{enumerate}
Since $G_{x}$ is transitive on the $\lambda (\lambda +1)$ blocks incident
with $x$, it follows that $2\left( 2^{d-1}-1\right) \left( 2^{d}-1\right)
\mid \left\vert G_{x}\right\vert $. Suppose that there is a prime $u$
dividing $\lambda/2=2^{d-1}-1$ but not dividing of the order of $G_{x}^{\Delta }$.
Hence, $u$ divides the order of $G(\Delta )$. Let $U$ be a Sylow $u$%
-subgroup of $G(\Delta )$, then $G_{x}=N_{G_{x}}(U)G(\Delta )$ by the Frattini argument.
Let $y \in \mathrm{Fix(U)} \setminus \{x\}$ and let $\Delta ^{\prime }$ be the element of $\Sigma$ containing $y$. Then $U \leq G_{y}$, and actually $U \leq G(\Delta^{\prime})$ since $G_{x}$ and $G_{y}$ are $G$-conjugate (clearly, $\Delta=\Delta^{\prime}$ is possible). Therefore $\left\vert\mathrm{Fix}(U) \right \vert = t(\lambda+2)$, where $t \geq 2$ since $\left\vert\Sigma \right\vert = \lambda^{2}$. Also, $U$ is a Sylow $u$-subgroup of $G(\Delta ,\Delta ^{\prime })$. Thus $%
G_{x,y}=N_{G_{x,y}}(U)G(\Delta ,\Delta ^{\prime })$ again by Frattini argument since $G(\Delta ,\Delta ^{\prime
})\vartriangleleft G_{x,y}$. Then%
\begin{equation*}
\left\vert G_{x}:G_{x,y}\right\vert = \frac{\left\vert
N_{G_{x}}(U):N_{G_{x,y}}(U)\right\vert \cdot \left\vert G(\Delta
):G(\Delta ,\Delta ^{\prime })\right\vert }{\left\vert N_{G(\Delta
)}(U):N_{G(\Delta ,\Delta ^{\prime })}(U)\right\vert }
\end{equation*}%
and hence $\lambda +1\mid \left\vert
N_{G_{x}}(U):N_{G_{x,y}}(U)\right\vert $, since $\lambda +1\mid
\left\vert G_{x}:G_{x,y}\right\vert $ by Lemma \ref{PP}, and since $\left( \lambda
+1,\left\vert G(\Delta )\right\vert \right) =1$ by Corollary \ref{CFixT} being $\lambda$ even, as $v=\lambda^{2}(\lambda+2)$ is even.
Therefore $\lambda +1\mid \left\vert y^{N_{G_{x}}(U)}\right\vert $ and $y^{N_{G_{x}}(U)} \subseteq \mathrm{Fix}(U)\setminus \{x \}$. Since $\mathrm{Fix}(U)\setminus \{x \}$ is a disjoint union of $N_{G_{x}}(U)$-orbits, it follows that $\lambda +1 \mid \left\vert\mathrm{Fix}(U) \right \vert-1$. Therefore, $\lambda+1 \mid t-1$ since $\left\vert\mathrm{Fix}(U) \right \vert = t(\lambda+2)$. Therefore, $\left\vert \mathrm{Fix}(\zeta)\right\vert \geq \left\vert\mathrm{Fix}(U) \right \vert \geq (\lambda+2)^{2}$ for any non-trivial element $\zeta \in U$ since $t \geq 2$, but
this contradicts Lemma \ref{Fixpoints}(1). Thus, each prime divisor of $%
2^{d-1}-1$ divides $\left\vert G_{x}^{\Delta }\right\vert $. Then $%
G_{x}^{\Delta }\cong SL_{7}(2)$ for $d=7$, whereas $2^{d-1}-1$ admits a
primitive prime divisor for $d\neq 7$ by \cite[Theorem II.6.2]{Lu}. A this point, it is easy to check that only $%
G_{x}^{\Delta }\cong SL_{d}(2)$, or $G_{x}^{\Delta }\cong A_{7}$ for $d \geq 4$ by using \cite[Proposition 5.2.15]{KL}.
The two possibilities for $G_{x}^{\Delta }$ together with Proposition \ref{DivanDan}(1) imply either $G(\Sigma )^{\Delta }=G_{\Delta }^{\Delta }$ or $%
G(\Sigma )^{\Delta }=Soc(G_{\Delta }^{\Delta })$. The former yields $%
G_{\Delta }=G(\Sigma )G(\Delta )$, and hence $\lambda +1\mid \left\vert G(\Sigma
)_{x}\right\vert $ since $\left( \lambda
+1,\left\vert G(\Delta )\right\vert \right) =1$ by Corollary \ref{CFixT}. However, this contradicts Lemma \ref{2S} since $\lambda +1$ odd. Thus $G(\Sigma )^{\Delta }=Soc(G_{\Delta }^{\Delta })$, and hence
$
\frac{G_{\Delta }^{\Sigma }}{G(\Delta )^{\Sigma }}\cong \frac{G_{\Delta }^{\Delta }}{G(\Sigma )^{\Delta }}%
\cong G_{x}^{\Delta }
$
by (\ref{salvaMAOL}).
Since $\frac{G(\Sigma )}{G(\Sigma )\cap G(\Delta )}\cong Soc(G_{\Delta
}^{\Delta })$, which is an elementary abelian $2$-group, we have $\Phi (G(\Sigma ))\leq G(\Sigma )\cap G(\Delta )$ for
each $\Delta \in \Sigma $. Thus $\Phi (G(\Sigma ))$ fixes each point of $%
\mathcal{D}$, hence $\Phi (G(\Sigma ))=1$, and so $G(\Sigma )$ is an
elementary abelian $2$-group.
If $C_{G}(G(\Sigma ))\neq G(\Sigma )$, then $%
G=C_{G}(G(\Sigma ))$ since $G^{\Sigma }$ is primitive on $\Sigma $ by
Proposition \ref{DivanDan}. Thus $G(\Sigma )\leq Z(G)$ acts
transitively on each $\Delta $, and hence $G_{x}\leq G(\Delta )$ for any $%
x\in \Delta $. However, this is a contradiction since $G_{\Delta }^{\Delta }$
is $2$-transitive on $\Delta $. Therefore $C_{G}(G(\Sigma ))=G(\Sigma )$,
and we obtain (2).
\end{proof}
\bigskip
\begin{theorem}
\label{veven}If $\mathcal{D}$ is of type 1 and $v$ is even, then Theorem \ref{AQP} holds.
\end{theorem}
\begin{proof}
Assume that that $\mathcal{D}$ is of type 1. The assertion follows from Proposition \ref{DivanDan}(2) for $G(\Sigma )=1$.
Hence, assume that $G(\Sigma )\neq 1$. Since $G^{\Sigma }$ acts
primitively on $\Sigma $ again by Proposition \ref{DivanDan} and $\left\vert
\Sigma \right\vert =2^{2}(2^{d-1}-1)^{2}$ with $d\geq 4$ by Lemma \ref%
{T1even}, by the O'Nan-Scott theorem (e.g. see \cite[Theorem 4.1A]{DM}) one
of the following holds:
\begin{enumerate}
\item[(i)] $Soc(G^{\Sigma })$ is almost simple.
\item[(ii)] $Soc(G^{\Sigma })\cong T^{2}$, where $T$ is a non-abelian simple
group such that $\left\vert T\right\vert =2^{2}(2^{d-1}-1)^{2}$.
\item[(iii)] $Soc(G^{\Sigma })\cong T^{2}$ and there is a
non-abelian almost simple group $Q$ with socle $T$ acting primitively on a set $\Theta$ of size $2(2^{d-1}-1)$ such that $\Sigma=\Theta^{2}$ and $G^{\Sigma}\leq Q\wr Z_{2}$.
\end{enumerate}
Assume that (ii) holds. Then $T\leq G_{\Delta }^{\Sigma }\leq Aut(T)\times
Z_{2}$, and hence $G_{\Delta }^{\Sigma }/T$ is solvable. Therefore $\left\vert
G_{x}^{\Delta }\right\vert \mid \left\vert T\right\vert $ since $G_{\Delta
}^{\Sigma }/G(\Delta )^{\Sigma }\cong G_{x}^{\Delta }$ with $G_{x}^{\Delta }$ isomorphic either to $SL_{d}(2)$, or to $A_{7}$ for $d=4$ by Lemma \ref{T1even}(2).
However, this is impossible since $\left\vert G_{x}^{\Delta }\right\vert \nmid 2^{2}(2^{d-1}-1)^{2}$, and (ii) is excluded.
Assume that (iii) holds. Since $G_{\Delta }^{\Sigma }/G(\Delta )^{\Sigma
}\cong G_{x}^{\Delta }$ by Lemma \ref{T1even}(2), then $\lambda +1=2^{d}-1$ divides the order of $G_{\Delta }^{\Sigma }$. If $d=6$ then $\lambda =2\cdot 31$, $\left\vert
\Sigma \right\vert =2^{2}31^{2}$ and $G_{x}^{\Delta }\cong SL_{6}(2)$. Since
$\lambda (\lambda +1)\mid \left \vert G_{\Delta } \right \vert $, it follows that $31^{3}\mid
\left\vert G\right\vert $. On the other hand, $G^{\Sigma }\leq Q\wr
Z_{2}$, where $\mathrm{Soc}(Q)$ is isomorphic to one of the groups $A_{62}$ or $PSL_{2}(61)$ by \cite[Table B.2]{DM}, and hence $31$ divides the order of $G(\Sigma
)$, which is not the case by Lemma \ref{T1even}(2). Thus $d\neq 6$, and hence there is a primitive prime divisor of $\lambda+1=2^{d}-1$, say $z$, by \cite[Theorem II.6.2]{Lu}. If $%
z\mid \left\vert G_{\Delta ,\Delta_{1}}\right\vert $ for some $\Delta
_{1}\in \Sigma \setminus \{\Delta\}$, it follows that there is a $u$-element $\varphi $
of $G_{\Delta ,\Delta _{1}}$ fixing a point $x$ in $\Delta $ and $%
x^{\prime }$ in $\Delta _{1}$. Therefore, it fixes at least one of the
$\lambda =2(2^{d-1}-1)$ blocks incident with $x$ and $x^{\prime }$, say $B$.
Hence, $\varphi $ fixes a further element in $B\cap \Delta $. Thus, $%
\varphi \in G(\Delta )$ by \cite[Theorem 3.5]{He} since $G_{\Delta}^{\Delta}$ is of affine type. However, this is
impossible by Corollary \ref{CFixT}(1) since $u\mid \lambda +1$. Then $z\nmid
\left\vert G_{\Delta ,\Delta _{1}}\right\vert $ and hence $z\mid
\left\vert G_{\Delta }^{\Sigma }:G_{\Delta ,\Delta ^{\prime }}^{\Sigma
}\right\vert $ for each $\Delta
^{\prime}\in \Sigma \setminus \{\Delta\}$ since $G(\Sigma )$ is a $2$-group by Lemma \ref{T1even}(2). Actually, $z\mid
\left\vert G_{\Delta }^{\Sigma }:G(\Delta )^{\Sigma }G_{\Delta ,\Delta
^{\prime }}^{\Sigma }\right\vert $ again by Corollary \ref{CFixT}(1), hence $z\mid \left\vert G_{\Delta
}^{\Sigma }/G(\Delta )^{\Sigma }:G(\Delta )^{\Sigma }G_{\Delta ,\Delta ^{\prime }}^{\Sigma
}/G(\Delta )^{\Sigma }\right\vert $. Thus
\begin{equation*}
P(G_{x}^{\Delta })\leq \left\vert G_{\Delta }^{\Sigma }/G(\Delta )^{\Sigma
}:G(\Delta )^{\Sigma }G_{\Delta ,\Delta ^{\prime }}^{\Sigma }/G(\Delta )^{\Sigma }\right\vert
\leq \left\vert G_{\Delta }^{\Sigma }:G_{\Delta ,\Delta ^{\prime }}^{\Sigma
}\right\vert \text{,}
\end{equation*}%
where $P(G_{x}^{\Delta })$ is the minimal primitive permutation
representation of $G_{x}^{\Delta }$, since $G_{\Delta }^{\Sigma }/G(\Delta
)^{\Sigma }\cong G_{x}^{\Delta }$ by Lemma \ref{T1even}(2). We may choose $\Delta
^{\prime}\in \Sigma \setminus \{\Delta\}$ such that
\begin{equation*}
P(G_{x}^{\Delta })\leq \left\vert G_{\Delta }^{\Sigma }:G_{\Delta ,\Delta
^{\prime }}^{\Sigma }\right\vert \leq 2\frac{2(2^{d-1}-1)}{s-1}
\end{equation*}%
where $s$ denotes the rank of $Q$ on $\Theta$ since $G^{\Sigma} \leq Q \wr S_{2}$ and $\Sigma=\Theta^{2}$. If $d\neq 4$ then $%
P(G_{x}^{\Delta })=2^{d}-1$ by \cite[Theorem 5.2.2]{KL}, hence $s=2$. Then $%
Q $ acts $2$-transitively on the set $\Theta $, with $\left\vert \Theta
\right\vert =2(2^{d-1}-1)$, and hence $Q\wr Z_{2}$ acts as a primitive rank $3$ group on $%
\Sigma =\Theta^{2}$. Moreover, the $\left( Q\wr Z_{2}\right) _{\Delta }$-orbits on $\Sigma \setminus \left\{ \Delta \right\} $ are two of length $%
\left\vert \Theta \right\vert -1$, one of $\left( \left\vert \Theta
\right\vert -1\right) ^{2}$. Each of
these orbits is a union of $G_{\Delta }^{\Sigma }$-orbits, and since each $%
G_{\Delta }^{\Sigma }$-orbits distinct from $\left\{ \Delta \right\} $ is
divisible by $z$, then $z\mid \left\vert \Theta \right\vert -1$. So $z\mid
2^{d}-3$, whereas $z$ is a divisor of $2^{d}-1$. Therefore $d=4$, $s=2,3,4$
and $G_{x}^{\Delta }\cong A_{m}$, where $m =7,8$, since $SL_{4}(2)\cong A_{8}$.
Suppose that $s\geq 3$. Then $\left\vert G_{\Delta }^{\Sigma }:G_{\Delta
,\Delta ^{\prime }}^{\Sigma }\right\vert \leq \frac{28}{s-1}\leq 14$, and
hence $\left\vert G_{\Delta }^{\Sigma }:G_{\Delta ,\Delta ^{\prime
}}^{\Sigma }\right\vert =m $ by \cite{At}. Then $m \mid \left\vert
\Sigma \right\vert -1$ with $m=7,8$, and we reach a contradiction since $\left\vert \Sigma
\right\vert -1=\allowbreak 3\cdot 5\cdot 13$. Thus $s=2$,
and hence $Q$ acts $2$-transitively on $\Theta$. As above, $Q\wr Z_{2}$ acts as a primitive rank $3$ group on $%
\Sigma =\Theta^{2}$, and the $\left( Q\wr Z_{2}\right) _{\Delta }$-orbits on $\Sigma \setminus \{\Delta\}$ are two of length $%
13$ and one of length $13^{2}$. None of these lengths is divisible by $z=5$, and so this case is excluded. Thus only (i) occurs, which is the assertion.
\end{proof}
\bigskip
\begin{remark}
If $\mathcal{D}$ is of type 1 with $v$ even and $G(\Sigma) \neq 1$, it follows from Lemma \ref{T1even}(2) and Theorem \ref{veven} that, $G^{\Sigma}$ is an almost simple subgroup of $GL_{d+t}(2)$ , where $\left\vert\Delta\right\vert =2^{d}$ with $\Delta \in \Sigma$, $\left\vert G(\Sigma)\right\vert=2^{d+t}$ and $\left\vert G(\Sigma) \cap G(\Delta)\right\vert=2^{t}$. However, it is not easy to fully exploit the previous property because it is not easy to control the order of $G(\Sigma) \cap G(\Delta)$ in this case. This motivates our choice of an alternative proof in which the embedding of $G^{\Sigma}$ in $GL_{d+t}(2)$ is partially considered.
\end{remark}
\bigskip
\begin{theorem}
\label{D1PqP}If $\mathcal{D}$ is of type 1 and $v$ is odd, then $G$ acts
point-quasiprimitively on $\mathcal{D}$.
\end{theorem}
\begin{proof}
Assume that $\mathcal{D}$ is of type 1. Recall that $G(\Sigma )$ is an
elementary abelian $p$-group acting regularly
on $\Delta $, $p$ is and odd prime and $G^{\Sigma
}\leq GL_{d}(p)$ by Lemma \ref{strike}. Thus $\lambda+2=\left \vert \Delta \right \vert=p^{d}$, $d \geq 1$, and hence $\lambda =p^{d}-2$. Therefore, $%
(p^{d}-1)(p^{d}-2)$ divides $\left\vert G_{x}\right\vert $, and hence $%
\left\vert G\right\vert $, as $k\mid \left\vert G_{x}\right\vert $.
Also, $(p^{d}-1)(p^{d}-2)^{3}\mid \left\vert G^{\Sigma }\right\vert $ since $\left\vert\Sigma \right\vert=(p^{d}-2)^{2}$.
If $d=2$ then $(p^{2}-1)(p^{2}-2)^{3}\mid \left\vert G^{\Sigma}\right\vert $ with $%
G^{\Sigma }\leq GL_{2}(p)$, which is a contradiction. Thus $d>2$, and hence $%
\Phi _{d}^{\ast }(p)>1$ by \cite[Theorem II.6.2]{Lu} since $p$ is odd. Then $%
G^{\Sigma }$ is an irreducible subgroup of $GL_{d}(p)$ by \cite[Theorem
3.5(iv)]{He}. For each divisor $m$ of $d$ the group $\Gamma L_{d/m}(p^{m})$
has a natural irreducible action. We may choose $m$ to be minimal such that $%
G^{\Sigma }\leq \Gamma L_{d/m}(p^{m})$. Set $K^{\Sigma }=G^{\Sigma }\cap
GL_{d/m}(p^{m})$. Then $%
\Phi _{d}^{\ast }(p)\frac{(p^{d}-2)^{3}}{((p^{d}-2)^{3},d)}\mid \left\vert
K^{\Sigma }\right\vert $ by \cite[Proposition 5.2.15.(ii)]{KL}. Easy
computations show that the order of $GL_{d/m}(p^{m})$, and hence that of $%
K^{\Sigma }$, is not divisible by $p^{d}\Phi _{d}^{\ast }(p)\frac{(p^{d}-2)}{%
((p^{d}-2),d)}$ for $(d/m,p^{m})=(3,5^{2}),(4,3),(6,3),(6,5),(9,3)$. So
these cases are excluded. Thus, bearing in
mind the minimality of $m$ and the fact that $p$ is odd, \cite[Theorem 3.1]{BP} implies that $K^{\Sigma }$ contains a normal subgroup isomorphic to one of the groups $%
SL_{d/m}(p^{m}),Sp_{d/m}(p^{m}),\Omega _{d/m}^{-}(p^{m})$, or $%
SU_{d/m}(p^{m/2})$ with $d/m$ odd. Since $\left\vert
G^{\Sigma }:G_{\Delta }^{\Sigma }\right\vert =\lambda ^{2}$, and $\lambda
=p^{d}-2$ with $p$ odd, it follows that $G_{\Delta }^{\Sigma }$ is a maximal
parabolic subgroup of $G^{\Sigma }$ by Proposition \ref{DivanDan} and \cite[Theorem 1.6]{Se}. Also, $\Phi _{d}^{\ast
}(p)\mid \left\vert G_{\Delta }^{\Sigma }\right\vert $ since $\left( \Phi
_{d}^{\ast }(p),p^{d}-2\right) =1$, but this contradicts \cite[Theorem
3.5(iv)]{He} applied to $G_{\Delta }^{\Sigma }$. Thus $G(\Sigma )=1$, and
hence $G$ acts point-quasiprimitively on $\mathcal{D}$ by Proposition \ref%
{DivanDan}(2).
\end{proof}
\bigskip
\subsection{Designs of type 2 and quasiprimitivity}
\bigskip
\begin{lemma}\label{firstpart}
If $\mathcal{D}$ is of type 2 with $\lambda =2w^{2}$, where $w$ is
odd, $w\geq 3$, and $2(w^{2}-1)$ is a square, then $G$ acts point-quasiprimitively on $\mathcal{D}$.
\end{lemma}
\begin{proof}
Suppose that $\mathcal{D}$ is of type 2 with $\lambda =2w^{2}$, where $u$ is
odd, $u\geq 3$, and $2(w^{2}-1)$ is a square. Then $\left\vert \Delta \right\vert=\lambda/2+1=w^{2}+1$ is even.
If $\mathrm{Soc}(G_{\Delta}^{\Delta})$ is an elementary abelian $2$-group, then $w^{2}+1=2^{s}$ for some $s\geq 1$. However, it has no integer solutions for \cite[B1.1]{Rib}. Thus $\mathrm{Soc}(G_{\Delta}^{\Delta})$ is non-abelian simple, and hence $\mathrm{Soc}(G_{\Delta}^{\Delta})$ is isomorphic to one of the groups $ A_{w^{2}+1}$, $PSL_{d}(s)$ with $w^{2}+1=\frac{s^{d}-1}{s-1}$, $d \geq 2$ and $(d,s)\neq (2,2),(2,3)$, or $PSU_{3}(w^{2/3})$ by \cite[List (A)]{Ka} since $\left\vert \Delta \right\vert=w^{2}+1$. In the second case one has $w^2=s\frac{s^{d-1}-1}{s-1}$, and so $s$ is an even power of an odd prime. Therefore $\left(\frac{w}{s^{1/2}}\right)^{2}=\frac{s^{d-1}-1}{s-1}$, and hence $d=2$ by \cite[A7.1, A8.1 and B1.1]{Rib}, and so $\mathrm{Soc}(G_{\Delta}^{\Delta}) \cong PSL_{2}(w^{1/2})$. Also, $w^{2}-1 \neq 2^{t}$ for some $t \geq 1$. Indeed, if not so, then $t=w=3$ by \cite[B1.1]{Rib}, and hence $(\lambda,\left\vert \Delta \right \vert, \left\vert \Sigma \right \vert)=(18,10,145)$ and $\mathrm{Soc}(G_{\Delta}^{\Delta}) \cong A_{5}$ and $\mathrm{Soc}(G^{\Sigma}) \cong A_{145}$ by \cite[Table B.4]{DM}. Then $A_{144} \trianglelefteq G_{\Delta}^{\Sigma} \leq S_{144}$ but this contradicts (\ref{salvaMAOL}) since $G_{\Delta}^{\Delta} \leq S_{5}$. Thus in each case $\mathrm{Soc}(G_{\Delta}^{\Delta})$ contains elements of order an odd prime divisor of $w^{2}-1$, and each of these elements fixes at least two points on $\Delta$.
If $G(\Sigma)\neq 1$ then $\mathrm{Soc}(G_{\Delta}^{\Delta}) \trianglelefteq G(\Sigma)^{\Delta}$ by Proposition \ref{DivanDan}(1). Now, let $\eta \in G(\Sigma)$ be an element of order an odd prime divisor of $w^{2}-1$, which exists by the previous argument. Then, for each $\Delta \in \Sigma$ either $\eta \in G(\Delta)$ or $\eta$ induces an element of $G(\Sigma)^{\Delta}$ fixing at least two points of $\Delta$. Therefore $\left\vert \mathrm{Fix}(\eta)\right\vert \geq 2\left\vert\Sigma \right\vert=\lambda^2-2\lambda+2$, but his contradicts Lemma \ref{Fixpoints}(2). Thus $G(\Sigma)= 1$, and hence $G$ acts point-quasiprimitively on $\mathcal{D}$ by
Proposition \ref{DivanDan}(2).
\end{proof}
\begin{theorem}
\label{GSigma}If $\mathcal{D}$ is of type 2 then $G$ acts
point-quasiprimitively on $\mathcal{D}$.
\end{theorem}
\begin{proof}
Suppose that $\mathcal{D}$ is of type 2. If $\lambda =2w^{2}$, where $w$ is
odd, $w\geq 3$, and $2(w^{2}-1)$ is a square, the assertion follows from Lemma \ref{firstpart}. Hence, we may assume that $\lambda \equiv 0 \pmod{4}$ by Theorem \ref{PZM}. Hence, $v$ is odd.
Suppose that $G(\Sigma )\neq 1$.
Then $G(\Sigma )\cong Soc(G_{\Delta })$, where $\Delta \in \Sigma $, is an
elementary abelian $p$-group of order acting regularly on $\Delta $, where $%
p $ is odd, and $G^{\Sigma }\leq GL_{d}(p)$, with $p^{d}=\left\vert \Delta
\right\vert $ by Lemma \ref{strike} since $v$ is odd. Then $\lambda =2(p^{d}-1)$ and $%
k=\lambda ^{2}/2=2(p^{d}-1)^{2}$. Then $\left( p^{d}-1\right) ^{2}\mid
\left\vert G_{B}\right\vert $, where $B$ is any block of $\mathcal{D}$,
since $G_{B}$ acts transitively on $B$. Then $\left( p^{d}-1\right) ^{2}$
divides $\left\vert G^{\Sigma }\right\vert $ since $\left\vert G(\Sigma
)\right\vert =p^{d}$. Now, we may proceed as in Theorem \ref{D1PqP} and we
see that no admissible groups arise in this case as well. Thus $G(\Sigma
)=1 $, and hence $G$ acts point-quasiprimitively on $\mathcal{D}$ by
Proposition \ref{DivanDan}(2).
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \protect\ref{AQP}]
The assertion follows from Theorems \ref{veven} and \ref{D1PqP} for $\mathcal{D}$ of type 1, Theorem \ref{GSigma} for $\mathcal{D}$ of type 2.
\end{proof}
\bigskip
\section{Reduction to the almost simple case}
In this section we analyze the case where $G$ acts point-quasiprimitively on
$\mathcal{D}$. In this case $G=G^{\Sigma }$. The main tool used to tackle this
case is the O'Nan-Scott Theorem for quasiprimitive groups proven in \cite{P1} and reported below for reader's convenience. We investigate
each of the seven possibilities for $G^{\Sigma }$ provided in the above mentioned theorem by adapting the techniques
developed in \cite{ZZ}, and we show that $\mathrm{Soc}(G^{\Sigma })$ is a
non-abelian simple group. This fact, together with the conclusions of
Theorem \ref{AQP}, yields the following result.
\begin{theorem}
\label{ASPQP}$G^{\Sigma }$ is an almost simple group acting primitively on \ $%
\Sigma $. Moreover, one of the following holds:
\begin{enumerate}
\item $G(\Sigma )=1$ and $G$ acts point-quasiprimitively on $\mathcal{D}$.
\item $G(\Sigma )$ is a self-centralizing elementary abelian $2$%
-subgroup of $G$. Also, the following hold:
\begin{enumerate}
\item $\mathcal{D}$ is a symmetric $2$-$(2^{d+2}(2^{d-1}-1)^{2},\allowbreak
2\left( 2^{d}-1\right) \left( 2^{d-1}-1\right) ,2(2^{d-1}-1))$ design with $%
d\geq 4$;
\item $G_{\Delta }^{\Sigma }/G(\Delta )^{\Sigma }\cong G_{x}^{\Delta }$ and
either $G_{x}^{\Delta }\cong SL_{d}(2)$, or $G_{x}^{\Delta }\cong A_{7}$ and
$d=4$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\bigskip
We only need to prove that $G^{\Sigma}$ is almost simple, the remaining parts of Theorem \ref{ASPQP} have been already proven in Theorem \ref{AQP}
\bigskip
In the sequel we denote $\mathrm{Soc}(G)$ simply by $L$ and let $x\in
\mathcal{P}$. Then $L\cong T^{h}$ with $h\geq 1$, where $T$ is a simple
group. By \cite[Theorem 1]{P1}, one of the following holds:
\begin{enumerate}
\item[I.] \emph{Affine groups.} Here $T\cong Z_{p}$ for some prime $p$, and $%
L$ is the unique minimal normal subgroup of $G$ and is regular on $\mathcal{P%
}$ of size $p^{h}$. The set $\mathcal{P}$ can be identified with $L\cong Z_{p}^{h}$ so that $%
G\leq AGL_{h}(p)$ with $L$ the translation group and $G_{x}=G\cap GL_{h}(p)$
acting irreducibly on $L$. Moreover, $G$ acts primitively on $\mathcal{P}$.
\item[II.] \emph{Almost simple groups.} Here $h=1$, $T$ is a non-abelian
simple group, $T\trianglelefteq G\leq \mathrm{Aut}(T)$ and $G=TG_{x}$.\emph{%
\ }
\item[III.] In this case $L\cong T^{h}$ with $h\geq 2$ and $T$ is a non-abelian
simple group. We distinguish three types:
\item[III(a).] \emph{Simple diagonal action.} Define%
\small
\begin{equation*}
W=\left\{ (a_{1},...,a_{h})\cdot \pi :a_{i}\in \mathrm{Aut}(T),\pi \in
S_{h},a_{i}\equiv a_{j}\pmod{\mathrm{Inn}(T)}\text{ for all }i,j\right\} \text{,%
}
\end{equation*}%
\normalsize
where $\pi \in S_{h}$ just permutes the components $a_{i}$ naturally. With
the usual multiplication, $W$ is a group with socle $L\cong T^{h}$, and $%
W=L.(\mathrm{Out}(T)\times S_{h})$. The action of $W$ on $\mathcal{P}$ is
defined by setting%
\begin{equation*}
W_{x }=\left\{ (a,...,a)\cdot \pi :a\in \mathrm{Aut}(T),\pi \in
S_{h}\right\} \text{.}
\end{equation*}%
Thus $W_{x }\cong \mathrm{Aut}(T)\times S_{h}$, $L_{x }\cong T$
and $\left\vert \mathcal{P}\right\vert =\left\vert T\right\vert ^{h-1}$.
For $1\leq i\leq h$ let $T_{i}$ be the subgroup of $L$ consisting of the $h$%
-tuples with $1$ in all but the $i$-th component, so that $T_{i}\cong T$ and
$L\cong T_{1}\times \cdots \times T_{h}$. Put $\mathcal{T}=\left\{
T_{1},...,T_{h}\right\} $, so that $W$ acts on $\mathcal{T}$. We say
that subgroup $G$ of $W$ is of type III(a) if $L\leq G$ and, letting $P$ the permutation group of $G^{%
\mathcal{T}}$, one of the following holds:
\begin{enumerate}
\item[(i)] $P$ is transitive on $\mathcal{T}$;
\item[(ii)] $h=2$ and $P=1$.
\end{enumerate}
We have $G_{x }\leq \mathrm{Aut}(T)\times P$ and $G\leq L.(\mathrm{Out}%
(T)\times P)$. Moreover, in case (i) $L$ is the unique minimal normal
subgroup of $G$ and $G$ is primitive on $\mathcal{P}$ if and only if $P$ is
primitive on $\mathcal{T}$. In case (ii) $G$ has two minimal normal
subgroups $T_{1}$ and $T_{2}$, both regular on $\mathcal{P}$, and $G$ is
primitive on $\mathcal{P}$.
\item[III(b).] \emph{Product action.} Let $H$ be a quasiprimitive permutation group on a set $%
\Gamma $ of type II or III(a). For $l>1$, let $W=H\wr S_{l}$, and take $W$
to act $\Lambda =\Gamma ^{l}$ in its natural product action. Then for $y\in
\Gamma $ and $z=(y,...,y)\in \Lambda $ we have $W_{z}=H_{y}\wr S_{l}$ and $%
\left\vert \Lambda \right\vert =\left\vert \Gamma \right\vert ^{l}$. If $K$
is the socle $H$, then the socle $L$ of $W$ is $K^{l}$.
Now $W$ acts naturally on the $l$ factors in $K^{l}$, and we say that a
subgroup $G$ of $W$ is of type III(b) if $L\leq G$, $G$ acts transitively on
these $l$ factors, and one of the following holds:
\begin{enumerate}
\item[(i)] $H$ is of type II, $K=T$, $h=l$, and $L$ is the unique minimal normal
subgroup $G$; further $\Lambda $ is a $G$ invariant partition of $\mathcal{P}
$ and, for $x$ in the part $z\in \Lambda $, $L_{z}=T_{y}^{h}<L$ an for some
non-trivial normal subgroup $R$ of $T_{y}$, $L_{x}$ is a subdirect product
of $R^{k}$, that is $L_{x}$ projects surjectively on each of the direct factors $%
R$.
\item[(ii)] $H$ is of type III(a), $\mathcal{P}=\Lambda $, $K\cong T^{h/l}$
and both $G$ and $H$ have $m$ minimal normal subgroups where $m\leq 2$ . If $%
m=2$ then each of the two minimal normal subgroups of $G$ is regular on $%
\mathcal{P}$.
\end{enumerate}
\item[III(c).] \emph{Twisted wreath action.} Here $G$ is a twisted wreath
action $T\wr _{\phi }P$ defined as follows. Let $P$ have a transitive action
on $\left\{ 1,...,h\right\} $ and let $Q$ be the stabilizer $P_{1}$ of
the point $1$ in this action. We suppose that there is an homomorphism $\phi
:Q\rightarrow \mathrm{Aut}(T)$ such that $\mathrm{core}_{P}\left( \phi
^{-1}(\mathrm{Inn}(T))\right) =\cap _{x\in P}\phi ^{-1}(\mathrm{Inn}%
(T))^{x}=\left\{ 1\right\} $. Define%
\begin{equation*}
L=\left\{ f:P\rightarrow T:f(\alpha \beta )=f(\alpha )^{\phi (\beta )}%
\text{ for all }\alpha \in P\text{, }\beta \in Q\right\} \text{.}
\end{equation*}%
Then $B$ is a group under the pointwise multiplication, and $L\cong T^{l}$.
Let $P$ act on $L$ by%
\begin{equation*}
f^{ }(\gamma )=f(\alpha \gamma )\text{ for }\alpha ,\gamma \in P\text{.%
}
\end{equation*}%
Define $T\wr _{\phi }P$ to be the semidirect product of $L$ by $P$ with this
action, and define the action on $\mathcal{P}$ by setting $G_{x}=P$. Then $%
\left\vert \mathcal{P}\right\vert =\left\vert T\right\vert ^{h}$, and $L$ is
the unique minimal normal subgroup of $G$ and acts regularly on $\mathcal{P}$%
. We say that $G$ is of type III(c).
\end{enumerate}
\bigskip
\begin{theorem}\label{ASt1}
If $\mathcal{D}$ is of type 1 then $G$ is almost simple.
\end{theorem}
\begin{proof}
Suppose that $\mathcal{D}$ is of type 1. Case (I) is ruled out since $G$
acts imprimitively on $\mathcal{P}$ by our assumptions. Suppose that $G$ is
of type III(a) or III(c). Then $(\lambda +2)\lambda ^{2}=\left\vert \mathcal{%
P}\right\vert =\left\vert T\right\vert ^{j}$ where $T$ is non-abelian simple and $j=h-1$ or $h$, with $%
h\geq 2$, respectively. Hence, $\lambda $ is even.
If $j=1$, then $%
h=j+1=2$ and hence $G$ acts primitively on $\mathcal{P}$ (see III(a)), which is a
contradiction. Thus $j>1$. Note that $\left\vert
x^{T_{i}}\right\vert =\left\vert T\right\vert $ since $T_{i}$ acts
semiregularly on $\mathcal{P}$, where $T_{i}$ is the subgroup of $L$ consisting of the $h$%
-tuples with $1$ in all but the $i$-th component, $T_{i}\cong T$. Moreover, $x^{T_{i_{1}}}\cap
x^{T_{i_{2}}}=\left\{ x\right\} $ for each $i_{1},i_{2}$ such that $%
i_{1}\neq i_{2}$. Let $w\in x^{T_{1}}\setminus \left\{ x\right\} $, then $w^{G_{x}}$ is the disjoint union of $w^{G_{x}}\cap x^{T_{i}}$ for $%
1\leq i\leq h$. Therefore $\left\vert w^{G_{x}}\right\vert =\left\vert
w^{G_{x}}\cap x^{T_{i}}\right\vert h$ since $G_{x}$ permutes transitively $T_{1},...,T_{h}$. On the other hand $\lambda+1 \mid \left\vert w^{G_{x}}\right\vert $ by Lemma \ref{PP}, hence $\lambda +1\leq \left\vert
w^{G_{x}}\cap x^{T_{i}}\right\vert h\leq \left\vert T\right\vert h$. Thus $(\lambda +2)\lambda ^{2}=\left\vert T\right\vert ^{j}$ implies
\begin{equation*}
\left\vert T\right\vert ^{j}\leq (\left\vert T\right\vert
h-1)^{2}(\left\vert T\right\vert h+1)\leq \left\vert T\right\vert ^{3}h^{3},
\end{equation*}%
and hence $j=2$ or $3$, or $j=4$ and $T \cong A_{5}$, as $j>1$ and $\left\vert T\right\vert \geq 60$. The latter is ruled out since it does not provide integer solutions for $(\lambda +2)\lambda ^{2}=\left\vert T\right\vert ^{4}$. If $j=3$ then $\left\vert T\right\vert
>\lambda >\left\vert T\right\vert -2$ since $(\lambda +2)\lambda
^{2}=\left\vert T\right\vert ^{3}$, and hence $\lambda =\left\vert
T\right\vert -1$, but this contradicts $\lambda$ even. Thus, $j=2$. If $h=3$ then $G$ acts primitively on $\mathcal{P}$. Indeed, $G$ is as in III(a) and $Z_{3} \leq P \leq S_{3}$. Thus $j=h=2$ and $G$ is as in III(c). Moreover, $\lambda +1\mid \left\vert
w^{G_{x}}\cap x^{T_{1}}\right\vert$ for each $w\in x^{T_{1}}\setminus \left\{ x\right\} $ since $\lambda$ is even. Then there is $\theta \geq 1$ such that $\left\vert T\right\vert =\theta
(\lambda +1)+1$ since $\left\vert x^{T_{1}}\right\vert=\left\vert T \right \vert$. Then $\lambda ^{2}(\lambda +2)=\left\vert T\right\vert ^{2}$
implies
\begin{equation}
\left( \lambda +1\right) \left( \left( \lambda +1\right) \lambda -1\right)
=\theta ^{2}(\lambda +1)^{2}+2\theta (\lambda +1) \label{theta}
\end{equation}%
and hence $\lambda (\lambda +1)-1=\theta ^{2}(\lambda +1)+2\theta $. Then
\[
\theta =\frac{(\lambda +1)t-1}{2}
\]%
for some $t\geq 1$. If $t\geq 2$ then $\theta >\lambda $ and we reach a
contradiction, thus $t=1$ and hence $\theta =\lambda /2$ which substituted in
(\ref{theta}) yields a contradiction too. Therefore, $G$ is not of type III(a) or III(c).
Suppose that $G$ is of type III(b.ii). Then $G_{z}\leq W_{z}=H_{y}\wr S_{l}$
where $z=(y,y,...,y)$, and denoted by $\mu $ the number of $W_{z}$-orbits on $%
\Delta $, we see that $\mu \geq 4$ by \cite[Corollary 1.9]{DGLPP}.
Let $y_{1}\in \Delta $, $y_{1}\neq y$ such that $\left\vert y_{1}^{H}\right\vert
\leq \frac{\left\vert \Delta \right\vert -1}{\mu -1}$, and let $z_{1}=(y_{1},y,\dots,y)$. Since $\left(H_{y,y_{1}}\times H_{y}\times \cdots \times H_{y} \right):S_{l-1}\leq
W_{z,z_{1}}$, it results
\begin{equation*}
\left\vert z_{1}^{G_{z}}\right\vert \leq \left\vert z_{1}^{W_{z}}\right\vert
\leq \frac{\left\vert H_{y}\right\vert ^{l}\left( l!\right) }{\left\vert
H_{y,y_{1}}\right\vert \left\vert H_{y}\right\vert ^{l-1}\left(
(l-1)!\right) }=l\left\vert y_{1}^{H}\right\vert \leq \frac{l\left(
\left\vert \Delta \right\vert -1\right) }{\mu -1}\text{,}
\end{equation*}%
and since $\lambda +1\mid \left\vert z_{1}^{G_{z}}\right\vert $ by Lemma \ref{PP}, it follows
that%
\begin{equation*}
\lambda +1\leq \frac{l\left( \left\vert \Delta \right\vert -1\right) }{\mu -1%
}\text{.}
\end{equation*}%
Then $\lambda $ is even since $(\lambda +2)\lambda ^{2}=\left\vert \mathcal{P}\right\vert =\left(
\left\vert T\right\vert ^{h/l-1}\right) ^{l}=\left\vert T\right\vert ^{h-l}$, $l>1$. Thus
\begin{equation*}
\left\vert T\right\vert ^{l(h/l-1)/3}-1\leq \frac{l\left( \left\vert T\right\vert
^{h/l-1}-1\right) }{\mu -1}< \frac{h\left( \left\vert T\right\vert
^{h/l-1}-1\right) }{3}\text{.}
\end{equation*}%
and hence $l=2,3$, as $l>1$. If $l=3$ then $\left\vert T\right\vert ^{h/3-1}>\lambda >\left\vert
T\right\vert ^{h/3-1}-2$ and hence $\lambda =\left\vert T\right\vert ^{h/3-1}-1$ with $h>l$, whereas $\lambda$ is even. Thus $l=2$, and since $\lambda$ is even and $ \left(\Delta\setminus \{y\} \times \{y \} \right) \cup \left( \{ y\} \times \Delta\setminus \{y\} \right)$ is union of some non-trivial $G_{z}$-orbits, being $G_{z} \leq H_{z}\wr S_{l}$, it follows from Lemma \ref{PP} that $ \lambda+1 \mid \left\vert T\right\vert ^{h/2-1}-1$. Now, we may apply the final argument used to rule out III(a) with $\left\vert T\right\vert ^{h/2-1}$ in the role of $\left\vert T\right\vert$ to exclude this case as well.
Finally, assume that $G$ is of type III(b.i). Then $h=l$ and $T^{h}$ is the
unique minimal normal subgroup of $G$. In this case $\Sigma$ can be identified with the Cartesian product $\Gamma^{h}$, hence each $\Delta \in \Sigma$ corresponds to a unique $h$-tuple of elements of $\Gamma$. Therefore, it results that $\left\vert
\Sigma\right\vert=\lambda ^{2}=\left\vert
\Gamma\right\vert ^{h}$. Let $y\in \Gamma $ and $\Delta=(y,...,y)\in \Sigma $. Then
$$ \bigcup_{i=1}^{h} \left( \{y\}\times \cdots \times \Gamma_{i}\setminus \{y\} \times \cdots \times \{y\}\right) $$
is a union of some non-trivial $G_{\Delta}$-orbits and ultimately of some non-trivial $G_{x}$-orbits, where $x \in \Delta$. Then $\lambda +1\mid (\lambda+2)h\left( \left\vert \Gamma\right\vert -1\right)$ by Lemma \ref{PP}. Thus $\lambda
+1\mid (\lambda +2)h\left( \lambda ^{2/h}-1\right) $, and hence $\lambda
+1\mid h\left( \lambda ^{2/h}-1\right) $.
If $h=2$ then $\lambda +1\mid 4$, whereas $\lambda >10$. Thus $%
h\geq 3$ and hence $\lambda ^{1/3}\leq \lambda ^{1-2/h}\leq h$. Therefore $%
5^{h/3}\leq \left\vert T:T_{\Delta }\right\vert ^{h/3}\leq h^{2}$ since $T$ is a non-abelian simple group and $\left\vert T:T_{\Delta }\right\vert=\lambda^{2}$, and so $%
h\leq 7$. Actually, $\left\vert T:T_{\Delta }\right\vert ^{h/2(1-2/h)}\leq h$
implies $h=3$, and hence $\lambda +1\mid 3\left( \lambda
^{2/3}-1\right) $ which is impossible since $\lambda>10$. This completes the proof.
\end{proof}
\bigskip
\begin{theorem}\label{ASt2}
If $\mathcal{D}$ is of type (2) then $G$ is almost simple.
\end{theorem}
\begin{proof}
If $\mathcal{D}$ is of type (2) then $v \not \equiv 0 \pmod{4}$ (see Theorem \ref{PZM}). Case (I) is ruled out since $G$ acts imprimitively on $\mathcal{P}$
by our assumptions. Also $v\neq $ $\left\vert T\right\vert ^{j}$ with $%
j\geq 1$, where $T$ is non-abelian simple, and hence $G$ is not of type
III(a) or III(c). Assume that $G$ is of type III(b.i). Arguing as in Theorem \ref{ASt1} we see that
\begin{equation*}
\frac{\lambda ^{2}-2\lambda +2}{2}=\left\vert \Sigma \right\vert
=\left\vert \Gamma \right\vert ^{h}\text{ and } \lambda +1\mid (\lambda /2+1)h\left( \left\vert
\Gamma \right\vert -1\right)
\end{equation*}%
where $h\geq 2$. It follows that $\lambda +1\mid h\left(
\left\vert \Gamma \right\vert -1\right) $. Since $\left\vert \Gamma
\right\vert ^{l}-1=\allowbreak \frac{1}{2}\lambda \left( \lambda -2\right) $%
, it follows $\lambda +1\mid 3h$ and hence $2\cdot 60^{h}\leq 2\left\vert
\Gamma \right\vert ^{h}<9h^{2}$, which is impossible for $h\geq 2$.
Assume that $G$ is of type III(b.ii). Then
\begin{equation}
\left( \frac{\lambda +2}{2}\right) \left( \frac{\lambda ^{2}-2\lambda +2}{2}%
\right) =\left\vert \mathcal{P} \right\vert =\left\vert \Gamma \right\vert ^{l}%
\text{,} \label{ded}
\end{equation}%
where $\left\vert \Gamma \right\vert =\left\vert T\right\vert ^{k/l-1}$
with $k/l\geq 2$, and from (\ref{ded}) we derive that $v \equiv 0 \pmod{4}$, a
contradiction. This completes the proof.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \ref{ASPQP}]
The assertion immediately follows from Theorems \ref{AQP}, \ref{ASt1} and \ref{ASt2}.
\end{proof}
\bigskip
\section{Reduction to the case $\lambda \leq 10$}
Let $L$ be the preimage in $G$ of $\mathrm{Soc}(G^{\Sigma })$. Hence $%
L^{\Sigma }\trianglelefteq G^{\Sigma }\leq \mathrm{Aut}(L^{\Sigma })$ with $L^{\Sigma }$ non-abelian simple by Theorem \ref{ASPQP}.
Moreover, $G=G^{\Sigma }$ and $L=L^{\Sigma }$ when $G(\Sigma)=1$. The first part of this section is devoted to prove that either $L_{\Delta
}^{\Delta }$ acts $2$-transitively on $\Delta $, where $\Delta \in \Sigma $, or $G(\Sigma)=1$, $Soc(G_{\Delta }^{\Delta })< L_{\Delta }^{\Delta } \leq G_{\Delta }^{\Delta } \leq A \Gamma L_{1}(u^{h})$, where $u^{h}=\left\vert \Delta \right\vert$. Then we prove that either $\left\vert L^{\Sigma }\right\vert \leq \left\vert L_{\Delta
}^{\Sigma }\right\vert ^{2}$ or $\left\vert L\right\vert \leq 4\left\vert L_{\Delta }^{\Delta
}\right\vert^{2} \left\vert \mathrm{Out}(L)\right\vert^{2}$ respectively.
As we will see, these constraints on $L_{\Delta }^{\Sigma }$ are combined with \cite{AB} and \cite{LS} allows us to to completely classify $\mathcal{D}$. Finally, the analysis of $2$-designs $\mathcal{D}$ of type 1 and 2 is carried out in separate subsections.
\bigskip
In the sequel the minimal primitive permutation representation of a non-abelian simple
group $\Gamma $ will be denoted by $P(\Gamma )$. It is known that $P(A_{\ell })=\ell \geq 5$, whereas $P(\Gamma )$ is determined in \cite{At}, \cite[Theorem 5.2.2]{KL}
and in \cite{Va1, Va2, Va3} according to whether $\Gamma $ is sporadic,
classical or exceptional of Lie type respectively. The following technical lemma is
useful to show that $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta
$ provided that $G_{\Delta}^{\Delta}$ is not a semilinear $1$-dimensional group.
\bigskip
\begin{lemma}
\label{MPPRG}If $\Gamma $ is a non-abelian simple group non-isomorphic to
$PSL_{2}(q)$ and such that $P(\Gamma )<2\left( \left\vert \mathrm{Out}(\Gamma
)\right\vert +1\right) \left\vert \mathrm{Out}(\Gamma )\right\vert $, then
one of the following holds:
\begin{enumerate}
\item $\Gamma \cong A_{\ell }$ with $\ell =7,8,9,10,11$;
\item $\Gamma \cong M_{11}$;
\item $\Gamma \cong PSL_{3}(q)$ with $q=4,7,8,16$;
\item $\Gamma \cong PSp_{4}(3)$;
\item $\Gamma \cong PSU_{n}(q)$ with $(n,q)=(3,5),(3,8),(4,3)$;
\item $\Gamma \cong P\Omega_{8}^{+}(3)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $\Gamma \cong A_{\ell }$ with $\ell \geq 5$. Then $\ell \neq 5,6$
since $A_{5} \cong PSL_{2}(5)$ and $A_{6} \cong PSL_{2}(9)$. Then $7 \leq \ell
=P(\Gamma )<12$ and we obtain (1).
If $\Gamma $ is sporadic then $P(\Gamma )<12$, and hence $\Gamma \cong
M_{11} $ by \cite{At}, which is (2).
If $\Gamma $ is a simple exceptional group of Lie type then $P(\Gamma)$ is provided in \cite{Va1, Va2, Va3}, and it is easy to check that no cases arise.
Finally, suppose that $\Gamma$ is a simple classical group.
Assume that $\Gamma \cong PSL_{n}(q)$, where $q=p^{f}$, $f \geq 1$, and $n \geq 3$. We may also assume that $(n,q) \neq (4,8)$ since $PSL_{4}(2)\cong
A_{8}$ was analyzed above. Then $P(\Gamma )=\frac{q^{n}-1}{q-1}$ by \cite[Theorem 5.2.2]{KL} since $n \geq 3$, and hence
\begin{equation}
\frac{q^{n}-1}{q-1}<2(2(n,q-1)f+1)\cdot 2(n,q-1)f\text{.} \label{prvaeq}
\end{equation}
Assume that $p^{f}\geq 2f(f+1)$. Then $\frac{q^{n}-1}{q-1}<4q(q-1)^{2}$ and hence $n\leq 4$. Actually, $(n,q)=(3,7)$ by (\ref{prvaeq}) since $n \geq 3$.
Assume that $p^{f}<2f(f+1)$. Then either $p=2$ and $2\leq f\leq 6$, or $p=3$
and $f=1,2$. Hence, $n=3$ and $q=4,8,16$ by (\ref{prvaeq}) since $n \geq 3$ and since $PSL_{3}(2) \cong PSL_{2}(7)$ cannot occur. Thus, we get (3).
We analyze the remaining classical groups by proceeding as in the $PSL_{n}(q)$-case. It is a routine exercise to show that one obtains $PSp_{4}(2)^{\prime }$, $%
PSp_{4}(3)$, $PSU_{n}(q)$ with $(n,q)=(3,5),(3,8),(4,2),(4,3)$, and $P\Omega_{8}^{+}(3)$. Since $PSp_{4}(2)^{\prime }\cong PSL_{2}(9)$ this
case is excluded, whereas the remaining ones yield (4), (5) and (6) respectively, bearing in mind that $PSU_{4}(2) \cong PSp_{4}(3)$.
\end{proof}
\bigskip
\begin{lemma}\label{njami}
Let $x \in \Delta$, if $G(\Sigma)=1$ and $\mathrm{Soc}(
G^{\Delta}_{\Delta}) \cong (Z_{u})^{h}$, where $u$ is an odd prime, then $u^{h} \neq 3^{2},5^{2},7^{2},11^{2}$, $19^{2},23^{2},29^{2},59^{2}$ and $3^{4}$
\end{lemma}
\begin{proof}
Suppose the $u^{h} = 3^{2},5^{2},7^{2},11^{2},19^{2},23^{2},29^{2},59^{2}$ or $3^{4}$. Then $\lambda=u^{h}-2$ or $2(u^{h}-1)$ according to whether $\mathcal{D}$ is of type $i=1$ or $2$ respectively. Now, Table \ref{tav0} contains the admissible pairs $(\lambda, \left\vert \Sigma \right\vert)_{i}$, where $i=1,2$.
\tiny
\begin{table}[h!]
\caption{{\protect\small {Admissible $(\lambda, \left\vert \Sigma \right\vert)_{i}$ for $\mathcal{D}$ of type $i=1,2$ }}}
\label{tav0}\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& $3^{2}$ & $5^{2}$ & $7^{2}$ & $11^{2}$ & $19^{2}$ \\
\hline
$(\lambda, \left\vert \Sigma \right\vert)_{1}$& $(7,7^{2})$ & $(23,23^{2})$& $(47,47^{2})$& $(7\cdot17,7^{2}\cdot17^{2})$ & $(17,17^{2})$ \\
\hline
$(\lambda, \left\vert \Sigma \right\vert)_{2}$ & $(16,113)$ & $(48,1105)$ & $(96,4513)$& $(7\cdot 103,53\cdot4877)$ & $(240,13^{4})$\\
\hline
\hline
& $23^{2}$ & $29^{2}$ & $59^{2}$ & $3^{4}$ & \\
\hline
$(\lambda, \left\vert \Sigma \right\vert)_{1}$ & $(17\cdot 31,17^{2}\cdot 31^{2})$& $(3^{3},3^{6})$ & $(3 \cdot 19, 3^{2} \cdot 19^{2})$ & $(79,79^{2})$&\\
\hline
$(\lambda, \left\vert \Sigma \right\vert)_{2}$ & $(1056,556513)$ & $(41^{2}, 17 \cdot 82913)$& $( 2^{4}\cdot 3\cdot 5 \cdot 29, 101 \cdot 149 \cdot 1609)$ & $(160,12641)$ &\\
\hline
\end{tabular}
\end{table}
\normalsize
From Table \ref{tav0} we deduce that $\left\vert \Sigma \right\vert$ is either a power of a prime or a squarefree number. It is straightforward to check that no admissible cases occur by \cite[Theorem 1]{Gu} and \cite[Theorem 1]{LiSe} respectively since $L$ is almost simple by Theorem \ref{ASPQP}, being $G(\Sigma)=1$, and since $G^{\Delta}_{\Delta}$ must act $2$-transitively on $\Delta$, where $\Delta \in \Sigma$.
\end{proof}
\bigskip
\begin{proposition}
\label{unapred} Let $\Delta \in \Sigma$, then $Soc(G_{\Delta }^{\Delta })\trianglelefteq L_{\Delta }^{\Delta }$ and $G=G_{x}L$, where $x$ is any point of $\mathcal{D}$. Moreover, one of the following holds:
\begin{enumerate}
\item $G(\Sigma)=1$, $\mathrm{Soc}(G_{\Delta }^{\Delta })< L_{\Delta }^{\Delta } \leq G_{\Delta }^{\Delta } \leq A \Gamma L_{1}(u^{h})$, where $u^{h}=\left\vert \Delta \right\vert$, and $\left \vert G_{\Delta }^{\Delta }: L_{\Delta }^{\Delta } \right \vert \mid \left \vert \mathrm{Out}(L) \right \vert $.
\item $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $.
\end{enumerate}
\end{proposition}
\begin{proof}
Since $L_{\Delta }\trianglelefteq G_{\Delta }$ and $G_{\Delta
}^{\Delta }$ acts primitively on $\Delta $, either $L_{\Delta }=L(\Delta )$
or $Soc(G_{\Delta }^{\Delta })\trianglelefteq L_{\Delta }^{\Delta }$ by \cite[Theorem 4.3B]{DM}.
Moreover, $G=G_{\Delta }L$ since $L^{\Sigma}=\mathrm{Soc}(G^{\Sigma})$ and $G^{\Sigma}$ acts primitively on $\Sigma$ by Theorem \ref{ASPQP}, and hence $G/L=G_{\Delta
}L/L\cong G_{\Delta }/L_{\Delta }$. Also, it results that $G/G(\Sigma)L\cong G^{\Sigma
}/L^{\Sigma }\leq \mathrm{Out}(L^{\Sigma })$.
Assume that $G(\Sigma) \neq 1$. Since $(G/L)/(G(\Sigma)L/L) \cong G/G(\Sigma)L$ is solvable and since $G(\Sigma)L/L \cong G(\Sigma)/L(\Sigma)$ is a $2$-group by Theorem \ref{ASPQP}(2), it follows that $G/L$ is solvable. Therefore, $G_{\Delta }/L_{\Delta}$, and hence $G^{\Delta}_{\Delta }/L^{\Delta}_{\Delta } \cong G_{\Delta }/G(\Delta)L_{\Delta }$ is solvable. Then $L_{\Delta }^{\Delta }=G_{\Delta }^{\Delta }$ since $G_{\Delta }^{\Delta } \cong E_{2^{d}}:G_{x}^{\Delta }$, where $G_{x}^{\Delta }$ is either $SL_{d}(2)$, or $G_{x}^{\Delta }\cong A_{7}$ for $d=4$ by Theorem \ref{ASPQP}(2.b), and the assertion (2) follows in this case.
Assume that $G(\Sigma) = 1$. Then $G=G^{\Sigma}$ and $L=L^{\Sigma}$, and hence $G_{\Delta }/L_{\Delta }$ is isomorphic to a subgroup of $\mathrm{Out}(L)$.
If $L_{\Delta }=L(\Delta )$ then $G_{\Delta }^{\Delta }=G_{\Delta }/G(\Delta )\cong \left(
G_{\Delta }/L_{\Delta }\right) /\left( G(\Delta )/L_{\Delta }\right) $.
Therefore $G_{\Delta }^{\Delta }$ is isomorphic to a quotient group of $%
G_{\Delta }/L_{\Delta }$, and hence $\mathrm{Out}(L)$ contains a subgroup with a quotient isomorphic to $G_{\Delta }^{\Delta }$. Then $\left\vert \Delta
\right\vert \left( \left\vert \Delta \right\vert -1\right) \mid \left\vert
\mathrm{Out}(L)\right\vert $ since $G_{\Delta }^{\Delta }$ acts $%
2 $-transitively on $\Delta $. Therefore
\begin{equation}
P(L)\leq \left\vert L:L_{\Delta }\right\vert =\left\vert \Sigma \right\vert \leq 2
\left\vert \Delta \right\vert \left( \left\vert \Delta \right\vert -1\right)
\leq 2\left\vert \mathrm{Out}(L)\right\vert \text{,} \label{vise}
\end{equation}%
where $P(L)$ is the minimal primitive permutation representation
of $L$. Then $L \cong PSL_{2}(q)$ by Lemma \ref{MPPRG}. If $q>11$ then $q+1\leq
4(2,q-1)\log q$ which has no solutions. So $q=4,5,7,8,9,11$, which are ruled out since their corresponding value of $\left\vert \Delta
\right\vert$ does not fulfill $\left\vert \Delta
\right\vert \left( \left\vert \Delta \right\vert -1\right) \mid \left\vert
\mathrm{Out}(L)\right\vert $. Thus $L_{\Delta }=L(\Delta )$ is ruled out. Therefore $Soc(G_{\Delta }^{\Delta })\trianglelefteq L_{\Delta }^{\Delta }$, and hence $L$ acts transitively on $\Delta$. Then $L$ acts point-transitively on $\mathcal{D}$ since $L$ acts transitively on $\Sigma$. Thus, $G=G_{x}L$ where $x$ is any point of $\mathcal{D}$.
Since the assertion (2) immediately follows if $Soc(G_{\Delta }^{\Delta })$ is non-abelian simple, we may assume that $Soc(G_{\Delta }^{\Delta })$ is an elementary abelian $u$-group for some prime $u$. Suppose that $L_{\Delta }^{\Delta } = Soc(G_{\Delta }^{\Delta })$. Since $%
G/L=G_{\Delta }L/L\cong G_{\Delta }/L_{\Delta }$, $G/L\leq \mathrm{Out}(L)$ and $G_{\Delta }/G(\Delta
)L_{\Delta }\cong G_{\Delta }^{\Delta }/L_{\Delta }^{\Delta }$, it follows
that $G_{\Delta }^{\Delta }/L_{\Delta }^{\Delta }$ is isomorphic to a
quotient subgroup of $\mathrm{Out}(L)$. Thus $\left\vert \Delta \right\vert
-1\mid \left\vert \mathrm{Out}(L)\right\vert $. Moreover,
\begin{equation}
P(L)\leq \left\vert L:L_{\Delta} \right\vert = \left\vert \Sigma \right\vert <2\left\vert \Delta \right\vert \left(
\left\vert \Delta \right\vert -1\right) \leq 2\left( \left\vert \mathrm{Out}(L)\right\vert +1\right) \left\vert \mathrm{Out}(L)\right\vert \label{najvise}
\end{equation}
Then either $L\cong PSL_{2}(q)$, or $L$ is as in Lemma \ref%
{MPPRG}. In the latter case $L$ is neither isomorphic to $A_{\ell
} $ with $7\leq \ell \leq 11$, nor to $M_{11}$, $PSp_{4}(3)$ or $PSU_{3}(5)$ since $\left\vert \Delta \right\vert
-1 \leq \left\vert \mathrm{Out}(L)\right\vert $ (we use this weaker constraint rather than $\left\vert \Delta \right\vert
-1\mid \left\vert \mathrm{Out}(L)\right\vert $ in order to apply the argument here to Theorem \ref{Large} as well), where $\left\vert \Delta \right\vert$ is $\lambda +2$ or $\lambda /2+1$ according to whether $\mathcal{D}$ is of type 1 or 2 respectively, and since $\lambda >10$
by our assumption. The unique admissible group among the remaining ones listed in Lemma \ref%
{MPPRG} is $L\cong PSU_{4}(3)$ and $(\left\vert \Delta \right \vert,\left\vert \Sigma \right \vert)=(20,18^{2})$ or $(38,36^{2})$ by \cite{At} and \cite[Tables 8.3--8.4]{BHRD} since $\left\vert \Sigma \right \vert$ is not of the form as in Theorem \ref{PZM}(V). However, both exceptional cases are ruled out since $\left\vert \Delta \right\vert
> \left\vert \mathrm{Out}(PSU_{4}(3))\right\vert $. Thus $L\cong PSL_{2}(q)$ and hence $%
\left\vert \Delta \right\vert \leq (2,q-1)f$. Then $%
f\geq 3$ for $q$ odd and $f\geq 6$ for $q$ even since $\lambda>10$ and since $\left\vert \Delta \right\vert$ is $\lambda +2$ or $%
\lambda /2+1$ according to whether $\mathcal{D}$ is of type 1 or 2 respectively. Thus $P(L)=q+1$, hence (\ref{najvise}) becomes $p^{f}+1\leq 2(2,q-1)f((2,q-1)f+1)\leq
4f(2f+1)$ which yields $q=2^{6},3^{3},3^{4}$ and $\mathcal{D}$ is of type 2
with $\lambda =12,12,16$ respectively. Thus $\left\vert \Sigma \right\vert
=61,61,113$ respectively, but $L\cong PSL_{2}(q)$ with $%
q=2^{6},3^{3},3^{4}$ has no such transitive permutation degrees. Thus $Soc(G_{\Delta }^{\Delta })<L_{\Delta }^{\Delta }$ and $\left \vert G_{\Delta }^{\Delta }: L_{\Delta }^{\Delta } \right \vert \mid \left \vert \mathrm{Out}(L) \right \vert $.
If $G_{\Delta }^{\Delta } \not \leq A \Gamma L_{1}(u^{h})$, then $G_{\Delta }^{\Delta }$ contains as a normal subgroup one of the groups $SL_{n}(u^{h/n})$, $Sp_{n}(u^{h/n})$, $G_{2}^{\prime}(2^{h/6})$ with $h \equiv 0 \pmod{6}$, $A_{6}$ or $A_{7}$ for $(u,h)=(2,4)$, or $SL_{2}(13)$ for $(u,h)=(3,6)$ by \cite[List(B)]{Ka} since $u^{h} \neq 3^{2},5^{2},7^{2},11^{2}$, $19^{2},23^{2},29^{2},59^{2}$ and $3^{4}$ by Lemma \ref{njami}, and $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta$ in these cases. This completes the proof.
\end{proof}
\bigskip
\begin{corollary}
\label{QuotL} If $G(\Sigma )\neq 1$, then $\lambda =2(2^{d-1}-1)$, $\left\vert \Sigma
\right\vert =\lambda ^{2}$ and a quotient group of $L_{\Delta }^{\Sigma }$
is isomorphic either to $SL_{d}(2)$ for $d\geq 4$, or to$A_{7}$ for $d=4$.
\end{corollary}
\begin{proof}
If $G(\Sigma )\neq 1$, then $G_{\Delta }^{\Sigma }/G(\Delta )^{\Sigma }\cong
G_{x}^{\Delta }$ where either $G_{x}^{\Delta }\cong SL_{d}(2)$ for $d\geq 4$, or $%
G_{x}^{\Delta }\cong A_{7}$ for $d=4$ by Theorem \ref{ASPQP}. On the other hand, $G_{x}/L_{x} \cong G_{x}L/L =G/L$ is isomorphic to a subgroup of $\mathrm{Out}(L)$ as a consequence of Proposition \ref{unapred}. Therefore $%
G_{x}/L_{x}$, and hence $G_{x}/G(\Delta )L_{x}$ is solvable. Since $G_{x}/G(\Delta )L_{x}\cong G_{x}^{\Delta }/L_{x}^{\Delta }$ and $G_{x}^{\Delta }$ is non-abelian simple, it follows that $%
L_{x}^{\Delta }=G_{x}^{\Delta }$. Thus $L_{\Delta }^{\Sigma } \not \leq G(\Delta )^{\Sigma}$, and hence $L_{\Delta }^{\Sigma } / \left(L_{\Delta }^{\Sigma } \cap G(\Delta )^{\Sigma}\right) \cong G_{\Delta }^{\Sigma } / G(\Delta )^{\Sigma} \cong G_{x}^{\Delta }$, which is the assertion.
\end{proof}
\bigskip
\begin{theorem}
\label{Large} Let $\Delta \in \Sigma$ then $L_{\Delta }^{\Sigma }$ is a
large subgroup of $L^{\Sigma }$. Moreover one of the following holds:
\begin{enumerate}
\item $\left\vert L^{\Sigma }\right\vert \leq \left\vert L_{\Delta
}^{\Sigma }\right\vert ^{2}$.
\item $G(\Sigma)=1$, $Soc(G_{\Delta }^{\Delta })< L_{\Delta }^{\Delta } \leq G_{\Delta }^{\Delta } \leq A \Gamma L_{1}(u^{h})$, where $u^{h}=\left\vert \Delta \right\vert$. Furthermore, the following holds:
\begin{enumerate}
\item $L_{\Delta}$ does not act $2$-transitively on $\Delta$;
\item $\left\vert L\right\vert \leq 4\left\vert L_{\Delta }^{\Delta
}\right\vert^{2} \left\vert \mathrm{Out}(L)\right\vert^{2}$;
\item $ \left\vert L(\Delta) \right\vert <2 \left\vert \mathrm{Out}(L) \right\vert < \left\vert L_{\Delta} \right\vert$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose that $G(\Sigma)=1$. Then $G=G^{\Sigma }$ and $%
L=L^{\Sigma }$. Assume that $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $. If $\mathcal{D}$ is of type 1 then $\left\vert L:L_{\Delta
}\right\vert =\left\vert \Sigma \right\vert \leq \left\vert \Delta
\right\vert \left( \left\vert \Delta \right\vert -1\right) \leq \left\vert
L_{\Delta }^{\Delta }\right\vert $, and hence $\left\vert L\right\vert \leq \left\vert L_{\Delta
}\right\vert ^{2}$, which is the assertion (1) in this case.
If $\mathcal{D}$ is of type 2, since $G$ acts flag-transitively on $\mathcal{D}$, it follows that $\lambda ^{2}/2\mid \left\vert G_{x}\right\vert $ and
hence $\lambda ^{2}/2\mid \left\vert L_{x}\right\vert \left\vert \mathrm{Out}%
(L)\right\vert $. On the other hand, $\left\vert L_{\Delta }\right\vert
=\left( \frac{\lambda }{2}+1\right) \frac{\lambda }{2}\left\vert
L_{x,y}\right\vert $ since $L_{\Delta }$ induces a $2$-transitive group on $%
\left\vert \Delta \right\vert $. In particular,
$\left\vert L_{x}:L_{x,y}\right\vert =\frac{\lambda }{2}$ and so $\lambda
\mid \left\vert L_{x,y}\right\vert \left\vert \mathrm{Out}(L)\right\vert $.
If $\lambda \mid \left\vert \mathrm{Out}(L)\right\vert $ then $P(L)\leq
\left\vert L:L_{\Delta }\right\vert =\frac{\lambda ^{2}-2\lambda +2}{2}%
<\left\vert \mathrm{Out}(L)\right\vert ^{2}$. Then $L\cong PSL_{2}(q)$ or $L$
is one of the groups listed in Lemma \ref{MPPRG}. Actually, in the latter
case only $L\cong PSL_{3}(q)$ with $(q,\lambda)=(4,12),(16,12),(16,24)$ are admissible since $\lambda \mid \left\vert \mathrm{Out}%
(L)\right\vert $ and $\lambda >10$. Then $\left\vert \Sigma \right\vert =61$
or $265$ respectively, but none of these divides the order of
the corresponding $L$. So these cases are excluded, and hence $L\cong
PSL_{2}(q)$ with $q=p^{f}$ and $f\geq 6$, as $\lambda \mid \left\vert \mathrm{%
Out}(L)\right\vert $ and $\lambda >10$ and $\left\vert \mathrm{Out}%
(L)\right\vert =(2,p^{f}-1)f$. However, $p^{f}+1\leq \left\vert L:L_{\Delta
}\right\vert \leq 2f^{2}-2f+1$ has no admissible solutions for $f\geq 6$.
Then $\left( \left\vert \mathrm{Out}(L)\right\vert ,\lambda \right) <\lambda
$ and hence $\left\vert L_{x,y}\right\vert \geq 2$ since $\lambda \mid
\left\vert L_{x,y}\right\vert \left\vert \mathrm{Out}(L)\right\vert $. Then $%
\left\vert L_{\Delta }\right\vert \geq \left( \frac{\lambda }{2}+1\right)
\lambda >\left\vert \Sigma \right\vert =\left\vert L:L_{\Delta }\right\vert $
and we obtain the assertion (1) also in this case.
Assume that $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $. Then $Soc(G_{\Delta }^{\Delta })< L_{\Delta }^{\Delta }$ and $G_{\Delta }^{\Delta }$ is a $2$-transitive subgroup of the semilinear $1$-dimensional group by Proposition \ref{unapred}. Note that $\left\vert L:L_{\Delta }\right\vert =\left\vert G:G_{\Delta
}\right\vert \leq 2\left\vert G_{\Delta }^{\Delta }\right\vert $. Also, it results that $G_{\Delta }/L_{\Delta } \cong G_{\Delta }L/L=G/L\leq \mathrm{Out}(L)$ and $G_{\Delta }/G(\Delta
)L_{\Delta }\cong G_{\Delta }^{\Delta }/L_{\Delta }^{\Delta }$. Hence, $G_{\Delta }^{\Delta }/L_{\Delta }^{\Delta }$ is isomorphic to a quotient group of a subgroup of $\mathrm{Out}(L)$. Therefore, $%
\left\vert L:L_{\Delta }\right\vert \leq 2\left\vert L_{\Delta }^{\Delta
}\right\vert \left\vert \mathrm{Out}(L)\right\vert$.
Assume that $\left \vert L(\Delta) \right \vert \geq 2\left\vert \mathrm{Out}(L)\right\vert$. Then $2\left\vert L_{\Delta }^{\Delta
}\right\vert \left\vert \mathrm{Out}(L)\right\vert \leq \left \vert L_{\Delta} \right \vert$, and we still obtain (1).
Assume that $\left \vert L(\Delta) \right \vert < 2\left\vert \mathrm{Out}(L)\right\vert$. Then $\left\vert L\right\vert \leq 4\left\vert L_{\Delta }^{\Delta
}\right\vert^{2} \left\vert \mathrm{Out}(L)\right\vert^{2}$. Suppose that $\left\vert
L_{\Delta }\right\vert \leq \left\vert \mathrm{Out}(L)\right\vert $. Then $%
2\left\vert \Delta \right\vert \leq \left\vert L_{\Delta }\right\vert \leq
2\left\vert \mathrm{Out}(L)\right\vert $ since $\mathrm{Soc}(G_{\Delta
}^{\Delta })<L_{\Delta }^{\Delta }$. Then $\left\vert \Delta \right\vert
\leq \left\vert \mathrm{Out}(L)\right\vert $, and hence $P(L) \leq \left\vert
L:L_{\Delta }\right\vert \leq 2\left\vert \mathrm{Out}(L)\right\vert
(\left\vert \mathrm{Out}(L)\right\vert -1)$. Then $L_{\Delta }$ is isomorphic either to $PSL_{2}(q)$ or to one of the groups listed in Lemma \ref{MPPRG}. However, the same argument used in Proposition \ref{unapred} can be also used here to rule out all these groups since $\left\vert \Delta \right\vert
\leq \left\vert \mathrm{Out}(L)\right\vert $. Thus $\left\vert L_{\Delta }\right\vert >2\left\vert \mathrm{Out}(L)\right\vert $ and hence $%
\left\vert L:L_{\Delta }\right\vert <\left\vert L_{\Delta }\right\vert ^{2}$, which means that $L_{\Delta }$ is a large subgroup of $L$, and we obtain assertions (2a)--(2c).
Suppose that $G(\Sigma) \neq 1$. Then $\lambda =2(2^{d-1}-1)$, $\left\vert \Sigma
\right\vert =\lambda ^{2}$ and a quotient group of $L_{\Delta }^{\Sigma}$
is isomorphic either to $SL_{d}(2)$ for $d\geq 4$, or to $A_{7}$ for $d=4$ by Corollary \ref{QuotL}. In either case one has $\left\vert \Delta
\right\vert \left( \left\vert \Delta \right\vert -1\right) \leq \left\vert
L_{\Delta }^{\Sigma }\right\vert $ since $\left\vert \Delta \right\vert
=2^{d}$. Since $G(\Sigma )\trianglelefteq L_{\Delta }<L$ and since $%
G^{\Sigma }$ acts primitively on $\Sigma $, it follows that $\left\vert
L^{\Sigma }:L_{\Delta }^{\Sigma }\right\vert =\left\vert L:L_{\Delta
}\right\vert =\left\vert \Sigma \right\vert \leq \left\vert \Delta
\right\vert \left( \left\vert \Delta \right\vert -1\right) \leq \left\vert
L_{\Delta }^{\Sigma }\right\vert $ and the assertion (1) follows in this
case.
\end{proof}
\bigskip
\section{Classification of the $2$-designs of type 1}
In this section we mainly use the constraints for $L^{\Sigma}$ provided in Proposition \ref{unapred}, Corollary \ref{QuotL} and Theorem \ref{Large} to prove Theorem \ref{T1} stated below. It is worth noting that, when $L^{\Sigma}$ is a Lie type simple group we show that $L^{\Sigma}_{\Delta}$ is a large subgroup of $L^{\Sigma}$ of order divisible by a suitable primitive prime divisor of $p^{\zeta}-1$, where $\zeta$ is determined in \cite[Proposition 5.2.16]{KL}. We combine this constraints on $L^{\Sigma}_{\Delta}$ to show that a small number of groups listed in \cite{AB} are admissible. These groups are then ruled out by exploiting the $2$-transitivity of $G_{\Delta}^{\Delta}$ on $\Delta$.
\begin{theorem}
\label{T1}If $\mathcal{D}$ is a symmetric $2$-$\left( \left( \lambda
+2\right) \lambda ^{2},(\lambda +1)\lambda ,\lambda \right) $ design admitting a flag-transitive and
point-imprimitive automorphism group, then $\lambda \leq 10$.
\end{theorem}
\bigskip
We analyze the cases where $L^{\Sigma }$ is sporadic, alternating, a Lie type simple classical or exceptional group separately.
\bigskip
\begin{lemma}
\label{spor1}$L^{\Sigma }$ is not sporadic.
\end{lemma}
\begin{proof}
Either $L_{\Delta }^{\Sigma }$ is maximal in $L^{\Sigma }$, or $%
\mathrm{Out}(L)\cong Z_{2}$ and $G_{\Delta }^{\Sigma }$ is a novelty. In the
latter case, the unique admissible case is $G^{\Sigma }\cong M_{12}:Z_{2}$ and $G_{\Delta
}^{\Sigma }\cong PGL_{2}(11)$ since $\left\vert \Sigma \right\vert $ is a square. Hence, $\left\vert \Sigma \right\vert =144$
by \cite[Table 1]{Wi1} (see also \cite{Wi2} for the cases $L^{\Sigma }\cong
Fi_{22}$ or $Fi_{24}^{\prime }$). Then $\lambda +2=14$ and hence $G(\Sigma
)=1$ by Theorem \ref{ASPQP}. Since $G_{\Delta }\cong PGL_{2}(11)$ has
no transitive permutation representations of degree $14$, the case is ruled out by
Proposition \ref{unapred}. Therefore, $%
L_{\Delta }$ is maximal in $L$.
Assume that $L^{\Sigma }\cong M_{i}$, where $i=11,12,22,23$ or $24$. Since $%
\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma }\right\vert =\lambda ^{2}$, it
follows from \cite[Table 5.1.C]{KL} that, $k^{2}=2^{a_{1}}3^{a_{2}}$ for
some $a_{1},a_{2}\geq 2$. Then $\lambda =12$ and either $L\cong M_{11}$ and $%
L_{\Delta }\cong F_{55}$, or $L^{\Sigma }\cong M_{12}$ and $L^{\Sigma }\cong
PSL_{2}(11)$ by \cite{At}. However, these cases are ruled out by Proposition %
\ref{unapred} since $\lambda +2$ does not divide the order of $L_{\Delta }$.
Assume that $L^{\Sigma }\cong J_{i}$, where $i=1,2,3$ or $4$. Then $\lambda
^{2}$ divides $2^{2}$, $2^{6}3^{2}5^{2}$, $2^{6}3^{4}$, or $%
2^{20}3^{2}11^{2} $, respectively, by \cite[Table 5.1.C]{KL}. Then $i=2$ and
either $\lambda =10$ and $L_{\Delta }^{\Sigma }\cong PSU_{3}(3)$, or $%
\lambda =60$ and $L_{\Delta }^{\Sigma }\cong PSL_{2}(7)$ by \cite{At}.
However these cases are ruled out, as $L_{\Delta }^{\Sigma }$ does not have
a $2$-transitive permutation representation of degree $12$ or $62$
respectively, and this contradicts Proposition \ref{unapred}.
Assume that $L^{\Sigma }$ is isomorphic to one of the groups $HS$ or $McL$.
By \cite[Table 5.1.C]{KL} $\lambda ^{2}$ divides $2^{8}3^{2}5^{2}$ or $%
2^{6}3^{6}5^{2}$ respectively. Then either $L^{\Sigma }\cong HS$, $L_{\Delta
}^{\Sigma }\cong M_{22}$ and $\lambda=10$, or $L^{\Sigma }\cong McL$, $L_{\Delta
}^{\Sigma }\cong M_{22}$ and $\lambda=45$. Both these cases are excluded since they contradict
Proposition \ref{unapred}.
It is straightforward to check that the remaining cases are ruled out
similarly, as they do not have transitive permutation representations of
degree $\lambda ^{2}$ by \cite{At} and \cite{Wi2}.
\end{proof}
\bigskip
\begin{lemma}
\label{AltLM}If $L^{\Sigma }\cong A_{\ell }$, $\ell \geq 5$, then $L^{\Sigma
}$ acts primitively on $\Sigma $.
\end{lemma}
\begin{proof}
Assume that $L \cong A_{6}$. Then $A_{6} \cong PSL_{2}(9) \trianglelefteq G \leq P\Gamma L_{2}(9)$, hence $G$ does not have a primitive permutation representation of degree $\lambda^{2}$ with $\lambda>10$ by \cite{At}. Thus $\ell \neq 6$ by Theorem \ref{ASPQP}, and hence $\mathrm{Out%
}(L)\cong Z_{2}$.
Suppose the contrary of the statement. Then there is a subgroup $M$ of $L$
containing $L_{\Delta }$ such that $L_{\Delta }^{\Sigma }<M^{\Sigma
}<L^{\Sigma }$ with $M^{\Sigma }$ maximal in $L^{\Sigma }$. Let $x\in \Delta
$, then $x^{M}$ is a union of $\theta$ elements of $\Sigma $, where $\theta = \left\vert M^{\Sigma }: L^{\Sigma }_{\Delta} \right\vert$. Therefore $\left\vert
x^{M}\right\vert =\theta (\lambda +2)$ with $\lambda ^{2}=a\theta $ for some
$a\geq 1$. Then $x^{M}\setminus \left\{ x\right\} $ is a union of $%
L_{x} $-orbits since $L_{x}<L_{\Delta }<L$. Therefore $\frac{\lambda +1}{%
\eta }\mid \theta (\lambda +2)-1$ by Lemma \ref{PP}, where $\eta =(\lambda +1,2)$ since $%
\mathrm{Out}(L)\cong Z_{2}$. Then $\theta =f\frac{\lambda +1}{\eta }+1$ for
some $f\geq 1 $, hence
\begin{equation}
\left( f\frac{\lambda +1}{\eta }+1\right) a-1=\lambda ^{2}-1 \label{en}
\end{equation}%
and so $a=e\frac{\lambda +1}{\eta }+1$ for some $e\geq 1$. Then (\ref{en})
becomes%
\begin{equation*}
ef\left( \frac{\lambda +1}{\eta }\right) ^{2}+(e+f)\left( \frac{\lambda +1}{%
\eta }\right) +1=\lambda ^{2}
\end{equation*}%
Then $ef<\eta ^{2}=(\lambda +1,2)^{2}$, and hence $\eta =2$ and $\lambda $ is odd. Therefore $%
ef=3$ and $\lambda =15$, $\left\vert \Delta \right \vert=17$ and $\left\vert\Sigma \right \vert=225$ since $\lambda>10$. Moreover, $G(\Sigma )=1$ by
Theorem \ref{ASPQP}. Then $L_{\Delta }\cong A_{224}$ by \cite[%
Table B.4]{DM}. However, this case cannot occur by Proposition \ref%
{unapred} since $A_{224}$ has no quotient groups with a transitive permutation
representations of degree $17$.
\end{proof}
\begin{lemma}
\label{Alt1}$L^{\Sigma }$ is not isomorphic to $A_{\ell }$, $\ell \geq 5$.
\end{lemma}
\begin{proof}
Since $L_{\Delta }^{\Sigma }$ is a large maximal subgroup of $L^{\Sigma }$
by Theorem \ref{Large} and Lemma \ref{AltLM}, and since $\left\vert \Sigma \right\vert =\lambda^{2}$ with $\lambda>10$, only the following cases are admissible by \cite[Theorem 2]{AB}:
\begin{enumerate}
\item[(i)] $L_{\Delta }^{\Sigma }\cong \left( S_{t}\times S_{\ell -t}\right)
\cap A_{\ell }$ with $1\leq t\leq \ell /2$;
\item[(ii)] $L_{\Delta }^{\Sigma }\cong \left( S_{t}\wr S_{\ell /t}\right)
\cap A_{\ell }$ with $2\leq t\leq \ell /2$.
\end{enumerate}
Suppose that (i) holds. Then $\binom{\ell }{t}=\lambda ^{2}$, and hence
either $t=1,2$, or $t=3$ and $\ell =50$ by \cite[Chapter 3]{AZ}.
Assume that $t=1$. Since $L_{\Delta }^{\Sigma }\cong A_{\lambda ^{2}-1}$ with $\lambda>10$, no quotient groups of $%
L_{\Delta }^{\Sigma }$ are isomorphic to $SL_{d}(2)$ for any $d \geq 4$. Also, the minimal transitive permutation representation of $%
A_{\lambda ^{2}-1}$ is strictly greater than $\lambda+2$. Thus, this case cannot occur by Proposition \ref{unapred} and Corollary \ref{QuotL}.
Assume that $t=2$. Then $L_{\Delta }^{\Sigma }\cong \left( S_{2}\times
S_{\ell -2}\right) \cap A_{\ell }$. Suppose that $G(\Sigma )\neq 1$. Then a
quotient group of $G_{\Delta }^{\Sigma }$ is isomorphic to $SL_{d}(2)$ with $d>4$ by
Theorem \ref{ASPQP} since $\binom{\ell }{2}\neq 14 ^{2}$, but this is clearly impossible. Thus $G(\Sigma
)=1$ and hence a quotient group of $L_{\Delta }\cong \left( S_{2}\times
S_{\ell -2}\right) \cap A_{\ell }$ must have a $2$-transitive permutation
representation of degree $\ell-2=\lambda +2$ by Proposition \ref{unapred} since $\lambda >10$. However, $\binom{\lambda +4 }{2}=\lambda ^{2}$ has no integer solutions. Thus, this case is
excluded.
Assume $t=3$ and $\ell =50$. Then $\lambda +2=142$, as $\lambda>10$, and hence $G(\Sigma )=1$
by Theorem \ref{ASPQP}. Then $\lambda +2\mid \left\vert G_{\Delta
}\right\vert $, whereas $G_{\Delta }=\left(
S_{3}\times S_{47}\right) \cap G$, which is a contradiction.
Suppose that (ii) holds. Hence, $L_{\Delta }^{\Sigma }\cong \left( S_{\ell /t}\wr
S_{t}\right) \cap A_{\ell }$ where $s/t,t>1$. Then $\left\vert \Sigma
\right\vert =\frac{\ell !}{((\ell /t)!)^{t}t!}$ and $A_{\ell /t}\wr
A_{t}\leq L_{\Delta }^{\Sigma }\leq \left( S_{\ell /t}\wr S_{t}\right) \cap
L $. Easy computations show that $\ell >25$ since $\left \vert \Sigma \right \vert =\lambda^{2}$. Moreover, $\left( A_{\ell /t}\right) ^{t}\trianglelefteq L_{\Delta
}^{\Sigma }$ and $A_{t} \leq L_{\Delta }^{\Sigma }/\left( A_{\ell /t}\right) ^{t}\leq
(Z_{2})^{t}:S_{t}$, where $(Z_{2})^{t}$ is a permutation module for $A_{t}$. Thus, by \cite[Lemma 5.3.4]{KL}, $L_{\Delta }^{\Sigma
}/\left( A_{\ell /t}\right) ^{t}$ is isomorphic to one of the following groups:
\begin{enumerate}
\item $A_{t},S_{t}$;
\item $A_{t}\times Z_{2},S_{t} \times Z_{2}$;
\item $(Z_{2})^{t-1}:A_{t},(Z_{2})^{t-1}:S_{t}$;
\item $(Z_{2})^{t}:A_{t},(Z_{2})^{t}:S_{t}$.
\end{enumerate}
Suppose that $G(\Sigma )\neq 1$. Then a quotient group of $L_{\Delta }^{\Sigma }$
is isomorphic either to $SL_{d}(2)$ for $d\geq 4$, or to $A_{7}$ for $d=4$ by Corollary \ref{ASPQP}. Matching such information with (1)--(4) one obtains $d=4$, $t=7,8$ and $\lambda =14$. Then $((\ell /t)!)^{t-1}<\frac{\ell !}{((\ell
/t)!)^{t}t!}=196$ as shown in \cite[(3.4)]{MF} and hence $\ell =14,16$. However, $\left\vert \Sigma \right\vert$ is not a square for such values of $t$ and $\ell$.
Suppose that $G(\Sigma )=1$. Then $L_{\Delta }/\left( A_{\ell /t}\right) ^{t}$ is one of the groups in (1)--(4). Assume that $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $. Then $\mathrm{Soc}(G_{\Delta }^{\Delta })< L_{\Delta }^{\Delta } \leq G_{\Delta }^{\Delta } \leq A \Gamma L_{1}(2^{h})$, where $\lambda+2=2^{h}$ and $\left \vert G_{\Delta }^{\Delta }: L_{\Delta }^{\Delta } \right \vert \leq 2$ by Proposition \ref{unapred}, as $\ell>6$, which force $L_{\Delta }^{\Delta }$ to act $2$-transitively on $\Delta $, and we reach a contradiction. Therefore, $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $.
If $L_{\Delta }^{\Delta }$ is of affine type then $\lambda+2 =2^{i}$ with $i \leq t$. Therefore $((\ell /t)!)^{t-1}<\frac{\ell !}{((\ell/t)!)^{t}t!}<2^{2t}$ and hence $\ell/t=2$, and $t>12$ since $\ell>25$. However, this is impossible by \cite[List (B)]{Ka} since $A_{t} \leq L_{\Delta }/\left( A_{\ell /t}\right) ^{t}$. Thus $L_{\Delta }^{\Delta }$ is almost simple and hence $%
A_{t}\trianglelefteq L_{\Delta }^{\Delta }\leq S_{t}$ with $t \geq 5$ by (1)--(4). Therefore, $t=\lambda
+2>12$ and $2^{t-1} \leq ((\ell
/t)!)^{t-1}<\frac{\ell !}{((\ell /t)!)^{t}t!}<t^{2}$, which is a contradiction. This completes the proof.
\end{proof}
\bigskip
\begin{lemma}
\label{nMag2}$L^{\Sigma }\ncong PSL_{2}(q)$.
\end{lemma}
\begin{proof}
Assume that $L^{\Sigma }\cong PSL_{2}(q)$, $q=p^f$, $p$ prime and $f \geq 1$. Then $q\neq 7$ since $PSL_{2}(7)$
does not have transitive permutation representations of square degree, and $q\neq 9$ by Lemma \ref{Alt1} since $PSL_{2}(9)\cong A_{6}$.
Moreover, if $q=p$, then $p\mid \left\vert L_{\Delta }^{\Sigma }\right\vert $
since $\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma }\right\vert =\lambda ^{2}$%
. Thus no novelties occur by \cite[Table 8.1%
]{BHRD}, hence $L_{\Delta }^{\Sigma }$ is maximal in $L^{\Sigma }$. Moreover, no quotients groups of $L_{\Delta }^{\Sigma }$ are isomorphic to $SL_{d}(2)$ with $d\geq 4$ or to $A_{7}$
and $d=4$ for $G(\Sigma ) \neq1$, hence $G(\Sigma )=1$ by Corollary \ref{QuotL}. Thus, $L_{\Delta }$ is maximal in $L$.
Assume that $L_{\Delta }$ is isomorphic to any of the groups $A_{4},S_{4}$ or $A_{5}$. Then $\lambda+2 \mid \left\vert L_{\Delta} \right \vert$, hence $\lambda+2$ is not a power of a prime since $\lambda >10$. Thus $L_{\Delta }$ is non-solvable and acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}, hence $L_{\Delta } \cong A_{5}$ and $\lambda+2=6$, whereas $\lambda>10$. Thus, these groups are ruled out.
Assume that $L_{\Delta }$ is isomorphic to $D_{\frac{q \pm 1}{(2,q-1)}}$. Then $\lambda \leq (q-1)/(2,q-1)$ since $\lambda+2 \mid \left\vert L_{\Delta} \right \vert$, and hence $q(q\mp 1)= \left\vert \Sigma \right \vert \leq \frac{(q-1)^{2}}{(2,q-1)^{2}}$, which has no admissible solutions.
Assume that $PSL_{2}(q^{1/m})\trianglelefteq
L_{\Delta }\leq PGL_{2}(q^{1/m})$. Then $L_{\Delta }$ acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}. Moreover, $\lambda +2=q^{1/m}+1$, since $q^{1/m}> 11$ being $\lambda>10$. Thus $\lambda= q^{1/m}-1$, and hence $L_{\Delta }$ must contain a Sylow $p$-subgroup of $L$, which is a contradiction.
Finally, assume that $L_{\Delta }\cong E_{q}:Z_{\frac{q-1}{(q-1,2)}}$. Then $\lambda
^{2}=q+1$, which has no solutions by \cite[A5.1]{Rib} since $\lambda >10$.
\end{proof}
\bigskip
\begin{lemma}
\label{ciciona}$L^{\Sigma }$ is not isomorphic to one of the groups $%
PSL_{3}(4)$, $PSU_{4}(2)$, $PSL_{6}(2)$, $PSp_{6}(2)$, $P\Omega _{8}^{+}(2)$, $G_{2}(2)^{\prime }$, $^{2}G_{2}(3)^{\prime }$ or $^{2}F_{4}(2)^{\prime }$.
\end{lemma}
\begin{proof}
Assume that $L^{\Sigma }\cong PSL_{3}(4)$. Then $35\mid \left\vert L_{\Delta
}^{\Sigma }\right\vert $ since $\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma
}\right\vert =\lambda ^{2}$, but $L^{\Sigma }$ does not contain such a group
by \cite{At}.
Assume that $L^{\Sigma }\cong PSU_{4}(2)$. Then $5\mid \left\vert L_{\Delta
}^{\Sigma }\right\vert $ since $\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma
}\right\vert =\lambda ^{2}$, and hence $L_{\Delta }^{\Sigma }\cong S_{6}$
and $\lambda =6$ by \cite{At}, whereas $\lambda >10$ by our assumptions.
Assume that $L^{\Sigma }\cong PSL_{6}(2)$. Thus $2\cdot 5\cdot 31\mid
\left\vert L_{\Delta }^{\Sigma }\right\vert $ since $\left\vert L^{\Sigma
}:L_{\Delta }^{\Sigma }\right\vert =\lambda ^{2}$, and hence either $%
L_{\Delta }^{\Sigma }\cong SL_{5}(2)$ or $L_{\Delta }^{\Sigma }\cong
E_{2^{5}}:SL_{5}(2)$ by \cite[Tables 8.24 and 8.25]{BHRD}. However, both cases are ruled out since $7$ divides $\left\vert L^{\Sigma
}:L_{\Delta }^{\Sigma }\right\vert$ but $7^{2}$ does not.
Assume that $L^{\Sigma }\cong PSp_{6}(2)$. Then $\mathrm{Out}(L^{\Sigma })=1$, $L_{\Delta }^{\Sigma }\cong S_{8}$
and $\lambda =6$ by \cite{At}, whereas $\lambda>10$.
Assume that $L^{\Sigma }\cong P\Omega _{8}^{+}(2)$. Therefore $21\mid
\left\vert L_{\Delta
}^{\Sigma }\right\vert $ since $\left\vert L^{\Sigma }:L_{\Delta
}^{\Sigma }\right\vert =\lambda ^{2}$, and hence $\lambda ^{2}\mid
2^{12}\cdot 3^{4}\cdot 5^{2}$. Then $\left\vert L^{\Sigma }:L_{\Delta
}^{\Sigma }\right\vert $ must be divisible by one among its primitive degrees $120$, $%
135$ or $960$ by \cite{At}. Thus $15^{2}$ divides $\left\vert L^{\Sigma
}:L_{\Delta }^{\Sigma }\right\vert $ in each case, and hence $\lambda=15j$ for some $j \geq 1$. If $G(\Sigma) \neq 1$ then $\lambda=15j=2^{d}-2$ with $d \geq 4$ by Theorem \ref{ASPQP}. Easy computations show that no admissible cases occur. Therefore $G(\Sigma)=1$, and hence $G$ is a subgroup of $P\Omega _{8}^{+}(2).S_{3}$. Moreover, the order of $G$ is divisible by $(15j)^{3}(15j+2)(15j+1)$ since $G_{\Delta}^{\Delta}$ acts $2$-transitively on $\Delta$, which has size $15j+2$, and since $G$ acts flag-transitively on $\mathcal{D}$ and $k=\lambda(\lambda+1)$. Since no groups occur, this case is excluded.
The case $L^{\Sigma }\cong G_{2}(2)^{\prime }$ is ruled out in Lemma \ref%
{nMag2} since $G_{2}(2)^{\prime }\cong PSL_{2}(8)$. Also, if $L^{\Sigma }\cong $ $%
^{2}G_{2}(3)^{\prime }\cong PSU_{3}(3)$ then $L_{\Delta }^{\Sigma }\cong
PSL_{2}(7)$ and $\left\vert \Sigma \right\vert =36$ by \cite{At}. So $%
\lambda =6$, whereas $\lambda >10$.
Finally, if $L^{\Sigma }\cong $ $%
^{2}F_{4}(2)^{\prime }$ then $L_{\Delta }^{\Sigma }\cong PSL_{3}(3):Z_{2}$
and $\left\vert \Sigma \right\vert =1600$ by \cite{At}. Then $\lambda
=40 $ and hence $\lambda +1=41$ must divide the order of $G$ by Lemma \ref{PP}. However, this is impossible since $G^{\Sigma }\leq $%
$^{2}F_{4}(2)$ and the order of $G(\Sigma)$ is either $1$ or a power of $2$ by Theorem \ref{ASPQP}. \end{proof}
\bigskip
A divisor $s$ of $q^{e}-1$ that is coprime to each $q^{i}-1$ for $i<e$ is
said to be a \emph{primitive divisor}, and we call the largest primitive
divisor $\Phi _{e}^{\ast }(q)$ of $q^{e}-1$ the \emph{primitive part} of $%
q^{e}-1$. One should note that $\Phi _{e}^{\ast }(q)$ is strongly related to
cyclotomy in that it is equal to the quotient of the cyclotomic number $\Phi
_{e}(q)$ and $(n,\Phi _{e}(q))$ when $e>2$. Also $\Phi _{e}^{\ast }(q)>1$
for $e>2$ and $(q,e)\neq (2,6)$ by Zsigmondy's Theorem (for instance, see
\cite[P1.7]{Rib}).
\bigskip
Let $p$ be a prime, $w$ a prime distinct from $p$, and $m$ an integer which
is not a power of $p$. Also let $\Gamma $ be a group which is not a $p$%
-group. Then we define%
\begin{eqnarray*}
\zeta _{p}(w) &=&\min \left\{ z:z\geq 1\text{ and }p^{z}\equiv 1\pmod{w}%
\right\} \\
\zeta _{p}(m) &=&\max \left\{ \zeta _{p}(w):w\text{ prime, }w\neq p\text{
and }w\mid m\right\} \\
\zeta _{p}(X) &=&\zeta _{p}(\left\vert X\right\vert )\text{.}
\end{eqnarray*}
If $L^{\Sigma }$ is isomorphic to a simple group of Lie type over $GF(q)$, $%
q=p^{f}$, then $\zeta _{p}(L^{\Sigma })$ is listed in \cite[Proposition 5.2.16 and Table
5.2.C]{KL}. In the sequel we will denote $\zeta _{p}(L^{\Sigma })$ simply by $\zeta $. It is worth noting that $\Phi _{\zeta }^{\ast }(p)>1$ by Lemmas \ref{nMag2} and \ref{ciciona}.
\bigskip
\begin{lemma}
\label{Phi0}$L_{\Delta}^{\Sigma }$ is a large subgroup of $L^{\Sigma }$ such that
$\left( \Phi _{\zeta }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$.
\end{lemma}
\begin{proof}
Suppose $\left( \Phi _{\zeta}^{\ast }(p), \left\vert
L_{\Delta }^{\Sigma }\right\vert \right) =1$. Note that $\left \vert G_{\Delta }^{\Sigma }: L_{\Delta }^{\Sigma } \right \vert \mid \left \vert \mathrm{Out}(L^{\Sigma
}) \right \vert$ since $\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma }\right\vert= \left\vert G^{\Sigma }:G_{\Delta }^{\Sigma }\right\vert$, as $G^{\Sigma}$ acts primitively on $\Sigma$. Thus $\left( \Phi _{\zeta }^{\ast }(p), \left\vert G_{\Delta}^{\Sigma }\right\vert \right) =1$ since $\left( \Phi _{\zeta }^{\ast }(p), \left\vert \mathrm{Out}(L^{\Sigma
})\right\vert \right) =1$ by \cite[Proposition 5.2.15(ii)]{KL}. Therefore $\Phi _{\zeta }^{\ast
}(p)\mid \left\vert G^{\Sigma }:G_{\Delta }^{\Sigma }\right\vert $, and hence $\Phi
_{\zeta }^{\ast }(p)\mid \lambda ^{2}$ since $\left\vert G^{\Sigma
}:G_{\Delta }^{\Sigma }\right\vert =\lambda ^{2}$. Then $(\Phi
_{\zeta }^{\ast }(p),\left\vert G_{x}\right\vert )>1$, where $x$ is any
point of $\mathcal{D}$, since $\lambda (\lambda +1)\mid \left\vert
G_{x}\right\vert $ being $G$ flag-transitive on $\mathcal{D}$. Therefore $%
(\Phi _{\zeta }^{\ast }(p),\left\vert G_{\Delta }\right\vert )>1$, where $%
\Delta $ is the element of $\Sigma $ containing $x$, and hence $(\Phi
_{\zeta }^{\ast }(p),\left\vert G_{\Delta }^{\Sigma }\right\vert )>1$ which is a contradiction. Thus $\left( \Phi _{\zeta}^{\ast }(p), \left\vert L_{\Delta }^{\Sigma }\right\vert \right) >1$, and $L_{\Delta }^{\Sigma }$ is large by Theorem \ref{Large}.
\end{proof}
\begin{lemma}
\label{Exc1}$L^{\Sigma }$ is not isomorphic to an exceptional simple group
of Lie type.
\end{lemma}
\begin{proof}
Let $M$ be a subgroup of $L$ such that $M^{\Sigma }$ is a maximal subgroup
of $L^{\Sigma }$ containing $L_{\Delta }^{\Sigma }$. Also $M^{\Sigma }$ is
large since $L_{\Delta }^{\Sigma }$ is so by Lemma \ref{Large}. Therefore $%
M^{\Sigma }$ is one of the groups classified in \cite[Theorem 5]{AB}.
Moreover, $\left( \Phi _{\zeta }^{\ast }(p), \left\vert M^{\Sigma
}\right\vert \right) >1$ by Lemma \ref{Phi0}.
Assume that $M^{\Sigma }$ is parabolic. If $L^{\Sigma }$ is untwisted then $%
M^{\Sigma }$ can be obtained by deleting the $i$-th node in the Dynkin
diagram of $L^{\Sigma }$, and we see that none of these groups is of order
divisible by a prime factor of $\Phi _{\zeta }^{\ast }(p)$. Indeed, for instance consider the Levi
factors of the maximal parabolic subgroups of $L^{\Sigma }\cong F_{4}(q)$, $%
q=p^{f}$ are of type $B_{3}(p^{f})$, $C_{3}(p^{f})$ or $A_{1}(p^{f})\times
A_{2}(p^{f})$ and none of these has order divisible by a prime factor of $\Phi
_{12f}^{\ast }(p)$.
If $L^{\Sigma }$ is twisted, that is $L^{\Sigma }$ is
centralized by an automorphism $\gamma $ of the corresponding untwisted
group and $\gamma$ induces a non-trivial symmetry $\rho $ on the Dynkin diagram. In
this case the $M^{\Sigma }$ exist only when deleting the resulting subset
obtained by deleting the $i$-th node in the Dynkin diagram of corresponding
untwisted group is $\rho $-invariant. The Levi factor of $M^{\Sigma }$ is
obtained by taking the fixed points of the automorphism $\gamma $ on the
Levi factor of the corresponding untwisted subgroup. Also in the twisted case the order of any maximal subgroups of $L^{\Sigma }$ is not divisible by a prime factor of $\Phi _{\zeta }^{\ast
}(p)$. Indeed, for instance, the Levi factors of the maximal parabolic
subgroups of $^{2}E_{6}(q)$, $q=p^{f}$, are of types $^{2}A_{5}(q)$, $%
^{2}D_{4}(q)$, $A_{1}(q)\times A_{2}(q^{2})$ and $A_{1}(q^{2})\times
A_{2}(q) $, and none of these has order divisible by a prime factor of $\Phi
_{18f}^{\ast }(p)$.
Assume that $M^{\Sigma }$ is not parabolic. Then $(L^{\Sigma },M^{\Sigma })$
is listed in \cite[Table 2]{AB}. Since $%
\left( \Phi _{\zeta }^{\ast }(p), \left\vert M^{\Sigma }\right\vert
\right) >1$, only the groups contained in Table \ref{tav2} are admissible by \cite{LSS}.
\begin{table}[h!]
\caption{\small {Admissible $(L^{\Sigma },M^{\Sigma })$}}\label{tav2}
\centering
\begin{tabular}{|l|l|l|}
\hline
$L^{\Sigma }$ & $M^{\Sigma }$ & Conditions \\ \hline \hline
$E_{7}(q)$ & $ (3,q+1).(^{2}E_{6}(q) \times \frac{q+1}{(3,q+1)}).(3,q+1).{2}$ & \\ \hline
$E_{6}(q)$ & $F_{4}(q)$ & \\
& $(q^{2}+q+1 \times$ $^{3}D_{4}(q)).3$ & \\ \hline
$F_{4}(q)$ & $^{3}D_{4}(q).3$ & \\
& $^{2}F_{4}(q)$ & \\ \hline
$G_{2}(q)$ & $SU_{3}(q):2$ & \\
& $^{2}G_{2}(q)$ & $q=3^{2e+1}>1$ \\
& $G_{2}(2)$ & $q=5$ \\
& $PSL_{2}(13)$ & $q=3,4$ \\
& $2^{3}.SL_{3}(2)$ & $q=3$ \\ \hline
$^{2}B_{2}(q)$ & $13:4$ & $q=8$ \\ \hline
$^{3}D_{4}(q)$ & $G_{2}(q)$ & \\ \hline
\end{tabular}%
\end{table}
$L^{\Sigma }$ is not isomorphic to any groups $G_{2}(3), G_{2}(5)$ and $^{2}B_{2}(8)$ since these do not have a transitive permutation representation of square degree by \cite{At}. Also, if $L^{\Sigma } \cong G_{2}(4)$ then $L_{\Delta}^{\Sigma } \cong PSL_{2}(13)$ again by \cite{At} and hence $\lambda=480$. Then $G(\Sigma)=1$ by Corollary \ref{QuotL}, and hence $PSL_{2}(13) \trianglelefteq G_{\Delta}^{\Delta} \leq PGL_{2}(13)$. However, $\left\vert \Delta \right \vert = \lambda+2=482$ does not divide the order of $G_{\Delta}^{\Delta}$, and we reach a contradiction.
Suppose that $ \left\vert M^{\Sigma }\right\vert
^{2} < \left\vert L^{\Sigma }\right\vert$. Then $G(\Sigma)=1$, $L_{\Delta }^{\Delta }$ is solvable, $\left\vert
L\right\vert \leq 4 \left\vert \mathrm{Out}(L) \right \vert ^{2} \left\vert L_{\Delta }^{\Delta}\right\vert ^{2}$ and $ \left\vert L(\Delta) \right\vert <2 \left\vert \mathrm{Out}(L) \right \vert$ by Theorem \ref{Large}. Thus $\left\vert M\right\vert^{2} < \left\vert L \right\vert \leq 4 \left\vert \mathrm{Out}(L) \right \vert ^{2} \left\vert L_{\Delta }\right\vert ^{2}$, and hence $\left\vert M:L_{\Delta}\right\vert < 2 \left\vert \mathrm{Out}(L) \right \vert$. In the remaining admissible groups of Table \ref{tav2} the last term of the derived series $M^{(\infty)}$ of $M$ is non-abelian simple. Let $P(M^{(\infty)})$ be the minimal primitive permutation representation of $M^{(\infty)}$. If $M^{(\infty)} \not \leq L_{\Delta}$, then
$$P(M^{(\infty)}) \leq \left\vert M^{(\infty)}:L_{\Delta} \cap M^{(\infty)} \right\vert \leq \left\vert M:L_{\Delta}\right\vert < 2 \left\vert \mathrm{Out}(L) \right \vert,$$
and we reach a contradiction by Lemma \ref{MPPRG}
Therefore $M^{(\infty)} \leq L_{\Delta}$, and hence $M^{(\infty)} \leq L(\Delta)$ since $L_{\Delta }^{\Delta }$ is solvable. Then $P(M^{(\infty)}) \leq \left\vert L(\Delta) \right \vert < 2 \left\vert \mathrm{Out}(L) \right \vert$ and we again reach a contradiction by Lemma \ref{MPPRG}.
Suppose that $\left\vert L^{\Sigma }\right\vert \leq \left\vert M^{\Sigma }\right\vert ^{2}$. Then one of the following holds by \cite{LSS}:
\begin{enumerate}
\item $L^{\Sigma }\cong E_{7}(q)$ and $L_{\Delta }^{\Sigma }=M^{\Sigma
}\cong (3,q+1).(^{2}E_{6}(q)\times (q-1)/(3,q+1)).(3,q+1).2$;
\item $L^{\Sigma }\cong E_{6}(q)$ and $L_{\Delta }^{\Sigma }=M^{\Sigma
}\cong F_{4}(q)$;
\item $L^{\Sigma }\cong F_{4}(q)$ and $L_{\Delta }^{\Sigma }=M^{\Sigma
}\cong $ $^{3}D_{4}(q).Z_{3}$;
\item $L^{\Sigma }\cong G_{2}(q)$ and $L_{\Delta }^{\Sigma }=M^{\Sigma
}\cong SU_{3}(q):Z_{2}$.
\end{enumerate}
Then $G(\Sigma)=1$ by Corollary \ref{QuotL}. If $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta$, then $L_{\Delta }^{\Delta }$ is solvable by Proposition \ref{unapred}, hence $ \left\vert L(\Delta) \right\vert <2 \left\vert \mathrm{Out}(L) \right \vert$ by Theorem \ref{Large}. However, this is impossible in cases (1)--(4). Then $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta$, and hence only (4) occurs with $\lambda+2
=q^{3}+1$. Then $\left\vert \Sigma \right\vert =(q^{3}-1)^{2}$, whereas $%
L_{\Delta }^{\Sigma }$ is a maximal non-parabolic subgroup of $L^{\Sigma }$.
So this case is excluded, and the proof is completed.
\end{proof}
\bigskip
Now, it remains to analyze the case where $L^{\Sigma}$ is a simple classical group.
\bigskip
\begin{proposition}
\label{Phi}$L_{\Delta}^{\Sigma }$ is a large maximal geometric subgroup of $L^{\Sigma }$. Moreover, it results that
$\left( \Phi _{\zeta }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$.
\end{proposition}
\begin{proof}
Recall that $L_{\Delta}^{\Sigma }$ is a large subgroup of $L^{\Sigma }$ such that
$\left( \Phi _{\zeta }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$ by Lemma \ref{Phi0}. Let $M$ be a subgroup of $L$ such that $M^{\Sigma }$ is a maximal subgroup of $L^{\Sigma }$ containing $L_{\Delta }^{\Sigma }$, then $M^{\Sigma }$ is large and $\left( \Phi _{nf}^{\ast }(p), \left\vert
M^{\Sigma }\right\vert \right) >1$. If $\left(L^{\Sigma}, L_{\Delta}^{\Sigma} \right)$ is not $\left(P \Omega^{+}_{8}(q),G_{2}(q) \right)$ and $\left(PSU_{4}(3) , A_{7}\right)$ then we may use the same argument of \cite[Theorem 7.1]{Mo1}, with $L^{\Sigma }, L^{\Sigma }_{\Delta}$ and $M^{\Sigma }$ in the role of $X,X_{x}$ and $Y$ respectively, to prove that $M^{\Sigma }$ is a geometric subgroup of $L^{\Sigma }$.
Assume that $L^{\Sigma} \cong P \Omega^{+}_{8}(q)$ and $L_{\Delta}^{\Sigma} \cong G_{2}(q)$. Then $G(\Sigma )=1$ by Corollary \ref{QuotL}. If $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $, then $\mathrm{Soc}(G_{\Delta }^{\Delta })<L_{\Delta }^{\Delta }\leq G_{\Delta}^{\Delta }\leq A\Gamma L_{1}(u^{h})$, where $u^{h}=\lambda +2$. Then $\lambda <f$, where $q=p^{f}$, and hence $p^{2f} \leq \left\vert L: L_{\Delta}\right \vert< f^{2}$ a contradiction. Thus $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $, and hence $q=2$. However, this is impossible by Lemma \ref{ciciona}.
Assume that $L^{\Sigma} \cong PSU_{4}(3)$ and $L_{\Delta}^{\Sigma} \cong A_{7}$. Then $\lambda=36$, and hence $G(\Sigma )=1$ by Corollary \ref{QuotL}. Therefore, one obtains $G \leq P \Gamma U_{4}(3)$. However, this is impossible by Lemma \ref{PP} since $\lambda+1=37$ does not divide the order of $G$. Thus, we have proven that $M^{\Sigma }$ is a geometric subgroup of $L^{\Sigma }$ containing $L_{\Delta}^{\Sigma}$.
If $L_{\Delta}^{\Sigma } \neq M^{\Sigma }$, then $G_{\Delta}^{\Sigma }$ is a novelty, and hence $L_{\Delta}^{\Sigma }$ is listed in \cite[Tables
3.5.H--I]{KL} for $n\geq 13$ and in \cite{BHRD} for $3 \le n\leq 12$. Now, the candidates for $L_{\Delta}^{\Sigma }$ must be large subgroups of $L^{\Sigma }$ and their order must have a factor in common with $\Phi _{\zeta}^{\ast }(p)$. For instance, if $L^{\Sigma } \cong PSL_{n}(q)$, $q=p^{f}$, then $\zeta=nf$ by \cite[Proposition 5.2.16]{KL}, and the unique admissible case is when $L_{\Delta}^{\Sigma }$ lies in a maximal member of $\mathcal{C}_{1}(L^{\Sigma})$. However, this is impossible by \cite[Theorem 3.5(iv)]{He}. Therefore, no novelties occur when $L^{\Sigma } \cong PSL_{n}(q)$. The remaining simple classical groups are analyzed similarly and it is straightforward to check that there are no novelties which are compatible with the constraints on $L_{\Delta}^{\Sigma}$. Thus $L_{\Delta}^{\Sigma } = M^{\Sigma }$, which is the assertion.
\end{proof}
\bigskip
\begin{lemma}
\label{PSL1}$L^{\Sigma }$ is not isomorphic to $PSL_{n}(q)$.
\end{lemma}
\begin{proof}
Assume that $L^{\Sigma }\cong PSL_{n}(q)$. Then $n\geq 3$ by Lemma \ref%
{nMag2}, and $L_{\Delta}^{\Sigma }$ is a large maximal geometric subgroup of $L^{\Sigma }$ such that
$\left( \Phi _{nf }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$ by Proposition \ref{Phi} and by \cite[Proposition 5.2.16]{KL}. Then $L_{\Delta}^{\Sigma } \notin \mathcal{C}_{1}(L^{\Sigma})$ by \cite[Theorem 3.5(iv)]{He}, and hence
one of the following holds by \cite[Propositions 4.7]{AB}:
\begin{enumerate}
\item[(i)] $\mathrm{Soc}(L_{\Delta }^{\Sigma })$ is one of the groups $%
PSp_{n}^{\prime }(q)$, $PSU_{n}(q^{1/2})$ and $n$ odd, or $P\Omega
_{n}^{-}(q)$;
\item[(ii)] $L_{\Delta }^{\Sigma }$ is a $\mathcal{C}_{3}$-group of type $%
GL_{n/t}(q^{t})$, where $t=2$, or $t=3$ and either $q=2,3$ or $q=5$ and $n$
is odd.
\end{enumerate}
Assume that (i) holds. Then $G(\Sigma )=1$ by Corollary \ref{QuotL}. If $%
L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $, then $%
\mathrm{Soc}(G_{\Delta }^{\Delta })<L_{\Delta }^{\Delta }\leq G_{\Delta
}^{\Delta }\leq A\Gamma L_{1}(u^{h})$, where $u^{h}=\lambda +2$ by
Proposition \ref{unapred}. If $\mathrm{Soc}(L_{\Delta }^{\Sigma })\cong
PSU_{n}(q^{1/2})$ then $\lambda +2\leq q-1$ by \cite[Proposition 4.8.5(II)]%
{KL}, and hence $q^{2}+q+1\leq P(L)\leq \left\vert \Sigma \right\vert \leq
(q-3)^{2}$ since $n\geq 3$, which is a contradiction. Thus, $L_{\Delta
}^{\Delta }$ acts $2$-transitively on $\Delta $ in this case. The previous
argument together with \cite[Propositions 4.8.3--4.8.4]{KL} can be used to
show that $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $ also in
the remaining cases. Furthermore, $(n,q)\neq (4,2),(6,2)$ by Lemmas \ref%
{Alt1} and \ref{Phi} respectively, and $n\geq 3$. Then one of the following
holds by \cite[Propositions 4.8.3--4.8.5]{KL}:
\begin{enumerate}
\item $n > 6$, $q=2$, $L_{\Delta } \cong Sp_{n}(2)$ and $\lambda=2^{2n-1}\pm
2^{n}-2$;
\item $n=3$, $\mathrm{Soc}(L_{\Delta }) \cong PSU_{3}(q^{1/2})$ and $%
\lambda=q^{3/2}-1$;
\item $n=4$, $\mathrm{Soc}(L_{\Delta }) \cong P\Omega _{4}^{-}(q)\cong
PSL_{2}(q^{2})$ and $\lambda=q^{2}-1$.
\end{enumerate}
Cases (2) and (3) are immediately ruled out since $(\lambda,p)=1$ but $%
L_{\Delta}$ does not contain a Sylow $p$-subgroup of $L$. In case (1) $%
2^{n(n-1)/2-2}$ must divide the order of a Sylow $2$-subgroup of $L_{\Delta}$
which is $2^{n^{2}/4}$, and we reach a contradiction since $n>6$.
Assume that (ii) holds. Then $L_{\Delta }^{\Sigma }\cong
Z_{a}.PSL_{n/t}(q^{t}).Z_{e}.Z_{t}$, where $a=\frac{(q-1,n/t)(q^{t}-1)}{%
(q-1)(q-1,n)}$ and $e=\frac{(q^{t}-1,n/t)}{(q-1,n/t)}$ by \cite[Proposition
4.3.6.(II)]{KL}, and $t=2,3$. Then $G(\Sigma )=1$ by Corollary \ref{QuotL}.
Recall that $\mathrm{Soc}(G_{\Delta }^{\Delta })\trianglelefteq L_{\Delta
}^{\Delta }$ by Proposition \ref{unapred}.
If $G_{\Delta }^{\Delta }$ is of affine type, then $L_{\Delta }^{\Delta
}\leq Z_{e}.Z_{t}$ for $n>t$. Thus $\lambda +2\leq e\leq n/t$, and hence
\begin{equation}
\frac{q^{n}-1}{q-1}=P(L)\leq \left\vert L:L_{\Delta }\right\vert =\left\vert
\Sigma \right\vert \leq n^{2}/2 \label{rain}
\end{equation}%
by \cite[Theorem 5.2.2]{KL} since $n\geq 3$ and $(n,q)\neq (4,2)$. However, (%
\ref{rain}) has no admissible solutions. Thus $n=t=3$ since $t=2,3$ and $%
n\geq 3$. Moreover, $L_{\Delta }\cong Z_{\frac{q^{2}+q+1}{(3,q-1)}}.Z_{3}$
and hence $\left\vert \Sigma \right\vert =\frac{1}{3}q^{3}(q+1)(q-1)^{2}$.
On the other hand, since $\lambda +2$ is a power of prime and divides the
order of $L_{\Delta }$, it follows that $\lambda \leq q^{2}+q-1$ and $%
\left\vert \Sigma \right\vert \leq (q^{2}+q-1)^{2}$, and we reach a
contradiction.
If $G_{\Delta }^{\Delta }$ is almost simple, then $t<n$ and $L_{\Delta
}^{\Delta }$ acts $2$-transitively on $\Delta $ by Proposition \ref{unapred}%
. Thus $L_{\Delta }^{\Delta }\cong PSL_{n/t}(q^{t}).Z_{e}.Z_{t}$ and hence
either $(n/t,q^{t})=(2,9)$ and $\lambda +2=6$, or $\lambda +2=\frac{q^{n}-1}{%
q^{t}-1}$ since $t=2,3$. The former contradicts $\lambda >10$, the latter
yields $\lambda =\frac{q^{n}-2q^{t}+1}{q^{t}-1}$ and hence $\left( \lambda
,p\right) =1$. Then $L_{\Delta }^{\Sigma }$ must contain a Sylow $p$%
-subgroup of $L^{\Sigma }$, which is a contradiction.
\end{proof}
\bigskip
\begin{lemma}
\label{PSU1}$L^{\Sigma }$ is not isomorphic to $PSU_{n}(q)$.
\end{lemma}
\begin{proof}
Assume that $L^{\Sigma }\cong PSU_{n}(q)$. Then $n\geq 3$ by Lemma \ref%
{nMag2}. Moreover, $L_{\Delta}^{\Sigma }$ is a large maximal geometric subgroup of $L^{\Sigma }$ such that
$\left( \Phi _{\zeta }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$, where $\zeta$ is either $nf$ or $(n-1)f$ according to whether $n$ is even or odd respectively, by Proposition \ref{Phi} and \cite[Proposition 5.2.16]{KL}. Then $L_{\Delta}^{\Sigma } \notin \mathcal{C}_{1}(L^{\Sigma})$ by \cite[Theorem 3.5(iv)]{He}, and hence
one of the following holds by \cite[Propositions 4.17]{AB}:
\begin{enumerate}
\item[(i)] $L_{\Delta }^{\Sigma }$ is a $\mathcal{C}_{1}$-subgroup of $%
L^{\Sigma }$;
\item[(ii)] $L_{\Delta }^{\Sigma }$ is a $\mathcal{C}_{3}$-subgroup of $%
L^{\Sigma }$ of type $GU_{n/3}(3^{3})$ with $n$ odd.
\end{enumerate}
Assume that (i) holds. Then $n$ is even by \cite[Theorem 3.5(iv)]{He}, and hence $L_{\Delta
}^{\Sigma } \cong Z_{\frac{q+1}{(q+1,n)}}.PSU_{n-1}(q).Z_{(q+1,n-1)}$ is the stabilizer of a non-isotropic point of $PG_{n-1}(q^{2})$
by \cite[Propositions 4.1.4.(II)-4.1.18.(II)]{KL}. Also, $G(\Sigma)=1$ by Corollary \ref{QuotL}. If $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $, then $%
\mathrm{Soc}(G_{\Delta }^{\Delta })<L_{\Delta }^{\Delta }\leq G_{\Delta
}^{\Delta }\leq A\Gamma L_{1}(u^{h})$ with $u^{h}=\lambda +2$ by
Proposition \ref{unapred}. Then $\lambda \leq (q+1,n-1) -2$ by \cite[Propositions 4.1.4.(II)]{KL}, and hence $q^{n-1} \leq \left\vert L:L_{\Delta} \right \vert <q$, which is impossible for $n \geq 4$. Then $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $, and hence $n=4$ and $\lambda^{2}=q^{3}(q-1)$. So $q-1$ is a square, which is impossible by \cite[B1.1]{Rib}.
Assume that (ii) holds. Then $L_{\Delta}^{\Sigma}\cong
Z_{7}.PSU_{n/3}(3^{3}).Z_{(n/3,7)}.Z_{3}$ by \cite[Proposition 4.3.6(II)]{KL}. Hence, $G(\Sigma)=1$ by Corollary \ref{QuotL}. Also, $L_{\Delta}^{\Delta}$ is forced to act $2$-transitively on $\Delta$ by Proposition \ref{unapred} since $\lambda>10$. Therefore, $n=9$ and $\lambda= 3^{9}-2$. However, $\left\vert L:L_{\Delta }\right\vert \neq \lambda ^{2}$ in this case, which is then ruled out.
\end{proof}
\bigskip
\begin{lemma}
\label{PSp1}$L^{\Sigma }$ is not isomorphic to $PSp_{n}(q)^{\prime}$.
\end{lemma}
\begin{proof}
Assume that $L^{\Sigma }\cong PSp_{n}(q)^{\prime}$. Then $n\geq 4$ by Lemma \ref%
{nMag2} since $n$ is even. Also $(n,q) \neq (4,2)$ by Lemma \ref{ASt1} since $PSp_{4}(2)^{\prime} \cong A_{6}$. Thus $L^{\Sigma }\cong PSp_{n}(q)$. Moreover, $L_{\Delta}^{\Sigma }$ is a large maximal geometric subgroup of $L^{\Sigma }$ such that
$\left( \Phi _{nf }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$ by Proposition \ref{Phi} and \cite[Proposition 5.2.16]{KL}. Then $L_{\Delta}^{\Sigma } \notin \mathcal{C}_{1}(L^{\Sigma})$ by \cite[Theorem 3.5(iv)]{He}, and hence
one of the following holds by \cite[Propositions 4.22]{AB}:
\begin{enumerate}
\item $L_{\Delta}^{\Sigma}$ is a $\mathcal{C}_{8}$-subgroup of $L^{\Sigma}$;
\item $L_{\Delta}^{\Sigma}$ is a $\mathcal{C}_{3}$-subgroup of $L^{\Sigma}$ of type $Sp_{n/2}(q^{2})$, $%
Sp_{n/3}(q^{3})$ or $GU_{n/2}(q)$;
\item $\left( L^{\Sigma}, L_{\Delta}^{\Sigma}\right)$ is either $(PSp_{4}(7),2^{4}.O_{4}^{-}(2))$ or $(PSp_{4}(3),2^{4}.\Omega_{4}^{-}(2))$.
\end{enumerate}
Assume that Case (i) holds. Then $L_{\Delta}^{\Sigma}\cong O_{n}^{\varepsilon}(q)$, with $\varepsilon= \pm$ and $q$ even, by \cite%
[Proposition 4.8.6.(II)]{KL}. Then $\lambda^{2}=\frac{q^{n/2}}{2}(q^{n/2}+ \varepsilon)$ since $\lambda^{2}=\left\vert L^{\Sigma}:L_{\Delta}^{\Sigma}\right\vert $, and hence $q^{n/2}+ \varepsilon$ is a square. Then $(n,q)=(6,2)$ by \cite[B1.1]{Rib} since $n\geq 4$. However, $L^{\Sigma} \cong PSp_{6}(2)$ cannot occur by Lemma \ref{ciciona}.
Assume the Case (ii) holds. Then either $L_{\Delta}^{\Sigma}$ is isomorphic either to $PSp_{n/t}(q^{t}).Z_{t}$ with $t=2,3$ or $Z_{(q+1)/2}.PGU_{n/2}(q).Z_{2}$ with $q$ odd by \cite[Propositions 4.3.7(II) and 4.3.10.(II)]{KL}. Also, in both cases it results $G(\Sigma)=1$ by Corollary \ref{QuotL}.
If $L_{\Delta}^{\Delta}$ does not act $2$-transitively on $\Sigma$, then $Z_{(q+1)/2}.PSU_{n/2}(q) \leq L(\Delta)$ and $\left \vert L_{\Delta}^{\Delta}\right \vert \mid 4f(n/2,q+1)$ since $\lambda>10$. By \cite[Corollary 4.3(ii)--(iii)]{AB} we obtain $$p^{f \frac{n^{2}+2n}{4}} \leq\left\vert L^{\Sigma}:L_{\Delta}^{\Sigma}\right\vert \leq 16f^{2}(n/2,q+1)^{2},$$ which has no solutions for $q$ odd and $n \geq 4$.
If $L_{\Delta}^{\Delta}$ acts $2$-transitively on $\Sigma$. Then $L_{\Delta}^{\Sigma}$ is isomorphic either to $PSp_{2}(q^{t}).Z_{t}$ with $t=2,3$ or $Z_{(q+1)/2}.PGU_{3}(q).Z_{2}$ with $q$ odd. Then either $\lambda =q^{t}-1$ with $t=2,3$ or $q^{3}-1$ respectively. In each case $(\lambda,q)=1$, and hence $L_{\Delta }^{\Sigma }$ must contain a Sylow $p$%
-subgroup of $L^{\Sigma }$, which is a contradiction.
Finally, Case (iii) cannot occur since $\left\vert L^{\Sigma}:L_{\Delta}^{\Sigma}\right\vert $ is a non-square.
\end{proof}
\bigskip
\begin{lemma}
\label{Cla1}$L^{\Sigma }$ is not isomorphic to simple classical group.
\end{lemma}
\begin{proof}
In order to prove the assertion we only need to tackle the case $L^{\Sigma
}\cong P\Omega _{n}^{\varepsilon }(q)$, where $\varepsilon \in \left\{ \circ
,\pm \right\} $ since the remaining classical groups are ruled out in
Lemmas \ref{PSL1}, \ref{PSU1} and \ref{PSp1}. Hence, assume that $L^{\Sigma
}\cong P\Omega _{n}^{\varepsilon }(q)$, where $\varepsilon \in \left\{ \circ
,\pm \right\} $. Then $n\neq 4$ by Lemma \ref{nMag2}. Moreover $n\neq 5,6$
by Lemmas \ref{PSL1}, \ref{PSU1} and \ref{PSp1}, since $P\Omega _{5}^{\circ
}(q)\cong PSp_{4}(q)$ with $q$ odd, $P\Omega _{6}^{+}(q)\cong PSL_{4}(q)$
and $P\Omega _{6}^{-}(q)\cong PSL_{6}(q)$. Thus $n\geq 7$. Moreover $%
(n,q,\varepsilon )\neq (8,2,+)$ by Lemma \ref{ciciona}. Finally, $L_{\Delta}^{\Sigma }$ is a large maximal geometric subgroup of $L^{\Sigma }$ such that $\left( \Phi _{\zeta }^{\ast }(p), \left\vert L_{\Delta
}^{\Sigma }\right\vert \right) >1$ by Proposition \ref{Phi}, where $\zeta$ is either $nf$, $(n-1)f$ or $(n-2)f$ by \cite[Proposition 5.2.16]{KL} according to whether $\varepsilon$ is $-,\circ$ or $+$ respectively. Thus, one of the following holds by \cite[Proposition 4.23]{AB}:
\begin{enumerate}
\item[(i)] $L_{\Delta }^{\Sigma }$ is $\mathcal{C}_{1}$-subgroup
of $L^{\Sigma }$;
\item[(ii)] Either $(n,q)=(7,3),(7,5)$ or $(n,q,\varepsilon)=(8,3,+)$ and $L_{\Delta }^{\Sigma }$ is of type $O_{1}(q) \wr S_{n}$;
\item[(iii)] $L_{\Delta }^{\Sigma }$ is a $\mathcal{C}_{3}$-subgroup
of $L^{\Sigma }$. Moreover, its type is either $O_{n/2}^{\varepsilon^{\prime}}(q^{2})$ with $(\varepsilon, \varepsilon^{\prime})=(-,-)$ and $n/2$ even or $(\varepsilon, \varepsilon^{\prime})=(+,\circ)$ and $n/2$ odd, or $GU_{n/2}(q)$ with $\varepsilon=-$ and $n/2$ odd or $\varepsilon=+$ and $n/2$ even;
\item[(iv)] $L^{\Sigma } \cong P\Omega^{+}_{8}(3)$ and $L_{\Delta }^{\Sigma
}\cong 2^{6}.\Omega_{6}^{+}(2)$.
\end{enumerate}
Assume that (i) holds. Then one of the following cases occurs by \cite[Propositions
4.1.6(II), 4.1.7(II) and 4.1.20(II)]{KL}:
\begin{enumerate}
\item $L_{\Delta }^{\Sigma }$ is the stabilizer in $L^{\Sigma }$ of a
non-singular point of $PG_{n-1}(q)$:
\begin{enumerate}
\item $\varepsilon =\circ $ and $L_{\Delta }^{\Sigma }\cong \Omega
_{n-1}^{-}(q).Z_{2}$
\item $\varepsilon =+$ and $L_{\Delta }^{\Sigma }\cong \Omega _{n-1}(q)$
with $q\equiv 1\pmod 4$, or $q\equiv 3\pmod 4$ and $n/2$ even;
\item $\varepsilon =+$ and $L_{\Delta }^{\Sigma }\cong \Omega _{n-1}(q).Z_{2}
$ with $q\equiv 3\pmod 4$ and $n/2$ odd;
\item $\varepsilon =+$ and $L_{\Delta }^{\Sigma }\cong Sp_{n-2}(q)$ with $q$
even.
\end{enumerate}
\item $\varepsilon =+$ and $L_{\Delta }^{\Sigma }$ is the stabilizer in $%
L^{\Sigma }$ of a non-singular line of type \textquotedblleft $-$" of $%
PG_{n-1}(q)$:
\begin{enumerate}
\item $L_{\Delta }^{\Sigma }\cong \left( Z_{\frac{q+1}{(q+1,2)}}\times
\Omega _{n-2}^{-}(q)\right) .Z_{2}$ with $q$ even or $q\equiv 1\pmod4$;
\item $L_{\Delta }^{\Sigma }\cong \left( Z_{\frac{q+1}{2}}\times \Omega
_{n-2}^{-}(q)\right) .[4]$ with $q\equiv 3\pmod 4$ and $n/2$ odd;
\item $L_{\Delta }^{\Sigma }\cong Z_{2}.\left( Z_{\frac{q+1}{4}}\times
P\Omega _{n-2}^{-}(q)\right) .[4]$ with $q\equiv 3\pmod 4$ and $n/2$ even.
\end{enumerate}
\end{enumerate}
In each case $G(\Sigma )=1$ by Corollary \ref{QuotL} since $n\geq 7$. Hence,
$L=L^{\Sigma }$ and $G=G^{\Sigma }$. Moreover, $L_{\Delta }^{\Delta }$ acts $%
2$-transitively on $\Delta $ by Proposition \ref{unapred} since $\lambda >10$%
. Then $\varepsilon=+$, $q=2$, $L_{\Delta }^{\Sigma }\cong Sp_{n-2}(2)$ and $\lambda =2^{2(n/2-1)}\pm 2^{n/2-2}-2$. On the
other hand,
\begin{equation*}
\left\vert L^{\Sigma }:L_{\Delta }^{\Sigma }\right\vert =\lambda
^{2}=2^{n/2-1}\left( 2^{n/2}-1\right)
\end{equation*}%
and hence $2^{2}(2^{2(n/2-1)-1}\pm 2^{n/2-1}-1)^{2}=2^{n/2-1}\left(
2^{n/2}-1\right) $, which has no admissible integer solutions for $n\geq 8$.
Assume that (iii) holds. The possibilities for $L_{\Delta }^{\Sigma }$ are
provided in \cite[Propositions 4.3.16(II), 4.3.18(II) and 4.3.20(II)]{KL}.
Thus $G(\Sigma )=1$ by Corollary \ref{QuotL}. If $%
L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta $, then $%
\mathrm{Soc}(G_{\Delta }^{\Delta })<L_{\Delta }^{\Delta }\leq G_{\Delta
}^{\Delta }\leq A\Gamma L_{1}(u^{h})$ with $u^{h}=\lambda +2$ by
Proposition \ref{unapred}. Thus $L_{\Delta }$ is forced to be
of type $GU_{n/2}(q)$ with $\left\vert L_{\Delta }^{\Delta }\right\vert
\mid (n/2,2,q)(q+1,n/2)$ since $\lambda >10$. Then $\lambda \leq n-2$ and so
$q^{\frac{1}{8}n\left( n+2\right) }\leq \left\vert L:L_{\Delta }\right\vert
\leq (n-2)^{2}$, which has no solutions for $n\geq 8$. Thus $L_{\Delta
}^{\Delta }$ acts $2$-transitively on $\Delta $, and hence $L \cong P\Omega^{-}_{8}(q)$ and $L_{\Delta } \cong P\Omega^{-}_{4}(q).Z_{4}\cong PSL_{2}(q^{4}).Z_{4}$ since $n \geq 8$. Therefore $\lambda=q^{4}-2$. If $q$ is odd then $L_{\Delta}$ must contain a Sylow $p$-subgroup of $L$ since $\left\vert L:L_{\Delta} \right\vert=\lambda^{2}$, which is not the case. So $q$ is even and $\left\vert \Sigma \right \vert = q^{12}(q^{6}-1)(q^{2}-1)$ which is different from $(q^{4}-2)^{2}$.
Finally, it is easy to check that $\left\vert L^{\Sigma }:L_{\Delta
}^{\Sigma }\right\vert $ is a non-square in (ii) and (iv), hence these are
ruled out. This completes the proof.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \protect\ref{T1}]
Since $L^{\Sigma }$ is almost simple by Theorem \ref{ASPQP}, the assertion
follows from Lemmas \ref{spor1}, \ref{Alt1}, \ref{Exc1} or \ref{Cla1}
respectively.
\end{proof}
\section{Classification of the $2$-designs of type 2}
In this section we assume that $\mathcal{D}$ is of type $2$. Recall that $G(\Sigma )=1$ and $G$ is an almost simple
group acting point-quasiprimitively on $\mathcal{D}$ by Theorem \ref{ASPQP}.
Thus, $G=G^{\Sigma }$ and $L=L^{\Sigma }$ where $L=\mathrm{Soc}(G)$.
Further constraints for $L$ are provided in Proposition \ref{unapred} and Theorem \ref{Large} which are then combined with the results
contained in \cite{AB, LS}. An important restriction is provided in Lemma \ref{Sevdah} where it is proven that, if $L$ is Lie type simple group, either $L_{\Delta }$ lies in a maximal parabolic subgroup of $L$ or $L_{\Delta }^{\Delta }$ is a non-solvable group acting $2$-transitively on $\Delta $. We use all these information to prove the
following result.
\bigskip
\begin{theorem}
\label{T2}If $\mathcal{D}$ is a symmetric $2$-$\left( \left( \frac{\lambda +2%
}{2}\right) \left( \frac{\lambda ^{2}-2\lambda +2}{2}\right) ,\frac{\lambda
^{2}}{2},\lambda \right) $ design admitting a flag-transitive and
point-imprimitive automorphism group, then $\lambda \leq 10$.
\end{theorem}
\bigskip
We analyze the cases where $L^{\Sigma }$ is sporadic, alternating, classical or exceptional of Lie type separately.
\bigskip
Recall that, when $\mathcal{D}$ is of type (2) either $\lambda \equiv 0 \pmod{4}$ and hence $\left\vert \Delta \right \vert =\lambda/2+1$ is odd, or $\lambda =2w^{2}$, where $w$ is odd, $w\geq 3$, $2(w^{2}-1)$ is a square and $\left\vert \Delta \right \vert=w^{2}+1$. In both cases it results that $\left\vert \Delta \right \vert \not \equiv 0 \pmod{4}$.
\medskip
A preliminary filter in the study of the the $2$-designs of type 2 is the following lemma.
\bigskip
\begin{lemma}
\label{Jedan}If $\mathcal{D}$ is of type 2, then the following hold:
\begin{enumerate}
\item $\left\vert \Sigma \right\vert $ is odd and $2\left\vert \Sigma \right\vert -1$ is a square.
\item If $u$ is any prime divisor of $\left\vert \Sigma \right\vert $, then $%
u\equiv 1\pmod{4}$.
\item If $\lambda=2w^{2}$, $w$ odd, $w \geq 3$ and such that $2(w^{2}-1)$ is a square, then $\mathrm{Soc}(G_{\Delta}^{\Delta})$ is isomorphic to one of the groups $ A_{w^{2}+1}$, $PSL_{2}(w^{2})$ or $PSU_{3}(w^{2/3})$.
\end{enumerate}
\end{lemma}
\begin{proof}
$\left\vert \Sigma \right\vert $ is clearly odd. If $\lambda =2w^{2}$, where $w$ is odd, $w\geq 3$, and $2(w^{2}-1)=x^{2}$ then $2w^{4}-2w^{2}t+\left( 1-\left\vert \Sigma \right\vert
\right) =0$, whereas if $\lambda =4t$ for some $t\geq 1$, then $16t^{2}-8t+\left( 2-2\left\vert \Sigma \right\vert
\right) =0$. In both cases $y^{2}=2\left\vert \Sigma \right\vert -1$ for some
positive integer $y$. Therefore, $\left\vert \Sigma \right\vert =\frac{y^{2}+1%
}{2}$ is odd. Moreover, if $u$ is any prime divisor of $\left\vert \Sigma
\right\vert $ then $y^{2}\equiv -1\pmod{u}$ and hence $u\equiv 1\pmod{4}$.
Thus, we obtain (1) and (2).
Finally, (3) follows from the first part of the proof of Lemma \ref{firstpart}.
\end{proof}
\bigskip
\begin{lemma}
\label{spor2}$L$ is not isomorphic to a sporadic group.
\end{lemma}
\begin{proof}
Assume that $L$ is sporadic. Then the possibilities for $G$ and $G_{x}$, where
$x$ is any point of $\mathcal{D}$, and hence for $\left\vert \Sigma \right\vert
=\left\vert G:G_{x}\right\vert $ are provided in \cite{As}. It is easy to
see that $2\left\vert \Sigma \right\vert -1$ is never square if $\left\vert \Sigma \right\vert$ is any
of such degrees. Thus $L$ cannot be a sporadic simple group by Lemma \ref%
{Jedan}(2).
\end{proof}
\bigskip
\begin{lemma}
\label{Alt2}If $L$ is not isomorphic to $A_{s}$ with $s \geq 5$.
\end{lemma}
\begin{proof}
Assume that $L\cong A_{s}$, where $s\geq 5$. Then one of the following holds by \cite{LS}:
\begin{enumerate}
\item $G_{\Delta }\cong \left(S_{t}\times S_{s-t} \right)\cap G$, $1\leq t<s/2$;
\item $G_{\Delta }\cong (S_{s/t}\wr S_{t})\cap G$, $s/t,t>1$;
\item $G_{\Delta }\cong A_{7}$ and $\left\vert \Sigma \right\vert =15$.
\end{enumerate}
Assume that (i) holds. Then $\left\vert \Sigma
\right\vert =\binom{s}{t}$ and $A_{t}\times A_{s-t}\trianglelefteq L_{\Delta
}\leq \left( S_{t}\times S_{s-t}\right) \cap L$. Suppose that $t\geq 5$.
Then $s-t>t\geq 5$. Hence, both $A_{t}$ and $A_{s-t}$ are simple groups. Moreover, $%
L_{\Delta }^{\Delta }$ is either $A_{t}$ or $A_{s-t}$ by Proposition \ref%
{unapred} since $\lambda>10$. If $%
L_{\Delta }^{\Delta } \cong A_{t}$ then either $\left\vert \Delta \right\vert =t$, or $t=6$ and $\left\vert \Delta \right\vert =10$, or $t=7,8$ and $\left\vert \Delta \right\vert =15$ since $t\geq 5$. Actually, $t=6$ and $\left\vert \Delta \right\vert=10$ imply $\lambda=2w^{2}=18$ and $\left\vert \Sigma \right \vert=145$, which is not of the form $\binom{s}{6}$ and hence it cannot occur. Also, if $t=7,8$ and $\left\vert \Delta
\right\vert =15$ then $\binom{s}{t}=\left\vert \Sigma \right\vert
=365$, and we reach a contradiction. Thus, $\lambda =2(t-1)$, and hence $\left\vert \Sigma \right\vert =\allowbreak 2t^{2}-6t+5$. Therefore, we have
\begin{equation*}
2^{t}<\left( \frac{s}{t}\right) ^{t}\leq \binom{s}{t}=2t^{2}-6t+5\text{,}
\end{equation*}%
which is impossible for $t\geq 5$. We reach the same contradiction for $L_{\Delta }^{\Delta }\cong A_{s-t}$.
Assume that $1\leq t\leq 4$. If $L_{\Delta }^{\Delta }$ is non-solvable,
then $s-t\geq 5$ and $L_{\Delta }^{\Delta }\cong A_{s-t}$ and the previous
argument rules out this case. Thus $L_{\Delta }^{\Delta }$ is solvable, and
hence $\left\vert \Delta \right\vert =3$ by Proposition \ref{unapred} since $L_{\Delta }\leq \left(
S_{t}\times S_{s-t}\right) \cap L$ and $\left\vert \Delta \right\vert $
is odd by Lemma \ref{Jedan}(3). Then $\lambda =4$, whereas $\lambda >10$ by our assumptions.
Assume that (ii) holds. Then
$\left\vert \Sigma \right\vert =\frac{s!}{((s/t)!)^{t}(t!)}$ and $A_{s/t}\wr
A_{t}\leq L_{\Delta }\leq \left( S_{s/t}\wr S_{t}\right) \cap L$. Moreover, $%
\left( A_{s/t}\right) ^{t}\trianglelefteq L_{\Delta }$ and $A_{t}\leq
L_{\Delta }/\left( A_{s/t}\right) ^{t}\leq (Z_{2})^{t}:S_{t}$, where the action
of $A_{t}$ on $(Z_{2})^{t}$ and on its permutation module are equivalent. Thus $%
L_{\Delta }/\left( A_{s/t}\right) ^{t}$ is isomorphic to one of the groups $A_{t}$, $S_{t}$, $Z_{2}\times
A_{t}$, $Z_{2}\times S_{t}$, $(Z_{2})^{t-1}(2):A_{t}$, $(Z_{2})^{t-1}(2):S_{t}$, $%
(Z_{2})^{t}:A_{t}$ or $(Z_{2})^{t}:S_{t}$ by \cite[Lemma 5.3.4]{KL}.
If $t \leq 4$ then $L_{\Delta}^{\Delta}$ is solvable, and hence $\lambda/2+1=\left\vert \Delta \right\vert =3$ by Proposition \ref{unapred} since $\left\vert \Delta \right\vert $
is odd by Lemma \ref{Jedan}(3). However, this is impossible since $\lambda>10$. Thus $t\geq 5$. Also, $L_{\Delta}^{\Delta}$ is non-solvable otherwise we reach a contradiction as above. Then $L_{\Delta}^{\Delta}$ acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}. Therefore $L_{\Delta }^{\Delta }\cong A_{t}$, and hence either $\left\vert \Delta
\right\vert =t$ for $t\geq 5$, or $t=6$ and $\left\vert \Delta \right\vert
=10$, or $t=7,8$ and $\left\vert \Delta \right\vert
=15$. On the other hand, $2^{t-1}\leq ((s/t)!)^{t-1}<\frac{s!}{((s/t)!)^{t}(t!)%
}<2t^{2}$ as shown in \cite[(34)]{MF}. Thus $(t,s)=(3,6),(5,10),(7,14)$ and
hence $\left\vert \Sigma \right\vert =15$, $945$, $135135$,
which are ruled out since they violate Lemma \ref{Jedan}(1). Then either $t=6$, $\left\vert \Delta \right\vert =15$ and $\left\vert \Sigma \right\vert =41$, or $t=7,8$, $%
\left\vert \Delta \right\vert =15$ and $\left\vert \Sigma \right\vert =365$. However both cases cannot occur since $\frac{(2t)!}{(2)^{t}(t!)}>
\left\vert \Sigma \right\vert$.
Finally, (iii) is excluded by Lemma \ref{Jedan}(1), as $2\left\vert \Sigma \right\vert -1$ is
not a square.
\end{proof}
\bigskip
\begin{lemma}
\label{Sevdah}The following hold:
\begin{enumerate}
\item If either $p\mid \lambda -\mu $ for some $\mu \in \left\{
0,1,2\right\} $, or $p\mid \lambda -3$ and $p\neq 5$, then $L_{\Delta }$
lies in a maximal parabolic subgroup of $L$.
\item If $L_{\Delta }$ does not lie in a maximal parabolic subgroup of $L$
then $L_{\Delta }^{\Delta }$ is a non-solvable group acting $2$-transitively
on $\Delta $.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\left\vert \Sigma \right\vert =\frac{\lambda ^{2}-2\lambda +2}{2}$ it
is immediate to see that $\left( \left\vert \Sigma \right\vert ,\lambda -\mu
\right) =1$ for either $\mu =0,1,2$, or $\mu =3$ and $p\neq 5$. In these
cases $L_{\Delta }$ contains a Sylow $p$-subgroup of $L$, and hence $%
L_{\Delta }$ lies in a maximal parabolic subgroup of \cite[Theorem 1.6]{Se}
since $L$ is a non-abelian simple group acting transitively on $\Sigma $,
and (1) holds.
Suppose to the contrary that $L_{\Delta }$ does not lie in a maximal
parabolic subgroup of $L$ and that $L_{\Delta }^{\Delta }$ is solvable. Then
$p\mid \left\vert \Sigma \right\vert $ by \cite[Theorem 1.6]{Se}. Also $%
p\neq 2,3$ by Lemma \ref{Jedan}(1)--(2), and $\mathrm{Soc}(G_{\Delta }^{\Delta
})<L_{\Delta }^{\Delta }<G_{\Delta }^{\Delta }\leq A\Gamma L_{1}(u^{h})$,
where $\frac{\lambda }{2}+1=\left\vert \Delta \right\vert =u^{h}$ for some
prime $u$ by Proposition \ref{unapred}. Also, $u$ is odd since either $\lambda \equiv 0 \pmod{4}$, or $\lambda=2w^{2}$ with $w$ odd and $w\ge 3$ by Theorem \ref{PZM}. Then, $\lambda =2(u^{h}-1)$
with $h>1$ since $\lambda >10$.
If $p=5$ divides $\lambda -3$, then $u=p$ since $\left( \left\vert \Sigma
\right\vert ,\left\vert \Delta \right\vert \right) =\left( \left\vert \Sigma
\right\vert ,\lambda -3\right) =\left( \left\vert \Delta \right\vert
,\lambda -3\right) \mid 5$. Thus $\left\vert \Sigma \right\vert =\allowbreak
2\cdot 5^{2h}-6\cdot 5^{h}+5$, and $5^{2}\nmid \left\vert \Sigma \right\vert
$ since $h>1$. Hence, $L_{\Delta }$ contains a subgroup of index $p$ of a
Sylow $p$-subgroup of $L$ with $p=5$. It is a straightforward check that
none of the groups listed in \cite{LS} fulfills the previous constraint. So
this case is excluded.
If $p\nmid \lambda -\mu $ for some $\mu \in \left\{ 0,1,2,3\right\} $. Then $%
\left( p,\left\vert L(\Delta )\right\vert \right) =1$ by Corollary \ref%
{CFixT}(2) since $p\neq 2,3$. Thus $\left\vert L_{\Delta }\right\vert
_{p}=\left\vert L_{\Delta }^{\Delta }\right\vert _{p}$, and hence $p\mid
\frac{\lambda }{2}\left( \frac{\lambda }{2}+1\right) h$ since $L_{\Delta
}^{\Delta }<G_{\Delta }^{\Delta }\leq A\Gamma L_{1}(u^{h})$. Actually, $%
p\nmid \frac{\lambda }{2}$ by our assumption. If $p\mid \frac{\lambda }{2}+1$%
, then $p=5$ divides $\lambda -3$ since $\left( \left\vert \Sigma
\right\vert ,\left\vert \Delta \right\vert \right) =\left( \left\vert \Sigma
\right\vert ,\lambda -3\right) =\left( \left\vert \Delta \right\vert
,\lambda -3\right) \mid 5$ which we saw being impossible. Therefore, $%
\left\vert L_{\Delta }\right\vert _{p}\mid h$ and any Sylow $p$-subgroup of $%
L_{\Delta }$ is cyclic since $L_{\Delta }^{\Delta }<G_{\Delta }^{\Delta
}\leq A\Gamma L_{1}(u^{h})$. However, this is impossible by \cite{LS}. Thus $%
L_{\Delta }^{\Delta }$ is non-solvable, and hence $L_{\Delta }^{\Delta }$
acts $2$-transitively on $\Delta $ by Proposition \ref{unapred}.
\end{proof}
\bigskip
\begin{lemma}
\label{Exc2}$L$ is not a simple exceptional group of Lie type.
\end{lemma}
\begin{proof}
Assume that $L_{\Delta}$ is parabolic. Then $L \cong E_{6}(q)$ and $L_{\Delta} \cong [q^{16}].d.(P\Omega_{10}^{+}(q) \times (q-1)/ed).d$, where $d=(2,q-1)$ e $e=(3,q-1)$ by \cite[Table 1]{LS}. If $L_{\Delta}^{\Delta}$ is solvable then the order of $L_{\Delta}^{\Delta}$ must divide $(q-1)/e$. So does $\left\vert \Delta \right \vert$, and hence
$$\frac{q^{9}-1}{q-1}\cdot (q^{8}+q^{4}+1)=\left\vert \Sigma \right \vert \leq 2\left\vert \Delta \right \vert^{2} \leq 2\frac{(q-1)^{2}}{e^{2}},$$
which is clearly impossible. Thus $L_{\Delta}^{\Delta}$ is non-solvable acting $2$-transitively on $\Delta$ by Proposition \ref{unapred}, and this is impossible too.
Assume that $L_{\Delta}$ is not parabolic. Then $L_{\Delta}^{\Delta}$ is non-solvable acting $2$-transitively on $\Delta$ by Lemma \ref{Sevdah} with $\left\vert \Delta \right \vert$ odd. Also, $\left\vert L\right\vert \leq \left\vert L_{\Delta}\right\vert^{2}$ by Theorem \ref{Large}. All these constraints together with \cite[Table 1]{LS} lead to the following admissible cases:
\begin{enumerate}
\item $L \cong E_{7}$ and $\mathrm{Soc}(L_{\Delta}^{\Delta}) \cong PSL_{2}(q)$;
\item $L \cong$ $^{3}D_{4}(q)$ and $\mathrm{Soc}(L_{\Delta}^{\Delta})$ is isomorphic to one of the groups $ PSL_{2}(q^{j})$ with $j=1$ or $3$, $PSL_{3}(q)$ or $PSU_{3}(q)$;
\item $L \cong$ $^{2}G_{2}(q)$, $q=3^{2m+1}$, $m \geq 1$, and $\mathrm{Soc}(L_{\Delta}^{\Delta}) \cong PSL_{2}(q)$.
\end{enumerate}
Then possible values for $\left\vert \Delta \right \vert$ are $q+1,q^{2}+1,q^{2}+q+1,q^{2}+1$, hence $q \mid \lambda$ since $\left\vert \Delta \right \vert =\lambda/2+1$. Therefore $q$ is coprime to $\left\vert \Sigma \right \vert= \frac{\lambda^{2}-2\lambda+2}{2}$, and hence $L_{\Delta}$ must contain a Sylow $p$-subgroup of $L$, which is not the case. This completes the proof.
\end{proof}
\bigskip
\begin{lemma}
\label{Filt}The following cases are admissible:
\begin{enumerate}
\item $q$ is even and $L_{\Delta }$ lies in a maximal parabolic subgroup of $%
L$;
\item $q$ is odd and one of the following holds:
\begin{enumerate}
\item $L_{\Delta }$ lies in maximal member of $\mathcal{C}_{1}(L)\cup \mathcal{C}_{2}(L)$, with $%
L_{\Delta }$ lying in a maximal parabolic subgroup of $L$ of type either $P_{i}$ or $P_{m,m-i}$ only for $%
L\cong PSL_{n}(q)$.
\item $L\cong PSL_{2}(q)$ and $L_{\Delta }$ is isomorphic to one of the
groups $D_{q\pm 1}$, $A_{4}$, $S_{4}$, $A_{5}$ or $PGL_{2}(q^{1/2})$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
Since $G$ acts primitively on $\Sigma$ and the size of this one is odd, then $G$ is one of the groups classified by \cite{LS}. Actually, $L\cong X(q)$, where $X(q)$ denotes any simple classical group as a consequence of Lemmas \ref{spor2}, \ref{Alt2} and \ref{Exc2}.
Assume that $G_{\Delta }=N_{G}(X(q_{0}))$ with $q=q_{0}^{c}$ and $q,c$ odd by \cite{LS}. Then $L_{\Delta }=X(q_{0})$ is maximal in $L$ by \cite[Tables H--I]{KL} for $n \geq 13$ and \cite[Section 8.2]{BHRD} for $2\leq n \leq 12$. Then $c=3$ and $L\cong PSL_{m}^{\epsilon }(q)$, where $\epsilon =\pm $ by \cite[Propositions 4.7, 4.17, 4.22 and 4.23]{AB} since $L_{\Delta }$ is a large subgroup of $L$ by Theorem \ref{Large}. Then $L_{\Delta }\cong \frac{j}{%
(q-\epsilon ,n)}.PGL_{n}^{\epsilon }(q^{1/3})$, where $j=\frac{q-\epsilon }{%
(q^{1/3}-\epsilon ,\frac{q-\epsilon }{(q-\epsilon ,n)})}$, by \cite[%
Proposition 4.5.3(II)]{KL}.
Assume that $L_{\Delta }^{\Delta}$ does not act $2$-transitively on $\Delta$. Then $L_{\Delta }^{\Delta}$ is solvable and $\lambda/2+1 \leq (n,q^{1/3}- \varepsilon) $ by Proposition \ref{unapred}. Moreover, it results that $\lambda \equiv 0 \pmod{4}$ by Lemma \ref{Jedan}(3). Therefore, $ q^{n(8n+3)/18} \leq \left\vert L: L_{\Delta}\right\vert <2(n,q^{1/3}- \varepsilon)^2 $, which is impossible for $n \geq 2$. Thus $L_{\Delta }^{\Delta}$ acts $2$-transitively on $\Delta$ and hence $n=2,3$. Then $\lambda=2q^{1/3}$, $2(q^{2/3}+q^{1/3})$ or $2q$ according to whether $n=2$, $(n,\varepsilon)=(3,+)$ or $(n,\varepsilon)=(3,-)$ respectively. Therefore, $\left\vert \Sigma \right \vert$ is coprime to $q$ and we reach a contradiction by \cite[Theorem 1.6]{Se} since $L^{\Delta}$ is a non-parabolic subgroup of $L$.
Assume that $L\cong \Omega _{7}(q)$ and $L_{\Delta }\cong \Omega _{7}(2)$. Then $q=3,5$ since $L_{\Delta}$ is a Large subgroup of $L$, and hence $\left\vert \Sigma \right\vert =3159$ or $157421875$ respectively. However, both contradict Lemma \ref{Jedan}(1), as $2\left\vert \Sigma \right\vert -1$ is
not a square.
Assume that $L\cong P\Omega _{8}^{+}(q)$, where $q$ is a prime and $q\equiv
\pm 3\pmod{8}$, and either $L_{\Delta }\cong \Omega _{8}^{+}(2)$ or $%
L_{\Delta }\cong 2^{3}\cdot 2^{6}\cdot PSL_{3}(2)$. In the former case $q=3,5$ since $L_{\Delta}$ is a Large subgroup of $L$, and hence $%
\left\vert \Sigma \right\vert =28431$ or $51162109375$. However, both contradict Lemma \ref{Jedan}(1). Then $L_{\Delta }\cong 2^{3}\cdot 2^{6}\cdot PSL_{3}(2)$, and hence $\left\vert \Sigma \right\vert =57572775$, but $2\left\vert \Sigma \right\vert -1$ is
not a square.
Finally, the case $L\cong PSU_{3}(5)$ and $L_{\Delta }\cong P\Sigma L_{2}(9)$
implies $\left\vert \Sigma \right\vert =175$, and we again reach a
contradiction by Lemma \ref{Jedan}(1).
\end{proof}
\bigskip
\begin{lemma}
\label{m=2}$L$ is not isomorphic to $PSL_{2}(q)$.
\end{lemma}
\begin{proof}
Assume that $L\cong PSL_{2}(q)$ and $L_{\Delta }$ is isomorphic to one of
the groups $D_{q\pm 1}$, $A_{4}$, $S_{4}$, $A_{5}$ or $PGL_{2}(q^{1/2})$. If
$L_{\Delta }\cong D_{q\pm 1}$ then $\left\vert \Sigma \right\vert =\frac{%
q(q\mp 1)}{2}$ and hence $2\left\vert \Sigma \right\vert -1=q^{2}+q+1$ or $%
(q-1)^{2}+(q-1)+1$ must be square by Lemma \ref{Jedan}(1). However this is
impossible by \cite[A7.1]{Rib}.
If $L_{\Delta }\cong PGL_{2}(q^{1/2})$ then $\left\vert \Sigma \right\vert
=q^{1/2}(q+1)/2$, and hence $2\left\vert \Sigma \right\vert
-1=q^{3/2}+q^{1/2}-1$. Moreover, $L_{\Delta }^{\Delta}$ acts $2$-transitively on $\Delta$. If it is not so, then $\lambda/2+1 \leq (2,q^{1/2}-1)$ as a consequence of Proposition \ref{unapred}, whereas $\lambda>10$. Thus, either $\left\vert \Delta
\right\vert =q^{1/2}+1$ or $\left\vert \Delta \right\vert =q^{1/2}$ and $q^{1/2}=7,11$ since either $\lambda \equiv 0 \pmod{4}$ and hence $\left\vert \Delta \right \vert =\lambda/2+1$ is odd, or $\lambda =2w^{2}$, where $w$ is odd, $w\geq 3$, $2(w^{2}-1)$ is a square and $\left\vert \Delta \right \vert=w^{2}+1$. The two numerical cases are ruled out since they
violate Lemma \ref{Jedan}(1), whereas the former yields $\lambda =2q^{1/2}$ with $q$ odd. Then $\left \vert \Sigma \right \vert$ is coprime to $q$ and hence $L_{\Delta}$ must contain a Sylow $p$-subgroup of $L$, which is not the case.
Finally, assume that $L_{\Delta }\cong A_{4}$, $S_{4}$ or $A_{5}$. In the first two cases $\lambda$ must be divisible by $4$ by Lemma \ref{Jedan}(3), hence $\lambda /2 +1=\left\vert \Delta \right\vert =3$ since $\left\vert \Delta \right\vert$ is odd,
and so $\lambda =4$, whereas $\lambda >10$. Thus $L_{\Delta } \cong A_{5}$. The previous argument can be applied to exclude the case $\lambda \equiv 0 \pmod{4}$. Therefore $\lambda=18$, $\left\vert \Delta \right\vert =10$ and $145=\left\vert \Delta \right\vert =\frac{q(q^2-1)}{120}$ which has no integer solutions.
\end{proof}
\bigskip
\begin{lemma}
\label{PSL2}$L$ is not isomorphic to $PSL_{n}(q)$, $n \geq 2$.
\end{lemma}
\begin{proof}
Assume that $L\cong PSL_{n}(q)$. Then $n\geq 3$ by Lemma \ref{m=2}, and hence
$M\in \mathcal{C}_{1}(L)\cup \mathcal{C}_{2}(L)$ by Lemma \ref{Filt}.
Assume that $L_{\Delta }$ lies in a maximal parabolic subgroup $M$ of $L$. If $L_{\Delta }$ is not of type $%
P_{h,n-h}$ then $L_{\Delta }=M$ by \cite[Table 3.5.H--I]{KL} for $n \geq 13$ and by \cite[Section 8.2]{BHRD} for $3\leq n \leq 12$. Also, $L_{\Delta }$ is as in \cite[Proposition 4.1.17.(II)]{KL} and $%
\left\vert \Sigma \right\vert ={n\brack h}_{q}$, where $h\leq n/2$.
Assume that $L_{\Delta }^{\Delta }$ is solvable. Then $ \left \vert L_{\Delta }^{\Delta }\right\vert \mid q-1$ by \cite[Proposition 4.17.(II)]{KL}, and hence
$$\frac{1}{2} q^{h(n-h)} \leq {n\brack h}_{q}=\left\vert \Sigma \right\vert < 2q^{2}.$$
Then either $n =3$ and $h=1$, or $(n,h,q)=(4,1,2),(4,2,2)$ since $\left\vert \Sigma \right\vert$. The numerical cases are immediately ruled by Lemma \ref{Jedan}(1). Therefore $n=3$, $\left\vert \Sigma \right\vert=q^{2}+q+1$ and hence $X^{2}=q^{2}+(q+1)^{2}$, where $X^{2}=2\left\vert \Sigma \right\vert-1$ again by Lemma \ref{Jedan}(1). Thus $(q,q+1,X)$ is primitive solution of the Pythagorean equation and hence it is of the form as in \cite[P3.1]{Rib}. Easy computations show that $q=3$. Then $\left\vert \Sigma \right\vert=13$ and hence $\lambda=6$, whereas $\lambda>10$.
Assume that $L_{\Delta}^{\Delta}$ is non-solvable. Then $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}. Then $\mathrm{Soc}(L_{\Delta }^{\Delta })$ is isomorphic to $PSL_{x}(q)$, where $x\in
\left\{ h,n-h\right\} $ and $x\geq 2$, by \cite[List (B)]{Ka} and by \cite[Proposition 4.17.(II)]{KL}. Note that $(x,q)\neq (2,5),(2,9)$ since $\lambda>10$. Moreover, $(x,q)\neq (2,7)$. It is not so, then $\lambda=12$ and hence $\left\vert \Sigma \right \vert =121$. However, this is impossible by \cite[Table B.4]{DM} since $L \cong PSL_{n}(7)$. Thus $\left\vert \Delta
\right\vert =\frac{q^{x}-1}{q-1}$, $\lambda =2q\frac{q^{x-1}-1}{q-1}$ and
hence%
\begin{equation}
8q^{2x-2}\geq 2q^{2}\left( \frac{q^{x-1}-1}{q-1}\right) ^{2}-2q\frac{%
q^{x-1}-1}{q-1}+1=\left\vert \Sigma \right\vert ={n\brack x}_{q}\geq \frac{%
\allowbreak q^{\frac{1}{2}x\left( 2n-x+1\right) }}{2q^{\frac{1}{2}x\left(
x+1\right) }}=\frac{1}{2}q^{x\left( n-x\right) } \label{ULB}
\end{equation}%
and so $2^{x\left( n-x\right) -2x+2}\leq q^{h\left( n-h\right) -2x+2}\leq 16$%
. Then $x(n-2-x) \leq 2$, and hence $x=h=2$ and $n=5$ since $x \geq 2$ and $h \leq n/2$. Also, $q=2,3$, and hence $\left\vert \Sigma \right\vert=155$ or $1210$ respectively, but both values of $\left\vert\Sigma\right\vert$ contradict Lemma \ref{Jedan}(1).
Assume that $L_{\Delta }$ is of type $P_{h,n-h}$, where $h<n/2$. If $L_{\Delta }^{\Delta }$ is solvable, then $ \left \vert L_{\Delta }^{\Delta }\right\vert \mid (q-1)^{2}$ by \cite[Proposition 4.1.22.(II)]{KL}. Also, $\left\vert \Sigma \right\vert \geq {n\brack h}_{q}$ since $L_{\Delta }$ lies in a maximal parabolic subgroup of type $P_{h}$. Hence,
$$\frac{1}{2} q^{h(n-h)} \leq {n\brack h}_{q}\leq\left\vert \Sigma \right\vert < 2(q-1)^{4}.$$
Then $n \leq 5$ and $h=1$ since $h<n/2$. We actually obtain $n=3$ since $\left\vert \Sigma
\right\vert =\frac{(q^{n}-1)(q^{n-1}-1)}{(q-1)^{2}}$. Hence, $%
2q^{3}+(2q+1)^{2}=2\left\vert \Sigma \right\vert -1=X^{2}$ for some positive odd integer $X$ by Lemma \ref{Jedan}(1). Then $%
2q^{3}=(X-2q-1)(X+2q+1)$ and hence $q=2^{t}$, $t\geq 1$. Then $2^{s}+2^{t+1}+1=X=2^{3t+1-s}-2^{t+1}-1$ for some integer $s$ such
that $0\leq s\leq 3t$. Thus $2^{3t+1-s}=2^{s}+2^{2(t+1)}+2$. If $s>1$ then $%
s=3t$, which is clearly impossible. Then $s=1$ and hence $2^{3t}=2^{2(t+1)}+4$%
, which has no integer solutions for $t\geq 1$. Thus, $L_{\Delta }^{\Delta }$ is a non-solvable group acting $2$-transitively on $\Delta$ by Proposition \ref{unapred}. Then $L_{\Delta }^{\Delta
} $ is isomorphic to $PSL_{x}(q)$, where $x\in \left\{ h,m-h\right\} $ and $%
x\geq 2$, by \cite[Proposition 4.1.22.II]{KL} and \cite[List (B)]{Ka}. Then the same conclusion of (\ref{ULB}) holds
since $\left\vert \Sigma \right\vert \geq {m\brack x}_{q}$. Thus $n=5$, $x=2$ and $q=2,3$. Then $\left\vert \Sigma \right\vert=1085$ or $7865$ respectively, but both values of $\left\vert\Sigma\right\vert$ contradict Lemma \ref{Jedan}(1).
Assume that $L_{\Delta}$ lies in a maximal member of $\mathcal{C}_{2}(L)$. Then $q$ is odd by \cite{LS}, and $L_{\Delta}$
is of type $GL_{n/t}(q)\wr S_{t}$, where either $t=2$, or $t=3$ and either $%
q\in \left\{ 5,9\right\} $ and $n$ odd, or $(n,q)=(3,11)$ by \cite[Proposition 4.7]{AB}. Moreover,
\begin{equation}\label{hocudaumrem}
L_{\Delta}\cong \left[ \frac{(q-1)^{t-1}(q-1,n/t)}{(q-1,n)}\right] .PSL_{n/t}(q)^{t}.%
\left[ (q-1,n/t)^{t-1}\right] .S_{t}
\end{equation}%
by \cite[Proposition 4.2.9.(II)]{KL}. In addition, $L_{\Delta}^{\Delta}$ is non-solvable and acts $2$-transitively on $\Delta$ by Lemma \ref{Sevdah}. Then $t<n$ by (\ref{hocudaumrem}) and $\left\vert L\right\vert
<\left\vert L_{\Delta }\right\vert ^{2}$ by Theorem \ref{Large}. Hence,
\begin{equation}\label{izbaciti}
q^{n^{2}-2}<\frac{(q-1)^{2t-2}(q-1,n/t)^{2t}}{(q-1,n)^{2}} \cdot (q^{2n^{2}/t-2t})\cdot (t!)^{2}
\end{equation}
by (\ref{hocudaumrem}) and \cite[Corollary 4.1.(i)]{AB}. If $t=3$ then $q^{n^{2}-2}<36q^{2n^{2}/3+2}$ and hence $q^{n^{2}/3-4}<36$, which has no solutions for $q \geq 3$ and $n \geq 6$. Thus $t=2$, and hence $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{n/2}(q)$ since $L_{\Delta}^{\Delta}$ is non-solvable and acts $2$-transitively on $\Delta$. Then either $%
\left\vert \Delta \right\vert =5$ and $(n,q)=(4,9)$, or $\left\vert \Delta
\right\vert =q$ and $(n,q)=(4,5),(4,7),(4,11)$, or $\left\vert \Delta
\right\vert =\frac{q^{n/2}-1}{q-1}$. In each case one has $q^{5n}\leq
q^{n(3n-2)/2}\leq \left\vert L:L_{\Delta }\right\vert =\left\vert \Sigma
\right\vert \leq 2\left\vert \Delta \right\vert ^{2}<2q^{n}$, which is a
contradiction. This completes the proof.
\end{proof}
\begin{lemma}
\label{PSU2}$L$ is not isomorphic to $PSU_{n}(q)$.
\end{lemma}
\begin{proof}
Assume that $L\cong PSU_{n}(q)$. Then $n\geq 3$ by Lemma \ref{m=2} since $%
PSU_{2}(q)\cong PSL_{2}(q)$. Moreover, one of the following holds by Lemma %
\ref{Filt} and by \cite[Proposition 4.17]{AB}:
\begin{enumerate}
\item[(i)] $q$ is even and $L_{\Delta }$ lies in a maximal parabolic
subgroup of $L$.
\item[(ii)] $q$ is odd and $L_{\Delta }$ is a maximal $\mathcal{C}_{1}$%
-subgroup of $L$ of type $GU_{t}(q)\perp GU_{n-t}(q)$;
\item[(iii)] $q$ is odd, $L_{\Delta }$ is a maximal $\mathcal{C}_{2}$%
-subgroup of $L$ of type $GU_{n/t}(q)\wr S_{t}$ and one of the following
holds:
\begin{enumerate}
\item $t=2$.
\item $t=3$ and $(q,d)=(5,3),(13,1)$, where $d=(n,q+1)$.
\item $t=n=4$ and $q=5$.
\end{enumerate}
\end{enumerate}
Suppose that (i) holds. Then $L_{\Delta }$ is a maximal parabolic subgroup
of $L$ by \cite[Table 3.5.H--I]{KL} for $n\geq 13$ and \cite[Section 8.2]%
{BHRD} for $3\leq n\leq 12$. If $L_{\Delta }^{\Delta }$ does not act $2$-transitively on $\Delta$, then $L_{\Delta }^{\Delta }$ is solvable by Proposition \ref{unapred}. Thus $\left\vert L_{\Delta }^{\Delta }\right\vert \mid
q^{2}-1$ by \cite[Proposition 4.18.(II)]{KL}, and hence
\[
(q-1)q^{n^{2}-3}<\left\vert L\right\vert <4\left\vert Out(L)\right\vert
^{2}\left\vert L_{\Delta }^{\Delta }\right\vert
^{2}=16f^{2}(n,q+1)^{2}(q^{2}-1)^{2}
\]%
by Theorem \ref{Large}(2b). Thus $q^{n^{2}-6}<16n^{2}(q+1)$, and hence $n=3$
and $q=4,8$ since $q$ is even. So, $\left\vert \Sigma \right\vert =65$ and $%
513$ respectively. However, both these contradict Lemma \ref{Jedan}(1). Thus $%
L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta $ by Proposition \ref%
{unapred}.
Assume that $L_{\Delta }^{\Delta }$ is non-solvable. Then either $t\geq 2$, $PSL_{t}(q^{2})%
\trianglelefteq L_{\Delta }^{\Delta }$ and $\left\vert \Delta \right\vert =%
\frac{q^{2t}-1}{q-1}$, or $n=4$, $t=1$, $PSL_{2}(q)\trianglelefteq L_{\Delta
}^{\Delta }$ and $\left\vert \Delta \right\vert =q+1$, or $n-2t=3$, $%
PSU_{3}(q)\trianglelefteq L_{\Delta }^{\Delta }$ and $\left\vert \Delta
\right\vert =q^{3}+1$ by \cite[Proposition 4.1.18(II)]{KL} and by \cite[List
(B)]{Ka} since $q$ is even. On the other hand, by \cite[Proposition 4.1.18.(II)%
]{KL} and \cite[Lemma 4.1]{AB}, one obtains
\begin{equation}\label{smrt}
\left\vert \Sigma \right\vert =\frac{\prod_{i=4}^{2t+3}\left(
q^{i}-(-1)^{i}\right) }{\prod_{j=1}^{t}\left( q^{2j}-1\right) }>\frac{q^{(2t+3)(t+2)-3}}{q^{3}+1} \cdot \frac{1}{(q^{2}-1)(q^{4}-1)q^{t(t+1)-6}} >q^{t^{2}+6t}.
\end{equation}
If $\left\vert \Delta \right\vert =\frac{q^{2t}-1}{q-1}$ then
\begin{equation}
2q^{4t-2}>2\left( q\frac{q^{2t-1}-1}{q-1}\right) ^{2}-2\left( q\frac{%
q^{2t-1}-1}{q-1}\right) +1=\left\vert \Sigma \right\vert >q^{t^{2}+6t}
\label{polud}
\end{equation}%
and we reach a contradiction.
In the remaining cases, we have $\lambda =2q^{i}$ and $\left\vert \Sigma
\right\vert =2q^{2i}-2q^{i}+1$ with $i=1,2$. Both these lead to $%
q^{t^{2}+6t}<\left\vert \Sigma \right\vert <8q^{4}$ and hence to a
contradiction.
Assume that $L_{\Delta }^{\Delta }$ is solvable. As above $\left\vert L_{\Delta }^{\Delta }\right\vert \mid
q^{2}-1$, hence $\left\vert \Delta \right \vert \mid q^{2}-1$ by Proposition \ref{unapred}. Then $q^{t^{2}+6t} \leq \left\vert \Sigma \right \vert \leq 2 \left\vert \Delta \right \vert^{2} \leq 2(q^{2}-1)^{2}$ by (\ref{smrt}), which is clearly impossible for $t\leq 2$.
Note that, $L_{\Delta }$ is clearly non-parabolic in the remaining cases.
Thus $p\mid \left\vert \Sigma \right\vert $, and hence $q\geq 5$ by Lemma %
\ref{Jedan}(3) since $q$ is odd.
Suppose that (ii) holds. Then
$L_{\Delta }^{\Delta }$ is non-solvable and acts $2$-transitively on $\Delta $ by Lemma \ref{Sevdah}(2). Then either $t=3$ or $n-t=3$ and in both cases $L_{\Delta }\cong
PSU_{3}(q)$ by \cite[Proposition 4.1.4.(II)]{KL}. Then $\lambda /2+1=q^{3}+1$
and so $\lambda =2q^{3}$. Then $\left\vert \Sigma \right\vert
=2q^{6}-2q^{3}+1$, whereas $p\mid \left\vert \Sigma \right\vert $.
Suppose that (iii) holds. Then
\begin{equation}
L_{\Delta }\cong \left[ \frac{(q+1)^{t-1}(q+1,n/t)}{(q+1,n)}\right]
.PSU_{n/t}(q)^{t}.\left[ (q+1,n/t)^{t-1}\right] .S_{t} \label{ciufciuf}
\end{equation}
by \cite[Proposition 4.2.9.(ii)]{KL}. Case (iii.c) implies $\left\vert \Sigma
\right\vert =5687500$ but this contradicts Lemma \ref{Jedan}(1). So, it does
not occur. Thus $t=2$ or $3$. Also $L_{\Delta }^{\Delta }$ is non-solvable and acts $2$-transitively on $\Delta$ by Lemma \ref{Sevdah}(2). Then $t<n$ by (\ref%
{ciufciuf}) since $t=2,3$, and
\begin{equation}
(q-1)q^{n^{2}-3}<\left\vert L\right\vert <\left\vert L_{\Delta }\right\vert
^{2}=\frac{(q+1)^{2t-2}(q+1,n/t)^{2t}\left( t!\right) ^{2}}{(q+1,n)^{2}}%
q^{n^{2}/t-t} \label{omg}
\end{equation}%
by Theorem \ref{Large} and \cite[Corollary (ii)]{AB}. If $t=3$ then $n\geq 6$%
, as $t<n$, and hence (\ref{omg}) implies $%
(q-1)q^{2n^{2}/3}<4(q+1)^{4}n^{4}/9$ which has no admissible solutions for $%
q\geq 5$. Then $t=2$, $n\geq 4$ and $q\geq 5$. Also (\ref{omg}) implies $%
(q-1)q^{n^{2}/2-1}<4(q+1)^{2}n^{2}$ and again no admissible solutions. This completes the proof.
\end{proof}
\begin{lemma}
\label{PSp2}If $L$ is not isomorphic to $PSp_{n}(q)^{\prime}$.
\end{lemma}
\begin{proof}
Assume that $L\cong PSp_{n}(q)^{\prime}$. Then $n\geq 4$ by Lemma \ref{m=2} and $(n,q) \neq (4,2)$ by Lemma \ref{Alt2} since $PSp_{4}(2)^{\prime} \cong A_{6}$. Thus $L\cong PSp_{n}(q)$. By Lemma \ref{Filt} and by \cite{LS} and \cite[Proposition 4.22]{AB} one of the following holds:
\begin{enumerate}
\item[(i)] $L_{\Delta }$ lies in a maximal parabolic subgroup of $L$ and $q$ is
even.
\item[(ii)] $L_{\Delta }$ is a maximal $\mathcal{C}_{1}$-subgroup of $L$ of type $Sp_{i}(q)\perp Sp_{n-i}(q)$ with $q$
odd
\item[(iii)] $L_{\Delta }$ is a maximal $\mathcal{C}_{2}$-subgroup of $L$ of type $Sp_{n/t}(q)\wr S_{t}$, where $t=2,3$,
or $(n,t)=(8,4)$, or $(n,t)=(10,5)$ and $q=3$.
\end{enumerate}
Suppose that (i) holds. Then $L_{\Delta }$ is a maximal parabolic subgroup
of $L$ by \cite[Table 3.5.H--I]{KL} for $n \geq 13$ and by \cite[Section 8.2]{BHRD} for $4 \leq n \leq 12$. Thus
$$\left\vert \Sigma \right\vert =\prod_{i=0}^{t-1}\frac{q^{n-2i}-1}{q^{i+1}-1}>\frac{1}{2}q^{(n-1)t-3t(t-1)/2}$$
If $L_{\Delta }^{\Delta }$ is solvable then $\left\vert L_{\Delta }^{\Delta } \right \vert \mid (q-1,t)$
by \cite[Proposition 4.1.19(II)]{KL} and hence $\left\vert \Delta \right \vert \mid (q-1,t)$ by Proposition \ref{unapred}. Therefore $ q^{(n-1)t-3t(t-1)/2/2}<\left\vert \Sigma \right\vert <2(q-1,t)^{2}$, which is clearly impossible.
If $L_{\Delta }^{\Delta }$ is non-solvable then $L_{\Delta }^{\Delta }$ acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}. Hence, by \cite[List (B)]{Ka}, one of the following holds:
\begin{enumerate}
\item[(I)]$\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{t}(q)$, either $t\geq 2$, or $t=1$ and $n=4$, and $%
\left\vert \Delta \right\vert =\frac{q^{t}-1}{q-1}$;
\item[(II)]$\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSp_{n-t}(2)$, $n-t\geq 6$ and $\left\vert \Delta
\right\vert =2^{2(n-t)-1}\pm 2^{n-t-1}$.
\end{enumerate}
Then (I) is ruled out since it implies $q^{(n-1)t-3t(t-1)/2/2}<\left\vert \Sigma \right\vert <2q^{2t}$, which is impossible; (II) is ruled out since $\left\vert \Delta \right\vert \not \equiv 0 \pmod{4}$.
Suppose that (ii) holds. Then $L_{\Delta }\cong Sp_{i}(q)\circ Sp_{n-i}(q)$
by \cite[Proposition 4.1.3.(II)]{KL}. Then $L_{\Delta }^{\Delta }\cong
PSp_{j}(q)$ with $j\in \left\{ i,n-i\right\} $ by Lemma \ref{Sevdah}. Hence, one of the following
holds (recall that $q$ is odd):
\begin{enumerate}
\item $j=2$ and $\left\vert \Delta \right\vert =6$ for $q=9$, $\left\vert
\Delta \right\vert =q$ and $q=5,7,11$, or $\left\vert \Delta \right\vert =q+1$;
\item $j\geq 6$ and $\left\vert \Delta \right\vert =2^{j-1}\pm 2^{j/2-1}$.
\end{enumerate}
Actually, in (1) $q=5$ and $\left\vert \Delta \right\vert =5$ cannot occur since $\lambda>10$, and $q\neq 7,9,11$ by Lemma \ref{Jedan}(2). Also (2) is ruled out since $\left\vert \Delta \right\vert \not \equiv 0 \pmod{4}$. Thus $\left\vert \Delta \right\vert =q+1$, $\lambda=2q$ and hence $\left\vert \Sigma \right\vert $ is coprime to $q$. So $L_{\Delta }$ must contain a Sylow $q$-subgroup of $L$, which is a contradiction.
Suppose that (iii) holds. Then $q \mid \left \vert \Sigma \right \vert$ and hence $q \neq 3$ by Lemma \ref{Jedan}(2). Thus either $t=2,3$, or $(n,t)=(8,4)$. Also, $L_{\Delta }^{\Delta }$ is non-solvable and acts $2$-transitively on $\Delta$ by Lemma \ref{Sevdah}(2)
If $t=2$, then either $n=4$ and $L_{\Delta }^{\Delta }\cong PSL_{2}(q)$ and
either $\left\vert \Delta \right\vert =5$ for $q=5$, or $\left\vert \Delta
\right\vert =6$ for $q=9$, $\left\vert \Delta \right\vert =q$ for $q=7,11$,
or $\left\vert \Delta \right\vert =q+1$, or $n\geq 12$, $L_{\Delta }^{\Delta
}\cong PSL_{n/2}(2)$ and $\left\vert \Delta \right\vert =2^{n-1}\pm 2^{n/2-1}
$. However, all these cases are excluded by the same argument previously used.
If $t=3,4$, by Theorem \ref{Large} and by \cite[Proposition 4.2.10.(II)]{KL} and \cite[Corollary 4.3(iii)]{AB}, we have
\begin{equation}
q^{\frac{n(n+1)}{2}}/4<\left\vert L\right\vert <\left\vert
L_{\Delta }\right\vert ^{2}<2^{2(t-1)}q^{n(n/t+1)}(t!)^{2} \label{ineq}
\end{equation}
which implies $q^{\frac{n(n+1)}{2}-\frac{n(n+t)}{t}}<2^{2t}(t!)^{2}$, and
hence $t=3$, $n=6$ and $q=5$ or $13$ since $q\equiv 1\pmod{4}$ by Lemma %
\ref{Jedan}(2). Therefore $\left\vert \Sigma \right\vert =44078125$ or $%
3929239732405$, but both contradict Lemma \ref{Jedan}(1).
\end{proof}
\begin{lemma}
\label{Cla2}$L$ is not isomorphic to a simple classical group.
\end{lemma}
\begin{proof}
In order to complete the proof we need to tackle the case $L\cong P\Omega
_{n}^{\varepsilon }(q)$, where $\varepsilon \in \left\{ \pm ,\circ \right\} $
since the other simple groups are analyzed in Lemmas \ref{PSL2}, \ref{PSU2} and \ref{PSp2}. Since $L$ is non-abelian
simple, $n>2$ and $(n,\varepsilon )\neq (4,+)$. Also $(n,\epsilon ,)\neq
(3,\circ )$ for $q$ odd, $(4,-),(6,+)$ by Lemma \ref{PSL2}, since in these
cases $L$ is isomorphic to $PSL_{2}(q)$, $PSL_{2}(q^{2})$ or $PSL_{4}(q)$
respectively. Finally $(n,\epsilon )\neq (4,-)$, and $(n,\epsilon )\neq
(5,\circ )$ for $q$ odd, otherwise $L$ would be isomorphic to $PSU_{4}(q)$ or $PSp_{4}(q)$ respectively, which are excluded in Lemmas \ref{PSU2} and \ref{PSp2} respectively. Thus, $n\geq 7$. By \cite{LS} and by \cite[Proposition 4.23]{AB} one of the following holds:
\begin{enumerate}
\item[(1)] Either $q$ is even and $L_{\Delta}$ lies in a maximal parabolic subgroup, or $q$ is odd and $L_{\Delta}$ is
the stabilizer in $L$ of a non-degenerate subspace of $PG_{n-1}(q)$.
\item[(2)] $L_{\Delta}$ is a $\mathcal{C}_{2}$-subgroup of $L$ of type $%
O_{n/t}^{\varepsilon ^{\prime }}(q)\wr S_{t}$, where $q$ is odd, and either $%
t=2$, or $n=t=7$ and $q=5$, or $7\leq n=t\leq 13$ and $q=3$.
\end{enumerate}
Assume that $q$ is even and that $L_{\Delta}$ lies in a maximal parabolic subgroup $M$ of type $%
P_{m}$. Thus $\varepsilon =\pm $ and hence $n \geq 8$. If $(\varepsilon ,m)\neq (+,n/2-1)$, then
$L_{\Delta }=M$ by \cite[Table 3.5.H--I]{KL} for $n\geq 13$ and by \cite[Section 8.2]{BHRD} for $7\leq n\leq
12$. Nevertheless, in each case we have that $L_{\Delta }\cong \lbrack q^{a}]:GL_{m}(q)\times
\Omega _{n-2m}^{\varepsilon }(q)$, where $a=nm-\frac{m}{2}(3m-1)$, by \cite[%
Proposition 4.1.20.II]{KL}. Therefore,
\begin{equation}
\left\vert \Sigma \right\vert ={ \frac{n-1+\varepsilon }{2} \brack m}%
_{q}\prod_{i=0}^{m-1}\left( q^{\frac{n-1-\varepsilon }{2}-i}+1\right)
>q^{m\left( \frac{n-1+\varepsilon -2m}{2}+\frac{n-\varepsilon -m}{2}\right)
}=q^{\left( n-\frac{3}{2}m-\frac{1}{2}\right) m} \label{nosfe}
\end{equation}%
by (\ref{ULB}) (see also \cite[Exercise 11.3]{Tay}).
Assume that $L_{\Delta}^{\Delta}$ is solvable. Then $\left\vert L_{\Delta}^{\Delta} \right\vert \mid q-1$, and hence $\left\vert \Delta \right\vert \mid q-1$. Then $q^{\left( n-\frac{3}{2}m-\frac{1}{2}\right) m} <\left\vert \Sigma \right\vert \leq 2 \left\vert \Delta \right\vert ^{2}=2(q-1)^{2}$, which is impossible for $n \geq 8$.
Assume that $L_{\Delta}^{\Delta}$ is non-solvable. Then $L_{\Delta}^{\Delta}$ acts $2$-transitively on $\Delta$ by Proposition \ref{unapred}, hence one of the following holds by \cite{Ka}:
\begin{enumerate}
\item[(I)] $\mathrm{Soc}(L_{\Delta }^{\Delta })$ is isomorphic to $PSL_{m}(q)$, $m \geq 2$, and either $\left\vert \Delta \right\vert =\frac{q^{m}-1}{q-1}$, or $\left\vert \Delta \right\vert =8$ for $(m,q)=(4,2)$.
\item[(II)] $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong \Omega _{4}^{-}(q)\cong
PSL_{2}(q^{2})$, $\varepsilon =-$, $n=2m+4$ and $\left\vert \Delta
\right\vert =q^{2}+1$.
\item[(III)] $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{2}(q)$, $\varepsilon =+$, $%
n=2m+4$ and $\left\vert \Delta \right\vert =q+1$.
\item[(IV)]$\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{4}(q)$, $\varepsilon =+$, $%
n=2m+6$ and either $\left\vert \Delta \right\vert =\frac{q^{4}-1}{q-1}$, or $\left\vert \Delta \right\vert =8$ for $%
q=2$.
\end{enumerate}
Assume that (I) or (IV) holds. Then $\left\vert \Delta \right\vert \neq 8$ since $\left\vert \Delta \right\vert$ is not divisible by $4$. Then $\left\vert \Delta
\right\vert =\frac{q^{e}-1}{q-1}$, where either $e=m$, or $e=4$ and $n=2m+6$ and $\varepsilon=+$. Furthermore, $%
m\geq 2$ in both cases since $n\geq 8$. Now, $\left\vert \Delta \right\vert= \lambda/2+1$ and $\left\vert \Delta \right\vert= (\lambda^{2}-2\lambda+2)/2$ imply
\begin{equation*}
\left\vert \Sigma \right\vert =2\left( q\frac{q^{e-1}-1}{q-1}\right)
^{2}-2\left( q\frac{q^{e-1}-1}{q-1}\right) +1\text{.}
\end{equation*}%
and so $q^{\left( n-\frac{3}{2}m-\frac{1}{2}\right) m}<\left\vert \Sigma
\right\vert <2q^{2e-2}$.
If $e=m$ and $q^{\left( n-\frac{3}{2}m-\frac{1}{2}\right) m-2m+2}<2$ then $n \leq %
10-\frac{8}{m}$, and hence $(n,m,\varepsilon)=(8,4,+)$ since $n$ is even, $n \geq 8$ and $m \leq n/2$. Then \begin{equation*}
2q^{6}+4q^{5}+6q^{4}+2q^{3}-2q+1=\left\vert \Sigma \right\vert =\allowbreak
\left( q^{2}+1\right) \left( q^{3}+1\right) \left( q+1\right)
\end{equation*}%
If $e=4$, $n=2m+6$ and $\varepsilon=+$, then $m=1$ and so%
\begin{equation*}
2q^{6}+4q^{5}+6q^{4}+2q^{3}-2q+1=\left\vert \Sigma \right\vert
=(q^{4}-1 )(q^{3}+1 )\text{,}
\end{equation*}%
which has not integer solutions.
Assume that case (II) or (III) holds. Then $n=2m+4$ and $\left\vert \Delta
\right\vert =q^{j}+1$ with $j=2,1$ respectively. Then $\left\vert \Sigma
\right\vert <8q^{2j}$. On the other hand $\left\vert \Sigma \right\vert
>q^{\left( n-\frac{3}{2}m-\frac{1}{2}\right) m}$ by (\ref{nosfe}) since $%
n\geq m/2$. Therefore $q=2$ and $n=8$ and either $m=1$ or $4$, but each of
these contradicts $n=2m+4$. This excludes case (1) for $q$
even.
In the remaining cases, namely (1) and (2) for $q$ odd, it results that $p \mid \left \vert \Sigma \right \vert $ by \cite[Theorem 1.6]{Se}. Also $p$ is odd, and $p \neq 3$ by Lemma \ref{Jedan}(2). Therefore, in the sequel we may assume that $q \geq 5$. Then, by \cite[Table 3.5.H--I]{KL} for $n\geq 13$ and by \cite[Section 8.2]{BHRD} for $7\leq n\leq 12$, either $L_{\Delta }$ is maximal in $L$, or $L \cong P\Omega_{n}^{+}(5)$, $L_{\Delta }$ is a $\mathcal{C}_{2}$-subgroup of $L$ of type $O_{2}^{+}(5) \wr S_{n/2}$ and $G_{\Delta}$ is a novelty. In the latter case $n$ is forced to be $4$ by (2), whereas $n \geq 8$. Therefore, $L_{\Delta }$ is maximal in $L$.
Assume that $L_{\Delta}$ is the stabilizer in $L$ of a non-degenerate subspace of $%
PG_{n-1}(q)$, $q$ odd. Since $L_{\Delta}^{\Delta}$ is non-solvable and acts $2$-transitively on $\Delta$ by Lemma \ref{Sevdah}, one of the following holds by \cite[%
Propositions 4.1.6.(II)]{KL} and \cite{Ka} and since $q$ is odd, $q\geq 5$
and $n\geq 7$:
\begin{enumerate}
\item[(i)] $L_{\Delta }$ preserves a non-degenerate $4$-subspace
of type $+$ and $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{2}(q)$;
\item[(ii)] $L_{\Delta }$ preserves a non-degenerate $4$-subspace
of type $-$ and $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{2}(q^{2})$;
\item[(iii)] $L_{\Delta }$ preserves a non-degenerate $6$%
-subspace of type $-$ and $\mathrm{Soc}(L_{\Delta }^{\Delta })\cong PSL_{4}(q)$.
\end{enumerate}
Assume that (i) or (ii) holds. Then either $\left\vert \Delta \right\vert =q^{j}+1$,
where $j=1$ or $2$ respectively, or $j=1$ and $\left\vert \Delta \right\vert
=q$ for $q=5,7,11$ or $\left\vert \Delta \right\vert =6$ for $q=9$, or $j=2$
and $\left\vert \Delta \right\vert =6$ for $q=3$. The first case implies $%
\lambda =2q^{j}$, therefore $\left\vert \Sigma \right\vert $ is coprime to $q$, whereas $L_{\Delta }$ must contain a Sylow $q$-subgroup of $L$, which is not the case. Also $q \neq 3,7,9,11$ by Lemma \ref{Jedan}(2). Finally, $\left\vert \Delta \right\vert
=q=5$ cannot occur since $\lambda>10$.
Assume that (iii) holds. Then $\left\vert \Delta \right\vert =\frac{%
q^{4}-1}{q-1}$ since $q$ is odd. Then $\lambda =2q\frac{q^{3}-1}{q-1}$, hence $\left\vert \Sigma \right\vert $ is coprime to $q$, whereas $L_{\Delta }$ must contain a Sylow $q$-subgroup of $L$, which is not the case. This excludes (1).
A similar argument to that used to rule out (iii) in the $PSp_{n}(q)$%
-case excludes $t=2$ in (2) as well (see Lemma \ref{PSp2}). Thus, $n=t=7$ and $q=5$ since we have seen that $q \geq 5$, and hence $L_{\Delta } \cong 2^{6}.A_{7}$ by \cite[Proposition 4.2.15(II)]{KL}. So $\left\vert \Sigma \right\vert=29752734375$, which contradicts Lemma \ref{Jedan}(1) and hence it is ruled out. This completes the proof.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \protect\ref{T2}]
Since $L^{\Sigma }$ is almost simple by Theorem \ref{ASPQP}, the assertion
follows from Lemmas \ref{spor2}, \ref{Alt2}, \ref{Exc2} and \ref{Cla2}.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \protect\ref{main}]
Since $\lambda \leq 10$ by Theorems \ref{T1} and \ref{T2}, the assertion
follows from Theorem \ref{MMS}.
\end{proof}
\bigskip
|
2,877,628,090,243 | arxiv | \section{Introduction}
The Bienaym\'e-Galton-Watson (BGW-) process is a basic model for the stochastic dynamics of the size of a population formed by independently reproducing particles. It has a long history \cite{HS} with its origin dating back to 1837. This paper is devoted to the BGW-processes with countably many types.
One of the founders of the theory of multi-type branching processes is B.A. Sevastyanov \cite{Sev}, \cite{Sew}.
A single-type BGW-process is a Markov chain $\{Z^{(n)}\}_{n=0}^\infty$ with countably many states $\{0,1,2,\ldots\}$. The evolution of the process is described by a probability generating function
\begin{equation}} \newcommand{\ee}{\end{equation}
f(s)=\sum_{k=0}^\infty p_ks^k, \qquad p_1<1,
\label{fs}
\ee
where $p_k$ stands for the probability that a single particle produces exactly $k$ offspring. If particles reproduce independently with the same reproduction law \eqref{fs}, then the chain $\{Z^{(n)}\}_{n=0}^\infty$ represents consecutive generation sizes. In this paper, if not specified otherwise, we assume that $Z^{(0)}=1$, the branching process stems from a single particle. Due to the reproductive independence it follows that
$f^{(n)}(s)=\mathbb E(s^{Z^{(n)}})$ is the $n$-th iteration of $f(s)$.
Since zero is an absorbing state of the BGW-process,
$q^{(n)}=\mathbb P(Z^{(n)}=0)$ monotonely increases to a limit $q$ called the extinction probability. The latter is implicitly determined as a minimal non-negative solution of the equation
\begin{equation}} \newcommand{\ee}{\end{equation}
f(x)=x.
\label{fx}
\ee
A key characteristic of the BGW-process is the mean offspring number $M=f'(1)$. In the subcritical ($M<1$) and critical ($M=1$) cases the process is bound to go extinct $q=1$,
while in the supercritical case ($M>1$) we have $q<1$. Clearly $q=0$ if and only if $p_0=0$.
In the supercritical case the number of descendants of the progenitor particle is either finite with probability $q$ or infinite with probability $1-q$. Recognizing that the same is true for any particle appearing in the BGW-process we can distinguish between {\it skeleton particles} having an infinite line of descent \cite{O} and {\it doomed particles} having a finite line of descent. Graphically we get a picture of the genealogical tree similar to that given in Figure 1.
If we disregard the doomed particles, the skeleton particles form a BGW-process with a transformed reproduction law excluding extinction
\begin{equation}} \newcommand{\ee}{\end{equation}
\tilde f(s)={f(s(1-q)+q)-q\over 1-q}\label{HST}
\ee
and having the same mean $\tilde M=M>1$. Formula \eqref{HST} is usually called the Harris-Sevastyanov transformation. On the other hand, the doomed particles form another branching process corresponding to the supercritical branching process conditioned on extinction. The doomed particles produce only doomed particles according to another transformation of the reproduction law $\hat f(s)={f(sq)/q}$, which is usually called the dual reproduction law and has mean $\hat M=f'(q)<1$.
The supercritical BGW-process as a whole can be viewed as a decomposable branching process with two subtypes of particles \cite[Ch. 1.12]{AN}.
Each skeleton particle must produce at least one new skeleton particle and also can give rise to a number of doomed particles.
In Section \ref{sHS} we describe in detail this decomposition for the single type supercritical BGW-processes.
\begin{figure}
\centering
\includegraphics[height=3.5cm]{skelet.pdf}
\caption{An example of a BGW-tree up to level $n=10$. Solid lines represent the infinite lines of descent and dotted lines represent the finite lines of descent.}
\label{fig1}
\end{figure}
In the special case when the reproduction generating function \eqref{fs} is linear-fractional many characteristics of the BGW-process can be computed in an explicit form \cite{KRS}. In Section \ref{sLF} we summarize explicit results concerning decomposition of a supercritical single-type BGW-processes.
Section \ref{sC} presents the BGW-processes with countably many types. Our focus is on the linear-fractional case recently studied in \cite{S}.
The main results of this paper are collected in Section \ref{sMR} and their derivation is given in Section \ref{sP}. The remarkable fact that a supercritical branching process conditioned on extinction is again a branching process was recently established in \cite{JL} in a very general setting. In general, the transformed reproduction laws are characterized in an implicit way and are difficult to analyse. This paper presents a case where the properties of the skeleton and doomed particles are very transparent.
\section{Decomposition of a supercritical single-type BGW-process}\label{sHS}
The BGW-process is a time homogeneous Markov chain with transition probabilities satisfying
\begin{align*}
\sum_{j=0}^\infty P^{(n)}_{ij}s^j=\left(f^{(n)}(s)\right)^i.
\end{align*}
In the supercritical case with mean $M>1$ and extinction probability $q<1$, using the property
$\sum_{j=0}^\infty P^{(n)}_{ij}q^j=q^i$,
we can get another set of transition probabilities putting
\begin{align*}
\hat P^{(n)}_{ij}:=P^{(n)}_{ij}q^{j-i}.
\end{align*}
The transformed transition probabilities also possess the branching property
\begin{align*}
\sum_{j=0}^\infty \hat P^{(n)}_{ij}s^j=\left(\hat f^{(n)}(s)\right)^i,
\end{align*}
where $\hat f^{(n)}(s)={f^{(n)}(sq)/ q}$ is the $n$-th iteration of the so-called dual generating function
\[\hat f(s)={f(sq)\over q}=\sum_{k=0}^\infty \hat p_ks^k, \qquad \hat p_k=p_kq^{k-1}, \ k\ge0.\]
The corresponding dual BGW-process is a subcritical branching process with offspring mean $\hat M=f'(q)<1$, see Figure \ref{fig2}.
The dual BGW-process is distributed as the original supercritical BGW-process conditioned on extinction:
\begin{eqnarray*}
\mathbb P(Z^{(n)}=j|Z^{(0)}=i,Z^{(\infty)}=0) &=&{\mathbb P(Z^{(n)}=j,Z^{(\infty)}=0|Z^{(0)}=i)\over\mathbb P(Z^{(\infty)}=0|Z^{(0)}=i)} \\
&=&q^{-i}\mathbb P(Z^{(\infty)}=0|Z^{(n)}=j)P^{(n)}_{ij} \\
&=& q^{j-i}P^{(n)}_{ij}\\
& =& \hat P^{(n)}_{ij}.
\end{eqnarray*}
\begin{figure}
\centering
\includegraphics[width=10cm]{dualpgf.pdf}
\caption{Duality between the subcritical and supercritical cases. Left: a supercritical generating function \eqref{fs} with two positive roots $(q,1)$ for the equation \eqref{fx}. Right: the dual generating function drawn on a different scale. }
\label{fig2}
\end{figure}
The two parts of the curve on the left panel of Figure \ref{fig2} represent two transformations of the supercritical branching process. The lower-left part of the curve, replicated on the right panel of Figure \ref{fig2} using a different scale, gives the generating function of the dual process. The upper-right of the curve on the left panel corresponds to the Harris-Sevastyanov transformation \eqref{HST}.
The function \eqref{HST} is the generating function for the probability distribution
\[\tilde p_0=0, \ \tilde p_k=\sum_{i=k}^\infty p_i {i\choose k}q^{i-k}(1-q)^{k-1}\]
with the same mean $\tilde M=M$ as the original offspring distribution. It is easy to see that the $n$-th iteration of $\tilde f(s)$ is given by
\begin{align*}
\tilde f^{(n)}(s)&={f^{(n)}(s(1-q)+q)-q\over 1-q}.
\end{align*}
Looking into the future of the system of reproducing particles we can distinguish between two subtypes of particles:
\begin{quote}
\begin{itemize}
\item skeleton particles with infinite line of descent (building the skeleton of the genealogical tree),
\item doomed particles having finite line of descent.
\end{itemize}
\end{quote}
These two subtypes form a decomposable two-type BGW-process $\{(S^{(n)},D^{(n)})\}_{n=0}^\infty$ with $S^{(n)}+D^{(n)}=Z^{(n)}$. The joint reproduction law for the skeleton particles has the following generating function
\begin{align*}
F(s,t)=\mathbb E(s^{S^{(1)}}t^{D^{(1)}})={f(s(1-q)+tq)-f(tq)\over 1-q}.
\end{align*}
A check on the branching property for the decomposed process is given by
\begin{align*}
\mathbb E(s^{S^{(n)}}t^{D^{(n)}}|Z^{(\infty)}>0)&={\mathbb E(s^{Z^{(n)}_1}t^{Z^{(n)}_2}1_{\{Z^{(\infty)}>0\}})\over \mathbb P(Z^{(\infty)}>0)}\\
&={\mathbb E(s^{S^{(n)}}t^{D^{(n)}})-\mathbb E(t^{D^{(n)}}1_{\{Z^{(\infty)}=0\}})\over1-q}\\
&={f^{(n)}(s(1-q)+tq)-f^{(n)}(tq)\over 1-q}=F^{(n)}(s,t).
\end{align*}
The original offspring distribution can be recovered as a mixture of the joint reproduction laws of the two subtypes
\begin{align*}
f^{(n)}(s)&=(1-q)F^{(n)}(s,s)+q\hat f^{(n)}(s).
\end{align*}
Observe also that the total number of offspring for a skeleton particle has a distribution given by
\[\bar f(s)=F(s,s)={f(s)-f(sq)\over 1-q}=\sum_{k=0}^\infty \bar p_ks^k,\quad \bar p_k=(1-q)^{-1}(1-q^{k})p_k,\]
with mean $\bar M={M-\hat Mq\over1-q}$. It follows,
\[\bar M=M+(M-\hat M){q\over1-q}\]
and we can summarize the relationship among different offspring means as
$$\hat M<1<\tilde M=M<\bar M.$$
\section{ Linear-fractional single-type BGW-process}\label{sLF}
An important example of BGW-processes is the linear-fractional branching process. Its reproduction law has a linear-fractional generating fuction
\begin{align}\label{lff}
f(s)=h_0+{h_1s\over1+m-ms}
\end{align}
fully characterized by two parameters: the probability $h_0=p_0$ of having no offspring, and the mean $m$ of the geometric number of offspring beyond the first one. Here $h_1=1-h_0$ stands for the probability of having at least one offspring.
Notice that with
$h_0={1\over 1+m}$, the generating function \eqref{lff} describes a Geometric $({1\over 1+m})$ distribution with mean $m$.
If
$h_0=0$ the generating function \eqref{lff} gives a Shifted Geometric $({1\over 1+m})$ distribution with mean $m+1$.
If $m=0$, we arrive at a Bernoulli $(h_1)$ distribution.
Since the iterations of the linear-fractional function are again linear-fractional, many key characteristics of the linear-fractional BGW-processes can be computed explicitly in terms of the parameters $(h_0,m)$.
For example, we have $M=h_1(1+m)$, and if $M>1$, we get
$$q=h_0(1+m^{-1})={1+m-M\over m}.$$
The dual reproduction law for \eqref{lff} is again linear-fractional
\begin{align*}
\hat f(s)&=\hat h_0+{\hat h_1s\over1+\hat m-\hat ms}, \qquad \hat h_0={m\over m+1}, \qquad \hat m=h_0/h_1,
\end{align*}
with $\hat M=1/M$.
The Harris-Sevastyanov transformation in the linear-fractional case corresponds to a shifted geometric distribution
\begin{align*}
\tilde f(s)&={s\over1+\tilde m-\tilde ms},\hspace{7mm} \tilde m=m(1-q)=M-1.
\end{align*}
Interestingly, the joint reproduction law of skeleton particles
\begin{align*}
F(s,t) &={h_1\over 1-q}\left({s(1-q)+tq\over 1+m-m(s(1-q)+tq)}-{tq\over 1+m-mtq}\right)\\
&={s\over1+m-m(s(1-q)+tq)}\cdot{1\over1+\hat m-\hat mt}
\\
&=s\cdot{1\over1+m-\tilde ms-(m-\tilde m)t}\cdot{1\over1+\hat m-\hat mt}
\end{align*}
has three independent components:
\begin{quote}
\begin{itemize}
\item one particle of type 1 (the infinite lineage),
\item a Geometric $({1\over 1+m})$ number of offspring each choosing independently between the skeleton and doomed subtypes with probabilities $\tilde m/m$ and $(m-\tilde m)/m$,
\item a Geometric $({1\over 1+\hat m})$ number of doomed offspring.
\end{itemize}
\end{quote}
Observe that even though both marginal distributions $\tilde f(s)$ and $\hat f(s)$
are linear-fractional, the decomposable BGW-process $(S^{(n)},D^{(n)})$ is not a two-type linear-fractional BGW-process. The distribution of the total number of offspring for the skeleton particles in not linear-fractional
\begin{align*}
\bar f(s) ={s\over1+m-ms}\cdot{1\over1+\hat m-\hat ms}
\end{align*}
and has mean
$$\bar M=1+m+\hat m=M+(M+1)\hat m.$$
\section{BGW-processes with countably many types}\label{sC}
A BGW-process with countably many types
\[\mathbf{Z}^{(n)}=(Z_1^{(n)},Z_2^{(n)},\ldots),\ n=0,1,2,\ldots\]
describes demographic changes in a population of particles with different reproduction laws depending on the type of a particle $i\in\{1,2,\ldots\}$. Here $Z_i^{(n)}$ is the number of particles of type $i$ existing at generation $n$.
In the multi-type setting we use the following vector notation:
\begin{align*}
\mathbf x&=(x_1,x_2\ldots), \ \mathbf 1=(1,1\ldots), \ \mathbf{e}_i=(1_{\{i=1\}},1_{\{i=2\}},\ldots), \\
\mathbf x\mathbf y&=(x_1y_1,x_2y_2,\ldots), \hspace{3mm}\mathbf x^{-1}=(x_1^{-1},x_2^{-1},\ldots), \hspace{3mm} \mathbf x^{\mathbf y}=x_1^{y_1}x_2^{y_2}\cdots,
\end{align*}
we write $\mathbf x^{\rm t}$, if we need a column version of a vector $\mathbf x$.
A particle of type $i$ may produce random numbers of particles of different types so that the corresponding joint reproduction laws are given by the multivariate generating functions
\begin{equation}} \newcommand{\ee}{\end{equation}
f_i(\mathbf{s})=\mathbb{E}( \mathbf{s}^{\mathbf{Z}^{(1)}}|\mathbf{Z}^{(0)}=\mathbf{e}_i).
\label{fis}
\ee
The offspring means
$$M_{ij}=\mathbb{E}(Z^{(1)}_j|\mathbf{Z}^{(0)}=\mathbf{e}_i)$$
are convenient to summarize in a matrix form $\mathbf{M}=\left(M_{ij}\right)_{i,j=1}^\infty$.
For the $n$-th generation the vector of generating functions $\mathbf{f}^{(n)}(\mathbf{s})$ with components
$$f_i^{(n)}(\mathbf{s})=\mathbb{E}( \mathbf{s}^{\mathbf{Z}^{(n)}}|\mathbf{Z}^{(0)}=\mathbf{e}_i)$$
are obtained as iterations of $\mathbf{f}(\mathbf{s})$ with components \eqref{fis}, and the matrix of means is given by $\mathbf{M}^n$.
The vector of extinction probabilities $\mathbf{q}=(q_1,q_2,\ldots)$ has its $i$-th component $q_i$ defined as the probability of extinction given that the BGW-process starts from a particle of type $i$. The vector $\mathbf{q}$ is found as the minimal solution with non-negative components of equation $\mathbf{f}(\mathbf{x})=\mathbf{x}$, which is a multidimensional version of \eqref{fx}.
From now on we restrict our attention to the positive recurrent (with respect to the type space) case when there exists a Perron-Frobenius eigenvalue $\rho$ for $\mathbf{M}$ with positive eigenvectors $\mathbf{u}$ and $\mathbf{v}$ such that
$$\mathbf{v}\mathbf{M}=\rho\mathbf{v}, \ \ \mathbf{M}\mathbf{u}^{\rm t}=\rho\mathbf{u}^{\rm t}, \ \ \mathbf v\mathbf u^{\rm t}=\mathbf v\mathbf 1^{\rm t}=1, \ \ \rho^{-n}\mathbf{M}^n\to \mathbf{u}^{\rm t}\mathbf{v}, \ n\to\infty.$$
In the supercritical case, $\rho>1$, all $q_i<1$ and we can speak about the decomposition
of a supercritical BGW-process with countably many types:
$(\mathbf{S}^{(n)},\mathbf{D}^{(n)})$. Now each type is decomposed in two subtypes: either with infinite or finite line of descent. The decomposed supercritical BGW-process is again a BGW-process with countably many types whose reproduction law is given by the expressions
\begin{align*}
F_{i}(\mathbf s,\mathbf t)={f_i(\mathbf s(\mathbf 1-\mathbf q)+\mathbf t\mathbf q)-f_i(\mathbf t\mathbf q)\over 1-q_i}, \ \ \ \hat f_{i}(\mathbf t)&={f_i(\mathbf t\mathbf q)\over q_i}.
\end{align*}
Linear-fractional BGW-processes with countably many types were studied recently in \cite{S}. In this case the joint probability generating functions \eqref{fis} have a restricted linear-fractional form
\begin{equation}} \newcommand{\ee}{\end{equation}\label{fislf}
f_i(\mathbf s)=h_{i0}+{\sum_{j=1}^\infty h_{ij}s_j\over 1+m-m\sum_{j=1}^\infty g_js_j}.
\ee
The defining parameters of this branching process form a triplet $(\mathbf H,\mathbf g,m)$, where $\mathbf H=(h_{ij})_{i,j=1}^\infty$ is a sub-stochastic matrix,
$\mathbf g=(g_1,g_2,\ldots)$ is a proper probability distribution, and $m$ is a positive constant. The free term in \eqref{fislf} is defined as
\[h_{i0}=1-\sum_{j=1}^\infty h_{ij}.\]
As in the case of finitely many types \cite {JL}, the denominators in \eqref{fislf} are necessarily independent of the mother type to ensure that the iterations are also linear-fractional. This is a major restriction of the multitype linear-fractional BGW-process excluding for example decomposable branching processes.
It is shown in \cite{S} that in the linear-fractional case the Perron-Frobenius eigenvalue $\rho$, if exists, is the unique positive solution of the equation
\begin{equation}} \newcommand{\ee}{\end{equation}
m\sum_{k=1}^\infty \rho^{-k}\mathbf{g}\mathbf H^k\mathbf{1}^{\rm t}=1.
\label{PF}
\ee
In the positive recurrent case, when the next sum is finite
\begin{equation}} \newcommand{\ee}{\end{equation}
\beta=m\sum_{k=1}^\infty k \rho^{-k}\mathbf{g}\mathbf H^k\mathbf{1}^{\rm t},
\label{beta}
\ee
the Perron-Frobenius eigenvectors $(\mathbf v,\mathbf u)$ can be normalized in such a way that $\mathbf v\mathbf u^{\rm t}=\mathbf v\mathbf 1^{\rm t}=1$. They are computed as
\begin{align}
\mathbf u^{\rm t}&=(1+m)\beta^{-1}\sum_{k=1}^\infty \rho^{-k}\mathbf H^k\mathbf{1}^{\rm t},\label{ut}\\
\mathbf v&={m\over 1+m}\sum_{k=0}^\infty \rho^{-k}\mathbf{g}\mathbf H^k.\label{vt}
\end{align}
In the supercritical positive recurrent case with $\rho>1$ and $\beta<\infty$ the extinction probabilities are given by
\begin{align}
\mathbf{q}&=\mathbf{1}-(\rho-1)(1+m)^{-1}\beta\mathbf{u}.
\label{gq}
\end{align}
Observe that $\mathbf g\mathbf u^{\rm t}={1+m\over m\beta}$ and
\begin{align}
\mathbf{g}\mathbf{q}^{\rm t}={1+m-\rho\over m}.\label{geq}
\end{align}
The total offspring number for a type $i$ particle has mean
\begin{equation}} \newcommand{\ee}{\end{equation}
M_i=(1-h_{i0})(1+m).\label{Mi}
\ee
\section{ Main results}\label{sMR}
In this section we summarize explicit formulae that we were able to obtain for the decomposition of the supercritical linear-fractional BGW-processes with countably many types. The derivation of these results is given in the next section.
Consider the positive recurrent supercritical case with $\rho>1$ and $\beta<\infty$. We demonstrate that the dual reproduction laws are again linear-fractional
\begin{equation}} \newcommand{\ee}{\end{equation}
\hat f_i(\mathbf s)=\hat h_{i0}+{\sum_{j=1}^\infty \hat h_{ij}s_j\over 1+\hat m-\hat m\sum_{j=1}^\infty \hat g_js_j},
\label{fi}
\ee
with
\begin{align}
\hat h_{i0}&={h_{i0}\over q_i}, \hspace{19mm}\hat h_{ij}={h_{ij}q_j\over q_i\rho}, \label{fi1}\\
\hat m&={1+m-\rho\over \rho}, \hspace{7mm} \hat g_{j}={g_{j}q_jm\over 1+m-\rho}.\label{fi2}
\end{align}
It turns out that the following remarkably simple formulae hold for the key characteristics of the dual branching process
\begin{align}
\hat \rho&=\rho^{-1}, \label{hr}\\
\hat \beta&={\mu-1\over \rho-1},\quad \mu=m\sum_{n=1}^\infty \mathbf{g}\mathbf H^n\mathbf{1}^{\rm t}. \label{br}
\end{align}
For the Perron-Frobenius eigenvectors we obtain the following expressions
\begin{align}
\hat {\mathbf u}&=\beta \hat \beta^{-1}\mathbf u\mathbf q^{-1}=(\mathbf q^{-1}-\mathbf 1)(1+m)( \mu-1)^{-1},\label{u}\\
\hat {\mathbf v}&=\frac{m}{1+m}\sum_{k=0}^\infty(\mathbf{g}\mathbf{H}^k)\mathbf q.\label{v}
\end{align}
We show that the Harris-Sevastyanov transformation results in multivariate shifted geometric distributions
\begin{equation}} \newcommand{\ee}{\end{equation}
\tilde f_i(\mathbf s)={\sum_{j=1}^\infty \tilde h_{ij}s_j\over 1+\tilde m-\tilde m\sum_{j=1}^\infty \tilde g_js_j}, \label{tfi}
\ee
where
\begin{align}
\tilde h_{ij}&={1-q_j\over 1-q_i}\left(h_{ij}+mg_j(q_i-h_{i0})\right),\label{mgh}\\
\tilde m&=\rho-1,\quad \tilde g_{j}={m\over \rho-1}g_{j}(1-q_j). \label{mgh1}
\end{align}
Moreover, we demonstrate that
\begin{equation}} \newcommand{\ee}{\end{equation}
\tilde \rho=\rho,\qquad \tilde \beta={\rho\over \rho-1},\label{tb}
\ee
and
\begin{align}
\tilde {\mathbf u}=\mathbf 1,\hspace{5mm}\tilde{\mathbf{v}}
&=m\sum_{k=0}^\infty\rho^{-1-k}\Big(\mathbf{g}\{\mathbf{H}+m\rho^{-1}\mathbf{H}\mathbf{q}^{\rm t}\mathbf{g}\}^k\Big)(\mathbf{1}-\mathbf{q}). \label{uv}
\end{align}
\begin{theorem}\label{Thm}
Consider a linear-fractional BGW-process characterized by a triplet $(\mathbf H,\mathbf g,m)$. Assume it is supercritical and positively recurrent over the state space, that is $\rho>1$ and $\beta<\infty$.
Its dual BGW-process and its skeleton are also linear-fractional BGW-processes with the transformed parameter triplets $(\hat{\mathbf H},\hat{\mathbf g},\hat m)$ and $(\tilde{\mathbf H},\tilde{\mathbf g},\tilde m)$ with components given by \eqref{fi1}, \eqref{fi2}, \eqref{mgh}, \eqref{mgh1}.
The joint offspring generating function for a skeleton particle of type $i$ has the form
\begin{align}
F_{i}(\mathbf s,\mathbf t)&=\sum_{j=1}^\infty { \tilde h_{ij} s_j\over 1+m-\tilde m\tilde{\mathbf g}\mathbf s^{\rm t}-(m-\tilde m)\hat{\mathbf g}\mathbf t^{\rm t}}\Big(h_{ij0}+{\sum_{k=1}^\infty h_{ijk}t_k\over 1+\hat m- \hat m\hat{\mathbf g}\mathbf t^{\rm t}}\Big),
\label{f1i}
\end{align}
where
\[h_{ij0}={h_{ij}\over h_{ij}+mg_j(q_i-h_{i0})},\quad h_{ijk}={mg_jq_i \hat h_{ik}\over h_{ij}+mg_j(q_i-h_{i0})}.\]
\end{theorem}
Similarly to the single-type case, we can distinguish in \eqref{f1i} three components but now with dependence:
\begin{itemize}
\item a ``reborn" skeleton particle of type $i$ may change its type to $j$ with probability $\tilde h_{ij}$,
\item independent of $i$ and $j$ a multivariate geometric number of offspring of both subtypes,
\item a linear-fractional number of doomed offspring with the fate of the first offspring being dependent on $(i,j)$ .
\end{itemize}
The total number of offspring of a skeleton particle of type $i$ has generating function $ \bar f_i(s)=F_{i}(s\mathbf 1,s\mathbf 1)$ of the next form
\begin{align*}
\bar f_i(s
&={s\over 1+m-ms}\Big(1-\alpha_i+{\alpha_is\over 1+\hat m- \hat ms}\Big),
\end{align*}
where $\alpha_i={\rho-1\over 1-q_i}(q_i-h_{i0})$ must belong to the interval $(0,1)$. The corresponding mean offspring number is larger than that given by \eqref{Mi}:
\begin{align*}
\bar M_i&=1+m+\alpha_i(1+\hat m)=M_i+(1+m)\Big(h_{i0}+{(\rho-1)(q_i-h_{i0})\over \rho(1-q_i)}\Big).
\end{align*}
\section{ Proof of Theorem \ref{Thm}}\label{sP}
In this section we derive the formulae stated in Section \ref{sMR}.
\vspace{5mm}{\bf Proof of \eqref{fi}.}
From
\[\hat f_i(\mathbf s)={f_i(\mathbf s\mathbf q)\over q_i}=\frac{h_{i0}}{q_i}+\frac{\sum_{j=1}^{\infty}h_{ij}s_j q_j/q_i}{1+m-m \sum_{j=1}^{\infty} g_j s_j q_j}\]
it is straightforward to obtain \eqref{fi} with \eqref{fi1} and \eqref{fi2}. We have to verify that $\hat{\mathbf g}\mathbf{1}^{\rm t}=1$ and
\[\hat h_{i0}=1-\sum_{j=1}^\infty\hat h_{ij}.\]
The first requirement follows from
\eqref{geq}. The second is obtained from
\begin{align}
\mathbf{H}\mathbf{q}^{\rm t}
&=\rho(\mathbf{H}\mathbf{1}^{\rm t}-\mathbf{1}^{\rm t}+\mathbf{q}^{\rm t}) \label{Hq}
\end{align}
which is proved next. We have (relation (6) in \cite{S})
\[\mathbf H=\mathbf M-{m\over 1+m}\mathbf M\mathbf{1}^{\rm t}\mathbf g\]
and therefore $\mathbf M\mathbf{1}^{\rm t}=(1+m)\mathbf H\mathbf{1}^{\rm t}$, which is \eqref{Mi}.
Using the last two equalities and \eqref{gq} we find first
\begin{align}
\mathbf{H}\mathbf{q}^{\rm t}
&={1\over 1+m}\mathbf{M}\mathbf{1}^{\rm t}-\rho(\mathbf{1}^{\rm t}-\mathbf{q}^{\rm t}) +{\rho-1\over 1+m}\mathbf{M}\mathbf{1}^{\rm t}\nonumber
\end{align}
and then obtain \eqref{Hq}.
\vspace{5mm}{\bf Proof of \eqref{hr}.} In view of equation \eqref{PF} determining the Perron-Frobenius eigenvalue for a linear-fractional BGW-process, to show \eqref{hr} it is enough to verify that
\begin{align*}
\hat{m} \sum_{n=1}^\infty \rho^{n} \hat{\mathbf{g}} \hat{\mathbf{H}}^n \mathbf{1}^{\rm t}=1.
\end{align*}
Observe that according to \eqref{fi1}
\begin{equation}} \newcommand{\ee}{\end{equation}
\hat{\mathbf{g}}\hat{\mathbf{H}}^k=\frac{m}{(1+m-\rho)\rho^k}(\mathbf{g}\mathbf{H}^k) \mathbf{q}.
\label{gH}
\ee
It follows,
\begin{align}
\hat{\mathbf{g}} \hat{\mathbf{H}}^{n} \mathbf{1}^{\rm t}=\frac{m}{(1+m-\rho)\rho^{n}} \mathbf{g}\mathbf{H}^{n}\mathbf{q}^{\rm t},
\label{gH1}
\end{align}
so that we have to check that
\begin{align}
m\sum_{n=1}^\infty \mathbf{g}\mathbf{H}^{n}\mathbf{q}^{\rm t}=\rho.\label{mur}
\end{align}
Turning to \eqref{Hq} we find
\begin{align}
\mathbf{H}^n\mathbf{q}^{\rm t}
&=\rho(\mathbf{H}^n\mathbf{1}^{\rm t}-\mathbf{H}^{n-1}\mathbf{1}^{\rm t}+\mathbf{H}^{n-1}\mathbf{q}^{\rm t}) \label{Hn}
\end{align}
yielding
\begin{align}
(\rho-1)\sum_{n=1}^\infty \mathbf{H}^{n}\mathbf{q}^{\rm t} =\rho(\mathbf{1}^{\rm t}-\mathbf{q}^{\rm t}).\label{qur}
\end{align}
This and \eqref{geq} entail \eqref{mur}.
\vspace{5mm}{\bf Proof of \eqref{br}.} Starting from a counterpart of \eqref{beta} we find using \eqref{gH1}
\begin{align*}
\hat{\beta}&=\hat{m}\sum_{n=1}^\infty n
\hat{\rho}^{-n}\hat{\mathbf{g}}\hat{\mathbf
H}^n\mathbf{1}^{\rm t}=\frac{m}{\rho}\sum_{n=1}^\infty n \mathbf{g} \mathbf{H}^{n} \mathbf{q}^{\rm t}.
\end{align*}
Rewrite \eqref{Hn} as
\begin{align*}
n\mathbf{H}^n\mathbf{q}^{\rm t}
&=\rho\Big(n\mathbf{H}^n\mathbf{1}^{\rm t}-(n-1)\mathbf{H}^{n-1}\mathbf{1}^{\rm t}+(n-1)\mathbf{H}^{n-1}\mathbf{q}^{\rm t}\Big)+\rho \mathbf{H}^{n-1}(\mathbf{q}^{\rm t}-\mathbf{1}^{\rm t}
\end{align*}
to obtain
$$\sum_{n=1}^\infty n\mathbf{H}^{n}\mathbf{q}^{\rm t}=\frac{\rho}{\rho-1}\sum_{n=0}^\infty \mathbf{H}^{n}(\mathbf{1}^{\rm t}-\mathbf{q}^{\rm t}).$$
Thus
$$\hat{\beta}=\frac{m}{\rho-1}\sum_{n=0}^\infty \mathbf{g}\mathbf{H}^{n}(\mathbf{1}^{\rm t}-\mathbf{q}^{\rm t})=\frac{\mu-1}{\rho-1}.$$
\vspace{5mm}{\bf Proof of \eqref{u} and \eqref{v}.} From \eqref{fi1} we derive $\hat{\mathbf{H}}^{n}\mathbf{1}^{\rm t}=(\mathbf{H}^{n}\mathbf{q}^{\rm t})\mathbf{q}^{-1}$.
This and a counterpart of \eqref{ut}
$$ \hat{\mathbf{u}}=(1+\hat{m}) \hat{\beta}^{-1}\sum_{n=1}^\infty \hat{\rho}^{-n}\hat{\mathbf{H}}^{n}\mathbf{1}^{\rm t}$$
in view of \eqref{qur} brings \eqref{u}
$$ \hat{\mathbf{u}}=
\frac{(1+m)(\rho-1)}{\rho(\mu-1) } \Big(\sum_{n=1}^\infty \mathbf{H}^{n}\mathbf{q}^{\rm t}\Big)\mathbf{q}^{-1}=(\mathbf q^{-1}-\mathbf 1)(1+m)( \mu-1)^{-1}.$$
On the other hand, a counterpart of \eqref{vt} together with \eqref{gH} yields
$$\hat{\mathbf{v}}=\frac{\hat{m}}{1+\hat{m}}
\sum_{k=0}^\infty \hat{\rho}^{-k}
\hat{\mathbf{g}}\hat{\mathbf{H}}^k=\frac{m}{1+m} \sum_{k=0}^\infty \mathbf{g}\mathbf{H}^k \mathbf{q}.$$
\vspace{5mm}{\bf Proof of \eqref{f1i}.} We have
\begin{align*}
&f_i(\mathbf s(\mathbf 1-\mathbf q)+\mathbf t\mathbf q)-f_i(\mathbf t\mathbf q)\\
&={\sum_{j=1}^\infty h_{ij}s_j(1-q_j)+\sum_{j=1}^\infty h_{ij}t_jq_j\over 1+m-m\sum_{k=1}^\infty g_ks_k(1-q_k)-m\sum_{k=1}^\infty g_kt_kq_k}-{\sum_{j=1}^\infty h_{ij}t_jq_j\over 1+m- m\sum_{k=1}^\infty g_kt_kq_k}\\
&={\sum_{j=1}^\infty h_{ij}s_j(1-q_j)
\over 1+m-\tilde m\sum_{k=1}^\infty \tilde g_ks_k-(m-\tilde m)\sum_{k=1}^\infty \hat g_kt_k}\\
&+{\big(\sum_{j=1}^\infty h_{ij}t_jq_j\big)\big(m\sum_{k=1}^\infty g_ks_k(1-q_k)\big)\over\big(1+m-\tilde m\sum_{k=1}^\infty \tilde g_ks_k-(m-\tilde m)\sum_{k=1}^\infty \hat g_kt_k\big)\rho\big(1+\hat m- \hat m\sum_{k=1}^\infty \hat g_kt_k\big)}.
\end{align*}
Replacing the last numerator by $m\sum_{j=1}^\infty g_js_j(1-q_j)\sum_{k=1}^\infty h_{ik}t_kq_k$ and dividing the whole expression by $1-q_i$ we get
\begin{align*}
F_{i}(\mathbf s,\mathbf t)&=\sum_{j=1}^\infty {s_j(1-q_j)(1-q_i)^{-1}
\over1+m-\tilde m\tilde{\mathbf g}\mathbf s^{\rm t}-(m-\tilde m)\hat{\mathbf g}\mathbf t^{\rm t}}\left(h_{ij}+{mg_jq_i\sum_{k=1}^\infty \hat h_{ik}t_k\over 1+\hat m- \hat m\hat{\mathbf g}\mathbf t^{\rm t}}\right)
\end{align*}
and the relation \eqref{f1i} follows.
\vspace{5mm}{\bf Proof of \eqref{tfi}, \eqref{tb}.} Putting $\mathbf t=\mathbf 1$ in \eqref{f1i} we arrive at \eqref{tfi} . Notice that according to definition \eqref{mgh} and relations \eqref{geq}, \eqref{Hq} we have
\[\tilde{\mathbf g}\mathbf 1^{\rm t}=1,\qquad \tilde{\mathbf H}\mathbf 1^{\rm t}=\mathbf 1^{\rm t}.\]
Since $\tilde{\rho}$ is the unique positive solution of
$$\tilde{m}\sum_{n=1}^\infty \tilde{\rho}^{-n} \tilde{\mathbf{g}}\tilde{\mathbf{H}}^n \mathbf{1}^{\rm t}=1$$
and $\tilde{\mathbf{g}}\tilde{\mathbf{H}}^{n}\mathbf{1}^{\rm t}=1$ we derive $$(\rho-1)\sum_{n=1}^\infty \tilde{\rho}^{-n}=1.$$
Thus $\tilde{\rho}=\rho$ and
$$\tilde{\beta}=\tilde{m}\sum_{n=1}^\infty n \tilde{\rho}^{-n}\tilde{\mathbf{g}}\tilde{\mathbf{H}}^n\mathbf{1}^{\rm t}=(\rho-1)\sum_{n=1}^\infty n
\rho^{-n}=\frac{\rho}{\rho-1}.$$
\section*{Acknowledgments}
SS was supported by the Swedish Research Council grant 621-2010-5623. AS was supported by the Scientific Committee of Kazakhstan's Ministry of Education and Science, grant 0732/GF 2012-14.
|
2,877,628,090,244 | arxiv | \section{Introduction}
\label{sec:introduction}
In this article we prove the Hardy--Stein type identity for a regular pure-jump Dirichlet form
\begin{align}
\label{eq:E2=iintFp}
\mathcal{E}(u,v)
=
\frac{1}{2}\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} (u(y)-u(x))(v(y)-v(x))\,J(dx,dy)
\end{align}
on a locally compact separable metric space $E$, with Borel $\sigma$-algebra $\mathcal{B}$ and a positive Radon measure $m$ such that $\supp(m)=E$.
Here $\mathrm{diag}$ is the diagonal of the cartesian product $E\times E$ and $J$ is the jumping measure on $E\times E\setminus\mathrm{diag}$.
Let $(P_t)_{t\geq0}$ be a strongly continuous semigroup of contractions associated to the Dirichlet form $\mathcal{E}$.
The following is the main result of this paper.
\begin{thm}
\label{thm:HSp-j}
Let $p\in(1,\infty)$.
Let $\mathcal{E}$ be a regular pure-jump Dirichlet form
given by \eqref{eq:E2=iintFp}.
Assume that:
\begin{enumerate}[label=(\roman*)]
\item For every $t>0$ and $u\inL^p\EAm$ we have $P_tu\inC(E)$.
\item For every $f\inL^p\EAm$
\begin{align}
\label{lim:|P_Tf|to0}
\|P_T f \|_p \to 0
\qquad \mbox{when }T\to\infty.
\end{align}
\end{enumerate}
If $f\inL^p\EAm$,
then
\begin{align}
\label{eq:HSpure-jump}
\int\limits_E |f|^p \,dm
=
\int\limits_0^{\infty} \quad\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
F_p(P_tf(x),P_tf(y)) \,J(dx,dy)dt.
\end{align}
Here
$
F_p(a,b):=
|b|^p-|a|^p - pa^{\langle p - 1 \rangle}(b-a)
$
is the Bregman divergence.
\end{thm}
The above theorem extends previous results in \cite{MR3556449}, \cite{HS}.
Along the way, in Theorem \ref{thm:HS} we prove a more general variant of the Hardy--Stein identity:
\begin{align*}
\int\limits_E |f|^p \,dm
=
p\int\limits_0^{\infty} \mathcal{E}_p[P_tf] \,dt,
\quad
f\inL^p\EAm,
\end{align*}
where $\mathcal{E}_p$ is the $p$-form defined as the limit of appropriate approximate forms $\mathcal{E}^{(t)}(u,u^{\langle p-1\rangle})=\langle u-P_tu,u^{\langle p-1\rangle} \rangle/t$. The $p$-form may be treated as an extension of the classical quadratic Dirichlet form.
In the case of pure-jump Dirichlet forms, in Theorem \ref{thm:Ep=iintFp} we derive the explicit formula for the $p$-form
\begin{align*}
\mathcal{E}_p[u] = \frac{1}{p} \iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} F_p(u(x),u(y)) \,J(dx,dy)
\end{align*}
for continuous functions $u$ from the domain $\D(\E_p)$ of $p$-form.
The Hardy--Stein identity was originally proved for the classical Laplace operator as a consequence of Green's theorem and the chain rule $\Delta u^p = p(p-1)u^{p-2}|\nabla u|^2+pu^{p-1}\Delta u$. We refer to Lemma 1 and Lemma 2 in Stein \cite[p. 86--88]{MR0290095}. The identity in the non-local case can be shown using analytic methods as in \cite{MR3556449} or in \cite{HS}. On the other hand, in \cite{MR3994925} the identity is derived from It\^{o}'s lemma.
The Hardy--Stein identity has found applications in the Littlewood--Paley theory, especially to the proof of $L^p$-boundedness of the square function and the Fourier multipliers. For more details, we refer to \cite{MR3556449} and \cite{MR3994925}. The Hardy--Stein identity also gives a characterisation of Hardy spaces \cite{MR3251822} and it is used to prove Douglas-type identities \cite{bogdan2020nonlinear}.
The goal of this paper is to extend the previous Hardy--Stein type identities to more general Dirichlet forms which correspond to pure-jump regular Dirichlet form under certain mild assumptions. Our main tool is the theory of Dirichlet forms and $p$-forms. To prove Theorem \ref{thm:HS} we adopt the approach from \cite[Theorem 15]{HS}.
The $p$-form (or Sobolev--Bregman form) corresponding to the fractional Laplacian (defined as a double integral of the Bregman divergence) was studied in \cite{lenczewska2021sharp}, \cite{bogdan2021optimal}, and for more general L\'{e}vy operators in \cite{HS}, but similar expression appeared already in \cite[Lemma 7.2]{MR1386760}. Similar object was frequently used also in \cite{bogdan2020nonlinear}.
In this work we begin with a non-standard definition of the $p$-form -- as a limit of appropriate approximating forms $\mathcal{E}^{(t)}(u,u^{\langle p-1\rangle})$. This gives access to an extended class of operators. The same definition was used in the case of the Brownian motion in \cite[Section 8]{HS}. At this moment it is known that this definition is equivalent to the commonly used definition in terms of the Bregman divergence for pure-jump L\'{e}vy operators \cite[Lemma 7]{bogdan2021optimal}, \cite[Proposition 13]{HS}. In Theorem \ref{thm:Ep=iintFp} we show that for more general pure-jump Dirichlet form the equivalence remains true for continuous functions.
Equivalence of the two definitions for discontinuous functions in the domain of the $p$-form remains an open problem.
Independently from the main subject of this article, the Hardy--Stein identity, we are interested in the relationship between the domains of Dirichlet form and the $p$-form. In \cite[Lemma 7]{bogdan2021optimal} and \cite[Proposition 13]{HS} it was shown that for the pure-jump L\'{e}vy operators, with certain assumptions about the L\'{e}vy measure,
the domain $\D(\E_p)$ of the $p$-form consists of functions of the form $u^{\langle p/2\rangle}$, where $u$ is a function in the domain $\mathcal{D}(\mathcal{E})$ of the Dirichlet form.
In Theorem \ref{thm:Ep=iintFp} we show the inclusion $\D(\E_p)\subseteq\mathcal{D}(\mathcal{E})^{\langle p/2\rangle}$ in the general pure-jump case, and we conjecture that in fact equality holds.
The structure of the article is as follows. In Section \ref{sec:prelimitaries} we introduce the notions of a Hunt process, its semigroup, the definition of the corresponding $p$-form and derivatives of $L^p$-valued functions and we discuss their basic properties. The Hardy--Stein identity in the general case is proved in Section \ref{sec:HS}. In Section \ref{sec:pure-jump} we consider pure-jump Dirichlet forms and we show that the corresponding $p$-form $\mathcal{E}_p$ for continuous functions is given by a double integral. This proves the Hardy--Stein identity for such Dirichlet forms (Theorem \ref{thm:HSp-j}).
\section{Preliminaries}
\label{sec:prelimitaries}
We consider a conservative symmetric Hunt process $(X_t)_{t\geq0}$ on a measure space $(E,\mathcal{B}, m)$ with associated regular Dirichlet form $\mathcal{E}$. We assume that $E$ is a locally compact separable metric space and $m$ is a positive Radon measure on $E$ such that $\supp(m)=E$, defined on the $\sigma$-algebra $\mathcal{B}$ of all Borel sets in $E$.
By $C(E)$ we denote the class of continuous functions on $E$.
For a Borel function $u$ we define
\begin{align*}
P_tu(x) := {\mathbb{E}^x} u(X_t) = \int\limits_E u(y)\,P_t(x,dy),
\qquad t\geq0, x\in E,
\end{align*}
whenever the integral is well-defined.
To emphasize symmetry, we write $P_t(dx,dy):=P_t(x,dy)m(dx)$; then $P_t(dx,dy)=P_t(dy,dx)$.
Recall that for every $p\in[1,\infty)$ $(P_t)_{t\geq0}$ is a strongly continuous semigroup of contractions on $L^p\EAm$.
We use the notation
\begin{align*}
a^{\langle \kappa \rangle} := \left|a \right|^\kappa \sgn a,
\end{align*}
whenever above expression makes sense.
Note that
\begin{align*}
(
|x|^\kappa
)' = \kappa x^{\langle \kappa-1 \rangle}
,\quad
\mbox{if }
x\in\mathbb{R},\ \kappa>1
\mbox{ or }
x\in\mathbb{R}\!\setminus\!\{0\},
\end{align*}
and
\begin{align*}
(
x^{\langle \kappa \rangle}
)' = \kappa |x|^{\kappa-1 }
,\quad
\mbox{if }
x\in\mathbb{R},\ \kappa\geq1
\mbox{ or }
x\in\mathbb{R}\!\setminus\!\{0\}.
\end{align*}
Let $p,q\in(1,\infty)$ with $p^{-1}+q^{-1}=1$. For $u\inL^p\EAm$, $v\in L^q(m)$ we use the notation
\begin{align*}
\langle u,v \rangle := \int\limits_E u(x) v(x) \,m(dx).
\end{align*}
For $t>0$ and $u\inL^p\EAm$, $v\in L^q(m)$ we define
\begin{align}
\mathcal{E}^{(t)}(u,v)
:=
\frac{1}{t}
\langle u-P_tu,v \rangle.
\end{align}
Let $u\inL^p\EAm$. We define the nonlinear functional
\begin{align*}
\mathcal{E}_p[u] := \lim_{t\to0^+} \mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right)
\end{align*}
with its natural domain
\begin{align*}
\D(\E_p) := \left\{u\inL^p\EAm\!: \mbox{finite } \lim_{t\to0^+} \mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right) \mbox{ exists}\right\}.
\end{align*}
We call $\mathcal{E}_p$ the \emph{$p$-form} corresponding to the Dirichlet form $\mathcal{E}$.
When $p=2$, then $\mathcal{E}_2$ is just the usual Dirichlet form $\mathcal{E}(u,u)$ of the process $(X_t)_{t\geq0}$, with domain $\mathcal{D}(\mathcal{E}_2)=\mathcal{D}(\mathcal{E})$.
It is well-known that if $u\in L^2(m)$, then $\mathcal{E}^{(t)}(u,u)$ is non-increasing as a function of $t$ \cite[Lemma 1.3.4]{MR2778606}.
We consider the infinitesimal generator $L_p$ of the semigroup $(P_t)_{t\geq0}$ on $L^p\EAm$:
\begin{align*}
L_p u := \lim_{t\to0^+} \frac{1}{t}(P_tu - u)
\qquad
\mbox{in }
L^p\EAm
\end{align*}
with the natural domain
\begin{align*}
\D(L_p) := \left\{u\inL^p\EAm\!: \lim_{t\to0^+} \frac{1}{t}(P_tu - u) \mbox{ exists in }L^p\EAm\right\}.
\end{align*}
Of course, $\D(L_p)\subseteq\D(\E_p)$ and
\begin{align*}
\mathcal{E}_p[u] = -\langle L_p u, u^{\langle p-1\rangle} \rangle,
\qquad
u\in\D(L_p).
\end{align*}
We use the \emph{Bregman divergence}: a function $F_p:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, where $p>1$, defined by
\begin{align*}
F_p(a,b)
:=
|b|^p-|a|^p - pa^{\langle p - 1 \rangle}(b-a).
\end{align*}
We also use \emph{symmetrized Bregman divergence}
\begin{align*}
H_p(a,b)
:=
\frac{1}{2}
\left(
F_p(a,b)+F_p(b,a)
\right)
=
\frac{p}{2}
(b-a)
\left(
b^{\langle p - 1 \rangle} - a^{\langle p - 1 \rangle}
\right).
\end{align*}
Note that $F_p(a,b)$ is the second-order Taylor remainder of the convex map $\mathbb{R}\ni a\mapsto |a|^p\in\mathbb{R}$, we have $F_p\ge 0$ and also $H_p\ge 0$. Furthermore, $F_2(a,b)=H_2(a,b)=(b-a)^2$.
The following estimate was proved in \cite[Lemma 1]{MR1160303} for $H_p$ in place of $F_p$.
\begin{lem}
\label{lem:FpsimD}
Let $p\in(1,\infty)$. There exist constants $c_p,C_p>0$ such that
\begin{align}
\label{neq:FpsimD}
c_p (b^{\langle p/2\rangle} - a^{\langle p/2\rangle})^2
\leq
F_p(a,b)
\leq
C_p (b^{\langle p/2\rangle} - a^{\langle p/2\rangle})^2
\end{align}
for all $a,b\in\mathbb{R}$.
\end{lem}
\begin{proof}
If $a=0$, we have $F_p(a,b)=|b|^p$ and the statement is obvious. If $a\neq0$, then we let $x:=b/a$ and we arrive
at the following equivalent formulation of \eqref{neq:FpsimD}:
\begin{align}
\label{neq:FpsimDx}
c_p (x^{\langle p/2\rangle} - 1)^2
\leq
F_p(x,1)
\leq
C_p (x^{\langle p/2\rangle} - 1)^2,
\end{align}
where $F_p(x,1) = |x|^p - 1 - p(x-1)$. The above expressions define continuous and positive functions of $x\neq1$. Therefore to prove \eqref{neq:FpsimDx} it is enough to notice that
\begin{align*}
\lim_{x\to\pm\infty}
\frac{F_p(x,1)}{\left(x^{\langle p/2\rangle} - 1\right)^2}
=
1
\end{align*}
and, by L'H\^{o}pital's rule,
\begin{align*}
\lim_{x\to1}
\frac{F_p(x,1)}{\left(x^{\langle p/2\rangle} - 1\right)^2}
=
\lim_{x\to1}
\frac{px^{p-1}-p}{2\left(x^{p/2} - 1\right)\cdot\frac{p}{2}x^{(p-2)/2}}
=
\lim_{x\to1}
\frac{x^{p-1} - 1}{x^{p/2} - 1}
\lim_{x\to1} x^{-\frac{p-2}{p}}
=
\frac{2(p-1)}{p}.
\end{align*}
The above limits are finite and positive. This completes the proof.
\end{proof}
For every $u\in L^1(m)$, by symmetry of $P_t$, we have
\begin{align}
\label{eq:intPtu=intu}
\int\limits_E P_tu(x) \,m(dx)
&=
\int\limits_E\left(\int\limits_E u(y) \,P_t(x,dy)\right)\,m(dx)
\\ \nonumber
&=
\int\limits_E u(y) \left(\int\limits_E \,P_t(y,dx)\right)\,m(dy)
=
\int\limits_E u(x) \,m(dx).
\end{align}
Let $u\inL^p\EAm$. Using \eqref{eq:intPtu=intu} for $|u|^p\in L^1(m)$ we can write
\begin{align}
\label{eq:Et=intintFp}
\mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right)
&=
\frac{1}{t} \langle u-P_tu,u^{\langle p-1\rangle} \rangle
\\
&= \nonumber
-\frac{1}{t} \iint\limits_{E\times E} u^{\langle p-1\rangle}(x) (u(y)-u(x)) \,P_t(dx,dy)
\\
&= \nonumber
\frac{1}{pt} \int\limits_E P_t(|u|^p)(x) \,m(dx)
-
\frac{1}{pt} \int\limits_E |u|^p(x) \,m(dx)
\\
&\quad- \nonumber
\frac{1}{t} \iint\limits_{E\times E} u^{\langle p-1\rangle}(x) (u(y)-u(x)) \,P_t(dx,dy)
\\
&= \nonumber
\frac{1}{pt} \iint\limits_{E\times E} F_p(u(x),u(y)) \,P_t(dx,dy).
\end{align}
In particular, we see that $\mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right)\geq0$, and so $\mathcal{E}_p[u]\geq0$ whenever $u\in\D(\E_p)$.
By symmetry of $P_t$ we can also write
\begin{align}
\label{eq:Et=intintHp}
\mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right)
=
\frac{1}{pt} \iint\limits_{E\times E} H_p(u(x),u(y)) \,P_t(dx,dy).
\end{align}
Let $p\in[1,\infty)$ and let $I\subseteq[0,\infty)$ be an interval. For a mapping
$I \ni t \mapsto u(t) \in L^p\EAm$ we denote
\begin{align*}
\Delta_hu(t) := u(t+h)-u(t) \quad \mbox{if }t,t+h\in I.
\end{align*}
We say that $u$ is \emph{continuous} on $I$ with values in $L^p\EAm$ if $\Delta_hu(t) \to 0$ in $L^p\EAm$ as $h\to 0$ for every $t\in I$,
and we say that $u$ is \emph{continuously differentiable} (or shortly $C^1$) on $I$ with values in $L^p\EAm$ if $u'(t) := \lim_{h \to 0} \frac{1}{h} \Delta_hu(t)$ exists in $L^p\EAm$ for every $t\in I$ and the mapping $I \ni t \mapsto u'(t) \in L^p\EAm$ is
continuous.
The following two elementary results are proved rigorously in \cite{HS}.
\begin{lem}[\cite{HS}]
Let $p\in(1,\infty)$. If $I\ni t \mapsto u(t)$ is $C^1$ on $I$ with values in $L^p\EAm$ and $\kappa\in(1,p]$, then $|u|^\kappa$ is $C^1$ on $I$ with values in $L^{\frac{p}{\kappa}}(m)$ and $(|u|^\kappa)' = \kappa u^{\langle \kappa - 1 \rangle} u'$.
\end{lem}
Let $f\inL^p\EAm$ and let $u(t) := P_tf\inL^p\EAm$.
If $f\in\D(L_p)$, then $u'(t) = L_p P_t f = P_t L_p f=L_p u(t)$.
We know that if $p>1$, then $(P_t)_{t\geq0}$ is an analytic semigroup on $L^p\EAm$ \cite[p. 67]{MR0252961}. In particular, for every $t>0$ and $f\inL^p\EAm$ the derivative
$\frac{d}{dt} P_t f = u'(t)$
exists in $L^p\EAm$. Hence $P_t f \in \D(L_p)$ and
$u'(t) = L_p P_t f = L_p u(t)$.
\begin{cor}[\cite{HS}]
\label{cor:ppp}
Let $f\in\D(L_p)$ and $u(t) := P_tf$.
If $\kappa\in(1,p]$, then $|u(t)|^\kappa$ is $C^1$ on $[0,\infty)$ with values in $L^{\frac{p}{\kappa}}(m)$, with derivative
\begin{equation}
\label{eq:ppp}
(|u(t)|^\kappa)' =\kappa u(t)^{\langle \kappa - 1 \rangle} u'(t)= \kappa u(t)^{\langle \kappa - 1 \rangle} L_p u(t), \quad t \geq 0.
\end{equation}
\end{cor}
\section{Hardy--Stein identity in the general case}
\label{sec:HS}
In this section we prove the Hardy--Stein identity for arbitrary regular Dirichlet form. The explicit form of the right-hand side of following identity depends on the specific Dirichlet form. In particular, in Section \ref{sec:pure-jump} we will give an explicit expression for pure-jump Dirichlet forms.
\begin{thm}
\label{thm:HS}
Let $p\in(1,\infty)$.
Assume that condition (ii) from Theorem \ref{thm:HSp-j} holds.
For every $f\inL^p\EAm$,
\begin{align}
\label{eq:HS}
\int\limits_E |f|^p \,dm
=
p\int\limits_0^{\infty} \mathcal{E}_p[P_tf] \,dt.
\end{align}
\end{thm}
\begin{proof}
Consider first $f\in\D(L_p)$ and fix $T>0$. Let $u(t):=P_tf$. By Corollary \ref{cor:ppp}, $|u|^p$ is $C^1$ on $[0,T]$ with values in $L^1(m)$ with derivative $(|u(t)|^p)'=p u(t)^{\langle p-1\rangle} L_pu(t)$.
The integral is a continuous linear functional on $L^1(m)$, hence $[0,T]\ni t\mapsto \int_E |u(t)|^p \,dm$ is $C^1$ and
\begin{align*}
\frac{d}{dt} \int\limits_E |u(t)|^p \,dm
&=
\int\limits_E \frac{d}{dt} |u(t)|^p \,dm
=
\int\limits_E pu(t)^{\langle p-1\rangle} L_pu(t) \,dm
\\
&=
p\langle L_p u(t), u(t)^{\langle p-1\rangle} \rangle
=
-p \mathcal{E}_p[u(t)].
\end{align*}
Therefore, we can write
\begin{align*}
\int\limits_E |f|^p \,dm
-
\int\limits_E |P_Tf|^p \,dm
&=
-\left(
\int\limits_E |u(T)|^p \,dm
-
\int\limits_E |u(0)|^p \,dm
\right)
\\
&=
-\int\limits_0^T \frac{d}{dt} \int\limits_E |u(t)|^p \,dm\,dt
\\
&=
p\int\limits_0^T \mathcal{E}_p[u(t)] \,dt.
\end{align*}
From assumption \eqref{lim:|P_Tf|to0}, we have $\int_E |P_Tf|^p \,dm\to0$ when $T\to\infty$.
Since $\mathcal{E}_p[u(t)]\geq0$, the right-hand side tends to $p\int_0^{\infty} \mathcal{E}_p[u(t)] \,dt$ as $T\to\infty$. Therefore
\begin{align*}
\int\limits_E |f|^p \,dm = p\int\limits_0^{\infty} \mathcal{E}_p[P_tf] \,dt.
\end{align*}
Next, we relax the assumption that $f\in\D(L_p)$. Let $f$ be an arbitrary function in $L^p\EAm$ and let $s>0$. Recall that $P_s f\in\D(L_p)$. Thus, \eqref{eq:HS} holds
for $P_sf$:
\begin{align*}
\int\limits_E |P_sf|^p \,dm = p\int\limits_s^{\infty} \mathcal{E}_p[P_tf] \,dt.
\end{align*}
Since $(P_t)_{t\geq0}$ is a strongly continuous semigroup on $L^p\EAm$ and $f \mapsto \int_E |f|^p\,dm$ is a continuous functional on $L^p\EAm$, the left-hand side tends to $\int_E |f|^p \,dm$ when $s\to0^+$.
Since $\mathcal{E}_p[P_tf]\geq0$,
the right-hand side tends to $p\int_0^{\infty} \mathcal{E}_p[u(t)] \,dt$ by the monotone convergence theorem.
\end{proof}
\section{Pure-jump Dirichlet forms}
\label{sec:pure-jump}
In this section we consider a pure-jump regular Dirichlet form: we assume that the Dirichlet form $\mathcal{E}$ is given by \eqref{eq:E2=iintFp}. Here and below, $\mathrm{diag}:=\{(x,y)\in E\times E\!: x=y\}$. Measure $J$, the so-called \emph{jumping measure}, is a symmetric positive Radon measure on the $E\times E\setminus\mathrm{diag}$.
Our goal is to propose explicit form of $p$-form for such Dirichlet form.
\begin{lem}
We have
\begin{align}
\label{lim:P_tvaguely}
\frac{1}{t}P_t(dx,dy) \to J(dx,dy)
\quad
\mbox{vaguely on } E\times E\setminus\mathrm{diag} \mbox{ when }t\to0^+.
\end{align}
\end{lem}
An analogous result for the resolvent
rather than the semigroup $(P_t)$
was showed in \cite[(3.2.7)]{MR2778606}. The proof is similar and we omit it.
We consider the class $\mathcal{U}$ of non-negative functions $f$ on $E\times E$ such that
\begin{align*}
\lim_{t\to0^+} \frac{1}{t} \iint\limits_{E\times E} f(x,y) \,P_t(dx,dy)
=
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,J(dx,dy) < \infty.
\end{align*}
We know that if $u\in\mathcal{D}(\mathcal{E})$ and $f(x,y)=(u(y)-u(x))^2$, then $f\in\mathcal{U}$ because
\begin{align*}
\lim_{t\to0^+} \frac{1}{t} \iint\limits_{E\times E} (u(y)-u(x))^2 \,P_t(dx,dy)
&=
\lim_{t\to0^+} 2\mathcal{E}^{(t)}(u,u)
=
2\mathcal{E}[u]
\\
&=
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} (u(y)-u(x))^2\,J(dx,dy).
\end{align*}
Here we used \eqref{eq:Et=intintHp}.
\begin{lem}
\label{lem:if_g_good_then_also_f}
Suppose that $0\leq f\leq g$, $f=g=0$ on $\mathrm{diag}$, $f,g\inC(E\times E)$ and $g\in\mathcal{U}$. Then $f\in\mathcal{U}$.
\end{lem}
\begin{proof}
Fix $\varepsilon>0$.
Since $g\in\mathcal{U}$, we have
\begin{align*}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} g(x,y) \,J(dx,dy) < \infty.
\end{align*}
Therefore,
\begin{align*}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,J(dx,dy)
\leq
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} g(x,y) \,J(dx,dy)
<
\infty
\end{align*}
and there is a compact subset $K\subseteq E\times E\setminus\mathrm{diag}$ such that
\begin{align}
\label{neq:Kclessepsilon}
\iint\limits_{K^c} g(x,y) \,J(dx,dy) < \varepsilon.
\end{align}
Let $\varphi\inC_c(E\times E\setminus\mathrm{diag})$ be such that $0\leq\varphi\leq1$ and $\varphi=1$ on $K$. Since $f$ is continuous, we have $\varphi\cdot f\inC_c(E\times E\setminus\mathrm{diag})$, hence from \eqref{lim:P_tvaguely} we obtain
\begin{align}
\label{lim:iintfvarphi}
\lim_{t\to0^+}
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
\varphi(x,y) f(x,y)\,P_t(dx,dy)
=
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
\varphi(x,y) f(x,y)\,J(dx,dy).
\end{align}
Using \eqref{neq:Kclessepsilon} we can write
\begin{align}
\label{neq:int(1-varphi)fJ}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))f(x,y) \,J(dx,dy)
&\leq
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))g(x,y) \,J(dx,dy)
\nonumber
\\
&\leq
\iint\limits_{K^c}
g(x,y) \,J(dx,dy)
<
\varepsilon.
\end{align}
We also have
\begin{align*}
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))f(x,y) \,P_t(dx,dy)
&\leq
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))g(x,y) \,P_t(dx,dy)
\\
&=
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
g(x,y) \,P_t(dx,dy)
\\
&\quad-
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
\varphi(x,y) g(x,y) \,P_t(dx,dy).
\end{align*}
Since $g\in\mathcal{U}$ and $g$ is continuous, we have $\varphi\cdot g\inC_c(E\times E\setminus\mathrm{diag})$, and thus, using \eqref{lim:P_tvaguely}, we find that as $t\to0^+$, the right-hand side converges to
\begin{align*}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
g(x,y) &\,J(dx,dy)
-
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
\varphi(x,y) g(x,y) \,J(dx,dy)
\\
&=
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))g(x,y) \,J(dx,dy)
\leq
\iint\limits_{K^c}
g(x,y) \,J(dx,dy)
<
\varepsilon.
\end{align*}
Here we used \eqref{neq:Kclessepsilon}. Therefore,
\begin{align}
\label{neq:int(1-varphi)fPt}
\limsup_{t\to0^+}
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))f(x,y) \,P_t(dx,dy)
<
\varepsilon.
\end{align}
Finally, from \eqref{lim:iintfvarphi}, \eqref{neq:int(1-varphi)fJ} and \eqref{neq:int(1-varphi)fPt} we get
\begin{align*}
\limsup_{t\to0^+}
&\left|\
\frac{1}{t}\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,P_t(dx,dy)
-
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,J(dx,dy)
\right|
\\
&\leq
\limsup_{t\to0^+}
\left|\
\frac{1}{t}\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} \varphi(x,y) f(x,y) \,P_t(dx,dy)
-
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} \varphi(x,y) f(x,y) \,J(dx,dy)
\right|
\\
&\quad+
\limsup_{t\to0^+}
\frac{1}{t}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))f(x,y) \,P_t(dx,dy)
\\
&\quad+
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
(1-\varphi(x,y))f(x,y) \,J(dx,dy)
<
0+\varepsilon+\varepsilon = 2\varepsilon.
\end{align*}
Since $\varepsilon>0$ is arbitrary,
\begin{align*}
\lim_{t\to0^+}\frac{1}{t}\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,P_t(dx,dy)
=
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} f(x,y) \,J(dx,dy).
\end{align*}
\end{proof}
\begin{thm}
\label{thm:Ep=iintFp}
Let $u\in\D(\E_p)$. Then $u^{\langle p/2\rangle}\in\mathcal{D}(\mathcal{E})$. Moreover, if $u\inC(E)$, then
\begin{align}
\label{eq:Ep=iintFp}
\mathcal{E}_p[u] = \frac{1}{p} \iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}} F_p(u(x),u(y)) \,J(dx,dy).
\end{align}
\end{thm}
\begin{proof}
Using \eqref{eq:Et=intintFp}, the fact that $F_2(a,b)=(b-a)^2$ and Lemma \ref{lem:FpsimD}, we find that
\begin{align*}
\mathcal{E}^{(t)}\left(u^{\langle p/2\rangle},u^{\langle p/2\rangle}\right)
&=
\frac{1}{2t}
\iint\limits_{E\times E}
\left(
(u(y))^{\langle p/2\rangle} - (u(x))^{\langle p/2\rangle}
\right)^2
\,P_t(dx,dy)
\\
&\leq
c_p^{-1} \frac{1}{2t}
\iint\limits_{E\times E}
F_p(u(x),u(y))
\,P_t(dx,dy)
\\
&=
\frac{p}{2c_p} \mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right).
\end{align*}
Since $u\in\D(\E_p)$, the right-hand side converges to a finite limit $\frac{p}{2c_p} \mathcal{E}_p[u]$ as $t\to0^+$, and since the left-hand side is non-increasing as a function of $t$, a finite limit $\lim_{t\to0^+} \mathcal{E}^{(t)}\left(u^{\langle p/2\rangle},u^{\langle p/2\rangle}\right)$ exists, i.e. $u^{\langle p/2\rangle}\in\mathcal{D}(\mathcal{E})$.
Let us additionally assume that $u\inC(E)$.
Denote
$f(x,y):=F_p(u(x),u(y))$,
and
$g(x,y):=C_p\left(
(u(y))^{\langle p/2\rangle} - (u(x))^{\langle p/2\rangle}
\right)^2$, where $C_p$ is as in \eqref{neq:FpsimD}.
Since $u^{\langle p/2\rangle}\in\mathcal{D}(\mathcal{E})$, we have $g\in\mathcal{U}$, and by \eqref{neq:FpsimD} we have $f\leq g$. Moreover, $f=g=0$ on $\mathrm{diag}$ and $f,g\inC(E\times E)$ because $u\inC(E)$. Therefore, we can use Lemma \ref{lem:if_g_good_then_also_f} and conclude that $f\in\mathcal{U}$. This means that
\begin{align*}
\mathcal{E}_p[u]
&=
\lim_{t\to0^+} \mathcal{E}^{(t)}\left(u,u^{\langle p-1\rangle}\right)
=
\lim_{t\to0^+} \frac{1}{pt}
\iint\limits_{E\times E}
F_p(u(x),u(y)) \,P_t(dx,dy)
\\
&=
\frac{1}{p}
\iint\limits_{\mathclap{E\times E\setminus\mathrm{diag}}}
F_p(u(x),u(y)) \,J(dx,dy).
\end{align*}
\end{proof}
Now, we are ready to prove the main result of the article.
\begin{proof}[Proof of Theorem \ref{thm:HSp-j}]
It is enough to note that, by assumption, for every $t>0$ we have $P_tf\inC(E)\cap\D(\E_p)$, and so we can rewrite formula \eqref{eq:HS} using \eqref{eq:Ep=iintFp} with $u=P_tf$.
\end{proof}
\bibliographystyle{alpha}
|
2,877,628,090,245 | arxiv | \section{Introduction}
This paper is a direct sequel to \cite{BP_FCOP},
assembling the model structures on the categories
$\sOp_{\mathfrak C}^G$
of equivariant simplicial operads with a fixed
$G$-set of colors $\mathfrak C$
\cite[Thm. \ref{OC-THMI}]{BP_FCOP}
to a Dwyer-Kan style model structure on the category
$\sOp_\bullet^G$
of equivariant simplicial operads with any set of colors.
More broadly, this paper follows
\cite{Per18,BP_geo,BP_edss,BP_FCOP}
as part of a larger project culminating in
the sequel \cite{BP_TAS}
with the existence of a Quillen equivalence
\begin{equation}\label{BPMAINTHM_EQ}
\mathsf{dSet}^G \rightleftarrows \mathsf{sOp}_{\bullet}^G
\end{equation}
where $\mathsf{dSet}^G$ is the category
of equivariant dendroidal sets with the model structure
from \cite{Per18}
and
$\mathsf{sOp}_{\bullet}^G$
has the model structure from
Theorem \ref{THMA} herein,
thereby generalizing the
analogous Cisinski-Moerdijk-Weiss project
\cite{MW09,CM11,CM13a,CM13b} to the equivariant context.
Much of the challenge in building the model structures
in \eqref{BPMAINTHM_EQ}
comes from the fact that they model the homotopy theory of
equivariant operads \emph{with norm maps},
which are an extra piece of data
not present non-equivariantly
(and recalled in \eqref{GRAPHEQ EQ},\eqref{ALGNORM EQ} below),
the importance of which was made clear by
Hill, Hopkins, Ravenel
in their solution to the Kervaire invariant one problem \cite{HHR16}.
Notably, the presence of norm maps causes the existence
of the model structures in \eqref{BPMAINTHM_EQ}
not to be a formal consequence of the
existence of their non-equivariant analogues
(this is further discussed at the end of this introduction).
We now briefly recall
the notion of norm maps.
For simplicity,
fix a finite group $G$
and consider the category
$\mathsf{sOp}^{G}_{\**} = \mathsf{Op}_{\**}(\mathsf{sSet}^G)$
of single colored (symmetric) operads
on $G$-equivariant simplicial sets $\mathsf{sSet}^G$.
Note that, for $\O \in \mathsf{sOp}^{G}_{\**}$,
the $n$-th operadic level $\O(n)$ has both a $\Sigma_n$-action and a $G$-action, commuting with each other, or, equivalently,
a $G \times \Sigma_n$-action.
A key upshot
of Blumberg and Hill's work \cite{BH15}
is then that
the preferred notion of weak equivalence in $\mathsf{sOp}^{G}_{\**}$
is that of \emph{graph equivalence},
i.e. those maps
$\O \to \mathcal{P}$
such that the fixed point maps
\begin{equation}\label{GRAPHEQ EQ}
\O(n)^{\Gamma} \xrightarrow{\sim} \mathcal{P}(n)^{\Gamma}
\qquad
\text{for }
\Gamma \leq G \times \Sigma_n
\text{ such that }
\Gamma \cap \Sigma_n = \**
\end{equation}
are Kan equivalences in $\mathsf{sSet}$.
Here, the term ``graph'' comes from a neat characterization of the $\Gamma$
as in \eqref{GRAPHEQ EQ}:
such a $\Gamma$ must be the graph of a partial homomorphism
$\phi \colon H \to \Sigma_n$ for some subgroup $H \leq G$,
i.e.
$\Gamma = \sets{(h,\phi(h))}{h \in H}$.
Note that one then has a canonical isomorphism $\Gamma \simeq H$.
Briefly, the need to consider such \emph{graph subgroups} $\Gamma$ comes from the study of algebras.
Suppose $X \in \mathsf{sSet}^G$ is an algebra over
$\O$,
so that one has algebra multiplication maps
as on the left below
\begin{equation}\label{ALGNORM EQ}
\O(n) \times X^n \to X
\qquad \qquad
\O(n)^{\Gamma} \times N_{\Gamma}X \to X
\end{equation}
which need to be
$G \times \Sigma_n$-equivariant
(the target $X$ is given the trivial $\Sigma_n$-action).
One then has induced
$H$-equivariant maps
on the right in \eqref{ALGNORM EQ},
where the \emph{norm object} $N_{\Gamma} X$
denotes $X^{n}$ with the $H$-action given by $\Gamma \simeq H$.
In particular, each point
$\rho \in \O(n)^{\Gamma}$
encodes a $H$-equivariant \emph{norm map}
$\rho \colon N_{\Gamma} X \to X$,
and such maps are a key piece of data
for algebras.
Thus, the advantage of the graph equivalences \eqref{GRAPHEQ EQ}
is that equivalent operads have
equivalent ``spaces of norm maps''.
In the single colored case,
the existence of a model structure
on $\mathsf{sOp}^G_{\**}$
with weak equivalences the graph equivalences
in \eqref{GRAPHEQ EQ}
was established in both
\cite[Thm. I]{BP_geo} and \cite[Thm. 3.1]{GW18}.
Our main result, Theorem \ref{THMA},
then extends the graph equivalence model structures of
\cite[Thm. I]{BP_geo}, \cite[Thm. 3.1]{GW18}
from the context of single colored operads
to the context of operads with a varying set of colors
(also known as multicategories).
Alternatively, one may view Theorem \ref{THMA} as extending the well known Dwyer-Kan model structures on all colored operads
in \cite[Thm 1.14]{CM13b}, \cite[Thm. 2]{Rob11}
from the non-equivariant context to the
equivariant context.
It is worth noting that
Theorem \ref{THMA} is non formal,
as discussed in below
(a similar discussion for the
single color and fixed color contexts
can be found in the introduction to \cite{BP_FCOP}).
First,
we note that our category of interest
is the category
$\mathsf{sOp}^G_{\bullet}=
\left(\mathsf{sOp}_{\bullet}\right)^G$
of $G$-objects on the usual category
$\mathsf{sOp}_{\bullet}$
of simplicial (symmetric) operads studied in
\cite{CM13b},\cite{Rob11},\cite{Cav}.
In particular, the group $G$ is allowed to act non-trivially
on the objects of an equivariant operad
$\O \in \mathsf{sOp}^G_{\bullet}$.
By contrast, in the category
$\mathsf{Op}_{\bullet}(\mathsf{sSet}^G)$
of colored operads in $\mathsf{sSet}^G$
the group $G$ never acts on objects (or, equivalently, acts trivially).
As such, one only has a proper inclusion
$\mathsf{Op}_{\bullet}(\mathsf{sSet}^G)
\subsetneq \mathsf{sOp}_{\bullet}^G$,
so a model structure on
$\mathsf{sOp}_{\bullet}^G$
can not be built using the enriched colored operad results of \cite{Cav}.
Alternatively, one could also try to build a model structure
on $\mathsf{sOp}^G_{\bullet}$
by applying the formalism in
\cite[Prop 2.6]{Ste16},
which builds model structures on $G$-objects,
to the model structure on $\mathsf{sOp}_\bullet$
from \cite{CM13b},\cite{Rob11}.
However, this approach does not produce
the desired notion of weak equivalence suggested by graph subgroups (more precisely, this approach ignores
``non trivial norm maps'', i.e.
it ignores any graph subgroups $\Gamma$ in \eqref{GRAPHEQ EQ}
associated to non trivial homomorphisms $\phi\colon H \to \Sigma_n$).
It is worth noting that the issue with this latter approach is intrinsically operadic
and does not occur when working with categories.
Indeed, the inclusion
$\mathsf{sCat}^G_{\bullet} \subset
\mathsf{sOp}^G_{\bullet}$
of colored $G$-categories into
colored $G$-operads identifies
$\mathsf{sCat}^G_{\bullet} =
\mathsf{sOp}^G_{\bullet} \downarrow \**$
as the overcategory over the terminal category $\**$.
Hence, our model structure on
$\mathsf{sOp}^G_{\bullet}$
induces a model structure on
$\mathsf{sCat}^G_{\bullet}$ which \emph{does}
in fact coincide with the model structure obtained by applying
\cite[Prop. 2.6]{Ste16}
to the usual model structure on $\mathsf{sCat}_{\bullet}$.
\subsection{Main Results}
Before stating our main result, Theorem \ref{THMA},
we require some preliminary setup.
First, as noted in \eqref{GRAPHEQ EQ},
our preferred notion of equivalence of equivariant operads is determined by the graph subgroups.
However, as in \cite{BP_FCOP}, we will work with general collections of subgroups.
\begin{definition}\label{FAM1ST DEF}
A \emph{$(G,\Sigma)$-family} is a
a collection
$\mathcal{F} = \{\mathcal{F}_n\}_{n \leq 0}$,
where each $\mathcal{F}_n$
is a family of subgroups of $G \times \Sigma_n^{op}$.
\end{definition}
The use of $\Sigma_n^{op}$ rather than $\Sigma_n$
in Definition \ref{FAM1ST DEF}
(and throughout)
is motivated by regarding $\Sigma$
as the category of corollas (i.e. trees with a single node;
see \eqref{OPSSYMS EQ},\eqref{CSYM EQ1},\eqref{CSYM EQ2}),
and the fact that the dendroidal nerve \cite[\S 1]{MW07} of an operad is contravariant on the category of trees.
In contrast to \cite[Thm. \ref{OC-THMI}]{BP_FCOP},
Theorem \ref{THMA} requires a minor restriction on the
$(G,\Sigma)$-family $\mathcal{F}$.
Noting that $\F_1$ is a family of subgroups of
$G \times \Sigma_1^{op} \simeq G$,
we regard
$H \in \F_1$ as a subgroup $H \leq G$.
\begin{definition}\label{FAMRESUNI DEF}
Write $G \times \Sigma_n^{op} \xrightarrow{\pi_n}
G$
for the natural projection.
We say that a $(G,\Sigma)$-family $\F$
\emph{has enough units} if,
for all $H \in \F_n$, $n\geq 0$,
it is $\pi_n(H) \in \F_1$.
\end{definition}
The motivation for the condition
in Definition \ref{FAMRESUNI DEF} is discussed in Remark \ref{WHYEU REM}.
Next, recall that a colored operad $\O$
with color set $\mathfrak{C}$ has levels
$
\O(\vect{C})=
\O(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)$
indexed by tuples
$\vect{C} = (\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0) = (\mathfrak c_i)_{0 \leq i \leq n}$
of elements in $\mathfrak{C}$, called \emph{$\mathfrak{C}$-signatures}.
If the operad is symmetric one has associative and unital isomorphisms
$
\O(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0) \to
\O(\mathfrak{c}_{\sigma(1)},\cdots,\mathfrak{c}_{\sigma(n)};\mathfrak{c}_0)
$
for each permutation $\sigma \in \Sigma_n$.
On the other hand, if
$\O \in \mathsf{sOp}^G_{\mathfrak{C}}$
is a $G$-equivariant operad,
the color set $\mathfrak{C}$ is itself a $G$-set,
and one similarly has associative and unital isomorphisms
$
\O(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0) \to
\O(g\mathfrak{c}_{1},\cdots,g\mathfrak{c}_{n};g\mathfrak{c}_0)
$ for $g \in G$.
All together, one thus has isomorphisms
\begin{equation}\label{OPSSYMS EQ}
\O(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)
\to
\O(g \mathfrak{c}_{\sigma(1)},\cdots,g \mathfrak{c}_{\sigma(n)};g\mathfrak{c}_0)
\end{equation}
for $(g,\sigma) \in G \times \Sigma_n^{op}$.
Note that these isomorphisms
are associated with an action of
$G \times \Sigma_n^{op}$
on the set $\mathfrak{C}^{n+1}$ of $n$-ary signatures via
$(g,\sigma) (\mathfrak{c}_i)_{0\leq i \leq n}
= (g \mathfrak{c}_{\sigma(i)})_{0\leq i \leq n}$,
where we implicitly write $\sigma(0)=0$.
As such, we say that a subgroup
$\Lambda \leq G \times \Sigma_n^{op}$
\emph{stabilizes} a signature $\vect{C}=(\mathfrak{c}_i)_{0 \leq i \leq n}$ if,
for any $(g,\sigma) \in \Lambda$,
it is
$\mathfrak{c}_i = g \mathfrak{c}_{\sigma(i)}$ for $0 \leq i \leq n$.
Note that,
for $\O \in \mathsf{sOp}^G_{\mathfrak{C}}$,
the level $\O(\vect{C})$ has a $\Lambda$-action.
Lastly, we need a notion of
\emph{essential surjectivity}.
For this purpose, we recall the following construction,
which associates to a $\V$-category $\mathcal{C}$
a \emph{category of components} $\pi_0 \mathcal{C}$.
\begin{definition}
Suppose $\V$ is as in Theorem \ref{THMA}
(in particular, $\V$ has a cofibrant unit).
Given $\mathcal C \in \Cat_{\mathfrak{C}}(\V)$,
define
$\pi_0 \mathcal C \in \Cat_{\mathfrak{C}} = \Cat_{\mathfrak{C}}(\mathsf{Set})$
to be the ordinary category with the same objects and
\[
\pi_0(\mathcal{C})(c,d)=
\Ho(\V)(1_\V, \mathcal C(c,c'))=
[1_\V, \mathcal{C}_f(c,c')]
\]
where $[-,-]$ denotes homotopy equivalence classes of maps
and $\mathcal{C}_f$ denotes a fibrant replacement of
$\mathcal C$ in the \emph{canonical model structure} on $\Cat_{\mathfrak{C}}(\V)$
\cite{BM13} (also, see Remarks
\ref{RESTTOCATS REM} and \ref{GTRIV REM}).
\end{definition}
Further writing
$j^{\**} \colon
\mathsf{Op}^G_{\bullet}(\V) \to \mathsf{Cat}^G_{\bullet}(\V)$
for the ``underlying category''
functor which forgets the non-unary operations,
we can now state the main result.
\begin{customthm}{A}\label{THMA}
Fix a finite group $G$
and a $(G,\Sigma)$-family $\F = \set{\F_n}_{n \geq 0}$
which has enough units.
Then there exists a model structure on
$\mathsf{sOp}^G_{\bullet} =
\mathsf{Op}^G_{\bullet}(\mathsf{sSet})$,
which we call the \emph{$\F$-model structure},
such that a map
$F\colon \mathcal{O} \to \mathcal{P}$
is a weak equivalence (resp. trivial fibration) if
\begin{itemize}
\item the maps
\begin{equation}\label{THMIII1ST EQ}
\O(\vect{C})^{\Lambda} \to \mathcal{P}(F(\vect{C}))^{\Lambda}
\end{equation}
are Kan equivalences (trivial Kan fibrations)
in $\mathsf{sSet}$
for all $\mathfrak{C}$-signatures $\vect{C}$
and $\Lambda \in \F$ which stabilizes $\vect{C}$;
\item
the maps of unenriched categories
\begin{equation}\label{THMIII2ND EQ}
\pi_0 j^{\**} \O^H
\to
\pi_0 j^{\**} \mathcal{P}^H
\end{equation}
are essentially surjective (surjective on objects)
for all $H \in \F_1$.
\end{itemize}
More generally, a $\F$-model structure on
$\Op^G_{\bullet}(\V)$
with weak equivalences/trivial fibrations as in
\eqref{THMIII1ST EQ},\eqref{THMIII2ND EQ}
exists provided $(\V,\otimes)$ satisfies:
\begin{enumerate}[label = (\roman*)]
\item $\V$ is a cofibrantly generated model category
such that the domains of the generating (trivial) cofibrations are small;
\item for any finite group $G$, the $G$-object category $\V^G$ admits the genuine model structure
\cite[Def. \ref{OC-GENMOD DEF}]{BP_FCOP};
\item $(\V, \otimes)$ is a closed symmetric monoidal model category with cofibrant unit;
\item $(\V, \otimes)$ satisfies the global monoid axiom \cite[Def. \ref{OC-GLOBMONAX_DEF}]{BP_FCOP};
\item $(\V, \otimes)$ has cofibrant symmetric pushout powers \cite[Def. \ref{OC-CSPP_DEF}]{BP_FCOP}.
\item[(vi)] $\V$ is right proper;
\item[(vii)]
for any finite group $G$, fixed points
$(-)^{G} \colon \V^G \to \V$
send genuine trivial cofibrations
(cf. \cite[Def. \ref{OC-GENMOD DEF}]{BP_FCOP})
to trivial cofibrations;
\item[(viii)] $(\V, \otimes)$ has a generating set of intervals
(Definition \ref{INTGENSET DEF}).
\end{enumerate}
\end{customthm}
Conditions (i) through (v) above
are the conditions in \cite[Thm. \ref{OC-THMI}]{BP_FCOP},
which builds the model structures on
fixed color operads
$\Op^G_{\mathfrak{C}}(\V)$,
one of the key ingredients in the proof
of Theorem \ref{THMA}.
In particular,
herein the technical conditions
(iv),(v) will only be needed to cite
results in \cite{BP_FCOP}.
\begin{remark}
Maps satisfying both of the weak equivalence conditions in
\eqref{THMIII1ST EQ},\eqref{THMIII2ND EQ}
are called \emph{Dwyer-Kan equivalences},
while maps satisfying only \eqref{THMIII1ST EQ}
are called \emph{local equivalences}.
\end{remark}
\begin{remark}\label{WHYEU REM}
The ``enough units'' condition in Definition \ref{FAMRESUNI DEF}
ensures compatibility of the \emph{local equivalences}
in \eqref{THMIII1ST EQ}
with the essential surjectivity in \eqref{THMIII2ND EQ}.
Informally, this guarantees that the spaces
$\O(\vect{C})^{\Lambda}$
are homotopically well behaved when replacing colors
in $\vect{C}$
(for details, see \S \ref{HMTYEQ SEC}).
\end{remark}
\begin{remark}\label{WETRFCAN REM}
The requirement that the maps in
\eqref{THMIII1ST EQ}
are weak equivalences implies that the maps in
\eqref{THMIII2ND EQ}
are fully faithful.
Therefore, the condition following \eqref{THMIII2ND EQ}
can be restated as saying that those maps are
equivalences of categories (resp. equivalences of categories that are surjective on objects) or, in other words,
that the maps in \eqref{THMIII2ND EQ}
are weak equivalences/trivial fibrations in the canonical model structure on the category $\mathsf{Cat}$ of unenriched categories
\cite{Rez}.
\end{remark}
\begin{remark}\label{FIBSALT REM}
In light of Remark \ref{WETRFCAN REM},
it is natural to ask if the fibrations in Theorem \ref{THMA}
admit an analogous description.
That is, we may ask if a map $F\colon \O \to \mathcal{P}$
is a fibration in the sense of Theorem \ref{THMA}
iff the maps in
\eqref{THMIII1ST EQ}
are Kan fibrations in $\mathsf{sSet}$
and the maps in
\eqref{THMIII2ND EQ}
are isofibrations (i.e. fibrations in the canonical model structure in $\mathsf{Cat}$).
However, at our level of generality we can only guarantee the
``only if'' direction of this characterization.
For the ``if'' direction to hold we need to either demand
that $\P$ itself is fibrant or
impose an extra condition on the unit of $\V$ (which happens to be satisfied by $\mathsf{sSet}$).
See Propositions \ref{ISOFIBEASY PROP} and \ref{ISOFIBHARD PROP} for more details.
\end{remark}
\begin{remark}\label{RESTTOCATS REM}
As noted at the end of the introduction,
there is an identification
$\Cat_\bullet^G(\V) \simeq \Op_\bullet^G(\V) \downarrow \**$,
where $\**$ denotes the terminal $\V$-category,
so the $\F$-model structure on $\Op_\bullet^G(\V)$
also induces a model structure on $\Cat_\bullet^G(\V)$.
Since $\Cat_\bullet^G(\V)$ contains only unary operations,
this latter model structure depends only on $\F_1$,
which is identified with a family of subgroups of $G$ itself.
In fact, the resulting model structure on
$\Cat_\bullet^G(\V)$ matches the model structure
obtained by applying \cite{Ste16}
to the family $\F_1$ and the canonical model structure on
$\Cat_\bullet(\V)$.
Moreover, we note that the analogues for $\Cat_\bullet^G(\V)$
of all three of \cite[Thms. \ref{OC-THMI} and \ref{OC-THMII}]{BP_FCOP} and Theorem \ref{THMA}
follow from our proofs without using either
the cofibrant pushout power condition (v)
or (vii) in Theorem \ref{THMA},
and without additional restrictions on $\F_1$
(i.e. no analogues of the pseudo indexing system
(cf. \cite[Thm. \ref{OC-THMII}]{BP_FCOP})
and ``enough units'' (cf. Definition \ref{FAMRESUNI DEF})
conditions are needed).
For details, see \cite[Rem. \ref{OC-CSPNTHI REM}]{BP_FCOP} and
Remark \ref{ALBEETA_REM}.
\end{remark}
\begin{remark}\label{SEMI_REM}
When working with operads, some authors (e.g. \cite{Spi,Whi17,WY18})
discuss \emph{semi-model structures}.
Briefly, these are a weakening of Quillen's original definition
where those factorization and lifting axioms
that involve trivial cofibrations
are only required to hold if the trivial cofibration
has cofibrant source \cite[\S 2.2]{WY18}.
We note that, in particular, semi-model structures suffice for
performing
bifibrant replacements.
The semi-model structure analogues of
\cite[Thms. \ref{OC-THMI} and \ref{OC-THMII}]{BP_FCOP}
and Theorem \ref{THMA}
can be obtained by slight variants of our proofs
without using the global monoid axiom (iv).
For details, see \cite[Rem. \ref{OC-THMISM REM}]{BP_FCOP}
and Remarks \ref{JCELLSM REM}, \ref{MONAXSUP REM}.
\end{remark}
\begin{remark}\label{GTRIV REM}
It seems tempting to think that, for trivial $G=\**$,
one can omit the existence of genuine model structures condition (ii) in Theorem \ref{THMA}.
However, this is not so, since
our work uses
\cite[Thm. \ref{OC-THMI}]{BP_FCOP},
whose proof needs
the cofibrant pushout powers condition (v)
\cite[Rem. \ref{OC-GTRIV REM}]{BP_FCOP}.
If one further specifies to
$G=\**$ and the categorical case $\Cat_\bullet(\V)$,
there is only one interesting choice of family $\F_1$,
i.e. the non-empty family of subgroups of $\Sigma_1$,
which recovers the \emph{canonical model structure} on
$\Cat_\bullet(\V)$ discussed in \cite{BM13}.
In this case, an analysis of our proofs shows that
one \emph{can}
drop assumptions (iii),(v)(vii) of Theorem \ref{THMA},
and replace the global monoid axiom in (iv)
with the usual Schwede-Shipley
monoid axiom \cite{SS00}
(see \cite[Rem. \ref{OC-MONAX_REM}]{BP_FCOP}),
so that our assumptions are then a close variation on those in \cite{BM13}.
\end{remark}
\subsection{Examples}\label{EXAMPLES SEC}
The examples of
model categories satisfying
all of conditions (i) through (vii)
in Theorem \ref{THMA}
are fairly limited,
mostly due to
the cofibrant pushout powers axiom (v),
which is rather restrictive.
For a discussion of the role of this condition,
see \cite[Rems. \ref{OC-CPPWHY REM} and
\ref{OC-SPNONEX REM}]{BP_FCOP}.
Below we list those examples of categories satisfying all conditions that we are aware of.
\begin{enumerate}[label = (\alph*)]
\item $(\mathsf{sSet},\times)$ or $(\mathsf{sSet}_{\**},\wedge)$
with the Kan model structure.
\item $(\mathsf{Top},\times)$ or $(\mathsf{Top}_{\**},\wedge)$
with the usual Serre model structure.
\item $(\mathsf{Set},\times)$ the category of sets with its canonical model structure,
where weak equivalences are the bijections and all maps are both cofibrations and fibrations.
\item $(\Cat,\times)$ the category of usual categories
with the ``folk'' or canonical model structure (e.g. \cite{Rez})
where weak equivalences are the equivalences of categories,
cofibrations are the functors which are injective on objects,
and fibrations are the isofibrations.
\end{enumerate}
In all these cases, conditions (i) through (v)
were discussed in \cite[\S \ref{OC-EXAMPLES SEC}]{BP_FCOP},
(vi) is well known, and
(viii) follows from either
\cite[Lemma 1.12]{BM13} or \cite[Lemma 2.1]{BM13}.
The following is a noteworthy
non-example.
\begin{remark}
The category $(\mathsf{Sp}^{\Sigma}(\mathsf{sSet}),\wedge)$
of symmetric spectra (on simplicial sets),
with the positive $S$ model structure,
satisfies most of axioms in Theorem
\ref{THMA},
with the exceptions being
the cofibrant unit requirement in (iii)
and the cofibrant pushout powers axiom in (v).
Nonetheless, $(\mathsf{Sp}^{\Sigma}(\mathsf{sSet}),\wedge)$
does seem to satisfy variants of axioms
(iii) and (v). For further discussion,
see \cite[Rem. \ref{OC-SPNONEX REM}]{BP_FCOP}.
\end{remark}
\subsection{Outline}
\S \ref{SUM SEC} mostly recalls the notions and results from
\cite{BP_FCOP} that we will use,
recalling in particular the model structures
on fixed color operads
$\mathsf{Op}_{\mathfrak{C}}(\V)$ from
\cite[Thm. \ref{OC-THMI}]{BP_FCOP},
which form the basis of the model structures on
$\mathsf{Op}_{\bullet}(\V)$
in Theorem \ref{THMA}.
\S \ref{MS_SEC} is dedicated to proving
Theorem \ref{THMA}.
In \S \ref{MAPSOPG_SEC} we identify the relevant classes of maps
in $\mathsf{Op}^G_{\bullet}(\V)$.
Then, in \S \ref{GENCOF SEC} we identify
the necessary sets of generating (trivial) cofibrations,
and outline the overall proof of Theorem \ref{THMA},
with \S \ref{TRIVCOF_SEC},\ref{EQUIVOBJ_SEC},\ref{HMTYEQ SEC}
concluding the proof by addressing the hardest steps.
Lastly, \S \ref{ISOFIB_SEC} discusses an alternative description
of the fibrations in
$\mathsf{Op}^G_{\bullet}(\V)$,
elaborating on
Remark \ref{FIBSALT REM}.
\section{Summary of previous work}
\label{SUM SEC}
This section is mostly expository,
recalling the key definitions and results
in \cite{BP_FCOP} that we need to
prove Theorem \ref{THMA},
while converting some technical results therein
to a more convenient format.
In \S \ref{COSYMSEQ SEC} and \S \ref{EQCOSYMSEQ SEC}
we recall the definitions
of the categories
$\mathsf{Sym}^G_{\bullet}(\V)$
of equivariant symmetric sequences and
$\mathsf{Op}^G_{\bullet}(\V)$
of equivariant operads.
Of particular importance is the discussion on
representable functors in
$\mathsf{Sym}^G_{\bullet}
=
\mathsf{Sym}^G_{\bullet}(\mathsf{Set})$,
culminating in \eqref{REPALTDESC EQ},
which are needed in \S \ref{GENCOF SEC}
when describing the generating (trivial) cofibrations
in $\mathsf{Op}^G_{\bullet}(\V)$.
\S \ref{COLFIXMOD SEC} then recalls
\cite[Thm. \ref{OC-THMI}]{BP_FCOP}
as Theorem \ref{THMIREST},
which discusses the model structures on the categories
$\mathsf{Op}^G_{\mathfrak{C}}(\V)$
of fixed color operads
that are one of the main ingredients to building
the model structure on
$\mathsf{Op}^G_{\bullet}(\V)$
in Theorem \ref{THMA}.
The (rather technical) condition in
Remark \ref{GOTC_REM} is of particular importance,
as it plays a key role in proving Theorem \ref{THMA}
(more concretely, it is needed in the
proof of Proposition \ref{J_CELL_PROP},
which is one of the key claims needed for Theorem \ref{THMA}).
\subsection{Colored symmetric sequences and colored operads}
\label{COSYMSEQ SEC}
\subsubsection*{Colored symmetric sequences}
\begin{definition}\label{CSYM DEF}
Let $\mathfrak {C} \in \mathsf{Set}$ be a fixed set of colors (or objects).
A tuple
$\vect C = (\mathfrak c_1, \dots, \mathfrak c_n; \mathfrak c_0) \in \mathfrak{C}^{\times n+1}$
is called a \textit{$\mathfrak {C}$-signature} of \textit{arity} $n$.
The \textit{$\mathfrak C$-symmetric category} $\Sigma_{\mathfrak C}$ is the category whose objects are the $\mathfrak{C}$-signatures and
whose morphisms are action maps
\begin{equation}\label{CSYM EQ1}
\vect{C} =
(\mathfrak c_1, \dots, \mathfrak c_n; \mathfrak c_0) \xrightarrow{\sigma} (\mathfrak c_{\sigma^{-1}(1)}, \dots, \mathfrak c_{\sigma^{-1}(n)}; \mathfrak c_0)
= \vect{C} \sigma^{-1}
\end{equation}
for each permutation $\sigma \in \Sigma_n$, with the natural notion of composition.
Alternatively, one can visualize signatures as corollas (i.e. trees with a single node)
with edges decorated by colors in $\mathfrak{C}$, as depicted below, so that the map labeled $\sigma$
is the unique map of trees indicated such that the coloring of an edge equals the coloring of its image.
\begin{equation}\label{CSYM EQ2}
\begin{tikzpicture}
[grow=up,auto,level distance=2.3em,every node/.style = {font=\footnotesize},dummy/.style={circle,draw,inner sep=0pt,minimum size=1.75mm}]
\node at (0,0) [font=\normalsize]{$\vect{C}$}
child{node [dummy] {}
child{
edge from parent node [swap,near end] {$\mathfrak c_n$} node [name=Kn] {}}
child{
edge from parent node [near end] {$\mathfrak c_1$}
node [name=Kone,swap] {}}
edge from parent node [swap] {$\mathfrak c_0$}
};
\draw [dotted,thick] (Kone) -- (Kn) ;
\node at (5,0) [font=\normalsize] {$\vect{C} \sigma^{-1}
$}
child{node [dummy] {}
child{
edge from parent node [swap,near end] {$\mathfrak c_{\sigma^{-1}(n)}$} node [name=Kn] {}}
child{
edge from parent node [near end] {$\mathfrak c_{\sigma^{-1}(1)}$}
node [name=Kone,swap] {}}
edge from parent node [swap] {$\mathfrak c_0$}
};
\draw [dotted,thick] (Kone) -- (Kn) ;
\draw[->] (1.5,0.8) -- node{$\sigma$} (3,0.8);
\end{tikzpicture}
\end{equation}
Given any map of color sets $\varphi \colon \mathfrak{C} \to \mathfrak{D}$,
there is a functor (abusively written)
$\varphi \colon \Sigma_{\mathfrak{C}} \to \Sigma_{\mathfrak{D}}$,
given by
$\varphi (\mathfrak c_1, \dots, \mathfrak c_n; \mathfrak c_0) = (\varphi(\mathfrak c_1),\cdots,\varphi(\mathfrak c_n);\varphi(\mathfrak c_0))$.
\end{definition}
\begin{remark}\label{GLOBSIG REM}
The notation $\vect{C} \sigma^{-1}$
in \eqref{CSYM EQ1},\eqref{CSYM EQ2}
reflects the fact that $\Sigma_n$
acts on the right on $\mathfrak{C}$-signatures of arity $n$
via
$\vect{C} \sigma = (\mathfrak{c}_i)\sigma =
(\mathfrak{c}_{\sigma(i)})$,
where we make the convention that $\sigma(0)=0$.
\end{remark}
\begin{definition}\label{CSSYM DEF}
Let $\mathcal{V}$ be a category.
The category $\mathsf{Sym}_\bullet(\mathcal{V})$ of
\textit{symmetric sequences on $\mathcal{V}$}
(on all colors) is the category with:
\begin{itemize}
\item objects given by pairs $(\mathfrak C, X)$ with
$\mathfrak{C} \in \mathsf{Set}$ a set of colors and
$X \colon \Sigma_{\mathfrak{C}}^{op} \to \mathcal{V}$ a functor;
\item arrows $(\mathfrak C, X) \to (\mathfrak D, Y)$ given by a map
$\varphi \colon \mathfrak{C} \to \mathfrak{D}$ of colors and a natural transformation $X \Rightarrow Y \varphi$ as below.
\begin{equation}\label{CSSYM EQ}
\begin{tikzcd}[row sep = tiny, column sep = 45pt]
\Sigma_{\mathfrak{C}}^{op} \arrow[dr, "X"{name=U}]
\arrow{dd}[swap]{\varphi}
\\
& \mathcal{V}
\\
|[alias=V]| \Sigma_{\mathfrak{D}}^{op} \arrow[ur, "Y"']
\arrow[Leftarrow, from=V, to=U,shorten >=0.25cm,shorten <=0.25cm]
\end{tikzcd}
\end{equation}
\end{itemize}
\end{definition}
\begin{notation}\label{SIGMABULL NOT}
We write
$\Sigma_{\bullet} \to \mathsf{Set}$
for the Grothendieck construction
\cite[Not. \ref{OC-GROTHCONS NOT}]{BP_FCOP}
of the functor
$\mathsf{Set} \to \mathsf{Cat}$ given by
$\mathfrak{C} \mapsto \Sigma_{\mathfrak{C}}$.
Explicitly,
the objects of $\Sigma_{\bullet}$
are the $\vect{C} \in \Sigma_{\mathfrak{C}}$
for some set of colors $\mathfrak{C}$
and an arrow from
$\vect{C} \in \Sigma_{\mathfrak{C}}$ to
$\vect{D} \in \Sigma_{\mathfrak{D}}$
over $\varphi \colon \mathfrak{C} \to \mathfrak{D}$
is an arrow
$\varphi \vect{C} \to \vect{D}$ in $\Sigma_{\mathfrak{D}}$.
\end{notation}
\begin{remark}\label{COLCHADJ REM}
We caution that
$\mathsf{Sym}_{\bullet}(\V)$
is quite different from the presheaf category
$\mathsf{Fun}(\Sigma_{\bullet}^{op},\V)$.
Instead,
$\mathsf{Sym}_{\bullet}(\V)$
can be regarded as a category of ``fibered presheaves''.
More precisely,
the \emph{color set functor}
$\mathsf{Sym}_{\bullet}(\V) \to \mathsf{Set}$
is both a Grothendieck fibration
and opfibration
(cf., e.g. \cite[\S \ref{OC-GROTFIB SEC}]{BP_FCOP}),
with fibers the
presheaf categories
$\Sym_{\mathfrak C}(\V)=
\mathsf{Fun}(\Sigma_{\mathfrak{C}}^{op},\mathcal{V})$
and cartesian (resp. cocartesian)
arrows the diagrams \eqref{CSSYM EQ}
which are natural isomorphisms (resp. left Kan extensions).
In particular \cite[Rem. \ref{OC-ALSOOPADJ REM}]{BP_FCOP},
for any map $\varphi \colon \mathfrak{C} \to \mathfrak{D}$
one has adjunctions
\begin{equation}\label{COLCHADJ EQ}
\varphi_! \colon \Sym_{\mathfrak C}(\V)
\rightleftarrows
\Sym_{\mathfrak D}(\V) \colon \varphi^{\**}
\end{equation}
where $\varphi^{\**}$
(resp. $\varphi_!$)
is precomposition with (resp. left Kan extension along)
$\varphi\colon
\Sigma^{op}_{\mathfrak{C}}
\to
\Sigma^{op}_{\mathfrak{D}}$.
\end{remark}
\subsubsection*{Representable functors}
The description of the model structures
on $\Sym_{\mathfrak C}(\V) $ in
\S \ref{COLFIXMOD SEC}
will require us to identify certain
representable functors in
$\mathsf{Sym}_{\bullet} = \mathsf{Sym}_{\bullet}(\mathsf{Set})$.
We start with the following.
\begin{notation}\label{FIBYON NOT}
Let $\mathfrak{C} \in \mathsf{Set}$, $\vect{C} \in \Sigma_{\mathfrak{C}}$.
We write
$\Sigma_{\mathfrak{C}}[\vect{C}]
\in \mathsf{Sym}_{\mathfrak{C}} = \mathsf{Set}^{\Sigma_{\mathfrak{C}}^{op}}$ for the representable presheaf
\[\Sigma_{\mathfrak{C}}[\vect{C}](-)
= \Sigma_{\mathfrak{C}}(-,\vect{C}).\]
Moreover, this defines a
\emph{fibered Yoneda functor}
$
\Sigma_{\bullet} \xrightarrow{\Sigma_{\bullet}[-]} \mathsf{Sym}_{\bullet}
$
by mapping an arrow
$\varphi \vect{C} \to \vect{D}$
over $\varphi \colon \mathfrak{C} \to \mathfrak{D}$
to the natural transformation
$\Sigma_{\mathfrak{C}}[\vect{C}]
\Rightarrow
\varphi^{\**}
\Sigma_{\mathfrak{D}}[\vect{D}]
$ given by the composites
\[\Sigma_{\mathfrak{C}}[\vect{C}](-)
= \Sigma_{\mathfrak{C}}(-,\vect{C})
\to
\Sigma_{\mathfrak{D}}(\varphi(-),\varphi\vect{C})
\to
\Sigma_{\mathfrak{D}}(\varphi(-),\vect{D})
=
\varphi^{\**} \Sigma_{\mathfrak{D}}[\vect{D}](-).
\]
\end{notation}
\begin{proposition}[{\cite[Prop. \ref{OC-FIBYONPUSH PROP}]{BP_FCOP}}]
\label{FIBYONPUSH PROP}
Let $\vect{C} \in \Sigma_{\mathfrak{C}}$,
$\varphi \colon \mathfrak{C} \to \mathfrak{D}$
be a map of colors.
Then there is an identification
$\varphi_{!} \Sigma_{\mathfrak{C}}[\vect{C}]
\xrightarrow{\simeq}
\Sigma_{\mathfrak{D}}[\varphi\vect{C}]$,
adjoint to the canonical map
$\Sigma_{\mathfrak{C}}[\vect{C}]
\to
\varphi^{\**}\Sigma_{\mathfrak{D}}[\varphi\vect{C}]$).
In other words,
$\Sigma_{\bullet}[-] \colon
\Sigma_{\bullet} \to \mathsf{Sym}_{\bullet}$
preserves cocartesian arrows.
\end{proposition}
The fibered Yoneda functor
$
\Sigma_{\bullet} \xrightarrow{\Sigma_{\bullet}[-]} \mathsf{Sym}_{\bullet}
$
does not quite suffice for our purposes,
due to the domain $\Sigma_{\bullet}$
lacking enough colimits.
To extend $\Sigma_{\bullet}[-]$,
we now discuss colored forests.
In the following, $\Phi$ denotes the category of forests
(i.e. formal coproducts of trees; see \cite[\S 5.1]{Per18}).
\begin{definition}\label{COLFOR DEF}
Let $\mathfrak{C}$ be a set of colors.
The category $\Phi_{\mathfrak{C}}$ of $\mathfrak{C}$-colored forests has
\begin{itemize}
\item objects pairs
$\vect{F} = (F,\mathfrak{c})$
where
$F\in \Phi$ is a forest
and
$\mathfrak{c}\colon \boldsymbol{E}(F) \to \mathfrak{C}$
is a coloring of its edges;
\item a map
$\vect{F}=(F,\mathfrak{c}) \to
(F',\mathfrak{c}') = \vect{F'}$
is a map $\rho \colon F \to F'$ in $\Phi$
such that
$\mathfrak{c} = \mathfrak{c}' \rho$.
\end{itemize}
For a map of colors,
$\varphi\colon \mathfrak{C} \to \mathfrak{D}$
we again write
$\varphi \colon \Phi_{\mathfrak{C}} \to \Phi_{\mathfrak{D}}$
for the functor
$\vect{F} = (F,\mathfrak{c})
\mapsto (F,\varphi\mathfrak{c}) = \varphi\vect{F}$.
Adapting Notation \ref{SIGMABULL NOT},
we likewise write
$\Phi_{\bullet} \to \mathsf{Set}$
for the Grothendieck constrution of the functor
$\mathfrak{C} \mapsto \Phi_{\mathfrak{C}}$.
Explicitly,
the objects of $\Phi_{\bullet}$
are the $\vect{F} \in \Phi_{\mathfrak{C}}$
for some set of colors $\mathfrak{C}$
and an arrow from
$\vect{F} \in \Phi_{\mathfrak{C}}$ to
$\vect{F'} \in \Phi_{\mathfrak{D}}$
over $\varphi \colon \mathfrak{C} \to \mathfrak{D}$
is an arrow
$\varphi \vect{F} \to \vect{F'}$ in $\Phi_{\mathfrak{D}}$.
\end{definition}
For each vertex $v \in \boldsymbol{V}(F)$ in a forest,
we write $F_v$ for the associated corolla.
Note that, given a $\mathfrak{C}$-coloring $\vect{F}$ on $F$,
one one likewise obtains colorings $\vect{F}_v$ on $F_v$.
\begin{notation}
Given $\vect{F} \in \Phi_{\mathfrak{C}}$
we define
\begin{equation}\label{GENSIGC EQ}
\Sigma_{\mathfrak{C}}[\vect{F}]=
\coprod\nolimits^{\mathfrak{C}}_{v \in \boldsymbol{V}(F)}
\Sigma_{\mathfrak{C}}[\vect{F}_v]
\end{equation}
where we highlight that the coproduct $\amalg^{\mathfrak{C}}$ is fibered, i.e. it takes place in $\mathsf{Sym}_{\mathfrak{C}}$
rather than $\mathsf{Sym}_{\bullet}$.
\end{notation}
\begin{example}\label{COLFORES EX}
Let
$\mathfrak{C} = \{ \mathfrak{a}, \mathfrak{b}, \mathfrak{c} \}$.
On the left below we depict a $\mathfrak{C}$-colored forest
$\vect{F} = \vect{T} \amalg \vect{S}$
with tree components $\vect{T},\vect{S}$.
\begin{equation}
\begin{tikzpicture}[auto,grow=up, level distance = 2.2em,
every node/.style={font=\scriptsize,inner sep = 2pt}
\tikzstyle{level 2}=[sibling distance=3em
\node at (0,0) [font = \normalsize] {$\vect{T}$
child{node [dummy] {
child[level distance = 2.9em]{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{c}$}
edge from parent node [swap,near end] {$\mathfrak{b}$}
child[level distance = 2.9em]{node [dummy] {
child[level distance = 2.9em]{node [dummy] {
edge from parent node [swap, near end] {$\mathfrak{a}\phantom{\mathfrak{b}}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{b}$}
edge from parent node [near end] {$\phantom{\mathfrak{b}}\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (2,0) [font = \normalsize] {$\vect{S}$
child{node [dummy] {
child[level distance = 2.9em]{node [dummy] {
child{node [dummy] {
edge from parent node [swap] {$\mathfrak{c}$}
edge from parent node [swap] {$\mathfrak{b}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (4.5,1.5) [font = \normalsize] {$\vect{T}_1$
child{node [dummy] {
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (6.5,1.5) [font = \normalsize] {$\vect{T}_2$
child{node [dummy] {
child{node {
edge from parent node [swap, near end] {$\mathfrak{a}\phantom{\mathfrak{b}}$}
child{node {
edge from parent node [near end] {$\mathfrak{b}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (8.5,1.5) [font = \normalsize] {$\vect{T}_3$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{c}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\node at (10.5,1.5) [font = \normalsize] {$\vect{T}_4$
child{node [dummy] {
child{node {
edge from parent node [swap, near end] {$\mathfrak{b}$}
child{node {
edge from parent node [near end] {$\phantom{\mathfrak{b}}\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (4.5,-0.7) [font = \normalsize] {$\vect{S}_1$
child{node [dummy] {
edge from parent node [swap] {$\mathfrak{c}$}}
\node at (6.5,-0.7) [font = \normalsize] {$\vect{S}_2$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{c}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\node at (8.5,-0.7) [font = \normalsize] {$\vect{S}_3$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{b}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\draw[decorate,decoration={brace,amplitude=2.5pt}] (2.1,-0.5) -- (-1.1,-0.5)
node[midway,inner sep=4pt,font=\normalsize]{$\vect{F}$};
\end{tikzpicture
\end{equation
Moreover, on the right we depict the $\mathfrak{C}$-signatures/corollas
$\vect{T}_i$ and $\vect{S}_j$
corresponding to the vertices of $T,S$, so that
\[
\Sigma_{\mathfrak{C}}[\vect{F}]
=
\Sigma_{\mathfrak{C}}[\vect{T}]
\amalg^{\mathfrak{C}}
\Sigma_{\mathfrak{C}}[\vect{S}]
=
\left(
\coprod_{1\leq i \leq 4}^{\mathfrak{C}}
\Sigma_{\mathfrak{C}}[\vect{T}_i]
\right)
\amalg^{\mathfrak{C}}
\left(
\coprod_{1\leq j \leq 3}^{\mathfrak{C}}
\Sigma_{\mathfrak{C}}[\vect{S}_j]
\right)
\]
\end{example}
\begin{remark}
Writing $\Phi^o_{\bullet} \hookrightarrow \Phi_{\bullet}$
for the wide subcategory whose arrows are the outer face maps
\cite[\S 3.2]{BP_geo} in each tree component
(these are the maps sending vertices to vertices),
\eqref{GENSIGC EQ} defines a functor
\begin{equation}\label{REPSFUN EQ}
\Phi_{\bullet}^o
\xrightarrow{\Sigma_{\bullet}[-]}
\mathsf{Sym}_{\bullet},
\end{equation}
and, by formula \eqref{GENSIGC EQ},
Proposition \ref{FIBYONPUSH PROP} immediately generalizes, i.e. one has identifications
\begin{equation}\label{PUSHEQAG EQ}
\varphi_! \Sigma_{\mathfrak{C}}[\vect{F}] \simeq
\Sigma_{\mathfrak{D}}[\varphi \vect{F}].
\end{equation}
\end{remark}
\begin{notation}\label{TAUTCOL NOT}
We write $(-)^{\tau} \colon \Phi \to \Phi_{\bullet}$
for the \emph{tautological coloring} functor
which sends $F \in \Phi$ to
$F^{\tau} \in \Phi_{\boldsymbol{E}(T)}$
where
$F^{\tau} = (F,\mathfrak{t})$ is the underlying forest $F$
together with the identity coloring
$\mathfrak{t} \colon \boldsymbol{E}(T) \xrightarrow{=} \boldsymbol{E}(T)$.
Moreover, we then abbreviate
$\Sigma_{\tau}[F] = \Sigma_{\boldsymbol{E}(F)}[F^{\tau}]$.
\end{notation}
\begin{remark}
For any colored forest $\vect{F}=(F,\mathfrak{c})$,
regarding $\mathfrak{c} \colon \boldsymbol{E}(T) \to \mathfrak{C}$
as a change of color map,
one has $\vect{F} = \mathfrak{c} F^{\tau}$,
so that \eqref{PUSHEQAG EQ} yields
\begin{equation}\label{CANPUSH EQ}
\Sigma_{\mathfrak{C}}[\vect{F}] =
\mathfrak{c}_! \Sigma_{\tau}[F]
\end{equation}
\end{remark}
\subsubsection*{Colored operads}
We now describe the category
$\mathsf{Op}_{\bullet}(\V)$
of colored operads as the fiber algebras
(cf. \cite[Def. \ref{OC-FIBMON DEF}]{BP_FCOP};
see Remark \ref{FIBMON REM} below) over a
certain fibered monad $\mathbb{F}$ on
$\mathsf{Sym}_{\bullet}(\V)$,
described using trees.
Following Definition \ref{COLFOR DEF},
we write
$\Omega_{\mathfrak{C}} \subset \Phi_{\mathfrak{C}}$
for the subcategory of $\mathfrak{C}$-colored forests which are trees,
as well as
$\Omega^0_{\mathfrak{C}} \subseteq \Omega_{\mathfrak{C}}$
for the wide subcategory whose arrows are the isomorphisms.
Next, as in \cite[Not. 3.38]{BP_geo},
there is an ``arity functor'',
which we call the \emph{leaf-root functor}, described as follows
\begin{equation}\label{LRDEF EQ}
\begin{tikzcd}[row sep = 0pt]
\Omega_{\mathfrak C}^0
\ar{r}{\mathsf{lr}} &
\Sigma_{\mathfrak{C}}
\\
\vect{T} =
(T,\boldsymbol{E}(T) \xrightarrow{\mathfrak{c}} \mathfrak{C})
\ar[mapsto]{r} &
\left(
\mathfrak{c}(l_1),\cdots,\mathfrak{c}(l_n);\mathfrak{c}(r)
\right)
\end{tikzcd}
\end{equation}
where $r$ is the root of $T$ and
$l_1,\cdots,l_n$ the leaves
(ordered left to right following the planarization).
\begin{example}
For $\vect{T},\vect{S}$ the trees in Example \ref{COLFORES EX}
we have
$\mathsf{lr}(\vect{T}) = (\mathfrak{b},\mathfrak{c};\mathfrak{a})$
and
$\mathsf{lr}(\vect{S}) = (;\mathfrak{a})$.
\end{example}
For each $\mathfrak{C}$-signature $\vect{C}$,
we write $\vect{C} \downarrow \Omega_{\mathfrak{C}}^0$
for the undercategory with respect to $\mathsf{lr}$,
whose objects consist of a tree $\vect{T}\in \Omega_{\mathfrak{C}}^0$
together with a choice of isomorphism
$\vect{C} \to \mathsf{lr}(\vect{T})$.
Morally, $\vect{C} \downarrow \Omega_{\mathfrak{C}}^0$
is the ``groupoid of trees with arity $\vect{C}$.
Adapting \cite[page 816]{BM07} we now have the following.
\begin{definition}\label{FREEOP DEF}
Let $\mathcal{V}$ be a closed symmetric monoidal category.
The \textit{fibered free operad monad} $\mathbb{F}$ on $\mathsf{Sym}_\bullet(\mathcal{V})$
assigns to
$X \colon \Sigma_{\mathfrak{C}}^{op} \to \mathcal{V}$
the functor
\begin{equation}\label{FROPEXP EQ}
\mathbb{F} X (\vect{C})
=
\coprod_{[\vect{T}] \in
\mathsf{Iso}(\vect{C} \downarrow \Omega^0_{\mathfrak{C}})}
\left(
\left(
\bigotimes_{v \in \boldsymbol{V}(T)} X(\vect{T}_v)
\right)
\cdot_{\mathsf{Aut}_{\Omega_{\mathfrak{C}}}(\vect{T})}
\mathsf{Aut}_{\Sigma_{\mathfrak{C}}}(\vect{C})
\right)
\end{equation}
where $\mathsf{Iso}(-)$ denotes isomorphism classes of objects.
\end{definition}
Formula \eqref{FROPEXP EQ}
is presented here only for the sake of completeness,
as this paper will not require a full understanding of
$\mathbb{F}$.
A complete description of the monad
$\mathbb{F}$
is given in the prequel \cite{BP_FCOP},
with \cite[Def. \ref{OC-FREEOP DEF}]{BP_FCOP}
providing an alternative description of
\eqref{FROPEXP EQ},
and the structure maps
$\mathbb{F}\mathbb{F} \Rightarrow \mathbb{F}$,
$id \Rightarrow \mathbb{F}$
discussed in \cite[App. \ref{OC-MONAD_APDX}]{BP_FCOP},
culminating in \cite[Def. \ref{OC-COLORMON_DEF}]{BP_FCOP}.
\begin{remark}\label{FIBMON REM}
Following the construction in \cite[Def. \ref{OC-COLORMON_DEF}]{BP_FCOP},
one has that the monad $\mathbb{F}$
on $\mathsf{Sym}_{\bullet}(\V)$
preserves color sets,
and that the structure maps
$\mathbb{F}\mathbb{F} \Rightarrow \mathbb{F}$,
$id \Rightarrow \mathbb{F}$
are the identity on colors.
In other words,
$\mathbb{F}$ is a \emph{fibered monad} for
$\mathsf{Sym}_{\bullet} \to \mathsf{Set}$
in the sense of
\cite[Def. \ref{OC-FIBMON DEF}]{BP_FCOP}.
In particular,
by restriction one obtains monads
$\mathbb{F}_{\mathfrak{C}}$
on the fixed color fibers
$\mathsf{Sym}_{\mathfrak{C}}(\V)$.
\end{remark}
Recall \cite[Def. \ref{OC-FIBMON DEF}]{BP_FCOP},
that an $\mathbb{F}$ algebra $X$
is called a \emph{fiber algebra}
if the multiplication
$\mathbb{F} X \to X$
is an identify on colors.
\begin{definition}
The category
$\mathsf{Op}_{\bullet}(\V)$
of colored operads
is the category of fiber algebras for
the fibered monad $\mathbb{F}$ on
$\mathsf{Sym}_{\bullet}(\V) \to \mathsf{Set}$.
\end{definition}
\subsection{Equivariant colored symmetric sequences and colored operads}
\label{EQCOSYMSEQ SEC}
We now extend the discussion in the previous section to the equivariant context.
Letting $G$ be a group,
we will write
$\mathsf{Sym}^G_{\bullet}(\V)$,
which we call the category of
\emph{equivariant symmetric sequences},
for the category of $G$-objects in $\mathsf{Sym}_{\bullet}(\V)$.
By abstract nonsense,
the color set functor
$\mathsf{Sym}^G_{\bullet}(\V)
\to \mathsf{Set}^G$
is again a Grothendieck fibration
\cite[Rem. \ref{OC-FUNISGROTH REM}]{BP_FCOP},
and one has a fibered monad
$\mathbb{F}^G$ on $\mathsf{Sym}^G_{\bullet}(\V)$
(explicitly, $\mathbb{F}^G$
is simply $\mathbb{F}$ applied to $G$-objects)
whose fiber algebras
are the category
$\mathsf{Op}^G_{\bullet}(\V)$
of $G$-objects on $\mathsf{Op}_{\bullet}(\V)$
\cite[Prop. \ref{OC-DIAGRAMFM_PROP}]{BP_FCOP}.
As a side note, we observe that,
though $\mathbb{F}^G$ is again described by
\eqref{FROPEXP EQ},
it can be tricky to describe the $G$-actions via that formula,
since those are in general not the identity on colors.
Alternative descriptions can be found in
\cite[Prop. \ref{OC-FGC PROP} and Rem. \ref{OC-FROPEXPG REM}]{BP_FCOP}.
For $\mathfrak{C} \in \mathsf{Set}^G$,
we then write
$\mathsf{Sym}^G_{\mathfrak{C}}(\V)$,
$\mathsf{Op}^G_{\mathfrak{C}}(\V)$
for the associated fibers of
$\mathsf{Sym}^G_{\bullet}(\V)$,
$\mathsf{Op}^G_{\bullet}(\V)$.
Extending \eqref{COLCHADJ EQ},
we then have the following,
cf. \cite[Rem. \ref{OC-OP_MAP REM}]{BP_FCOP}.
\begin{remark}\label{OP_MAP REM}
For any map of $G$-sets
$\varphi \colon \mathfrak C \to \mathfrak D$
one has a pair of adjunctions
\begin{equation}\label{GC_CHANGE_EQ}
\begin{tikzcd}
\Op^G_{\mathfrak C}(\V)
\arrow[shift left]{r}{\check{\varphi}_!}
\arrow[d, "\mathsf{fgt}"']
&
\Op^G_{\mathfrak D}(\V)
\arrow[shift left]{l}{\varphi^{\**}}
\arrow[d, "\mathsf{fgt}"]
\\
\Sym^G_{\mathfrak C}(\V)
\arrow[shift left]{r}{\varphi_!}
&
\Sym^G_{\mathfrak D}(\V)
\arrow[shift left]{l}{\varphi^{\**}}
\end{tikzcd}
\end{equation}
where the right adjoints $\varphi^{\**}$ are
both given by precomposition with
$\varphi \colon
\Sigma_{\mathfrak{C}} \to \Sigma_{\mathfrak{D}}$,
and are thus compatible with the forgetful functors,
i.e. $\varphi^{\**} \circ \mathsf{fgt} =
\mathsf{fgt} \circ \varphi^{\**}$,
while the left adjoints are not:
$\varphi_!$ is simply a left Kan extension, while $\check{\varphi}_!$ is given by the coequalizer
\begin{equation}\label{CFS_EQ}
\check{\varphi}_! \O \simeq \mathop{\mathrm{coeq}}(\mathbb F_{\mathfrak D} \varphi_! \mathbb F_{\mathfrak C}\O \rightrightarrows \mathbb F_{\mathfrak D} \varphi_! \O).
\end{equation}
In general, we do not have a more explicit description of $\check{\varphi}_!$.
However, when $\varphi$ is injective,
$\varphi_!X$ is the extension by $\emptyset$,
from which it follows that
$\mathbb F_{\mathfrak D} \varphi_! = \varphi_! \mathbb F_{\mathfrak C}$,
and \eqref{CFS_EQ}
then says that
$\check{\varphi}_! \O
\simeq
\mathop{\mathrm{coeq}} \left( \varphi_! \mathbb{F}_{\mathfrak{C}} \mathbb{F}_{\mathfrak{C}} \O
\rightrightarrows
\varphi_! \mathbb{F}_{\mathfrak{C}} \O
\right)
\simeq
\varphi_! \left( \mathop{\mathrm{coeq}} \left(
\mathbb{F}_{\mathfrak{C}} \mathbb{F}_{\mathfrak{C}} \O
\rightrightarrows
\mathbb{F}_{\mathfrak{C}} \O
\right) \right)
\simeq
\varphi_! \O$,
so that
$\varphi_! \circ \mathsf{fgt} \simeq
\mathsf{fgt} \circ \check{\varphi}_!$.
\end{remark}
In \S \ref{GENCOF SEC} we will make use of the following,
which is an instance of
\cite[Rem. \ref{OC-LIMINFIBSUP REM}]{BP_FCOP}.
\begin{remark}
Given a diagram
$I \xrightarrow{X_{\bullet}}
\mathsf{Sym}_{\bullet}^G(\V)$,
and writing
$\mathfrak{C} = \colim_{i \in I} \mathfrak{C}_{X_i}$
and
$\varphi_i \colon \mathfrak{C}_{X_i} \to \mathfrak{C}$
for the canonical maps,
one has
$
\colim_{i \in I} X_i =
\colim_{i \in I} \varphi_{i,!} X_i,
$
where the second colimit is computed in
$\mathsf{Sym}_{\mathfrak{C}}^G(\V)$.
Thus, for an arbitrary cocone
$\psi_i \colon \mathfrak{C}_{X_i} \to \mathfrak{D}$
with
$\psi \colon \mathfrak{C} \to \mathfrak{D}$ the induced map, one has
\begin{equation}\label{LIMINFIBSUP EQ}
\psi_!\left(\colim_{i \in I} X_i\right)
=
\psi_!
\left(\colim_{i \in I} \varphi_{i,!} X_i\right)
=
\colim_{i \in I} \psi_! \varphi_{i,!} X_i
=
\colim_{i \in I} \psi_{i,!} X_i.
\end{equation}
\end{remark}
To recall the model structures on
$\mathsf{Sym}^G_{\mathfrak{C}}(\V)$,
$\mathsf{Op}^G_{\mathfrak{C}}(\V)$
in \S \ref{COLFIXMOD SEC}
we will need a more explicit description of
$\mathsf{Sym}^G_{\mathfrak{C}}(\V)$
(the discussion above
only describes $\mathsf{Sym}^G_{\mathfrak{C}}(\V)$
abstractly as a fiber of $\mathsf{Sym}^G_{\bullet}(\V)$;
note that this
\emph{is not} the category
of $G$-objects on $\mathsf{Sym}_{\mathfrak{C}}(\V)$
unless the $G$-action on $\mathfrak{C}$
is trivial).
By \cite[Prop. \ref{OC-EQUIVFNCON PROP}]{BP_FCOP},
there is an identification
$\mathsf{Sym}^G_{\mathfrak{C}}(\mathcal{V})
\simeq
\V^{G \ltimes \Sigma^{op}_{\mathfrak{C}}}$
as the functors from a certain groupoid
$G \ltimes \Sigma^{op}_{\mathfrak{C}}$,
which can be described as an instance of
\cite[Ex. \ref{OC-GLTIMES EQ}]{BP_FCOP}.
Here we prefer an alternative description
of $G \ltimes \Sigma^{op}_{\mathfrak{C}}$,
which follows from
\cite[Rem. \ref{OC-SIGACT REM}]{BP_FCOP},
and adapts Definition \ref{CSYM DEF} and
Remark \ref{GLOBSIG REM}.
\begin{remark}\label{GLTIMESSIG REM}
Let $\mathfrak {C} \in \mathsf{Set}^G$ be a fixed $G$-set of colors.
The groupoid $G \ltimes \Sigma^{op}_{\mathfrak{C}}$
has objects the $\mathfrak {C}$-signatures
$\vect C = (\mathfrak c_1, \dots, \mathfrak c_n; \mathfrak c_0)$
and morphisms the action maps
\begin{equation}\label{CSYMG EQ}
\vect{C} =
(\mathfrak c_1, \dots, \mathfrak c_n; \mathfrak c_0) \xrightarrow{(g,\sigma)}
(g \mathfrak c_{\sigma(1)}, \dots,
g \mathfrak c_{\sigma(n)};
g \mathfrak c_0)
= g \vect{C} \sigma
\end{equation}
for $(g,\sigma) \in G \times \Sigma_n^{op}$,
with the natural notion of composition.
\end{remark}
\begin{remark}
Setting $G=\**$ in Remark \ref{GLTIMESSIG REM}
recovers the \emph{opposite}
$\Sigma_{\mathfrak{C}}^{op}$ of
Definition \ref{CSYM DEF}.
\end{remark}
\begin{remark}\label{SIGACT REM}
Extending Remark \ref{GLOBSIG REM},
the notation $g \vect{C} \sigma$ in \eqref{CSYMG EQ}
encodes a $(G \times \Sigma_n^{op})$-action
(i.e. $G$ acts on the left and $\Sigma_n$ on the right)
on the set of $n$-ary $\mathfrak{C}$-signatures,
via
$g(\mathfrak{c}_i)\sigma =
(g\mathfrak{c}_{\sigma(i)})$.
\end{remark}
\begin{definition}\label{STABS DEF}
If a subgroup $\Lambda \leq G \times \Sigma_n^{op}$
fixes a signature
$\vect{C} = (\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)$,
i.e. if
$g\mathfrak{c}_{\sigma(i)} = \mathfrak{c}_i$
for all $(g, \sigma) \in \Lambda, 0 \leq i \leq n$,
we say that \textit{$\Lambda$ stabilizes $\vect C$}.
\end{definition}
\begin{remark}\label{CHOOSESIGN REM}
For $\Lambda \leq G \times \Sigma_n^{op}$
the projection to $\Sigma_n^{op}$ yields a
right action of $\Lambda$ on
$\underline{n}_+ = \{0,1,\cdots,n\}$.
Writing $\Lambda_i\leq \Lambda$ for the stabilizer of $i \in \underline{n}_{+}$ and $H_i = \pi_G(\Lambda_i)$
for its projection onto $G$,
one then has $H_i = g H_{\sigma(i)} g^{-1}$ for all
$(g, \sigma) \in \Lambda$.
Moreover, the signatures $\vect{C}$
stabilized by $\Lambda$ are in bijection with choices of
$H_i$-fixed colors $\mathfrak{c}_i$
for $i$ ranging over a set of representatives of
the orbits $\underline{n}_+ /\Lambda$.
\end{remark}
We now discuss the representable functors in
$\mathsf{Sym}^G_{\mathfrak{C}} =
\mathsf{Sym}^G_{\mathfrak{C}}(\mathsf{Set}) \simeq
\mathsf{Set}^{G \ltimes \Sigma^{op}_{\mathfrak{C}}}$.
However, some caution is needed,
as though $G \ltimes \Sigma^{op}_{\mathfrak{C}}$
and $\Sigma^{op}_{\mathfrak{C}}$
have the same objects $\vect{C}$,
the representable functor for $\vect{C}$ in
$\mathsf{Sym}^G_{\mathfrak{C}}$
is \emph{not} simply
$\Sigma_{\mathfrak{C}}[\vect{C}]
\in \mathsf{Sym}_{\mathfrak{C}}$.
We first need the following construction,
where
$(-)^{\tau}\colon \Phi \to \Phi_{\bullet}$
is the tautological coloring,
cf. Notation \ref{TAUTCOL NOT}.
\begin{definition}
Let $G$ be a group,
$\mathfrak{C} \in \mathsf{Set}^G$
be a $G$-set of colors,
and $\vect{C} \in \Sigma_{\mathfrak{C}}$ be a $\mathfrak{C}$-signature/corolla.
Write
$\vect{C} = (C,\mathfrak{c})$
with $C\in \Sigma$ the underlying corolla
and
$\mathfrak{c} \colon \boldsymbol{E}(T) \to \mathfrak{C}$.
Writing
$G \cdot \mathfrak{c} \colon G \cdot \boldsymbol{E}(T) \to \mathfrak{C}$
for the adjoint map,
$G \cdot C \in \Phi^G$
for the $G$-free forest determined by $C$,
and noting that
$\boldsymbol{E}(G \cdot C) \simeq
G \cdot \boldsymbol{E}(C)$,
we define $G \cdot_{\mathfrak{C}} \vect{T} \in \Phi^G_{\mathfrak{C}}$ by
\begin{equation}\label{GCDTCC EQ}
G \cdot_{\mathfrak{C}} \vect{C} =
(G \cdot \mathfrak{c})(G \cdot C)^{\tau}.
\end{equation}
\end{definition}
\begin{remark}
Writing $g \colon \mathfrak{C} \to \mathfrak{C}$
for the $G$-action maps,
one has the more explicit formula
(see Example \ref{GDOTCC EX})
\[
G \cdot_{\mathfrak{C}} \vect{C}
=
\coprod_{g \in G}
g \vect{C}
\]
However, in practice we will prefer to use
\eqref{GCDTCC EQ} for technical purposes.
\end{remark}
\begin{remark}
\eqref{GCDTCC EQ}
extends to a functor
$\Phi_{\mathfrak{C}}
\xrightarrow{G \cdot_{\mathfrak{C}} (-)}
\Phi_{\mathfrak{C}}^G$,
left adjoint to the forgetful functor
$ \Phi_{\mathfrak{C}}^G
\to \Phi_{\mathfrak{C}}$.
\end{remark}
\begin{example}\label{GDOTCC EX}
Let $G = \{1,i,-1,-i\} \simeq \mathbb{Z}_{/4}$
be the group of quartic roots of unit and
$\mathfrak{C} = \{\mathfrak{a}, -\mathfrak{a}, i\mathfrak{a}, -i\mathfrak{a}, \mathfrak{b}, i \mathfrak{b} \}$ where we implicitly have
$-\mathfrak{b} = \mathfrak{b}$.
The following depicts the forest (of corollas)
$G \cdot_{\mathfrak{C}} \vect{C}$
in $\Phi_{\mathfrak{C}}^G$
for $\vect{C}$ in $\Sigma_{\mathfrak{C}}$ the leftmost corolla.
\begin{equation}
\begin{tikzpicture}[auto,grow=up, level distance = 2.2em,
every node/.style={font=\scriptsize,inner sep = 2pt}
\tikzstyle{level 2}=[sibling distance=3em
\node at (0,0) [font = \normalsize] {$\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$i\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$i\mathfrak{b}$}
child{node {
edge from parent node {$\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\node at (3.5,0) [font = \normalsize] {$i\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-i\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{b}$}
child{node {
edge from parent node {$i\mathfrak{a}$}
edge from parent node [swap] {$i\mathfrak{b}$}}
\node at (7,0) [font = \normalsize] {$-\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$i\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$i\mathfrak{b}$}
child{node {
edge from parent node {$-\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\node at (10.5,0) [font = \normalsize] {$-i\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$i\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{b}$}
child{node {
edge from parent node {$-i\mathfrak{a}$}
edge from parent node [swap] {$i\mathfrak{b}$}}
\draw[decorate,decoration={brace,amplitude=2.5pt}] (11.2,-0.4) -- (-0.7,-0.4)
node[midway,inner sep=4pt,font=\normalsize]{$G \cdot_{\mathfrak{C}} \vect{C}$};
\end{tikzpicture
\end{equation
Note that the pairs $\vect{C},-\vect{C}$
and $i\vect{C},-i\vect{C}$ are isomorphic in $\Sigma_{\mathfrak{C}}$
while any other pair,
such as $\vect{C},i\vect{C}$, is not.
In general, it is moreover possible for two or more tree components of
$G \cdot_{\mathfrak{C}} \vect{C}$ to be equal.
\end{example}
Applying \eqref{REPSFUN EQ} to $G$-objects,
\cite[Prop. \ref{OC-REPALTDESC PROP}]{BP_FCOP}
gives, for each $\mathfrak{C}$-signature $\vect{C}$,
an identification
\begin{equation}\label{REPALTDESC EQ}
(G \ltimes \Sigma^{op}_{\mathfrak{C}})(\vect{C},-)
\simeq
\Sigma_{\mathfrak{C}} [G \cdot_{\mathfrak{C}} \vect{C}].
\end{equation}
In other words,
$\Sigma_{\mathfrak{C}} [G \cdot_{\mathfrak{C}} \vect{C}]$
is the representable functor for $\vect{C}$ in
$\mathsf{Sym}^G_{\mathfrak{C}} \simeq
\mathsf{Set}^{G \ltimes \Sigma^{op}_{\mathfrak{C}}}$.
\begin{remark}\label{GCDOTCATS REM}
If $C \in \Sigma$ is the $n$-corolla,
one has a natural identification
$\boldsymbol{E}(G\cdot C) = G \times \underline{n}_+$
where
$\underline{n}_+ = \{0,1,\cdots,n\}$.
The automorphisms of
$G \cdot C$ in $\Phi^G$
are then naturally identified with the group
$G^{op} \times \Sigma_n$,
with the automorphism
${(g,\sigma)} \colon
G \cdot C \to G \cdot C$
given on edges by
$(\bar{g},i) \mapsto (\bar{g}g,\sigma(i))$.
\end{remark}
\begin{remark}\label{COLCHSQ REM}
Let $g\vect{C} \sigma = \vect{C'}$
be as in \eqref{CSYMG EQ}.
Then $\vect{C},\vect{C'}$
have the same underlying corolla $C$ and,
writing $\mathfrak{c},\mathfrak{c}'\colon
\boldsymbol{E}(C)=\underline{n}_+ \to \mathfrak{C}$
for the colorings,
one can rewrite $g\vect{C} \sigma = \vect{C'}$
as
$g \mathfrak{c} \sigma = \mathfrak{c}'$.
\eqref{GCDTCC EQ} then induces a diagram in $\Phi^G_{\bullet}$
as below, with the vertical maps given by
$G \cdot C \xrightarrow{(g,\sigma)} G \cdot C$
on the underlying forest,
\begin{equation}\label{COLCHSQ EQ}
\begin{tikzcd}
G \cdot C^{\tau} \ar{d}[swap]{(g,\sigma)}
\ar{r}{G \cdot \mathfrak{c}'}
&
G \cdot_{\mathfrak{C}} \vect{C'}
\ar{d}{(g,\sigma)}
\\
G \cdot C^{\tau} \ar{r}[swap]{G \cdot \mathfrak{c}}
&
G \cdot_{\mathfrak{C}} \vect{C}
\end{tikzcd}
\end{equation}
and where the right vertical map
is in $\Phi^G_{\mathfrak{C}}$,
i.e. it respects colors.
Note that this reflects \eqref{REPALTDESC EQ},
which identifies a map $\vect{C} \to \vect{C'}$ in
$G \ltimes \Sigma_{\mathfrak{C}}^{op}$
with a map
$\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C'}]
\to
\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]$
in $\mathsf{Sym}^G_{\mathfrak{C}}$.
\end{remark}
\begin{example}
In Example \ref{GDOTCC EX}
the permutation $(14)(23) \in \Sigma_4$
gives a map $\vect{C} \to -\vect{C}$ in $\Sigma_{\mathfrak{C}}$,
and thus induces an automorphism of
$\vect{C}$ in $G \ltimes \Sigma_{\mathfrak{C}}^{op}$.
\end{example}
\subsection{Homotopy theory of equivariant operads with fixed colors}\label{COLFIXMOD SEC}
In this section we recall the model structures on fixed color operads
$\mathsf{Sym}^G_{\mathfrak{C}}(\V)$
in
\cite[Thm. \ref{OC-THMI}]{BP_FCOP},
which was the main result therein.
We first recall and elaborate on the $(G,\Sigma)$-families in Definition \ref{FAM1ST DEF}.
\begin{definition}\label{GSFAM_DEF}
A \emph{$(G,\Sigma)$-family} $\mathcal{F}$ is
a collection $\{\mathcal{F}_n\}_{n \geq 0}$ of families $\F_n$ of the groups
$G \times \Sigma_n^{op}$.
Further, for $G \ltimes \Sigma^{op}_{\mathfrak{C}}$
and $n$-ary $\mathfrak{C}$-signature $\vect{C}$,
we write
\begin{equation}\label{FVECTC EQ}
\mathcal{F}_{\vect{C}} =
\{\Lambda \in \mathcal{F}_n \ | \ \Lambda \text{ stabilizes } \vect{C}\}.
\end{equation}
\end{definition}
\begin{remark}
The collection $\mathcal{F} = \{\mathcal{F}_n\}_{n \geq 0}$
can also be viewed as a family of subgroups
in the groupoid $G \times \Sigma^{op}$,
cf. \cite[Def. \ref{OC-FAMGROUPOID DEF}]{BP_FCOP}.
Similarly,
by \eqref{CSYMG EQ},
the subgroups in $\mathcal{F}_{\vect{C}}$
can be regarded as automorphisms of $\vect{C}$
in $G \ltimes \Sigma^{op}_{\mathfrak{C}}$,
so that
$\mathcal{F}_{\mathfrak{C}}
=\{ \mathcal{F}_{\vect{C}}\}_{\vect{C} \in \Sigma_{\mathfrak{C}}}$
similarly defines a family in the groupoid
$G \ltimes \Sigma^{op}_{\mathfrak{C}}$.
For further discussion,
see \cite[Def. \ref{OC-GSFAM_DEF} and Rem. \ref{OC-FAMC_DEF_EQ}]{BP_FCOP}.
\end{remark}
In this paper and the sequel \cite{BP_TAS},
we are interested in three main examples of $(G,\Sigma)$-families:
\begin{enumerate}[label = (\alph*)]
\item First, there is the family $\F_{all}$ of all the subgroups of $G \times \Sigma^{op}$
(in which case the $\F_{all,\mathfrak{C}}$ are also the families of all subgroups), which is useful mainly for technical purposes.
\item Secondly, there is the family of $\F^{\Gamma}$
of $G$-graph subgroups (e.g. \cite[Def. 6.36]{BP_geo}),
where $\F^{\Gamma}_n$ consists of the subgroups
$\Gamma \leq G \times \Sigma_n^{op}$
such that $\Gamma \cap \Sigma_n^{op} = \{\**\}$.
We note that the elements of such $\Gamma$
have the form $(h,\phi(h)^{-1})$
for $h$ ranging over some subgroup $H \leq G$
and $\phi \colon H \to \Sigma_n$
a homomorphism,
motivating the ``graph subgroup'' terminology.
Though secondary for the current paper,
we regard $\F^{\Gamma}$ as the ``canonical choice'' of
$(G,\Sigma)$-family,
as it is the family featured in the Quillen equivalence
$W_! \colon
\mathsf{dSet}^G \rightleftarrows
\mathsf{dSet}^G \colon hcN$
in \cite[Thm. I]{BP_TAS}
(see \eqref{BPMAINTHM_EQ}).
\item Lastly, there are the indexing systems of Blumberg and Hill,
which are special subfamilies of $\F^{\Gamma}$
which share the key technical properties of
$\F^{\Gamma}$ itself,
and are discussed in
\cite[\S \ref{OC-INDSYS_SEC}]{BP_FCOP}.
\end{enumerate}
\begin{example}
Let $G = \mathbb{Z}_{/2} = \{\pm 1\}$ and
$\mathfrak{C} = \{\mathfrak{a}, -\mathfrak{a}, \mathfrak{b}\}$ where we implicitly have
$-\mathfrak{b} = \mathfrak{b}$.
Consider the two $\mathfrak{C}$-corollas
$\vect{C},\vect{D} \in \Sigma_{\mathfrak{C}}$ below.
\begin{equation}
\begin{tikzpicture}[auto,grow=up, level distance = 2.2em,
every node/.style={font=\scriptsize,inner sep = 2pt}
\tikzstyle{level 2}=[sibling distance=3em
\node at (0,0) [font = \normalsize] {$\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{b}$}
child{node {
edge from parent node {$\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\node at (7,0) [font = \normalsize] {$\vect{D}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$-\mathfrak{a}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{a}$}
child{node {
edge from parent node {$\mathfrak{a}$}
edge from parent node [swap] {$\mathfrak{b}$}}
\end{tikzpicture
\end{equation
The non-trivial $G$-graph subgroups of
$\F^{\Gamma}_{\vect{C}}$,
$\F^{\Gamma}_{\vect{D}}$
correspond to the possible $\mathbb{Z}_{/2}$-actions on the underlying trees $C,D$ that are compatible with the action on labels
(in that the composites
$\boldsymbol{E}(C) \xrightarrow{-1} \boldsymbol{E}(C) \to \mathfrak{C}$
and
$\boldsymbol{E}(C) \to \mathfrak{C} \xrightarrow{-1} \mathfrak{C}$ coincide).
In this case, both
$\F^{\Gamma}_{\vect{C}}$,
$\F^{\Gamma}_{\vect{D}}$
have exactly two non-trivial groups,
corresponding to the $\mathbb{Z}_{/2}$-actions on the underlying corollas depicted below.
\begin{equation}
\begin{tikzpicture}[auto,grow=up, level distance = 2.2em,
every node/.style={font=\scriptsize,inner sep = 2pt}
\tikzstyle{level 2}=[sibling distance=3em
\node at (-1.6,0) [font = \normalsize] {$C_1$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-a$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$c\phantom{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$b$}
child{node {
edge from parent node {$a$}
edge from parent node [swap] {$r$}}
\node at (1.6,0) [font = \normalsize] {$C_2$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-a$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$-b$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$b$}
child{node {
edge from parent node {$a$}
edge from parent node [swap] {$r$}}
\node at (5.4,0) [font = \normalsize] {$D_1$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-a$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$-b$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$b$}
child{node {
edge from parent node {$a$}
edge from parent node [swap] {$r$}}
\node at (8.6,0) [font = \normalsize] {$D_2$
child{node [dummy] {
child{node {
edge from parent node [swap] {$-b$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$-a$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$b$}
child{node {
edge from parent node {$a$}
edge from parent node [swap] {$r$}}
\end{tikzpicture
\end{equation
\end{example}
As discussed in
\cite[Def. \ref{OC-SYMGFV DEF} and Rem. \ref{OC-VGSIGF REM}]{BP_FCOP},
we then have the following instance of
\cite[Prop. \ref{OC-ALLEQ PROP}]{BP_FCOP},
where
$\mathcal{I}$ (resp. $\mathcal{J}$)
denotes the generating (trivial) cofibrations of $\V$.
\begin{proposition}\label{SYMGFV PROP}
Let $\V$ satisfy
(i),(ii) in Theorem \ref{THMA}.
Fix $\mathfrak{C} \in \mathsf{Set}^G$
and $(G,\Sigma)$-family $\F$.
Then there exists a model structure on
$\mathsf{Sym}^G_{\mathfrak{C}}(\mathsf{\V})$,
which we call the \emph{$\mathcal{F}$-model structure}
and denote $\mathsf{Sym}^G_{\mathfrak{C},\F}(\V)$,
such that a map $X \to Y$
is a weak equivalence (resp. fibration) if the maps
\begin{equation}
X(\vect{C})^{\Lambda} \to Y(\vect{C})^{\Lambda}
\end{equation}
are weak equivalences (fibrations) in $\V$
for all $\mathfrak{C}$-signatures $\vect{C}$
and $\Lambda \in \F_{\vect{C}}$.
Moreover, the generating (trivial) cofibrations of
$\Sym^{G}_{\mathfrak{C},\F}(\V)$
are the sets of maps
\begin{equation}\label{VGSIGF EQ}
\mathcal{I}_{\mathfrak{C},\mathcal{F}}
=
\left\{
\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda \cdot i
\right\}
\qquad \qquad
\mathcal{J}_{\mathfrak{C},\mathcal{F}}
=
\left\{
\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda \cdot j
\right\}
\end{equation}
where $\vect{C}$ ranges over $\Sigma_{\mathfrak{C}}$,
$\Lambda$ ranges over $\F_{\vect{C}}$,
$i$ ranges over $\mathcal{I}$ and
$j$ ranges over $\mathcal{J}$.
\end{proposition}
Transfer along the adjunction
$
\mathbb{F}^G_{\mathfrak{C}} \colon
\mathsf{Sym}^G_{\mathfrak{C}}(\V)
\rightleftarrows
\mathsf{Op}^G_{\mathfrak{C}}(\V)
\colon \mathsf{fgt}
$
then yields the following.
\begin{theorem}
[{\cite[Thm. \ref{OC-THMI}]{BP_FCOP}}]
\label{THMIREST}
Let $\V$ satisfy (i),(ii),(iii),(iv),(v) in Theorem \ref{THMA}.
Fix $\mathfrak{C} \in \mathsf{Set}^G$
and $(G,\Sigma)$-family $\F$.
Then there exists a model structure on
$\mathsf{Op}^G_{\mathfrak{C}}(\mathsf{\V})$,
which we call the \emph{$\mathcal{F}$-model structure}
and denote $\mathsf{Op}^G_{\mathfrak{C},\F}(\V)$,
such that a map
$\mathcal{O} \to \mathcal{P}$
is a weak equivalence (resp. fibration) if the maps
\begin{equation}\label{THMI_EQ}
\O(\vect{C})^{\Lambda} \to \mathcal{P}(\vect{C})^{\Lambda}
\end{equation}
are weak equivalences (fibrations)
in $\V$ for all $\mathfrak{C}$-signatures $\vect{C}$
and $\Lambda \in \F_{\vect{C}}$.
Moreover, the generating (trivial) cofibrations in
$\mathsf{Op}^G_{\mathfrak{C},\F}(\V)$
are the sets
\begin{equation}\label{FVGSIGF EQ}
\mathbb{F}^G_{\mathfrak{C}}\mathcal{I}_{\mathfrak{C},\mathcal{F}}
=
\left\{
\mathbb{F}^G_{\mathfrak{C}}
\left(\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda \cdot i \right)
\right\}
\qquad \qquad
\mathbb{F}^G_{\mathfrak{C}}\mathcal{I}_{\mathfrak{C},\mathcal{F}}
=
\left\{
\mathbb{F}^G_{\mathfrak{C}}
\left(\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda \cdot j \right)
\right\}
\end{equation}
where $\vect{C}$ ranges over $\Sigma_{\mathfrak{C}}$,
$\Lambda$ ranges over $\F_{\vect{C}}$,
$i$ ranges over $\mathcal{I}$,
and $j$ ranges over $\mathcal{J}$.
\end{theorem}
\begin{remark}\label{FALLMAXMIN REM}
When $\mathcal{F}=\mathcal{F}_{all}$ is the family of all subgroups,
we refer to these model structures on
$\mathsf{Sym}^G_{\mathfrak{C}}(\V)$, $\mathsf{Op}^G_{\mathfrak{C}}(\V)$
as the \emph{genuine model structures}.
Further, note that the genuine model structures
minimize the classes of weak equivalences and fibrations
and thus, conversely,
they maximize the classes of cofibrations and trivial cofibrations.
\end{remark}
We now recall the following,
where the $\phi^{\**} \bar{\F}$ families
are as defined in \cite[Rem. \ref{OC-PULLFAM REM}]{BP_FCOP}.
\begin{corollary}[{\cite[Cor. \ref{OC-OPADJ_COR}]{BP_FCOP}}]
\label{OPADJ_COR}
\begin{enumerate}[label=(\roman*)]
\item \label{OPCOCHADJ_LBL}
For any $(G,\Sigma)$-family $\F$ and map of colors
$\varphi \colon \mathfrak C \to \mathfrak D$, the induced adjunction
\[
\check{\varphi}_! \colon \mathsf{Op}^G_{\mathfrak{C},\F}
\rightleftarrows
\mathsf{Op}^G_{\mathfrak{D},\F} \colon \varphi^{\**}
\]
is a Quillen adjunction.
\item \label{OPFIXSETCHGR_LBL}
For any homomorphism $\phi \colon G \to \bar G$,
$(G,\Sigma)$-family $\F$ and $(\bar G,\Sigma)$-family $\bar{\F}$,
and $\bar G$-set of colors $\mathfrak C$,
the adjunction
\[
\check{\phi}_! \colon \mathsf{Op}^G_{\mathfrak{C},\F}
\rightleftarrows
\mathsf{Op}^{\bar{G}}_{\mathfrak{C},\bar{\F}} \colon \phi^{\**}
\]
is a Quillen adjunction whenever $\F \subseteq \phi^{\**} \bar{\F}$, i.e.
if $\Lambda \in \F_n$ implies
$\phi(\Lambda) \in \bar{\F}_n$.
\item \label{OPCOMBADJ_LBL}
For any homomorphism $\phi \colon G \to \bar G$,
$(G,\Sigma)$-family $\F$ and $(\bar G,\Sigma)$-family $\bar{\F}$,
and $G$-set of colors $\mathfrak C$,
the adjunction
\[
\bar{G} \cdot_G (-) \colon \mathsf{Op}^G_{\mathfrak{C},\F}
\rightleftarrows
\mathsf{Op}^{\bar{G}}_{\bar{G} \cdot_G \mathfrak{C},\bar{\F}} \colon \mathsf{fgt}
\]
is a Quillen adjunction whenever $\F \subseteq \phi^{\**} \bar{\F}$, i.e.
if $\Lambda \in \F_n$ implies
$\phi(\Lambda) \in \bar{\F}_n$.
\end{enumerate}
\end{corollary}
The proof of Theorem \ref{THMA}
(cf. Proposition \ref{J_CELL_PROP})
will use an additional class of maps in
$\mathsf{Sym}^G_{\mathfrak C}(\V)$.
\begin{definition}[{\cite[Def. \ref{OC-GGENOTITC DEF}]{BP_FCOP}}]
\label{GGENOTITC DEF}
We write
$\mathcal{J}^{\otimes}_{\mathfrak{C}}=
\{j \otimes X | j \in \mathcal{J}, X \in \mathsf{Sym}^G_{\mathfrak C}(\V)\}$,
and refer to the saturation
$\mathcal{J}^{\otimes}_{\mathfrak{C}}$-cof
as the \emph{genuine $\otimes$-trivial cofibrations}
in $\mathsf{Sym}^G_{\mathfrak C}(\V)$.
\end{definition}
\begin{remark}[{\cite[Rem. \ref{OC-GOTC_REM}]{BP_FCOP}}]
\label{GOTC_REM}
$\F$-trivial cofibrations in $\Op^G_{\mathfrak C}(\V)$ are underlying genuine $\otimes$-trivial cofibrations
in $\mathsf{Sym}^G_{\mathfrak C}(\V)$.
\end{remark}
\begin{proposition}\label{GOTC_PROP}
\begin{itemize}
\item [(i)]
If $\V$ satisfies the global monoid axiom
((iv) in Theorem \ref{THMA}),
genuine $\otimes$-trivial cofibrations
in $\mathsf{Sym}^G_{\mathfrak C}(\V)$
are genuine weak equivalences.
\item[(ii)]
The functors
$\varphi_! \colon
\mathsf{Sym}^G_{\mathfrak C}(\V)
\to
\mathsf{Sym}^G_{\mathfrak D}(\V)$
and
$\varphi^{\**} \colon
\mathsf{Sym}^G_{\mathfrak D}(\V)
\to
\mathsf{Sym}^G_{\mathfrak C}(\V)$
preserve genuine $\otimes$-trivial cofibrations
for any color map
$\varphi \colon \mathfrak{C} \to \mathfrak{D}$.
\end{itemize}
\end{proposition}
\begin{proof}
(i) just restates the global monoid axiom
\cite[Def. \ref{OC-GLOBMONAX_DEF}]{BP_FCOP} in light of the observation that genuine $\otimes$-trivial cofibrations/genuine weak equivalences
are characterized levelwise (see
\cite[Rem. \ref{OC-SIGMACOF_REM}, Def. \ref{OC-GGENOTITC DEF}]{BP_FCOP}).
Part (ii) is \cite[Prop. \ref{OC-REGEOTCOF PROP}]{BP_FCOP}.
\end{proof}
\begin{remark}
In practice,
the properties of genuine $\otimes$-trivial cofibrations
given above
serve a similar function to the compact generation condition
in \cite[Def. 1.2]{BM13}.
\end{remark}
\section{Model structures on all equivariant colored operads
}\label{MS_SEC}
\renewcommand{\C}{\mathfrak C}
Theorem \ref{THMIREST} provides,
for each $(G,\Sigma)$-family $\F$,
a model structure on each category
$\mathsf{Op}_{\mathfrak{C}}^G(\V)$
of $G$-equivariant operads with a fixed $G$-set of colors $\mathfrak{C}$.
Adapting \cite{BM13,Cav,CM13b},
our main goal in this section is
to prove our main result, Theorem \ref{THMA},
which uses the model structures of Theorem \ref{THMIREST}
to build,
for suitable $(G,\Sigma)$-families $\F$
(see Definition \ref{FAMRESUNI DEF}),
a model structure on the full category $\mathsf{Op}^G_\bullet(\mathcal{V})$
of $G$-equivariant operads with varying $G$-sets of colors.
As stated in the formulation of Theorem \ref{THMA},
the weak equivalences and trivial fibrations
in $\mathsf{Op}^G_\bullet(\V)$
are described by combining
a ``local condition'' which involves the fixed color categories $\mathsf{Op}_{\mathfrak{C}}^G(\V)$,
as in \eqref{THMIII1ST EQ},
with a form of ``surjectivity on objects'',
as in \eqref{THMIII2ND EQ}.
However, the essential surjectivity condition for
weak equivalences in \eqref{THMIII2ND EQ}
has a key technical drawback:
when using this condition
it is unclear how to select a generating set of trivial cofibrations
for $\mathsf{Op}^G_\bullet(\V)$.
For this reason,
throughout the bulk of this section we will actually work with
an alternate (and a priori distinct)
notion of weak equivalence,
defined using a more abstract notion of essential surjectivity,
which will also allow us to characterize the fibrations (Definition \ref{MODEL_DEFN}).
This model structure will be built in sections \ref{MAPSOPG_SEC} through \ref{HMTYEQ SEC}.
\S \ref{MAPSOPG_SEC} introduces the relevant classes of maps of operads,
\S \ref{GENCOF SEC} produces a generating set of (trivial) cofibrations in Definition \ref{OPGENCOF DEF},
and \S \ref{TRIVCOF_SEC} proves in Proposition \ref{J_CELL_PROP} that trivial cofibrations are in fact weak equivalences.
Sections \ref{EQUIVOBJ_SEC} and \ref{HMTYEQ SEC} explore several notions of essential surjectivity in order to prove 2-out-of-3
(Proposition \ref{2OUTOF3 PROP})
and show that the weak equivalences in Definition \ref{MODEL_DEFN}
and Theorem \ref{THMA} indeed match
(Corollary \ref{WEDKEQ COR}).
Lastly, \S \ref{ISOFIB_SEC}
gives (under mild conditions) a more familiar description of fibrations in $\mathsf{Op}^G_\bullet(\V)$.
\subsection{Classes of maps in $\mathsf{Op}^G_\bullet(\mathcal V)$}
\label{MAPSOPG_SEC}
We now discuss the several types of maps in
$\mathsf{Op}^G_\bullet(\V)$ we will be interested in,
starting with the ``local'' notions,
i.e. those notions determined by the fixed color categories $\mathsf{Op}_{\mathfrak{C}}^G(\V)$.
\begin{definition}
Let $\F$ be a $(G, \Sigma)$-family.
We say a map $F: \O \to \P$ in $\mathsf{Op}^G_\bullet(\V)$
is a \emph{local $\F$-weak equivalence (resp. local $\F$-fibration, local $\F$-trivial fibration)}
if the induced fixed color map
($F^{\**}$ is as in \eqref{GC_CHANGE_EQ})
\[\O \to F^{\**} \P\]
is a $\F_{\mathfrak{C}_{\O}}$-weak equivalence (resp. $\F_{\mathfrak{C}_{\O}}$-fibration, $\F_{\mathfrak{C}_{\O}}$-trivial fibration) in the fiber $\mathsf{Op}^G_{\mathfrak{C}_{\O}}(\V)$.
\end{definition}
Local (trivial) $\F$-fibrations admit the following alternative characterization.
\begin{proposition}\label{LOCALTCHAR PROP}
Suppose $\V$ is as in Theorem \ref{THMIREST}.
The local $\F$-fibrations
and local trivial $\F$-fibrations
in $\mathsf{Op}^G_\bullet(\V)$
are characterized as the maps with the right lifting property against the generating sets of maps
$\mathbb{F}^G_{\mathfrak{C}}\mathcal{J}_{\mathfrak{C},\mathcal{F}}$ and $\mathbb{F}^G_{\mathfrak{C}}\mathcal{I}_{\mathfrak{C},\mathcal{F}}$
(cf. \eqref{FVGSIGF EQ})
of the fibers
$\mathsf{Op}^G_{\mathfrak{C}}(\V) \hookrightarrow \mathsf{Op}^G_\bullet(\V)$
for all $\mathfrak{C} \in \mathsf{Set}^G$.
\end{proposition}
\begin{proof}
Note first that,
for a square in $\mathsf{Op}^G_{\bullet}(\V)$
as on the left below and where $A_1 \to A_2$ is a color fixed map,
the lifting problems for all three squares given below are equivalent.
\begin{equation
\begin{tikzcd}
A_1 \arrow[d] \arrow[r, "a"]
&
\mathcal{O} \arrow[d, "F"]
&&
A_1 \arrow[d] \arrow[r, "a"]
&
\mathcal{O} \arrow[d]
&&
A_1 \arrow[d] \arrow[r]
&
a^{\**} \mathcal{O} \arrow[d]
\\
A_2 \arrow[r]
&
\P
&&
A_2 \arrow[r]
&
F^{\**} \P
&&
A_2 \arrow[r]
&
a^{\**} F^{\**} \P
\end{tikzcd}
\end{equation}
Writing $\mathfrak{C}$ for the colors of the $A_i$ and
$\mathfrak{D}$ for the colors of $\O$,
the result follows since the pullback functors
$a^{\**} \colon \mathsf{Op}^G_{\mathfrak{D},\F}(\V)
\to \mathsf{Op}^G_{\mathfrak{C},\F}(\V)$
preserve (trivial) fibrations.
\end{proof}
We next turn to the homotopical notions of essential surjectivity and isofibration,
which concern equivalences between objects within some
$\O \in \mathsf{Op}^G_\bullet(\V)$.
We first recall some notions from \cite{BM13}.
As usual, we let $1_\V$ and $\emptyset$ denote, respectively, the unit object and initial object of $\V$.
\begin{notation}\label{1_NOT}
We write $\mathbbm{1}$ (resp. $\widetilde{\mathbbm{1}}$)
for the $\V$-category which represents arrows (resp. isomorphisms):
it has two objects $0,1$
and mapping objects
$\mathbbm{1}(i,j)= 1_{\V}$
if $i \leq j$
and
$\mathbbm{1}(1,0)= \emptyset$
(resp. $\widetilde{\mathbbm{1}}(i,j)= 1_{\V}$ for all $i,j$)
and composition defined by the unit isomorphisms of $\otimes$.
Further, we write $\eta$ for the $\V$-category which represents objects,
with a single object $\**$ and $\eta(\**,\**) = 1_\V$.
\end{notation}
In the following, and throughout,
we give $\Cat_{\set{0,1}}(\V)$
its projective model structure.
\begin{definition}
A {\em $\V$-interval} is a cofibrant object $\mathbb{J}$ in $\Cat_{\set{0,1}}(\V)$
which is equivalent to $\widetilde{\mathbbm{1}}$.
\end{definition}
\begin{example}
The prototypical example of a $\V$-interval is the simplicial category $W_!J \in \Cat_{\set{0,1}}(\sSet)$,
where
$J = N \widetilde{[1]} = N(0 \rightleftarrows 1) $ is the nerve of the walking isomorphism category
$\widetilde{[1]} = (0 \rightleftarrows 1)$ and
$W_! \colon \sSet \to \Cat_{\bullet}(\sSet)$ is the left adjoint to the homotopy coherent nerve
of \cite{Cor82} (see e.g. \cite[\S 1]{Joy02}).
\end{example}
\begin{remark}
We note that, since $\widetilde{\mathbbm{1}}$ is typically not fibrant,
an arbitrary interval $\mathbb{J}$
needs not admit a map to $\widetilde{\mathbbm{1}}$,
but only a map $\mathbb{J} \to \widetilde{\mathbbm{1}}_f$,
where $\widetilde{\mathbbm{1}}_f$ denotes some fixed chosen fibrant replacement.
\end{remark}
Informally, $\V$-intervals detect ``homotopical isomorphisms'' in a $\V$-category $\mathcal{C}$
(this idea is formalized in Definition \ref{EQUIV_DEF} below).
Mimicking the definitions of isofibration and essentially surjective functor of (unenriched) categories, we have the following.
\begin{definition}\label{PL_ES_DEFN}
We say a functor $F: \mathcal C \to \mathcal D$ in $\Cat(\V)$ is
\begin{itemize}
\item \textit{path-lifting}
if it has the right lifting property against all maps of the form
$\eta \xrightarrow{0} \mathbb{J}$, $\eta \xrightarrow{1} \mathbb{J}$
where $\mathbb{J}$ is a $\V$-interval;
\item \textit{essentially surjective}
if, for any object $d \in \mathcal{D}$,
there is an object $c \in \mathcal{C}$,
$\V$-interval $\mathbb{J}$,
and map $i \colon \mathbb{J} \to \mathcal D$
such that $i(0) = F(c)$ and $i(1)=d$.
\end{itemize}
\end{definition}
We now adapt the previous definition for $G$-operads.
Recall that $j^{\**} \colon \mathsf{Op}^G_\bullet(\V) \to \mathsf{Cat}^G_{\bullet}(\V)$
denotes the functor that forgets all non-unary operations and that, moreover,
$j^{\**}$ commutes with all fixed points $(-)^H$.
\begin{definition}\label{FESSENSURJ DEF}
Let $\F$ be a $(G, \Sigma)$-family which has enough units
(Definition \ref{FAMRESUNI DEF}).
We say a map $F: \O \to \P$ in $\mathsf{Op}^G_\bullet(\V)$
is $\F$-essentially surjective (resp. $\F$-path-lifting)
if the maps
$j^{\**}\O^H \to j^{\**} \P^H$
in $\mathsf{Cat}(\V)$ are essentially surjective (path-lifting) for all $H \in \F_1$.
\end{definition}
We can finally define the classes of maps in the desired model structures on $\mathsf{Op}^G_\bullet(\V)$.
\begin{definition}\label{MODEL_DEFN}
Let $\F$ be a $(G, \Sigma)$-family which has enough units
(Definition \ref{FAMRESUNI DEF}).
We say a map $F: \O \to \P$ in $\mathsf{Op}^G_\bullet(\V)$ is:
\begin{itemize}
\item a {\em $\F$-fibration} if it is both a local $\F$-fibration and $\F$-path lifting;
\item a {\em $\F$-weak equivalence} if it is both a local $\F$-weak equivalence and $\F$-essentially surjective;
\item a \textit{$\F$-cofibration} if it has the left lifting property against all trivial $\F$-fibrations (i.e. $\F$-fibrations which are also $\F$-weak equivalences).
\end{itemize}
\end{definition}
Throughout the remainder of \S \ref{MS_SEC}
we will prove that
Definition \ref{MODEL_DEFN} describes the model structure on
$\mathsf{Op}^G_\bullet(\V)$
in Theorem \ref{THMA},
which we denote by
$\mathsf{Op}^G_{\bullet, \F}(\V)$.
We first show that $\F$-trivial fibrations
indeed satisfy the characterization given in Theorem \ref{THMA},
adapting \cite[4.8]{Cav}, \cite[2.3]{BM13}, \cite[1.18]{CM13b}.
\begin{proposition}\label{FTRIVCHAR PROP}
A map in $F: \O \to \P$ in $\mathsf{Op}^G_\bullet(\V)$ is
a $\F$-trivial fibration (i.e. both a $\F$-fibration and a $\F$-weak equivalence)
iff it is a local $\F$-trivial fibration
such that the induced map on $H$-fixed colors is surjective
for all $H \in \F_1$.
\end{proposition}
\begin{proof}
It is enough to show that,
if $F \colon \O \to \P$ is a local $\F$-trivial fibration,
then $F$ is both $\F$-path lifting and $\F$-essentially surjective
iff the induced map on $H$-fixed colors is surjective for all $H \in \F_1$.
For the ``if'' direction, it is immediate that $F$ is
$\F$-essentially surjective, so it remains to show that
the maps $\O^H\to \P^H$ for $H \in \F_1$
have the right lifting property against the maps
$\eta \to \mathbb{J}$.
But this follows by factoring the latter maps as
$\eta \to \eta \amalg \eta \to \mathbb{J}$,
since the lifting property against
$\eta \to \eta \amalg \eta$ follows from surjectivity on $H$-fixed objects while the lifting property against
$\eta \amalg \eta \to \mathbb{J}$
follows from Proposition \ref{LOCALTCHAR PROP}
and the fact that
$\eta \amalg \eta \to \mathbb{J}$
is a cofibration in $\mathsf{Cat}_{\{0,1\}}(\V)$
(given that $\eta \amalg \eta$ is the initial object of $\mathsf{Cat}_{\{0,1\}}(\V)$
while $\mathbb{J}$ is cofibrant by definition of $\V$-interval).
For the ``only if'' direction, let $y \in \P^H$ be an $H$-fixed object with $H \in \F_1$.
$\F$-essential surjectivity yields an $x \in \O^H$ and map
$i \colon \mathbb{J} \to \P^H$
with $i(0)=F(x)$, $i(1)=y$.
The $\F$-path lifting property then gives a lift $\tilde{i}$ as below,
so that $\tilde{i}(1)$ gives the desired lift of $y$.
\[
\begin{tikzcd}
\eta \ar{d}[swap]{0} \ar{r}{x}
&
\O^H \arrow{d}{F}
\\
\mathbb{J} \ar{r}[swap]{i} \ar[dashed]{ru}[swap]{\tilde{i}}
&
\P^H
\end{tikzcd}
\]
\end{proof}
\begin{proposition}\label{FIBERGLMOD PROP}
A fixed color map
$\O \to \P$ in
$\mathsf{Op}^G_{\mathfrak{C}}(\V)$ is:
\begin{enumerate}[label=(\roman*)]
\item a $\F$-weak equivalence in the fiber
$\mathsf{Op}_{\mathfrak{C}}^G(\V)$
iff it is a $\F$-weak equivalence in
$\mathsf{Op}_{\bullet}^G(\V)$;
\item a $\F$-cofibration in the fiber
$\mathsf{Op}_{\mathfrak{C}}^G(\V)$
iff it is a $\F$-cofibration in
$\mathsf{Op}_{\bullet}^G(\V)$;
\item a $\F$-fibration in the fiber
$\mathsf{Op}_{\mathfrak{C}}^G(\V)$
whenever it is a $\F$-fibration in
$\mathsf{Op}_{\bullet}^G(\V)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(i) follows since fixed color maps are certainly essentially surjective while (iii) is tautological since
$\F$-fibrations in $\mathsf{Op}_{\bullet}^G(\V)$ must be local $\F$-fibrations.
As for (ii),
Proposition \ref{LOCALTCHAR PROP} yields the ``if'' direction
while the ``only if'' direction follows from
Proposition \ref{FTRIVCHAR PROP},
which implies that all $\F$-trivial fibrations in $\mathsf{Op}_{\mathfrak{C}}^G(\V)$
are $\F$-trivial fibrations in
$\mathsf{Op}_{\bullet}^G(\V)$.
\end{proof}
\begin{remark}
In contrast to the other parts of
Proposition \ref{FIBERGLMOD PROP},
the implication in part (iii)
only holds in one direction.
As a counterexample to its converse,
consider the map $\eta \amalg \eta \to \widetilde{\mathbbm{1}}$
in $\mathsf{Cat}_{\bullet}(\mathsf{sSet})$. This is a local fibration,
and thus a fibration in $\mathsf{Cat}_{\{0,1\}}(\mathsf{sSet})$,
but not path-lifting, and thus not a fibration in $\mathsf{Cat}_{\bullet}(\mathsf{sSet})$.
Nonetheless, Proposition \ref{FTRIVCHAR PROP}
guarantees that the analogue of Proposition \ref{FIBERGLMOD PROP}
for $\F$-trivial fibrations is indeed an iff.
\end{remark}
\begin{corollary}
$\O \in \Op^G_{\mathfrak C}(\V)$ is cofibrant in
$\O \in \Op^G_{\mathfrak C,\mathcal{F}}(\V)$
iff $\O$ is cofibrant in $\Op^G_{\bullet,\F}(\V)$.
\end{corollary}
\begin{proof}
The ``if'' direction follows since
$\F$-trivial fibrations in $ \Op^G_{\mathfrak C}(\V)$
are
$\F$-trivial fibrations in $ \Op^G_{\bullet}(\V)$.
The ``only if'' direction follows since
$\F$-trivial fibrations in $ \Op^G_{\bullet}(\V)$
are local $\F$-trivial fibrations.
\end{proof}
The proof of Theorem \ref{THMA} will occupy most of
the remainder of \S \ref{MS_SEC},
where we will show that the maps in Definition \ref{MODEL_DEFN}
do indeed define a model structure on $\mathsf{Op}^G{\V}$.
The following is the outline of the proof.
\begin{proof}[Proof of Theorem \ref{THMA}]
As $\sSet$ satisfies all
hypotheses in Theorem \ref{THMA}, we prove the general case.
We will verify the conditions in
\cite[Theorem 2.1.19]{Hov99}, and we write (1),(2),(3),etc for the conditions therein.
Firstly, in \S \ref{GENCOF SEC} we identify
the generating (resp. trivial) cofibrations of $\mathsf{Op}^G_{\F}(\V)$,
which are given by the sets
(C1) and (C2) (resp. (TC1) and (TC2))
found in Definition \ref{OPGENCOF DEF}.
The implicit claim that the maps
with the right lifting property against
(TC1) and (TC2)
are the $\F$-fibrations as given by
Definition \ref{MODEL_DEFN}
follows from Propositions \ref{LOCALTCHAR PROP} and \ref{GENIN PROP}.
Likewise, the fact that the maps
with the right lifting property against
(C1) and (C2) are the
$\F$-trivial fibrations as given by
Definition \ref{MODEL_DEFN}
is Proposition \ref{FTRIVCHAR PROP},
establishing conditions (5),(6).
Lemma \ref{POINT_4_LEMMA} and Proposition \ref{J_CELL_PROP} establishes (4).
(2),(3) follow since colimits in $\mathsf{Op}^G_\bullet(\V)$ are created in $\Op_{\bullet}(\V)$, and it holds non-equivariantly.
Condition (1), i.e. the $2$-out-of-$3$ condition for $\F$-weak equivalences,
is Proposition \ref{2OUTOF3 PROP}.
Lastly, the fact that the weak equivalences in
Theorem \ref{THMA}
match the weak equivalences in
Definition \ref{MODEL_DEFN} is given by
Corollary \ref{WEDKEQ COR}.
\end{proof}
\subsection{Generating cofibrations and trivial cofibrations}
\label{GENCOF SEC}
We next turn to the task of identifying sets of generating cofibrations and generating trivial cofibrations
for the desired model structures
on $\mathsf{Op}^G_{\bullet}(\V)$ determined by Definition \ref{MODEL_DEFN}.
Proposition \ref{LOCALTCHAR PROP} suggests that
the generating sets of maps
$\mathbb{F}^G_{\mathfrak{C}} \mathcal{I}_{\mathfrak{C},\mathcal{F}},
\mathbb{F}^G_{\mathfrak{C}} \mathcal{J}_{\mathfrak{C},\mathcal{F}}$
(cf. \eqref{FVGSIGF EQ}) of the fibers
$\mathsf{Op}^G_{\mathfrak{C},\F}(\V)$
should be included
in the generating sets of maps for
$\mathsf{Op}^G_{\bullet,\F}(\V)$.
However, it is inefficient to include all such maps,
as there is a subset of those maps
that generates the remaining maps under pushouts along change of colors.
To see why,
consider a representable functor
$\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]$
in
$\mathsf{Sym}^G_{\mathfrak{C}}$
(cf. \eqref{REPALTDESC EQ}),
and write
$\mathfrak{c} \colon
\boldsymbol{E}(G \cdot C) \to \mathfrak{C}$
for the coloring on the underlying forest $G \cdot C$.
By Remark \ref{GCDOTCATS REM},
the group $G \times \Sigma^{op}$
has a right action on $G \cdot C$ and,
moreover, a subgroup
$\Lambda \leq G \times \Sigma^{op}$ stabilizes $\vect{C}$
precisely if $\mathfrak{c}$ is $\Lambda$-equivariant,
i.e. if it induces a map
$\bar{\mathfrak{c}} \colon
\boldsymbol{E}(G \cdot C)/\Lambda \to \mathfrak{C}$
on orbits
(indeed, this is simply the observation that the right vertical map in
\eqref{COLCHSQ EQ} respects colors, specified to the case
$\vect{C'} = \vect{C}$).
Combining the identification
$\mathfrak{c}_! \Sigma_{\tau}[G \cdot C]
=
\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]$
in
\eqref{CANPUSH EQ}
with
\eqref{LIMINFIBSUP EQ},
we now obtain that
\begin{equation}\label{CANPUSHQ EQ}
\bar{c}_{!}
\left(
\Sigma_{\tau}[G \cdot C]/\Lambda
\right)
=
\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda
\end{equation}
where we note that the quotient
$\Sigma_{\mathfrak{C}}[G \cdot_{\mathfrak{C}} \vect{C}]/\Lambda$
occurs in the fiber $\mathsf{Sym}^G_{\mathfrak{C}}$
while $\Sigma_{\tau}[G \cdot C]/\Lambda$
is not a fiber quotient.
In particular, the colors of the latter are
$\boldsymbol{E}(G \cdot C)/\Lambda$
rather than $\boldsymbol{E}(G \cdot C)$.
\eqref{CANPUSHQ EQ} now readily implies similar identifications for the generating sets
$\mathbb{F}^G_{\mathfrak{C}} \mathcal{I}_{\mathfrak{C},\mathcal{F}},
\mathbb{F}^G_{\mathfrak{C}} \mathcal{J}_{\mathfrak{C},\mathcal{F}}$.
Before describing the generating sets for
$\mathsf{Op}^G_{\bullet,\mathcal{F}}(\V)$, however,
we need also address the path-lifting condition,
requiring fibrations in $\mathsf{Op}^G_{\bullet,\F}(\V)$ to have the right lifting property against all maps
$G/H \cdot (\eta \to \mathbb J)$
with $\mathbb{J}$ a $\V$-interval and $H \in \F_1$.
As the collection of all intervals form a class, one must be able to select a suitable representative set of intervals, leading to the following (cf. \cite{BM13}).
\begin{definition}\label{INTGENSET DEF}
A set $\mathscr{G}$ of $\V$-intervals is \textit{generating} if,
in the projective model category on $\Cat_{\set{0,1}}(\V)$,
any $\V$-interval $\mathbb{I}$ is
a retract of a trivial extension of some element
$\mathbb{G} \in \mathscr{G}$.
More explicitly, this means that there is a diagram in
$\Cat_{\set{0,1}}(\V)$ as below,
where the left arrow is a trivial cofibration and
$ri = id_{\mathbb{I}}$.
\begin{equation}\label{GTILGI EQ}
\begin{tikzcd}
\mathbb{G} \arrow[r,rightarrowtail, "\sim"]
&
\widetilde{\mathbb{G}} \arrow[r,yshift=-.3em, "r"']
&
\mathbb{J} \arrow[l,yshift=.3em, "i"']
\end{tikzcd}
\end{equation}
\end{definition}
The following essentially recalls \cite[1.20]{CM13b}, \cite[\S 4.3]{Cav}.
\begin{remark}
\label{SSETINT_REM}
When $\V$ is either $\mathsf{sSet}$ or $\mathsf{sSet}_{\**}$
one can take $\mathscr{G}$ to be a set of representatives of isomorphism classes of intervals with countably many cells.
Indeed, since in both cases the mapping spaces of a $\V$-interval
$\mathbb{J}$ are a simplicial set with (either one or two) contractible components,
a standard argument (see e.g. the argument between \cite[Lemmas 4.2,4.3]{Ber07b})
shows that $\mathbb{J}$ has a countable subcomplex $\mathbb{G}$ with contractile components and
for which the inclusion
$\mathbb{G} \to \mathbb{J}$
is an equivalence in $\mathsf{Cat}_{\{0,1\}}(\V)$.
But then, forming the cofibration followed by trivial fibration factorization
$\mathbb{G} \rightarrowtail \widetilde{\mathbb{G}}
\overset{\sim}{\twoheadrightarrow} \mathbb{J}$
in $\mathsf{Cat}_{\{0,1\}}(\V)$,
one has that the first map is a trivial cofibration by $2$-out-of-$3$
and that the second has a section since $\mathbb{J}$ is cofibrant by assumption, yielding \eqref{GTILGI EQ}.
More generally, a more careful argument \cite[Lemma 1.12]{BM13}
shows that every combinatorial monoidal model category
has a generating set of intervals.
\end{remark}
\begin{proposition}\label{GENIN PROP}
If $\V$ has a generating set of intervals $\mathscr{G}$ then a local $\F$-fibration
$F \colon \O \to \P$ in
$\mathsf{Op}^G_{\mathfrak{C}}(\V)$
is $\F$-path lifting iff it has the right lifting property against the maps
$\{G/H \cdot (\eta \to \mathbb{G})\}_{\mathbb{G}\in \mathscr{G},H \in \mathcal{F}_1}$.
\end{proposition}
\begin{proof}
Given some chosen interval $\mathbb{J}$,
let $\mathbb{G}, \widetilde{\mathbb{G}}$
be as in \eqref{GTILGI EQ}.
A standard argument concerning retractions shows that,
to solve a lifting problem against $\eta \to \mathbb{J}$,
it suffices to solve the induced lifting problem against
$\eta \to \widetilde{\mathbb{G}}$.
But now given a lifting problem against
$G/H \cdot \left(\eta \to \widetilde{\mathbb{G}}\right)$,
we consider the diagram below, where the solid lift exists by hypothesis on $F$.
\[
\begin{tikzcd}
\eta \ar{d} \ar{rr}
&&
\O^H \ar{d}{F}
\\
\mathbb{G} \ar[rightarrowtail]{r}{\sim} \ar{rru}
&
\widetilde{\mathbb{G}} \ar{r} \ar[dashed]{ru}
&
\P^H
\end{tikzcd}
\]
But then, since $\mathbb{G} \overset{\sim}{\rightarrowtail} \widetilde{\mathbb{G}}$
is a trivial cofibration in $\mathsf{Cat}_{\{0,1\}}(\V)$
and $F$ is a local fibration,
the desired dashed lift exists
by Proposition \ref{LOCALTCHAR PROP}.
\end{proof}
We can now finally identify the generating (trivial) cofibrations of
$\mathsf{Op}^G_{\bullet,\F}$.
In the following we write $C_n \in \Sigma$ for the $n$-corolla.
\begin{definition}\label{OPGENCOF DEF}
Suppose that $\V$ has a generating set of intervals $\mathscr{G}$.
Then the generating cofibrations in $\mathsf{Op}^G_{\F}$
are the maps
\begin{itemize}
\item[(C1)] $\emptyset \to G/H \cdot \eta$ for $H \in \F_1$,
\item[(C2)] $\mathbb{F} \left( \Sigma_{\tau}[G \cdot C_n]/\Lambda \cdot i\right)$
for $n \geq 0$, $\Lambda \in \F_n$ and $i \in \mathcal{I}$,
\end{itemize}
while the generating trivial cofibrations are the maps
\begin{itemize}
\item[(TC1)]
$G/H \cdot \left(\eta \to \mathbb{G}\right)$ for $H \in \F_1$ and $\mathbb{G} \in \mathscr{G}$,
\item[(TC2)]
$\mathbb{F} \left( \Sigma_{\tau}[G \cdot C_n]/\Lambda \cdot j\right)$
for $n \geq 0$, $\Lambda \in \F_n$ and $j \in \mathcal{J}$.
\end{itemize}
\end{definition}
\begin{lemma}[{cf. \cite[1.19]{CM13b}}]\label{POINT_4_LEMMA}
The maps in (TC1),(TC2) are in the saturation of (C1),(C2).
\end{lemma}
\begin{proof}
Clearly (TC2) is in the saturation of (C2).
As for (TC1), one has factorizations
\begin{equation}
\begin{tikzcd}
G/H \cdot \eta \arrow[r, rightarrowtail]
&
G/H \cdot (\eta \amalg \eta) \arrow[r, rightarrowtail]
&
G/H \cdot \mathbb{G}
\end{tikzcd}
\end{equation}
with the first map a pushout of a map in (C1) and
the second map in the saturation of (C2).
\end{proof}
\subsection{Interval cofibrancy and trivial cofibrations}
\label{TRIVCOF_SEC}
In this section we establish Proposition \ref{J_CELL_PROP},
stating that maps built cellularly out of
(TC1) and (TC2) are $\F$-weak equivalences.
We first recall the following
technical result from \cite{BM13}.
\begin{theorem}
[Interval Cofibrancy Theorem {\cite[Thm. 1.15]{BM13}}]
\label{INTCOF THM}
Let $(\V,\otimes)$ be a cofibrantly generated monoidal model category which
satisfies the monoid axiom and has cofibrant unit.
If $\mathbb{J} \in \mathsf{Cat}_{\{0,1\}}(\V)$
is cofibrant then
$\mathbb{J}(0,0)$
is a cofibrant monoid, i.e. cofibrant
in $ \mathsf{Cat}_{\{0\}}(\V)$.
\end{theorem}
Our assumptions on $\V$
in Theorem \ref{INTCOF THM} differ slightly
from those in the original formulation \cite[Thm. 1.15]{BM13},
as we replace the \emph{adequacy} condition in
\cite[Def. 1.1]{BM13} with the monoid axiom.
Nonetheless, the proof therein (occupying \S 3.6,\S 3.7,\S 3.8 in \cite{BM13}) still follows as written.
This is because adequacy is never used directly,
serving only to guarantee existence of the
model structures
on $\mathsf{Cat}_{\{0,1\}}(\V),\mathsf{Cat}_{\{0\}}(\V)$
and
on modules $\mathsf{Mod}_{R}(\V)$, $_R\mathsf{Mod}(\V)$
over a monoid $R$.
However, the monoid axiom suffices for these claims
\cite[Thm. 1.3]{Mur11},\cite[Thm. 4.1]{SS00}.
\begin{remark}
By symmetry, one also has that $\mathbb{J}(1,1)$ is cofibrant.
Moreover, the formulation in \cite[Thm. 1.15]{BM13}
includes additional cofibrancy conditions for
$\mathbb{J}(0,1),\mathbb{J}(1,0)$
as modules over $\mathbb{J}(0,0),\mathbb{J}(1,1)$.
These conditions are essential for their proof,
but not needed for our application.
\end{remark}
We note that the Interval Cofibrancy Theorem is a particular case of the following conjecture when $\mathfrak{C} \to \mathfrak{D}$
is the inclusion $\{0\} \to \{0,1\}$.
\begin{conjecture}\label{CATOP CONJ}
Let $\varphi \colon \mathfrak{C} \to \mathfrak{D}$
be an injection of colors.
Then the pullback functors
\[
\mathsf{Cat}_{\mathfrak{D},\F}^G(\V)
\xrightarrow{\varphi^{\**}}
\mathsf{Cat}_{\mathfrak{C},\F}^G(\V)
\qquad
\mathsf{Op}_{\mathfrak{D},\F}^G(\V)
\xrightarrow{\varphi^{\**}}
\mathsf{Op}_{\mathfrak{C},\F}^G(\V)
\]
preserve cofibrations between cofibrant objects.
\end{conjecture}
\begin{remark}
To see why Conjecture \ref{CATOP CONJ} is at least plausible,
we argue that $\varphi^{\**}$ sends free objects to free objects,
which is essentially tantamount to
sending generating cofibrations \eqref{VGSIGF EQ} to generating cofibrations.
To see this, consider the simplest example
with
$\varphi \colon \{0\} \to \{0,1\}$
and a free $\mathbb{F}_{\{0,1\}}X $ in $\mathsf{Cat}_{\{0,1\}}(\V)$.
Then one can check that
$\varphi^{\**} \left( \mathbb{F}_{\{0,1\}}X \right)$ in $\mathsf{Cat}_{\{0\}}(\V)$ is the free monoid
\begin{equation}\label{PULLFREEEX EQ}
\varphi^{\**} \left( \mathbb{F}_{\{0,1\}}X \right)
\simeq
\mathbb{F}_{\{0\}}
\left(X(0,0) \amalg
\coprod_{n \geq 0}
X(0,1)\otimes X(1,1)^{\otimes n} \otimes X(1,0)
\right)
\end{equation}
where we note that the expression inside
$\mathbb{F}_{\{0\}}$
in \eqref{PULLFREEEX EQ}
can be intuitively described as the formal composites
$0 \to 1 \to 1 \to \cdots \to 1 \to 0$
of ``arrows'' in $X$ which start and end at $0$ and where all intermediate objects are $1$.
More generally, for an inclusion of colors
$\varphi \colon \mathfrak{C} \to \mathfrak{D}$
one has that
$\varphi^{\**} \left(\mathbb{F}_{\mathfrak{D}} X\right)$
is similarly free on formal composites
$c_0 \to d_1 \to d_2 \to \cdots \to d_n \to c_{n+1}$
of arrows in $X$
where $c_i \in \mathfrak{C}$
and $d_j \in \mathfrak{D} \setminus \mathfrak{C}$,
while for operads the analogue claim involves labeled trees whose root and leaves are labeled by $\mathfrak{C}$
and whose inner edges are labeled by
$\mathfrak{D} \setminus \mathfrak{C}$.
It is then straightforward to check that, under mild assumptions on $\V$,
$\varphi^{\**} \left(\mathbb{F}_{\mathfrak{D}} X \right)$
will be a (trivial) cofibration in
$\mathsf{Cat}^G_{\mathfrak{C},\F}(\V)$
(resp. $\mathsf{Op}^G_{\mathfrak{C},\F}(\V)$))
when $\mathbb{F}_{\mathfrak{D}} X$
is a generating (trivial) cofibration
in $\mathsf{Cat}^G_{\mathfrak{D},\F}(\V)$
(resp. $\mathsf{Op}^G_{\mathfrak{D},\F}(\V)$)).
However, the argument just given \emph{does not} outline a proof of Conjecture \ref{CATOP CONJ},
due to $\varphi^{\**}$ not preserving pushouts,
so that, to actually prove Conjecture \ref{CATOP CONJ},
one would need a careful analysis of the interaction of $\varphi^{\**}$ with pushouts of free categories/operads,
as in the proof of \cite[Thm. 1.15]{BM13}.
Lastly, we make note of a very similar conjecture:
it is natural to ask if the restriction functor
$j^{\**} \colon \mathsf{Op}_{\mathfrak{C},\F}^G
\to \mathsf{Cat}_{\mathfrak{C},\F}^G$
preserves cofibrations between cofibrant objects,
and again one has that $j^{\**}$
sends generating (trivial) cofibrations to (trivial) cofibrations.
However, since our operads have $0$-ary operations,
$j^{\**}$ does not preserve pushouts
(indeed, this would be tantamount to the claim that trees with a single leaf are linear trees, which is not true if we allow for trees with stumps).
\end{remark}
In the next result we write
$\partial_i \colon \{0,1\} \to \{0,1,2\}$
for the ordered inclusion that omits $i$,
and
$\widetilde{\mathbbm{2}} \in
\mathsf{Cat}_{\{0,1,2\}}(\V)$
for the ``double isomorphism category''
where all mapping objects are $\widetilde{\mathbbm{2}}(i,j)=1_{\V}$.
In the following note that, since the $\partial_i$
are injective, one has
$\check{\partial}_{i,!} \simeq \partial_{i,!}$,
cf. Remark \ref{OP_MAP REM}.
\begin{lemma}[Interval Amalgamation Lemma {\cite[Lemma 1.16]{BM13}}]
\label{AMALGLEM LEM}
Let $\V$ be as in Theorem \ref{INTCOF THM} and
$\mathbb{I},\mathbb{J}$ be $\V$-intervals.
Then the coproduct
$\mathbb{K} = \partial_{2,!} \mathbb{I} \amalg \partial_{0,!} \mathbb{J}$
in $\mathsf{Cat}_{\{0,1,2\}}(\V)$
is cofibrant and weakly equivalent to $\widetilde{\mathbbm{2}}$.
In particular,
$\mathbb{I} \star \mathbb{J} = \partial_1^{\**}\mathbb{K}$
is weakly equivalent to $\widetilde{\mathbbm{1}}$
in $\mathsf{Cat}_{\{0,1\}}(\V)$.
\end{lemma}
\begin{remark}
$\mathbb{I} \star \mathbb{J} =
\partial_1^{\**} \mathbb{K} =
\partial_1^{\**} \left(\partial_{2,!} \mathbb{I} \amalg \partial_{0,!} \mathbb{J}\right)$
is called the \emph{amalgamation} of $\mathbb{I}$ and $\mathbb{J}$,
so that Lemma \ref{AMALGLEM LEM} can be phrased as saying that an amalgamation of intervals is,
up to cofibrant replacement, again an interval
(Conjecture \ref{CATOP CONJ} would imply that $\mathbb{I} \star \mathbb{J}$ is already cofibrant,
but we will not need to know this).
\end{remark}
The original proof of this result \cite[Lemma 1.16]{BM13} uses the cofibrancy of modules conditions on
$\mathbb{J}(0,1),\mathbb{J}(1,0)$
in \cite[Thm 1.15]{BM13}.
Here we present an alternative argument requiring only the cofibrancy of $\mathbb{J}(0,0)$ as a monoid,
as stated in our formulation of Theorem \ref{INTCOF THM}.
\begin{proof}
Since $\mathbb{I},\mathbb{J}$ are cofibrant and the $\partial_{i,!}$
preserve cofibrations by
(the category version of)
Corollary \ref{OPADJ_COR}(i),
the coproduct
$\partial_{2,!} \mathbb{I} \amalg \partial_{0,!} \mathbb{J}$
is a homotopy coproduct,
so we are free to replace $\mathbb{I},\mathbb{J}$
with any chosen intervals.
In particular, we may thus assume there are (local) trivial fibrations
$\mathbb{I} \overset{\sim}{\twoheadrightarrow} \widetilde{\mathbbm{1}}$,
$\mathbb{J} \overset{\sim}{\twoheadrightarrow} \widetilde{\mathbbm{1}}$.
One then has a map
\[
\mathbb{K}=
\partial_{2,!} \mathbb{I} \amalg \partial_{0,!} \mathbb{J}
\to
\partial_{2,!} \widetilde{\mathbbm{1}} \amalg \partial_{0,!} \widetilde{\mathbbm{1}}
= \widetilde{\mathbbm{2}}
\]
which we will show to be a weak equivalence.
Firstly, by applying \cite[Cor. \ref{OC-FGTPUSH_COR}]{BP_FCOP} twice,
one has that $\mathbb{K}(1,1)= \mathbb{I}(1,1) \amalg \mathbb{J}(0,0)$,
where the coproduct is taken in $\Cat_{\{1\}}(\V)$.
Since Theorem \ref{INTCOF THM} says $\mathbb{I}(1,1),\mathbb{J}(0,0)$
are acyclic cofibrant
(i.e. the map from the initial object
$\eta \in \mathsf{Cat}_{\{0\}}(\V)$
is a trivial cofibration),
so is $\mathbb{K}(1,1)$.
Next, the trivial fibrations
$\mathbb{I} \overset{\sim}{\twoheadrightarrow} \widetilde{\mathbbm{1}}$,
$\mathbb{J} \overset{\sim}{\twoheadrightarrow} \widetilde{\mathbbm{1}}$
allow us to find factorizations
of the identity $1_{\V} \xrightarrow{=} 1_{\V}$
\[
1_{\V} \xrightarrow{\alpha}
\mathbb{I}(0,1)
\to
\widetilde{\mathbbm{1}}(0,1)
\quad
1_{\V} \xrightarrow{\beta}
\mathbb{I}(1,0)
\to
\widetilde{\mathbbm{1}}(1,0)
\quad
1_{\V} \xrightarrow{\bar{\alpha}}
\mathbb{J}(0,1)
\to
\widetilde{\mathbbm{1}}(0,1)
\quad
1_{\V} \xrightarrow{\bar{\beta}}
\mathbb{I}(1,0)
\to
\widetilde{\mathbbm{1}}(1,0).
\]
For any choice of $i,j$ in $\{0,1,2\}$,
by pre and postcomposing with $\alpha,\beta,\bar{\alpha},\bar{\beta}$
as appropriate, one gets a commutative diagram
\begin{equation}\label{INTAM EQ}
\begin{tikzcd}[column sep = 45pt]
\mathbb{K}(1,1)
\arrow{d}[swap]{\sim}
\arrow{r
&
\mathbb{K}(i,j)
\arrow[d]
\\
\widetilde{\mathbbm{2}}(1,1)
\arrow[equal]{r}
&
\widetilde{\mathbbm{2}}(i,j).
\end{tikzcd}
\end{equation}
We now note that
$\alpha,\beta,\bar{\alpha},\bar{\beta}$
are homotopy equivalences
in the sense of \cite[Def. 2.6]{BM13}
(or Definition \ref{EQUIV_DEF} below),
since they represent the
identity homotopy class of maps
$[id_{1_{\V}}] \in \Ho \V(1_{\V},\widetilde{\mathbbm {1}}(k,l))$,
which are isomorphisms between $0,1$ in the category
$\pi_0 \widetilde{\mathbbm {1}}$
of \cite[Rem. 2.7]{BM13}
(or Definition \ref{HTPY_DEFN} below).
See also Remark \ref{NATISO REM}.
But now \cite[Lemma 2.12]{BM13}
(or its generalization Corollary \ref{ALBEETA COR})
implies that the top horizontal map in
\eqref{INTAM EQ} is a weak equivalence,
and thus so are the maps
$\mathbb{K}(i,j) \to \widetilde{\mathbbm{2}}(i,j)$,
establishing that $\mathbb{K} \to \widetilde{\mathbbm{2}}$
is indeed a weak equivalence.
\end{proof}
\begin{lemma}[cf. {\cite[4.17]{Cav}}] \label{TRANSCOMP_ES_LEM}
A transfinite composition of $\F$-essentially surjective maps in $\Op^G_\bullet(\V)$ is $\F$-essentially surjective.
\end{lemma}
\begin{proof}
Let $\kappa$ be a limit ordinal and consider a transfinite composition
$\O_0 \to \O_1 \to \cdots
\to \colim_{\alpha < \kappa} \O_{\alpha} = \O_{\kappa}$
of $\F$-essentially surjective maps
$F_{\alpha} \colon \O_{\alpha} \to \O_{\alpha +1}$, where, as usual,
we assume that $\O_{\alpha} = \colim_{\beta < \alpha} \O_{\beta}$ whenever $\alpha < \kappa$ is a limit ordinal.
We argue by transfinite induction on $\alpha \leq \kappa$
that the composite maps $\bar{F}_{\alpha} \colon \O_0 \to \O_{\alpha}$
are $\F$-essentially surjective. Fix $H \in \F_1$.
For a successor ordinal $\alpha+1$, given $c \in \O_{\alpha+1}^H$
one can find $b \in \O_\alpha^H$ and map from an interval
$\mathbb{J} \xrightarrow{j} \O_{\alpha+1}^H$ with $j(0) = F_{\alpha}(b)$, $j(1)=c$
and, by the induction hypothesis, can likewise find
$a \in \O_0^H$ and map from an interval $\mathbb{I} \xrightarrow{i} \O_\alpha^H$
with $i(0)=\bar{F}_{\alpha}a$, $i(1)=b$.
The amalgamated map
$\mathbb{I} \star \mathbb{J} \xrightarrow{F_{\alpha}i \star j} \O_{\alpha+1}^H$
now connects $\bar{F}_{\alpha+1}(a)$ and $c$, as desired.
The case of a limit ordinal $\alpha$ is immediate: any object $b \in \O_{\alpha}^H$
lifts to an object $\bar{b} \in \O_{\bar{\alpha}}^H$ for some $\bar{\alpha} < \alpha$,
so noting that by induction there exists $a \in \O_0^H$ and map from an interval $\mathbb{I} \xrightarrow{i} \O_{\bar{\alpha}}^H$
with $i(0) = F_{\bar{\alpha}}a$, $i(1) = \bar{b}$ yields the result.
\end{proof}
\begin{remark}\label{LOCOTIMESTRI REM}
Say a map $F \colon X \to Y$ in
$\mathsf{Sym}^{G}_{\bullet}(\V)$
is a \emph{local genuine $\otimes$-trivial cofibration}
if
$X \to F^{\**}Y$ is a
genuine $\otimes$-trivial cofibration in
$\mathsf{Sym}^{G}_{\mathfrak{C}_X}(\V) = \V^{G \ltimes \Sigma^{op}_{\mathfrak{C}_{X}}}$
(cf. Definition \ref{GGENOTITC DEF}).
One then has that local genuine $\otimes$-trivial cofibrations are closed under transfinite composition.
Indeed, given such a transfinite composite as on the left
\[
X_0 \xrightarrow{F_1}
X_1 \xrightarrow{F_2}
X_2 \xrightarrow{F_3}
X_3 \to \cdots
\qquad \qquad
X_0 \to
F_1^{\**} X_1 \to
F_1^{\**}F_2^{\**}X_2 \to
F_1^{\**}F_2^{\**}F_3^{\**}X_3 \to \cdots
\]
the induced transfinite composite in $\mathsf{Sym}^{G}_{\mathfrak{C}_{X_0}}(\V)$
on the right consists of
genuine $\otimes$-trivial cofibrations,
since these are preserved under pullback
(Proposition \ref{GOTC_PROP}).
\end{remark}
\begin{proposition}[{c.f. \cite[4.20]{Cav}}]\label{J_CELL_PROP}
Suppose $\V$ satisfies the conditions in Theorem \ref{THMIREST}.
Then maps in the saturation of (TC1),(TC2)
are $\F$-weak equivalences in $\mathsf{Op}^G_{\bullet}(\V)$.
\end{proposition}
\begin{proof}
We reduce to the case $\F=\F_{all}$,
as that makes (TC1),(TC2) in Definition \ref{OPGENCOF DEF} as large as possible
and $\mathcal{F}$-weak equivalences as small as possible,
cf. Remark \ref{FALLMAXMIN REM}.
By Proposition \ref{GOTC_PROP}(i)
and the closure under transfinite composition properties in Lemma \ref{TRANSCOMP_ES_LEM} and Remark \ref{LOCOTIMESTRI REM},
it suffices to show that, for every pushout
\begin{equation}\label{JUSTAPUSH EQ}
\begin{tikzcd}
J_1 \ar{r}{a} \arrow{d}[swap]{j}
&
\O \arrow{d}
\\
J_2 \ar{r}
&
\P
\end{tikzcd}
\end{equation}
where $j$ is one of the generating trivial cofibrations
in (TC1),(TC2),
one has that
$\O \to \mathcal{P}$ is both a local genuine $\otimes$-trivial cofibration and $\F_{all}$-essentially surjective.
Firstly, if $j$ happens to be a map in (TC2),
then this pushout
can be alternatively calculated as the pushout below
in the fixed color category $\mathsf{Op}^{G}_{\mathfrak C_{\O}}(\V)$.
\begin{equation}
\begin{tikzcd}
\check{a}_!J_1 \ar{r} \arrow{d}[swap]{\check{a}_!j}
&
\O \arrow{d}
\\
\check{a}_!J_2 \ar{r}
&
\P
\end{tikzcd}
\end{equation}
And, since $\check{a}_!$ is left Quillen (cf. Corollary \ref{OPADJ_COR}\ref{OPCOCHADJ_LBL}),
this is the pushout of a trivial cofibration in the fiber model structure
$\mathsf{Op}^{G}_{\mathfrak C_{\O},\F}(\V)$.
The essential surjectivity claim is then obvious,
while the $\otimes$-trivial cofibration
claim follows from
Remark \ref{GOTC_REM} and Proposition \ref{GOTC_PROP}.
Secondly, in the more interesting case of $j$ a map in (TC1),
i.e. of the form $G/H \cdot (\eta \to \mathbb{G})$ for $\mathbb{G}$ a generating $\V$-interval,
we split the pushout \eqref{JUSTAPUSH EQ} as a composition of two pushouts
\begin{equation}\label{BROKENPUSH EQ}
\begin{tikzcd}
G/H \cdot \eta \arrow[r, "a"] \arrow[d, "G/H \cdot \phi"']
&
\O \arrow[d,"\phi'"]
\\
G/H \cdot \mathbb{G}_{\set{0}} \arrow[r] \arrow[d, "G/H \cdot \psi"']
&
\O' \arrow[d,"\psi'"]
\\
G/H \cdot \mathbb{G} \arrow[r]
&
\P
\end{tikzcd}
\end{equation}
where $\mathbb{G}_{\set{0}}$ is the full $\V$-subcategory of $\mathbb{G}$ spanned by the object $0$.
It now suffices to show that the desired properties hold individually for $\phi'$ and $\psi'$.
For the top pushout in \eqref{BROKENPUSH EQ}, Theorem \ref{INTCOF THM} implies that $\eta \to \mathbb{G}$
is a trivial cofibration in $\mathsf{Cat}_{\{0\}}(\V)$
so that, since Corollary \ref{OPADJ_COR}\ref{OPCOMBADJ_LBL}
says that $G/H \cdot (-) \colon \mathsf{Cat}_{\{0\}}(\V) \to \mathsf{Cat}^G_{G/H,\F_{all}}(\V)$ is left Quillen,
we have that $G/H \cdot \phi$ is a $\F_{all}$-trivial cofibration with fixed objects,
and thus the (TC2) argument above implies $\phi'$ is
a local genuine $\otimes$-trivial cofibration and
$\F_{all}$-essentially surjective.
Now consider the bottom pushout in \eqref{BROKENPUSH EQ}.
Since $\psi$ is a local isomorphism (i.e. $\mathbb{G}_{\{0\}} \to \psi^{\**}\mathbb{G}$ is an isomorphism),
so is $G/H \cdot \psi$ and thus, by
\cite[Cor. \ref{OC-LOCALISO_COR}]{BP_FCOP}
(see also \cite[Prop. B.22]{Cav}),
the map $\psi' \colon \O' \to \mathcal{P}$
is itself a local isomorphism, and thus certainly a local genuine $\otimes$-trivial cofibration.
To address $\F_{all}$-essential surjectivity,
we write $[g]_0$ and $[g]_1$ for $[g] \in G/H$
to denote the objects of $G/H \cdot \mathbb{G}$,
so that $\C_{\P} = \C_{\O} \amalg \{[g]_1\}_{[g] \in G/H}$.
Clearly one needs only verify the essential surjectivity condition for the $[g]_1$.
Given $K \leq G$,
$[g]_1$ is $K$-fixed in $\P$ iff it is $K$-fixed in $G/H \cdot \mathbb{G}$,
in which case $[g]$ induces a map
$\mathbb{G} \xrightarrow{[g]} \left(G/H \cdot \mathbb{G}\right)^K$.
Writing $i$ for the composite
$\mathbb{G} \xrightarrow{[g]} \left(G/H \cdot \mathbb{G}\right)^K \to \P^K$,
one then has $i(0) = a([g]_0)$ and $i(1)=[g]_1$,
establishing $\F_{all}$-essential surjectivity.
\end{proof}
\begin{remark}\label{JCELLSM REM}
If $\O \in \mathsf{sOp}^G_{\bullet}$
in \eqref{JUSTAPUSH EQ}
is underlying cofibrant in $\mathsf{Sym}^G_{\mathfrak{C}_{\O}}(\V)$
then, by \cite[Rem. \ref{OC-THMISM REM}]{BP_FCOP},
the map $\O \to \P$ in \eqref{JUSTAPUSH EQ}
is actually a local genuine trivial cofibration
(rather than just a local genuine $\otimes$-trivial cofibration).
Hence, the claim that
``a map \emph{with $\F$-cofibrant domain} in the saturation
of (TC1),(TC2) is a $\F$-weak equivalence in
$\mathsf{sOp}^G_{\bullet}$''
does not require the global monoid axiom
in \cite[Def. \ref{OC-GLOBMONAX_DEF}]{BP_FCOP}
and in (iv) of Theorem \ref{THMA}.
\end{remark}
\subsection{Equivalences of objects}\label{EQUIVOBJ_SEC}
Our next task is to show that the
$\F$-weak equivalences
in Definition \ref{MODEL_DEFN}
satisfy $2$-out-of-$3$,
with the main difficulty coming
from the fact that essential surjectivity is
defined using $\V$-intervals.
To address this, this section relates the $\F$-weak equivalences
in Definition \ref{MODEL_DEFN} with the $\F$-Dwyer-Kan equivalences
in the statement of Theorem \ref{THMA},
for which $2$-out-of-$3$ is easier to establish
(though we note that this claim is more subtle in
the equivariant setting;
see Proposition \ref{23HARDCASE PROP}).
\begin{definition}\label{HTPY_DEFN}
Suppose that $(\V,\otimes)$ has a cofibrant unit.
Given $\mathcal C \in \Cat_{\mathfrak{C}}(\V)$,
we define $\pi_0 \mathcal C \in \Cat_{\mathfrak{C}}(\mathsf{Set})$
to be the ordinary category with the same objects and
\[
\pi_0(\mathcal{C})(c,d)=
\Ho(\V)(1_\V, \mathcal C(c,c'))=
[1_\V, \mathcal{C}_f(c,c')]
\]
where $[-,-]$ denotes homotopy equivalence classes of maps,
and $\mathcal{C}_f$ denotes some fibrant replacement of
$\mathcal C$ in $\Cat_{\mathfrak{C}}(\V)$.
The composition $[g]\circ [f]$
in $\pi_0(\mathcal{C})$
of classes $[f],[g]$
represented by
$1_{\mathcal{V}} \xrightarrow{f} \mathcal{C}_f({c,c'})$
and
$1_{\mathcal{V}} \xrightarrow{g} \mathcal{C}_f({c',c''})$
is given by the class $[gf]$, where $gf$ denotes the composite
\begin{equation}\label{COMPI0 EQ}
1_{\mathcal{V}} \simeq
1_{\mathcal{V}} \otimes 1_{\mathcal{V}} \xrightarrow{g \otimes f}
\mathcal{C}_f({c',c''}) \otimes \mathcal{C}_f({c,c'}) \xrightarrow{\circ}
\mathcal{C}_f({c,c''}).
\end{equation}
\end{definition}
The assumption that $1_{\mathcal{V}}$ is cofibrant
is needed to prove that \eqref{COMPI0 EQ}
respects equivalence classes.
Moreover, since any two fibrant replacements are connected by a zigzag of weak equivalences,
(the isomorphism class of) $\pi_0 \mathcal{C}$ does not depend on the choice of fibrant replacement $\mathcal{C}_f$.
\begin{remark}
The assignment
$\pi_0\colon \mathsf{Cat}_{\mathfrak{C}}(\V)
\to \mathsf{Cat}_{\mathfrak{C}}(\mathsf{Set})$ is functorial,
i.e. a $\V$-functor
$\mathcal{C} \to \mathcal{D}$
induces a functor
$\pi_0\mathcal{C} \to \pi_0\mathcal{D}$.
Moreover, $\pi_0$ sends weak equivalences to isomorphisms.
\end{remark}
\begin{remark}\label{NATISO REM}
The map $\widetilde{\mathbbm{1}} \to \widetilde{\mathbbm{1}}_f$
shows that the identity
$id_{1_{\V}}\colon 1_{\V} \xrightarrow{=} 1_{\V} = \widetilde{\mathbbm{1}}(0,1) = \widetilde{\mathbbm{1}}(1,0)$
induces two inverse arrows
$[id_{1_{\V}}] \in \pi_0 \widetilde{\mathbbm{1}}(0,1)$
and
$[id_{1_{\V}}] \in \pi_0 \widetilde{\mathbbm{1}}(1,0)$.
We refer to these arrows as the
\emph{natural isomorphisms} between $0$ and $1$ in $\pi_0 \widetilde{\mathbbm{1}}$.
\end{remark}
Following \cite[Def. 2.6]{BM13} (also \cite{Cav}),
we make the following definitions.
\begin{definition}\label{EQUIV_DEF}
Given $\mathcal{C}$ in $\Cat(\V)$ and $c,c'\in \mathcal C$, we say $c$ and $c'$ are
\begin{itemize}
\item {\em equivalent} if there exists a $\V$-interval $\mathbb{J}$
and map $i: \mathbb{J} \to \mathcal C$ such that
$i(0)= c$, $i(1)= c'$;
\item {\em virtually equivalent} if $c,c'$ are equivalent in some fibrant replacement
$\mathcal C_f$ of $\mathcal C$ in $\Cat_{\mathfrak{C}_{\mathcal{C}}}(\V)$;
\item {\em homotopy equivalent} if $c,c'$ are isomorphic in the unenriched category $\pi_0 \mathcal C$.
Explicitly, this means there are maps
$1_\V \xrightarrow{\alpha} \mathcal C_f(c,c')$,
$1_\V \xrightarrow{\beta} \mathcal C_f(c',c)$ such that
$1_{\V} \xrightarrow{\beta \alpha} \mathcal C_f(c,c)$,
$1_{\V} \xrightarrow{\alpha \beta} \mathcal C_f(c',c')$
are homotopic to the identities
$1_{\V} \xrightarrow{id_c} \mathcal C_f(c,c)$,
$1_{\V} \xrightarrow{id_{c'}} \mathcal C_f(c',c')$.
\end{itemize}
\end{definition}
\begin{remark}\label{VIRTEQRESTA REM}
Given $c,c' \in \mathcal{C}$, write $\iota_{c,c'} \ \colon \{0,1\} \to \mathfrak{C}_{\mathcal{C}}$
for the induced map
(for $\mathfrak{C}_{\mathcal{C}}$ the object set of
$\mathcal{C}$).
The condition that $c,c'$ are equivalent can then be restated as saying that
there is a $\V$-interval $\mathbb{J}$
together with some map $\mathbb{J} \to \iota^{\**}_{c,c'}\mathcal{C}$ in $\mathsf{Cat}_{\{0,1\}}(\V)$.
We can similarly restate the notion of virtual equivalence.
Since a $\V$-interval $\mathbb{J}$ is a cofibrant replacement of $\widetilde{\mathbbm{1}}$ in $\mathsf{Cat}_{\{0,1\}}(\V)$
while $\iota^{\**}_{c,c'} \mathcal{C}_f$ is a fibrant replacement of $\iota^{\**}_{c,c'} \mathcal{C}$,
the condition that $c,c'$ are virtually equivalent is precisely the statement that
\[
\Ho \left(\mathsf{Cat}_{\{0,1\}}(\V)\right)\left(\widetilde{\mathbbm{1}},\iota^{\**}_{c,c'} \mathcal{C}\right)
=
[\mathbb{J},\iota^{\**}_{c,c'} \mathcal{C}_f]
\neq
\emptyset
\]
i.e. that, up to homotopy, there is at least one map from $\widetilde{\mathbbm{1}}$ to $\iota^{\**}_{c,c'} \mathcal{C}$
in $\mathsf{Cat}_{\{0,1\}}(\V)$.
Note in particular that this does not depend on the choice of
replacements
$\mathbb{J}$ and $\mathcal{C}_f$.
\end{remark}
\begin{remark}
That homotopy equivalence of objects is an equivalence relation
follows from the fact that composites of isomorphisms are isomorphisms.
On the other hand, when checking that equivalence and virtual equivalence are likewise equivalence relations,
the transitive property is not elementary, being instead
a straightforward consequence of Interval Amalgamation, cf. Lemma \ref{AMALGLEM LEM}.
\end{remark}
\begin{remark}\label{EQUIVNEST_REM}
The notions in Definition \ref{EQUIV_DEF}
are nested: the map $\mathcal{C} \to \mathcal{C}_f$
yields that equivalence implies virtual equivalence;
a map $\mathbb{J} \to \mathcal{C}_f$
with $\mathbb{J}$ a $\mathcal{V}$-interval
induces a map
$\pi_0 \widetilde{\mathbbm{1}} \simeq \pi_0 \mathbb{J} \to \pi_0 \mathcal{C}$,
so that (since $0,1 \in \pi_0 \widetilde{\mathbbm{1}}$ are isomorphic)
virtual equivalence implies homotopy equivalence.
Moreover, \cite{BM13,Cav} show that, under suitable assumptions on $\V$, the converse implications also hold, as summarized below.
We discuss these converse results in what follows.
\begin{equation}\label{BASCOMP EQ}
\begin{tikzcd}[column sep = large]
\mbox{equivalent}
\arrow[r, Rightarrow, shift left=2, "\mathrm{always}"]
&
\mbox{ virtually equivalent}
\arrow[r, Rightarrow, shift left=2, "\mathrm{always}"]
\arrow[l, Rightarrow, shift left = 2, "\substack{$\V\phantom{ }$\mathrm{right} \\ \mathrm{proper}}"]
&
\mbox{ homotopy equivalent}
\arrow[l, Rightarrow, shift left = 2, "\substack{\mathrm{coherence} \\ \mathrm{condition}}"]
\end{tikzcd}
\end{equation}
\end{remark}
\begin{proposition}[{cf. \cite[4.12]{Cav}, \cite[2.10]{BM13}}] \label{RIGHTPROPER PROP}
If $\V$ is right proper, then two colors in a $\V$-category $\mathcal{C}$ are virtually equivalent iff they are equivalent.
\end{proposition}
\begin{proof}
Let $c,c'\in \mathcal{C}$ be two colors and (following Remark \ref{VIRTEQRESTA REM})
let $\mathbb{J} \overset{\sim}{\twoheadrightarrow} \iota_{c,c'} \mathcal{C}_F$
exhibit a virtual equivalence between them, where we note
that we can assume the map is a fibration in $\mathsf{Cat}_{\{0,1\}}(\mathcal{C})$
by using the factorization of any map as a "trivial cofibration followed by a fibration".
Forming the pullback
\begin{equation}\label{RIGHTPROPER EQ}
\begin{tikzcd}
\mathbb{J}' \ar{r} \arrow{d}
&
\iota_{c,c'} \mathcal{C} \arrow{d}{\sim}
\\
\mathbb{J} \ar[twoheadrightarrow]{r}
&
\iota_{c,c'}\mathcal{C}_f
\end{tikzcd}
\end{equation}
right properness of $\V$ implies
that $\mathbb{J}' \to \mathbb{J}$ is a weak equivalence in
$\mathsf{Cat}_{\{0,1\}}(\V)$,
and thus choosing a cofibrant replacement
$\mathbb{J}'_c \to \mathbb{J}'$
yields that $c,c'$ are equivalent.
\end{proof}
We now discuss the requirement for homotopy equivalence and virtual equivalence to coincide.
Informally, this will be the case provided any isomorphism
$c \to c'$ in $\pi_0 \mathcal{C}$
can be suitably lifted to a map
$\mathbb{J} \to \mathcal C_f$ for some $\V$-interval $\mathbb{J}$.
The following makes this idea precise, cf. \cite[\S 2]{BM13}.
\begin{definition}\label{COH DEF}
Let $\mathbbm{1}$ be the free $\V$-arrow category and $\mathbb{J}$ a $\V$-interval.
A cofibration $\mathbbm{1} \to \mathbb{J}$ in $\Cat_{\set{0,1}}(\V)$
is called \textit{natural} if it fits into a commutative diagram in $\Cat_{\set{0,1}}(\V)$ as on the left below.
\begin{equation}
\begin{tikzcd}
\mathbbm{1} \arrow[d, tail] \arrow[r]
&
\widetilde{\mathbbm{1}} \arrow[d, "\sim"]
&
\mathbbm{1} \arrow[r, "\alpha"] \arrow[d, tail, dashed]
&
\mathcal C_f
\\
\mathbb{J} \arrow[r, "\sim"']
&
\widetilde{\mathbbm{1}}_f
&
\mathbb{J} \arrow[ur, dashed]
\end{tikzcd}
\end{equation}
A homotopy equivalence between two objects in a $\V$-category $\mathcal C$ is called \textit{coherent} if
there is a representative $\alpha: \mathbbm{1} \to \mathcal C_f$ that factors along a natural cofibration,
as on the right above.
Lastly, the monoidal model category $\V$ is said to satisfy the \textit{coherence axiom} if
all homotopy equivalences in every $\V$-category are coherent.
\end{definition}
\begin{remark}
\label{COH_EX_REM}
The coherence axiom originates with the work of Boardman-Vogt, who showed that it holds for compactly-generated weak Hausdorff spaces $(\Top, \times)$ \cite[Lem. 4.16]{BV73}.
The coherence axiom is also a consequence of Lurie's \textit{invertibility hypothesis} \cite[A.3.2.12]{Lur09}, a stronger hypothesis, by an argument of Berger-Moerdijk \cite[Rem. 2.19]{BM13}.
\end{remark}
\begin{remark}
Ignoring the more technical requirements,
Definition \ref{COH DEF} loosely says that
$\mathbbm{1} \rightarrowtail \mathbb{J}$
is natural if the map $1_{\V} = \mathbbm{1}(0,1) \to \mathbb{J}(0,1)$
represents the natural isomorphism
$[id_{1_{\V}}]$ from $0$ to $1$ in $ \pi_0 \widetilde{\mathbbm{1}} \simeq \pi_0 \mathbb{J}$ (cf. Remark \ref{NATISO REM}),
and that the homotopy equivalence $\alpha$ is coherent
if there exists a map $\mathbb{J} \to \mathcal C_f$
such that
$\pi_0 \widetilde{\mathbbm{1}} \simeq
\pi_0 \mathbb{J} \to \pi_0 \mathcal C$
sends the natural isomorphism $[id_{1_{\V}}]$ to $[\alpha]$.
If, in addition, $\V$ is also right proper, we can slightly strengthen this observation, as follows.
\end{remark}
\begin{proposition}\label{ALTCOH PROP}
Suppose $\V$ is right proper and satisfies the coherence axiom.
Then, for any $\V$-category $\mathcal{C}$ and isomorphism
$[\alpha]$ in $\pi_0 \mathcal{C}$,
there exists a map from an interval
$\mathbb{J} \to \mathcal{C}$
such that
$\pi_0 \mathbb{J} \to \pi_0\mathcal{C}$
sends the natural isomorphism $[id_{1_{\V}}]$ to $[\alpha]$.
\end{proposition}
\begin{proof}
By coherence, we can find a factorization of
$\alpha$ as
$\1 \rightarrowtail \mathbb J \twoheadrightarrow \mathcal C_f$
where, just as in the proof of
Proposition \ref{RIGHTPROPER PROP},
we are free to assume the second map is a fibration.
The result then follows by forming the pullback \eqref{RIGHTPROPER EQ} and arguing as in the proof of Proposition \ref{RIGHTPROPER PROP}.
\end{proof}
Berger and Moerdijk then prove the following,
which depends on a careful technical analysis of the $W$-construction applied to free isomorphism $\V$-category
$\widetilde{\mathbbm{1}}$.
\begin{proposition}[{cf. \cite[Prop. 2.24]{BM13}}]
\label{COHAX PROP}
If a cofibrantly generated monoidal model category $\V$
satisfies the monoid axiom and has cofibrant unit,
then $\V$ satisfies the coherence axiom.
\end{proposition}
Just as in Theorem \ref{INTCOF THM},
we have replaced the adequacy assumption \cite[Def. 1.1]{BM13}
with the monoid axiom, which suffices for the existence of the relevant model structures.
Yet again the proof in loc. cit. makes no direct use of adequacy,
though special mention should be made of \cite[Lemma 2.23]{BM13},
which builds the \emph{interval} \cite[Def. 4.1]{BM06} in $\V$
required to define the $W$-construction.
The pointed category $\V_{\**}=\V_{1_{\V}//1_{\V}}$
of factorizations
$1_{\V} \to X \to 1_{\V}$
of the identity
$1_{\V} \xrightarrow{=} 1_{\V}$
has a monoidal structure
$X \wedge Y = \mathop{\mathrm{coeq}} \left(1_{\V} \leftarrow X \amalg_{1_{\V}} Y \to X \otimes Y\right)$ with unit $1_{\V} \amalg 1_{\V}$,
which is readily seen to define a cofibrantly generated monoidal model category with cofibrant unit whenever $(\V,\otimes)$ is one.
A \emph{segment} \cite[Def. 4.1]{BM06} is then a monoid
in $(\V_{\**},\wedge)$,
while an interval is a segment $H$ for which the two natural maps
$1_{\V} \amalg 1_{\V} \rightarrowtail H \xrightarrow{\sim} 1_{\V}$
are a cofibration and weak equivalence in $\V$.
Our hypothesis are not quite strong enough
to guarantee that the category of segments $\mathsf{Seg}_{\V}$
has a full model structure
(the monoid axiom for $(\V,\otimes)$ does not imply the monoid axiom for $(\V_{\**},\wedge)$) but,
by Remarks \ref{SEMI_REM}, \ref{GTRIV REM}
we nonetheless have a semi-model structure. This is enough to build an interval as a cofibrant replacement
$H \xrightarrow{\sim} 1_{\V}$
of the terminal segment $1_{V}$, with the fact that the forgetful functor
$\mathsf{Seg}_{\V} \to \V_{\**}$
preserves cofibrations between cofibrant objects
(by the semi-model structure version of \cite[Thm. \ref{OC-THMII}]{BP_FCOP};
see Remarks \ref{SEMI_REM} and \ref{GTRIV REM})
implying that
$1_{\V} \amalg 1_{\V} \rightarrowtail H$
is indeed a cofibration in $\V$.
\vskip 10pt
Replacing the notion of equivalence of objects
in Definition \ref{MODEL_DEFN}
with that of homotopy equivalence,
one obtains the notion of Dwyer-Kan equivalence
in the formulation of Theorem \ref{THMA}.
\begin{definition}\label{DKEQUIV_DEF}
Let $\F$ be a $(G,\Sigma)$-family with enough units.
We say a map $\O \to \P$ in $\Op^G_\bullet(\V)$ is:
\begin{itemize}
\item \textit{$\F$-$\pi_0$-essentially surjective} if
$j^{\**}\pi_0 \O^H \to j^{\**}\pi_0 \P^H$
is essentially surjective for $H \in \F_1$;
\item a \textit{$\F$-Dwyer-Kan equivalence} if
it is a local $\F$-weak equivalence and $\F$-$\pi_0$-essentially
surjective.
\end{itemize}
\end{definition}
\begin{remark}\label{CATEQUIV REM}
The requirement that
a $\F$-Dwyer-Kan equivalence $\O \to \P$
be a local $\F$-weak equivalence
implies that the maps of categories
$j^{\**}\pi_0 \O^H \to j^{\**}\pi_0 \P^H$
must be local isomorphisms,
and thus equivalences in the category $\mathsf{Cat}$
of (unenriched) categories in the usual sense.
\end{remark}
\begin{corollary}\label{WEDKEQ COR}
$\F$-weak equivalences in $\mathsf{Op}^G_\bullet(\V)$
are $\F$-Dwyer-Kan equivalences.
Further, the converse holds provided that
$\V$ is a cofibrantly generated monoidal model category
which satisfies the monoid axiom, is right proper,
and has a cofibrant unit.
\end{corollary}
\begin{proof}
This follows by \eqref{BASCOMP EQ} for the first claim
and by Propositions \ref{RIGHTPROPER PROP} and \ref{COHAX PROP}
for the converse.
\end{proof}
\begin{proposition}\label{2OUTOF3 PROP}
Suppose that $\V$ satisfies conditions
(i) through (v) and (vii) in Theorem \ref{THMA} and
let $\F$ be a $(G,\Sigma)$-family which has enough units.
Then:
\begin{enumerate}[label=(\roman*)]
\item $\F$-weak equivalences between fibrant objects in $\mathsf{Op}^G_\bullet(\V)$
satisfy $2$-out-of-$3$;
\item if $\V$ is right proper, $\F$-weak equivalences in $\mathsf{Op}^G_\bullet(\V)$
satisfy $2$-out-of-$3$;
\item $\F$-Dwyer-Kan equivalences in $\mathsf{Op}^G_\bullet(\V)$
satisfy $2$-out-of-$3$.
\end{enumerate}
\end{proposition}
\begin{proof}
Consider the diagram in $\mathsf{Op}^G_\bullet(\V)$ on the left below,
and the induced maps on the right for each
$\mathfrak{C}_{\O}$-signature $\vect{C}$ and
$\Lambda \in \F_{\vect{C}}$.
\begin{equation}\label{23EASYDIAG EQ}
\begin{tikzcd}[row sep=5pt]
\O \arrow{rr}{\bar{F}F}
\arrow{dr}[swap]{F}
&&
\mathcal{Q}
&
\O(\vect{C})^{\Lambda} \arrow{rr
\arrow{dr
&&
\mathcal{Q} (\bar{F}F(\vect{C}))^{\Lambda}
\\
&
\mathcal{P} \ar{ru}[swap]{\bar{F}}
&
&
&
\mathcal{P}(F(\vect{C}))^{\Lambda} \ar{ru
&
\end{tikzcd}
\end{equation}
We first address (i) and (iii) in parallel
(where for (i) we assume $\O,\P,\mathcal{Q}$ are fibrant).
Suppose first that $F,\bar{F}$ are $\F$-weak equivalences/$\F$-Dwyer Kan equivalences.
Then $2$-out-of-$3$ applied to the right diagram in
\eqref{23EASYDIAG EQ}
implies that $\bar{F}F$ is a
local $\F$-weak equivalence.
Moreover, in the $\F$-weak equivalence case
Lemma \ref{TRANSCOMP_ES_LEM}
implies $\bar{F}F$ is $\F$-essentially surjective,
while in the $\F$-Dwyer-Kan equivalence case
Remark \ref{CATEQUIV REM}
and $2$-out-of-$3$ for (unenriched categories)
implies
$\bar{F}F$ is $\F$-$\pi_0$-essentially surjective.
Suppose next that $\bar{F},\bar{F}F$ are $\F$-weak equivalences/$\F$-Dwyer Kan equivalences.
It is again immediate that $F$ is a
local $\F$-weak equivalence and that,
in the $\F$-Dwyer Kan equivalence case,
$F$ is $\F$-$\pi_0$-essentially surjective.
It remains to establish the
$\F$-essential surjectivity of $F$ in the $\F$-weak equivalence case.
Given $b \in \mathcal{P}^H$,
the $\F$-essential surjectivity of $\bar{F}F$
yields $a \in \O^H$ and a map
$\mathbb{J}
\to
\iota^{\**}_{\bar{F}Fa,\bar{F}b}
\left( j^{\**} \mathcal{Q}^H \right)
=
\iota^{\**}_{Fa,b}
\left( j^{\**} \bar{F}^{\**}\mathcal{Q}^H \right)$
in
$\mathsf{Cat}_{\{0,1\}}(\V)$
(see Remark \ref{VIRTEQRESTA REM}).
But since
$\P \to \bar{F}^{\**} \mathcal{Q}$
is a weak equivalence of fibrant objects in
$\mathsf{Op}^G_{\mathfrak{C}_{\P}}(\V)$,
the map
$
\iota^{\**}_{Fa,b}
\left( j^{\**} \mathcal{P}^H \right)
\to
\iota^{\**}_{Fa,b}
\left( j^{\**} \bar{F}^{\**}\mathcal{Q}^H \right)$
is likewise a weak equivalence of
fibrant objects in $\mathsf{Cat}_{\{0,1\}}(\V)$,
and one thus also has a map
$\mathbb{J}
\to
\iota^{\**}_{Fa,b}
\left( j^{\**} \mathcal{P}^H \right)
$,
showing that $F$ is $\F$-essential surjective.
Consider now the remaining case where
$F,\bar{F}F$ are
$\F$-weak equivalences/$\F$-Dwyer Kan equivalences.
The $\F$-essential surjectivity/$\F$-$\pi_0$-essential surjectivity
of $\bar{F}F$
states that, for any $c \in \mathcal{Q}^H$,
there exists $a \in \mathcal{O}^H$ and a map
$\mathbb{I} \to \iota^{\**}_{\bar{F}Fa,c} j^{\**}\mathcal{Q}^H$
/isomorphism between $\bar{F}Fa$ and
$c$ in $j^{\**}\pi_0\mathcal{Q}^H$.
Hence, setting $b = Fa \in \mathcal{P}^H$,
we see that $\bar{F}$ is also
$\F$-essential surjective/$\F$-$\pi_0$-essential surjective.
It remains to show that
$F$ is a local $\F$-weak equivalence.
In contrast with the previous cases,
applying $2$-out-of-$3$ to the right diagram in \eqref{23EASYDIAG EQ}
only yields equivalences
$\P(\vect{D})^{\Lambda} \to \mathcal{Q}(\bar{F}(\vect{D}))^{\Lambda}$
for $\mathfrak{C}_{\P}$-signatures of the form
$\vect{D} = \bar{F} (\vect{C})$ for some $\mathfrak{C}_{\O}$-signature $\vect{C}$
and $\Lambda \in \F_{\vect{C}}$.
To show that we have equivalences for all
$\mathfrak{C}_{\P}$-signatures $\vect{D}$
and all $\Lambda \in \F_{\vect{D}}$
(note that, even if $\vect{D} = F(\vect{C})$,
one can only guarantee $\F_{\vect{C}} \subseteq \F_{\vect{D}}$,
rather than $\F_{\vect{C}} = \F_{\vect{D}}$),
one needs a careful modification of the analogous non-equivariant argument \cite[Lemma 4.14]{Cav}.
As such, we postpone this claim to
Proposition \ref{23HARDCASE PROP} below
(note that, by Corollary \ref{WEDKEQ COR},
only the $\F$-Dwyer Kan equivalence case needs be considered),
and dedicate the entirety of \S \ref{HMTYEQ SEC}
to proving that result.
We note that this is the case which requires that $\F$ has enough units (Definition \ref{FAMRESUNI DEF}).
Lastly, we address (ii).
Given a map of operads $F\colon \O \to \P$ and a color fixed fibrant replacement
$F_f \colon \O_f \to \P_f$,
it is immediate that
$F$ is a $\F$-local weak equivalence iff $F_f$ is.
Moreover, if $\V$ is right proper, Proposition \ref{RIGHTPROPER PROP}
implies that $F$ is $\F$-essentially surjective iff $F_f$ is.
In other words, if $\V$ is right proper,
$F$ is a $\F$-weak equivalence iff $F_f$ is.
But (ii) is now immediate from (i).
\end{proof}
\subsection{Homotopy equivalences and fully faithfulness}
\label{HMTYEQ SEC}
This section is dedicated to proving the following,
which is the remaining claim in the proof of Proposition \ref{2OUTOF3 PROP},
and is the claim requiring
the "enough units" condition in Definition \ref{FAMRESUNI DEF}.
\begin{proposition}\label{23HARDCASE PROP}
Suppose $\V$ satisfies the conditions
(i) through (v) and (vii) in Theorem \ref{THMA},
and let $\F$ be a $(G,\Sigma)$-family which has enough units.
Consider the diagram below
in $\mathsf{Op}^G_\bullet(\V)$.
\begin{equation}\label{23HARDDIAG EQ}
\begin{tikzcd}[row sep=5pt]
\O \arrow{rr}{\bar{F}F}
\arrow{dr}[swap]{F}
&&
\mathcal{Q}
\\
&
\mathcal{P} \ar{ru}[swap]{\bar{F}}
\end{tikzcd}
\end{equation}
If $F$ and $\bar{F}F$ are $\F$-Dwyer-Kan equivalences
then $\bar{F}$ is a local $\F$-weak equivalence.
\end{proposition}
The proof of this result will adapt the proof of the non-equivariant analogue \cite[Lemma 4.14]{Cav},
but one must be more careful since equivariance introduces a number of subtleties.
For the sake of motivation,
we first discuss a concrete example.
\begin{example}
Let $\V=\mathsf{sSet}$,
$G = \{1,i,-1,-i\} \simeq \mathbb{Z}_{/4}$
be the group of quartic roots of unit,
and $\mathfrak{C} = \{\mathfrak{a}, \mathfrak{b}, i \mathfrak{b},
\mathfrak{c} \}$,
where we implicitly have
$i\mathfrak{a} = \mathfrak{a}$,
$-\mathfrak{b} = \mathfrak{b}$,
$i\mathfrak{c} = \mathfrak{c}$.
Consider the $\mathfrak{C}$-corollas below.
\begin{equation}
\begin{tikzpicture}[auto,grow=up, level distance = 2.2em,
every node/.style={font=\scriptsize,inner sep = 2pt}
\tikzstyle{level 2}=[sibling distance=3em
\node at (0,0) [font = \normalsize] {$\vect{B}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$i\mathfrak{b}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$i\mathfrak{b}$}
child{node {
edge from parent node {$\mathfrak{b}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\node at (4.5,0) [font = \normalsize] {$\vect{C}$
child{node [dummy] {
child{node {
edge from parent node [swap] {$\mathfrak{c}$}
child[level distance = 2.9em]{node {
edge from parent node [swap, near end] {$\mathfrak{c}$}
child[level distance = 2.9em]{node {
edge from parent node [near end] {$\mathfrak{c}$}
child{node {
edge from parent node {$\mathfrak{c}$}
edge from parent node [swap] {$\mathfrak{a}$}}
\end{tikzpicture
\end{equation
Let
$\O \xrightarrow{F} \P \xrightarrow{\bar{F}} \mathcal{Q}$
be as in \eqref{23HARDDIAG EQ},
and suppose
$\mathfrak{C}_{\O} = \{\mathfrak{a},\mathfrak{c}\}$,
$\mathfrak{C}_{\P} = \mathfrak{C} = \{\mathfrak{a},\mathfrak{b},i\mathfrak{b},\mathfrak{c}\}$.
Then, if $F$ and $\bar{F}F$ are local Kan equivalences,
it is clear that
$\P(\vect{C}) \to \mathcal{Q}(\bar{F}(\vect{C}))$
is a Kan equivalence, since
$\vect{C}$ is in the image of $F$.
However, to ensure that
$\P(\vect{B}) \to \mathcal{Q}(\bar{F}(\vect{B}))$
is also a Kan equivalence,
we need an essential surjectivity condition on $F$.
For concreteness,
suppose there was a homotopy equivalence
$\alpha \colon \mathfrak{b} \to \mathfrak{c}$
in $\P$
(cf. Definition \ref{EQUIV_DEF}; note that $\alpha \in \P(\mathfrak{b};\mathfrak{c})$).
Then, by $G$-equivariance, one also has a homotopy equivalence
$i\alpha \colon i\mathfrak{b} \to \mathfrak{c}$ in $\P$,
so that
(cf. Corollary \ref{ALBEETA COR};
see also \cite[Lemma 2.1]{BM13},
\cite[Lemma 4.14]{Cav})
precomposing
with $\alpha$, $i \alpha$ gives a string of Kan equivalences
\begin{equation}\label{ITERWE EQ}
\P(\vect{C})
=
\P(\mathfrak{c},\mathfrak{c},\mathfrak{c},\mathfrak{c};\mathfrak{a})
\sim
\P(\mathfrak{b},\mathfrak{c},\mathfrak{c},\mathfrak{c};\mathfrak{a})
\sim
\P(\mathfrak{b},i\mathfrak{b},\mathfrak{c},\mathfrak{c};\mathfrak{a})
\sim
\P(\mathfrak{b},i\mathfrak{b},i\mathfrak{b},\mathfrak{c};\mathfrak{a})
\sim
\P(\mathfrak{b},i\mathfrak{b},i\mathfrak{b},\mathfrak{b};\mathfrak{a})
=
\P(\vect{B})
\end{equation}
and similarly
$\mathcal{Q}(\bar{F}(\vect{C}))
\sim
\cdots
\sim
\mathcal{Q}(\bar{F}(\vect{B}))$,
so that
$\P(\vect{B}) \to \mathcal{Q}(\bar{F}(\vect{B}))$
is indeed a Kan equivalence.
Our discussion thus far has ignored a key feature of the equivariant setting: the choice of $(G,\Sigma)$-family $\F$.
Given a general such $\F$,
and assuming $F$ and $\bar{F} F$ are local $\F$-Kan equivalences,
it is again clear that
$\P(\vect{C})^{\Lambda} \to \mathcal{Q}(\bar{F}(\vect{C}))^{\Lambda}$
is a Kan equivalence for all $\Lambda \in \F_{\vect{C}}$.
And, yet again, to conclude that
$\P(\vect{B})^{\Lambda} \to \mathcal{Q}(\bar{F}(\vect{B}))^{\Lambda}$
is also a Kan equivalence for $\Lambda \in \F_{\vect{B}}$ we further need an essential surjectivity requirement on $F$.
As it turns out, this requirement on $F$ depends on
$\Lambda \leq G \times \Sigma_4^{op}$ itself so that, for concreteness, we set (writing $(g,\sigma) \in G \times \Sigma_4^{op}$ simply as $g\sigma$)
\[
\Lambda = \langle (14)(23), i (12)(34) \rangle.
\]
and assume that $\Lambda \in \F_4$.
Note that $\Lambda$ stabilizes both
$\vect{B}$ and $\vect{C}$
(cf. Definition \ref{STABS DEF}),
so that
$\Lambda \leq \mathsf{Aut}_{G \ltimes \Sigma^{op}_{\mathfrak{C}}}(\vect{B})$,
$\Lambda \leq \mathsf{Aut}_{G \ltimes \Sigma^{op}_{\mathfrak{C}}}(\vect{C})$
and $\P(\vect{B})^{\Lambda}, \P(\vect{C})^{\Lambda}$
are well defined.
At this point it may be tempting to think that,
given a homotopy equivalence
$\alpha \colon \mathfrak{b} \to \mathfrak{c}$,
one may simply apply $\Lambda$-fixed points
to \eqref{ITERWE EQ}
to obtain a Kan equivalence
$\P(\vect{B})^{\Lambda} \simeq \P(\vect{C})^{\Lambda}$.
However, this argument fails: not only are the intermediate objects
in \eqref{ITERWE EQ} not $\Lambda$-equivariant, so that the intermediate maps therein can not possibly be $\Lambda$-equivariant,
neither is it necessarily the case that the composite Kan equivalence
$\P(\vect{C}) \simeq \P(\vect{B})$ is $\Lambda$-equivariant,
unless one makes a further assumption on the homotopy equivalence
$\alpha\colon \mathfrak{b} \to \mathfrak{c}$.
Namely, since $\Lambda$
must contain the square of the element $i(12)(34)$,
which is $-1 \in G \leq G \times \Sigma_4^{op}$
one must have that
$-\alpha =\alpha$
(note that $-\alpha \colon \mathfrak{b} \to \mathfrak{c}$
since $-\mathfrak{b}=\mathfrak{b},-\mathfrak{c}=\mathfrak{c}$).
It is then easy to check that, if $-\alpha = \alpha$,
the composite \eqref{ITERWE EQ}
is indeed $\Lambda$-equivariant
(this also follows from Lemma \ref{LAMBEQMAPS LEM},
which covers the general case), so that one indeed has
$\P(\vect{C})^{\Lambda} \simeq \P(\vect{B})^{\Lambda}$.
\end{example}
In the following,
recall from Definition \ref{STABS DEF} that
for
$\vect{C} = (\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)$
one has that $\Lambda$ stabilizes $\vect{C}$
iff
$g \mathfrak{c}_{\sigma(i)} = \mathfrak{c}_i$
for all $(g,\sigma) \in \Lambda, 0\leq i \leq n$,
motivating the condition $g \kappa_{\sigma(i)} = \kappa_i$
in Lemma \ref{LAMBEQMAPS LEM}(i).
\begin{lemma}\label{LAMBEQMAPS LEM}
Fix a $G$-set of colors $\mathfrak{C}$.
Let $\P \in \mathsf{Op}^G_{\mathfrak{C}}(\V)$ be an operad,
$\vect{B}=(\mathfrak{b}_1,\cdots,\mathfrak{b}_n;\mathfrak{b}_0),\vect{C}=(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)$ be $\mathfrak{C}$-signatures,
and suppose
$\Lambda \leq G \times \Sigma_n^{op}$
stabilizes $\vect{B},\vect{C}$, i.e.
$\Lambda \leq \mathsf{Aut}_{G\ltimes \Sigma_{\mathfrak{C}}^{op}}(\vect{B})$,
$\Lambda \leq \mathsf{Aut}_{G\ltimes \Sigma_{\mathfrak{C}}^{op}}(\vect{C})$.
Moreover, suppose that for some $\mathbb{K} \in \V$
one has maps
$\mathbb{K} \xrightarrow{\kappa_i} \P(\mathfrak{b}_i;\mathfrak{c}_i)$, $0\leq i \leq n$
such that $g\kappa_{\sigma(i)} = \kappa_i$ for all $(g,\sigma) \in \Lambda$, $0\leq i \leq n$. Then:
\begin{enumerate}[label=(\roman*)]
\item if $\vect{B},\vect{C}$ have a common target $\mathfrak{b}_0=\mathfrak{c}_0$,
one has
$\Lambda$-equivariant maps as below, where the right map is the composition in $\P$
and the action of $(g,\sigma) \in \Lambda$ on $\mathbb{K}^{\otimes n}$
is the permutation action of $\sigma$.
\begin{equation}\label{LAMBEQMAPS EQ}
\P(\vect{C}) \otimes
\mathbb{K}^{\otimes n}
\xrightarrow{id \otimes \underset{1\leq i \leq n}{\bigotimes} \kappa_i}
\P(\vect{C}) \otimes \bigotimes_{1\leq i \leq n} \P(\mathfrak{b}_i;\mathfrak{c}_i)
\xrightarrow{\circ}
\P(\vect{B})
\end{equation}
\item
if $\vect{B},\vect{C}$ have common sources $\mathfrak{b}_i=\mathfrak{c}_i,1\leq i \leq n$,
one has
$\Lambda$-equivariant maps as below, where the right map is the composition in $\P$.
\[
\mathbb{K} \otimes \P(\vect{B})
\xrightarrow{\kappa \otimes id}
\P(\mathfrak{b}_0;\mathfrak{c}_0) \otimes \P(\vect{B})
\xrightarrow{\circ}
\P(\vect{C})
\]
\end{enumerate}
\end{lemma}
Note that, if $\mathbb{K}=1_{\V}$
in Lemma \ref{LAMBEQMAPS LEM},
we get $\Lambda$-equivariant maps
$\P(\vect{C}) \to \P(\vect{B})$,
$\P(\vect{B}) \to \P(\vect{C})$.
\begin{proof}
We discuss only (i), with (ii) following from a similar but easier argument.
The fact that the left map in \eqref{LAMBEQMAPS EQ} is
$\Lambda$-equivariant can be deduced
from the $g \kappa_{\sigma(i)} = \kappa_i$ requirement
via direct calculation, but we prefer a more abstract argument.
The maps
$\P(\mathfrak{b}_i;\mathfrak{c}_i) \to
\P(g\mathfrak{b}_{\sigma(i)};g\mathfrak{c}_{\sigma(i)}) $
for $(g,\sigma) \in \Lambda$
make the tuple
$\left(\P(\mathfrak{b}_i;\mathfrak{c}_i) \right)_{1\leq i \leq n}$
into a $\Lambda$-equivariant object of
$(\Sigma_n \wr \V^{op})^{op}$,
and likewise
$(\mathbb{K})_{1\leq i \leq n}
\xrightarrow{(\kappa_i)}
\left(\P(\mathfrak{b}_i;\mathfrak{c}_i) \right)_{1\leq i \leq n}
$
into a $\Lambda$-equivariant map in $(\Sigma_n \wr \V^{op})^{op}$.
Hence $\Lambda$-equivariance of the left map in
\eqref{LAMBEQMAPS EQ}
follows by functoriality of
$(\Sigma_n \wr \V^{op})^{op} \xrightarrow{\otimes} \V$.
To check that the right map in \eqref{LAMBEQMAPS EQ}
is also $\Lambda$-equivariant,
consider the $\mathfrak{C}$-trees below, including the signatures
$\vect{B},\vect{C}$.
\[
\begin{tikzpicture}
[grow=up,auto,level distance=2.3em,every node/.style = {font=\footnotesize},dummy/.style={circle,draw,inner sep=0pt,minimum size=1.75mm}]
\node at (0,0) [font=\normalsize]{$\vect{B}$}
child{node [dummy] {}
child{
edge from parent node [swap,near end] {$\mathfrak{b}_n$} node [name=Kn] {}}
child{
edge from parent node [near end] {$\mathfrak{b}_1$}
node [name=Kone,swap] {}}
edge from parent node [swap] {$\mathfrak{c}_0$}
};
\draw [dotted,thick] (Kone) -- (Kn) ;
\node at (4.5,0) [font=\normalsize]{$\vect{C}$}
child{node [dummy] {}
child{
edge from parent node [swap,near end] {$\mathfrak{c}_n$} node [name=Kn] {}}
child{
edge from parent node [near end] {$\mathfrak{c}_1$}
node [name=Kone,swap] {}}
edge from parent node [swap] {$\mathfrak{c}_0$}
};
\draw [dotted,thick] (Kone) -- (Kn) ;
\draw [dotted,thick] (Kone) -- (Kn) ;
\node at (9,0) [font=\normalsize]{$\vect{T}$}
child{node [dummy] {}
child{node [dummy] {}
child{
edge from parent node [swap] {$\mathfrak{b}_n$} node {}}
edge from parent node [swap,near end] {$\mathfrak{c}_n$} node [name=Kn] {}}
child{node [dummy] {}
child{
edge from parent node {$\mathfrak{b}_1$} node {}}
edge from parent node [near end] {$\mathfrak{c}_1$}
node [name=Kone,swap] {}}
edge from parent node [swap] {$\mathfrak{c}_0$}
};
\draw [dotted,thick] (Kone) -- (Kn) ;
\end{tikzpicture}
\]
Clearly the fact that $\Lambda$ stabilizes
$\vect{B},\vect{C}$
implies that $\Lambda$ stabilizes $\vect{T}$ as well.
Thus, the $\Lambda$-equivariance of the right map in
\eqref{LAMBEQMAPS EQ}
follows by noting that said map is the multiplication
$\bigotimes_{v \in \boldsymbol{V}(T)} \P(\vect{T}_v)
\to
\mathcal{P}(\mathsf{lr}(\vect{T}))
=
\mathcal{P}(\vect{B})$
encoded by the tree $\vect{T}$.
\end{proof}
\begin{remark}\label{CHOOSEKAPPA REM}
Generalizing Remark \ref{CHOOSESIGN REM},
a choice of
$\kappa_i$ as in Lemma \ref{LAMBEQMAPS LEM}
is in bijection with a
choice of $H_i$-equivariant maps
$\kappa_i \colon \mathbb{K} \to \mathcal{P}(\mathfrak{b}_i;\mathfrak{c}_i)$ for $i$ ranging over a set of representatives of
$\underline{n}_+/\Lambda$.
\end{remark}
In the next result we let $\mathbb{C}$ denote a good cylinder object for $1_{\V}$ (cf. \cite[Def. 4.2]{DS95}),
meaning that one has a factorization
$1_{\V} \amalg 1_{\V} \rightarrowtail \mathbb{C} \xrightarrow{\sim} 1_{\V}$
of the fold map, where the first map is a cofibration
and the second map is a weak equivalence.
\begin{corollary}\label{ALBEETA COR}
Assume that $\V$ is a closed monoidal model category with cofibrant pushout powers,
and such that fixed points in $\V^G$ send genuine trivial cofibrations to trivial cofibrations
(i.e. $\V$ satisfies (ii),(iii),(v),(vii) in Theorem \ref{THMA}). Additionally, suppose
that $\V$ satisfies the usual monoid axiom of \cite{SS00} (see also \cite[Rem. \ref{OC-MONAX_REM}]{BP_FCOP}).
Let $\P$,
$\vect{B}=(\mathfrak{b}_1,\cdots,\mathfrak{b}_n;\mathfrak{b}_0)$,
$\vect{C}=(\mathfrak{c}_1,\cdots,\mathfrak{c}_n;\mathfrak{c}_0)$
and $\Lambda$ be as in
Lemma \ref{LAMBEQMAPS LEM}.
Moreover, suppose
$\vect{B},\vect{C}$
are ``$\Lambda$-homotopy equivalent'',
by which we mean that there exist
\[
\alpha_i \colon 1_{\V} \to \P(\mathfrak{b}_i;\mathfrak{c}_i)
\qquad
\beta_i \colon 1_{\V} \to \P(\mathfrak{c}_i;\mathfrak{b}_i)
\qquad
\eta_i \colon \mathbb{C} \to \P(\mathfrak{b}_i;\mathfrak{b}_i)
\qquad
\bar{\eta}_i \colon \mathbb{C} \to \P(\mathfrak{c}_i;\mathfrak{c}_i)
\qquad
0\leq i \leq n
\]
with $\eta_i$ (resp. $\bar{\eta}_i$) a left homotopy between
$\beta_i\alpha_i$ and $id_{\mathfrak{b}_i}$
(resp.
$\alpha_i\beta_i$ and $id_{\mathfrak{c}_i}$)
and such that
\[
g \alpha_{\sigma(i)} = \alpha_i \qquad
g \beta_{\sigma(i)} = \beta_i \qquad
g \eta_{\sigma(i)} = \eta_i \qquad
g \bar{\eta}_{\sigma(i)} = \bar{\eta}_i \qquad
(g,\sigma) \in \Lambda,0\leq i \leq n.
\]
Then:
\begin{enumerate}[label=(\roman*)]
\item if $\vect{B},\vect{C}$ have a common target $\mathfrak{b}_0=\mathfrak{c}_0$, the precomposition maps
\[
\P(\vect{C})^{\Lambda} \xrightarrow{(\alpha_i)^{\**}} \P(\vect{B})^{\Lambda}
\qquad
\P(\vect{B})^{\Lambda} \xrightarrow{(\beta_i)^{\**}}
\P(\vect{C})^{\Lambda}
\]
induced by $\alpha_i,\beta_i,1\leq i \leq n$
(cf. Lemma \ref{LAMBEQMAPS LEM}(i)) are weak equivalences in $\V$;
\item if $\vect{B},\vect{C}$ have common sources $\mathfrak{b}_i=\mathfrak{c}_i, 1 \leq i \leq n$, the postcomposition maps
\[
\P(\vect{B})^{\Lambda} \xrightarrow{(\alpha_0)_{\**}}
\P(\vect{C})^{\Lambda}
\qquad
\P(\vect{C})^{\Lambda} \xrightarrow{(\beta_0)_{\**}}
\P(\vect{B})^{\Lambda}
\]
induced by $\alpha_0,\beta_0$
(cf. Lemma \ref{LAMBEQMAPS LEM}(ii))
are weak equivalences in $\V$.
\end{enumerate}
\end{corollary}
\begin{proof}
We address only (i), with (ii) being similar but easier.
Applying Lemma \ref{LAMBEQMAPS LEM},
one obtains diagrams as below,
which will show that
$(\alpha_i)^{\**}$ and
$(\beta_i)^{\**}$
are inverse up to homotopy provided we show
$\P(\vect{B})^\Lambda \otimes
\left(\mathbb{C}^{\otimes n}\right)^{\Lambda}$
is a cylinder on $\P(\vect{B})^\Lambda$, and likewise for
$\P(\vect{C})^\Lambda$.
\begin{equation}\label{ALBEETA EQ}
\begin{tikzcd}[column sep=31pt]
\P(\vect{B})^\Lambda \amalg \P(\vect{B})^\Lambda
\arrow[d] \arrow[r, "{((\beta_i)^{\**}(\alpha_i)^{\**}, id)}"]
&
\P(\vect{B})^\Lambda
\P(\vect{C})^\Lambda \amalg \P(\vect{C})^\Lambda
\arrow[d] \arrow[r, "{((\alpha_i)^{\**}(\beta_i)^{\**}, id)}"]
&
\P(\vect{C})^\Lambda
\\
\P(\vect{B})^\Lambda \otimes
\left(\mathbb{C}^{\otimes n}\right)^{\Lambda}
\arrow[r]
&
\left(\P(\vect{B}) \otimes
\mathbb{C}^{\otimes n}\right)^{\Lambda}
\arrow{u}[swap]{(\eta_i)^{\**}}
\P(\vect{C})^\Lambda \otimes
\left(\mathbb{C}^{\otimes n}\right)^{\Lambda}
\arrow[r]
&
\left(\P(\vect{C}) \otimes
\mathbb{C}^{\otimes n}\right)^{\Lambda}
\arrow{u}[swap]{(\bar{\eta}_i)^{\**}}
\end{tikzcd}
\end{equation}
This amounts to checking that the canonical map
$\P(\vect{B})^\Lambda \otimes
\left(\mathbb{C}^{\otimes n}\right)^{\Lambda}
\to \P(\vect{B})^\Lambda$
induced by $\mathbb{C} \to 1_{\V}$
is a weak equivalence, and by $2$-out-of-$3$ it suffices to show this for one of the sections
$\P(\vect{B})^\Lambda \to
\P(\vect{B})^\Lambda \otimes
\left(\mathbb{C}^{\otimes n}\right)^{\Lambda}$.
But, since
$1_{\V} \simeq 1_{\V}^{\otimes} \to \mathbb{C}^{\otimes n}$
is a $\Sigma_n$-genuine trivial cofibration
\cite[Prop. \ref{OC-SIGMAWRGF PROP}(ii)]{BP_FCOP},
the fixed point map
$1_{\V} \to \left(\mathbb{C}^{\otimes n}\right)^{\Lambda}$
is a trivial cofibration in $\V$ by assumption on $\V$,
and thus the required claim follows by the monoid axiom.
\end{proof}
\begin{remark}\label{MONAXSUP REM}
The monoid axiom assumption in Corollary \ref{ALBEETA COR}
is actually superfluous.
To see this, note first that if
$\mathcal{P}(\vect{B})^{\Lambda}$ happens to be cofibrant in $\V$,
then the claim that
$\mathcal{P}(\vect{B})^{\Lambda} \otimes (\mathbb{C}^{\otimes n})^{\Lambda}$
is a cylinder follows from $\V$ being a monoidal model category.
But then, writing $Q \xrightarrow{\sim} \mathcal{P}(\vect{B})^{\Lambda}$
for a cofibrant replacement,
by prepending
$Q \amalg Q \to Q \otimes (\mathbb{C}^{\otimes n})^{\Lambda}$
to the left square in \eqref{ALBEETA EQ}
one still has that
$(\beta_i)^{\**}(\alpha_i)^{\**}$
and
$id$
are homotopic, and similarly for the right square in \eqref{ALBEETA EQ}.
\end{remark}
\begin{remark}\label{ALBEETA_REM}
If in Corollary \ref{ALBEETA COR}
one has that $\P \in \mathsf{Cat}^G_{\mathfrak{C}}(\V)$
is a $\V$-category,
the only interesting case is that of
$\vect{B},\vect{C}$ unary signatures.
But then in the proof it is $n=1$,
and $\Lambda$ necessarily acts trivially on
$\mathbb{C}^{\otimes n} = \mathbb{C}$,
so in this case the result follows without using either condition (v) or (vii) in Theorem \ref{THMA}.
\end{remark}
\begin{proof}[Proof of Proposition \ref{23HARDCASE PROP}]
Set $\mathfrak{C} = \mathfrak{C}_{\P}$.
We need to show that,
for every $\mathfrak{C}$-signature $\vect{C}$ and
$\Lambda \in \F_{\vect{C}}$,
the map
$\P(\vect{C})^{\Lambda} \to \mathcal{Q}(\bar{F}(\vect{C}))^{\Lambda}$
is a weak equivalence in $\V$.
Moreover, since one has a functorial $\F$-fibrant replacement functor
(fixing object sets1)
and natural transformation $\O \to \O_f$,
we reduce to the case where all of $\O,\P,\mathcal{Q}$
are $\F$-fibrant.
As in Remarks \ref{CHOOSESIGN REM} and \ref{CHOOSEKAPPA REM},
write $\Lambda_i$ for the stabilizer of $i \in \underline{n}_+$ under the action of $\Lambda$,
and $H_i =\pi_G(\Lambda_i)$ for the projection.
Note that the requirement that $\F$ has enough units
says precisely that $H_i \in \F_1$ for all $i$.
Using $\F$-$\pi_0$-essential surjectivity allows us to,
for $i$ ranging over a set of representatives of
$\underline{n}_+/\Lambda$,
find $H_i$-fixed $\mathfrak{a}_i\in \mathfrak{C}_{\O}$
together with $H_i$-fixed isomorphisms
$F(\mathfrak{a}_i) = \mathfrak{b}_i \xrightarrow{[\alpha_i]} \mathfrak{c}_i$
in $\pi_0 j^{\**}\P^H$.
These now yield $\alpha_i,\beta_i,\eta_i,\bar{\eta}_i$
as in Corollary \ref{ALBEETA COR}
(note that we first choose these for $i$ in the set of representatives of $\underline{n}_+/\Lambda$,
then extend them to all $i$ by conjugation,
cf. Remark \ref{CHOOSEKAPPA REM})
so that by Corollary \ref{ALBEETA COR}
we have the following commutative square,
where the horizontal maps are weak equivalences in $\V$.
\begin{equation}\label{23SQDIAG EQ}
\begin{tikzcd}[column sep = 45pt]
\P(\vect{C})^{\Lambda}
\arrow[d]
\arrow{r}{\sim}[swap]{(\alpha_i)^{\**}(\beta_0)_{\**}}
&
\P(\vect{B})^{\Lambda}
\arrow[d]
\\
\mathcal{Q}\left(\bar{F}(\vect{C})\right)^{\Lambda}
\arrow{r}{\sim}[swap]{(\bar{F}\alpha_i)^{\**}(\bar{F}\beta_0)_{\**}}
&
\mathcal{Q}\left(\bar{F}(\vect{B})\right)^{\Lambda}
\end{tikzcd}
\end{equation}
Next write $\vect{A} = \{\mathfrak{a}_1,\cdots,\mathfrak{a}_n;\mathfrak{a}_0\}$
for the $\mathfrak{C}_{\O}$-signature determined by the
$\mathfrak{a}_i$, where we again use Remark \ref{CHOOSESIGN REM}
(recall that the $\mathfrak{a}_i$ were only chosen for $i$ in a set of representatives of $\underline{n}_+/\Lambda$),
which moreover implies that $\Lambda$ stabilizes $\vect A$, and thus $\Lambda \in \F_{\vect{A}}$.
The result now follows by applying $2$-out-of-$3$ to both the following diagram (where $\vect{B} = F(\vect{A})$, so that the arrows marked $\sim$ are weak equivalences by assumption) and \eqref{23SQDIAG EQ},
yielding that $\P(\vect{C})^{\Lambda} \to \mathcal{Q}(\bar{F}(\vect{C}))^{\Lambda}$
is also a weak equivalence, as desired.
\begin{equation}
\begin{tikzcd}[row sep=0]
\O(\vect{A})^{\Lambda} \arrow{rr}{\sim}
\arrow{dr}[swap]{\sim}
&&
\mathcal{Q}(\bar{F}(\vect{B}))^{\Lambda}
\\
&
\P(\vect{B}) \ar{ru}
\end{tikzcd}
\end{equation}
\end{proof}
\subsection{Characterizing fibrations}\label{ISOFIB_SEC}
In addition to the fibrations in Definition \ref{MODEL_DEFN},
and as noted in Remark \ref{FIBSALT REM},
there is another natural notion of fibration in $\mathsf{Op}^G_{\bullet}(\mathcal{V})$,
which parallels the notion of Dwyer-Kan equivalence
by replacing the
$\F$-path lifting condition
with the analogue condition on the
$j^{\**} \pi_0(-)$ categories.
\begin{definition}
Let $\F$ be a $(G,\Sigma)$-family which has enough units
(Definition \ref{FAMRESUNI DEF}).
We say a map $\O \to \P$ in $\Op_\bullet^G(\V)$ is an \textit{$\F$-isofibration} if
it is a local $\F$-fibration and
$j^{\**}\pi_0 \O^H \to j^{\**}\pi_0\P^H$
is an isofibration of categories for all $H \in \F_1$.
\end{definition}
Our goal in this section is to compare the
notions of $\F$-fibration and $\F$-isofibration.
We start with the easier direction.
\begin{proposition}[{cf. \cite[Prop. 2.3]{Ber07b}}]
\label{ISOFIBEASY PROP}
$\F$ be a $(G,\Sigma)$-family,
suppose $\V$ satisfies the coherence axiom,
and let
$F: \O \to \P$ in $\Op^G_\F(\V)$
be $\F$-path lifting.
Then $F$ is also an $\F$-isofibration
provided that either $\mathcal{P}$ is $\F$-fibrant
or $\V$ is right proper.
\end{proposition}
\begin{proof}
Considering each map
$j^{\**} \pi_0\left(\O \to \P \right)^H$,
we reduce to
the case of $F \colon \mathcal{C} \to \mathcal{D}$
a map in $\mathsf{Cat}_{\bullet} (\V)$.
Given $a \in \mathcal{C}$
and isomorphism $[\alpha]$ in $\pi_0 \mathcal{D}$
with $[\alpha](0)=F(a)$,
using either the definition of coherence
if $\mathcal{D}$ is fibrant or
Proposition \ref{ALTCOH PROP},
we obtain a bottom horizontal map below
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
\eta \ar{d} \ar{r}{a}
&
\mathcal{C} \ar{d}{F}
\\
\mathbb{J} \ar{r} \ar[dashed]{ru}
&
\mathcal{D}
\end{tikzcd}
\end{equation}
such that $\pi_0 \mathbb{J} \to \pi_0\mathcal{D}$
maps the natural isomorphism $[id_{1_{\V}}]$ to $[\alpha]$.
And,
since the lift in the diagram exists due to $F$ being path lifting,
the image $[\bar{\alpha}]$
of the natural isomorphism $[id_{1_{\V}}]$
under $\pi_0 \mathbb{J} \to \pi_0\mathcal{C}$
lifts $[\alpha]$
and satisfies $[\bar{\alpha}](0) = a$,
showing that
$\pi_0 \mathcal{C} \to \pi_0 \mathcal{D}$
is indeed an isofibration.
\end{proof}
Proposition \ref{ISOFIBEASY PROP}
implies that, if $\V$ is right proper,
all $\F$-fibrations $F \colon \mathcal{O} \to \mathcal{P}$ are $\F$-isofibrations.
We now turn to the converse direction, which
is given by Proposition \ref{ISOFIBHARD PROP},
but will require some preparation.
We first list two necessary lemmas.
\begin{lemma}[{cf. \cite[Lemma 2.6]{Ber07b}}]
\label{TRANSFLIFT LEMMA}
Let $\V$ be a model category and consider the diagram below,
where the map marked $\sim$ is a weak equivalence,
$\rightarrowtail$ is a cofibration
and $\twoheadrightarrow$ is a fibration.
\begin{equation}\label{TRANSFLIFT EQ}
\begin{tikzcd}[column sep = 45pt]
A
\ar{d}
\arrow[equal]{r}
&
A \ar{r}
\arrow[rightarrowtail]{d}
&
X \arrow[twoheadrightarrow]{d}
\\
B'
\arrow{r}[swap]{\sim}
&
B\ar{r}
&
Y
\end{tikzcd}
\end{equation}
Then a lift $B \to X$
exists iff
a lift $B' \to X$
exists.
\end{lemma}
\begin{remark}\label{UNDEROVER REM}
Recall (cf. \cite[Rem. 3.10]{DS95})
that for any $A \in \V$ the undercategory $\V_{A/}$
has a model structure such that a map is a
weak equivalence or (co)fibration iff it becomes one under the
forgetful functor
$\V_{A/} \to \V$ given by $(A \to X) \mapsto X$,
and dually for overcategories $\V_{/Y}$.
Hence, for any map $A \to Y$,
the category
$\V_{A//Y} = \left(\V_{A/}\right)_{/(A\to Y)}$
of factorizations $A \to C \to Y$
likewise has a model structure determined by the forgetful functor
$\V_{A//Y} \to \V$.
\end{remark}
\begin{proof}[Proof of Lemma \ref{TRANSFLIFT LEMMA}]
We will argue using Remark \ref{UNDEROVER REM}
(\cite{Ber07b} gives a more explicit argument).
Only the ``if'' direction requires proof.
A lift $B' \to X$ implies that,
in the homotopy category of $\V_{A//Y}$, it is
$\Ho \V_{A//Y}(B',X) \neq \emptyset$.
And since $B,B'$ are weak equivalent in
$\V_{A//Y}$, it is also
$\Ho \V_{A//Y}(B,X) \neq \emptyset$.
But the (co)fibrancy assumptions in \eqref{TRANSFLIFT EQ}
say that $B$ is cofibrant in $\V_{A//Y}$ while
$X$ is fibrant, so any map in $\Ho \V_{A//Y}(B,X) \neq \emptyset$
is represented by an actual lift $B \to X$.
\end{proof}
\begin{remark}\label{NOTMATCH REM}
We caution that, in the previous proof, when given a lift $f'\colon B' \to X$,
the induced lift $f \colon B \to X$
needs not be such that the composite $B' \to B \xrightarrow{f} X$
equals $f'$. Rather, these need only be homotopic in
$\V_{A//Y}$.
\end{remark}
\begin{lemma}\label{HOMINPULL LEM}
Consider a diagram
in $\mathsf{Cat}_{\{0,1\}}(\V)$
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
\mathbbm{1}
\arrow{d}[swap]{\bar{\alpha}}
\arrow{r}{\alpha}
&
\mathcal{C}
\arrow[twoheadrightarrow]{d}{F}
\\
\bar{\mathcal{C}}
\arrow{r}[swap]{\bar{F}}
&
\mathcal{D}
\end{tikzcd}
\end{equation}
such that $\alpha,\bar{\alpha}$ encode homotopy equivalences in
$\mathcal{C}, \bar{\mathcal{C}}$ and
$F \colon \mathcal{C} \twoheadrightarrow \mathcal{D}$
is a (local) fibration.
Then the induced map
$\mathbbm{1} \xrightarrow{(\alpha,\bar{\alpha})}
\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$
encodes a homotopy equivalence in $\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$
provided that either:
\begin{enumerate*}[label = (\roman*)]
\item $\mathcal{D}$, $\bar{\mathcal{C}}$ are fibrant or;
\item $\V$ is right proper.
\end{enumerate*}
\end{lemma}
The proof of Lemma \ref{HOMINPULL LEM} requires preparation,
and is postponed to the end of the section.
We now discuss a further assumption on
the unit $1_{\V}$ which is needed to guarantee that
\emph{all} $\F$-isofibrations are $\F$-fibrations.
\begin{definition}
Let $\V$ be a model category with $A$ a cofibrant object and
$X$ any object.
We say \emph{$X$ is fibrant with respect to $A$}
if a fibrant replacement $X\to X_f$
induces an isomorphism
\[
[A,X] \xrightarrow{\simeq} [A,X_f] = \Ho \V (A,X)
\]
of left homotopy classes of maps.
Explicitly, this means that any
map $A \xrightarrow{[f]} X$ in $\Ho \V (A,X)$
is represented by a map
$A \xrightarrow{f} X$ in $\V$
and that, if $[f]=[g]$, there is an exhibiting left homotopy
$A \otimes \mathbb{C} \xrightarrow{H} X$
(where, cf. Corollary \ref{ALBEETA COR},
$\mathbb{C}$ denotes a good cylinder object for $1_{\mathcal{V}}$ \cite[Def. 4.2]{DS95}).
\end{definition}
\begin{example}
In the Kan model structure on $\V=\mathsf{sSet}$
all objects are fibrant with respect to $\** = 1_{\V}$.
Further, fibrant objects are always fibrant with respect to any cofibrant object.
Thus, in the canonical model structures on
$\mathsf{Set}, \mathsf{Cat}, \mathsf{Top}$,
all objects are fibrant with respect to the unit $1_{\V}$.
\end{example}
\begin{remark}\label{RELFIBLIFT REM}
Suppose $F\colon X \to Y$ is a fibration between two objects
which are fibrant with respect to $A$.
Then, given
$f\colon A \to X$ and $g \colon A \to Y$
such that
$[Ff] = [g]$
one can find
$f'\colon A \to X$ such that $Ff' =g$.
Indeed, this follows from the existence of a
lift in
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
A \ar[>->]{d}[swap]{\sim} \ar{r}{f}
&
X \ar[twoheadrightarrow]{d}{F}
\\
A \otimes \mathbb{C} \ar{r}[swap]{H} \ar[dashed]{ru}
&
Y
\end{tikzcd}
\end{equation}
where $H$ is a left homotopy between $Ff$ and $g$.
\end{remark}
We can now prove a partial converse to Proposition \ref{ISOFIBEASY PROP},
adapting \cite[Prop. 2.5]{Ber07b}.
\begin{proposition}
\label{ISOFIBHARD PROP}
Let $\F$ be a $(G,\Sigma)$-family which has enough units.
Moreover, suppose $\V$ has cofibrant unit
and satisfies the coherence axiom,
and let $F: \O \to \P$ in $\Op^G_\F(\V)$
be an $\F$-isofibration.
Then $F$ is also an $\F$-fibration
provided that either:
\begin{enumerate*}[label = (\roman*)]
\item $\P$ is fibrant or;
\item $\V$ is right proper and
all objects in $\V$ are fibrant with respect to $1_{\V}$.
\end{enumerate*}
\end{proposition}
\begin{proof}
As in the proof of Proposition \ref{ISOFIBEASY PROP},
we reduce to the case of $F \colon \mathcal{C} \to \mathcal{D}$
an isofibration in $\mathsf{Cat}_{\bullet}(\V)$.
And, since $F$ is a local fibration by assumption,
the task is to show that if
$F$ is a $\pi_0$-isofibration then it is also path lifting,
i.e. that we can solve any lifting problem as on the left below
(where $\mathbb{J}$ is an interval, as usual).
\begin{equation}\label{FIRSTRED EQ}
\begin{tikzcd}[column sep = 45pt]
\eta \ar{d} \ar{r}{a}
&
\mathcal{C} \ar{d}{F}
&
&
\iota^{\**}_{a,b} \mathcal{C} \ar[twoheadrightarrow]{d}
\\
\mathbb{J} \ar{r} \ar[dashed]{ru}
&
\mathcal{D}
\mathbb{J} \ar[rightarrowtail]{r}{\sim}
&
\mathbb{J}' \ar[twoheadrightarrow]{r} \ar[dashed]{ru}
&
\iota^{\**}_{Fa,Fb} \mathcal{D}
\end{tikzcd}
\end{equation}
Writing $[\alpha]$ for the image in
$\pi_0 \mathcal{D}$
of the natural isomorphism (cf. Remark \ref{NATISO REM})
in $\pi_0 \mathbb{J} \simeq \pi_0 \widetilde{\mathbbm{1}}$,
the fact that $F$ is an isofibration
yields a lifted isomorphism
$[\bar{\alpha}]$ in $\pi_0 \mathcal{C}$
between $a \in \mathcal{C}$ and some other object $b \in \mathcal{C}$.
This allows us to form the lifting problem
in $\mathsf{Cat}_{\{0,1\}}(\V)$
on the right of
\eqref{FIRSTRED EQ}
(where we factor the bottom map as a trivial cofibration followed by a fibration),
and it clearly suffices to solve this alternate problem.
We now claim that we can form a diagram in $\mathsf{Cat}_{\{0,1\}}(\V)$ as below
\begin{equation}\label{GETMAP EQ}
\begin{tikzcd}
\mathbbm{1} \ar{d} \ar{r}
&
\iota^{\**}_{a,b} \mathcal{C} \ar[twoheadrightarrow]{d}
\\
\mathbb{J}' \ar[twoheadrightarrow]{r}
&
\iota^{\**}_{Fa,Fb} \mathcal{D}
\end{tikzcd}
\end{equation}
and for which $\mathbbm{1} \to \mathbb{J}'$
represents the natural isomorphism $[id_{1_{\V}}]$ of
$\pi_0 \mathbb{J}' \simeq \pi_0 \widetilde{\mathbbm{1}}$
and
$\mathbbm{1} \to \iota^{\**}_{a,b} \mathcal{C}$
represents the isomorphism $[\bar{\alpha}]$ in $\mathcal{C}$.
Indeed, in either case (i) or (ii)
our assumptions guarantee that
the mapping objects of all three of
$\mathbb{J}',\iota^{\**}_{Fa,Fb} \mathcal{D},\iota^{\**}_{a,b} \mathcal{C}$
are fibrant with respect to $1_{\V}$,
so one can certainly choose maps
$\mathbbm{1} \to \mathbb{J}'$,
$\mathbbm{1} \to \iota^{\**}_{a,b} \mathcal{C}$
representing the natural isomorphism $[id_{1_{\V}}]$ and $[\bar{\alpha}]$,
though a priori one has no guarantee that the composites
$\mathbbm{1} \to \mathbb{J}' \to \iota^{\**}_{Fa,Fb} \mathcal{D}$,
$\mathbbm{1} \to \iota^{\**}_{a,b} \mathcal{C} \to
\iota^{\**}_{Fa,Fb} \mathcal{D}$
coincide.
Nonetheless, since both composites represent
$[\alpha]$,
by Remark \ref{RELFIBLIFT REM}
we can make the square commute by replacing one of the maps up to homotopy.
We now form the following solid square
\begin{equation}
\begin{tikzcd}
\mathbb{J}'' \ar[dashed]{r}
&
\mathcal{E} \ar{d} \ar{r}
&
\iota^{\**}_{a,b} \mathcal{C} \ar[twoheadrightarrow]{d}
\\
&
\mathbb{J}' \ar[twoheadrightarrow]{r}
&
\iota^{\**}_{Fa,Fb} \mathcal{D}
\end{tikzcd}
\end{equation}
where $\mathcal{E}$ is simply the pullback.
The diagram
\eqref{GETMAP EQ}
yields a map $\mathbbm{1} \to \mathcal{E}$ which,
by Lemma \ref{HOMINPULL LEM},
encodes a homotopy equivalence.
Therefore, using either the definition of coherence in case (i) or Proposition \ref{ALTCOH PROP} in case (ii),
we obtain a map from an interval
$\mathbb{J}'' \to \mathcal{E}$
with the property that
the composite
$\mathbb{J}'' \to \mathcal{E} \to \mathbb{J}'$
sends the class of the natural isomorphism to itself.
In particular, this
$\mathbb{J}''(0,1) \xrightarrow{\sim} \mathbb{J}'(0,1)$
is a weak equivalence in $\V$
so that \cite[Lemma 2.12]{BM13}
(or its generalization Corollary \ref{ALBEETA COR})
implies
$\mathbb{J}'' \xrightarrow{\sim} \mathbb{J}'$
is itself a weak equivalence.
The required lift in \eqref{FIRSTRED EQ} now follows from
Lemma \ref{TRANSFLIFT LEMMA}
applied to the category
$\mathsf{Cat}_{\{0,1\}}(\V)$
with $A$ the initial object,
$B' \xrightarrow{\sim} B$ the map
$\mathbb{J}'' \xrightarrow{\sim} \mathbb{J}'$,
and $X \twoheadrightarrow Y$
the map
$\iota^{\**}_{a,b} \mathcal{C}
\twoheadrightarrow
\iota^{\**}_{Fa,Fb} \mathcal{D}$.
\end{proof}
The remainder of this section addresses the postponed proof of
Lemma \ref{HOMINPULL LEM}.
We first make some remarks about
the model structures
on $\V_{A/}$, $\V_{A//Y}$ in Remark \ref{UNDEROVER REM}.
\begin{remark}\label{UNDFGT REM}
Since the forgetful functor $\V_{A//Y} \to \V$
preserves all weak equivalences,
it preserves left and right homotopies \cite[\S 4.1,\S 4.12]{DS95} between maps.
\end{remark}
\begin{remark} \label{LEFTQUILUND REM}
For any map $A \to A'$ the induced adjunction
$A' \amalg_A (-) \colon \V_{A/}
\rightleftarrows
\V_{A'/} \colon \mathsf{fgt}$
is Quillen.
In particular,
given a cofibration
$A \rightarrowtail B$
and map $A' \to X$ with fibrant $X$, one has
\[
\Ho \V_{A/}\left(A \rightarrowtail B, A \to X\right)
\simeq
\Ho \V_{A'/}\left(A' \rightarrowtail A' \amalg_A B, A' \to X\right).
\]
\end{remark}
\begin{lemma}\label{LIFTEQUIV LEM}
Let $\V$ be a model category and
consider the lifting problems below, were $A,B$ are cofibrant,
the common map $A \rightarrowtail B$ is a cofibration, and $X$ is fibrant.
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
A
\ar[rightarrowtail]{d}
\arrow{r}{f}
&
X
A
\ar[rightarrowtail]{d}
\arrow{r}{g}
&
X
\\
B \ar[dashed]{ru}[swap]{F}
&
B \ar[dashed]{ru}[swap]{G}
&
\end{tikzcd}
\end{equation}
Then, if $f$ and $g$ are homotopic
(i.e. they coincide in $\Ho \V(A,X)$),
a lift $F$ exists iff a lift $G$ exists.
\end{lemma}
\begin{proof}
Let
$X \overset{\sim}{\rightarrowtail} PX
\twoheadrightarrow X \times X$
be a choice of path object for $X$ \cite[\S 4.12]{DS95},
and $A \xrightarrow{H} PX$
be a right homotopy between $f,g$,
i.e. writing $p_1,p_2 \colon PX \to X$ for the two projections,
one has
$p_1H=f$, $p_2H=g$.
The result now follows from the dual of Lemma \ref{TRANSFLIFT LEMMA}
applied to the following
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
A
\ar[rightarrowtail]{d}
\arrow{r}{H}
&
PX
\arrow[twoheadrightarrow]{d} \ar{r}{\sim}[swap]{p_1}
&
X \ar{d}
A
\ar[rightarrowtail]{d}
\arrow{r}{H}
&
PX
\arrow[twoheadrightarrow]{d} \ar{r}{\sim}[swap]{p_2}
&
X \ar{d}
\\
B
\arrow{r}
&
\**\ar{r}
&
\**
B
\arrow{r}
&
\**\ar{r}
&
\**
\end{tikzcd}
\end{equation}
which shows that lifts $B \to X$ in either diagram
exist iff a lift $B \to PX$ exists.
\end{proof}
In the remainder of the section we write
$\mathbb{C}_{\bullet} \in \V^{\Delta}$
for a cosimplicial frame on $1_{\V}$
\cite[Def. 16.6.1]{Hir03}.
In particular, this means that
$\mathbb{C}_0 = 1_{\V}$
and that the degeneracy maps
$\mathbb{C}_n \to \mathbb{C}_0$
are weak equivalences.
Moreover, $\mathbb{C}_{\bullet} \in \V^{\Delta}$ is Reedy cofibrant so that, writing
$\mathbb{C}_{K} = \colim_{[n] \to K} \mathbb{C}_n$
for $K \in \mathsf{sSet}$,
one has that
$\mathbb{C}_{K} \to \mathbb{C}_{L}$
is a cofibration in $\V$
whenever $K\to L$ is a monomorphism in $\mathsf{sSet}$.
In addition, $\mathbb{C} = \mathbb{C}_1$
is then a good cylinder on $1_{\V}$, in the sense of \cite[Def. 4.2(i)]{DS95}.
To avoid the need to label arrows,
we will write
$\mathbb{C}_{\{i,j\}} \to \mathbb{C}_{\{0,1,2\}}$
to denote the map
$\mathbb{C}_{1} \to \mathbb{C}_{2}$
induced by the inclusion $\{i,j\} \subset \{0,1,2\}$,
and similarly for $\mathbb{C}_{\{i\}} \to \mathbb{C}_{\{0,1,2\}}$.
The following is the key to proving Lemma \ref{HOMINPULL LEM}.
\begin{lemma}\label{HOMTPOFHOMTP LEM}
Let $\mathcal{D} \in \mathsf{Cat}_{\{0,1\}}(\V)$
be fibrant and suppose
$\alpha \colon 1_{\V} \to \mathcal{D}(0,1)$
is a homotopy equivalence.
Moreover, let
\[\beta \colon 1_{\V} \to \mathcal{D}(1,0), \qquad
\bar{\beta} \colon 1_{\V} \to \mathcal{D}(1,0),\qquad
H \colon \mathbb{C} \to \mathcal{D}(0,0), \qquad
\bar{H} \colon \mathbb{C} \to \mathcal{D}(0,0)
\]
be two left homotopy inverses $\beta,\bar{\beta}$ to $\alpha$
together with exhibiting homotopies
$H,\bar{H}$
between $id_0$ and $\beta \alpha$, $\bar{\beta}\alpha$.
Then there exists a commutative diagram (with $\alpha^{\**}$ the precomposition with $\alpha$)
\begin{equation}
\begin{tikzcd}[column sep = 45pt]
\mathbb{C}_{\{1,2\}}
\ar{d}
\arrow{r}{B}
&
\mathcal{D}(1,0)
\arrow{d}{\alpha^{\**}}[swap]{\sim}
\\
\mathbb{C}_{\{0,1,2\}}
\arrow{r}[swap]{\mathcal{H}}
&
\mathcal{D}(0,0)
\end{tikzcd}
\end{equation}
which satisfies the following compatibilities with restrictions
\[
B|_{\mathbb{C}_{\{1\}}} = \beta, \qquad
B|_{\mathbb{C}_{\{2\}}} = \bar{\beta}, \qquad
\mathcal{H}|_{\mathbb{C}_{\{0,1\}}} = H, \qquad
\mathcal{H}|_{\mathbb{C}_{\{0,2\}}} = \bar{H}.
\]
\end{lemma}
We note that, informally, $B$ is an homotopy between $\beta$ and $\bar{\beta}$
while $\mathcal{H}$
is a compatible homotopy of homotopies
between $H$ and $\bar{H}$
(except encoded by a ``triangle'' rather than a ``square'').
\begin{proof}
We first build the dashed lift $\bar{\mathcal{H}}$
as on the left below (which exists since we are lifting a trivial cofibration against a fibrant object).
Using this $\bar{\mathcal{H}}$ we now consider the right diagram,
\begin{equation}
\begin{tikzcd}[column sep = 35pt]
\mathbb{C}_{\{0,2\}}
\amalg_{\mathbb{C}_{\{0\}}}
\mathbb{C}_{\{0,1\}}
\ar[rightarrowtail]{d}[swap]{\sim}
\arrow{r}{(\bar{H},H)}
&
\mathcal{D}(0,0)
\mathbb{C}_{\{1\}} \amalg \mathbb{C}_{\{2\}}
\ar[rightarrowtail]{d}
\arrow{r}{(\beta,\bar{\beta})}
&
\mathcal{D}(1,0)
\ar{r}{\sim}[swap]{\alpha^{\**}}
&
\mathcal{D}(0,0)
\\
\mathbb{C}_{\{0,1,2\}}
\arrow[dashed]{ur}[swap]{\widetilde{\mathcal{H}}}
&
\mathbb{C}_{\{1,2\}}
\ar[dashed]{ru}{B}
\ar[bend right=2]{rru}[swap]{\bar{\mathcal{H}}|_{\mathbb{C}_{\{1,2\}}}}
&
&
\end{tikzcd}
\end{equation}
where $B$ making the top left triangle commute exists by the dual of Lemma \ref{TRANSFLIFT LEMMA}.
Moreover, note that while
$\alpha^{\**}B$ and $\bar{\mathcal{H}}_{\mathbb{C}_{\{1,2\}}}$
need not match (cf. Remark \ref{NOTMATCH REM}),
we nonetheless know that these are homotopic maps
in the undercategory $\V_{\left(\mathbb{C}_{\{1\}} \amalg \mathbb{C}_{\{2\}}\right)/}$.
Now consider the following lifting problems,
where we note that $\mathbb{C}_{\partial \Delta[2]}$
can be thought of as the informal ``union''
$\mathbb{C}_{\{1,2\}} \cup \mathbb{C}_{\{0,2\}} \cup \mathbb{C}_{\{0,1\}}$.
\begin{equation}\label{GOTTALIFT EQ}
\begin{tikzcd}[column sep = 65pt]
\mathbb{C}_{\partial \Delta[2]}
\ar[rightarrowtail]{d}
\arrow{r}{(\bar{\mathcal{H}}|_{\mathbb{C}_{\{1,2\}}},\bar{H},H)}
&
\mathcal{D}(0,0)
\mathbb{C}_{\partial \Delta[2]}
\ar[rightarrowtail]{d}
\arrow{r}{(\alpha^{\**}B,\bar{H},H)}
&
\mathcal{D}(0,0)
\\
\mathbb{C}_{\{0,1,2\}} \ar{ru}[swap]{\bar{\mathcal{H}}}
&
\mathbb{C}_{\{0,1,2\}} \ar[dashed]{ru}[swap]{\mathcal{H}}
&
\end{tikzcd}
\end{equation}
Since $\alpha^{\**}B$ and $\bar{\mathcal{H}}|_{\mathbb{C}_{\{1,2\}}}$
are homotopic in
$\V_{\left(\mathbb{C}_{\{1\}} \amalg \mathbb{C}_{\{2\}}\right)/}$,
Remark \ref{LEFTQUILUND REM}
implies that the top maps in \eqref{GOTTALIFT EQ}
are homotopic in
$\V_{\left(
\mathbb{C}_{\{0,2\}} \amalg_{\mathbb{C}_{\{0\}}} \mathbb{C}_{\{0,1\}}
\right)/}$ and thus,
by Remark \ref{UNDFGT REM}, also homotopic in
$\V$.
Lemma \ref{LIFTEQUIV LEM} applied to \eqref{GOTTALIFT EQ}
now gives the desired lift $\mathcal{H}$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{HOMINPULL LEM}]
Note first that
$\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$
is a homotopy pullback
\cite[Prop. A.2.4.4]{Lur09},
so we may assume that
$\mathcal{C},\bar{\mathcal{C}},\mathcal{D}$
are fibrant and both maps $F,\bar{F}$ are fibrations.
We need to show that $(\alpha,\bar{\alpha})$
admits left and right inverses up to homotopy.
By symmetry, we need only address the left inverse case.
Let
$\beta \colon 1_{\V} \to \mathcal{C}(1,0)$,
$\bar{\beta} \colon 1_{\V} \to \bar{\mathcal{C}}(1,0)$
be left homotopy inverses to $\alpha,\bar{\alpha}$,
with
$H \colon \mathbb{C} \to \mathcal{C}(0,0)$,
$\bar{H} \colon \mathbb{C} \to \bar{\mathcal{C}}(0,0)$
the homotopies between
$id_0$ and $\beta\alpha$, $\bar{\beta}\bar{\alpha}$.
We now note that, for $\beta$ and $\bar{\beta}$ to induce the desired
left homotopy inverse to
$(\alpha,\bar{\alpha})$,
we would need to know not just that
$F\beta=\bar{F}\bar{\beta}$
but also
$FH=\bar{F}\bar{H}$.
The proof will follow by showing that one can modify
$\beta,H$ so as to achieve this.
We now consider the diagram below,
where $B, \mathcal{H}$ in the bottom square
are obtained by applying Lemma \ref{HOMTPOFHOMTP LEM}
to the homotopy equivalence
$F\alpha = \bar{F} \bar{\alpha}$ in $\mathcal{D}$,
its two homotopy left inverses
$F\beta, \bar{F} \bar{\beta}$,
and exhibiting homotopies
$F H, \bar{F} \bar{H}$.
\begin{equation}\label{BIGISHDIAG EQ}
\begin{tikzcd}[row sep = 15pt,column sep = 40pt]
\mathbb{C}_{\{1\}} \ar{rr}{\beta} \ar{rd} \ar{dd}&&
\mathcal{C}(1,0) \ar[twoheadrightarrow]{dd} \ar{dr}{\alpha^{\**}}
\ar[dashed,leftarrow,bend left =15]{ddll}
\\
&
\mathbb{C}_{\{0,1\}} \arrow[crossing over]{rr}[near end]{H} &&
\mathcal{C}(0,0)
\ar[twoheadrightarrow]{dd}
\\
\mathbb{C}_{\{1,2\}} \ar{rr}[swap,near end]{B} \ar{rd} &&
\mathcal{D}(1,0) \ar{rd}{(F\alpha)^{\**}}
\\
&
\mathbb{C}_{\{0,1,2\}}
\ar[dashed,bend right =15,crossing over]{uurr}
\ar{rr}[swap]{\mathcal{H}} \arrow[uu, leftarrow, crossing over] &&
\mathcal{D}(0,0)
\end{tikzcd}
\end{equation}
We now claim that the curved dashed arrows in
\eqref{BIGISHDIAG EQ}
making the diagram commute exist
(explicitly, this means the dashed arrows are lifts in the front and back squares, and that the slanted square with two dashed sides commutes).
To see this, regarding the diagonal $\searrow$ arrows as objects in the arrow category $\V^{\bullet \to \bullet}$,
\eqref{BIGISHDIAG EQ}
is reinterpreted as a square in $\V^{\bullet \to \bullet}$,
with the desired dashed arrows being precisely a lift of said square.
But the right vertical arrow in that square
(i.e. the right side face of \eqref{BIGISHDIAG EQ}) is a projective fibration
while the left vertical arrow
(i.e. the left side face)
is a projective trivial cofibration
(this amounts to the claim that the maps
$\mathbb{C}_{\{1\}} \overset{\sim}{\rightarrowtail} \mathbb{C}_{\{1,2\}}$ and
$\mathbb{C}_{\{0,1\}} \amalg_{\mathbb{C}_{\{1\}}} \mathbb{C}_{\{1,2\}} \overset{\sim}{\rightarrowtail} \mathbb{C}_{\{0,1,2\}}$
are trivial cofibrations),
so the dashed lifts indeed exist.
Writing
$B'\colon \mathbb{C}_{\{1,2\}} \to \mathcal{C}(1,0)$,
$\mathcal{H}' \colon \mathbb{C}_{\{0,1,2\}} \to \mathcal{C}(0,0)$
for the lifts,
we now set
$\beta' = B'|_{\mathbb{C}_{\{2\}}}$,
which is a left inverse of $\alpha$
as exhibited by $H' = \mathcal{H}'|_{\mathbb{C}_{\{0,2\}}}$.
Moreover, by construction, we now have
$F\beta' = B|_{\mathbb{C}_{\{2\}}} = \bar{F} \bar{\beta}$
and
$FH' = \mathcal{H}|_{\mathbb{C}_{\{0,2\}}} = \bar{F} \bar{H}$,
hence
$(\beta',\bar{\beta})\colon \mathbbm{1} \to
\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$
now defines the desired left homotopy inverse to
$(\alpha,\bar{\alpha}) \colon \mathbbm{1} \to
\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$,
as exhibited by
$(H',\bar{H}) \colon \mathbb{C} \to
\mathcal{C} \times_{\mathcal{D}}\bar{\mathcal{C}}$.
\end{proof}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\doi}[1]
doi:\href{https://dx.doi.org/#1}{#1}}
\providecommand{\arxiv}[1]
arXiv:\href{https://arxiv.org/abs/#1}{#1}}
\providecommand{\href}[2]{#2}
|
2,877,628,090,246 | arxiv | \section{Introduction\label{sec:Introduction}}
Meshless methods have a rich variety of applications in hydrodynamic
modeling, particularly in situations that are challenging to mesh
initially or where complex flows make it difficult to maintain a good
meshed description. Examples include problems with large deformations,
fractures, unstable flows, and a variety of astrophysical problems
(where such methods originated) \cite{belytschko1996meshless}. However,
many astrophysical problems in the high energy density physics regime
also require a thermal radiation transfer treatment \cite{castor2004radiation},
and to date there has been much less work on meshfree treatments for
radiation transport. One fundamental choice one must make is what
sort of angular representation of the radiation is appropriate for
the physics problem at hand. The few prior meshfree thermal radiative
transfer treatments have generally focused on radiation diffusion
and similar approximations \cite{whitehouse2004smoothed,whitehouse2005faster,viau2006implicit,mayer2007fragmentation,petkova2009implementation},
wherein the angular distribution of the radiation is neglected. This
is appropriate for situations dominated by scattering and absorption
(such as deep in stellar interiors), but many interesting problems
include both transparent and opaque regions. To model these problems
accurately, a more complicated angular discretization of the radiation
transport equation is needed. The goal of this work is to develop
a discrete ordinates radiation transport implementation that is compatible
with a method such as smoothed particle hydrodynamics \cite{monaghan2005smoothed}
and variations of the reproducing kernel particle method \cite{liu1995reproducing,frontiere2017crksph}.
For this paper, the transport does not include radiation hydrodynamics
effects and the nonlinear emission terms in thermal radiative transfer.
The discrete ordinates radiation transport equation has been solved
previously using meshless methods, including for collocation methods
\cite{sadat2006use,sadat2012meshless,kindelan2010application,liu2007least,zhao2013second,kashi2017mesh}
and the meshless local Petrov-Galerkin (MLPG) method \cite{liu2007meshless,bassett2019meshless}.
The collocation methods often use the second-order form of the radiation
transport equation, such as the self-adjoint angular flux equation
\cite{morel1999self}, while the MLPG methods use a background mesh
for the integration.
This paper extends the past MLPG discretizations of the radiation
transport equation in several ways. First, reproducing kernels \cite{liu1995reproducing}
are used instead of moving least squares, which reduces the number
of linear solves needed to evaluate the kernels. Using these RK functions,
higher-than-second-order convergence is demonstrated for certain choices
of RK correction order. Second, the implementation is time-dependent,
which adds additional complexity and means that the integration, which
for for some problems may be performed at each time step, cannot require
manual adjustment and needs to be efficient. Third, the integration
is done using a Voronoi decomposition, which meets both of these criteria.
Fourth, the angular quadrature can be refined, which permits consideration
of problems that require high angular resolution in specific directions.
Finally, the implementation of the code has been done inside a code
that already includes radiation hydrodynamics with diffusion \cite{bassett2020efficient,bassett2020efficienta}
and reproducing kernel hydrodynamics \cite{frontiere2017crksph},
which should permit future consideration of meshless radiation hydrodynamics.
In many cases, including the meshless Galerkin approach \cite{li2002meshfree},
the integration is done using a background mesh \cite{nguyen2008meshless}.
In the original MLPG paper, the authors recommend using circular (2D)
or spherical (3D) domains of integration to retain a truly meshless
method \cite{atluri1998new}. Integration can also be performed by
introducing a quadrature into the lens-shaped intersection of two
kernels \cite{racz2012novel} or by reducing the dimensionality of
the integrals \cite{khosravifard2010new}. One method of avoiding
any evaluations outside of the kernel centers is by using nodal integration,
which may require stabilization for the derivatives \cite{chen2001stabilized}
and cannot be further refined to increase accuracy \cite{zhou2003nodal},
although using a Voronoi tessellation to create the volumes can increase
accuracy \cite{puso2008meshfree}. By using a quadrature within a
Voronoi tessellation to integrate the kernels, the resolution of the
integrals follows the resolution of the particles and that the integration
mesh depends only on the location of the particles and the boundary
surfaces of the problem. This, in effect, makes the integration invariant
to the rotation or translation of the points within the domain.
The paper is structured as follows. In Sec. \ref{sec:Interpolation},
the smoothed particle hydrodynamics (SPH) and reproducing kernels
(RK) are introduced, along with methods for calculating derivatives.
Next, the integration method using a Voronoi tessellation is discussed
in Sec. \ref{sec:Integration} and transformations from local to global
coordinates are derived in App. \ref{sec:Transformations}. The time-dependent
transport equation is introduced and discretized using these kernels
and integration methods in Sec. \ref{sec:Transport}. The discretization
is then tested in Sec. \ref{sec:Results} for a purely absorbing problem,
two manufactured solutions, a purely-scattering problem with disparate
cross sections, and a problem with a source in a void far from a strong
absorber whose solution is derived in App. \ref{sec:Analytic-asteroid}.
Finally, conclusions and future work are discussed in Sec. \ref{sec:Conclusions}.
\section{Interpolation methods\label{sec:Interpolation}}
This section introduces the smoothed particle hydrodynamics (SPH)
and reproducing kernel (RK) functions. Here and throughout this paper,
Greek superscripts represent dimensional indices, with repeated indices
representing summation, e.g. $x^{\alpha}$ is the component $\alpha$
of the vector $x$ and $X^{\alpha\beta}$ is the component $\alpha,\beta$
of the tensor $X$. The notation $\partial_{x}^{\alpha}$ denotes
the partial derivative with respect to $x^{\alpha}$. Subscripts denote
evaluation of a function at a discrete point, e.g. $f_{j}=f\left(x_{j}\right)$,
unless otherwise noted.
\subsection{Introduction to smoothed particle hydrodynamics\label{subsec:SPH}}
In SPH, spatial fields are interpolated from the discrete values at
individual points using functions referred to as kernels, which are
symmetric functions (typically of vector distance) with compact support,
i.e., they fall to zero at some range. The support of the kernel is
determined by a smoothing parameter, which translates from physical
space to the reference space for the kernel. This smoothing parameter
can either be a scalar $h$, as in standard SPH, or a symmetric tensor
$H^{\alpha\beta}$, as in adaptive smoothed particle hydrodynamics
(ASPH) \cite{owen1998adaptive}. This smoothing parameter is generally
allowed to vary point to point, so $H_{i}^{\alpha\beta}$in general.
Note that when using SPH, the interpolation kernel is radially symmetric,
while under ASPH this is not necessarily true. Because the ASPH $H^{\alpha\beta}$
tensor is symmetric, the corresponding smoothing scale isocontours
around a point are elliptical (2D) or ellipsoidal (3D). Using the
tensor smoothing parameter (which has units of inverse length), the
transformed distance vector in ASPH reference space $\eta\left(x\right)$
and the scaled distance $\chi\left(\eta\right)$ are
\begin{gather}
\eta^{\alpha}=H^{\alpha\beta}x^{\beta},\label{eq:ASPH-eta}\\
\chi=\sqrt{\eta^{\alpha}\eta^{\alpha}}.
\end{gather}
Note that SPH is a simply a special case of ASPH, wherein $H^{\alpha\beta}=h^{-1}\delta^{\alpha\beta}$,
so in the SPH case Eq. (\ref{eq:ASPH-eta}) reduces to
\begin{equation}
\eta^{\alpha}=x^{\alpha}/h.
\end{equation}
With these conventions in mind the kernel equations can be written
in terms of $H^{\alpha\beta}$ and apply equally to SPH or ASPH. The
kernel $W\left(x\right)$ and its derivatives can be defined in terms
of the base kernel in reference space $W^{b}\left(\chi\right)$ as
\begin{gather}
W=W^{b},\\
\partial_{x}^{\gamma}W=\partial_{\eta}^{\alpha}\chi\partial_{x}^{\gamma}\eta^{\alpha}\partial_{\chi}W^{b}.
\end{gather}
The derivative equation can be simplified by inserting the derivatives
of $\chi$ and $\eta$,
\begin{gather}
\partial_{\eta}^{\alpha}\chi=\frac{\eta^{\alpha}}{\chi},\\
\partial_{x}^{\gamma}\eta^{\alpha}=H^{\alpha\gamma},
\end{gather}
which results in
\begin{gather}
\partial_{x}^{\gamma}W=\frac{\eta^{\alpha}}{\chi}H^{\alpha\gamma}\partial_{\chi}W^{b}.
\end{gather}
As the kernels approximate delta functions ($W\left(x\right)\to\delta\left(x\right)$
as $H^{\alpha\beta}\to\infty$), they can be used in interpolation,
\begin{align}
f\left(x\right) & =\int_{V}\partial\left(x-x'\right)f\left(x'\right)dV'\nonumber \\
& \approx\int_{V}W\left(x-x'\right)f\left(x'\right)dV'.\label{eq:sph-delta-property}
\end{align}
The kernels are normalized such that formally the volume integral
is unity,
\begin{equation}
\int_{V}W\left(x-x'\right)dV'=1,
\end{equation}
though this property is only approximately true in the discrete case
for SPH. The interpolant can be discretized for a set of these kernels
with discrete positions $x_{j}$ with associated volumes $V_{j}$,
\begin{equation}
\overline{f}\left(x\right)=\sum_{j}V_{j}W_{j}\left(x\right)f_{j},
\end{equation}
where
\begin{equation}
W_{i}\left(x\right)=W\left(x-x_{i}\right)
\end{equation}
and $\overline{f}$ denotes the discrete interpolant of $f$.
\subsection{Introduction to reproducing kernels\label{subsec:Reproducing}}
In general, the standard SPH kernels cannot reproduce even a constant
solution exactly,
\begin{equation}
\sum_{j}V_{j}W_{j}\left(x\right)\neq\text{const}.
\end{equation}
The SPH kernels can be augmented with RK functions \cite{liu1995reproducing},
which permit exact interpolation of functions up to a certain polynomial
order. Interpolation with RK functions $U_{i}\left(x\right)$ works
the same as in SPH,
\begin{equation}
\overline{f}\left(x\right)=\sum_{j}V_{j}U_{j}f_{j},\label{eq:rk-interpolant}
\end{equation}
with the caveat that the RK functions have the property that
\begin{equation}
\sum_{j}V_{j}U_{j}\left(x\right)f_{j}=f\left(x\right),\quad f\left(x\right)\in\mathbb{P}_{n}
\end{equation}
where $\mathbb{P}_{n}$ is the space of polynomials with degree less
than or equal to $n$. These functions and their derivatives are defined
in terms of the SPH functions as
\begin{gather}
U_{i}=P_{i}^{\top}CW_{i},\\
\partial^{\gamma}U_{i}=\left(\partial_{x}^{\gamma}P_{i}^{\top}C+P_{i}\partial_{x}^{\gamma}C\right)W_{i}+P_{i}^{\top}C\partial_{x}^{\gamma}W_{i},
\end{gather}
where $P\left(x\right)$ is the polynomial basis vector,
\begin{gather}
P\left(x\right)=\left[1,x^{\alpha},x^{\alpha}x^{\beta},\cdots\right]^{\top},\\
P_{i}=P\left(x-x_{i}\right),\\
P_{ij}=P\left(x_{j}-x_{i}\right),
\end{gather}
and $C\left(x\right)=\left[C^{0}\left(x\right),C^{1}\left(x\right),C^{2}\left(x\right),\cdots\right]^{\top}$
is a corrections vector of the same size as the polynomial vector
with coefficients $C^{k}$ (i.e. the component $k$ of $C$) to be
determined. Suppose that $F$ is a vector of arbitrary coefficients.
The RK method calculates $C$ such that the reproducing kernels can
exactly represent $F^{\top}P$. The term $F^{\top}P_{i}$ is interpolated
as
\begin{align}
F^{\top}P_{i} & =\sum_{j}V_{j}F^{\top}P_{ji}U_{j}\nonumber \\
& =F^{\top}\sum_{j}V_{j}P_{ji}P_{j}^{\top}CW_{j}.
\end{align}
This equation must be true for each component of $F$,
\begin{equation}
P_{i}=\sum_{j}V_{j}P_{ji}P_{j}^{\top}CW_{j},
\end{equation}
and can be evaluated at the point $x_{i}$ to produce simple conditions
for the coefficients,
\begin{equation}
\sum_{j}V_{j}P_{ji}P_{ji}^{\top}W_{ji}C_{i}=G,
\end{equation}
where $W_{ji}=W_{j}\left(x_{i}\right)$ and $G=\left[1,0,0,\cdots\right]^{\top}$.
The matrix for this linear system and its derivatives can be written
explicitly as
\begin{gather}
M_{i}=\sum_{j}V_{j}P_{ji}P_{ji}^{\top}W_{ji},\label{eq:rk-matrix}\\
\partial^{\gamma}M_{i}=\sum_{j}V_{j}\left[\left(\partial^{\gamma}P_{ji}P_{ji}^{\top}+P_{ji}\partial^{\gamma}P_{ji}^{\top}\right)W_{ji}+P_{ji}P_{ji}^{\top}\partial^{\gamma}W_{ji}\right].
\end{gather}
In terms of these matrices, the linear systems to solve for $C_{i}$
and its derivatives can be written as
\begin{gather}
M_{i}C_{i}=G,\\
\partial^{\gamma}M_{i}C_{i}+M_{i}\partial^{\gamma}C_{i}=0.
\end{gather}
By first solving for $C_{i}$ and then $\partial^{\gamma}C_{i}$,
the only matrix that needs to be inverted is $M_{i}$,
\begin{gather}
C_{i}=M_{i}^{-1}G,\\
\partial^{\gamma}C_{i}=-M_{i}^{-1}\partial^{\gamma}M_{i}C_{i},
\end{gather}
which lets us reuse its factorization.
\section{Meshless integration\label{sec:Integration}}
In this section, the methodology for creating a Voronoi tessellation
is introduced, the meshless integration process is described, and
the connectivity for the weak-form kernels is derived in terms of
a similar strong-form connectivity.
\subsection{Process of creating the Voronoi tessellation}
As discussed in the introduction (Sec. \ref{sec:Introduction}), there
have been several methods developed to integrate radial basis functions.
For these results, the problem is decomposed using what is essentially
a Voronoi tessellation constructed using the PolyClipper library \cite{owen2020polyclipper},
with one line segment (1D), polygon (2D), or polyhedron (3D) per meshfree
point. Each cell is then further decomposed into triangles (2D) or
tetrahedra (3D), with surfaces defined by points (1D), line segments
(2D) or triangles (3D). It is worth pointing out that the decomposition
is not truly the Voronoi. Rather the decomposition begins with an
initial polytope for each point that encompasses the finite kernel
extent of that point (i.e., the space over which its kernel value
is non-zero), which is progressively clipped by planes halfway between
the point in question and each neighbor point it interacts with. In
the end, this results in a tiling of space with these polytopes per
point that exactly constructs a partition of unity in space for all
points that overlap.
Note that the topological connection between the polytopes for each
point is not computed, but only a unique polygon or polyhedron for
each point independently. Figure \ref{fig:polygon-volume}a shows
a cartoon of this process. The goal is to construct the Voronoi-like
polygon for the central red point, which has a set of neighbor points
it overlaps (in blue), and a non-zero kernel extent represented as
the gray region. The starting polygon for this point is the bounding
surface of this gray region, which is progressively clipped by planes
half-way between the central red point and each of its neighbors.
In the end all that is left is the central light red polygon, which
is the unique volume closer to the red point than any of its neighbors.
To facilitate simple integration quadratures, these polytopes for
each point are further broken down into triangles (in 2D) and tetrahedra
(in 3D). Fig. \ref{fig:polygon-decomposition} shows this procedure
for the polygon generated in Fig. \ref{fig:polygon-volume}, where
the cross markers denote the centroid of each of the sub-triangles
in the polygon. Note the centroid of the polygon does not necessarily
coincide with the original point used to construct it.
\subsection{Meshless integration quadrature\label{subsec:Meshless-quadrature}}
The set of integration quadratures over the base shapes represents
a contiguous, non-overlapping quadrature that covers the domain. A
Gauss-Legendre quadrature is used for integration of the line segments,
while symmetric quadrature rules as described in Ref. \cite{witherden2015identification}
are used to integrate the triangles and tetrahedra. Appendix \ref{sec:Transformations}
presents information on how the integrals are transformed from physical
to reference space.
At each integration point, all the functions whose support includes
the integration point must be evaluated. The RK functions (Sec. \ref{subsec:Reproducing})
are expensive to evaluate relative to a standard SPH kernel and the
evaluation of one such function depends on the values of all other
functions at that point. As such, a large amount of computation can
be saved by making the quadrature the outermost loop in the code,
as shown in Alg. \ref{alg:integration}. For each quadrature point,
all functions whose support includes the integration quadrature point
are evaluated. Then these values are used to perform each integral.
It is worth comparing this approach with prior background integration
methodologies, wherein a traditional background mesh (often some sort
of orthogonal Cartesian grid aligned with the lab frame) is placed
independently of the meshless points, even if the point locations
inform characteristics of the background mesh. The approach in this
section, using a Voronoi tessellation for the integration, produces
an integration mesh whose properties are solely determined by the
volume and relative positioning of the meshless points, which makes
it invariant to rotation or translation. This integration does not
meet the strictest of meshless criteria, in which no mesh is allowed
\cite{atluri1999critical}, but does meet a looser criterion in that
all information can be derived directly from the meshless points.
The geometry of each point's unique volume is constructed based solely
on the positions of surrounding points without storing, evolving,
or specifying anything except the the point positions and kernel extents.
\subsection{Strong and weak forms for reproducing kernels}
For the following sections, to simplify the notation, volume and surface
integrals will be written as
\begin{gather}
\left\langle f,g\right\rangle =\int_{V}fgdV,\\
\left(f,g\right)=\int_{S}fgdS.
\end{gather}
For many applications, such as hydrodynamics and diffusion, SPH and
RK can be used to directly discretize the equations via collocation
\cite{monaghan2005smoothed}. The equation, in this example
\begin{equation}
a^{\alpha}\partial_{x}^{\alpha}f+bf=0,
\end{equation}
is first integrated by parts,
\begin{equation}
\left(U_{i},n^{\alpha}a^{\alpha}f\right)-\left\langle \partial_{x}^{\alpha}U_{i},a^{\alpha}f\right\rangle +\left\langle U_{i},bf\right\rangle =0,
\end{equation}
(with $n$ denoting the surface normal), the surface term is discarded,
and interpolants {[}Eq. (\ref{eq:rk-interpolant}){]} and the delta
function property {[}Eq. (\ref{eq:sph-delta-property}){]} are used
to simplify the equation to
\begin{equation}
-\sum_{j}V_{j}a_{j}^{\alpha}f_{j}\partial_{x_{i}}^{\alpha}U_{ji}+b_{i}f_{i}=0.
\end{equation}
While this form of the equation has the advantage of simplicity, it
depends on a one-point quadrature rule for the integration,
\begin{equation}
\int_{V}U_{i}f\left(x\right)\approx f_{i},
\end{equation}
and generally throws away surface terms. Another option, and the one
used in this paper, is to insert a basis function expansion,
\begin{equation}
f\left(x\right)=\sum_{j}V_{j}U_{j}g_{j}\label{eq:basis-expansion}
\end{equation}
(with coefficients $g_{j}$),
\begin{equation}
\sum_{j}V_{j}\left[\left(U_{i},n^{\alpha}a^{\alpha}U_{j}\right)-\left\langle \partial_{x}^{\alpha}U_{i},a^{\alpha}U_{j}\right\rangle +\left\langle U_{i},bU_{j}\right\rangle \right]g_{j}=0,
\end{equation}
and perform the integrals directly using a quadrature like the one
described in Sec. \ref{subsec:Meshless-quadrature},
\begin{equation}
\left\langle \partial_{x}^{\alpha}U_{i},a^{\alpha}U_{j}\right\rangle \approx\sum_{k}w_{k}\partial_{x_{k}}^{\alpha}U_{ik}a_{k}^{\alpha}U_{jk},
\end{equation}
where $w_{k}$ are the weights of a quadrature spanning the integration
volume. For functions with compact support, this is the Meshless-Local
Petrov-Galerkin (MLPG) method.
\subsection{Meshless connectivity}
Two types of connectivity are used in evaluating and storing the MLPG
integrals. In a standard SPH code, the connectivity is the sets of
points whose evaluation is nonzero at the center of the other, so
points $i$ and $j$ are neighbors if the support of $W_{i}$ includes
the point $x_{j}$ or vice versa, or
\begin{equation}
\chi\left(\eta\left(x_{i}-x_{j}\right)\right)\leq r,\label{eq:sph-connectivity}
\end{equation}
where $r$ is the dimensionless support radius of the kernel. For
MLPG, the connectivity (or sets of points for which the bilinear integrals
are nonzero) is determined by whether points have overlapping support,
so points $i$ and $j$ are neighbors if the support regions of $W_{i}$
and $W_{j}$ intersect. The overlap connectivity is a subset of the
set of points for which
\begin{equation}
\chi\left(\eta\left(x_{i}-x_{j}\right)\right)\leq2r.\label{eq:mlpg-connectivity}
\end{equation}
To show this, suppose that there is a point $x_{k}$ that is in the
support radius of $W_{i}$ and $W_{j}$. It follows from Eq. (\ref{eq:sph-connectivity})
and the triangle inequality in Euclidean space that $\chi\left(\eta\left(x_{i}-x_{j}\right)\right)\leq\chi\left(\eta\left(x_{i}-x_{k}\right)\right)+\chi\left(\eta\left(x_{k}-x_{j}\right)\right)\leq2r$.
Standard SPH connectivity information can be used to both create the
overlap connectivity and calculate which functions are nonzero at
each integration point, which is similar to standard SPH connectivity.
If the radius of the MLPG kernels is doubled (or the smoothing length
altered to produce a similar effect), then by Eqs. (\ref{eq:sph-connectivity})
and (\ref{eq:mlpg-connectivity}), the standard SPH connectivity of
the doubled-radius will include all overlap neighbors. As each cell
produced by the Voronoi tessellation is completely contained within
the support of its associated MLPG point, this same double-radius
SPH connectivity for $i$ and $j$ will include all integration points
in the cell $i$ for which the kernel $W_{j}$ is nonzero. This is
why the same connectivity can be used for both the overlap of two
kernels and the overlap of a kernel with a Voronoi cell associated
with a kernel in Alg. \ref{alg:integration}.
\section{Radiation transport\label{sec:Transport}}
In this section, the radiation transport equation is introduced and
discretized using MLPG with RK functions. Then, the iterative methods
that are used for the solution of the discretized equation are described.
Finally, the angular quadrature with selective refinement is introduced.
\subsection{Discretization of the transport equation}
The gray radiation transport equation, which is the transport equation
integrated over all energies, is
\begin{equation}
\partial_{t}\psi+\Omega^{\alpha}\partial_{x}^{\alpha}\psi+\sigma_{t}\psi=\frac{1}{4\pi}\sigma_{s}\phi+q,\label{eq:transport-equation}
\end{equation}
with the boundary condition
\begin{equation}
\psi=\psi^{b},\quad x\in\partial V,\ \Omega^{\alpha}n^{\alpha}>0,\label{eq:rad-boundary}
\end{equation}
and initial condition
\[
\psi=\psi^{\text{init}},\quad t=0,
\]
where $\Omega$ is the radiation propagation direction, $\psi$ is
the angular flux, $\phi=\int_{4\pi}\psi d\Omega$ is the scalar flux,
$\sigma_{s}$ is the scattering cross section, $\sigma_{a}$ is the
absorption cross section, $\sigma_{t}=\sigma_{a}+\sigma_{s}$ is the
total cross section, $q$ is a source that may include physics such
as thermal emission, $\psi^{b}$ is the incoming flux at the boundary,
$\psi^{\text{init}}$ is the initial angular flux, and $\partial V$
denotes the boundary of the domain. The discrete-ordinates approximation
evaluates this equation at discrete angles that are ordinates of a
quadrature over a unit sphere. Denoting these ordinates as $\Omega_{m}$
for the angular index $m$, integrals over the unit sphere become
\begin{equation}
\int_{4\pi}fd\Omega\approx\sum_{m}w_{m}f_{m},
\end{equation}
where $w_{m}$ are the weights of the quadrature. The transport equation
evaluated at the discrete ordinate $m$ becomes
\begin{equation}
\partial_{t}\psi_{m}+\Omega_{m}^{\alpha}\partial_{x}^{\alpha}\psi_{m}+\sigma_{t}\psi_{m}=\frac{1}{4\pi}\sigma_{s}\phi+q_{m},
\end{equation}
with the scalar flux $\phi=\sum_{m}w_{m}\psi_{m}$. The backward Euler
(or fully-implicit) method is used to discretize in time,
\begin{equation}
\frac{1}{c\Delta t}\psi_{m}+\Omega_{m}^{\alpha}\partial_{x}^{\alpha}\psi_{m}+\sigma_{t}\psi_{m}=\frac{1}{c\Delta t}\psi_{m}^{n-1}+\frac{1}{4\pi}\sigma_{s}\phi+q_{m},
\end{equation}
where all variables are evaluated at time index $n$ except where
noted otherwise as a superscript.
A standard Galerkin approach to transport would be to multiply the
transport equation by $U_{i}$ and then integrate over the support
of $U_{i}$. For streamline-upwind Petrov-Galerkin (SUPG) stabilization,
the transport equation is instead multiplied by $U_{i}+\tau_{i}\Omega^{\alpha}\partial_{x}^{\alpha}U_{i}$,
where $\tau$ is a proportionality constant with unit length that
is chosen to be constant for each trial function. Performing this
operation, the transport equation becomes
\begin{align}
& \frac{1}{c\Delta t}\left\langle U_{i},\psi_{m}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\psi_{m}\right\rangle +\left(U_{i},\Omega_{m}^{\alpha}n^{\alpha}\psi_{m}\right)_{\Omega^{\alpha}n^{\alpha}>0}-\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\psi_{m}\right\rangle \nonumber \\
& \quad+\tau_{i}\Omega_{m}^{\alpha}\Omega_{m}^{\beta}\left\langle \partial_{x}^{\alpha}U_{i},\partial_{x}^{\beta}\psi_{m}\right\rangle +\left\langle U_{i},\sigma_{t}\psi_{m}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{t}\psi_{m}\right\rangle \nonumber \\
& \qquad=\frac{1}{c\Delta t}\left\langle U_{i},\psi_{m}^{n-1}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\psi_{m}^{n-1}\right\rangle +\left(U_{i},\left|\Omega_{m}^{\alpha}n^{\alpha}\right|\psi_{m}^{b}\right)_{\Omega^{\alpha}n^{\alpha}<0}\nonumber \\
& \qquad\quad+\frac{1}{4\pi}\left\langle U_{i},\sigma_{s}\phi\right\rangle +\frac{1}{4\pi}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{s}\phi\right\rangle +\left\langle U_{i},q_{m}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},q_{m}\right\rangle .
\end{align}
The surface integral term produced by integration by parts has been
split into known (incoming) and unknown (outgoing) parts and the boundary
condition {[}Eq. (\ref{eq:rad-boundary}){]} has been applied. Inserting
a basis function expansion for the scalar and angular flux {[}as in
Eq. (\ref{eq:basis-expansion}){]}, \begin{subequations}\label{eq:flux-expansion}
\begin{gather}
\phi=\sum_{j}V_{j}U_{j}\Phi_{j},\\
\psi=\sum_{j}V_{j}U_{j}\Psi_{j},
\end{gather}
\end{subequations}the equation becomes
\begin{align}
& \sum_{j}V_{j}\left[\frac{1}{c\Delta t}\left\langle U_{i},U_{j}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle +\left(U_{i},\Omega_{m}^{\alpha}n^{\alpha}U_{j}\right)_{\Omega^{\alpha}n^{\alpha}>0}-\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle \right.\nonumber \\
& \quad\left.+\tau_{i}\Omega_{m}^{\alpha}\Omega_{m}^{\beta}\left\langle \partial_{x}^{\alpha}U_{i},\partial_{x}^{\beta}U_{j}\right\rangle +\left\langle U_{i},\sigma_{t}U_{j}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{t}U_{j}\right\rangle \right]\Psi_{m,j}\nonumber \\
& \qquad=\sum_{j}V_{j}\left[\frac{1}{c\Delta t}\left\langle U_{i},U_{j}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle \right]\Psi_{m,j}^{n-1}+\left(U_{i},\left|\Omega_{m}^{\alpha}n^{\alpha}\right|\psi_{m}^{b}\right)_{\Omega^{\alpha}n^{\alpha}<0}\nonumber \\
& \qquad\quad+\frac{1}{4\pi}\sum_{j}V_{j}\left[\left\langle U_{i},\sigma_{s}U_{j}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{s}U_{j}\right\rangle \right]\Phi_{j}+\left\langle U_{i},q_{m}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},q_{m}\right\rangle .\label{eq:weak-transport}
\end{align}
Note that, in general, $\Phi_{i}\neq\phi_{i}$ and $\Psi_{i}\neq\psi_{i}$.
Once Eq. (\ref{eq:weak-transport}) is solved for $\Phi$ and $\Psi$,
the solution at each point must be recovered using Eqs. (\ref{eq:flux-expansion}).
The effect of the SUPG stabilization is to reduce oscillations and
make the system of equations easier to solve iteratively while not
affecting global particle balance \cite{bassett2019meshless}. For
the results in Sec. \ref{sec:Results}, the stabilization parameter
is set to be
\[
\tau_{i}=\frac{h_{i}}{k},
\]
where $k$ is approximately the number of points across the kernel
radius. This results in a $\tau_{i}$ that is approximately equal
to the spacing between the points.
Because RK permits interpolation, the initial condition can be set
using initial values of the angular flux instead of needing to interpolate
coefficients, or $\Psi_{i}\bigg|_{t=0}=\psi^{\text{init}}\bigg|_{x=x_{i}}$.
This is what is done for the results in Sec. \ref{sec:Results}.
\subsection{Methods for solution of the transport equation\label{subsec:Iterative-methods}}
The transport equation can be written in operator form as
\begin{equation}
\mathcal{L}\Psi=\mathcal{T}\Psi^{n-1}+\mathcal{M}\mathcal{S}\Phi+r,
\end{equation}
or in terms of $\Phi$ as
\begin{equation}
\left(\mathcal{I}-\mathcal{D}\mathcal{L}^{-1}\mathcal{M}\mathcal{S}\right)\Phi=\mathcal{D}\mathcal{L}^{-1}\left(\mathcal{T}\Psi^{n-1}+r\right),
\end{equation}
with the operators defined as
\begin{align}
\left(\mathcal{L}\Psi\right)_{m,i}= & \sum_{j}V_{j}\left[\frac{1}{c\Delta t}\left\langle U_{i},U_{j}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle +\left(U_{i},\Omega_{m}^{\alpha}n^{\alpha}U_{j}\right)_{\Omega^{\alpha}n^{\alpha}>0}-\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle \right.\nonumber \\
& \quad\left.+\tau_{i}\Omega_{m}^{\alpha}\Omega_{m}^{\beta}\left\langle \partial_{x}^{\alpha}U_{i},\partial_{x}^{\beta}U_{j}\right\rangle +\left\langle U_{i},\sigma_{t}U_{j}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{t}U_{j}\right\rangle \right]\Psi_{m,j},\\
\left(\mathcal{T}\Psi^{n-1}\right)_{m,i} & =\sum_{j}V_{j}\left[\frac{1}{c\Delta t}\left\langle U_{i},U_{j}\right\rangle +\frac{1}{c\Delta t}\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},U_{j}\right\rangle \right]\Psi_{m,j}^{n-1},\\
\left(\mathcal{M}\mathcal{S}\Phi\right)_{i,m} & =\frac{1}{4\pi}\sum_{j}V_{j}\left[\left\langle U_{i},\sigma_{s}U_{j}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},\sigma_{s}U_{j}\right\rangle \right]\Phi_{j},\\
\left(\mathcal{D}\Psi\right)_{i} & =\sum_{m}w_{m}\Psi_{m,i},\\
\left(r\right)_{i,m} & =\left(U_{i},\left|\Omega_{m}^{\alpha}n^{\alpha}\right|\psi_{m}^{b}\right)_{\Omega^{\alpha}n^{\alpha}<0}+\left\langle U_{i},q_{m}\right\rangle +\tau_{i}\Omega_{m}^{\alpha}\left\langle \partial_{x}^{\alpha}U_{i},q_{m}\right\rangle .
\end{align}
Note that using this notation, $\Phi=\mathcal{D}\Psi$. The notation
$\mathcal{L}^{-1}$ denotes the linear inverse of the $\mathcal{L}$
operator, which is block diagonal in angle. With the first-flight
source defined as
\begin{gather}
b_{\psi}=\mathcal{L}^{-1}\left(\mathcal{T}\Psi^{n-1}+r\right),\\
b_{\phi}=\mathcal{D}b_{\psi},
\end{gather}
the equation can be simplified to
\begin{equation}
\left(\mathcal{I}-\mathcal{D}\mathcal{L}^{-1}\mathcal{M}\mathcal{S}\right)\Phi=b_{\phi}.\label{eq:krylov-operator}
\end{equation}
Equation (\ref{eq:krylov-operator}) can be solved directly using
a matrix-free linear solver such as GMRES or iteratively using a method
such as fixed point iteration. Once the scattering source is converged,
the angular flux is recovered for use in the next time step by performing
an additional solve using the converged scalar flux,
\begin{equation}
\Psi=\mathcal{L}^{-1}\mathcal{M}\mathcal{S}\Phi+b_{\psi}.
\end{equation}
For the results in Sec. \ref{sec:Results}, the $\mathcal{L}^{-1}$
operation is performed using two packages from Trilinos \cite{heroux2005overview},
the Belos package for GMRES and the Ifpack2 package for the ILUT preconditioner.
The ILUT factorizations for each angle are precomputed at the start
of the time step and reused to minimize computation. The iterations
to converge the scattering source {[}the solution of Eq. (\ref{eq:krylov-operator}){]}
are also performed using GMRES from Belos without a preconditioner.
For a discussion on preconditioners for the scattering iterations,
see Sec. \ref{sec:Conclusions}.
\subsection{Refinement of the angular quadrature\label{subsec:Angular-refinement}}
For the problems in Sec. \ref{sec:Results}, the Gauss-Legendre quadrature
is used in 1D, while the LDFE (linear discontinuous finite element)
quadrature \cite{jarrell2011discrete} is used in 2D and 3D. The LDFE
quadrature is hierarchal, meaning that each octant of the unit sphere
can be further subdivided into four ordinates, which can themselves
be subdivided and so on. This can be used to produce a high density
of angular ordinates in chosen directions object to prevent ray effects,
which as shown in Sec. \ref{subsec:Asteroid}. Given a goal quadrature,
the refined angular discretization keeps all ordinates from the goal
quadrature that hit the object and combines the ordinates that do
not hit the object inasmuch as is possible (Alg. \ref{alg:angular-refinement}).
This significantly reduces the number of angles needed for a given
number of rays from a small source to hit a distant object.
\section{Results\label{sec:Results}}
In this section, five problems are considered to test the discretization
described in Sec. \ref{sec:Transport} with the integration in Sec.
\ref{sec:Integration}. The first two problems use the method of manufactured
solutions. The third problem considers a purely absorbing medium,
while the fourth problem considers a purely scattering medium. The
final problem shows a possible application of the code to simulate
an asteroid absorbing a large quantity of radiation from a distant
source. All the problems use a kernel sampling radius of 4 neighbors
(i.e., the equivalent smoothing scale is 4 times the local particle
spacing) and an RK order of one, as described in Sec. \ref{sec:Integration},
except for the purely absorbing problem, which also explores other
combinations of RK order and neighbors. Note that for uniformly spaced
points, a sampling radius of 4 neighbors implies a total number of
overlap neighbors for each point of 8 (1D), 50 (2D), and 268 (3D).
The first two sections use the relative error as a measure for convergence.
This is defined as
\begin{equation}
\epsilon_{rel}=\dfrac{\sum_{i}\left|\phi_{i}^{\text{numeric}}-\phi_{i}^{\text{analytic}}\right|}{\sum_{i}\phi_{i}^{\text{analytic}}}\label{eq:relative-err}
\end{equation}
for the numeric and analytic scalar fluxes at the MLPG centers $i$.
\subsection{Manufactured problems}
The method of manufactured solutions works by selecting a solution
for $\psi$, solving for a source $q$ by inserting this solution
into the continuous transport equation {[}Eq. (\ref{eq:transport-equation}){]},
assigning the boundary source $\psi_{b}$ to be equal to the solution,
and then calculating a numerical solution using the discretized transport
equation {[}Eq. (\ref{eq:weak-transport}){]} for comparison to the
original solution.
The spatial and time convergence of the manufactured problems is considered
in 1D, 2D, and 3D. In 1D, due to the low cost of integration and because
the integration cells are not subdivided, the integration is performed
using a 64-point Gauss-Legendre quadrature. In 2D, the integration
of the subcell triangles is performed using a tenth-order symmetric
quadrature with 25 points, while in 3D, the integration of the surface
triangles and the subcell tetrahedra is performed using a third-order
symmetric quadrature with 8 points. For more information on the integration
quadratures, see Sec. \ref{sec:Integration}.
The spatially-dependent results are run for several cases between
$8^{d}$ and $128^{d}$ points (for the dimension $d$) without time
dependence. For the time-dependent case, the simulation is run until
$t=1$ with time steps between 0.001 and 1.0 and $64^{d}$ points.
The minimum time step is increased to 0.01 in 2D and 0.1 in 3D. In
each case the points are laid down in a spatially uniform lattice
configuration. For the non-uniform cases, the point positions are
randomly perturbed by up to 0.2 times the point distance in each dimension,
or $x^{\alpha}=x^{\alpha}\pm\Delta x^{\alpha}\gamma^{\alpha}$, where
$-0.2\leq\gamma^{\alpha}\leq0.2$ is randomly generated for each point
and dimension independently and $\Delta x^{\alpha}$ is the point
spacing for the given dimension.
\subsubsection{Sinusoidal manufactured problem}
The first manufactured solution,
\begin{equation}
\psi_{\text{sinusoidal}}=1+\frac{1}{2\pi}\prod_{\alpha}\cos\left(\pi\left(x^{\alpha}+t\right)\right),
\end{equation}
is designed to test convergence of the discretized transport equation
{[}Eq. (\ref{eq:weak-transport}){]}. The solution is chosen such
that the manufactured source never becomes negative, which would be
unphysical. To ensure that the integration of the cross sections {[}Sec.
\ref{sec:Integration}{]} works correctly, the scattering and absorption
opacities are also chosen to have sinusoidal values that are out of
phase with the solution and one another,
\begin{gather}
\sigma_{a}=1+\frac{2}{3}\prod_{\alpha}\cos\left(3x^{\alpha}\right),\\
\sigma_{s}=1+\frac{3}{4}\prod_{\alpha}\sin\left(2x^{\alpha}\right).
\end{gather}
The domain is $-1\leq x^{\alpha}\leq1$. For the steady-state case,
the manufactured solution is fixed at $t=0$.
The spatial convergence results in Fig. \ref{fig:man-sin-steady}
indicate second-order convergence in 1D, 2D, and 3D, as expected for
linear RK corrections. The 3D results eventually plateau around 96
points. It is likely that the difference between the numeric and analytic
solution is reaching the accuracy limit of the third-order quadrature
in 3D, which appears to reduce the convergence order to first-order.
The inclusion of spatially-dependent cross sections does not appear
to hinder convergence.
When the point positions are randomly perturbed (Fig. \ref{fig:man-sin-steady-perturb}),
the convergence rate stays the same, with a caveat. The algorithm
for calculating kernel extents is designed for hydrodynamics and requires
that the average level of support is above a certain threshold, not
the support for each point. It is possible that in 2D and 3D, the
solution reaches the accuracy of the poorly-supported RK calculation
and stops converging for certain points. Before this occurs, the convergence
is second-order. When the calculation is run with a higher kernel
extent (not pictured), the quantitative behavior is similar but the
error levels off at a lower value. The other difference between dimensions
is the integration quadrature, which is coarser and less accurate
between dimensions. This could be contributing to the leveling off
of the error.
The temporal convergence rate is first-order (Fig. \ref{fig:man-sin-time}),
as expected from the backward Euler time discretization. The 2D and
3D results have similar or lower error than the 1D results for a similar
number of points, which may be due to the higher level of connectivity
in 2D and 3D. For this problem, the time discretization error even
with the smallest time step considered (0.001) is similar to the spatial
discretization error with $32^{d}$ points. It is expected that to
increase the accuracy of a time-dependent simulation at the point
where the temporal and spatial discretization errors are similar,
the time step would need to be decreased as the distance between points
squared.
\subsubsection{Outgoing wave manufactured problem}
The second manufactured problem represents a wave traveling from the
origin outward,
\[
\psi_{\text{wave}}=1+\frac{1}{t^{2}+6}\exp\left(-10\left[\left|x\right|-t\right]^{2}\right),
\]
with constant opacities of $\sigma_{a}=0.5$ and $\sigma_{s}=2.0$
and a domain of $-1\leq x^{\alpha}\leq1$. For the steady-state case,
the manufactured solution is fixed at $t=0.5$.
As in the sinusoidal case, the spatial convergence is second-order
(Fig. \ref{fig:man-wave-steady}) and the temporal convergence is
first-order (Fig. \ref{fig:man-wave-time}). The magnitude of the
error is similar in the wave and the sinusoidal case, and just as
in the sinusoidal case, the error in 3D plateaus on the spatial convergence
plot, probably due to the integration error. The perturbed version
of the steady-state problem (Fig. \ref{fig:man-wave-steady-perturb})
has similar behavior to the sinusoidal case described above, with
second-order convergence until reaching issues with either RK kernel
support or integration.
\subsection{Purely absorbing problem\label{subsec:Purely-absorbing}}
One challenge in transport is handling highly absorptive regions without
incurring negative fluxes. In this problem, a single ray with $\Omega=\left\{ 1,0,0\right\} $
is incident on a purely absorbing slab with a domain $0\leq x\leq1$,
which is modeled in 1D. First, a constant cross section of $\sigma_{a}=5$
is considered with a variable number of points between 8 and 64. Then,
the number of points is held constant at $32$ and the cross section
is varied between $1$ and $64$. The points are again placed uniformly.
Convergence results are shown in Fig. \ref{fig:purely-absorbing-err}
for a few cases of the RK order and the number of neighbors, which
is the number of other points across a kernel radius for the base
connectivity used to create the overlap connectivity. The number of
neighbors for the reduced-radius kernels should be at least one higher
than the RK order to prevent the system in Eq. (\ref{eq:rk-matrix})
from being singular, which means that the number of neighbors for
the original kernels (which is the number reported here) should be
two times the RK order plus one. For zeroth-order RK corrections,
the solution converges with approximately second-order accuracy. For
first-order corrections, the solution converges with approximately
second-order accuracy for 4 neighbors and between second and third
order for 6 neighbors. With second-order corrections and 6 neighbors,
the convergence order is between third and fourth. In general, for
a smooth solution, the expected convergence order is one greater than
the RK order.
Results for the case with a constant number of points and a changing
cross section are shown in Fig. \ref{fig:purely-absorbing-xs}. For
this problem, in which the primary gradient is at the edge of the
problem with the lowest point density, the MLPG approach requires
around one point per mean free path of the material to avoid negativities,
which is reflected in the results. The solution begins to exhibit
negativities at $\sigma_{a}=32$ for the case with 6 neighbors, while
for 4 neighbors, the negativities show up for $\sigma_{a}=64$ and
above. Note that the error is an absolute error, since the normalization
would otherwise skew the results. As in the convergence study, an
increasing number of neighbors and RK order decrease the solution
error.
Based on these results, it may be tempting to use kernels with large
radii and high RK order for other problems to increase solution accuracy.
One issue with this is computational cost, which increases significantly
in 2D and 3D as the function radii increase. Since the RK order is
limited for kernels with small radii, this also limits the RK order.
For instance, in 3D, the number of neighbors increases as $r^{3}$,
where $r$ is the kernel radius, so moving from 4 neighbors across
the kernel radius to 6 will more than triple the cost. This also increases
the difficulty of solving the transport system (the $\mathcal{L}^{-1}$
operation). Another issue is negativities, which are present for all
the 32-point simulations with 6 neighbors but not for any 32-point
simulations with 4 neighbors. These negativities can become amplified
in time-dependent problems, where a negative absorption becomes an
unphysical source of particles. For more discussion on negative fluxes,
see Sec. \ref{sec:Conclusions}.
\subsection{Crooked pipe problem}
This problem is described in Ref. \cite{smedley-stevenson2015benchmark}
as a test of diffusion synthetic acceleration (DSA). Results for the
MLPG code with the Krylov iteration as described in Sec. \ref{subsec:Iterative-methods}
are compared to those from a code based on the discontinuous finite
element method (DFEM) with acceleration based on a variable Eddington
factor (VEF), as described in Ref. \cite{yee2020quadratic}.
The geometry for the problem is shown in Fig. \ref{fig:crooked-geometry},
with $\sigma_{s}=200$ in the wall and $\sigma_{s}=0.2$ in the pipe.
The absorption cross section is zero in both regions. The relatively
coarse angular discretization with 16 ordinates is identical between
the MLPG and DFEM codes. The MLPG results have a spatial discretization
with 716,800 equally-spaced points (or 160 by 160 points per unit
area), while the DFEM results are calculated on a mesh with 1,335,296
elements, with a higher density of elements placed near the pipe-wall
boundary. The MLPG points and DFEM mesh at quarter resolution (or
16 times fewer points) is shown in Fig. \ref{fig:crooked-mesh}. The
units for the problem are set such that the speed of light is $c=1$.
The problem is run until $t=20$ with a fixed time step of $\Delta t=0.1$.
A comparison of the crooked pipe results at $t=10$ and $t=20$ is
shown in Fig. \ref{fig:crooked-comparison}. The propagation speed
of the radiation appears is nearly identical between the two codes.
For the $t=10$ plot, the radiation would have traveled 10 unit distance
at most in the 10 unit time (since $c=1$). The minimum path the radiation
could take to reach the plane at $x=2$ from the source at $x=-3.5$
is $6.5$ unit distance. Depending on the direction, the actual distance
the radiation would need to travel to reach the plane $x=2$ would
be at between 7 and 12 unit distance. The strongest visible ray is
at $\Omega^{\alpha}=1/\sqrt{3}$, which would have reached $x=2$
at $t=9.5$ if scattering and corners were neglected. As the radiation
appears to have just reached $x=2$ at $t=10$, the calculated time
of arrival is close to the distance the radiation would have traveled
in that time.
The results show significant ray effects, but because the two codes
use the same angular quadrature, the effects appear to be the same.
Before reaching the crooked part of the problem, there are no significant
differences visible between the two solutions. After the radiation
has gone around the obstacle, the MLPG solution is higher in magnitude,
which is visible in the $t=10$ plot near the right edge of the obstacle
or in the $t=20$ plot at the exiting surface of the pipe. Part of
this could be due to the higher resolution along the pipe-wall interface
in the DFEM simulation, which could affect the scattering rate at
the interface. The contours of the solution near the interfaces also
line up very closely, except at the wall edge at $x=0.5$, where the
MLPG solution reaches a further through the wall, which could again
be due to the lower resolution near the interface.
As mentioned before, this problem has been used as a test of DSA.
The Krylov solution procedure {[}for the solution of the MLPG system
in Eq. (\ref{eq:krylov-operator}){]} works for this case without
preconditioning, with an average of 64 iterations to converge. As
the ILUT factorization of the matrices representing the $\mathcal{L}^{-1}$
operation is performed once and stored, this doesn't increase the
total simulation time by nearly 64 times more than a single iteration,
as the factorization is a far larger cost than a single solve. Within
each scattering iteration, the transport GMRES solver converges to
a tolerance of $10^{-15}$ in 55 iterations on average. The DFEM solution,
however, required only one transport solve and two VEF solves per
time step, which if applied to the MLPG solution could open up more
cost-effective methods for solving for the scattering source.
For general reference, the RK transport code takes 15,943 seconds,
or 79 seconds per time step, to run the crooked pipe problem with
16 ordinates and 716,800 points on 288 processors, which equates to
2,488 points or 39,808 unknowns per processor on average. This includes
the time for integration, computation of the ILUT preconditioners
for all 16 directions, and convergence of the solution and scattering
source. With an effective preconditioner and possibly avoiding ILUT
decompositions, the solve time would decrease significantly. The need
for appropriate preconditioning is discussed further in Sec. \ref{sec:Conclusions}.
\subsection{Asteroid problem\label{subsec:Asteroid}}
One motivation for combining radiation transport with a smoothed particle
hydrodynamics code is for a planetary defense application: the deflection
of an asteroid due to radiation from a standoff nuclear burst. In
this scenario, the absorbed radiation energy ablates the surface of
the asteroid, causing material to blow off and alter the orbit of
the asteroid via momentum conservation. This problem is a simplified
version of that scenario, a spherical rock ``asteroid'' that absorbs
radiation from a distant point source in 2D. This problem is similar
to the purely-absorbing version of Kobayashi benchmarks \cite{kobayashi20013d},
which also have features that are difficult to angularly resolve and
a small source emitting particles into a void. The asteroid has an
absorption cross section of $\sigma_{a}=10.0$, while the medium surrounding
the asteroid has an absorption cross section of $\sigma_{a}=0.001$.
Neither material includes scattering. The asteroid has a radius of
35 and is centered at the origin. The point source is located a distance
of 70 away from the surface of the asteroid. The asteroid can be modeled
by a shell, since almost all of the radiation is absorbed at the surface
of the asteroid.
The shell of the asteroid is set to be 20 mean free paths thick. The
distance between points is set to be 0.2 at the inside of the shell
of the asteroid, 0.1 at the outside of the shell, 1.0 halfway between
the source and the asteroid, and 0.1 near the source. The initial
angular quadrature of 4,096 ordinates is refined as described in Sec.
\ref{subsec:Angular-refinement} down to 550 ordinates, of which 480
hit the asteroid. The problem is run with a single time step large
enough for the radiation to propagate throughout the domain. Afterward,
the numeric solution is compared to the analytic solution, which is
derived in App. \ref{sec:Analytic-asteroid}. The solution points
and integration mesh for this problem are shown in Fig. \ref{fig:asteroid-geometry}.
Note that the integration mesh is further broken down into triangles
for use with standard quadratures (Sec. \ref{sec:Integration}).
The analytic and numeric solutions to the asteroid problem are shown
in Fig. \ref{fig:asteroid-solution}. The most obvious difference
at first glance is the large areas at the top of the numeric solution
where the solution is close to zero. These are areas where, by design,
the angular refinement has not put a sufficient number of angles to
resolve the solution. These should not affect the solution at the
asteroid, since there, the solution should be sufficiently resolved
in the angular domain. While there are 480 rays that hit the asteroid
from the point source, the radiation is still not angularly uniform
in the region that is resolved, as can be seen by the more intense
ray that hits the asteroid around the point $\left\{ 11,33\right\} $.
Near the $y=0$ plane, where there is approximately one point per
mean free path in the direction the solution is changing, the contours
of the analytic and numeric solutions line up very well, with the
exception of the aforementioned oscillations. This agrees with the
purely absorbing results (Sec. \ref{subsec:Purely-absorbing}), in
which the solutions with around one point per mean free path showed
higher accuracy and few oscillations compared to those with more than
one point per mean free path.
There are two connected difficulties in this problem, which are ray
effects and negativities. Oscillations can be seen toward the inner
surface of the asteroid at all positions, but the oscillations are
by far the worst where the radiation is traveling nearly parallel
to the surface of the asteroid. At these points, the solution will
change from the vacuum solution just outside of the surface to nearly
zero inside of the surface. This causes oscillations that lead to
negativities. The SUPG stabilization does a good job of handling the
oscillations that may develop in the direction of radiation propagation
(near the $y=0$ plane), but more consideration is needed to prevent
the oscillations that develop perpendicular to the radiation propagation
direction or when the solution changes discontinuously.
This problem is designed to stress the code and show opportunities
for future work. If the results needed to be accurate, the source
could be analytically calculated just before it hits the asteroid
and inserted as a boundary source there, which would reduce much of
the need for a refined and specialized quadrature. The negatives,
however, would persist, which is something that would need to be addressed
before this calculation would work well in a time-dependent or thermal
radiative transfer scenario, as discussed in Sec. \ref{sec:Conclusions}.
\section{Conclusions and future work\label{sec:Conclusions}}
The MLPG discretization in this paper simplifies the process running
a problem with meshless transport. The fully-implicit time differencing
is stable for large time steps. The SUPG stabilization works to prevent
oscillations and increase the efficiency of inverting the transport
matrix. The integration with a Voronoi diagram is robust and follows
the resolution of the meshless points without user input. The SUPG
stabilization and Voronoi integration add complexity to the code,
which could be a barrier to entry, but reduces the need for specialized
solvers or repeated adjustments of a background mesh.
The RK functions used in the discretization permit higher-than-second-order
convergence, but practically, the radii of the kernels should often
be minimized to reduce negativities and computation cost, which constrains
the RK order. With the fully-implicit time discretization and first-order
RK corrections, the results are consistent with second-order convergence
in space and first-order convergence in time. For a purely absorbing
problem with an incoming source, higher-order convergence is achieved
in space with second-order RK corrections and larger kernel radii.
The two manufactured solutions show the capability to represent spatially-dependent
cross sections and converge in 1D, 2D, and 3D.
There are at least three issues that remain to be resolved. The first
is negativities. In the purely-absorbing problem, around one point
is needed per mean free path to avoid negativities. In the asteroid
problem, negativities are difficult to avoid due to ray effects, as
the SUPG stabilization applies numerical diffusion only in the direction
of radiation propagation. A negative flux fixup method such as the
zero-and-rescale approach \cite{hamilton2009negative} may work for
meshless transport, but care would need to be taken to rescale a quantity
that should always be positive and not the expansion coefficients.
While this could inhibit the effects of oscillations on time-dependent
problems and perhaps keep them from growing, it would be much more
difficult to remove oscillations entirely.
The second issue is preconditioning. In the crooked pipe problem,
the solution converged when using only GMRES to converge the scattering
source and agreed well with a DFEM solution, but this required many
iterations to achieve. Combining the Krylov solve with a method such
as DSA \cite{warsa2004krylov} could significantly reduce the number
of iterations needed to converge the scattering source. The MLPG transport
equation without SUPG should have the diffusion limit, similar to
a high-order DFEM discretization \cite{haut2018dsa}, but it is not
apparent whether the same is true with SUPG. The process of deriving
DSA for the SUPG system should give information on whether it has
the diffusion limit and if not, what changes may be made to the discretized
system to ensure it has the diffusion limit.
The third issue is ensuring proper support for the RK kernels. As
shown in the manufactured problems, randomly perturbing the point
positions can lead to a limit on convergence. Ensuring that every
point has the needed support individually through a more robust calculation
may resolve these issues.
The current meshless discretization works well for problems in which
the solution does not go negative. Once a negative flux fixup treatment
is applied, the addition of additional physics to the transport discretization
such as thermal radiative transfer and radiation hydrodynamics would
be more achievable. In a radiation hydrodynamics simulation, where
the meshless topology is constantly changing, the consistently discretized
transport with a Voronoi integration approach eliminates mapping to
and from a mesh for radiation transport and is far cheaper than placing
a non partition-of-unity quadrature for each kernel.
\section*{Acknowledgements}
This work was performed under the auspices of the U.S. Department
of Energy by Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344. This document was prepared as an account of work
sponsored by an agency of the United States government. Neither the
United States government nor Lawrence Livermore National Security,
LLC, nor any of their employees makes any warranty, expressed or implied,
or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product,
or process disclosed, or represents that its use would not infringe
privately owned rights. Reference herein to any specific commercial
product, process, or service by trade name, trademark, manufacturer,
or otherwise does not necessarily constitute or imply its endorsement,
recommendation, or favoring by the United States government or Lawrence
Livermore National Security, LLC. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the
United States government or Lawrence Livermore National Security,
LLC, and shall not be used for advertising or product endorsement
purposes.
\bibliographystyle{unsrt}
|
2,877,628,090,247 | arxiv | \section{Introduction}
For most of the mathematical constants used today, like $\pi$, $e$ or $\zeta$, very efficient
algorithms to approximate these constants are known. There are, however, rare cases such as Bloch's, Landau's or the Hayman-Wu constant,
where conjectures and rough bounds on such constants are known but the exact value is unknown. Recently, an algorithm to approximate
Bloch's constant was given in \cite{rett08}. We continue this line of research and present a similar algorithm to approximate Landau's constant up
to any precision.
Because of the close connection between the Landau and Bloch constants we can reuse many of the ideas (and motivations)
on the computation of the Bloch constant in this paper. A core component of the algorithm, namely the computation of maximum discs in the
image of a (normalised) holomorphic function, is totally different. So injectivity does not play any role any longer. What is more, however,
homotopy methods used in the algorithm for Bloch's constant do not work any longer. This is due to the fact that for discs which need not be schlicht,
several overlapping parts of the pre-image come into play. To decide whether the images do indeed overlap is in general not even computable.
Though the ideas of the algorithm are inspired by Type-2 theory of effectivity, we formulate our results totally independently of this theory.
In this way we hope that our result is accessible to a wider audience. Finally, we improve even on the
parts which could be taken literally from the algorithm for Bloch's constant to narrow the gap between the theory and possible implementations.
Landau's constant (see \cite{lan29}) gives a quantitative version of the fact that non-constant holomorphic functions are open.
More precisely, it states that for any $r>0$ and any holomorphic function $f$ defined on
a disc $\mathbb{D}_r(z_0):=\{ z\in\mathbb{C}\mid |z-z_0|<r\}$ with $f'(z_0)\not=0$ there exists a disc of
radius $|f'(z_0)|\cdot r\cdot c$ inside the image $f(\mathbb{D}_r(z_0))$, where the constant $c>0$ does not depend on $f$!
Obviously $c$ is bounded from above; thus the supremum, the so called Landau constant $\lambda$, exists. The best upper
bound known for $\lambda$,
\[ \lambda \leq \frac{\Gamma(\frac{1}{3})\cdot\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6})},
\]
is at the same time conjectured to be the exact value of
$\lambda$. However the best lower bound known so far is
\[ \frac{1}{2}< \lambda .
\]
(see \cite{rob38}\ and \cite{rad43}).
Putting this in decimal representation gives
\[0.5<\lambda\leq 0.54325...\]
i.e. all we know is the constant up to $4\cdot 10^{-2}$.
In this paper we will give an algorithm to compute Landau's constant $\lambda$ up to any precision in
the sense that on input $n\in\mathbb{N}$ some rational number $q$ with
$|q-\lambda|<2^{-n}$ can be computed.
The main idea of our algorithm is to compute for several normalised functions the corresponding
$\lambda$-values. Following the definition, it seems that we have to take the infimum for all normalised
functions, which could not be done in finitely many steps. We will overcome this problem by a compactness argument in Section 5.
In the next section we recall a few notations and fix the machine model which we will use throughout this paper.
In Section 3 we introduce a subset of all holomorphic functions such that $\lambda$ is already determined by this smaller, compact class.
In Section 4 we will then show how $\lambda_f$ of a single function can be approximated and, finally, in Section 5 we prove the computability
of Landau's constant.
\section{Preliminaries}\label{sec_prel}
We begin this section with a few remarks on the model underlying our algorithms. Our main algorithm (Section 5) will only compute on finite words, where we assume some straightforward, standard encodings (representations) of the dyadic numbers, i.e. numbers of the form $m\cdot 2^n$ with $m,n\in\mathbb{Z}$, by such words. We do not need any
operations on infinite words or other structures like $\mathbb{R}$ or $\mathbb{C}$, and therefore our algorithms could be implemented by classical Turing machines.
If some value is given as input to our machine, we also say that the machine {\it computes}\ something {\it on}\ this value.
In addition, to simplify things, we use a second model for intermediate results where we allow infinite sequences as input.
One can think of this model as a classical Turing machine which can in addition ask some kind of oracle for single elements of the input. The output, however, will always be a finite word which will be returned after a finite number of steps.
This second model is only used for simplification reasons, and our final algorithm to compute Landau's constant
will not depend on this model.
Let $\mathbb{D}_r(z_0):=\{ z\in\mathbb{C}\mid |z-z_0|<r\}$ denote the open disc of radius $r$ with
centre $z_0$ and let $\overline{\mathbb{D}}_r(z_0)$ denote its topological closure, i.e. $\overline{\mathbb{D}}_r(z_0):=\{ z\in\mathbb{C}\mid |z-z_0|\leq r\}$. To simplify notation we use $\mathbb{D}_r:=\mathbb{D}_r(0)$ and $\mathbb{D}=\mathbb{D}_1$. A
{\it normalised (holomorphic) function}\ on a domain $D$, $0\in D$, is a holomorphic function $f$ with $f'(0)=1$. The space of normalised functions on $D$ is denoted by ${\mathcal{N}}(D)$.
For given domain $A\subseteq \mathbb{C}$ we denote the radius of the largest disc in $A$ by $l(A)$, i.e.
\[l(A)=\sup \{ r\mid \exists z. \mathbb{D}_r(z)\subseteq A\}.\]
Given a holomorphic function $f$ with $D\subseteq\mbox{dom}(f)$ let $\lambda_f(D)$ denote the radius of the largest disc in $f(D)$, i.e. \[\lambda_f(D)= l(f(D)).\] Finally let $\lambda_f:=\lambda_f(\mathbb{D})$ and the {\it Landau constant}\ {$\lambda$} be the infimum of all $\lambda_f$ with $f\in{\mathcal{N}}(D)$.
Obviously, $\lambda$ is already the infimum of all $\lambda_f$ for normalized $f$ with the additional condition $f(0)=0$.
Throughout this paper we use most of the time only very basic results of complex analysis covered by most textbooks (see e.g. \cite{ahl66}).
The only somewhat more advanced result from complex analysis is Koebe's $1/4$ Lemma (see e.g. \cite{rud87}):
\begin{theorem}[Koebe's $1/4$ Lemma]\label{theo_koebe}
Let $f$ be an injective holomorphic function on some disc $\mathbb{D}_r$. Then $\mathbb{D}_{r\cdot |f'(0)|\cdot 1/4}(f(0))\subseteq f(\mathbb{D}_r)$.
\end{theorem}
The representations of the objects we use throughout this paper are introduced next, where
we implicitly use some kind of efficiently computable pairing function without further mention.
Let $\mathbb{Y}$ denote the class of dyadic numbers, i.e. \[\mathbb{Y} := \{ m\cdot 2^{n}\mid n,m\in\mathbb{Z}\}.\] We could use rational numbers as well but having implementations in
mind, we stick to the efficiently implementable dyadic numbers.
Identifying complex numbers $z=x+\iota\cdot y$ and pairs $(x,y)$, where we denote the imaginary unit by $\iota$, we can represent
elements of the set $\mathbb{Y}[\iota]$ of complex dyadics simply by pairs of dyadic numbers.
In this way, it should be clear that there are efficient algorithms to approximate operations like $\cdot,/,+,-$ on the complex dyadics,
in the sense that, given complex dyadics $z$, $z'\not=0$ and some $n\in\mathbb{N}$, we can easily compute a dyadic number $y$ such that
$|y-z/z'|<2^{-n}$ etc.
To simplify notation, we say that we can {\it compute a value $z$ up to precision $2^{-n}$}\ if we can, on input $n$ and $z$, compute a (complex) dyadic $y$ such that $|y-z|<2^{-n}$.
If we can compute $z$ up to precision $2^{-n}$ for all $n$, then we say that we can compute $z$ up to any precision.
Let furthermore $\mathbb{I}$ denote the class of intervals $[c,c']\times [d,d']$ where $c,c',d,d'$ are dyadics. We consider these intervals as sets of complex numbers.
Let $\Pi$ be a finite alphabet. Then $\Pi^\ast$ denotes the set of finite sequences (words) over $\Pi$, $\varepsilon$ the empty sequence, and $\Pi^\infty$ the set of infinite sequences over $\Pi$. As usually, we think of $\Pi^\infty$ as a topological space where $\{ w\Pi^\infty\mid w\in\Pi^\ast\}$
is a basis of the topology. It is well known that $\Pi^\infty$ with the above topology is a compact space.
In our algorithm we need to approximate holomorphic functions in a certain function space.
We will do this by representing this function class by $\Pi^\infty$ for $\Pi = \{ 1,2,3,4\}$.
To
simplify things we will first define, for given interval $I\in\mathbb{I}$, the representation $\Psi_I^\infty$ of the complex numbers in
$I$. To this end, let for a given interval $I=[c,c']\times [d,d']$, the intervals
\[[c,c'']\times [d,d''],\quad [c'',c']\times [d,d''],\quad [c'',c']\times [d'',d'],\quad [c,c'']\times [d'',d'],\] where $c''=(c+c')/2$ and $d''=(d+d')/2$,
be denoted by $I_1$, $I_2$, $I_3$ and $I_4$, respectively.
With these preliminaries, we use sequences $\alpha_0$, $\alpha_1$, $\ldots$ of numbers in $\Pi:=\{1,2,3,4\}$ to represent complex numbers where we
define $\Psi_I:\Pi^\ast\rightarrow \mathbb{I}$ by
\[ \begin{array}{lll}\Psi_I(\varepsilon) = I,\\
\Psi_I(w\alpha) = (\Psi_I(w))_\alpha
\end{array}\]
for all $I\in\mathbb{I}$, $w\in\Pi^\ast$ and $\alpha\in\Pi$, and finally \[\Psi_I^\infty(\alpha_0\alpha_1\ldots)=\bigcap_i \Psi_I(\alpha_0\alpha_1\ldots\alpha_i),\] i.e.
we define complex numbers by suitable nested intervals.
Let now $m_1,m_2,\ldots$ denote a sequence of positive dyadics such that $\sum_{i\geq 1} m_i\cdot\varepsilon^i$ converges for all $\varepsilon\in [0,1)$.
Then the set of sequences $a_1,a_2,\ldots$ of complex numbers where the absolute values of the real and imaginary parts of $a_i$ are bounded by $m_i$ for all $i\geq 1$ defines a class $F_{m_1,m_2,\ldots}$ of holomorphic functions
on $\mathbb{D}$ by identifying a sequence $a_1$, $a_2$, $\ldots$ with the power series $1+\sum_{i\geq 1} a_i\cdot z^i$.
We are interested only in power series where the first coefficient is 1 because this class represents exactly the derivatives of normalized holomorphic functions. We use $\Psi$ to define a representation of
these holomorphic functions: Let $t_1$, $t_2$, $\ldots$ be a sequence of natural numbers such that any natural number is encountered infinitely often,
e.g. $0$, $0$, $1$, $0$, $1$, $2$, $0$, $1$, $2$, $3$, $0$, $\ldots$. Then for a given sequence $\alpha_1$, $\alpha_2$, $\ldots$ with $\alpha_i\in\Pi$ and
any $n\in\mathbb{N}$ the sequence $t_1$, $t_2$, $\ldots$ defines the subsequence $\alpha_{n_1}$, $\alpha_{n_2}$, $\ldots$ of all those elements $\alpha_{n_j}$
such that $t_{n_j}=n$. In this way we can define a mapping \[\Psi_{m_1,m_2,\ldots}^\infty:\Pi^\infty\rightarrow F_{m_1,m_2,\ldots}\] by
\[ \Psi_{m_1,m_2,\ldots}^\infty(\alpha_1\alpha_2\ldots)=1+\sum_{n\geq 1} \Psi_{[-m_n,m_n]\times [-m_n,m_n]}^\infty(\alpha_{n_1}\alpha_{n_2}\ldots)\cdot z^n.\]
To simplify notations we denote, for given holomorphic function $f:\mathbb{D}\rightarrow \mathbb{C}$, the anti-derivative of $f$, which maps $0$ to $0$, by $\int f$,
i.e. $\int f:\mathbb{D}\rightarrow\mathbb{C}$, $(\int f)(0)=0$ and $(\int f)' = f$.
Notice that $\int \Psi_{m_1,m_2,\ldots}^\infty(\alpha_1\alpha_2\ldots)\in{\mathcal{N}}(\mathbb{D})$ with the above settings.
A proof of the following theorem can for example be found in \cite{muel87}\ or \cite{rett08b}.
\begin{theorem}\label{theo_L1}
Let $\alpha_1$, $\alpha_2$, $\ldots$, $m_1$, $m_2$, $\ldots$ and $t_1$, $t_2$, $\ldots$ be as above. Furthermore let $b_1$, $b_2$, $\ldots$ be
a sequence of positive dyadic numbers such that with \[f=\Psi_{m_1,m_2,\ldots}^\infty(\alpha_1\alpha_2\ldots),\] for all $n\geq 1$ we have
\[ \forall z\in\mathbb{D}_{1-2^{-n}}.|f(z)|<b_n.\]
Then, on input $r\in\Dyn\cap(0,1)$, $z\in\Dyn[\iota]$ and \[\alpha_1, \alpha_2, \ldots,\quad m_1, m_2, \ldots,\quad t_1, t_2, \ldots,\quad b_1, b_2, \ldots\mbox{ and }n\in\mathbb{N},\] we can compute $\int f(z)$, $f(z)$ and $f^{(n)}(z)$
on $\mathbb{D}$ up to any precision.
\end{theorem}
\section{$\lambda$-bounding Functions}\label{sec_3_2}
Similar to the $\beta$-bounding functions in \cite{rett08}\ we define here a class $F_\lambda$ of holomorphic functions which
determines $\lambda$ in the sense that $\lambda=\inf_{f\in F_\lambda}\lambda_f$. Unlike \cite{rett08}\ we use different bounds on
the coefficients of the corresponding power series and, furthermore, use the compactness of $\Pi^\infty$ together with the
mapping $\Psi^\infty$ rather than compactness of the class $F_\lambda$ itself.
To start with we will prove suitable upper bounds on (the derivative of) functions $f$ whose $\lambda$-values approximate $\lambda$:
\begin{lemma}\label{lemma_3_2}
Let $c>1$ be given. Then for any $f\in{\mathcal{N}}(\mathbb{D})$ and $w\in\mathbb{D}$ with
$|f'(w)|\geq c /(1-|w|^2)$ we have
\begin{equation*} \lambda_f\geq c\cdot \lambda.\end{equation*}
\end{lemma}
\begin{proof}
Let $w\in\mathbb{D}$ and $f\in{\mathcal{N}}(\mathbb{D})$ with $|f'(w)|\geq c /(1-|w|^2)$ be given.
Let $g:\mathbb{D}\rightarrow\mathbb{D}$ be the automorphism \[z\mapsto (z+w)/(1+\overline w z).\]
Then $f\circ g(0) = f(w)$, $(f\circ g)'(0) = f'(w)\cdot (1-|w|^2)$, and
\[h:=\frac{1}{f'(w)\cdot (1-|w|^2)} f\circ g\] is a normalised
function. Therefore
\[\frac{1}{c}\cdot \lambda_{f\circ g}\geq \frac{1}{|f'(w)|\cdot (1-|w|^2)}\cdot \lambda_{f\circ g}\geq \lambda_h\geq \lambda.\]
As $f(\mathbb{D})=f\circ g(\mathbb{D})$ the statement of the lemma follows.
\end{proof}
To simplify things we fix some value $c>1$, say $c=1+2^{-100}$ for the time being. To find $\lambda$ it suffices to consider all $\lambda_f$
of all normalised functions $f$ with $|f'(z)|\leq c/(1-|z|^2)$ for all $z\in\mathbb{D}$. This immediately gives us a bound on the coefficients of
the corresponding power series:
\begin{lemma}\label{lemma_3_3}
Let $f$ be a normalised function such that
$|f'(z)|\leq c/(1-|z|^2)$ for all $z\in\mathbb{D}$. Then $f'(z)=\sum_n a_n\cdot z^n$ with
\begin{enumerate}[(a)]
\item $a_0=1$ and
\item $|a_n|\leq c\cdot e\cdot (n+2)/2$ for all $n\geq 1$.
\end{enumerate}
(Here $e$ denotes the Euler constant.)
\end{lemma}
\proof
Item (a) is obvious because $f$ is assumed to be normalised.
By the Cauchy inequality we have for given $n$ and $r\in (0;1)$
\[ |a_n| \leq \frac{1}{r^n} \sup_{|z|=r} |f'(z)|\leq \frac{1}{r^n}\cdot \frac{c}{1-r^2}. \]
Choosing $r:=\sqrt{n/(n+2)}$ we get
\[ |a_n| \leq \frac{1}{r^n}\cdot \frac{c}{1-r^2}\leq c\cdot\left(1+\frac{2}{n}\right)^{\frac{n}{2}}\cdot \frac{n+2}{2}\leq c\cdot e\cdot \frac{n+2}{2}.\eqno{\qEd} \]
Let, for $n\geq 1$, $m_n$ be some dyadic approximation such that \[c\cdot e\cdot (n+2)/2\leq m_n \leq c\cdot e\cdot (n+2)/2 + 2^{-n}.\]
Then we have for given $\varepsilon\in (0;1)$
\[ \begin{array}{lll}1+\sum_{n\geq 1} m_n\cdot \varepsilon^n & \leq & \frac{c\cdot e}{2} \cdot (\sum_n \varepsilon^n + \sum_n (n+1)\cdot\varepsilon^n) + \sum_n 2^{-n} -c\cdot e\\[0.2cm]
& \leq & \frac{c\cdot e}{2} \cdot
\left(\frac{1}{1-\varepsilon}+\frac{1}{(1-\varepsilon)^2}\right) + (2-c\cdot e).\end{array}\]
A similar bound does also hold for the corresponding derivatives:
\[ \sum_{n\geq 1} n\cdot m_n\cdot \varepsilon^{n-1}\leq \frac{c\cdot e}{(1-\varepsilon)^3} + \frac{c\cdot e}{2\cdot (1-\varepsilon)^2} + 2.
\]
Thus we can easily compute a sequence $b_1$, $b_2$, $\ldots$
of bounds such that Theorem \ref{theo_L1}\ can be applied. To simplify notations let \[\Psi^\infty:=\Psi_{m_1,m_2,\ldots}^\infty.\]
In the definition of
$\Psi^\infty$ above the coefficients of the corresponding power series in $\Psi^\infty(\Pi^\infty)$ can get larger than we considered so far
because, to simplify things, we bound the real and imaginary parts of the $n$-th coefficient by $m_n$, not the absolute value of the coefficient itself. Thus an additional factor $\sqrt{2}$ is sufficient in the following corollary.
Due to the structure of our algorithm, however, we can get rid of this additional factor in implementations by
restricting $\Pi^\infty$ to a proper subclass by a simple test for the absolute value of the coefficients.
We can summarize the essence of this
section by the following corollary:
\begin{corollary}\label{cor_3_2}
Let $c>1$ and let, for $n\geq 1$, $m_n$ be some dyadic number such that
\[c\cdot e\cdot \frac{n+2}{2}\leq m_n \leq c\cdot e\cdot \frac{n+2}{2} + 2^{-n}.\]
Then
\begin{enumerate}[(a)]
\item $\lambda = \inf \{ \lambda_{\int f}\mid f\in \Psi^\infty(\Pi^\infty)\}$,
\item we can compute on given dyadic complex number $z\in\mathbb{D}$, $\alpha\in\Pi^\infty$, $n\in\mathbb{N}$ and $r\in\Dyn\cap(0;1)$
\begin{enumerate}[(i)]
\item $f^{(n)}(z)$ and
\item upper bounds \[\mu_r' := \sqrt{2}\cdot\left(c\cdot \frac{e}{2} \cdot \left(\frac{1}{1-r}+\frac{1}{(1-r)^2}\right) + (2-c\cdot e)\right)\]
and
\[\mu_r'' := \sqrt{2}\cdot\left(\frac{c\cdot e}{(1-r)^3} + \frac{c\cdot e}{2\cdot (1-r)^2} + 2\right)\]
of $\sup \{ |f'(z)|\mid z\in \mathbb{D}_r\}$ and $\sup \{ |f''(z)|\mid z\in \mathbb{D}_r\}$, respectively,\\
\end{enumerate}
up to any precision, where $f:=\int \Psi^{\infty}(\alpha)$.
\end{enumerate}
\end{corollary}
\section{The $\lambda$-value of a single Function}\label{sec_3_3}
In this section we will show how $\lambda_f$ can be approximated. More precisely, we will give lower bounds on $\lambda_f$
which are at the same time approximately upper bounds for $\lambda$. This will be enough to compute $\lambda$ up to any precision in the end.
To this end, let $\alpha_1\alpha_2\ldots$ be a fixed sequence in $\Pi^\infty$ which will be fed to all algorithms considered in this section as input. Furthermore, we
will denote the corresponding functions $\Psi^\infty(\alpha_1\ldots)$ and $\int\Psi^\infty(\alpha_1\ldots)$ by $f'$ and $f$, respectively.
As in the case of Bloch's constant, it suffices to search large discs in the image of a proper sub-domain of the unit disc to approximate $\lambda_f$.
One advantage of this is that we have to search only on a bounded image for such discs. The following lemma can be proven by a simple transformation
of holomorphic functions $f(z)\mapsto \frac{1}{r} f(r\cdot z)$ (see e.g. \cite{con78}).
\begin{lemma}\label{lemma_3_1}
Let $r$ with $0<r<1$ and a domain $D$ with $\mathbb{D}_r\subseteq
D\subseteq\mathbb{D}$ be given. Then
\[ r \lambda\leq \lambda_f(D)\leq\lambda_f.
\]
\end{lemma}
Now, approximating the image $f(\mathbb{D}_r)$ can be done straightforwardly.
Furthermore, with quite basic methods, the largest disc inside such an approximation could be found easily.
We are doing exactly this by the $\varepsilon$-covering grids which we introduce next.
However, an approximation will not necessarily mean that we have an approximation of $\lambda_f$:
there indeed exist examples, where small changes of $f$ can lead to a large change in $\lambda_f$.
That means that the homotopic methods used in \cite{rett08}\ cannot be used here. Instead, we will show that the approximations
we get by $\varepsilon$-covering grids for small $\varepsilon$ are still suitable to bound $\lambda$ from above.
\begin{definition}\label{def_3_1}
Let $\varepsilon>0$ and a bounded subset $A\subseteq\mathbb{C}$ be given. Then an
{\it $\varepsilon$-covering grid}\ of $A$ is a tuple $(\varepsilon,\delta, G)$ where $0<\delta\leq\varepsilon/4$ is some dyadic number and
$G$ is a non-empty, finite subset of $\delta\mathbb{Z}\times\delta\mathbb{Z}$ such that
\begin{enumerate}[(a)]
\item $A\cap \left(\delta\mathbb{Z}\times\delta\mathbb{Z}\right) \subseteq G$ and
\item $\forall z\in G.\exists {a\in A}.|z-a|\leq\varepsilon/4$.\\
\end{enumerate}
\end{definition}
Notice, that for any $\varepsilon > \varepsilon' >0$ and any $\varepsilon'$-covering grid $(\varepsilon',\delta, G)$ of a set $A$, the
tuple $(\varepsilon, \delta, G)$ is an $\varepsilon$-covering grid of $A$. In this sense any $\varepsilon'$-covering grid of a set $A$ is
also an $\varepsilon$-covering grid.
Following the notation of Section \ref{sec_prel}, $l(A)$ denotes the radius of the largest disc inside a domain $A\subseteq\mathbb{C}$. Furthermore let \[l(\varepsilon,\delta, G):=\delta+\max_{z\in G}\min_{y\in (\delta\mathbb{Z}\times\delta\mathbb{Z})\setminus G} |z-y|\] for
an $\varepsilon$-covering grid $(\varepsilon,\delta, G)$. Then the easy to compute value $l(\varepsilon,\delta, G)$ gives us an approximation
of the largest disc inside the "covered set" \[D(\varepsilon,\delta, G) := \bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z).\] More precisely we get the following result:
\begin{lemma}\label{lemma_3_2}
Let $(\varepsilon,\delta, G)$ be an $\varepsilon$-covering grid of $A\subseteq \mathbb{C}$. Then
\[ l(A)\leq l(\varepsilon,\delta, G)\leq l\left(D(\varepsilon,\delta,G)\right).\]
\end{lemma}
\begin{proof}
We start by proving the left inequality: Let $\mathbb{D}_r(z)$ be some disc with $\mathbb{D}_r(z)\subseteq A$.
We can assume that $r>\delta$ because otherwise we have $r\leq\delta\leq l(\varepsilon,\delta, G)$ anyway.
There exists some $y\in G$ such that $|y-z|\leq \delta$.
Let furthermore $x\in \delta\mathbb{Z}\times\delta\mathbb{Z}$ be some point with $x\not\in G$. Then we have $\overline{\mathbb{D}}_{|x-y|}(y)\not\subseteq A$ and
thus $\overline{\mathbb{D}}_{|x-y|+\delta}(z)\not\subseteq A$. Therefore $|x-y|+\delta\geq r$ and the statement follows.
The right inequality follows immediately from the following claim:
\begin{claim} \label{claim_n_1}
Let $x\in G$ be given and $r:= \delta+\min_{z\in (\delta\mathbb{Z}\times\delta\mathbb{Z})\setminus G} |x-z|$. Then
$ \mathbb{D}_r(x)\subseteq D(\varepsilon,\delta,G)$.
\end{claim}
Let us assume that there exist $y\not\in D(\varepsilon,\delta,G) = \bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z)$
such that $|x-y|< r$.
Then there also exists some point $z'\in\delta\mathbb{Z}\times\delta\mathbb{Z}$ such that $|x-z'|<r$ and $|z'-y|< \sqrt{2} \delta$. As $\sqrt{2} \delta <3/4\cdot \varepsilon$, $z'$ cannot belong to $G$ because otherwise $y$ would belong to $\bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z)$.
\begin{figure}[h]
\unitlength1cm
\includegraphics[scale=0.3]{Lemma4.pdf}
\caption{Proof of Claim \ref{claim_n_1}}
\end{figure}
Furthermore, as $z'\not= x$ and $x,z'\in\delta\mathbb{Z}\times\delta\mathbb{Z}$ there
exists some $z\in \delta\mathbb{Z}\times\delta\mathbb{Z}$ such that $|z-z'|\leq\sqrt{2}\cdot\delta$ and $|z-x|\leq |z'-x|-\delta < r-\delta$, which means
$z\in G$ by the definition of $r$ and furthermore \[|z-y|\leq 2\cdot\sqrt{2}\cdot\delta \leq 2\cdot\frac{\sqrt{2}}{4} \cdot \varepsilon < \frac{3}{4}\cdot\varepsilon\] in contradiction
to $y\not\in \bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z)$.
\end{proof}
Besides, an $\varepsilon$-covering grid of $f(\mathbb{D}_r)$ can be easily computed:
\begin{lemma}\label{lemma_3_3}
For given dyadic numbers $\varepsilon>0$ and $0<r<1$ as input, we can compute an $\varepsilon$-covering grid of $f(\mathbb{D}_r)$.
\end{lemma}
\proof
Following the notation of Theorem 2 we can compute a dyadic upper bound $\mu_{r}'$ on the maximum
of the values $|f'(z)|$ with $z\in\mathbb{D}_{r}$. Furthermore let $\delta_D>0$ be a dyadic
lower bound on $\varepsilon/(16\cdot\mu_{r}')$ and \[G_D:=\mathbb{D}_r\cap (\delta_D\mathbb{Z}\times\delta_D\mathbb{Z}).\] Notice that $0\in G_D$ and thus $G_D\not=\emptyset$. Furthermore we can compute for every $z\in G_D$ some approximation $d_z$ with $|d_z-f(z)|\leq\varepsilon/16$.
Then $(\varepsilon, \varepsilon/4, G)$ is an $\varepsilon$-covering grid of $f(\mathbb{D}_r)$, where
\[ G = \bigcup_{z\in G_D} \{ y\in \frac{\varepsilon}{4}\mathbb{Z}\times\frac{\varepsilon}{4}\mathbb{Z}\mid |d_z-y|\leq \frac{3}{2} \cdot \frac{\varepsilon}{8}\}. \]
To see item (a) of Definition \ref{def_3_1}\ let $y\in f(\mathbb{D}_r)\cap \left(\frac{\varepsilon}{4}\mathbb{Z}\times\frac{\varepsilon}{4}\mathbb{Z}\right)$ be given. Furthermore let $x\in\mathbb{D}_r$ such that $f(x)=y$. Then there exists some $z\in G_D$ with $|z-x|<\sqrt{2}\cdot\delta_D$ and we get
\[ \begin{array}{lll}|y-d_z|& = & |f(x)-d_z|\\
& \leq & |f(x)-f(z)|+|f(z)-d_z|\\
& \leq & \mu_{r}'\cdot |x-z| + \frac{\varepsilon}{16}\\
& \leq & \sqrt{2}\cdot\frac{\varepsilon}{16} + \frac{\varepsilon}{16}\\
& \leq & \frac{3}{2}\cdot\frac{\varepsilon}{8} \end{array}\]
To see item (b) of Definition \ref{def_3_1}\ let now $y\in G$ be given. Then there exist $z\in G_D$ such that $|d_z-y|\leq (3/2)\cdot (\varepsilon/8)$ and
we have \[ \inf_{a\in f(\mathbb{D}_r)} | y - a| \leq |y-f(z)|\leq |d_z-y| + | f(z)-d_z| \leq \frac{3}{2}\cdot\frac{\varepsilon}{8} + \frac{\varepsilon}{16} = \frac{\varepsilon}{4}.\eqno{\qEd}\]
Up to this point we have followed the naive way of computing large discs in approximations of $f(\mathbb{D}_r)$. We have already argued that this
will not necessarily give us appropriate bounds on $\lambda_f$. The main step we will take next is to show that for suitable $\hat r > r$
and suitably small $\varepsilon$ we get that for any $\varepsilon$-covering grid $(\varepsilon,\delta,G)$ of $f(\mathbb{D}_r)$ we have $D(\varepsilon,\delta,G)\subseteq f(\mathbb{D}_{\hat r})$, which in the end allows us to show that the bound given by $l(\varepsilon,\delta,G)$ is actually not too bad.
\begin{lemma}\label{lemma_3_4}
On given dyadic number $0<r<1$ we can compute dyadic numbers $\varepsilon>0$ and $\hat r$ with $r<\hat r<1$ such that for any $\varepsilon$-covering grid $(\varepsilon,\delta,G)$ of $f(\mathbb{D}_r)$ we have
\[ \bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z) \subseteq f(\mathbb{D}_{\hat r}).\]
\end{lemma}
\begin{proof}
As $f$ is not constant, we can compute $\rho>0$ and $\hat{r}$ such that $0<r<\hat{r}<1$ and $|f'(z)|\geq\rho$ for all $z\in\mathbb{C}$ with $|z|=\hat r$. Furthermore, following the notation of Corollary 1, we can compute a dyadic upper bound $\mu_{\hat r}''$ on the maximum of $|f''(z)|$ for $z\in \mathbb{D}_{\hat r}$. Let $\Delta$ be a dyadic number with \[0 < \Delta \leq \rho/(4\mu_{\hat r}'')\mbox{ and }2\cdot \Delta < \hat r - r.\]
Then we have $|f'(z)|\geq \rho/2$ for all $z\in \mathbb{D}_{\hat{r}}\setminus\mathbb{D}_{\hat r -2\Delta}$ and, with $\overline r := \hat{r} - \Delta$ we have $0<r<\overline r< \hat r <1$. By choice of $\Delta$ we do have actually more: $f$ is injective on $\mathbb{D}_{\Delta/2}(z)$ for each
$z\in \mathbb{C}$ with \[\overline r -\Delta/2\leq |z|\leq \overline r+\Delta/2.\] Thus we can apply Koebe's $1/4$-Lemma to get the following claim:
\begin{claim}\label{claim_1}
For each $z\in \mathbb{C}$ with $\overline r -\Delta/2 \leq |z| \leq \overline r + \Delta/2$ and each $y$ with $|f(z)-y|< \rho\cdot \Delta/16$ we have $y\in f(\mathbb{D}_{\hat r})$.
\end{claim}
Assume now, that there exists some $z_0$ with $|z_0|<\overline r - \Delta/2$ and some $y_0$ with $|y_0-f(z_0)|<\rho\cdot \Delta/16$ and
$y_0\notin f(\mathbb{D}_{\overline r})$. Then for the holomorphic function $h:\mathbb{D}\rightarrow\mathbb{C}$, determined by $h(z):= y_0 - f(z)$ for all $z\in\mathbb{D}$, we have $h(z)\not= 0$ for all $z\in\mathbb{D}_{\overline r}$, i.e. $|h|$ takes its minimum on the boundary $\partial \mathbb{D}_{\overline r}=\{ z\mid |z|=\overline r\}$. Therefore there exists some $z$ with $|z|=\overline r$ and \[|h(z)|\leq |y_0 - f(z_0)|< \rho\cdot \frac{\Delta}{16}.\] Thus by Claim \ref{claim_1}\ we have $y_0\in f(\mathbb{D}_{\hat r})$,
i.e. we can extend Claim \ref{claim_1}\ as follows:
\begin{claim}\label{claim_2}
For each $z\in \mathbb{C}$ with $ |z| \leq \overline r + \Delta/2$ and each $y$ with $|f(z)-y|< \rho\cdot \Delta/16$ we have $y\in f(\mathbb{D}_{\hat r})$.
\end{claim}
Let now $\varepsilon:=\rho\cdot \Delta/16$, $(\varepsilon,\delta,G)$ be some $\varepsilon$-covering grid of $f(\mathbb{D}_{r})$ and
$x\in \bigcup_{z\in G} \mathbb{D}_{\frac{3}{4}\varepsilon}(z)$. Then there exists some $z\in G$ with $|z-x|<(3/4)\cdot \varepsilon$ and,
by condition (b) in Definition 1, there exist $y\in f(\mathbb{D}_r)$ such that $|z - y|\leq (1/4)\cdot\varepsilon$.
Thus we have $|x-y|< (1/4)\cdot\varepsilon + (3/4)\cdot\varepsilon =
\rho\cdot \Delta/16$ and, by Claim \ref{claim_2}\ we have $x\in f(\mathbb{D}_{\hat r})$ which proves the lemma.
\end{proof}
Finally, we can combine the results of this section as follows:
\begin{corollary}
Given $\alpha_1\alpha_2\ldots$ in $\Pi^\infty$ and $n\in\mathbb{N}$, we can compute a dyadic number $l$ such that $(1-2^{-n})\lambda\leq l\leq \lambda_f$, where $f=\int \Psi^\infty(\alpha_1\alpha_2\ldots)$.
\end{corollary}
\proof
Given some $n\in\mathbb{N}$ we can compute by Lemma \ref{lemma_3_4}\ first some $\varepsilon>0$ and $\hat r<1$ with $\hat{r}>1-2^{-n}$
such that for any $\varepsilon$-covering grid $(\varepsilon,\delta,G)$ of
$f(\mathbb{D}_{1-2^{-n}})$ we have $D(\varepsilon,\delta,G)\subseteq f(\mathbb{D}_{\hat r})$. Then, by Lemma \ref{lemma_3_3}\ we can compute an $\varepsilon$-covering grid $(\varepsilon,\delta,G)$ of $f(\mathbb{D}_{1-2^{-n}})$
and compute $l:=l(\varepsilon,\delta,G)$. Finally, by Lemma \ref{lemma_3_2}\ and Lemma \ref{lemma_3_1}\ we get
\[ (1-2^{-n})\cdot \lambda\leq \lambda_f(\mathbb{D}_{1-2^{-n}})\leq l\leq l(D(\varepsilon,\delta,G))\leq \lambda_f(\mathbb{D}_{\hat r})\leq \lambda_f.\eqno{\qEd}
\]
\section{The Main Theorem}\label{sec_3_4}
The proof of our main theorem can now be simply done by covering the space of $\lambda$-bounding
functions by neighbourhoods given by the algorithm of the previous section.
\begin{theorem}\label{theo_3_1}
We can compute approximations to the Landau constant up to any precision.
\end{theorem}
\begin{proof}
Let $n\in\mathbb{N}$ be given. We proceed in steps $t=0,1,\ldots$ as follows. For each $t$ we
consider the set \[W_t:=\{ w1^\infty\mid w\in \Pi^t\},\] i.e. the set of infinite words starting with an
arbitrary (finite) word followed by infinitely many 1. Now we apply the algorithm given in Corollary 2 to each word
$\omega$ in $W_t$ and receive an value $l(\omega)$ such that \[(1-2^{-n})\lambda\leq l(\omega)\leq \lambda_{\int \Psi^{\infty}(\omega)}.\]
If for any $\omega\in W_t$ information on any symbol outside the leading $t$ symbols of $\omega$ are asked by the algorithm, then
we go to the next step $t+1$. Otherwise $l:=\inf_{\omega\in W_t} l(\omega)$ is an appropriate approximation of $\lambda$, i.e.
$(1-2^{-n})\lambda\leq l\leq\lambda$.
The latter is obvious, because, as none of the algorithms asks any information outside the leading words $w\in\Pi^t$, the
results on $w1^\infty$ equals the results on $w\nu$ for all $\nu\in\Pi^\infty$. Thus $l$ is indeed the infimum over all approximations
to $\lambda_{\int f}$ of all $f\in\Psi^\infty(\Pi^\infty)$.
It remains, therefore, to prove that for each $n$ there does indeed exist some $t$ such that our algorithm stops after this step.
This can be seen as follows: As the algorithm of Corollary 2 stops on all $\alpha\in\Pi^\infty$, there exist for
each $\alpha$ some $n(\alpha)$ such that no information is asked by the algorithm on any symbol beside the leading $n(\alpha)$ symbols
of $\alpha$. Thus the computation of the algorithm is identical for all words which coincides with $\alpha$ on the first $n(\alpha)$
symbols. This means that for each $\alpha$ there exists an open neighbourhood of $\alpha$ in $\Pi^\infty$ such that the algorithm
is identical on any word in this neighbourhood. As $\Pi^\infty$ is compact, there is a finite sub-covering of $\Pi^\infty$ by such
neighbourhoods. Let $\alpha^1$,...,$\alpha^m$ be the corresponding words. Then the algorithm will stop in step \[t:=\max \{ n(\alpha^i)\mid 1\leq i\leq m\}\] or before.
\end{proof}
The most interesting open problem is whether the conjectures on the Landau and Bloch constant hold.
If so, the constant can clearly be computed in polynomial time. To this end, our algorithm and the algorithm given in \cite{rett08}\
can present holomorphic functions which are very near the optimum for $\beta$- and $\lambda$-values, thus
giving possibly new insights on the kind of functions involved.
Concerning our algorithm the main intriguing problem is to improve the complexity bound, which is
roughly double exponential, to an acceptable running time. The problem on reducing the time
complexity of our algorithm is that the functions we have to consider can explode when reaching the
boundary $\partial\mathbb{D}$, where evaluation can be quite expensive.\\[3cm]
|
2,877,628,090,248 | arxiv | \section{Introduction}
On March 11, 2020 World health organization (WHO) declared the novel coronavirus disease, COVID-19, as a pandemic. More than a million people across 203 countries have already been affected by this disease, with more than 50,000 lives lost globally (as of April 2, 2020). In addition, daily lives of millions of people have been impacted because of the mandatory lock-downs observed across the world, let alone the economic cost of this adversity. The COVID-19 disease is caused by a new coronavirus SARS-CoV-2, belonging to the SARS family (SARS-CoV). SARS-CoV-2 has already been sequenced and several studies focused on understanding its interaction with the human cells (or receptors) are ongoing \cite{wu2020new, yu2020decoding, tang2020origin, sun2020understanding, bai2020presumed, gralinski2020return,wang2020}. Screening of small-molecules or biomolecules with potential therapeutic ability against COVID-19 is also being conducted using theoretical and machine learning methods \cite{smith2020repurposing, nguyen2020potentially, xu2020nelfinavir, beck2020predicting, bung2020novo}.
Initial reports on SARS-CoV-2, and previous works on the general SARS coronavirus, have suggested close interactions between the viral spike protein (S-protein) of coronavirus with specific human host receptors, such as the Angiotensin-converting enzyme 2 (ACE2) receptor. It has been hypothesized that compounds that can reduce interactions between S-protein:ACE2 receptors could limit viral recognition of the host (human) cells and/or disrupt the host-virus interactions. To this end, Smith et al. \cite{smith2020repurposing} recently conducted virtual high-throughput screening of nearly 9000 small-molecules that bind strongly to either 1) the isolated S-protein of SARS-CoV-2 at its host receptor region (thus, hindering the viral recognition of the host cells) or 2) to the S-protein:human ACE2 receptor interface (thus, reducing the host-virus interactions). They successfully identified 77 ligands (24 of which have regulatory approval from the Food and Drug Administration, FDA, or similar agencies) that satisfied one of the above two criteria. Despite the vast chemical space (millions to billions of biomolecules) that can be potentially explored, they were severely limited by the number of candidate compounds (nearly 9000) that were considered in their work owing to the high computational cost of the ensemble docking simulations employed in their methodology.
Here, we build on the exemplary work of Smith et al. \cite{smith2020repurposing} and use their data set generated from autodocking/molecular modeling for training and validating machine learning (ML) models. This allows us to significantly expand the search space and screen millions of potential therapeutic agents against COVID-19. We use the binding affinities (or their Vina scores) of the ligand to the S-protein:ACE2 interface complex and the isolated S-protein as the screening criteria to identify promising candidates using our ML models. The ML models were applied to two drug datasets, namely, CureFFI and DrugCentral containing $\sim$1500 and $\sim$4000 ligands present in known drugs, respectively. We also deploy the ML model to rapidly screen through a huge BindingDB dataset that contains over millions of small bio-molecules, to identify candidates that bind strongly to either the isolated S-protein or the S-protein:human ACE2 receptor interface complex.
The screening workflow adopted in this work
is presented in Figure \ref{fig:figure1}, while an illustration of the interface between coronavirus SARS-CoV-2 and the ACE2 receptor in presented in Figure \ref{fig:interface}. Our work is based on combining ML methods with ensemble docking simulations to screen promising ligands.
Two independent random forest (RF) regression models were trained to quickly estimate the Vina scores of a given candidate drug molecule (or ligand) for the isolated S-protein and the S-protein:human ACE2 receptor interface using the datasets provided by Smith et al. \cite{smith2020repurposing}. The Vina score is a hybrid (empirical and knowledge-based) scoring function that ranks molecular conformations and predicts the free energy of binding based on inter-molecular contributions (\textit{e.g.}, steric, hydrophobic, and hydrogen bonding, etc.)\cite{trott2010autodock}. A set of hierarchical descriptors (or features/fingerprints) that capture different geometric and chemical information at multiple length-scales (atomic and morphological) were used to represent the molecules for successful application of the ML models. The models were validated by monitoring their performance on the validation set, and against ensemble docking simulations for hundreds of promising candidate ligands. Overall, we identified several hundreds of new ligand candidates with potential therapeutic ability against COVID-19, 75 of which are FDA approved. A list of $\sim$19,000 bio-molecules (from BindingDB dataset) satisfying the same screening criteria is also provided using the developed ML models. Based on the feature importance revealed by the ML models, we also provide analysis of key chemical trends that are common across the identified promising candidates.
We note that this work not only expands our knowledge of potential small-molecule treatment against COVID-19, but also provides a powerful and efficient pathway, i.e., training ML on results of computationally expensive simulations, using ML to cast a wider net, down-selection followed by targeted computational simulations, and finally chemical guidelines, for accelerating the therapeutic cure of other diseases.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figure1_v4.pdf}
\caption{Overview of the workflow adopted to screen drug active ingredients with potential therapeutic capability for COVID-19. The white box denotes the stages performed in this work. The numbers within the bracket indicate the count of ligands in various datasets or stages of the workflow.}
\label{fig:figure1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{coronavirus.pdf}
\caption{A representation of the interface between the coronavirus n-CoV (or SARS-CoV-2) and the human ACE2 receptor is shown, with the virus in blue and the human receptor in red. The mutations at the particular virus site are shown in CPK.}
\label{fig:interface}
\end{figure}
\section{Methods}
\subsection{Machine learning}
As depicted in Figure \ref{fig:figure1}, starting from datasets of Vina scores from \cite{smith2020repurposing} for small molecules (for S-protein and S-protein:ACE2 interface), we first fingerprint the molecules using their SMILES representation; the fingerprinting procedure is discussed in detail below and in \cite{kim2018polymer}. The obtained molecular descriptors along with their respective Vina scores were then used to train two independent random forest ML models for each property. These ML models were next used to make Vina score predictions for the 1495 FDA approved drugs from CureFFI database, and 3967 other drugs from DrugCentral database. 187 candidate ligands with significantly low Vina score predictions (or high binding affinity) for both the S-protein and S-protein:ACE2 interface systems were screened for further validations using ensemble docking simulations. Our ML predictions were found to be in good agreement with the expensive docking simulations, thus, validating the developed ML model for its accurate Vina score predictions and identifying several tens of new FDA approved (or otherwise) ligands with high binding affinity to both the S-protein and S-protein:ACE2 interface. With the ML models validated, we apply them to an extensive small bio-molecule dataset, i.e., BindingDB with over millions of entries and screen many more potential candidates. Below we discuss in detail the training set, the molecular descriptors and the random forest algorithm used to develop the ML models.
\subsubsection{Training Dataset and Features}
Two training datasets were obtained from Smith et al. \cite{smith2020repurposing}, one corresponding to Vina score of a molecule with the S-protein and other for the S-protein:ACE2 interface complex. Each of the datasets contains 9127 molecules from the SWEETLEAD database \cite{novick2013sweetlead} along with their SMILES representations, which were used as input for our fingerprinting algorithm. For many molecules the Vina scores were reported to be extremely high (as much as 1,000,000 kcal/mol), while those with favorable binding energetics have Vina scores roughly in the -7 to 0 kcal/mol range. To remove such skewness in the data and train our models geared towards identifying favorable molecules with lower Vina scores, data points with only negative Vina scores were considered in this study. Further for a few cases, the SMILES representation could not be resolved correctly and were filtered out. Overall, this results in 5478 and 8120 data points (from the original number of 9127) for the S-protein:ACE2 interface and the isolated S-protein system, respectively. Henceforth, we refer to this cleaned dataset as the Smith dataset.
To build accurate and reliable ML models, it is important to include relevant features that collectively capture the trends in the Vina scores of different molecules towards S-protein and S-protein:ACE2 interface complex. The features should uniquely represent a molecule, and be readily available for new cases. Based on our past experience on fingerprinting organic materials including polymers\cite{kim2018polymer}, three hierarchical levels of features were considered capturing different geometric and chemical information about ligands at multiple length-scales. At the atomic scale, a count of a predefined set of motifs is included. The motifs are specified by the generic label ``A$_i$B$_j$C$_k$", representing an i-fold coordinated A atom, a j-fold coordinated B atom, and a k-fold coordinated C atom, connected in the specified order. For example, N3-C3-C4 represents a three-fold coordinated N, a three-fold coordinated carbon and a four-fold coordinated carbon \cite{huan2015accelerated}
At a slightly larger length-scale, quantitative structure-property relationship (QSPR) descriptors, often used in chemical and biological sciences, and implemented in the RDKit Python library, were used\cite{isarankura2009practical,nantasenamat2010advances,rdkit}. Lastly, at the highest length-scale, `morphological descriptors', such as length of the largest side-chain, shortest topological distance between rings, etc. were considered. More details on the different hierarchical descriptors can be found in our previous works \cite{kim2018polymer}.
\subsubsection{Machine learning Model}
The random forest (RF) regression algorithm, as implemented in the scikit-learn Python package\cite{scikitlearn}, was used to train the two Vina score models (S-protein and S-protein:ACE2 interface) using the Smith dataset. RF is an ensemble of decision trees, that averages predictions from a large group of `weak models' to overall result in a better prediction. It falls under the umbrella of ensemble methods, which are often the winning solutions in machine learning competitions.
The RF hyperparameters, i.e., the number of weak estimators, were estimated by maximizing the validation error during 5-fold cross-validation (CV), which leads to better generalization of the models and avoids overfitting. The performance of the ML models was evaluated using root mean square error (RMSE), mean absolute error (MAE) and correlation coefficient (R$^2$). To estimate prediction errors on unseen data, learning curves were generated by varying the sizes of the training and test sets, with results included in the supplementary information (SI). For learning curves, the test sets were obtained by excluding the training points from the Smith dataset. Additionally, for each random test-train split, statistically meaningful results were obtained by averaging over 10 runs. The final ML models used for prediction on the CureFFI, DrugCentral and BindingDB datasets were trained on the entire Smith dataset using 5-fold CV, and consisted of 400 and 700 estimators for the isolated S-protein and the S-protein:ACE2 interface datasets, respectively.
\subsection{Docking Computations}
To validate our ML models, we performed docking calculations of the top candidates identified by the models based on their low Vina scores. In-line with the works of Smith et al.\cite{smith2020repurposing}, we used the Autodock Vina software \cite{trott2010autodock} to compute binding affinities between the top candidates and the SARS-CoV-2 S-protein:ACE2 complex. The structure of the SARS-CoV-2 S-protein has a NCBI Reference Sequence YP\verb!_!009724390.1 and the ACE2 receptor has a protein data bank ID PDB:2AJF. Details regarding the construction and modeling of the SARS-CoV-2 S-protein:ACE2 complex are described here \cite{smith2020repurposing}. The SARS-CoV-2 S-protein has the necessary mutations from its predecessor SARS variety SARS-CoV, namely at L(455), F(486), Q(493), S (494), and N(501), respectively, which is illustrated in Figure \ref{fig:interface}. Smith et al. \cite{smith2020repurposing} focused on this binding pocket region and evaluated the binding affinities of different molecules from the SWEETLEAD library, as discussed earlier.
Following the procedure described by Smith et al. \cite{smith2020repurposing}, we also rank-ordered our top candidates based on their Vina scores, which is correlated to their free energy of binding to SARS-CoV-2 S-protein:ACE2. The docking receptors obtained from Smith et al. \cite{smith2020repurposing} consists of six conformations of SARS-CoV-2 S-protein:ACE2 interface as well as isolated SARS-CoV-2 S-protein receptor (\textit{i.e.}, with the ACE2 receptor removed), which were sampled using root mean squared displacement (RMSD) based clustering from 1.3 microsecond long all-atom Temperature Replica Exchange GROMACS simulations of the SARS-CoV-2 S-protein:ACE2 complex in water. The docking ligands were prepared from SMILES strings of the candidates using the Open Babel software \cite{o2011open}. Setup of the docking calculations is similar to that described by Smith et al. \cite{smith2020repurposing}, which defines a $1.2$ nm $\times$ $1.2$ nm $\times$ $1.2$ nm search space that encompasses the binding pocket located at the SARS-CoV-2 S-protein:ACE2 interface shown in Figure \ref{fig:interface}. The same search space was explored for the isolated SARS-CoV-2 S-protein receptor cases as well. For each candidate, the docking procedure finds the top $10$ optimized docking configurations and selects one with the best Vina score.
As described in more details below, the candidates identified by our ML model were obtained from the DrugCentral database as well as the FDA approved CureFFI database.
\subsection{Ligand Datasets}
While the Smith dataset\cite{smith2020repurposing} was used to train and validate the ML models, three additional drug datasets were used to make predictions and identify ligand candidates that show high binding affinity to the viral S-protein or the S-protein:ACE2 interface. These include 1) an all FDA approved CureFFI dataset \cite{cureffi}, 2) a dataset of common active ingredients from DrugCentral \cite{ursu2016drugcentral}, and a BindingDB dataset \cite{gilson2016bindingdb} of small molecules. SMILES representation of molecules were obtained from each of these datasets, and with some unprocessed candidates removed, resulted in 1495, 3967 and 985,756 entries, respectively. The CureFFI datasets consists of ligands approved by FDA and specifically contains central nervous system drugs. The drug list was parsed from an EPA (Environmental Protection Agency) suite and appropriately curated.
DrugCentral
is an open-access online drug compendium. It integrates the structure, bioactivity, regulatory, pharmacologic actions and indications for active pharmaceutical ingredients approved by FDA and other regulatory agencies. The BindingDB is a database publicly accessible based on measured binding affinities of drug-like molecules interacting with various protein targets and consists of more than a million entries of binding data and molecule datasets.
The first two datasets were exclusively used to validate the ML models against docking simulations, while the BindingDB dataset was used only for ML predictions.
\section{Results and Discussion}
\subsection{ML based Screening of FDA Approved and Other Ligands}
Figure \ref{fig:figure2}(a) and (b) present the performance results of the S-protein and S-protein:ACE2 interface RF models for the case when 75\% of Smith's dataset was used for training (with 5-fold CV), and the remaining 25\% as test set. The overall model performance on the test set is a good indicator of the expected errors on new candidate drugs with unknown Vina scores. Both the models can be seen to have good performance on the test set---a MAE of 0.21 kcal/mol was achieved for the S-protein model, while the S-protein:ACE2 model was only marginally worse with a MAE of 0.57 kcal/mol. Both these errors are well within typical chemical accuracy of 1 kcal/mol, and we believe the ML models are acceptable for screening purposes. Even for the S-protein:ACE2 model, relatively lower errors are observed for cases with low Vina scores, which are particularly more relevant to this study. See SI for more detailed validation of the ML models using learning curves, including error convergence studies on the training and test sets.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Figure2_v2.pdf}
\caption{Parity plot of the S-protein and interface ML models for the (a) training and the (b) test set, demonstrating the good prediction accuracy achieved by both the ML models. (c) ML predictions of Vina scores for the isolated S-protein and S-protein:ACE2 receptor interface for FDA approved (left panel) and other drug (right panel) candidates obtained from CureFFI and DrugCentral databases. Candidate predictions below the dashed line were selected for further validation using docking simulations.}
\label{fig:figure2}
\end{figure}
These results clearly indicate that the developed surrogate ML models could be used to quickly screen new ligand candidates with low S-protein or S-protein:ACE2 interface Vina scores without exclusively performing computationally demanding docking simulations. To this end, we use the ML models to make predictions for the FDA approved active ingredients in the CureFFI dataset and other ligands from the DrugCentral dataset, presented in Figure \ref{fig:figure2}(c). Since, the true Vina scores of these ligands is not known, here we only show their ML predictions. It has been hypothesized that a ligand could be effective against coronavirus if it either form S-protein:ACE2 interface-ligand binding complexes (low S-protein:ACE2 Vina score) to disrupt the host-virus interaction, or it binds to the receptor recognition region of the S-protein (low S-protein Vina score) to reduce viral recognition of the host. Thus, we define a simple screening criteria to select top candidates having low Vina scores on both the accounts. The dashed line in Figure \ref{fig:figure2}(c) depict the chosen screening criteria (given by equation $y = -\frac{x}{2} - 7.5$ with $x$, $y$ representing Vina scores for the S-protein:ACE2 interface-ligand complex and the S-protein-ligand system, respectively). All candidates having scores lower than this are screened for further validations. We note that 187 ligands were selected, from which 80 are FDA approved (CureFFI dataset), 107 are other drugs (DrugCentral dataset), and 29 are common to the Smith dataset. List of all 187 drugs (including their generic name and SMILES representation) and their Vina score predictions are provided in the SI.
\subsection{Validation using Docking Computations on FDA Approved and Other Ligands}
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{dockResults_v2.pdf}
\caption{Vina scores for the isolated S-protein and interface between ACE2 receptor and S-protein using docking calculations. The 187 candidates selected using ML are shown in blue, and are concentrated within region of low Vina scores. For comparison, previously considered candidates from an exhaustive past work are also represented in red.}
\label{fig:figure3}
\end{figure}
Next, ensemble docking simulations were performed for the selected 187 drug candidates, with results presented in Figure \ref{fig:figure3}. For comparison, results from the Smith dataset are also included. The purpose of these simulations was three-fold. First, a more accurate estimate of the Vina scores was obtained from these high-fidelity computations for the identified promising candidates; second, they provided new data points for further validation of the ML models, and lastly, for the 29 common candidate ligands (common to our top list and that of Smith), they help us validate our simulations against the simulations performed in the Smith paper\cite{smith2020repurposing}. From Figure \ref{fig:figure3} its evident that the ML models indeed helped us screen candidates with favorable Vina scores; almost all screened candidates can be seen to be below the ML screening criteria line (dashed line), while only 12 of the identified 187 candidates were found to have Vina scores greater than 0 and did not show any binding affinity to the S-protein:ACE2 interface complex---such cases have relatively much higher Vina score ($>$10) and are excluded from the plots for better readability. Thus, 175 of the 187 screened candidates were indeed favorable. In comparison, Smith et al. have to do expensive docking simulations for a large set of candidates, with many falling outside the screening boundary. This not only captures the efficiency of the procedure adopted here, i.e., the use of cheap surrogate models for quick screening followed by expensive high-fidelity docking simulations for validation, but also provides further validation of the prediction accuracy of the developed ML models. Parity plots directly comparing the Vina score predictions from the ML models against their respective docking simulation results are also provided in SI. Example illustrations of the S-protein:ACE2 interface-ligand complex for the top candidates are also included in the SI.
More importantly, our trained ML model predicts several ligands (including several FDA approved active ingredients) with favorable Vina scores. The top 6 among the 187 candidates are presented in Figure \ref{fig:figure4} while the complete list of 187 candidates, along with their respective Vina scores, are provided in SI. The top FDA approved ligand candidates include Pemirolast (INN), which is a mast cell stabilizer used as an anti-allergic drug therapy. It is marketed under the tradenames Alegysal and Alamast. Sulfamethoxazole (SMZ or SMX), another FDA approved ligand, is an antibiotic used for bacterial infections such as urinary tract infections, bronchitis, and prostatitis. Valaciclovir is another top candidate identified from our screening and is an anti-viral drug used to treat herpes virus infections, including shingles, cold sores, genital herpes and chickenpox. Sulfanilamide is used typically as an antibacterial agent to treat bronchitis, prostatitis and urinary tract infections. Tzaobactum is another FDA approved antibiotic and is typically combined with piperacillin to treat anti-bacterial infections such as cellulitis, diabetic foot infections, appendicitis, and postpartum endometritis infection. Nitrofurantoin is also an antibiotic and used to treat urinary tract infections.
Amongst the non-FDA approved ligands, we find that the top candidate is Protirelin which is a synthetic analogue of the endogenous peptide thyrotropin-releasing hormone (TRH). Benserazide (also called Serazide) is another top ligand and is a peripherally acting aromatic L-amino acid decarboxylase or DOPA decarboxylase inhibitor that is used for Parkinson’s disease. Other top candidates include Sulfaperin (or sulfaperine), which is a sulfonamide antibacterial agent, and Succinylsulfathiazole which is a sulfonamide used for intestinal bacteriostatic agent. Interestingly, one of the top candidates to emerge from our screening is uridine triphosphate (UTP), which is a nucleotide tri-phosphate and source of energy or an activator of substrates in metabolic reactions.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figure4_v3.pdf}
\caption{Top candidates identified from this work along with their Vina scores for the S-protein:ACE2 interface (labeled, interface) and the S-protein systems using the ensemble docking simulations.}
\label{fig:figure4}
\end{figure}
\subsection{Additional Screening Criteria: Lipinski/Pfizer Rules}
In addition to the binding energies, one can also use other thermodynamic criteria to further screen the ligand candidates. For instance, although the binding energy of a ligand is the primary criterion and related to the binding affinity measured from the AutoDock simulations based on free energy, other metrics developed by Lipinski and co-workers \cite{lipinski1997experimental, lipinski2004lead} also have certain implications on the efficacy of the drug, and could be used to for further screening of the identified candidates. A ligand is most likely to have poor absorption when its n-octanol/water partition coefficient (log P) is $>$5, its molecular weight (MW) is $>$500, the number of H-bond donors is $>$5 and the number of H-bond acceptors is $>$10.
Figure \ref{fig:lipinski} shows the log P of the top 50 candidates identified (based on lowest value of Vina scores) from the CureFFI and DrugCentral databases. Most of the top 50 ligands can be seen to have log P $<$5, which is consistent with Lipinski rules of five. Further, the molecular weights of the compounds are lower than 500 Da, as provided in SI along with other properties, such as Henry's constant, and number of hydrogen bond acceptors and donors.
Henry's constant (or log H) is another important property, measuring the solubility of the compound in water. For a drug to be up-taken by the cellular membrane,
it is desirable for the drug to be soluble in water.
The more negative the Henry's constant the more soluble is the drug in aqueous phase. However, a balance between desirable partitioning between the membrane and aqueous phase is generally sought. Thus, as presented in Table \ref{tab:lipinski}, the identified top candidates continue to satisfy all of the above additional criteria. Importantly, we note that more such constraints can be introduced in future work to further screen desirable candidate ligands. For instance, molecules with log P below 0 are known to have high affinity towards the aqueous media and are poorly absorbed by the lipid bilayer of the cellular membranes. Many of the top candidates can be seen to fall under this category, for e.g., Amiloride which has a log P of $-1.45$ is water friendly and used as water-pills/diuretic. Nitrofurantoin with log P of $-0.47$ is used to treat urinary tract infections.
\begin{table}
\centering
\caption{n-octanol/water partition coefficient (log P), Henry's constant (log H), average molecular weight, and number of hydrogen bond donors and acceptors for the top ligands identified in this work. These values were obtained from \url{www.chemspider.com}}
\begin{tabular}{cccccc}
\hline
\hline
FDA Approved Ligands & log P & log H & MW (Da) & \# of H-bond donors & \# of H-bond acceptors \\
\hline
Pemirolast & -1.12 & -12.313 & 228.21 & 1 & 7 \\
Sulfamethoxazole & 0.89 & -10.408 & 253.278 & 3 & 6\\
Valaciclovir & -3.41 & -17.578 & 324.336 & 5 & 10\\
Sulfamerazine & 0.14 & -8.145 & 264.304 & 3 & 3\\
Tazobactam & -1.72 & -14.714 & 300.291 & 1 & 9 \\
\hline
Other Ligands & log P & log H & MW (Da) & \# of H-bond donors & \# of H-bond acceptors \\
\hline
Proterelin & -2.46 & -22.799 & 362.384 & 5& 10\\
Acitazanolast & -1.95 & -16.014 & 233.184 &3 & 8\\
Sulfaperin & 0.34 & -8.145 & 264.304 &3 &6\\
Benserazide & -1.49 & -28.420 & 257.243& 8&8\\
SuccinylSulfathiozole & 1.18 &-19.117 & 355.389&3 &8\\
Uridine triphosphate & -4.09 &-38.070 &484.141 & 7&17\\
\hline
\hline
\end{tabular}
\label{tab:lipinski}
\end{table}
\subsection{Learned Chemical knowledge from ML model}
Beyond serving as a more computationally efficient alternative to drug docking simulations, learnt ML models can also be utilized to mine important chemical trends and extract simple chemical rules from the data. In this regard, the developed RF models can be used to identify important features/descriptors utilized in this work. In RF, the relative importance of a feature can be defined using the the relative rank (or depth) of that feature when used as a decision node in a tree, since features used at the top of a tree contribute towards the final prediction for a larger fraction of the input samples. Based on this philosophy, we provide a list of top 20 features that were found to be most relevant for the S-protein and the S-protein:ACE2 interface models in SI. Importantly, we found that the $^2\chi_{n}$ score of a molecule is very well (with Pearson correlation coefficient, R$^2$ = -0.67) correlated with its S-protein Vina score; higher the $^2\chi_{n}$ score, lower is the Vina score of the molecule:S-protein complex \cite{hall1991molecular, rdkit}. A variety of molecular quantum numbers (MQNs) were also found be highly relevant---those that captured the number of 5 or 6 membered rings, the topological surface area, cyclic trivalent and tetravalent nodes, nodes and edges shared by more than 2 rings. Count of aliphatic rings was also among important descriptors.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{logp.pdf}
\caption{1-octanol/water partition coefficient (log P) of the top candidates. These values are obtained from \url{www.chemspider.com}. The green dashed line indicates a value for log P of $5$. Most of the screened top candidates have log P $<$5.}
\label{fig:lipinski}
\end{figure}
\subsection{ML based screening of non-FDA approved biomolecules}
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figure5_v1.pdf}
\caption{Vina scores predictions for the isolated S-protein and S-protein:ACE2 receptor complex for all the molecules in BindingDB dataset using ML models. Over 19,000 molecules were found to satisfy the chosen screening criteria, shown using the dashed line in the plot.}
\label{fig:figure5}
\end{figure}
Having validated the ML models, we significantly expanded the search space of candidate molecules and made predictions for roughly 1 million molecules in the BindingDB dataset, with the Vina score predictions presented in Figure \ref{fig:figure5}(a). Nearly 19,000 molecules were found to satisfy the earlier chosen screening criteria (see SI for the complete list), and a few exemplary cases are illustrated in Figure \ref{fig:figure5}(b). These results clearly demonstrate the power and efficiency of using surrogate models for preliminary screening. For instance, the docking simulations for the identified 187 candidate active ingredients were completed in a period of around 2 days. In contrast, Vina score predictions from the ML model for the entire BindingDB dataset were obtained within a day using similar computational resources, including the time required for fingerprinting and making the model predictions. Evidently, our ML strategy is efficiently able to screen millions of candidate biomolecules and make useful suggestions to aid the decision making process for expert biologists and medical professionals. More robust high-fidelity computations, followed by synthesis and trial experiments should be performed to confirm the validity of these selected molecules.
Amongst the screened non-FDA biomolecules, the top candidates include Fidarestat (SNK-860) which is an aldose reductase inhibitor and is under investigation for treatment of diabetic neuropathy. Quercetin is a plant flavonol from the flavonoid group of polyphenols, which also displayed high Vina scores amongst screened candidates. Other top candidates include Myricetin which is a member of the flavonoid class of polyphenolic compounds, with antioxidant properties, S-columbianetin which is used as anti-inflammatory, Indirubin that has anti-inflammatory and anti-angiogenesis properties in vitro and Cupressuflavone with anti-inflammatory and analgesic properties.
\section{Conclusions}
In conclusion, we present an efficient virtual screening strategy to identify ligands that can potentially limit and/or disrupt the host-virus interactions. Our hypothesis is that ligands that bind strongly to the isolated SARS-CoV-2 S-protein at its host(human) receptor region and to the Sprotein-human ACE2 interface complex are likely to be most effective. Our high-throughput screening strategy is based on using a combination of machine learning (ML) and high-fidelity docking simulations to identify candidates that display such high binding affinities. We first train ML on results of computationally expensive simulations, and subsequently use the validated ML to search a much larger chemical space ($\sim$1000's of FDA approved ligands and subsequently $\sim$millions of biomolecules). We down-select based on ML predicted Vina scores, and finally mine chemical guidelines to accelerate the therapeutic cure of diseases.
Two random forest models were trained to quickly predict Vina scores of molecules with isolated S-protein and Sprotein:human ACE2 interface complex, using dataset from the past work. To train the ML models, a comprehensive set of chemical features was compiled, based on our past experience on fingerprinting organic materials, to capture geometric and chemical information about ligands at multiple length-scales. The ML models were first used to screen 187 ligands from two drug datasets (CureFFI and DrugCentral), 175 of which were indeed found to bind strongly to the isolated S-protein and to the Sprotein-human ACE2 interface complex using the expensive AutoDock simulations. This not only validates the accuracy of the ML models developed here, but also helped to identify 75 promising FDA approved ligands. Many of the identified top ligands were also found to satisfy Lipinski's rule of five. With the ML models validated, we used them to screen $\sim$19,000 candidates from a large dataset of bio-molecules ($\sim$ 1 million) from the BindingDB dataset. A rank-ordered list of promising candidates from the different datasets are provided for further theoretical or experimental validation.
\section{Acknowledgements}
This work was performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, and supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357. We thank Naina Zachariah for useful discussions and for reviewing the manuscript. SKRS thanks UIC Start-up faculty grant for supporting this work.
\bibliographystyle{unsrt}
|
2,877,628,090,249 | arxiv |
\section{Structure of claw-free graphs}
In this section, we prove that for every $k$, every claw-free graph with bounded maximum degree and with no induced subgraph isomorphic to the line graph of a $(k\times k)$-wall has bounded treewidth. Our proof relies on a structural description of claw-free graphs due to the second author and Seymour. In particular, the theorem we apply here is a straightforward corollary of the main result of \cite{Claw5}. To state this theorem, we need a couple of definitions from \cite{Claw5}.
Given a graph $H$, a set $F$ of unordered pairs of vertices of $H$ is called a \emph{valid set} for $H$ if every vertex of $H$ belongs to at most one member of $F$. For a graph $H$ and a valid set $F$ of $H$, we say that a graph $G$ is a \emph{thickening} of $(H,F)$ if for every $v \in V(H)$ there is a nonempty subset $X_v \subseteq V(G)$, all pairwise disjoint and with union $V(G)$, for which the following hold.
\begin{itemize}
\itemsep0em
\item For each $v \in V(H)$, the set $X_v$ is a clique of $G$,
\item if $u, v \in V(H)$ are adjacent in $H$ and $\{u,v\} \notin F$, then $X_u$ is complete to $X_v$ in $G$,
\item if $u, v \in V(H)$ are non-adjacent in $H$ and $\{u,v\} \notin F$, then $X_u$ is anticomplete to $X_v$ in $G$,
\item if $\{u, v\} \in F$, then $X_u$ is neither complete nor anticomplete to $X_v$ in $G$.
\end{itemize}
Let $\Sigma$ be a circle and let $I = \{I_1, \dots, I_k\}$ be a collection of subsets of $\Sigma$, such that each $I_i$ is homeomorphic to the interval $[0,1]$, no two of $I_1, \dots, I_k$ share an endpoint, and no three of them have union $\Sigma$. Let $H$ be a graph whose vertex set is a finite subset of $\Sigma$, and distinct vertices $u, v \in V(H)$ are adjacent precisely if $u, v \in I_i$ for some $i=1,\ldots, k$. The graph $H$ is called a \emph{long circular interval graph}. Let $F'$ be the set of all pairs $\{u, v\}$ such that $u, v \in V(H)$ are distinct endpoints of $I_i$ for some $i$ and there exists no $j \neq i$ for which $u, v \in I_j$. Also, let $F \subseteq F'$. Then, for every such $H$ and $F$, every thickening $G$ of $(H, F)$ is called a \emph{fuzzy long circular interval graph}.\\
Given a graph $G$, a \textit{strip-structure} of $G$ is a pair $(H,\eta)$, where $H$ is a graph with no isolated vertices and possibly with loops or parallel edges, and $\eta$ is a function mapping each $e\in E(H)$ to a subset $\eta(e)$ of $V(G)$, and each pair $(e,u)$ consisting of an edge $e\in E(H)$ and an end $u$ of $e$ to a subset $\eta(e,u)$ of $\eta(e)$, with the following specifications.
\begin{itemize}
\item[(S1)] The sets $(\eta(e): e\in E(H))$ are non-empty and partition $V (G)$.
\item[(S2)] For each $v\in V (H)$, the union of sets $\eta(e,v)$ for all $e\in E(H)$ incident with $v$ is a clique of $G$. In particular, $\eta(e,v)$ is a clique of $G$ for all $e\in E(H)$ and $v\in V(H)$ an end of $e$.
\item[(S3)] For all distinct $e_1,e_2 \in E(H)$, if $x_1\in \eta(e_1)$ and $x_2\in \eta(e_2)$ are adjacent, then there exists $v\in V(H)$ with $v$ an end of both $e_1$ and $e_2$, such that $x_i \in \eta(e_i,v)$ for $i=1,2$.
\end{itemize}
We say a strip-structure $(H,\eta)$ is \textit{non-trivial} if $|E(H)|\geq 2$. The following can be derived from Theorem 7.2 in \cite{Claw5}.
\begin{theorem}[Corollary of Theorem 7.2 from \cite{Claw5}]\label{clawstructure2}
Let $G$ be a connected claw-free graph. Then one of the following holds.
\begin{itemize}
\item We have $\alpha(G)\leq 3$.
\item $G$ is a fuzzy long circular interval graph.
\item $G$ admits a non-trivial strip structure $(H,\eta)$, such that for every $e \in E(G)$ with ends $u$ and $v$,
\begin{itemize}
\item either $\alpha( \eta(e))\leq 4$ or $ \eta(e)$ is a fuzzy long circular interval graph; and
\item there exists a path $P_e$ in $ \eta(e)$ (possibly of length zero) with an end in $\eta(e,u)$ and an end in $\eta(e,v)$ whose interior is disjoint from $\eta(e,u)\cup \eta(e,v)$.
\end{itemize}
\end{itemize}
\end{theorem}
To begin with, we show that every fuzzy long circular interval graph with bounded maximum degree has bounded treewidth. Indeed, the proof is almost immediate from the following well-known fact about \textit{chordal} graphs, i.e. graphs with no induced cycle of length at least four.
\begin{theorem}[folklore] \label{twchordal}
A graph $G$ is chordal if and only if it admits a tree decomposition $(T,\beta)$ where for every $t\in V(T)$, the set $\beta(t)$ is a clique of $G$. Consequently, if $G$ is chordal, then $\text{tw}(G)= \omega(G)-1$.
\end{theorem}
\begin{theorem} \label{circtw}
Let $G$ be a fuzzy long circular interval graph of maximum degree $\Delta$. Then we have $\text{tw}(G) \leq 4\Delta+3$.
\end{theorem}
\begin{proof}
Suppose that $G$ is a thickening of $(H,F)$, where $H$ is a long circular interval graph with $\Sigma, I_1, \dots, I_k$ as in the definition, and $F$ is valid set for $H$ as in the definition. Let $G^*$ be the graph with $V(G^*)=V(G)$ and
\[E(G^*)=E(G)\cup \left(\bigcup_{\{u,v\}\in F}\{ab:a\in X_u,b\in X_v\}\right).\]
Then $G^*$ is a long circular interval graph (the same interval representation $\Sigma, I_1, \dots, I_k$ works for $G^*$, as well). In addition, we may easily observe that
\begin{itemize}
\item $\omega(G^*)\leq 2\omega(G)\leq 2(\Delta+1)$;
\item for all $i=1,\ldots, k$, the set $C_i=\bigcup_{u\in V(H)\cap I_i}X_u$ is a clique of $G^*$; and
\item for all $i=1,\ldots, k$, the graph $G-C_i$ is a chordal.
\end{itemize}
By Theorem \ref{twchordal} and the third bullet above, $G-C_1$ admits a tree decomposition $(T,\beta)$ of width $\omega(G^*)-1$. Now, for every $t\in V(T)$, let $\beta^*(t)=\beta(t)\cup C_1$. Then it is readily seen that $(T,\beta^*)$ is a tree decomposition of $G^*$ of width $\omega(G^*)+|C_1|-1\leq 2\omega(G^*)-1$, where the last inequality follows from the the second bullet above. Hence, since $G$ is a subgraph of $G^*$, we have $\text{tw}(G)\leq \text{tw}(G^*)\leq 2\omega(G^*)-1\leq 4\Delta+3$, where the last inequality follows from the first bullet above. This proves Theorem \ref{circtw}.
\end{proof}
The following is an easy observation.
\begin{observation}\label{subdtw}
Let $H$ be a graph and $H'$ be a subdivision of $H$. Then $\text{tw}(H)=\text{tw}(H')$.
\end{observation}
We also use Theorem \ref{wallminor} with an explicit value of $f(k)$. In fact, a considerable amount of work has been devoted to understanding the order of magnitude of $f(k)$, and as of now, the following result of Chuzhoy and Tan provides the best known bound.
\begin{theorem}[\cite{chuzhoy}]\label{RSwalltw}
There exist universal constants $c_1$ and $c_2$ such that for every integer $k$, every graph with no subgraph isomorphic to a subdivision of the $(k\times k)$-wall has treewidth at most $c_1k^9\log^{c_2}k$.
\end{theorem}
Now we are in a position to prove the main result of this section.
\begin{theorem}\label{clawfreelinewalltw}
Let $\Delta,k$ be integers and $c_1$ and $c_2$ be as in Theorem \ref{RSwalltw}. Let
\[w(\Delta,k)=\max\{c_1k^9\log^{c_2}k(\Delta +1)^2,6(\Delta+1)\}-1.\]
Then for every claw-free graph $G$ of maximum degree $\Delta$ and with no induced subgraph isomorphic to the line graph of a subdivision of the $(k\times k)$-wall, we have $\text{tw}(G)\leq w(\Delta,k)$.
\end{theorem}
\begin{proof}
We may assume that $G$ is connected, and so we may apply Theorem \ref{clawstructure2}. Note that if we allow for trivial strip-structures, then the first two bullets of Theorem \ref{clawstructure2} will be absorbed into the first dash of the third bullet. In other words, we have
\sta{\label{pawelclaw}$G$ admits a (possibly trivial) strip structure $(H,\eta)$, such that for every $e \in E(G)$ with ends $u$ and $v$,
\begin{itemize}
\item either $\alpha( \eta(e))\leq 4$ or $ \eta(e)$ is a fuzzy long circular interval graph; and
\item if $(H,\eta)$ is non-trivial, then there exists a path $P_e$ in $ \eta(e)$ (possibly of length zero) with an end in $\eta(e,u)$ and an end in $\eta(e,v)$ whose interior is disjoint from $\eta(e,u)\cup \eta(e,v)$.
\end{itemize}}
We also deduce:
\sta{\label{etaclique} For every $e\in E(H)$ and every $v\in V(H)$ incident with $e$, we have $|\eta(e,v)|\leq \Delta+1$.}
By (S2), $\eta(e,v)$ is a clique of $G$. So from $G$ being of maximum degree $\Delta$, we have $|\eta(e,v)|\leq \Delta+1$. This proves \eqref{etaclique}.
\sta{\label{Hbnddeg} For every $v\in V(H)$, the number of edges $e\in E(H)$ incident with $v$ for which $\eta(e,v)\neq \emptyset$ is at most $\Delta+1$.}
For otherwise by (S2), the union of sets $\eta(e,v)$ for all $e\in E(H)$ with $v\in e$ contains a clique of $G$ size of at least $\Delta+2$, which is impossible. This proves \eqref{Hbnddeg}.
\sta{\label{Hbnddtw} $H$ admits a tree decomposition $(T_0,\beta_0)$ of width at most $c_1k^9\log^{c_2}k$.}
If $|E(H)|\leq 1$, then we are done. So we may assume that $(H,\eta)$ is non-trivial. Let the paths $\{P_e : e\in E(H)\}$ be as promised in the second bullet of \eqref{pawelclaw}, and let $H^-$ be the graph obtained from $H$ by removing its loops. Then, $G'=\bigcup_{e\in E(H^-)}V(P_e)$ is isomorphic to the line graph of a subdivision $H'$ of $H^-$. Now, since $G$ has no induced subgraph isomorphic to the line graph of a subdivision of the $(k\times k)$-wall, neither does $G'$, and so $H'$ has no subgraph isomorphic to a subdivision of the $(k\times k)$-wall. Thus, by Theorem \ref{RSwalltw}, we have $\text{tw}(H')\leq c_1k^9\log^{c_2}k$, and so by Observation \ref{subdtw}, we have $\text{tw}(H^-)\leq c_1k^9\log^{c_2}k$. This, along with the fact that every tree decomposition of $H^-$ is also a tree decomposition of $H$, proves \eqref{Hbnddtw}.
\sta{\label{etabnddtw} For every $e\in E(H)$, $ \eta(e)$ admits a tree decomposition $(T_e,\beta_e)$ of width at most $4(\Delta+1)$.}
Note that $ \eta(e)$ is of maximum degree $\Delta$. So if $\alpha( \eta(e))\leq 4$, then we have $\text{tw}( \eta(e))\leq |\eta(e)|\leq \alpha( \eta(e))(\Delta+1)\leq 4(\Delta+1)$, as desired. Otherwise, by the first bullet of \eqref{pawelclaw}, $ \eta(e)$ is a fuzzy long circular interval graph, and so by Theorem \ref{circtw}, we have $\text{tw}(G)\leq 4\Delta+3$. This proves \eqref{etabnddtw}.\vspace*{3mm}
Recall that for a tree decomposition $(T,\beta)$ of a graph $J$ and $v\in V(J)$, we denote the set $\{t \in V(T) \mid v \in \beta(t)\}$ by $\beta^{-1}(v)$. Let $(T_0,\beta_0)$ be as in \eqref{Hbnddtw}, and for every $e\in E(H)$, let $(T_e,\beta_e)$ be as promised by \eqref{etabnddtw}. We assume $T_0$ and $T_e$'s have mutually disjoint vertex sets and edge sets. Now, we construct a tree $T$ as follows. For every $e \in E(H)$ with ends $u$ and $v$, choose a vertex $s_e\in \beta^{-1}_0(u)\cap \beta^{-1}_0(v)$, which exists by definition of tree decomposition, and pick $t_e\in V(T_e)$ arbitrarily. Let $V(T)=V(T_0)\cup (\bigcup_{e\in E(H)}V(T_e))$, and $E(T)= \{s_et_e:e\in E(H)\}\cup E(T_0)\cup (\bigcup_{e\in E(H)}E(T_e))$. We also define $\beta:V(T)\rightarrow 2^{V(G)}$ as follows. Let $t\in V(T)$. If $t\in V(T_0)$, then
\[\beta(t)=\bigcup_{u \in \beta_0(t)} \bigcup_{\substack{e \in E(H):\\ \text{$u$ is an end of $e$}}} \eta(e,u).\] Otherwise, if $t\in V(T_e)$ for some $e \in E(H)$ with ends $u$ and $v$, then $\beta(t)=\beta_e(t)\cup \eta(e,u) \cup \eta(e,v)$.
\pagebreak
\sta{\label{finaltd} $(T,\beta)$ is a tree decomposition of $G$.}
By (S1), for every vertex $x\in V(G)$, there exists $e\in E(H)$ such that $x\in \eta(e)$, and so $(T_e,\beta_e)$ being a tree decomposition of $ \eta(e)$, there exists $t\in V(T_e)\subseteq V(T)$ with $x\in \beta_{e}(t)\subseteq \beta(t)$.
Also, by (S3), for every edge $x_1x_2\in E(G)$, either $x_1x_2\in E( \eta(e))$ for some $e\in E(H)$, or there exists $v\in V(H)$ and $e_1,e_2\in E(H)$ with $v$ an end of $e_1$ and $e_2$ such that $x_i\in \eta(e_i,v)$ for $i=1,2$. In the former case, since $(T_e,\beta_e)$ is a tree decomposition of $ \eta(e)$, there exists $t\in V(T_e)\subseteq V(T)$ with $x_1,x_2\in \beta_{e}(t)\subseteq \beta(t)$. In the latter case, since $(T_0,\beta_0)$ is a tree decomposition of $H$, there exists $t\in V(T_0)\subseteq V(T)$ with $v\in \beta_0(t)$. Therefore, for $i=1,2$, we have
\[x_i\in \eta(e_i,v)\subseteq \bigcup_{\substack{e \in E(H):\\\text{ } v \text{ is an end of $e$}}} \eta(e,v)\subseteq \bigcup_{u \in \beta_0(t)} \bigcup_{\substack{e \in E(H):\\\text{ } u \text{ is an end of $e$}}} \eta(e,u)=\beta(t),\]
and so $x_1,x_2\in \beta(t)$.
It remains to show that for every $x\in V(G)$, the graph $T|\beta^{-1}(x)$ is connected. By (S1), there exists a unique edge $e \in E(H)$ with ends $u$ and $v$, with $x\in \eta(e)$. First, suppose that either $x\in \eta(e,u)$ or $x\in \eta(e,v)$, say the former. Then we have $\beta^{-1}(x)=\beta_0^{-1}(u)\cup V(T_e)$. Also, since $s_e\in \beta_0^{-1}(u)$ and $t_e\in V(T_e)$, we have $E(T|\beta^{-1}(x))=\{s_et_e\}\cup E(T_0|\beta_0^{-1}(u))\cup E(T_e)$. Now, from $(T_0,\beta_0)$ being a tree decomposition of $H$, we deduce that $T_0|\beta_0^{-1}(u)$ is connected, and so $T|\beta^{-1}(x)$ is connected, as well.
Next, suppose that $x\in \eta(e)\setminus (\eta(e,u)\cup \eta(e,v))$. Then we have $\beta^{-1}(x)=\beta_e^{-1}(x)$. So from $(T_e,\beta_e)$ being a tree decomposition of $ \eta(e)$, we deduce that $T|\beta^{-1}(x)=T_e|\beta_e^{-1}(x)$ is connected. This proves \eqref{finaltd}.\vspace*{3mm}
Now, let $t\in V(T)$. If $t\in V(T_0)$, then by \eqref{etaclique}, \eqref{Hbnddeg} and \eqref{Hbnddtw}, we have
\[|\beta(t)|=\sum_{u \in \beta_0(t)} \sum_{\substack{e \in E(H):\\\text{ $u$ is an end of $e$}}} |\eta(e,u)|\leq |\beta_0(t)|(\Delta+1)^2\leq c_1k^9\log^{c_2}k(\Delta +1)^2\leq w(\Delta,k)+1.\]
Also, if $t\in V(T_e)$ for some $e\in E(H)$, then by \eqref{etaclique} and \eqref{etabnddtw}, we have
\[|\beta(t)|\leq |\beta_e(t)|+|\eta(e,u)|+|\eta(e,v)|\leq 6(\Delta+1)\leq w(\Delta,k)+1.\]
Hence, by \eqref{finaltd}, $(T,\beta)$ is a tree decomposition of $G$ of width at most $w(\Delta,k)$. This proves Theorem \ref{clawfreelinewalltw}.
\end{proof}
\section{Balanced separators and treewidth}
\section{Introduction}
All graphs in this paper are finite and simple. Let $G = (V(G), E(G))$ be a graph. A {\em tree decomposition $(T, \beta)$} of $G$ consists of a tree $T$ and a map $\beta: V(T) \to 2^{V(G)}$, with the following properties:
\begin{enumerate}[(i)]
\item For every $v \in V(G)$, there exists $t \in V(T)$ such that $v \in \beta(t)$.
\item For every $v_1v_2 \in E(G)$, there exists $t \in V(T)$ such that $v_1, v_2 \in \beta(t)$.
\item For every $v \in V(G)$, the subgraph of $T$ induced on the set $\beta^{-1}(v)=\{t \in V(T) \mid v \in \beta(t)\}$ is connected.
\end{enumerate}
The {\em width} of the tree decomposition $(T, \beta)$ is $\max_{v \in V(T)} |\beta(v)| -1$. The {\em treewidth} of a graph $G$, denoted by $\text{tw}(G)$, is the minimum width of a tree decomposition of $G$. Treewidth, originally introduced by Robertson and Seymour in their study of graph minors, is widely considered to be an important graph parameter, both from a structural
\cite{RS-GMXVI} and algorithmic \cite{Bodlaender1988DynamicTreewidth} point of view. Roughly, the treewidth of a graph measures how ``close to a tree'' it is: trees have treewidth one, and in general, the larger the treewidth of a graph, the less ``tree-like'', and hence the more complicated it is. So it is natural to ask how one would certify whether a graph is of large treewidth, and in particular, what can we say about the unavoidable substructures emerging in graphs of large treewidth. As an example, for each $k$, the {\em $(k \times k)$-wall}, denoted by $W_{k \times k}$, is a planar graph with maximum degree three and with treewidth $k$ (the formal definition is provided at the end of Subsection~\ref{sec:defns}; see Figure \ref{fig:5x5wall}). Every subdivision of $W_{k \times k}$ is also a
graph of treewidth $k$. The Grid Theorem of Robertson and Seymour, slightly reformulated below, gives a complete characterization of the unavoidable subgraphs of graphs with large treewidth.
\begin{theorem}[\cite{RS-GMV}]\label{wallminor}
There is a function $f: \mathbb{N} \rightarrow \mathbb{N}$
such that every graph of treewidth at least $f(k)$
contains a subdivision of $W_{k \times k}$ as a subgraph.
\end{theorem}
\begin{figure}
\centering
\begin{tikzpicture}[scale=2,auto=left]
\tikzstyle{every node}=[inner sep=1.5pt, fill=black,circle,draw]
\centering
\node (s10) at (0,1.2) {};
\node(s12) at (0.6,1.2){};
\node(s14) at (1.2,1.2){};
\node(s16) at (1.8,1.2){};
\node(s18) at (2.4,1.2){};
\node (s20) at (0,0.9) {};
\node (s21) at (0.3,0.9) {};
\node(s22) at (0.6,0.9){};
\node (s23) at (0.9,0.9) {};
\node(s24) at (1.2,0.9){};
\node (s25) at (1.5,0.9) {};
\node(s26) at (1.8,0.9){};
\node (s27) at (2.1,0.9) {};
\node(s28) at (2.4,0.9){};
\node (s29) at (2.7,0.9) {};
\node (s30) at (0,0.6) {};
\node (s31) at (0.3,0.6) {};
\node(s32) at (0.6,0.6){};
\node (s33) at (0.9,0.6) {};
\node(s34) at (1.2,0.6){};
\node (s35) at (1.5,0.6) {};
\node(s36) at (1.8,0.6){};
\node (s37) at (2.1,0.6) {};
\node(s38) at (2.4,0.6){};
\node (s39) at (2.7,0.6) {};
\node (s40) at (0,0.3) {};
\node (s41) at (0.3,0.3) {};
\node(s42) at (0.6,0.3){};
\node (s43) at (0.9,0.3) {};
\node(s44) at (1.2,0.3){};
\node (s45) at (1.5,0.3) {};
\node(s46) at (1.8,0.3){};
\node (s47) at (2.1,0.3) {};
\node(s48) at (2.4,0.3) {};
\node (s49) at (2.7,0.3) {};
\node (s51) at (0.3,0.0) {};
\node (s53) at (0.9,0.0) {};
\node (s55) at (1.5,0.0) {};
\node (s57) at (2.1,0.0) {};
\node (s59) at (2.7,0.0) {};
\foreach \from/\to in {s10/s12, s12/s14,s14/s16,s16/s18}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s20/s21, s21/s22, s22/s23, s23/s24, s24/s25, s25/s26,s26/s27,s27/s28,s28/s29}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s30/s31, s31/s32, s32/s33, s33/s34, s34/s35, s35/s36,s36/s37,s37/s38,s38/s39}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s40/s41, s41/s42, s42/s43, s43/s44, s44/s45, s45/s46,s46/s47,s47/s48,s48/s49}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s51/s53, s53/s55,s55/s57,s57/s59}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s10/s20, s30/s40}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s21/s31,s41/s51}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s12/s22, s32/s42}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s23/s33,s43/s53}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s14/s24, s34/s44}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s25/s35,s45/s55}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s16/s26,s36/s46}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s27/s37,s47/s57}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s18/s28,s38/s48}
\draw [-] (\from) -- (\to);
\foreach \from/\to in {s29/s39,s49/s59}
\draw [-] (\from) -- (\to);
\end{tikzpicture}
\caption{$W_{5 \times 5}$}
\label{fig:5x5wall}
\end{figure}
While tree decompositions and classes of graphs with bounded treewidth are central concepts in the study of graphs with forbidden minors \cite{RS-GMXVI}, the problem of connecting tree decompositions with forbidden induced subgraphs had largely remained uninvestigated until very recently. In accordance, this work is a step toward understanding the unavoidable induced subgraphs of graphs with large treewidth. Formally, let us say a family $\mathcal{F}$ of graphs is
{\em useful} if there exists $c$ such that every graph $G$ with $\text{tw}(G) > c$ contains a memeber of $\mathcal{F}$ as an induced subgraph. Then our work is motivated by the
goal of characterizing useful families. For instance, Lozin and Razgon \cite{LR} have recently proved the following theorem, which gives a complete description of all finite useful families. Given a graph $F$, the {\em line graph} $L(F)$ of $F$ is the graph with vertex set $E(F)$, such that two vertices of $L(F)$ are adjacent if the corresponding edges of $G$ share an end.
\begin{theorem}[\cite{LR}]\label{Lozinfinite}
Let $\mathcal{F}$ be finite family of graphs. Then $\mathcal{F}$ is useful if and only if it contains a complete graph, a complete bipartite graph, a forest in which each component has at most three leaves, and the line graph of such a forest.
\end{theorem}
In fact, it is easy to see that the complete graph $K_t$ has treewidth $t-1$ and the complete bipartite graph $K_{t,t}$ has treewidth $t$. Also, as mentioned above, every subdivision of $W_{k \times k}$ is also of treewidth $k$, and crucially, no two non-isomorphic subdivisions of $W_{k \times k}$ are induced subgraphs of each other. The line graph of a subdivision of $W_{k \times k}$ is
another
example of a graph with large treewidth. Note that $L(W_{k \times k})$ does not contain
$W_{k \times k}$ as an induced subgraph.
In summary, if a family of graphs is useful, then it contains a complete graph, a complete bipartite graph, and for some $k$, an induced subgraph of every subdivision of $W_{k \times k}$, and an induced subgraph of the line graph of every subdivision of $W_{k \times k}$. Therefore, it would be natural to ask whether the converse of the latter statement is also true:
\begin{question}\label{usefulQ}
Let $\mathcal{F}$ be a family of graphs containing a complete graph, a complete bipartite graph, and for some $k$, an induced subgraph of every subdivision of $W_{k \times k}$, and an induced subgraph of the line graph of every subdivision of $W_{k \times k}$. Then is $\mathcal{F}$ useful?
\end{question}
It turns out that the answer to Question~\ref{usefulQ} is negative. To elaborate on this, we need a couple of definitions. By a {\em hole} in a graph we mean an induced cycle of length at least four, and an {\em even hole} is a hole on an even number of vertices. For graphs $G$ and $F$, we say that $G$ is {\em $F$-free} if $G$ does not contain an induced subgraph isomorphic to $F$. If $\mathcal{F}$ is a family of graphs, a graph $G$ is {\em $\mathcal{F}$-free} if $G$ is $F$-free for every $F \in \mathcal{F}$.
It is not difficult to show that for large enough $k$, subdivisions of $W_{k\times k}$, line graphs of subdivisions of $W_{k\times k}$, and the complete bipartite graph $K_{k,k}$ all
contain even holes. Therefore, the following theorem provides a negative answer to Question~\ref{usefulQ}.
\begin{theorem}[\cite{ST}]
\label{thm:layered_wheel}
For every integer $\ell \geq 1$, there exists an (even hole, $K_4$)-free graph $G_{\ell}$ such that $\text{tw}(G_\ell) \geq \ell$.
\end{theorem}
Observing that graphs $G_{\ell}$ in Theorem \ref{thm:layered_wheel} have vertices of arbitrarily large degree, the following conjecture was made (and proved for the case $\Delta\leq 3$) in \cite{Aboulker2020OnGraphs}:
\begin{conjecture}[\cite{Aboulker2020OnGraphs}] \label{evenholestw}
For every $\Delta > 0$ there exists $c_{\Delta}$ such that even-hole-free graphs
with maximum degree $\Delta$ have treewidth at most $c_{\Delta}$.
\end{conjecture}
Conjecture~\ref{evenholestw} was proved in \cite{ACV} by three of
the authors of the present paper. More generally, it is conjectured in \cite{{Aboulker2020OnGraphs}} that there is an affirmative asnwer to Question~\ref{usefulQ} in the bounded maximum degree case (note that bounded maximum degree automatically implies that a large complete graph and a large complete bipartite graph are excluded).
\begin{conjecture}[\cite{Aboulker2020OnGraphs}]\label{conj:wall}
For every $\Delta>0$ there is a function
$f_{\Delta}:\mathbb{N} \rightarrow \mathbb{N}$ such that
every graph with maximum degree at most $\Delta$ and treewidth at least
$f_{\Delta}(k)$
contains a subdivision of $W_{k \times k}$ or the line graph of a subdivision of $W_{k \times k}$ as an induced
subgraph.
\end{conjecture}
This remains open, though in \cite{Aboulker2020OnGraphs} it is proved for proper minor-closed classes of graphs (in which case the bound on the maximum degree is not needed anymore).
\begin{theorem}[\cite{Aboulker2020OnGraphs}]
For every
graph H there is a function $f_H : \mathbb{N} \to \mathbb{N}$ such that every graph of treewidth at least $f_H(k)$ and with no $H$-minor
contains a subdivision of $W_{k \times k}$ or the line graph of a subdivision of $W_{k \times k}$ as an induced subgraph.
\end{theorem}
In this paper we prove several
theorems supporting
Conjecture~\ref{conj:wall}. In order to state our main results, we need a few more definitions.
A {\em path} is a graph $P$ with vertex set $\{p_1, \hdots, p_k\}$ and edge set $\{p_1p_2, p_2p_3, \hdots, p_{k-1}p_k\}$. We write $P = p_1 \hbox{-} \hdots \hbox{-} p_k$, and we say $p_1$ and $p_k$ are the {\em ends} of $P$. The {\em length} of the path $P$ is the number of edges in $P$. We say that $P$ is a path {\em from $p_1$ to $p_k$}, where $p_1$ and $p_k$ are the vertices of degree one in $P$. The {\em interior of $P$} is denoted $P^*$ and is defined as $P \setminus \{p_1, p_k\}$.
Let $G$ be a graph and let $X, Y \subseteq V(G)$ be disjoint. Then, $X$ is {\em complete to $Y$} if for every $x \in X$ and $y \in Y$, we have $xy \in E(G)$, and $X$ is {\em anticomplete to $Y$} if there are no edges from $X$ to $Y$ in $G$.
The {\em claw} is the graph with vertex set $\{a, b, c, d\}$ and edge set $\{ab, ac, ad\}$. For nonnegative integers $t_1, t_2, t_3$, an $S_{t_1, t_2, t_3}$, also called a {\em long claw} or a {\em subdivided claw}, consists of a vertex $v$ and three paths $P_1, P_2, P_3$, where $P_i$ is of length $t_i$, with one end $v$, such that $V(P_1) \setminus \{v\}$, $V(P_2) \setminus \{v\}$, and $V(P_3) \setminus \{v\}$ are pairwise disjoint and anticomplete to each other. Note that for every $t$, every subdivision of $W_{k\times k}$ for large enough $k$ contains $S_{t,t,t}$ as an induced subgraph. Our first result is the following.
\begin{restatable}{theorem}{clawfree}
\label{thm:claw-free_nonspecific}
Let $\Delta, t, k$ be positive integers. There exists $c_{k, t, \Delta}$ such that for every $S_{t,t,t}$-free graph $G$ with maximum degree $\Delta$ and no induced subgraph isomorphic to the line graph of a subdivision of $W_{k \times k}$, we have $\text{tw}(G) \leq c_{k, t, \Delta}$.
\end{restatable}
A {\em theta} is a graph consisting of three internally vertex-disjoint paths $P_1 = a\hbox{-} \hdots \hbox{-} b$, $P_2 = a \hbox{-} \hdots \hbox{-} b$, and $P_3 = a \hbox{-} \hdots \hbox{-} b$ of length at least 2, such that no edges exist between the paths except the three edges incident with $a$ and the three edges incident with $b$. A {\em $t$-theta} is a theta such that each of $P_1, P_2, P_3$ has length at least $t$. A {\em pyramid} is a graph consisting of three paths $P_1 = a \hbox{-} \hdots \hbox{-} b_1$, $P_2 = a \hbox{-} \hdots \hbox{-} b_2$, and $P_3 = a \hbox{-} \hdots \hbox{-} b_3$ of length at least 1, two of which have length at least 2, pairwise vertex-disjoint except at $a$, and such that $b_1b_2b_3$ is a triangle and no edges exist between the paths except those of the triangle and the three edges incident with $a$. A {\em $t$-pyramid} is a pyramid such that each of $P_1, P_2, P_3$ has length at least $t$.
Note that the complete bipartite graph $K_{2,3}$ is in fact a theta. Also, for large enough $k$, every subdivision of $W_{k\times k}$ contains a theta as an induced subgraph, and the line graph of every subdivision of $W_{k\times k}$ contains a triangle. Therefore, the following theorem gives another reason why the answer to Question~\ref{usefulQ} is negative.
\begin{theorem}[\cite{ST}]
\label{thm:layered_wheel_theta}
For every integer $\ell \geq 1$, there exists a
(theta, triangle)-free graph $G_{\ell}$ such that $\text{tw}(G_\ell) \geq \ell$.
\end{theorem}
In analogy to the situation with Theorem \ref{thm:layered_wheel}, the graphs $G_{\ell}$ in
Theorem~\ref{thm:layered_wheel_theta} contain vertices of arbitrary large degree. So it is asked in \cite{PSTT} whether (theta, triangle)-free graphs of bounded maximum degree have bounded treewidth (while it is proved in \cite{PSTT} that (theta, triangle,$S_{t,t,t})$-free graphs, without a bound on the maximum degree, have bounded treewidth). We give an affirmative answer to this question. Indeed, our second result, the following, establishes a far-reaching generalization of this question. It also generalizes Theorem \ref{thm:claw-free_nonspecific}, and strongly addresses Conjecture \ref{conj:wall}.
\begin{restatable}{theorem}{pyramid-theta}
\label{thm:pyramid_theta-nonspecific}
Let $\Delta, t, k$ be positive integers with $t \geq 2$. Then, there exists $c_{t, k, \Delta}$ such that for every ($t$-theta, $t$-pyramid)-free graph $G$ with maximum degree $\Delta$ and no induced subgraph isomorphic to the line graph of a subdivision of $W_{k \times k}$, we have $\text{tw}(G)\leq c_{t, k, \Delta}$.
\end{restatable}
A tree $T$ is a {\em subdivided caterpillar} if there is a path $P$ in $T$ such that $P$ contains every vertex of $T$ of degree at least three in $T$. The {\em spine} of $T$ is the shortest path containing all vertices of degree at least three in $T$. A {\em leg} of a subdivided caterpillar $T$ is a path in $T$ from a vertex of degree one in $T$ to a vertex of degree at least three in $T$. A graph $G$ is {\em subcubic} if every vertex of $G$ has degree at most three.
Note that for every subcubic subdivided caterpillar $T$ and for large enough $k$, every subdivision of $W_{k\times k}$ contains a subdivison of $T$ as an induced subgraph, and the line graph of every subdivision of $W_{k\times k}$ contains the line graph of a subdivision of $T$ as an induced subgraph. Our third result is the following.
\begin{restatable}{theorem}{caterpillar}
\label{thm:caterpillar-non_specific}
Let $\Delta$ be a positive integer and let $T$ be a subcubic subdivided caterpillar. There exists $c_{T, \Delta}$ such that for every graph $G$ with maximum degree $\Delta$ and no induced subgraph isomorphic to a subdivision of $T$ or the line graph of a subdivision of $T$, we have $\text{tw}(G) \leq c_{T, \Delta}$.
\end{restatable}
Let us now roughly discuss the proofs.
Usually, to prove that a certain graph family has bounded treewidth, one attempts to construct a collection of ``non-crossing decompositions,'' which roughly means that the decompositions
``cooperate'' with each other, and the pieces that are obtained when the graph
is simultaneously decomposed by all the decompositions in the collection
``line up'' to form a tree structure. Such collections of decompositions are
called ``laminar.''
In all the cases above, there is a natural family of decompositions to turn to,
sharing a certain structural property: all the decompositions arise from
removing from the graph the neighborhood of a small connected subgraph.
Unfortunately,
these natural decompositions are very far from being non-crossing, and therefore
they cannot be used in traditional ways to get tree-decompositions. What turns out to be true, however, is that, due to the bound on the maximum degree of the graph, these collections of decompositions can be partitioned into a bounded number
of laminar collections (where the bound on the number of collections depends on the maximum degree and on the precise nature of the decomposition). We will explain how to make use of this fact in Section \ref{sec:central_bags}.
\begin{comment}
such that $S \subseteq N^d[v_1]$.
Let $G$ be a graph and let $w: V(G) \to [0, 1]$ be a weight function of $G$ such that $w(G) = 1$.
A set $Y \subseteq V(G)$ is a \emph{$(w, c, d)$-balanced separator} of $G$ if $Y$ is $d$-bounded and if $w(Z) \leq c$ for every component $Z$ of $G \setminus Y$.
One can then proceed as follows. Let $G$ be a graph satisfying the
assumptions of one our theorems, with maximum degree $\Delta$, and let $w:V(G) \to [0, 1]$ be such that
$w(G) = 1$. By a theorem of \cite{HarveyWood}, to
it is enough to show that for
certain $c$ and $d$ (that do not depend on $G$ and work for all graphs in the class), $G$ has a $(w, c, d)$-balanced separator;
we may assume that no such separator exists.
First,
$G$ is decomposed, simultaneously, by all the decompositions in $X_1$. Since $X_1$ is a
laminar collection, results of \cite{RS-GMX} (in fact, a very simple of version of them) imply the existence of a
tree-decomposition of $G$, and
one of the bags of this decomposition is identified as the ``central bag'' for $X_1$; denote it by $\beta_1$. Then, $\beta_1$ is an induced subgraph of $G$, and it is shown that
$\beta_1$ has no $(w_1,c,d_1)$-balanced separator for certain $w_1$ and $d_1$ that depend on $w$ and $d$.
Now the proof focuses on $\beta_1$, decomposing it using $X_2$, and so on.
At step $i$, having decomposed by $X_1,\ldots, X_i$, the focus is on a
central bag $\beta_i$ that does not have a $(w_i,c,d_i)$-separator
for suitably chosen $w_i, d_i$.
A key point here is that the decompositions in $X_1, \ldots, X_p$
are forced by the presence of certain induced subgraphs that we call
``forcers.'' Roughly speakoing, an induced subgraph $H$ of $G$ is a forcer (for a
particular kind of decomposition) if
$H$ admits a decomposition of that kind, and the decomposition of $H$
extends to $G$. Thus a forcer is a local ``predictor'' of a global behavior of
$G$.
It is ensured that at step $i$,
after decomposing by $X_1, \ldots, X_i$, none of the forcers that were
``responsible'' for the decompositions in $X_1, \ldots, X_i$ are present in
the central bag $\beta_i$. It then follows that, when $\beta_p$ is reached,
it is a ``much simpler'' graph, where one can find
a $(w_p,c,d_p)$-separator directly, thus obtaining a contradiction, and
proving the theorem.
\end{comment}
\subsection*{Structure of the paper}
We begin in Section \ref{sec:defns} with a review of relevant definitions and notation. In Section \ref{sec:balanced-separators}, we define an important graph parameter tied to treewidth called separation number. In Section \ref{sec:central_bags} we prove Theorem~\ref{thm:centralbag}, which summarizes our main proof method. In Section \ref{sec:tw_of_clawfree}, we bound the treewidth of claw-free graphs with no line graph of a subdivision of a wall, and in Section \ref{sec:claw_free_result}, we apply the results of Section \ref{sec:tw_of_clawfree} to prove Theorem \ref{thm:claw-free_nonspecific}. In Section \ref{sec:theta_pyramid}, we prove Theorem \ref{thm:pyramid_theta-nonspecific}, and in Section \ref{sec:caterpillar}, we prove Theorem \ref{thm:caterpillar-non_specific}.
\subsection{Definitions and Notation}
\label{sec:defns}
Let $G$ be a graph. In this paper, we use vertex sets and their induced subgraphs interchangeably. Let $H$ be a graph. We say that $X \subseteq V(G)$ {\em is an $H$ in $G$} if $X$ is isomorphic to $H$. We say that $G$ {\em contains} $H$ if there exists $X \subseteq V(G)$ such that $X$ is an $H$ in $G$.
The {\em open neighborhood} of a vertex $v \in V(G)$, denoted $N(v)$, is the set of all vertices adjacent to $v$. The {\em degree} of $v \in V(G)$ is the size of its open neighborhood. We say a graph $G$ has {\em maximum degree $\Delta$} if the degree of every vertex $v \in V(G)$ is at most $\Delta$. The {\em closed neighborhood} of a vertex $v \in V(G)$ is denoted $N[v]$ and is defined as $N[v] = N(v) \cup \{v\}$. Let $X \subseteq V(G)$. The {\em open neighborhood of $X$}, denoted $N(X)$, is the set of all vertices of $G \setminus X$ with a neighbor in $X$. The {\em closed neighborhood of $X$} is denoted $N[X]$ and is defined as $N[X] = N(X) \cup X$.
A set $X \subseteq V(G)$ is {\em connected} if for every $x, y \in X$, there is a path $P$ in $X$ from $x$ to $y$. A set $C \subseteq V(G)$ is a {\em cutset} of a connected graph $G$ if $G \setminus C$ is not connected. A set $D$ is a {\em connected component of $G$} if $D$ is inclusion-wise maximal such that $D \subseteq V(G)$ and $D$ is connected.
Let $u, v \in V(G)$ and let $X \subseteq V(G)$. The {\em distance between $u$ and $v$} is the length of a shortest path from $u$ to $v$ in $G$. The {\em distance between $u$ and $X$} is the length of a shortest path from $u$ to a vertex $x \in X$ in $G$. We denote by $N^d(v)$ the set of vertices at distance exactly $d$ from $v$ in $G$, and by $N^d[v]$ the set of vertices at distance at most $d$ from $v$ in $G$. Similarly, we denote by $N^d[X]$ the set of vertices of distance at most $d$ from $X$ in $G$. The {\em diameter} of a connected set $X \subseteq V(G)$ is the maximum distance in $G$ between two vertices of $X$.
A {\em clique} is a set $K \subseteq V(G)$ such that every pair of vertices in $K$ is adjacent. An {\em independent set} is a set $I \subseteq V(G)$ such that every pair of vertices in $I$ is non-adjacent. The {\em clique number of $G$}, denoted $\omega(G)$, is the size of a largest clique in $G$. The {\em independence number of $G$}, denoted $\alpha(G)$, is the size of a largest independent set in $G$.
A {\em weight function on $G$} is a function $w:V(G) \to \mathbb{R}$ that assigns a non-negative real number to every vertex of $G$. A weight function is {\em normal} if $w(V(G)) = 1$. Unless otherwise specified, we assume all weight functions are normal. We denote by $w^{\max}$ the maximum weight of a vertex; i.e. $w^{\max} = \max_{v \in V(G)} w(v)$.
Finally, let us include the precise definition of a wall. The {\em $(n \times m)$-wall}, denoted $W_{n \times m}$, is the graph $G$ with vertex set
\begin{align*}
V(G) =&
\{(1, 2j - 1) \mid 1 \leq j \leq m \} \\
&\cup \{(i, j) \mid 1 < i < n, 1 \leq j \leq 2m\} \\
&\cup \{(n, 2j - 1) \mid 1 \leq j \leq m, \text{ if $n$ is even}\} \\ & \cup \{(n, 2j) \mid 1 \leq j \leq m, \text{ if $n$ is odd }\}
\end{align*}
and edge set
\begin{align*}
E(G) =& \{(1, 2j - 1),(1, 2j + 1) \mid 1 \leq j \leq m - 1\}\\
& \cup \{(i, j),(i, j + 1) \mid 2 \leq i < n, 1 \leq j < 2m \} \\
&\cup \{(n, 2j),(n, 2j + 2)) \mid 1\leq j < m \text{ if $n$ is odd} \} \\
&\cup \{(n, 2j - 1),(n, 2j + 1) \mid 1 \leq j < m \text{ if $n$ is even} \} \\
&\cup \{(i, j),(i + 1, j) \mid 1 \leq i < n, 1 \leq j \leq 2m, i, j \text{ odd} \} \\
&\cup \{(i, j),(i + 1, j) \mid 1 \leq i < n, 1 \leq j \leq 2m, i, j \text{ even} \}.
\end{align*}
Again, see Figure \ref{fig:5x5wall} for an example.
\subsection{Balanced separators and treewidth}
\label{sec:balanced-separators}
Treewidth is tied to a parameter called the separation number. Let $G$ be a graph, let $S \subseteq V(G)$, let $k$ be a positive integer, and let $c \in [\frac{1}{2}, 1)$. A set $X \subseteq V(G)$ is a {\em $(k, S, c)^*$-separator} if $|X| \leq k$ and for every component $D$ of $G \setminus X$, it holds that $|D \cap S| \leq c|S|$. The {\em separation number} $\text{sep}_c^*(G)$ is the minimum $k$ such that $G$ has a $(k, S, c)^*$-separator for every $S \subseteq V(G)$. The following lemma states that the separation number gives an upper bound for the treewidth of a graph.
\begin{lemma}[\cite{HarveyWood}]
\label{lemma:harvey-wood}
For every $c \in [\frac{1}{2}, 1)$ and every graph $G$, we have $\text{tw}(G) + 1 \leq \frac{1}{1-c} \text{sep}_c^*(G)$.
\end{lemma}
Now, we redefine $(k, S, c)^*$-separators using weight functions. Given a normal weight function $w$ on a graph $G$ and a constant $c \in [\frac{1}{2}, 1)$, a set $X \subseteq V(G)$ is a {\em $(w, c)$-balanced separator of $G$} if $w(D) \leq c$ for every component $D$ of $G \setminus X$.
We call a weight function $w$ on $G$ a {\em uniform weight function} if there exists $Y \subseteq V(G)$ such that $w(v) = \frac{1}{|Y|}$ if $v \in Y$, and $w(v) = 0$ if $v \not \in Y$. Lemma \ref{lemma:harvey-wood} implies the following:
\begin{lemma}
\label{lemma:harvey-wood-weights}
Let $c \in [\frac{1}{2}, 1)$ and let $G$ be a graph. If $G$ has a $(w, c)$-balanced separator of size at most $k$ for every uniform weight function $w$, then $\text{tw}(G) \leq \frac{1}{1-c}k$.
\end{lemma}
\begin{proof}
We prove that $\text{sep}_c^*(G) \leq k$. Let $S \subseteq V(G)$ and let $w_S$ be the weight function on $G$ such that $w_S(v) = \frac{1}{|S|}$ if $v \in S$, and $w_S(v) = 0$ otherwise. Since $w_S$ is a uniform weight function, it follows that $G$ has a $(w_S, c)$-balanced separator $X$ such that $|X| \leq k$. Let $D$ be a component of $G \setminus X$, so $w(D) \leq c$. Consequently, $|D \cap S| \leq c|S|$, and so $X$ is a $(k, S, c)^*$-separator. Therefore, $\text{sep}_c^*(G) \leq k$, and the result follows from Lemma \ref{lemma:harvey-wood}.
\end{proof}
Lemma \ref{lemma:harvey-wood-weights} implies that if for some fixed $c \in [\frac{1}{2}, 1)$, $G$ has a balanced separator of size $k$ for every weight function $w$, then the treewidth of $G$ is bounded by a function of $k$. The next lemma states the converse.
\begin{lemma}[\cite{PA}]
\label{lemma:bounded-tw-balanced-separator}
If $\text{tw}(G) \leq k$, then $G$ has a $(w, c)$-balanced separator of size at most $k+1$ for every normal weight function $w$ and for every $c \in [\frac{1}{2}, 1)$.
\end{lemma}
Together, Lemmas \ref{lemma:harvey-wood-weights} and \ref{lemma:bounded-tw-balanced-separator} show that treewidth is tied to the size of balanced separators. In this paper, we rely on balanced separators to prove that graphs have bounded treewidth. In what follows, we will often assume that $G$ has no $(w, c)$-balanced separator of size $d$ for some normal weight function $w$, $c \in [\frac{1}{2}, 1)$, and positive integer $d$, since otherwise, in light of Lemma \ref{lemma:harvey-wood-weights}, we are done.
\section{Central bags and forcers}
\label{sec:central_bags}
\input{central_bags}
\section{Treewidth of claw-free graphs}
\label{sec:tw_of_clawfree}
\input{clawfree}
\section{Long claws and line graphs of walls}
\label{sec:claw_free_result}
Here, we apply the results of Sections \ref{sec:tw_of_clawfree} to prove Theorem \ref{thm:claw-free_nonspecific}, that excluding a long claw and the line graphs of all subdivisions of $W_{k \times k}$ gives bounded treewidth.
Let $t_1, t_2, t_3$ be integers, with $t_1 \geq 0$ and $t_2, t_3 \geq 1$. Recall from the introduction that a {\em long claw}, also called a {\em subdivided claw}, denoted $S_{t_1, t_2, t_3}$, is a vertex $v$ and three paths $P_1$, $P_2$, $P_3$, of length $t_1$, $t_2$, and $t_3$, respectively, with one end $v$, such that $P_1 \setminus \{v\}$, $P_2 \setminus \{v\}$, and $P_3 \setminus \{v\}$ are pairwise disjoint and anticomplete to each other. We call $P_1, P_2, P_3$ the {\em legs} of $S_{t_1, t_2, t_3}$. The vertex $v$ is called the {\em root} of $S_{t_1, t_2, t_3}$.
For two graphs $H_1,H_2$, we denote by $H_1+H_2$ the graph with vertex set
$V(H_1) \cup V(H_2)$ and edge set $E(H_1) \cup E(H_2)$.
We start with a lemma.
\begin{lemma}
Let $t_1, t_2, t_3$ be positive integers with $t_1 \geq 2$. Let $G$ be an $S_{t_1, t_2, t_3}$-free graph. Then, $S_{t_1-1, t_2, t_3} + K_1$ is an $S_{t_1-2, t_2, t_3}$-forcer for $G$.
\label{lemma:claw_forcers}
\end{lemma}
\begin{proof}
Let $H$ be an $S_{t_1-1, t_2, t_3}$ in $G$, and let $u \in V(G)$ be anticomplete to $H$, so that $H \cup \{u\}$ is an $S_{t_1-1, t_2, t_3} + K_1$. Let $H = P_1 \cup P_2 \cup P_3$, where $P_1 = v \hbox{-} x_1 \hbox{-} \hdots \hbox{-} x_{t_1-1}$, $P_2 = v \hbox{-} y_1 \hbox{-} \hdots \hbox{-} y_{t_2}$, and $P_3 = v \hbox{-} z_1 \hbox{-} \hdots \hbox{-} z_{t_3}$.
Let $X = H \setminus x_{t_1-1}$. Let $D$ be a connected component of $G \setminus N[X]$. Suppose $u, x_{t_1 -1} \in N[D]$. It follows that there exists a path $P = x_{t_1 - 1} \hbox{-} p_1 \hbox{-} \hdots \hbox{-} p_k \hbox{-} u$ from $x_{t_1 - 1}$ to $u$ with $P^* \subseteq D$, so $X$ is anticomplete to $P^*$. Then, $H \cup \{p_1\}$ is isomorphic to $S_{t_1, t_2, t_3}$, a contradiction. Therefore, $X$ breaks $\{u, x_{t_1 - 1}\}$, and it follows that $S_{t_1-1, t_2, t_3} + K_1$ is an $S_{t_1-2, t_2, t_3}$-forcer for $G$. \end{proof}
Now we can prove Theorem \ref{thm:claw-free_nonspecific}, which we restate.
\begin{theorem}
\label{thm:claw_free}
Let $\Delta, t_1,t_2,t_3, k$ be positive integers with $t = \max(t_1, t_2, t_3)$. Let $\mathcal{C}$ be the class of all $S_{t_1, t_2, t_3}$-free graph with maximum degree $\Delta$ and no induced subgraph isomorphic to the line graph of a subdivision of $W_{k \times k}$. There exists an integer $N_{k,t,\Delta}$ such that $\text{tw}(G) \leq N$ for every $G \in \mathcal{C}$.
\end{theorem}
\begin{proof}
The proof is by induction on $t_1+t_2+t_3$. If $t_1=t_2=t_3=1$, the result
follows from Theorem~\ref{clawfreelinewalltw}. Thus we may assume that
$t_1 \geq 2$.
By Theorem~\ref{thm:centralbag} and Lemma~\ref{lemma:claw_forcers}, it is enough to find a bound on the treewidth of $(S_{t_1-1, t_2, t_3}+K_1)$-free graphs in $\mathcal{C}$.
Let $H \in \mathcal{C}$ be $(S_{t_1-1, t_2, t_3}+K_1)$-free.
By the inductive hypothesis we may assume that there exists
$X \subseteq V(H)$ such that $X$ is an
$S_{t_1-1, t_2, t_3}$ in $H$.
Since $H$ does not contain $S_{t_1-1, t_2, t_3} + K_1$, it follows that
$V(H) \subseteq N[X]$, and therefore $\text{tw}(H) \leq |V(H)| \leq (t_1+t_2+t_2)\Delta$.
\end{proof}
\section{$t$-thetas, $t$-pyramids, and line graphs of walls}
\label{sec:theta_pyramid}
In this section, we prove Theorem \ref{thm:pyramid_theta-nonspecific}, that for all $k,t$, excluding $t$-thetas, $t$-pyramids, and the line graphs of all subdivisions of $W_{k \times k}$ in graphs with bounded degree gives bounded treewidth. The proof involves an application of Theorem \ref{thm:claw_free}. We also need the following lemma.
\begin{lemma}
Let $x_1, x_2, x_3$ be three distinct vertices of a graph $G$. Assume that $H$ is a connected induced subgraph of $G \setminus \{x_1, x_2, x_3\}$ such that $H$ contains at least one neighbor of each of $x_1$, $x_2$, $x_3$, and that subject to these conditions $V(H)$ is minimal subject to inclusion. Then, one of the following holds:
\begin{enumerate}[(i)]
\item For some distinct $i,j,k \in \{1,2,3\}$, there exists $P$ that is either a path from $x_i$ to $x_j$ or a hole containing the edge $x_ix_j$ such that
\begin{itemize}
\item $H = P \setminus \{x_i,x_j\}$, and
\item either $x_k$ has at least two non-adjacent neighbors in $H$ or $x_k$ has exactly two neighbors in $H$ and its neighbors in $H$ are adjacent.
\end{itemize}
\item There exists a vertex $a \in H$ and three paths $P_1, P_2, P_3$, where $P_i$ is from $a$ to $x_i$, such that
\begin{itemize}
\item $H = (P_1 \cup P_2 \cup P_3) \setminus \{x_1, x_2, x_3\}$, and
\item the sets $P_1 \setminus \{a\}$, $P_2 \setminus \{a\}$ and $P_3 \setminus \{a\}$ are pairwise disjoint, and
\item for distinct $i,j \in \{1,2,3\}$, there are no edges between $P_i \setminus \{a\}$ and $P_j \setminus \{a\}$, except possibly $x_ix_j$.
\end{itemize}
\item There exists a triangle $a_1a_2a_3$ in $H$ and three paths $P_1, P_2, P_3$, where $P_i$ is from $a_i$ to $x_i$, such that
\begin{itemize}
\item $H = (P_1 \cup P_2 \cup P_3) \setminus \{x_1, x_2, x_3\} $, and
\item the sets $P_1$, $P_2$ and $P_3$ are pairwise disjoint, and
\item for distinct $i,j \in \{1,2,3\}$, there are no edges between $P_i$ and $P_j$, except $a_ia_j$ and possibly $x_ix_j$.
\end{itemize}
\end{enumerate}
\label{lem:three_leaves}
\end{lemma}
\begin{proof}
For some distinct $i,j,k \in \{1,2,3\}$, let $P$ be a path from $x_i$ to $x_j$ with $V(P^*) \subseteq V(H)$ (in the graph where the edge $x_ix_j$ is deleted if it exists). Such a path exists since $x_i$ and $x_j$ have neighbors in $H$ and $H$ is connected. Assume that $x_k$ has neighbors in $P^*$. Then, by the minimality of $V(H)$, we have $H = P^*$. If $x_k$ has two non-adjacent neighbors in $P^*$, or $x_k$ has two neighbors in $P^*$ and its neighbors in $P^*$ are adjacent, then outcome (i) holds. If $x_k$ has a unique neighbor in $P^*$, then outcome (ii) holds. Thus, we may assume that $x_k$ is anticomplete to $P^*$.
Let $Q$ be a path with $Q \setminus \{x_k\} \subseteq H$ from $x_k$ to a vertex $w \in H \setminus P$ (so $x_k \neq w$) with a neighbor in $P^*$. Such a path exists since $x_k$ has a neighbor in $H$, $x_k$ is anticomplete to $P^*$, and $H$ is connected. By the minimality of $V(H)$, we have $H = (P \cup Q) \setminus \{x_1, x_2, x_3\}$ and no vertex of $Q \setminus w$ has a neighbor in $P^*$. Moreover, by the argument of the previous paragraph, we may assume that $x_i$ and $x_j$ are anticomplete to $Q \setminus \{x_k\}$.
Now, if $w$ has a unique neighbor in $P^*$, then outcome (ii) holds. If $w$ has two neighbors in $P^*$ and its neighbors in $P^*$ are adjacent, then outcome (iii) holds. Therefore, we may assume that $w$ has two non-adjacent neighbors in $P^*$. Let $y_i$ and $y_j$ be the neighbors of $w$ in $P^*$ that are closest in $P^*$ to $x_i$ and $x_j$, respectively. Let $R$ be the subpath of $P^*$ from $y_i$ to $y_j$. Now, the graph $H'$ induced by $\left((P \cup Q) \setminus R^* \right) \setminus \{x_1, x_2, x_3\}$ is a connected induced subgraph of $G \setminus \{x_1, x_2, x_3\}$ and it contains at least one neighbor of $x_1$, $x_2$, and $x_3$. Moreover, $H' \subset H$ since $R^* \neq \emptyset$. This contradicts the minimality of $V(H)$.
\end{proof}
Now we are ready to prove Theorem \ref{thm:pyramid_theta-nonspecific}, which we restate.
\begin{theorem}
Let $\Delta, t,k$ be positive integers with $t \geq 2$. Let $\mathcal{C}$ be the class of
graphs of maximum degree $\Delta$ with no $t$-theta, no $t$-pyramid, and no induced subgraph isomorphic to the line graph of a subdivision of $W_{k \times k}$. There exists an integer $M_{k,t,\Delta}$ such that $\text{tw}(G) \leq M_{k,t,\Delta}$ for every $G \in \mathcal{C}$.
\label{thm:theta_pyramid}
\end{theorem}
\begin{proof}
We start by proving a result about the existence of forcers for $\mathcal{C}$.
\sta{\label{claws_are_forcers} $S_{t, t, t}$ is an $S_{t-1, t-1, t-1}$-forcer for
$\mathcal{C}$.}
Let $G \in \mathcal{C}$, and let $Y$ be an $S_{t, t, t}$ in $G$, let $r$ be the root of $Y$, let $x, y, z$ be the leaves of $Y$, and let $X = Y \setminus \{x, y, z\}$. Let $D$ be a connected component of $G \setminus N[X]$, and suppose $\{x, y, z\} \subseteq N[D]$. Let $Z \subseteq D$ be an inclusion-wise minimal connected subset of $D$ such that $x, y, z$ each have a neighbor in $Z$. By Lemma \ref{lem:three_leaves}, one of three cases holds. If case (ii) or case (iii) holds, then it is clear that $Y \cup Z$ is either a $t$-theta or a $t$-pyramid, so we may assume case (i) holds. Then, up to symmetry between $x, y$, and $z$, we have that $Z \cup \{x, z\}$ is a path from $x$ to $z$. Suppose $y$ has two non-adjacent neighbors in $Z$. Let $p, q$ in $Z$ be the first and last neighbors of $y$ in $Z$, such that $x, p, q, z$ appear in $x \hbox{-} Z \hbox{-} z$ in that order. Then $G$ contains a theta between $r$ and $y$ through $r \hbox{-} Y \hbox{-} y$, $r \hbox{-} Y \hbox{-} x \hbox{-} Z \hbox{-} p \hbox{-} y$, and $r \hbox{-} Y \hbox{-} z \hbox{-} Z \hbox{-} q \hbox{-} y$. Since each of the paths of the theta contains a leg of $Y$, it follows that every path of the theta has length at least $t$, a contradiction. Therefore, $y$ has exactly two adjacent neighbors $p, q$ in $Z$ such that $x, p, q, z$ appear in $x \hbox{-} Z \hbox{-} z$ in that order. But now $G$ contains a pyramid from $r$ to $\{y, p, q\}$ through $r \hbox{-} Y \hbox{-} y$, $r \hbox{-} Y \hbox{-} x \hbox{-} Z \hbox{-} p$, and $r \hbox{-} Y \hbox{-} z \hbox{-} Z \hbox{-} q$. Since each of the paths of the pyramid contains a leg of $Y$, it follows that every path of the pyramid has length at least $t$, a contradiction. Therefore, $X$ breaks $\{x, y, z\}$, so $S_{t, t, t}$ is an $S_{t-1, t-1, t-1}$-forcer for $G$. This proves \eqref{claws_are_forcers}. \vspace*{3mm}
Now by Theorem~\ref{thm:centralbag}, the result follows immediately
from Theorem~\ref{thm:claw_free}.
\end{proof}
\section{Subcubic subdivided caterpillars and their line graphs}
\label{sec:caterpillar}
\input{caterpillars_sec}
Next we prove a lemma.
\begin{lemma}
\label{lem:wallandcreature}
Let $\Delta, b,k,t$ be positive integers where $k \geq 3$.
Let $\mathcal{C}$ be the class of graphs with maximum degree $\Delta$
that do not contain a $(k,t)$-creature or the line graph of a subdivision of $W_{b\times b}$.
There exists $R_{b, t,k, \Delta}$ such that $\text{tw}(G) \leq R_{b,t,k,\Delta}$ for
every $G \in \mathcal{C}$.
\end{lemma}
\begin{proof}
Let $t_i = t(1 + \Delta)^{k-i}$. Let $\mathcal{C}_i$ be the class of graphs with maximum degree $\Delta$
that do not contain an $(i,t_i)$-creature and have no induced subgraph isomorphic to the line graph of a subdivision of $W_{b\times b}$.
We will prove by induction that there exists $R_{b, t,k, i, \Delta}$ such that $\text{tw}(G) \leq R_{b,t,k,i,\Delta}$ for every $G \in \mathcal{C}_i$. Since $S_{t_3, t_3, t_3}$ is a $(3, t_3)$-creature, for $i=3$ the result follows from \ref{thm:claw_free}. Next we prove a result about the existence of forcers in graphs in $\mathcal{C}_i$.
\sta{\label{claws_are_forcers_caterpillar} $S_{t_i+1,t_i+1,t_i+1} + H$ is a $S_{t_i, t_i+1, t_i+1}$-forcer for $\mathcal{C}_i$ for every $(i-1, t_{i-1})$-creature $H$.}
Let $G \in \mathcal{C}_i$ and let $H$ be an $(i-1, t_{i-1})$-creature. Let $Y$ be an $S_{t_i+1, t_i+1, t_i+1} +H$ in $G$, let $Y' = Y \setminus H$, let $x \in Y'$ be a leaf of $Y'$, and let $X = Y' \setminus \{x\}$. Let $D$ be a connected component of $G \setminus N[X]$. Suppose $x \in N[D]$. Then, by Theorem \ref{creature}, it follows that $D$ has no $(i-1, t_{i-1})$-creature. Since $H$ is anticomplete to $Y'$, we have that $H \not \subseteq N[D]$. Therefore, $X$ breaks $\{x\} +H$, so $S_{t_i+1, t_i+1, t_i+1} +H$ is a $S_{t_i, t_i+1, t_i+1}$-forcer for $G$. This proves \eqref{claws_are_forcers_caterpillar}.
By Theorem~\ref{thm:centralbag}, it is now enough to bound the treewidth of
$\{(S_{t_i+1, t_i+1, t_i+1} +H) : H \text{ is an}$ \\ $(i-1, t_{i-1})\text{-creature}\}$-free graphs in $\mathcal{C}_i$. Let $F$ be a graph with no $(i, t_{i})$-creature.
If $F$ is $S_{t_i+1, t_i+1, t_i+1}$-free, the result follows from Theorem
\ref{thm:claw_free}. Thus, let $Q \subseteq V(F)$ be an $S_{t_i+1, t_i+1, t_i+1}$ in $F$. Then, $F \setminus N[Q]$ has no $(i-1, t_{i-1})$-creature, so by the inductive hypothesis, we deduce that $\text{tw}(F \setminus N[Q]) \leq R_{b,t,k,i-1,\Delta}$.
But $|Q|=3t_i+4$, and therefore $|N[Q]| \leq (3t_i+4) \Delta$.
Consequently, $\text{tw}(F) \leq R_{b,t,k,i-1,\Delta}+(3t_i+4) \Delta$, and we can set
$R_{b,t,k,i,\Delta}=R_{b,t,k,i-1,\Delta}+(3t_i+4) \Delta$.
\end{proof}
We can now prove Theorem \ref{thm:caterpillar-non_specific}, which we restate.
\begin{theorem}
Let $\Delta$ be a positive integer and let $T$ be a subcubic subdivided caterpillar. Let $\mathcal{C}$ be the class of graphs with maximum degree $\Delta$ which do not contain a subdivision of $T$ or the line graph of a subdivision of $T$. Then there exists $R_{\Delta, T}$ such that $\text{tw}(G) \leq R_{\Delta, T}$ for
every $G \in \mathcal{C}$.
\end{theorem}
\begin{proof}
Let $G \in \mathcal{C}$.
By Theorem \ref{cater}, there exist integers $k,t$ such if $G \in \mathcal{C}$
then $G$ does not contain a $(k, t)$-creature.
Next we observe:
\sta{\label{no_line-graph-wall} Let $G \in \mathcal{C}$. Then $G$ does not contain the line graph of a subdivision of $W_{|T| \times |T|}$.}
Let $H$ be the line graph of a subdivision of $W_{|T| \times |T|}$. Then, $H$ contains the line graph of a subdivision of $T$. It follows that if $G$ contains $H$, then $G$ contains the line graph of a subdivision of $T$, a contradiction. This proves \eqref{no_line-graph-wall}. \vspace*{3mm}
Now the result follows from Lemma \ref{lem:wallandcreature}.
\end{proof}
|
2,877,628,090,250 | arxiv | \section{\bf Introduction}
\par The
ongoing experiments like ultra-relativistic heavy-ion
collision at BNL and the large hadron collider at CERN have
focussed on the search of
the QCD phase structure and the formation of mini big bang.
The experiments at BNL and
CERN will provide the best platform to study
the creation and evolution of such mini big bang called
Quark-Gluon Plasma (QGP), which
is perhaps believed
to be formed in the expansion of the early universe~\cite{karch}.
Since we believe that
the matter existed
only for a few microseconds after the big-bang,
its direct detection is very difficult even in these experiments. There
are indirect possibilities for detection like strangeness enhancement~\cite{rafelski}, $J/\psi$
suppression~\cite{matsui} and radiation of
dileptons and photons~\cite{shuryak,kajantie,srivastava} etc.
Among these indirect probes,
dileptons and photons are considered to be the most promising
signals for its detection of QGP formation created in relativistic
heavy-ion collision (RHIC). It is due to the fact that the
dilepton driving out of the collisions among the quarks, antiquarks and
gluons brings the whole informations about the existence of the plasma
fireball and tell the properties of the fireball to the detector.
They interact through electromagnetic force due to the
large mean free path in their production. In order to see the
production of dilepton for
the signal of QGP formation, we look
at the process of annihilation of quark and antiquark and they produce
virtual
photons which subsequently decay
into dileptons
such as $l^{-}l^{+},~\mu^{+}\mu^{-}$.
\par
Many theoretical and experimental researchers have
calculated dilepton and photon emissions
at finite temperature and at quark chemical potential.
The experiments at AGS
and SPS energies~\cite{nagamiya} have reported the presence of significant
amount of baryon chemical potential and
even at RHIC energies~ $\sqrt{s} \leq 200\, A GeV$ there has been
the detection of such baryon chemical potential.
These informations indicate
~\cite{gustafson,mohring} that the colliding heavy ions may not
be fully transparent in the centrality region of the colliding
particles and the region may have significant amount of
dense nuclear matter. The work of Hammon and coworkers~\cite{hammon}
supported these arguments of existing the chemical potential and predicted
the initial nonequilibrium QGP produced at RHIC energies, indicating that
the system has finite baryon density or chemical potential.
So, in the theoretical study, dilepton emissions in finite baryonic chemical potential has been calculated
through various distribution functions and perhaps, the
work of Dumitru et al.~\cite{dumitru} gives the first signal
of dilepton emission at finite baryonic chemical potential with
Fermi distribution functions. Then this work is further studied
by Strickland using
quark and gluon fugacities in j$\ddot{u}$ttner distribution function.
He
showed another promising result calculated
from the non-equilibrium
quark-gluon plasma ~\cite{strickland}.
The recent work of Majumder et al.~\cite{majumder} have indicated
the emission of
dileptons from QGP at the RHIC energies at
finite baryon density. Bass et al. ~\cite{bass} idea of
parton rescattering and fragmentation leads to a substantial
increase in the net-baryon density
at midrapidity region. Besides these works, we have the reports of
other authors on dilepton production at low mass region~\cite{rapp}.
These works suggest the importance of chemical
potential in the dilepton calculation. To produce
such emission, we consider the system in the pionic medium in which
the equilibrium thermodynamic QGP is
a function of temperature~ $T$ and chemical potential$~\mu~$ and the
potential itself as function of temperature .
\par In this brief article, we choose the baryonic chemical potential which
is considered to be temperature dependent chemical potential (TDCP)
and
the value has a change on the quark and antiquark distribution functions.
We take the value of the chemical potential
in the scale of QCD parameter of dense nuclear matter. The
chemical potential considered is obtained through~\cite{hamieh}:
\begin{equation}
\mu(T)=2\pi\beta^{-1}\sqrt(1+\frac{1}{\pi^{2}}\ln^{2}\lambda_{q})
\end{equation}
where $T=\frac{1}{\beta}$ taken in the scale of QCD and $\lambda_{q}=e^{\frac{\mu_{q}}{T}}~$is
quark fugacity.
However, we consider
the massless dynamical quark as a finite
value and it is called thermal dependent quark mass (TDQM)
obtained through the parametrization value and temperature. The finite
value of
the quark mass is defined as
\begin{equation}
m^{2}_{q}= \frac{8 \pi}{(33-3n_{f})}\frac{T^{2}}{\ln(1+(\frac{\gamma N^{\frac{1}{3}} T^{2}}{2 \Lambda^{2}})^{\frac{1}{2}})}
\end{equation}
with the QCD parameter~ $\Lambda = 150~MeV$
and normalising $~ N=\frac{16 \pi}{27}$. $ \gamma $ is parametrization
factor which is like the Reynold's number to take care of
the hydrodynamical aspects of the hot QGP flow. Its value is determined
in the most effective way of the flow parameter of quarks~ $\gamma_{q}$
and gluons~ $\gamma_{g}$ and it enhances in the growth of
free energy of QGP droplet.
It is expressed as
\begin{equation}
\gamma=\sqrt 2( \frac{1}{\gamma_{q}^2}+\frac{1}{\gamma_{g}^2})
\end{equation}
with the value of~ $\gamma_{g}= 6 \gamma_{q}~~ or~~ 8 \gamma_{q}$~
and~
$\gamma_{q}=1/6$~\cite{yen}.
\par
Using all such parameters, we create the QGP droplet at TDCP
incooperating the quark mass as a finite value in the system and look
at the improvement produced by the TDCP in the growth of QGP droplet.
Then we calculate dilepton emission at the TDCP from such a system of
QGP and see its emission rate.
\par The paper is organised as follows: In Sec.II we recall a
brief highlight of the
evolution/growth of QGP fireball through the statistical model in
the pionic medium.
In Sec.III we look at the dilepton emission and intgrated yields
at temperature dependent
chemical potential (TDCP). In the last Sec.IV
we conclude and present our results.
\par
\section{\bf The free energy growth of QGP droplet at TDCP}
\par We use the statistical model at the TDCP for
the growth of QGP droplet in the pionic medium. The system
is considered
to be constituted by the free quarks, antiquarks, gluons and pions.
We construct
the free energies of the particles using the model of mean
field potential in their density of state.
The constructed
free energies of these
noninteracting fermions and bosons at the temperature dependent
chemical potential are defined as~$F_{i}$
in which contribution of quarks and gluons are
indicated by the upper sign or bosons by the lower sign.
It is expressed as:
\begin{equation}
F_{i}=\mp T g_{i}\int dp \rho(p_{i})ln(1\pm \exp(-\sqrt{m_{i}^{2}+p_{i}^{2}}+\mu/T))
\end{equation}
where $g_{i}$ is the appropriate colour and particle-antiparticle
degeneracy factor. Its value is taken as $6$ and $8$ for quarks and
gluons~\cite{neergaard}. $\rho(p_{i})$
is the density of states of the
particular particle~$i$~
(quarks,~gluons) based on the effective potential
among the interacting particles and it is defined within the range of
momentum space $dp_{i}$ in a
spherically symmetric
system. It is given by~\cite{ramanathan}
\begin{equation}
\rho(p_{i})=v/\pi^{2}[(-V_{eff}(p))^{2}(-\frac{dV_{eff}(p)}{dp})]
\end{equation}
where, $V_{eff} =(1/2p)\gamma g^{2}(p)T^{2}-m^{2}_{0}/2p$, known as
mean field effective
potential among the quarks, antiquarks and quarks-gluons.
$ g^{2}(p)$ is the
first order QCD running coupling constant. It is given by
\begin{equation}
g^{2}(p)=(4/3)(12\pi/27)[1/\ln(1+p^2/\Lambda^2)]
\end{equation}
The free energy of pion contributed to the total free
energies of quarks and gluons is given as:
\begin{equation}
F_{\pi}=(3T/2\pi^{2}) v \int_{0}^{\infty} p_{\pi}^{2}dpln(1- \exp(-\sqrt{m_{\pi}^{2}+p_{\pi}^{2}}/T))
\end{equation}
The contribution of the pion energy is due to the fact that
the transformation of the phase is slightly dominanted by the pions over the
other hadronic particles. In addition to these free energies there is
another contribution to the total free energy and it takes the role of
bag constant in confining the system. It is called the interfacial energy given by~\cite{weyl}
\begin{equation}
F_{interface}=\frac{1}{4}R^{2}T^{3}\gamma
\end{equation}.
Therefore, the total energy of QGP fireball is:
\begin{equation}
F=\sum_{j} F_{j}
\end{equation}
where j stands for the different particles viz quark, gluon, pion
and interface. The free energy can indicate the nature of QGP fireball
evolution and its transition. It also
explains the
creation of the plasma formation with the size of the droplets.
\par
\section{\bf Dilepton emission at TDCP from QGP}
\par
The calculation of dilepton emission at finite temperature
and at finite baryon chemical potential have been done by many authors.
These calculations are performed on the basis of the
expected results coming out from the heavy-ion collision experiments.
The experiments expect more productions of lepton than of other
particles produced.
The possible sources of dilepton are from the annihilation
$q\bar{q}\rightarrow l^{+}l^{-}$, compton like scattering,
$q(\bar{q})g\rightarrow q(\bar{q})l^{+}l^{-}$
and $gg\rightarrow q\bar{q}l^{+}l^{-} $fusion processes.
Among the processes, Drell-Yan reaction is mostly used
for thermal emission of dilepton pairs~\cite{ruuskanen}
and the compton scattering such as
$q(\bar{q})g \rightarrow q(\bar{q})+ l^{+}l^{-}$ follows after
Drell-Yan reaction. In this article, we
exclusively engage in quark-antiquark annihilation such as
$q\bar{q}\rightarrow l^{+}l^{-}$ reaction
for the dilepton
emission. This is due to fact that it produces larger amount of lepton
pair in comparison to the other collisions. In the process, we consider
only the dominant production of dilepton in the intermediate mass region
neglecting the dilepton spectra
from the low mass region. This is
fact that the contribution of dilepton through the deacy of mesons in the
system is negligence.
So the dilepton emission rate produced $\frac{dN}{d^{4}x}$ is given
by~\cite{gale}:
\begin{eqnarray}
\frac{dN}{d^{4}x}&=&\int \frac{d^{3}p_{1}}{(2 \pi)^{3}}
\frac{d^{3}p_{2}}{(2 \pi)^{3}} n_{q}(p_{1},\mu)
n_{\bar{q}}(p_{2},\mu)\nonumber
\\
&\times &v_{q\bar{q}} \sigma_{q\bar{q}}(M^{2})
\end{eqnarray}
where,
\begin{equation}
n_{q}(p_{1},\mu)=\frac{\lambda_{q}}{\exp^\frac{(p_{1}-\mu)}{T}+\lambda_{q}},
n_{\bar{q}}(p_{2},\mu)=\frac{\lambda_{\bar{q}}}{\exp^\frac{(p_{2}+\mu)}{T}+\lambda_{\bar q}}
\end{equation}
are Fermi-Dirac distribution functions for
quarks and antiquarks~\cite{biro}
with their corresponding parton fugacities~$\lambda_{q(\bar{q})}=e^{\frac{\mu}{T}}$.
For gluon, the Bose-Einstein distribution function is:
\begin{equation}
n_{g}(p,\mu)=\frac{\lambda_{g}}{\exp^\frac{p_{g}}{T}-\lambda_{g}}
\end{equation}
with parton gluon fugacity $\lambda_{g}$. The function for gluon
can be used in the collision of $qg\rightarrow l^{+}l^{-}$ or $gg$
fusion reaction.
$v_{q\bar{q}}$ is the relative velocity of annihilating quark
pair and $p_{\mu}$ is lepton pair four momentum.($M^{2}=p^{\mu}p_{\mu}$
invariant lepton pair mass ). $\sigma_{{q\bar{q}}
{\rightarrow}{l\bar{l}}}$ is the electromagnetic
annihilation cross section. Substituting
the distribution functions for quark
and antiquark in the equation $(10)$ using $(11)$, and integrate
over $q$ and $\bar{q}$ momentum, we obtain dilepton emission rate at TDCP as:
\begin{eqnarray}
\frac{dN}{dM^{2}d^{4}x} & = & \frac{5 \alpha^{2}}{18 \pi^{3}}
T M e^{4 \sqrt{\pi^{2}+\ln^{2}\lambda_{q}}} \nonumber \\
&\times &(1+\frac{2 m_{q}^2}{M^2}) K_{1}(M/T)
\end{eqnarray}
In the above solution, $ K_{1}(M/T) = G(z) $ which is known as the modified
Bessel's function and
volume element is $d^4x =d^2x_{T}dy\tau d\tau$.
We expand longitudinally the above expression and finally we have emission
rate as:
\begin{eqnarray}
\frac{dN}{dM^{2}dy} & = & \frac{5 \alpha^{2} R^2}{18 \pi^{2}}
M(1+\frac{2 m_{q}^2}{M^2}) \nonumber \\
&\times &\int e^{4 \sqrt{\pi^{2}+\ln^{2}\lambda_{q}}/T(\tau)} G(z,\tau)T(\tau) \tau d\tau
\end{eqnarray}
where,~ $ T(\tau)= T_{0}(\frac{\tau_{0}}{\tau})^{1/3}$.
\section{\bf Results and Conclusions}
\par The results of evolution of QGP
fireball are
shown in the figures~$(1-3)$. Figure~$(1)$ shows the evolution
of free energy at zero chemical potential at the parametrization value
of $~\gamma_{q}=1/6,~\gamma_{g}= 6 \gamma_{q}$. It shows
very much stability in the formation of droplets for the various
values of temperature. Figure~$(2)$ shows the change of evolution
of the free energy for these various
values of chemical potential and indicate the decreasing size of droplets
with the increasing chemical potential. The figure~$(3)$ shows
the change of free energy
with the size of droplet at temperature
dependent chemical potential with the finite value of quark chemical
potential$~\mu_{q}= 47 MeV$. The result indicates that the evolution of
QGP through the model has first order shift at the
temperature~$T=(150-170)~MeV$ with the increase of chemical potential
and the size of the droplet is increased with the increase of the chemical
potential. It implies that the evolution of QGP droplet is enhanced
with the effect of temperature dependent chemical potential.
Moreover the shift in the first order is further explained
in figure~$(4)$ by the behaviour of the entropy v/s chemical potential.
There is a clear jump
in the continuity of the entropy curve at the
chemical potential~$\mu=(990-1005)~MeV$ at which we found the corresponding
temperature as~$T=(150-170)~MeV$.
\par In figure~$(5)$, we show
dilepton emission for various values of
initial
temperature~$T_{0}$
and at transition temperature~$T_{c}=0.17~GeV$ without the chemical
potential and compared the results with other theoretical calculations
of dilepton emission at $~\mu=0$. The results are same over the range
of lepton pair mass $~M~$. In the figure~$(6)$, we show the comparison of
emission rates of dilepton at
the temperature
dependent chemical potential and at finite chemical potential.
The emission rate increases with the increase
of temperature dependent chemical potential at
the transition temperature~$T_{c}=0.17~GeV$ over the finite chemical potential.
The emission rate is much higher
at the temperature dependent chemical potential than the emission at finite
chemical potential.
\par Now, if we look dilepton yields with the change of the lepton pair mass
in above figures of dilepton production, we obtain
a sudden fall in the production rate of dilepton with increase in
lepton pair mass~$M$ upto~$3~GeV$. It indicates that at higher
lepton pair mass more suppression is obtained in comparison to the
low mass region.
\par
We look again at dilepton integrated yields with the evolution time of the
QGP droplet. The dilepton integrated yield is exponentially
increasing with the evolution time and after a certain time, it becomes
constant for these different values of chemical potential. The
plots are shown in the figures~$(7-8)$ for the chemical potential
with their corresponding initial
temperatures at transition temperature~$T=0.17~GeV~$.
Figure~$(7)$ shows the integrated yields at transition
temperature $T_{c}=0.17~GeV$ without the finite value of chemical
potential and again compared the results with other results. They are same
at this zero chemical potential.
In figure~$(8)$, we plot the
integrated yield for the temperature dependent chemical potential and
finite chemical potential at
the transition temperature~$T=0.17~GeV$. The integrated yield is
very large as compared to the results of the earlier work of integrated
yeild at finite value of chemical potential. This is due to the
fact that both temperature and chemical potential enhance in the
interaction of the particles of the system and yields more dileptons.
\par
Now it indicates that the emission and integrated yield are higher at TDCP
as compared to the subsequently fixed value of chemical potential and at
zero chemical potential.
It means in the QGP phase where the
temperature and chemical potential coexist together then the interacions
among
the partciles are more and dileptons are produced more.
So the model with the temperature dependent chemical potential produced
improvement over results of finite baryonic chemical potential and zero
chemical potential.
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig1.eps}}}
\vspace*{0.5cm}
\caption[]{
Free energy with the change of droplet size for various values of
temperature.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig2.eps}}}
\vspace*{0.5cm}
\caption[]{
Free energy with the change of droplet size at the particlur
transition temperature~$ T_{c}=0.17~GeV $ for variuos values of
chemical potential.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig3.eps}}}
\vspace*{0.5cm}
\caption[]{
Free energy with the change of droplet size
at the particular quark chemical potential~$\mu_{q}=0.47~GeV$
for different initial temperatures.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig4.eps}}}
\vspace*{0.5cm}
\caption[]{
Entropy v/s, change of chemical potential~$\mu $ and first order transition
at the temperature~$T_{c}=
0.17~GeV$.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig5.eps}}}
\vspace*{0.5cm}
\caption[]{
The dilepton emission rate,~$\frac{dN}{dM^{2} dy}~(GeV^{-2}) $,
at transition temperature~$T_{c}
=0.17~GeV$ and at zero chemical potential for different initial temperatures
and its compared curve.
}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig6.eps}}}
\vspace*{0.5cm}
\caption[]{
The dilepton emission rate,~$\frac{dN}{dM^2 dy}~(GeV^{-2}) $,
at transition temperature~$T_{c}
=0.17~GeV$ for the different values of
~$\mu $ with different initial temperatures and its compared curves.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig7.eps}}}
\vspace*{0.5cm}
\caption[]{
The dilepton integrated yields,~$\frac{dN}{dM dy}~(GeV^{-1}) $,
at transition temperature~$T_{c}
=0.17~GeV$ and at zero chemical potential
~$(\mu=0) $ with different initial temperatures and its compared curves.}
\label{scaling}
\end{figure}
\begin{figure}
\resizebox*{3.0in}{3.0in}{\rotatebox{360}{\includegraphics{nnfig8.eps}}}
\vspace*{0.5cm}
\caption[]{
The dilepton integrated yield,~$\frac{dN}{dM dy}~(GeV^{-1}) $,
at transition temperature~$T_{c}
=0.17~GeV$ for the different values of
~$\mu $ with different initial temperatures and its compared curves.}
\label{scaling}
\end{figure}
\acknowledgements
We are very thankful to Dr. R. Ramanathan for his
constructive
suggestions and discussions. The author (YK) would like to thank the
Department for research facility and highly oblized to express his gratitute
to Rajiv Gandhi Fellowship, UGC, New Delhi for the financial support.
|
2,877,628,090,251 | arxiv | \section{Introduction}
In order to test the wide-band imaging capabilities of the
EVLA\footnote{The EVLA is operated by the National Radio Astronomy
Observatory (NRAO) for Associated Universities Inc. under a license
from the National Science Foundation of the USA.}, we have
observed several known and possible Galactic supernova remnants
(SNRs). There are 274 Galactic SNRs catalogued
\citep{SNRCatGreen2009}, the majority of which are classified as
`shell' remnants, showing limb-brightened synchrotron radio emission,
typically with $S \propto \nu^{\sim -0.5}$. Some SNRs are classed as
`filled-center' remnants (or `plerions'), as they show emission at
radio wavelengths that is brightest in the center, due to the power
input from a central pulsar. These filled-center remnants generally
have much flatter radio spectra at GHz-frequencies, with $S \propto
\nu^{\sim -0.0{-}0.3}$. Finally, there are some remnants that are
classed as composite remnants, as they show a mixture of
limb-brightened shell-like emission, with a filled-center-like core
(i.e.,\ a pulsar wind nebula) showing a much flatter spectrum. In
addition to the known remnants, there are many other possible and
probable Galactic SNRs that have been proposed, for which additional
observations are needed to confirm their nature. In many regions of
the Galactic plane, particularly near to $b=0^\circ$ and $l=0^\circ$,
radio emission from SNRs is easily confused with thermal radio
emission (typically with $S \propto \nu^{\sim -0.1}$) from {\sc H\,ii}
regions. The wide-band of the EVLA allows a variety of spectral
studies of SNRs to be made, in order to: (1) investigate spectral
variations across the face of shell remnants, in order to study the
differences in the shock acceleration mechanism at work in different
regions of the remnants; (2) study the flat spectrum cores of
composite remnants; and (3) clarify the nature of proposed possible
SNRs, by disentangling the non-thermal synchrotron emission from
confusing thermal emission.
We have observed two smaller known remnants G16.7$+$0.1 and
G21.5$-$0.9. G16.7$+$0.1 is a `composite' remnant, $\sim 3\arcmin$
across (e.g.,\ \citealt{1989ApJ...341..151H}). G21.5$-$0.9 was classed
as a filled-center remnant, only $\sim 1.5\arcmin$ in extent, until a
faint X-ray shell $\sim 4\arcmin$ in diameter was identified by
\cite{2000ApJ...533L..29S}. Since then it has been classed as a
composite remnant, although a radio counterpart to the X-ray shell has
not been identified (see \citealt{2011MNRAS.412.1221B}). G21.5$-$0.9
contains a pulsar, which has a high spin-down luminosity
(\citealt{2005CSci...89..853G}). We also observed a field centered at
$l=19.6^\circ$, $b=-0.2^\circ$, which contains `high-probability' SNR
candidate, G19.66$-$0.22, identified by \cite{HelfandMAGPIS}.
Finally, we observed one larger known but poorly studied shell
remnant, G55.7$+$3.4, which is $\sim 20 \times 25$~arcmin$^2$ in
extent (see \cite{Goss_G55.7+3.4}). An old, probably unrelated pulsar
lies within the northern edge of this remnant.
\section{Observations and Calibration}
The observations were made using the EVLA in D-array. The {\it L}-Band
observations used the full 1~GHz bandwidth covering the frequency
range of $1-2$~GHz. The {\it C}-Band observations used 2~GHz bandwidth in
two separate $\sim1$~GHz bands. The observations parameters for all
the fields are listed in Table~\ref{Tab:ObsParams}. At the time of
observations, not all antennas in the array were outfitted with the
new {\it L}- and {\it C}-Band receivers. Due to this and other failures during
the commissioning phase of the EVLA, typically 20 antennas were used
for these observations. The {\it L}-Band data was recorded in 8 or 16
spectral windows (SPW) at a frequency resolution of 2 and 1~MHz
respectively with an integration time of 1~s. The {\it C}-Band data
was recorded in 16 SPWs at a resolution of 2~MHz. High time and
frequency resolution was required to allow effective removal of radio
frequency interference (RFI), accurate bandpass calibration and
corrections for wideband group delay present in the data from EVLA
WIDAR correlator. To reduce the data volume, after initial data
flagging the data were averaged to 10~s resolution in time. The
standard flux calibrators 3C286 and 3C147 were observed for flux
calibration and the compact sources J1822$-$0938 (for G16.7$+$0.1,
G19.6$-$0.2 and G21.5$-$0.9) and J1925$+$2106 (for G55.7$+$3.4) were
observed at intervals of 30~min for phase and bandpass calibration.
Using known frequency dependent model for the flux calibrators
(including spectral index) antenna gains for each frequency channel
across the observed bandwidth were determined for flux and group delay
calibration. The antenna gains for each SPW were also determined
using the phase calibraters. In order to account for the spectral
index of the calibraters, a model including the frequency dependent
flux was determined using the multi-scale multi-frequency synthesis
(MS-MFS) imaging algorithm which simultaneously makes Stokes-{\it I} and
spectral index maps. The final complex bandpass shape was determined
using this model and applied to the data for the target fields. This
calibration procedure allowed calibration of time-varying direction
independent gains without transferring the effects of the frequency
dependent flux of the calibraters to the target fields.
\section{Wide-band Wide-field Imaging}
RFI affected frequency channels were flagged throughout the observing
band. Full SPWs covering the frequency range $1.516 \sim
1.643$~GHz had to be dropped due to the presence of strong RFI at
{\it L}-Band. Ten channels at the edges of the SPWs were also flagged due
to the roll-off of the digital filters in the signal chain. The rest
of the frequency channels were used for wide-band imaging.
Conventional image reconstruction techniques for interferometric
imaging fundamentally models the brightness distribution as a
collection of scaleless components (amplitude per pixel) leaving
errors which limit the imaging performance. Scale-sensitive methods
\citep{Asp_Clean,MSCLEAN} significantly reduce the magnitude of such
errors by explicitly solving for the scale size of emission across the
field of view. Imaging performance using wide-band interferometric
observations of fields with extended emission is additionally limited
by the fact that not only does the scale of emission vary across the
field of view, the spectral properties also vary across the field.
The dominant source of these spectral variations is the change of
antenna primary beam with frequency and spatial dependence of the
spectral index of the radio emission. Both effects result in
variation of the strength of emission as a function of frequency in a
direction dependent manner. Furthermore, antenna primary beams are
typically rotationally asymmetric which result in time-varying gains
due to the rotation of the primary beams with parallactic angle for
El-Az mount antennas. Scale-sensitive imaging reconstruction
algorithms that can also {\it simultaneously} account for time and
frequency-dependent effects are therefore required for wide-band
wide-field imaging with the EVLA, particularly in the Galactic plane.
For this, two new algorithms, namely the MS-MFS
\citep{RAU_THESIS,MSMFCLEAN} and the A-Projection \citep{AWProjection}
algorithm, have been developed. The MS-MFS algorithm combines the
scale-sensitive deconvolution with explicit frequency dependent
modeling of the emission to map the spatial distribution of the
spectral index in a scale-sensitive manner. The A-Projection
algorithm on the other hand corrects for the time and frequency
dependence of the antenna primary beams as part of the iterative image
reconstruction.
We used the MS-MFS \citep{RAU_THESIS,MSMFCLEAN} algorithm implemented
in the $\mathcal{CASA}$\footnote{{\tt http://casa.nrao.edu}}
post-processing package, for wide-band imaging. The spectral index
information derived using the traditional method of using images made
at different frequencies make spectral index images at the {\it
lowest} resolution and at the sensitivity of {\it single} channel
bandwidth. Spectral index images made using the MS-MFS algorithm on
the other hand does not require smoothing the images to the lowest
resolution and produces the spectral index images at the full
sensitivity and resolution offered by the data. This must be combined
with techniques to correct for various time variable PB-effects during
image deconvolution to achieve noise limited imaging performance.
While the A-Projection algorithm can be used to correct for these
PB-effects during image deconvolution, for the images presented here
we used only the time-averaged PBs ({see Figure~\ref{Fig:AvgPB}}) to
apply correction post deconvolution. Work towards a fully integrated
wide-field wide-band imaging algorithms is in progress.
\begin{table*}
\begin{center}
\caption{List of Observations Parameters}
\begin{tabular}{clccl}
\tableline\tableline
Field & Pointing Center & Date & Frequency Range & Calibrators \\
& (J2000) & & (GHz) & Flux, Phase \\
\tableline
G16.7$+$0.1 & RA=18:20:57 & 2010 Aug 12 & 4.49--5.45 & J1331$+$3030 and J0137$+$331,\\
\nodata & Dec=-14:19:30 & \nodata & 6.89--7.85\tablenotemark{a} & J1822$-$0938 \\
G19.6$-$0.2 & RA=18:27:38 & 2010 Jul 19 & 1.00--2.03 & J1331$+$3030, J1822$-$0938 \\
\nodata & Dec=-11:56:32 & \nodata & \nodata & \\
G21.5$-$0.9 & RA=18:33:32 & 2010 Aug 12 & 4.49--5.45 & J1331$+$3030 and J0137$+$331, \\
\nodata & Dec=-10:34:10 & \nodata & 6.89--7.85\tablenotemark{a} & J1822$-$0938 \\
G55.7$+$3.4 & RA=19:21:40 & 201 Aug 23 & 1.00--2.03 & J1331$+$3030 and J0542$+$498,\\
\nodata & Dec=+21:45:00 & \nodata & \nodata & J1925$+$2106\\
\tableline
\end{tabular}
\tablenotetext{a}{The two ranges correspond to the two 1-GHz wide
frequency bands covering total of 2~GHz bandwidth at {\it C}-Band.}
\label{Tab:ObsParams}
\end{center}
\end{table*}
\section{Results}
\begin{figure*}
\centering
\vbox{
\hbox{
\includegraphics[width=6.2cm]{f1a1}
\includegraphics[width=6.0cm]{f1b1}
\includegraphics[width=6.0cm]{f1c1}
}
\hbox{
\includegraphics[width=6.2cm]{f1a2}
\includegraphics[width=6.2cm]{f1b2}
\includegraphics[width=6.0cm]{f1c2}
}
}
\caption{Stokes-{\it I} (top row) and Spectral Index images
(bottom row). The first two columns show {\it C}-Band images of the SNRs
G16.7$+$0.1 and G21.5$-$0.9 respectively. The resolution in these
images is 10\arcsec\ and RMS noise is $11\mu Jy~beam^{-1}$ and
$30\mu~Jy~beam^{-1}$ respectively. The last column shows the {\it L}-Band image
of G19.6$-$0.2 with a compact source of thermal emission of positive
spectral index surrounded by non-thermal emission with a negative
spectral index. The resolution in this image is 40\arcsec\ and RMS
noise of $0.2m~Jy~beam^{-1}$. The circle indicates the half-power point of
the antenna primary beam at the reference frequencies of 6.0~GHz for
{\it C}-Band images and 1.5~GHz for the {\it L}-Band image.}
\label{Fig:SNRs_NarrowField}
\end{figure*}
\subsection{G16.7$+$0.1, G21.5$-$0.9, G19.6$-$0.2}
Figure~\ref{Fig:SNRs_NarrowField} shows the Stokes-{\it I} and Spectral
Index images of two catalogued SNRs (G16.7$+$0.1 and G21.5$-$0.9)
\citep{SNRCatGreen2009} and one candidate SNR (G19.6$-$0.2). The top
row shows the Stokes-{\it I} images and the bottom row shows the spectral
index images, with the Stokes-{\it I} contours overlaid on the spectral
images. Correction for the primary beam response raises the noise
away from the center. In order to show the detailed brightness
distribution, we chose to show the Stokes-{\it I} images without correcting
for the primary beam response. The integrated flux reported below was
determined from PB-corrected images. The spectral index images have
been corrected for the average primary beam effects, since the effect
of primary beam response in the spectral index images is stronger.
Image of all the fields covering the full wide-band sensitivity show
many more compact and diffused sources. Due to space limitation, here
we show central portions of these images.
The first column has the images at a reference frequency of 6~GHz, of
the composite SNR G16.7$+$0.1 at a resolution of 10\arcsec. The SNR
has a flatter spectrum core with an average spectral index of
$-0.54\pm0.02$ ($S \propto \nu^\alpha$) surrounded by a relatively
steeper spectrum nebula with the spectral index ranging from $-0.8$ to
$1.1$. The total continuum integrated flux is 1.2~Jy with an RMS
noise of 11~$\mu~Jy~beam^{-1}$. The diffused sources on east of SNR are
unrelated but real sources of emission. The images in the second
column are of the filled-centered SNR G21.5$-$0.9, also at the
reference frequency of 6~GHz. The RMS noise in this image is
$\sim30\mu~Jy~beam^{-1}$ and the resolution is 10\arcsec. The spectral
index across this SNR is uniform across with a value of $+0.12\pm
0.03$ and integrated flux density of 6~Jy. Also visible in the image
is the compact source called the ``northern knot'' about 2\arcmin\ to
the north. The long extended feature extending towards the north-west
direction is part of the separate catalogued SNR G21.6$+$0.84. The
third column contains the {\it L}-Band images at a reference frequency of
1.5~GHz, of the region centered on the Galactic co-ordinates
$l=19.6^\circ, b=-0.2^\circ$. This field contains a candidate SNR
G19.66$-$0.22 \citep{HelfandMAGPIS}. Stokes-{\it I} image shown in the top
row shows weak diffused emission surrounding two compact strong
sources, making it difficult to determine the nature of the objects
based on Stokes-{\it I} morphology alone. The spectral index image however
clearly separates sources of thermal and non-thermal emission. The
spectral index of the diffused emission range from $-0.4$ to $-0.5$.
The spectral index of the compact source (G19.67$-$0.15) towards the
norther edge of the diffused nebula has a spectral index of $-0.3$.
The superimposed compact thermal source near the center is the known
ultra-compact \HII region G19.61$-$0.23
(e.g., \cite{UCHII_MORPHOLOGIES,
Hofner_Churchwell_1996A&AS..120..283H}) also with a spectral index
of $+0.3$. Based on morphology of the spectral distribution we
conclude that this field contains an SNR, possibly of filled-centered
type.
\subsection{G55.7$+$3.4}
The SNR G55.7+3.4 is classified as an ``incomplete-shell SNR'' with a
pulsar within the boundary of the remnant based on the only existing
radio image at 610 and 1.415~GHz by \cite{Goss_G55.7+3.4}. These are
the only image published in the literature of this SNR made over three
decades ago using the Westerbork Synthesis Radio Telescope. The full
$2^\circ \times 2^\circ$ Stokes-{\it I} {\it L}-Band image centered on
this SNR at the reference frequency of 1.5~GHz covering the full
wide-band sensitivity pattern of the EVLA antenna is shown in
Figure~\ref{Fig:G55.7+3.4}. This is the highest resolution and
sensitivity image of this region to date. The wide-band sensitivity
pattern of the EVLA extended significantly beyond even the first
sidelobe of the PB at the lowest frequency in the band. {The
time-averaged wide-band sensitivity pattern of the EVLA for the
equivalent frequency range and field-of-view is shown in the left
panel of Figure~\ref{Fig:AvgPB}. The one-dimensional slice through
this pattern in the right panel, shows the sensitivity pattern
extending significantly beyond the ``main-lobe'' at the level of
$5-15\%$ of the peak value.} With sources, some strong, detected up
to $1^\circ$ away from the center of the image, imaging the full field
was required to reach noise-limited imaging at the center of the
field. Due to this wide field of view, even for D-array observations,
correction for the effects of non-coplanar baselines had to be
applied. This was done using the W-Projection algorithm
\citep{WProjection}. The Stokes-{\it I} image of the SNR shows a
nearly complete shell-type SNR. At a resolution of 30\arcsec, this
image for the first time, shows a network of filamentary structures
along with smoother diffused emission filling the boundary of the
shell. The spectral index image of this SNR however was not reliable
at all scales since the large scale emission is not well constrained
by the shortest measured baselines at all frequencies. The filaments
however show up as regions of steeper spectral index compared to an
average spectral index of $-0.6$ of the surrounding regions. The
strong compact source (G55.78$+$3.5) on the northern edge of the shell
in the Stokes-{\it I} image is an older unrelated pulsar (PSR
J1921$+$2153). In the spectral index map (not shown here), this
pulsar shows up as a steep spectrum compact source with a spectral
index of $-2.2$. The integrated flux of the shell, including the
pulsar and other foreground or background sources superimposed across
the shell is 1.0~Jy. The rms noise in the image is
10~$\mu Jy~beam^{-1}$.
\begin{figure*}
\includegraphics[width=17cm]{f2}
\caption{Stokes-{\it I} image at a reference frequency of
1.5~GHz. Data in the frequency range 1.256--1.905~GHz were
included for this image. For this frequency range, the HPBW of
the primary beam changes from 32\arcmin\ to 21\arcmin. This
$2^\circ \times 2^\circ$ image covers the full-field corresponding
to the wide-band sensitivity pattern of the EVLA ({see
Figure~\ref{Fig:AvgPB}}) of the field containing the SNR G55.7+3.4.}
\label{Fig:G55.7+3.4}
\end{figure*}
\begin{figure*}
\hbox{
\includegraphics[width=9cm]{f3a}
\includegraphics[width=9cm,height=8.2cm]{f3b}
}
\caption{The EVLA time-averaged wide-band sensitivity
pattern covering the same frequency range and the area shown in
Figure~\ref{Fig:G55.7+3.4}. The left panel shows the pattern in gray
scale with overlaid contours at 50\%, 25\%, 15\%, 10\% and 6\% of
the peak value. The right panel shows a one-dimentional cut through
the pattern shown on the left.}
\label{Fig:AvgPB}
\end{figure*}
\section{Discussion}
The technical goals of this pilot project were to assess our
low-frequency wide-band imaging capabilities for targets in the
Galactic plane. At {\it L}-Band and {\it C}-band, the Galaxy is replete with
compact and diffuse emission of both thermal and non-thermal
origin. Sources often have a mixture of thermal and non-thermal
emission, resulting in spectral indices that vary with position across
a given source and with spatial scale. Conventional imaging
algorithms that rely on multiple narrow-band images to derive
spectral-index information are limited to the angular resolution
allowed by the lowest measured frequency, and the single-channel
sensitivity. The MS-MFS algorithm was designed to optimally use the
multi-frequency measurements to do spectral-index mapping across
diffused sources {\it at the highest sensitivity and angular
resolution} offered by the data.
The following is a summary of our current status, based on the imaging
results shown in the previous section. Our targets were chosen based
on existing information about their spatial and spectral structure, in
order to test the analysis methods in preparation for a Galactic plane
survey.
\begin{enumerate}
\item Separation of thermal versus non-thermal; steep versus steeper
spectra : we see clear morphological separations between regions
expected to have different spectral structure from the
surroundings. These differences range from
$\triangle\alpha\approx0.6$ between thermal and non-thermal regions,
down to $\triangle\alpha\approx0.2$ between structures thought to be
cores and shells. In regions with high S/N ratios ($>100$), the
estimated uncertainty in the spectral index is $\pm0.03$. In
regions with low signal-to-noise ($<100$), the uncertainty in the
spectral index rises rapidly, and all the images presented here used
a Stokes-{\it I} S/N threshold of 100.
\item Image-reconstruction errors : errors in the spectral
reconstructions are dominated by multi-scale deconvolution artifacts
that decrease with appropriate choices of multi-scale imaging
parameters (a set of scale-sizes with which to model the spatial
structure). {Error estimates from Monte-Carlo simulations (by
varying the choice of spatial scales) range from $\pm0.02$ when
scales are chosen appropriately, up to 0.2 in the extreme case
where diffuse emission is modeled by a series of
$\delta$-functions. These simulations were done for the EVLA
D-configuration and the errors depend on the precise uv-coverage
just as they depend for the standard imaging algorithms. The
MS-MFS algorithm reconstructs the frequency dependence of the flux
by a polynomial fit across the frequency range on a per component
basis. The errors on the estimated spectral index therefore also
depend on the errors of the fitted coeffcients. These errors
depend on variations in S/N ratio (across the
frequency range or due to the scale of the emission) in the same
way as the errors on the coefficients of any least-squares
polynomial fitting algorithm (see
~\cite{Numerical_Recipes,MSMFCLEAN} for further details)}.
Further, at the largest spatial scales sampled by the interferometer
{(which, of course, is telescope dependent)}, the spectrum is
unconstrained by the data and the MS-MFS model does not provide
adequate additional constraints. Out of our examples, G55.7$+$3.4
and G16.7$+$0.1 contain structure at spatial scales that fall within
the unsampled central region of the interferometer sampling
function, aleading to overall systematic errors in the spectral
index across the source. These errors are limited to the largest
scales, but in the current implementation of the algorithm, it is
not possible to analyse the scales separately. In our examples,
G55.7$+$3.4 is affected significantly by this error and spectra of
only the most compact emission can be trusted. Finally, in addition
to errors in the spectral index, off-source artifacts are present in
the Stokes-{\it I} images beyond dynamic-ranges of a few thousand
(measured as the ratio of the peak brightness to the peak residual
near the source). Dynamic-ranges for the images presented here range
from 2000 for G16.7$+$0.1 to 10000 for G19.6$-$0.2. In most cases,
wide-band self-calibration was required to achieve these limits.
\item Wide-field wide-band sensitivity of the instrument, and related
errors : across a 2:1 bandwidth, the size of the antenna
power-pattern changes enough to produce an average beam that is
significantly different from the beams at individual
frequencies. The most-prominent difference is the near lack of a
first null, with sensitivity at the few percent level out to almost
three times the HPBW. The G55.7$+$3.4 field is an example of this
wide-field sensitivity pattern (the reference-frequency primary beam
has a HPBW of 30\arcmin). For sources far from the phase-center,
the $w$-term causes the dominant wide-field error, and these images
(in particular G55.7+3.4) are a result of W-Projection combined with
MS-MFS. The next dominant error (in Stokes-{\it I} and spectral index) is
due to the frequency dependence of the primary beam. All the
spectral-index maps shown here have been corrected for this effect
using the time-averaged PB-model, and can be trusted out to the 25\%
point of the reference-frequency primary-beam. The time-variability
of the primary-beams has not been accounted for, and is thought to
be the cause of the next dominant error (visible as artifacts around
bright sources about $1^\circ$ from the phase center in
G55.7$+$3.4).
\end{enumerate}
The next steps in our analysis include combining multi-configuration
and multi-band data for the G16.7$+$0.1 and G19.6$-$0.2 fields to
introduce more constraints during image-reconstruction, to complete
the software integration required to combine MS-MFS and W-Projection
with the A-Projection algorithm to correct for time-varying
primary-beams, and further tests with mosaicking observations and how
the additional information aids the image reconstruction. These steps
are in accordance with our long-term goal of doing a wide-band
mosaiced survey of the Galactic plane at low frequencies.
\acknowledgments
We used the ATNF Pulsar Catalogue \citep{PSRCat2005}, accessible via
the URL {\tt http://www.atnf.csiro.au/people/pulsar/psrcat/} for the
co-ordinates of the pulsar in the G55.7$+$3.4 field.
|
2,877,628,090,252 | arxiv | \section{Introduction}
In the last decades, there has been a lot of progress on the two-dimensional steady water wave problem with vorticity (see for example \cite{C, EEW2011, Varholm20, Wahlen06, WahlenWeber21} and references therein). The corresponding three-dimensional problem is significantly more challenging, due to the lack of a general formulation which is amenable to methods from nonlinear functional analysis. This is related to the fact that in two dimensions, the vorticity is a scalar field which is constant along streamlines, while in three dimensions it is a vector field which satisfies the vorticity equation, including the vortex stretching term. One approach to at least gain some insight is to investigate flows under under certain geometrical assumptions to fill the gap between two-dimensional and three-dimensional flows. This is one of the motivations for studying the axisymmetric Euler equations, which in many ways behave like the two-dimensional equations. Indeed, for the time-dependent problem, in the swirl-free case, these possess a global existence theory for smooth solutions similar to two-dimensional flows; see \cite{Abidi, UkhovskiiIudovich68} and references therein (note however the recent remarkable result \cite{Elgindi21} on singularity formation of non-smooth solutions). The steady axisymmetric problem is also of considerable physical importance, as it can be used to model phenomena such as jets, cavitational flows, bubbles and vortex rings (see for example \cite{AltCaffarelliFriedman83, CaffarelliFriedman82, CaoWanZhan21, DoakVandenBroeck18, FraenkelBerger74, Friedman83, Ni, Saffman92, VarvarucaWeiss14} and references therein).
In this paper, we study axisymmetric water waves with surface tension, modelled by assuming that the domain is bounded by a free surface on which capillary forces are acting, and that in cylindrical coordinates $(r,\vartheta,z)$ the domain and flow are independent of the azimuthal variable $\vartheta$.
In the irrotational and swirl-free setting, such waves were studied numerically by Vanden-Broeck et al.~\cite{VB-M-S} and Osborne and Forbes \cite{OsborneForbes01}, who found similarities to two-dimensional capillary waves, including overhanging profiles and limiting configurations with trapped bubbles at their troughs.
The small-amplitude theory is intimately connected to Rayleigh's instability criterion for a liquid jet \cite{Rayleigh1879} (see also \cite{HancockBush02, VB-M-S}), which says that a circular capillary jet is unstable to perturbations whose wavelength exceed the circumference of the jet. Indeed, this instability criterion is satisfied precisely when the dispersion relation for small-amplitude waves has purely imaginary solutions, while steady waves are obtained when the solutions are real \cite{VB-M-S} (that is, for smaller wavelengths). According to Hancock and Bush \cite{HancockBush02} a stationary form of such steady waves may be observed at the base of a jet which is impacting on a reservoir of the same fluid. If the reservoir is contaminated, the wave field is moved up the jet and a so called `fluid pipe' with a quiescent surface is formed at the base. We also note that in recent years there has been increased interest in waves on jets in other physical contexts, such as electrohydrodynamic flows \cite{GrandisonEtAl08} and ferrofluids \cite{BlythParau14, DoakVandenBroeck19, GrovesNilsson18}.
In this paper, we consider liquid jets with both vorticity and swirl. A motivation for this is that a viscous boundary layer in a pipe typically gives rise to vorticity, which may have a significant effect on the jet flowing out of the pipe. As an idealisation, we assume that the jet extends indefinitely in the $z$-direction and ignore viscosity and gravity.
In the irrotational swirl-free case, the problem can be formulated in terms of a harmonic velocity potential.
In contrast, we formulate the problem in terms of Stokes' stream function, which satisfies a second-order semilinear elliptic equation known alternatively in the literature as the \textit{Hicks equation}, the \textit{Bragg--Hawthorne equation} or the \textit{Squire--Long equation}, cf.~\cite{Saffman92}. This equation is also known from plasma physics as the \textit{Grad--Shafranov equation}, cf. \cite{ConstantinP}. The first aim of the paper is to construct small-amplitude solutions using local bifurcation theory in this more general context. In contrast to \cite{VB-M-S}, this means that the bifurcation conditions are much less explicit and that we require qualitative methods. The second aim is to construct large-amplitude solutions using global bifurcation theory and a reformulation of the problem inspired by the recent paper \cite{WahlenWeber21} on the two-dimensional gravity-capillary water wave problem with vorticity.
We now describe the plan of the paper. First in Section~\ref{sec:description} we start by introducing the main problem we are going to study. This means we start with the incompressible Euler equations and recall its axisymmetric version. In Section~\ref{sec:preliminaries}, we discuss regularity issues and trivial solutions of the axisymmetric incompressible Euler equations. Regarding regularity issues and in order to reformulate the problem in a secure functional-analytic setting, we avoid coordinate-induced singularities by introducing a new variable in terms of the Stokes stream function and view it (partly) as a function on five-dimensional space; this trick to overcome this kind of coordinate singularities is well-known and goes back to Ni \cite{Ni}. Then, we study local bifurcations in Section~\ref{sec:localbif} in the spirit of the theorem by Crandall--Rabinowitz, mainly by introducing the so-called good unknown; the main result of this section is Theorem \ref{thm:LocalBifurcation}. In addition, in Section~\ref{sec:conditions} we take a closer look at the conditions for local bifurcation. First we establish spectral properties of the corresponding Sturm--Liouville problem of limit-point type and with boundary condition dependent on the eigenvalue. After that, we investigate some specific examples in more detail. Finally, in Section~\ref{sec:GlobalBifurcation} we close the paper by investigating global bifurcations; see Theorem \ref{thm:GlobalBifurcation}.
Since we require the radius $r$ to be a graph of the longitudinal position $z$ along the water surface, our theoretical framework, in contrast to \cite{WahlenWeber21}, does not allow for overhanging waves, and we leave it to further research to include this possibility. This would clearly be a desirable extension in view of the numerical results in \cite{OsborneForbes01, VB-M-S}.
\section{Description of the problem and the governing equations}\label{sec:description}
We consider periodic axisymmetric capillary waves travelling at constant speed along the $z$ axis. The fluid is assumed to be inviscid and incompressible. In a frame moving with the wave, the flow is therefore governed by the steady incompressible Euler equations
\begin{align}
\begin{split}
(\vec{u}\cdot\nabla)\vec{u}&=-\nabla p,
\\
\hfill\nabla\cdot\vec{u}&=0,
\end{split}\quad \vec{x}=(x,y,z)^T\in\Omega\subseteq\mathbb R^3\label{Euler}
\end{align}
where $\vec{u}=\vec{u}(\vec{x})$ and $p=p(\vec{x})$ denote the velocity and the pressure, respectively, and $\Omega$ is the fluid domain.
In cylindrical coordinates $(r,\vartheta,z)$, that is, $x=r\cos\vartheta$, $y=r\sin\vartheta$ and $z=z$, the velocity field $\vec{u}$ is expressed as
$$
\vec{u}=u^r(r,z)\vec{e}_r+u^\vartheta(r,z)\vec{e}_\vartheta+u^z(r,z)\vec{e}_z,
$$
where the vectors
$$
\vec{e}_r=\left(\frac{x}{r},\frac{y}{r},0\right)^T,\quad\vec{e}_\vartheta=\left(-\frac{y}{r},\frac{x}{r},0\right)^T\quad\text{and}\quad\vec{e}_z=\left(0,0,1\right)^T
$$
form an orthonormal basis. Note that we allow for non-zero swirl, $u^\vartheta\ne 0$. From the incompressibility and the axisymmetry of the flow it follows that we can introduce Stokes' stream function $\Psi(r,z)$, such that
$$
u^r=\frac{1}{r}\frac{\partial\Psi}{\partial z}\quad\text{and}\quad u^z=-\frac{1}{r}\frac{\partial\Psi}{\partial r}.
$$
Moreover, the quantity $ru^\vartheta$ is constant along streamlines, which we express as $ru^\vartheta=F(\Psi)$ where $F$ is an arbitrary function. The steady Euler equations are then equivalent to the \textit{Bragg--Hawthorne equation}
$$
-\Delta^*\Psi=r^2\gamma(\Psi)+F(\Psi)F'(\Psi),
$$
where $\gamma$ is an arbitrary function and
\[
\Delta^*\Psi:=\Psi_{rr}-\frac1r\Psi_r+\Psi_{zz}
\]
cf.~\cite[Chapter 3.13]{Saffman92}.
Note that the corresponding vorticity vector is given by
\begin{align*}
\vec{\omega}&=
-\frac{\partial u^\vartheta}{\partial z}\vec{e}_r
+\left(\frac{\partial u^r}{\partial z}-\frac{\partial u^z}{\partial r}
\right) \vec{e}_\vartheta+\frac{1}{r} \frac{\partial (r u^\vartheta)}{\partial r}\vec{e}_z,
\\
&=-\frac{1}{r} F'(\Psi)\Psi_z \vec{e}_r+\frac1{r} \Delta^* \Psi \vec{e}_\vartheta+\frac{1}{r} F'(\Psi)\Psi_r\vec{e}_z.
\end{align*}
We next consider the boundary conditions. Assume that the fluid domain is given by $\Omega=\{(r,z)\in\mathbb R^2:~0<r<d+\eta(z)\}$ and its boundaries by $\partial\Omega_\mathcal{S}=\{(r,z)\in\mathbb R^2:~r=d+\eta(z)\}$ (free surface) and $\partial\Omega_\mathcal{C}=\{(r,z)\in\mathbb R^2:~r=0\}$ (center line). Although the latter could be considered as part of the domain, it is sometimes convenient to consider it as a boundary due to the appearance of inverse powers of $r$ in the equations. On the free surface $r=d+\eta(z)$
we have the kinematic boundary condition $\vec{u}\cdot\vec{n}=0$, where $\vec{n}=\vec{e}_r - \eta'(z) \vec{e}_z$ denotes a normal vector.
Expressed in terms of $\Psi$, this takes the form $\Psi_z+\eta_z\Psi_r=0$ on $\partial\Omega_\mathcal{S}$.
In addition, we have the dynamic boundary condition $p=-\sigma\kappa$ on $\partial\Omega_\mathcal{S}$
where
\[\kappa=\kappa[\eta]=\frac{\eta_{zz}}{(1+\eta_z^2)^{3/2}}-\frac{1}{(d+\eta)\sqrt{1+\eta_z^2}}\]
is the mean curvature of $\partial\Omega_\mathcal{S}$ and $\sigma>0$ is the coefficient of surface tension. Using Bernoulli's law we can eliminate the pressure and express this as
$$
\frac{\Psi_r^2+\Psi_z^2+F(\Psi)^2}{2r^2}-\sigma\kappa=Q
$$
on $\partial\Omega_\mathcal{S}$, where $Q$ is the Bernoulli constant.
At the center line $\partial\Omega_\mathcal{C}$ the identity $\Psi_z=ru^r$ shows that $\Psi_z=0$.
Summarising, we have following boundary value problem:
\begin{equation}
\begin{aligned}
\Delta^*\Psi+r^2\gamma(\Psi)+F(\Psi)F'(\Psi)&=0 && \text{in } \Omega,
\\
\frac{\Psi_r^2+\Psi_z^2+F(\Psi)^2}{2r^2}-\sigma\kappa&=Q &&\text{on }\partial\Omega_\mathcal{S},
\\
\Psi_z+\eta_z\Psi_r&=0 && \text{on } \partial\Omega_\mathcal{S},
\\
\Psi_z&=0 && \text{on } \partial\Omega_\mathcal{C},
\end{aligned}
\label{stationary_axisymmetric_problem}
\end{equation}
where $F$ and $\gamma$ are arbitrary functions of $\Psi$.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{The equations} The last two boundary conditions in \eqref{stationary_axisymmetric_problem} mean that $\Psi$ is constant on both $\partial\Omega_\mathcal{S}$ and $\partial\Omega_\mathcal{C}$. We normalise $\Psi$ such that it vanishes on $\partial\Omega_\mathcal{C}$ and assign the name $m$ to its value on $\partial\Omega_\mathcal{S}$. Thus, we shall deal with the equations
\begin{subequations}
\begin{align}
\Psi_{rr}-\frac1r\Psi_r+\Psi_{zz}&=-r^2\gamma(\Psi)-F(\Psi)F'(\Psi) &&\text{in }\Omega,\\
\frac{\Psi_r^2+\Psi_z^2+F(\Psi)^2}{2r^2}-\sigma\kappa&=Q && \text{on }\partial\Omega_\mathcal{S},\\
\Psi&=m &&\text{on }\partial\Omega_\mathcal{S},\\
\Psi&=0 &&\text{on }\partial\Omega_\mathcal{C},\label{eq:Psi_r=0}
\end{align}
\end{subequations}
where $Q$ and $m$ are constants. The fluid velocity is given by
\begin{align}\label{eq:relation_velocity_Psi}
\vec{u}=\frac{F(\Psi)}{r}\vec{e}_\vartheta-\nabla\times(\Psi\vec{e}_\vartheta/r)=\frac{\Psi_z}{r}\vec{e}_r+\frac{F(\Psi)}{r}\vec{e}_\vartheta-\frac{\Psi_r}{r}\vec{e}_z.
\end{align}
Following a trick of Ni \cite{Ni}, we introduce the function $\psi$ via
\begin{align}\label{eq:relation_Psi_psi}
\Psi=r^2\psi,
\end{align}
In terms of $\psi$, the equations read
\begin{subequations}\label{eq:OriginalEquation_psi}
\begin{align}
\psi_{rr}+\frac3r\psi_r+\psi_{zz}&=-\gamma(r^2\psi)-\frac{1}{r^2}F(r^2\psi)F'(r^2\psi)&&\text{in }\Omega,\label{eq:OriginalEquation_psi_PDE}\\
\frac{r^2(\psi_r^2+\psi_z^2)}{2}+\frac{F(r^2\psi)^2}{2r^2}+\frac{2m\psi_r}{r}+\frac{2m^2}{r^4}-\sigma\kappa&=Q &&\text{on }\partial\Omega_\mathcal{S},\label{eq:OriginalEquation_psi_Bernoulli}\\
\psi&=\frac{m}{r^2} &&\text{on }\partial\Omega_\mathcal{S}.\label{eq:OriginalEquation_psi_top}
\end{align}
\end{subequations}
Notice that we no longer need to impose a condition on $r=0$, provided $\psi$ is continuous at $r=0$, since then \eqref{eq:Psi_r=0} is automatically satisfied for $\Psi$ given by \eqref{eq:relation_Psi_psi}.
\subsection{Regularity issues}
Quite naturally, the fluid velocity $\vec{u}$ should be at least of class $C^1$ (in Cartesian coordinates). Written in terms of $\psi$, \eqref{eq:relation_velocity_Psi} reads
\begin{align}\label{eq:relation_velocity_psi}
\vec{u}=\frac{F(r^2\psi)}{r}\vec{e}_\vartheta-\nabla\times(r\psi\vec{e}_\vartheta)=r\psi_z\vec{e}_r+\frac{F(r^2\psi)}{r}\vec{e}_\vartheta-(2\psi+r\psi_r)\vec{e}_z.
\end{align}
Due to \cite{Liu2009}, $\vec{u}$ is of class $C^1$ provided $F(r^2\psi)/r$ is of class $C^1$ and $r\psi$ is of class $C^2$, both viewed as functions on $\{(r,z)\in[0,\infty)\times\mathbb R:r\le d+\eta(z)\}$, and, moreover, $F(r^2\psi)/r$, $r\psi$, and $(r\psi)_{rr}$ vanish at $r=0$. In view of
\[(r\psi)_r=\psi+r\psi_r,\quad(r\psi)_{rr}=2\psi_r+r\psi_{rr},\]
and
\[(F(r^2\psi)/r)_r=F'(r^2\psi)(2\psi+r\psi_r)-F(r^2\psi)/r^2,\]
it is therefore sufficient to assume
\begin{align}\label{eq:regularity_condition_psi}
\psi\in C^2(\overline\Omega),\quad \psi_r\big|_{r=0}=0,\
\end{align}
and
\[F\in C^1(\mathbb R),\quad F(0)=0.\]
Furthermore, we shall need that the right-hand side of \eqref{eq:OriginalEquation_psi_PDE} is in a Hölder class $C^{0,\alpha}$ if $\psi$ is $C^{0,\alpha}$. To this end, it is sufficient that both $F'$ and $G(x)\coloneqq F(x)/x$ (continuously extended to $x=0$ by $G(0)\coloneqq F'(0)$) are locally Lipschitz continuous in view of
\begin{align}\label{eq:relFG}
\frac{1}{r^2}F(r^2\psi)=G(r^2\psi)\psi.
\end{align}
Moreover, the nonlinear operator $\mathcal F$ introduced later should be of class $C^2$. Hence, we need that $\gamma$, $F$, and $F'$ are locally of class $C^{2,1}$; notice that this condition on $F$ already implies the desired property of $G$ as above. Also, in order to construct trivial solutions, we will need a Lipschitz property of $\gamma$ and $FF'$. Overall we impose the following assumptions on $\gamma$ and $F$:
\begin{align}\label{eq:assumptions_gamma_F}
\gamma\in C_{\text{loc}}^{2,1}(\mathbb R),\quad F\in C_{\text{loc}}^{3,1}(\mathbb R),\quad\|\gamma'\|_\infty<\infty,\quad\|(FF')'\|_\infty<\infty,\quad F(0)=0.
\end{align}
\subsection{Trivial solutions}\label{sec:trivial}
We now have a look at trivial solutions of \eqref{eq:OriginalEquation_psi}, that is, solutions of \eqref{eq:OriginalEquation_psi} independent of $z$. Therefore, we consider the (singular) Cauchy problem
\begin{subequations}\label{eq:trivial}
\begin{align}
\psi_{rr}+\frac3r\psi_r&=-\gamma(r^2\psi)-\frac{1}{r^2}F(r^2\psi)F'(r^2\psi)\quad\text{on }(0,d],\label{eq:trivial_ode}\\
\psi(0)&=\lambda,\label{eq:trivial_initial_value}\\
\psi_r(0)&=0.\label{eq:trivial_initial_derivative}
\end{align}
\end{subequations}
Here, $\lambda\in\mathbb R$ is a parameter, which will later serve as the bifurcation parameter, and \eqref{eq:trivial_initial_derivative} is imposed due to \eqref{eq:regularity_condition_psi}. Notice that, in view of \eqref{eq:relation_velocity_psi}, there is a one-to-one correspondence of the parameter $\lambda$ and the velocity at the symmetry axis via $\vec{u}=-2\lambda\vec{e}_z$ at $r=0$.
In order to solve \eqref{eq:trivial}, we rewrite \eqref{eq:trivial}, making use of \eqref{eq:trivial_ode}, \eqref{eq:trivial_initial_value}, and $\partial_{rr}+\frac{3}{r}\partial_r=r^{-3}\partial_r(r^3\partial_r)$, as the integral equation
\begin{align}\label{eq:trivial_integral_equation}
\psi(r)=\lambda-\int_0^rt^{-3}\int_0^t\left(s^3\gamma(s^2\psi(s))+s(FF')(s^2\psi(s))\right)\,ds\,dt.
\end{align}
By Lipschitz continuity of $\gamma$ and $FF'$, it is straightforward to see that the right-hand side of \eqref{eq:trivial_integral_equation} gives rise to a contraction on $C([0,\varepsilon])$ if $\varepsilon>0$ is small enough. Thus, \eqref{eq:trivial_integral_equation} has a unique continuous solution on such $[0,\varepsilon]$ by Banach's fixed point theorem. It is clear that, by virtue of \eqref{eq:relFG} and \eqref{eq:trivial_integral_equation}, this solution is of class $C^2$ on $[0,\varepsilon]$, and satisfies \eqref{eq:trivial_ode} on $(0,\varepsilon]$ and \eqref{eq:trivial_initial_value}, \eqref{eq:trivial_initial_derivative}. Now, once having left the singular point $r=0$, it is obvious that $\psi$ can be uniquely extended to a $C^2$-solution of \eqref{eq:trivial_ode} on $(0,d]$, since $\gamma$ and $FF'$ are Lipschitz continuous. Moreover, $\psi\in C^{2,1}([0,d])$ in view of \eqref{eq:trivial_ode}, \eqref{eq:trivial_initial_derivative}.
Finally, motivated by the flattening considered below, we define
\begin{align}\label{eq:psi^lambda}
\psi^\lambda(s)\coloneqq\psi(sd),\quad s\in[0,1]
\end{align}
where $\psi$ is the unique solution of \eqref{eq:trivial} as obtained above.
\subsection{Working in 5D and flattening}
In the following, for a function $\psi=\psi(r,z)$ on some $\Omega\subset\mathbb R^2$ we denote by $\mathcal I\psi$ the function given by
\[\mathcal I\psi(x,z)=\psi(|x|,z)\]
and defined on the set $\Omega^\mathcal I$, which results from rotating $\Omega$ around the $z$-axis in $\mathbb R^5=\{(x,z)\in\mathbb R^4\times\mathbb R\}$. Conversely, any axially symmetric set in $\mathbb R^5$ can be written as $\Omega^\mathcal I$ for a suitable $\Omega\subset\mathbb R^2$, and any axially symmetric function $\tilde\psi$ on $\Omega^\mathcal I$ equals $\mathcal I\psi$ for a certain function $\psi$ on $\Omega$, i.e, $\psi=\mathcal I^{-1}\tilde\psi$, where $\mathcal I^{-1}$ is defined on the set of axially symmetric functions. Thus, it is easy to see that $\psi$ satisfies \eqref{eq:regularity_condition_psi} and solves \eqref{eq:OriginalEquation_psi_PDE}, \eqref{eq:OriginalEquation_psi_top} if and only if $\mathcal I\psi\in C^2(\overline{\Omega^\mathcal I})$ and solves
\begin{subequations}\label{eq:PDE_in_5D}
\begin{align}
\Delta_5\mathcal I\psi&=-\gamma(|x|^2\mathcal I\psi)-\frac{1}{|x|^2}F(|x|^2\mathcal I\psi)F'(|x|^2\mathcal I\psi)&\text{in }\Omega^\mathcal I,\\
\mathcal I\psi&=\frac{m}{|x|^2}&\text{on }\partial\Omega_\mathcal{S}^\mathcal I,
\end{align}
\end{subequations}
with $\Delta_5$ denoting the Laplacian in five dimensions. Therefore, no longer a term which is singular on the symmetry axis appears (cf. \eqref{eq:relFG}) -- this is the main motivation for working with $\psi$ instead of $\Psi$. In order to transform \eqref{eq:PDE_in_5D} into a fixed domain, we consider the flattening $(x,z)\mapsto(y,z)=(x/(d+\eta(z)),z)$ -- from now on, we shall always assume that $\eta>-d$. Thus, introducing $\tilde\psi$ via $\tilde\psi(y,z)=\mathcal I\psi(x,z)$, \eqref{eq:PDE_in_5D} is transformed into
\begin{subequations}\label{eq:PDE+Dirichlet_flattened_tildepsi}
\begin{align}
L^\eta\tilde\psi&=-\gamma((d+\eta)^2|y|^2\tilde\psi)-\frac{1}{(d+\eta)^2|y|^2}(FF')((d+\eta)^2|y|^2\tilde\psi)&\text{in }\Omega_0^\mathcal I,\\
\tilde\psi&=\frac{m}{(d+\eta)^2}&\text{on }|y|=1,
\end{align}
\end{subequations}
where $\Omega_0\coloneqq [0,1)\times\mathbb R$ and
\begin{align*}
L^\eta\tilde\psi\coloneqq\tilde\psi_{zz}+\frac{1}{(d+\eta)^2}\left(\tilde\psi_{y_iy_i}-2(d+\eta)\eta_zy_i\tilde\psi_{y_iz}+\eta_z^2y_iy_j\tilde\psi_{y_iy_j}-((d+\eta)\eta_{zz}-2\eta_z^2)y_i\tilde\psi_{y_i}\right);
\end{align*}
here and throughout this paper, repeated indices are summed over. It is straightforward to see that $L^\eta$ is a uniformly elliptic operator, provided $d+\eta$ is uniformly bounded from below by a positive constant.
As for Bernoulli's equation \eqref{eq:OriginalEquation_psi_Bernoulli}, we do not have to take a detour and increase the dimension, since in \eqref{eq:OriginalEquation_psi_Bernoulli} no singular term appears, at least whenever the surface does not intersect the symmetry axis. Therefore, here we consider the flattening \[H[\eta]^{-1}\colon\Omega\to\Omega_0,\quad(r,z)\mapsto(s,z)=(r/(d+\eta(z)),z);\]
we shall call $H[\eta]$ the inverse map. Then, with $\bar\psi(s,z)=\psi(r,z)$, that is,
\begin{align}\label{eq:barpsi_tildepsi}
\mathcal I\bar\psi=\tilde\psi,
\end{align}
\eqref{eq:OriginalEquation_psi_Bernoulli} is transformed into
\begin{align}\label{eq:Bernoulli_flattened_barpsi}
\frac{\bar\psi_s^2+((d+\eta)\bar\psi_z-\eta_z\bar\psi_s)^2}{2}+\frac{F((d+\eta)^2\bar\psi)^2}{2(d+\eta)^2}+\frac{2m\bar\psi_s}{(d+\eta)^2}+\frac{2m^2}{(d+\eta)^4}-\sigma\kappa[\eta]=Q\quad\text{on }s=1.
\end{align}
\subsection{Reformulation}
For later reasons, it is convenient to work with functions $\phi$ satisfying $\phi=0$ on $s=1$ instead of functions $\bar\psi$ with variable boundary condition at $s=1$. Thus, we introduce, for any $\lambda\in\mathbb R$, the function \[\phi=\bar\psi-\frac{d^2}{(d+\eta)^2}\psi^\lambda.\]
In terms of $\phi$, \eqref{eq:PDE+Dirichlet_flattened_tildepsi} and \eqref{eq:Bernoulli_flattened_barpsi}, furnished with \eqref{eq:barpsi_tildepsi} and $m=m(\lambda)\coloneqq d^2\psi^\lambda(1)$, read
\begin{subequations}\label{eq:PDE+Dirichlet_flattened}
\begin{align}
L^\eta\mathcal I\phi&=-\gamma\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\nonumber\\
&\phantom{=\;}-\frac{1}{(d+\eta)^2|y|^2}(FF')\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)-L^\eta\frac{d^2\mathcal I\psi^\lambda}{(d+\eta)^2}&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\phi&=0&\text{on }|y|=1,
\end{align}
\end{subequations}
and
\begin{align}\label{eq:Bernoulli_flattened}
&\frac{\left(\phi_s+\frac{d^2}{(d+\eta)^2}\psi^\lambda_s\right)^2+\left((d+\eta)\phi_z-\eta_z\left(\phi_s+\frac{2m(\lambda)+d^2\psi^\lambda_s}{(d+\eta)^2}\right)\right)^2}{2}\nonumber\\
&+\frac{F\left((d+\eta)^2\left(\phi+\frac{m(\lambda)}{(d+\eta)^2}\right)\right)^2}{2(d+\eta)^2}+\frac{2m(\lambda)\left(\phi_s+\frac{d^2}{(d+\eta)^2}\psi^\lambda_s\right)}{(d+\eta)^2}+\frac{2m(\lambda)^2}{(d+\eta)^4}-\sigma\kappa[\eta]=Q\quad\text{ on }s=1.
\end{align}
Henceforth, we search for solutions $(\lambda,\eta,\phi)$ of \eqref{eq:PDE+Dirichlet_flattened}, \eqref{eq:Bernoulli_flattened}.
Our goal is to rewrite \eqref{eq:PDE+Dirichlet_flattened}, \eqref{eq:Bernoulli_flattened} in the form \enquote{identity plus compact}, namely, as $(\eta,\phi)=\mathcal M(\lambda,\eta,\phi)$ with $\mathcal M$ compact. Meanwhile, we shall also clarify what $Q$ exactly is, namely, we define it as an expression in $(\lambda,\eta,\phi)$. To this end, we first fix $0<\alpha<1$ and introduce the Banach space
\[X\coloneqq\left\{(\eta,\phi)\in C_{0,\mathrm{per},\mathrm{e}}^{2,\alpha}(\mathbb R)\times C_{\mathrm{per},\mathrm{e}}^{0,\alpha}(\overline{\Omega_0}):\phi=0\text{ on }s=1,\ \mathcal I\phi\in H^1_{\mathrm{per},\mathrm{e}}(\Omega_0^\mathcal I)\right\},\]
equipped with the canonical norm
\[\|(\eta,\phi)\|_X=\|\eta\|_{C^{2,\alpha}_\mathrm{per}(\mathbb R)}+\|\phi\|_{C^{0,\alpha}_\mathrm{per}(\Omega_0)}+\|\mathcal I\phi\|_{H^1_\mathrm{per}(\Omega_0^\mathcal I)}.\]
Here, the indices \enquote{$\mathrm{per}$}, \enquote{$\mathrm{e}$}, and \enquote{$0$} denote $L$-periodicity ($\nu\coloneqq 2\pi/L$ in the following), evenness (in $z$ with respect to $z=0$), and zero average over one period.
First, for
\[(\lambda,\eta,\phi)\in\mathbb R\times\mathcal U\coloneqq\{(\lambda,\eta,\phi)\in\mathbb R\times X:d+\eta>0\text{ on }[0,L]\},\]
we let $\mathcal A(\lambda,\eta,\phi)=\mathcal I^{-1}\varphi$, where $\varphi\in C^{2,\alpha}(\overline{\Omega_0^\mathcal I})$ is the unique solution of
\begin{subequations}\label{eq:A_system}
\begin{align}
L^\eta\varphi&=-L^\eta\frac{d^2\mathcal I\psi^\lambda}{(d+\eta)^2}-\gamma\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\nonumber\\
&\phantom{=\;}-\frac{1}{(d+\eta)^2|y|^2}(FF')\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)&\text{in }\Omega_0^\mathcal I,\label{eq:A_PDE}\\
\varphi&=0&\text{on }|y|=1;
\end{align}
\end{subequations}
here, notice that the right-hand side of \eqref{eq:A_PDE} is an element of $C^{0,\alpha}(\overline{\Omega_0^\mathcal I})$ (cf. \eqref{eq:relFG} and the discussion there) and that \eqref{eq:A_system} is invariant under rotations about the $z$-axis, so that $\varphi$ has to be axially symmetric.
Second, we rewrite \eqref{eq:Bernoulli_flattened} as an equation for $\eta_{zz}$, using $\mathcal A=\mathcal A(\lambda,\eta,\phi)$ instead of $\phi$ -- notice that this change does not affect the equivalence of the whole reformulation to the original equations since clearly \eqref{eq:PDE+Dirichlet_flattened} is equivalent to $\phi=\mathcal A(\lambda,\eta,\phi)$:
\begin{align*}
\eta_{zz}&=\sigma^{-1}(1+\eta_z^2)^{3/2}\Bigg(\frac{\sigma}{(d+\eta)\sqrt{1+\eta_z^2}}+\frac{\left(\mathcal A_s+\frac{d^2}{(d+\eta)^2}\psi^\lambda_s\right)^2+\left((d+\eta)\mathcal A_z-\eta_z\left(\mathcal A_s+\frac{2m(\lambda)+d^2\psi^\lambda_s}{(d+\eta)^2}\right)\right)^2}{2}\\
&\omit\hfill$\displaystyle+\frac{F\left((d+\eta)^2\left(\mathcal A+\frac{m(\lambda)}{(d+\eta)^2}\right)\right)^2}{2(d+\eta)^2}+\frac{2m(\lambda)\left(\mathcal A_s+\frac{d^2}{(d+\eta)^2}\psi^\lambda_s\right)}{(d+\eta)^2}+\frac{2m(\lambda)^2}{(d+\eta)^4}-Q\Bigg)$
\end{align*}
on $s=1$. In order to apply $\partial_z^{-2}\colon C_{0,\mathrm{per}}^{0,\alpha}(\mathbb R)\to C_{0,\mathrm{per}}^{2,\alpha}(\mathbb R)$, the inverse operation to twice differentiation, to this relation, the right-hand side needs to have zero average over one period. Therefore, we view $Q$ as a function of $(\lambda,\eta,\phi)$ via
\begin{align*}
Q(\lambda,\eta,\phi)&\coloneqq\frac{1}{\langle(1+\eta_z^2)^{3/2}\rangle}\Bigg\langle(1+\eta_z^2)^{3/2}\Bigg(\frac{\sigma}{(d+\eta)\sqrt{1+\eta_z^2}}\\
&\omit\hfill$\displaystyle+\frac{\left(\mathcal S\mathcal A_s+\frac{d^2}{(d+\eta)^2}\mathcal S\psi^\lambda_s\right)^2+\left((d+\eta)\mathcal S\mathcal A_z-\eta_z\left(\mathcal S\mathcal A_s+\frac{2m(\lambda)+d^2\mathcal S\psi^\lambda_s}{(d+\eta)^2}\right)\right)^2}{2}$\\
&\omit\hfill$\phantom{\coloneqq\;}\displaystyle+\frac{F\left((d+\eta)^2\left(\mathcal S\mathcal A+\frac{m(\lambda)}{(d+\eta)^2}\right)\right)^2}{2(d+\eta)^2}+\frac{2m(\lambda)\left(\mathcal S\mathcal A_s+\frac{d^2}{(d+\eta)^2}\mathcal S\psi^\lambda_s\right)}{(d+\eta)^2}+\frac{2m(\lambda)^2}{(d+\eta)^4}\Bigg)\Bigg\rangle.$
\end{align*}
Here and in the following, $\langle f\rangle$ denotes the average of a $L$-periodic function $f$ over one period, and $\mathcal S f$ denotes the evaluation of a function $f$ at $s=1$.
Putting everything together, we reformulate \eqref{eq:PDE+Dirichlet_flattened}, \eqref{eq:Bernoulli_flattened} as
\begin{align}\label{eq:F=0}
\mathcal F(\lambda,\eta,\phi)=0
\end{align}
for $(\lambda,\eta,\phi)\in\mathbb R\times\mathcal U$, where
\[\mathcal F\colon\mathbb R\times\mathcal U\to X,\;\mathcal F(\lambda,\eta,\phi)=(\eta,\phi)-\mathcal M(\lambda,\eta,\phi),\]
with $\mathcal M=(\mathcal M^1,\mathcal M^2)$,
\begin{align*}
&\mathcal M^1(\lambda,\eta,\phi)\coloneqq\\
&\partial_z^{-2}\Bigg(\sigma^{-1}(1+\eta_z^2)^{3/2}\Bigg(\frac{\sigma}{(d+\eta)\sqrt{1+\eta_z^2}}\\
&\omit\hfill$\displaystyle+\frac{\left(\mathcal S\mathcal A_s+\frac{d^2}{(d+\eta)^2}\mathcal S\psi^\lambda_s\right)^2+\left((d+\eta)\mathcal S\mathcal A_z-\eta_z\left(\mathcal S\mathcal A_s+\frac{2m(\lambda)+d^2\mathcal S\psi^\lambda_s}{(d+\eta)^2}\right)\right)^2}{2}$\\
&\phantom{\partial_z^{-2}\Bigg(}+\frac{F\left((d+\eta)^2\left(\mathcal S\mathcal A+\frac{m(\lambda)}{(d+\eta)^2}\right)\right)^2}{2(d+\eta)^2}+\frac{2m(\lambda)\left(\mathcal S\mathcal A_s+\frac{d^2}{(d+\eta)^2}\mathcal S\psi^\lambda_s\right)}{(d+\eta)^2}+\frac{2m(\lambda)^2}{(d+\eta)^4}-Q(\lambda,\eta,\phi)\Bigg)\Bigg),\\
&\mathcal M^2(\lambda,\eta,\phi)\coloneqq\mathcal A(\lambda,\eta,\phi).
\end{align*}
Notice that $\mathcal F$ is well-defined; in particular, both $\mathcal M^1(\lambda,\eta,\phi)$ and $\mathcal M^2(\lambda,\eta,\phi)$ are periodic and even in $z$. We summarise our reformulation in the following lemma.
\begin{lemma}
A tuple $(\lambda,\eta,\phi)\in\mathbb R\times X$ satisfying $\eta>-d$ solves \eqref{eq:F=0} if and only if
\begin{enumerate}[label=(\roman*)]
\item $\eta$ and $\phi$ are of class $C^{2,\alpha}$; and
\item the tuple
\[(\eta,\psi)=\left(\eta,\left(\phi+\frac{d^2}{(d+\eta)^2}\psi^\lambda\right)\circ H[\eta]^{-1}\right)\]
solves \eqref{eq:OriginalEquation_psi} with $\Omega=H[\eta](\Omega_0)$, $Q=Q(\lambda,\eta,\phi)$, and $m=m(\lambda)$; and
\item $\psi\in C^{2,\alpha}_{\mathrm{per},\mathrm{e}}(\overline\Omega)$ and satisfies \eqref{eq:regularity_condition_psi}.
\end{enumerate}
\end{lemma}
\begin{proof}
We only need to take care of the regularity properties and \eqref{eq:regularity_condition_psi}. However, to this end it is sufficient to notice that $\eta\in C^{2,\alpha}(\mathbb R)$ and $\mathcal I\phi\in C^{2,\alpha}(\overline{\Omega_0^\mathcal I})$ provided $\mathcal F(\lambda,\eta,\phi)=0$. Indeed, we have $\mathcal I\psi\in C^{2,\alpha}(\overline{\Omega^\mathcal I})$ in this case since $\mathcal I\psi^\lambda\in C^{2,1}(\overline{\Omega_0^\mathcal I})$; in particular, $\psi$ satisfies \eqref{eq:regularity_condition_psi}.
\end{proof}
By construction, all points of the form $(\lambda,0,0)$ are solutions of \eqref{eq:F=0} -- they make up the curve of trivial solutions. An inspection of $\mathcal M$ shows the following; in particular, \eqref{eq:F=0} has the form \enquote{identity plus compact}.
\begin{lemma}\label{lma:M_prop}
$\mathcal M$ and thus $\mathcal F$ is of class $C^2$ on $\mathbb R\times\mathcal U$. Moreover, $\mathcal M$ is compact on
\[\mathbb R\times\mathcal U_\varepsilon\coloneqq\{(\lambda,\eta,\phi)\in\mathbb R\times X:d+\eta\ge\varepsilon\text{ on }\mathbb R\}\]
for each $\varepsilon>0$.
\end{lemma}
\begin{proof}
The other operations in the definition of $\mathcal M$ being smooth, the property that $\mathcal M$ is of class $C^2$ follows from the property that $\mathcal A$ is of class $C^2$; this, in turn, is guaranteed by the assumption \eqref{eq:assumptions_gamma_F}. Now let $(\lambda,\eta,\phi)\in\mathbb R\times\mathcal U_\varepsilon$ be arbitrary. In the following, the quantities $C$ can change from line to line, but are always shorthand for a certain expression in its arguments which remains bounded for bounded arguments. Moreover, let $R>0$ and suppose $\|(\lambda,\eta,\phi)\|_{\mathbb R\times X}\le R$. Since $\psi^\lambda$ is of class $C^1$ with respect to $\lambda$ and $L^\eta$ is elliptic uniformly in $\eta$ due to $\eta+d\ge\varepsilon$, we see that
\begin{align*}
\|\mathcal I\mathcal A(\lambda,\eta,\phi)\|_{C^{2,\alpha}_\mathrm{per}(\overline{\Omega_0^\mathcal I})}\le C\left(R,\varepsilon^{-1},\|\gamma'\|_{L^\infty([-C(R),C(R)])},\|GF'\|_{C^{0,1}([-C(R),C(R)])}\right)
\end{align*}
by applying a standard Schauder estimate. This shows that $\mathcal M^2$ is compact on $\mathbb R\times\mathcal U_\varepsilon$ because of the compact embedding of $C^{2,\alpha}_\mathrm{per}(\overline{\Omega_0^\mathcal I})$ in $H^1_\mathrm{per}(\Omega_0^\mathcal I)$ and in $C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0^\mathcal I})$ combined with \[\|f\|_{C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0})}\le\|\mathcal I f\|_{C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0^\mathcal I})},\quad f\in C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0}).\]
As for $\mathcal M^1$, we immediately find, in view of the obtained estimates for $\mathcal A$,
\begin{align*}
&\|\mathcal M^1(\lambda,w,\phi)\|_{C^{3,\alpha}([0,L])}\\
&\le C\left(R,\varepsilon^{-1},\|\gamma'\|_{L^\infty([-C(R),C(R)])},\|GF'\|_{C^{0,1}([-C(R),C(R)])},\|FF'\|_{L^\infty([-C(R),C(R)])}\right).
\end{align*}
Hence, also $\mathcal M_1$ is compact on $\mathbb R\times\mathcal U_\varepsilon$ since $C^{3,\alpha}([0,L])$ is compactly embedded in $C^{2,\alpha}([0,L])$.
\end{proof}
\section{Local bifurcation}\label{sec:localbif}
\subsection{Computing derivatives}
We now want to calculate the partial derivative $\mathcal F_{(\eta,\phi)}$ and, in particular, its evaluation at a trivial solution. For simplicity, we shall always write $\mathcal A_\eta$ for $\mathcal A_\eta(\lambda,\eta,\phi)\delta\eta$, that is, the partial derivative of $\mathcal A$ with respect to $\eta$ evaluated at $(\lambda,\eta,\phi)$ and applied to a direction $\delta\eta$. The same applies similarly to expressions like $\mathcal A_\phi$, $L_\eta^\eta$ etc.
Linearizing the operator $L^\eta$, which only depends on $\eta$ and not on $\phi$, leads to
\begin{align*}
L^\eta_\eta\varphi&=-\frac{2\delta\eta}{(d+\eta)^3}\left(\varphi_{y_iy_i}-2(d+\eta)\eta_zy_i\varphi_{y_iz}+\eta_z^2y_iy_j\varphi_{y_iy_j}-((d+\eta)\eta_{zz}-2\eta_z^2)y_i\varphi_{y_i}\right)\\
&\phantom{=\;}+\frac{1}{(d+\eta)^2}\Big(-2(\eta_z\delta\eta+(d+\eta)\delta\eta_z)y_i\varphi_{y_iz}+2\eta_z\delta\eta_zy_iy_j\varphi_{y_iy_j}\\
&\omit\hfill$\displaystyle-(\eta_{zz}\delta\eta+(d+\eta)\delta\eta_{zz}-4\eta_z\delta\eta_z)y_i\varphi_{y_i}\Big)$.
\end{align*}
Since formally linearizing an equation like $L\varphi=f$ gives $L\delta\varphi+\delta L\varphi=\delta f$, we see that $\mathcal I\mathcal A_\eta$ is the unique solution of
\begin{align*}
L^\eta\mathcal I\mathcal A_\eta&=-L^\eta_\eta\mathcal I\mathcal A-L^\eta_\eta\frac{d^2\mathcal I\psi^\lambda}{(d+\eta)^2}+2L^\eta\frac{d^2\mathcal I\psi^\lambda\delta\eta}{(d+\eta)^3}\\
&\phantom{=\;}-2\gamma'\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)(d+\eta)\delta\eta|y|^2\mathcal I\phi\\
&\phantom{=\;}+\frac{2\delta\eta}{(d+\eta)^3|y|^2}(FF')\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\\
&\phantom{=\;}-2(FF')'\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\frac{\delta\eta\mathcal I\phi}{d+\eta}&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal A_\eta&=0&\text{on }|y|=1.
\end{align*}
Similarly, $\mathcal I\mathcal A_\phi$ is the unique solution of
\begin{align*}
L^\eta\mathcal I\mathcal A_\phi&=-\gamma'\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)(d+\eta)^2|y|^2\mathcal I\delta\phi\\
&\phantom{=\;}-(FF')'\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\mathcal I\delta\phi&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal A_\phi&=0&\text{on }|y|=1.
\end{align*}
Evaluated at a trivial solution $(\lambda,0,0)$, we can simplify as follows:
\begin{align*}
L^\eta\varphi&=\varphi_{zz}+\frac{1}{d^2}\varphi_{y_iy_i},\\
L^\eta_\eta\varphi&=-\frac{2\delta\eta}{d^3}\varphi_{y_iy_i}-\frac{2\delta\eta_z}{d}y_i\varphi_{y_iz}-\frac{\delta\eta_{zz}}{d}y_i\varphi_{y_i}.
\end{align*}
In the following, we denote
\[\Delta_d\coloneqq\partial_{zz}+\frac{\partial_{y_iy_i}}{d^2}.\]
Moreover, since $\mathcal A=0$ here, we have
\begin{subequations}\label{eq:Aeta_tr}
\begin{align}
\Delta_d\mathcal I\mathcal A_\eta&=-L^\eta_\eta\mathcal I\psi^\lambda+\frac2d\Delta_d(\mathcal I\psi^\lambda\delta\eta)+\frac{2}{d^3|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\delta\eta\nonumber\\
&=\frac{4(\mathcal I\psi^\lambda)_{y_iy_i}}{d^3}\delta\eta+\frac{2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}}{d}\delta\eta_{zz}+\frac{2}{d^3|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\delta\eta&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal A_\eta&=0&\text{on }|y|=1,
\end{align}
\end{subequations}
and
\begin{subequations}\label{eq:Aphi_tr}
\begin{align}
\Delta_d\mathcal I\mathcal A_\phi&=-\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\delta\phi&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal A_\phi&=0&\text{on }|y|=1.
\end{align}
\end{subequations}
Next, we turn to $\mathcal M^1$. After a lengthy computation we get the following results for the partial derivatives of $\mathcal M^1$ evaluated at a trivial solution $(\lambda,0,0)$, noticing that $\mathcal S\mathcal A_\eta=\mathcal S\mathcal A_\phi=0$ at such points:
\begin{align*}
\mathcal M^1_\eta&=-\sigma^{-1}\left(\frac{\sigma}{d^2}+\frac{2}{d}(\mathcal S\psi^\lambda_s)^2+\frac{F(m(\lambda))^2}{d^3}+\frac{8m(\lambda)\mathcal S\psi^\lambda_s}{d^3}+\frac{8m(\lambda)^2}{d^5}\right)\partial_z^{-2}\delta\eta\\
&\phantom{=\;}+\sigma^{-1}\left(\mathcal S\psi^\lambda_s+\frac{2m(\lambda)}{d^2}\right)\partial_z^{-2}\mathcal P\mathcal S\mathcal A_{\eta s},\\
\mathcal M^1_\phi&=\sigma^{-1}\left(\mathcal S\psi^\lambda_s+\frac{2m(\lambda)}{d^2}\right)\partial_z^{-2}\mathcal P\mathcal S\mathcal A_{\phi s},
\end{align*}
where $\mathcal P$ is the projection onto the space of functions with zero average.
It will be convenient to introduce the abbreviation
\[c(\lambda)\coloneqq\mathcal S(2\psi^\lambda+s\psi^\lambda_s)=\mathcal S(2\psi^\lambda+\psi^\lambda_s)=\frac{2m(\lambda)}{d^2}+\mathcal S\psi^\lambda_s.\]
Notice that $-c(\lambda)$ is the $z$-component of the velocity at the surface of the trivial laminar flow corresponding to $\lambda$ in view of \eqref{eq:relation_velocity_psi}. With this, we can rewrite
\begin{align}
\mathcal M^1_\eta&=-\frac{1}{\sigma d^3}\left(\sigma d+2d^2c(\lambda)^2+F(m(\lambda))^2\right)\partial_z^{-2}\delta\eta+\sigma^{-1}c(\lambda)\partial_z^{-2}\mathcal P\mathcal S\mathcal A_{\eta s},\label{eq:M1eta_tr}\\
\mathcal M^1_\phi&=\sigma^{-1}c(\lambda)\partial_z^{-2}\mathcal P\mathcal S\mathcal A_{\phi s}\label{eq:M1phi_tr}.
\end{align}
\subsection{The good unknown}
Before we proceed with the investigation of local bifurcation, we first introduce an isomorphism, which facilitates the computations later and is sometimes called $\mathcal T$-isomorphism in the literature (for example, in \cite{EEW2011, Varholm20}). The discovery of the importance of such a new variable (here $\theta$) goes back to Alinhac \cite{Alinhac89}, who called it the \enquote{good unknown} in a very general context, and Lannes \cite{Lannes05}, who introduced it in the context of water wave equations.
\begin{lemma}
Let
\[Y\coloneqq\left\{\theta\in C_{\mathrm{per},\mathrm{e}}^{0,\alpha}(\overline{\Omega_0}):\mathcal S\theta\in C_{0,\mathrm{per},\mathrm{e}}^{2,\alpha}(\mathbb R),\ \mathcal I\theta\in H^1_{\mathrm{per},\mathrm{e}}(\Omega_0^\mathcal I)\right\}\]
and assume that $c(\lambda)\neq 0$. Then
\[\mathcal T(\lambda)\colon Y\to X,\quad\mathcal T(\lambda)\theta=\left(-\frac{d\mathcal S\theta}{c(\lambda)},\theta-\frac{2\psi^\lambda+s\psi_s^\lambda}{c(\lambda)}\mathcal S\theta\right)\]
is an isomorphism. Its inverse is given by
\[[\mathcal T(\lambda)]^{-1}(\delta \eta,\delta\phi)=\delta\phi-\frac{2\psi^\lambda+s\psi_s^\lambda}{d}\delta\eta.\]
\end{lemma}
\begin{proof}
Both $\mathcal T(\lambda)$ and $[\mathcal T(\lambda)]^{-1}$ are well-defined, and a simple computation shows that they are inverse to each other.
\end{proof}
Let us now consider a trivial solution $(\lambda,0,0)$. In view of the $\mathcal T$-isomorphism, we introduce
\[\mathcal L(\lambda)\coloneqq [\mathcal F_{(\eta,\phi)}(\lambda,0,0)]\circ [\mathcal T(\lambda)]\colon Y\to X\]
whenever $c(\lambda)\neq 0$. Now recall that
\[\mathcal F_{(\eta,\phi)}=(\delta\eta-\mathcal M_\eta^1-\mathcal M_\phi^1,\delta\phi-\mathcal A_\eta-\mathcal A_\phi).\]
For given $\eta$ we denote by $V=V[\eta]$ the unique solution of
\[\Delta_d\mathcal I V=0\text{ in }\Omega_0^\mathcal I,\quad \mathcal I V=\eta\text{ on }|y|=1.\]
We notice that
\[-\frac{2\psi^\lambda+s\psi^\lambda_s}{c(\lambda)}\mathcal S\theta+\frac{d}{c(\lambda)}\mathcal A_\eta\mathcal S\theta+\mathcal A_\phi\left(\frac{2\psi^\lambda+s\psi^\lambda_s}{c(\lambda)}\mathcal S\theta\right)=-V[\mathcal S\theta].\]
Indeed, from
\begin{align*}
\partial_{y_iy_i}\mathcal I(2\psi^\lambda+s\psi^\lambda_s)&=\partial_{y_iy_i}(2\mathcal I\psi^\lambda+y_j(\mathcal I\psi^\lambda)_{y_j})=2(\mathcal I\psi^\lambda)_{y_iy_i}+\partial_{y_j}(\mathcal I\psi^\lambda)_{y_j}+\partial_{y_i}(y_j(\mathcal I\psi^\lambda)_{y_iy_j})\\
&=4(\mathcal I\psi^\lambda)_{y_iy_i}+y_j(\mathcal I\psi^\lambda)_{y_iy_iy_j}\nonumber\\
&=4(\mathcal I\psi^\lambda)_{y_iy_i}-y_i\partial_{y_i}\left(d^2\gamma(d^2|y|^2\mathcal I\psi^\lambda)+\frac{1}{|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\right)\\
&=4(\mathcal I\psi^\lambda)_{y_iy_i}-y_i\Bigg(d^4\gamma'(d^2|y|^2\mathcal I\psi^\lambda)(2y_i\mathcal I\psi^\lambda+|y|^2(\mathcal I\psi^\lambda)_{y_i})\\
&\phantom{=\;}-\frac{2y_i}{|y|^4}(FF')(d^2|y|^2\mathcal I\psi^\lambda)+\frac{d^2}{|y|^2}(FF')'(d^2|y|^2\mathcal I\psi^\lambda)(2y_i\mathcal I\psi^\lambda+|y|^2(\mathcal I\psi^\lambda)_{y_i})\Bigg)\\
&=4(\mathcal I\psi^\lambda)_{y_iy_i}+\frac{2}{|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\\
&\phantom{=\;}-d^2\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\left(2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}\right)
\end{align*}
we infer that the function $f\coloneqq-\frac{2\psi^\lambda+s\psi^\lambda_s}{c(\lambda)}\mathcal S\theta+\frac{d}{c(\lambda)}\mathcal A_\eta\mathcal S\theta+\mathcal A_\phi\left(\frac{2\psi^\lambda+s\psi^\lambda_s}{c(\lambda)}\mathcal S\theta\right)+V[\mathcal S\theta]$ satisfies
\begin{align*}
\Delta_d\mathcal I f&=-\frac{2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}}{c(\lambda)}\mathcal S\theta_{zz}-\frac{1}{d^2c(\lambda)}\Bigg(4(\mathcal I\psi^\lambda)_{y_iy_i}+\frac{2}{|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\\
&\omit\hfill$\displaystyle-d^2\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\left(2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}\right)\Bigg)\mathcal S\theta$\\
&\phantom{=\;}+\frac{d}{c(\lambda)}\left(\frac{4(\mathcal I\psi^\lambda)_{y_iy_i}}{d^3}\mathcal S\theta+\frac{2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}}{d}\mathcal S\theta_{zz}+\frac{2}{d^3|y|^2}(FF')(d^2|y|^2\mathcal I\psi^\lambda)\mathcal S\theta\right)\\
&\phantom{=\;}-\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\frac{2\mathcal I\psi^\lambda+y_i(\mathcal I\psi^\lambda)_{y_i}}{c(\lambda)}\mathcal S\theta\\
&=0
\end{align*}
and $\mathcal I f=0$ at $|y|=1$. Thus, recalling \eqref{eq:Aeta_tr}, \eqref{eq:Aphi_tr}, \eqref{eq:M1eta_tr}, and \eqref{eq:M1phi_tr}, we can rewrite
\begin{align}\label{eq:L2}
\mathcal L_2(\lambda)\theta=\theta-(\mathcal A_\phi\theta+V[\mathcal S\theta])
\end{align}
and
\begin{align}
\mathcal L_1(\lambda)\theta&=-\frac{d}{c(\lambda)}\mathcal S\theta-\frac{1}{\sigma d^2c(\lambda)}(\sigma d+2d^2c(\lambda)^2+F(m(\lambda))^2)\partial_z^{-2}\mathcal S\theta\nonumber\\
&\phantom{=\;}+\sigma^{-1}c(\lambda)\partial_z^{-2}\mathcal P\mathcal S\partial_s\left(\frac{d}{c(\lambda)}\mathcal A_\eta\mathcal S\theta-\mathcal A_\phi\theta+\mathcal A_\phi\left(\frac{2\psi^\lambda+s\psi^\lambda_s}{c(\lambda)}\mathcal S\theta\right)\right)\nonumber\\
&=-\frac{d}{c(\lambda)}\mathcal S\theta-\sigma^{-1}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\partial_z^{-2}\mathcal S\theta\nonumber\\
&\phantom{=\;}-\sigma^{-1}c(\lambda)\partial_z^{-2}\mathcal P\mathcal S\partial_s(\mathcal A_\phi\theta+V[\mathcal S\theta])\label{eq:L1}
\end{align}
because of
\begin{align*}
\mathcal S\partial_s(2\psi^\lambda+s\psi^\lambda_s)&=\mathcal S(3\psi_s^\lambda+s\psi^\lambda_{ss})=\mathcal S\left(\frac3s\psi^\lambda_s+\psi^\lambda_{ss}\right)=\mathcal S\left(-d^2\gamma(d^2s^2\psi^\lambda)-\frac{1}{s^2}(FF')(d^2s^2\psi^\lambda)\right)\\
&=-(d^2\gamma+FF')(m(\lambda)).
\end{align*}
Notice that, under the assumption $\theta\in C_\mathrm{per}^{2,\alpha}(\overline{\Omega_0})$, $\mathcal L_2(\lambda)\theta$ is the unique solution of
\begin{subequations}\label{eq:L2theta_rewritten}
\begin{align}
\Delta_d[\mathcal I\mathcal L_2(\lambda)\theta]&=\Delta_d\theta+\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\theta&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal L_2(\lambda)\theta&=0&\text{on }|y|=1,
\end{align}
\end{subequations}
and $\mathcal L_1(\lambda)\theta$ is (in the set of $L$-periodic functions with zero average) uniquely determined by
\begin{align}
[\mathcal L_1(\lambda)\theta]_{zz}&=-\frac{d}{c(\lambda)}\mathcal S\theta_{zz}-\sigma^{-1}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\mathcal S\theta\nonumber\\
&\phantom{=\;}-\sigma^{-1}c(\lambda)\mathcal P\mathcal S\partial_s(\mathcal A_\phi\theta+V[\mathcal S\theta])\label{eq:L1theta_rewritten_2}\\
&=-\frac{d}{c(\lambda)}\mathcal S\theta_{zz}-\sigma^{-1}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\mathcal S\theta\nonumber\\
&\phantom{=\;}-\sigma^{-1}c(\lambda)\mathcal P\mathcal S\partial_s(\theta-\mathcal L_2(\lambda)\theta).\label{eq:L1theta_rewritten}
\end{align}
\subsection{Kernel}
We now turn to the investigation of the kernel of $\mathcal F_{(\eta,\phi)}(\lambda,0,0)$. Clearly, in view of the $\mathcal T$-isomorphism it suffices to study the kernel of $\mathcal L$; here and in the following, we will suppress the dependency of $\mathcal L$ on $\lambda$. From \eqref{eq:L2} we infer that $\theta\in C^{2,\alpha}(\overline{\Omega_0})$ provided $\mathcal L\theta=0$. Thus, combining \eqref{eq:L2theta_rewritten} and \eqref{eq:L1theta_rewritten} yields
\begin{align*}
\mathcal L\theta=0\;&\Longleftrightarrow\;\theta\in C_\mathrm{per}^{2,\alpha}(\overline{\Omega_0})\text{, and}\\
&\Delta_d\mathcal I\theta+\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\theta=0\text{, and}\\
&\frac{d\mathcal S\theta_{zz}}{c(\lambda)}+\sigma^{-1}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\mathcal S\theta+\sigma^{-1}c(\lambda)\mathcal P\mathcal S\theta_s=0.
\end{align*}
Let us now write $\theta(s,z)=\sum_{k=0}^\infty\theta_k(s)\cos(k\nu z)$ as a Fourier series. Then we easily see that
\begin{align*}
\mathcal L\theta=0\quad\Longleftrightarrow\quad\eqref{eq:LLtheta0=0}\quad\text{and}\quad\forall k\ge 1:\eqref{eq:LLthetak=0},
\end{align*}
where
\begin{align}\label{eq:LLtheta0=0}
\left(\frac{1}{d^2}\partial_{y_iy_i}+d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\theta_0=0,
\end{align}
noticing that $\theta_0(1)=0$ is already included in the definition of $Y$, and
\begin{subequations}\label{eq:LLthetak=0}
\begin{align}
\left(\frac{1}{d^2}\partial_{y_iy_i}+d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)-(k\nu)^2\right)\mathcal I\theta_k&=0,\label{eq:LL2thetak=0}\\
\left(\frac{\sigma}{dc(\lambda)}(1-(k\nu)^2d^2)+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\theta_k(1)+c(\lambda)\partial_s\theta_k(1)&=0.\label{eq:LL1thetak=0}
\end{align}
\end{subequations}
For $\mu\in\mathbb R$, let us now introduce the function $\tilde\beta=\tilde\beta^{\mu,\lambda}$, which is defined to be the unique solution of the singular Cauchy problem
\begin{align}\label{eq:tildebeta}
\left(\partial_s^2+\frac3s\partial_s+d^4s^2\gamma'(d^2s^2\psi^\lambda)+d^2(FF')'(d^2s^2\psi^\lambda)+\mu d^2\right)\tilde\beta=0\text{ on }(0,1],\quad\tilde\beta_s(0)=0,\quad\tilde\beta(0)=1.
\end{align}
Indeed, this problem has a unique solution $\tilde\beta\in C^{2,\alpha}([0,1])$ by the same argument as in Section \ref{sec:trivial}.
Henceforth, we shall assume that
\begin{align}\label{ass:SL-spectrum}
\tilde\beta^{0,\lambda}(1)\ne 0.
\end{align}
Thus, we see that \eqref{eq:LLtheta0=0} only has the trivial solution $\theta_0=0$. Indeed, if $\theta_0$ solves \eqref{eq:LLtheta0=0}, we have $\mathcal I\theta_0\in C^{2,\alpha}(\overline{\Omega_0})$, and therefore $\partial_s\theta_0(0)=0$. Hence, $\theta_0$ is a multiple of $\tilde\beta^{0,\lambda}$. But since necessarily $\theta_0(1)=0$, $\theta_0$ has to vanish identically in view of \eqref{ass:SL-spectrum}.
Let us now turn to $k\ge 1$ and notice as above that $\partial_s\theta_k(0)=0$ provided \eqref{eq:LL2thetak=0}. Thus, $\theta_k$ is a multiple of $\tilde\beta^{-(k\nu)^2,\lambda}$ if and only if \eqref{eq:LL2thetak=0} holds. First suppose that $\tilde\beta^{-(k\nu)^2,\lambda}(1)=0$ and that \eqref{eq:LLthetak=0} is satisfied. Then necessarily $\theta_k(1)=0$. Since therefore also $\partial_s\theta_k(1)=0$ by virtue of \eqref{eq:LL1thetak=0}, we conclude $\theta_k=0$. On the other hand, suppose that $\tilde\beta^{-(k\nu)^2,\lambda}(1)\ne 0$ and define $\beta^{-(k\nu)^2,\lambda}\coloneqq\tilde\beta^{-(k\nu)^2,\lambda}/\tilde\beta^{-(k\nu)^2,\lambda}(1)$. Hence, \eqref{eq:LLthetak=0} has a nontrivial solution $\theta_k$ if and only if the dispersion relation
\[\mathcal D(-(k\nu)^2,\lambda)=0,\]
where
\begin{align}\label{eq:d(k,lambda)}
\mathcal D(\mu,\lambda)\coloneqq\beta_s^{\mu,\lambda}(1)+\frac{\sigma}{dc(\lambda)^2}(1+\mu d^2)+2+\frac{F(m(\lambda))^2}{d^2c(\lambda)^2}+\frac{(d^2\gamma+FF')(m(\lambda))}{c(\lambda)},
\end{align}
is satisfied, and in this case $\theta_k$ is a multiple of $\beta^{-(k\nu)^2,\lambda}$. We summarise our results concerning the kernel:
\begin{lemma}\label{lma:kernel}
Given $\lambda\in\mathbb R$ with $c(\lambda)\ne 0$ and under the assumption \eqref{ass:SL-spectrum}, a function $\theta\in Y$, admitting the Fourier decomposition $\theta(s,z)=\sum_{k=0}^\infty\theta_k(s)\cos(k\nu z)$, is in the kernel of $\mathcal L(\lambda)$ if and only if $\theta_0=0$ and for each $k\ge 1$
\begin{enumerate}[label=(\alph*)]
\item $\theta_k=0$, or
\item $\tilde\beta^{-(k\nu)^2,\lambda}(1)\ne 0$, $\theta_k$ is a multiple of $\tilde\beta^{-(k\nu)^2,\lambda}$, and the dispersion relation
\[\mathcal D(-(k\nu)^2,\lambda)=0\]
holds, with $\mathcal D$ given in \eqref{eq:d(k,lambda)}.
\end{enumerate}
\end{lemma}
\begin{remark}
Clearly, $\mathcal D(\mu,\lambda)$ is at first only defined if $\tilde\beta^{\mu,\lambda}(1)\ne 0$. If this property fails to hold, we set $\mathcal D(\mu,\lambda)\coloneqq\infty$ in the following.
\end{remark}
\subsection{Range}
Before we proceed with the investigation of the transversality condition, we first prove that the range of $\mathcal L$ can be written as an orthogonal complement with respect to a suitable inner product. This will be helpful later. To this end, we introduce the inner product
\[\langle(f_1,g_1),(f_2,g_2)\rangle\coloneqq\langle f_1',f_2'\rangle_{L^2([0,L])}+\langle\nabla_d \mathcal I g_1,\nabla_d \mathcal I g_2\rangle_{L^2(\tilde\Omega_0^\mathcal I)}\]
for $f_1,f_2\in H^1_{0,\mathrm{per}}(\mathbb R)$, $g_1,g_2\colon\tilde\Omega_0\to\mathbb R$ with $\mathcal I g_1,\mathcal I g_2\in H^1_\mathrm{per}(\tilde\Omega_0^\mathcal I)$, where $\tilde\Omega_0\coloneqq[0,1)\times(0,L)$ is one periodic instance of $\Omega_0$ and $\nabla_d\coloneqq(\partial_{y_1}/d,\ldots,\partial_{y_4}/d,\partial_z)^T$; in order to avoid misunderstanding, we point out that the index \enquote{0} in $H^1_{0,\mathrm{per}}(\mathbb R)$ means \enquote{zero average} as before and not \enquote{zero boundary values}. This inner product is positive definite on the space
\[H^1_{0,\mathrm{per}}(\mathbb R)\times\left\{g\colon\tilde\Omega_0\to\mathbb R:\mathcal I g\in H^1_\mathrm{per}(\tilde\Omega_0^\mathcal I),\langle\mathcal S g\rangle=0\right\}.\]
Notice that
\[\langle f_1',f_2'\rangle_{L^2([0,L])}=-\langle f_1,f_2''\rangle_{L^2([0,L])}\]
if $f_2\in H^2_\mathrm{per}(\mathbb R)$ and that
\[\langle\nabla_d g_1,\nabla_d g_2\rangle_{L^2(\tilde\Omega_0^\mathcal I)}=-\langle g_1,\Delta_d g_2\rangle_{L^2(\tilde\Omega_0^\mathcal I)}+\frac{2\pi^2}{d^2}\langle\mathcal S g_1,\mathcal S\partial_s g_2\rangle_{L^2([0,L])}\]
if $\mathcal I g_2\in H^2_\mathrm{per}(\tilde\Omega_0)$, using that $2\pi^2$ is the surface area of the $3$-sphere.
Using \eqref{eq:L2}, \eqref{eq:L2theta_rewritten}, and \eqref{eq:L1theta_rewritten_2} we now compute for smooth $\theta,\vartheta\in Y$
\begin{align*}
&\left\langle\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right),\mathcal L\vartheta\right\rangle\\
&=\frac{2\pi^2\sigma}{d^2c(\lambda)}\Bigg\langle\mathcal S\theta,\frac{d}{c(\lambda)}\mathcal S\vartheta_{zz}+\sigma^{-1}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\mathcal S\vartheta\\
&\omit\hfill$\displaystyle+\sigma^{-1}c(\lambda)\mathcal P\mathcal S\partial_s(\mathcal A_\phi\vartheta+V[\mathcal S\vartheta])\Bigg\rangle_{L^2([0,L])}$\\
&\phantom{=\;}-\left\langle\mathcal I\theta,\Delta_d\mathcal I\vartheta+\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\vartheta\right\rangle_{L^2(\tilde\Omega_0^\mathcal I)}\\
&\phantom{=\;}+\frac{2\pi^2}{d^2}\langle\mathcal S\theta,\mathcal S\partial_s(\vartheta-(\mathcal A_\phi\vartheta+V[\mathcal S\vartheta]))\rangle_{L^2([0,L])}\\
&=-\frac{2\pi^2\sigma}{dc(\lambda)^2}\langle\mathcal S\theta_z,\mathcal S\vartheta_z\rangle_{L^2([0,2\pi])}+\langle\nabla_d\mathcal I\theta,\nabla_d\mathcal I\vartheta\rangle_{L^2(\tilde\Omega_0^\mathcal I)}\\
&\phantom{=\;}+\frac{2\pi^2}{d^2c(\lambda)}\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\langle\mathcal S\theta,\mathcal S\vartheta\rangle_{L^2([0,L])}\\
&\phantom{=\;}-\left\langle\mathcal I\theta,\left(d^2|y|^2\gamma'(d^2|y|^2\mathcal I\psi^\lambda)+(FF')'(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\vartheta\right\rangle_{L^2(\tilde\Omega_0^\mathcal I)}
\end{align*}
making use of $\langle\mathcal S\theta\rangle=0$. Noticing that the terms at the beginning and at the end of this computation only involve at most first derivatives of $\theta$ and $\vartheta$, an easy approximation argument shows that this relation also holds for general $\theta,\vartheta\in Y$. Moreover, since the last expression is symmetric in $\theta$ and $\vartheta$, we can also go in the opposite direction with reversed roles and arrive at the symmetry property
\[\left\langle\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right),\mathcal L\vartheta\right\rangle=\left\langle\mathcal L\theta,\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\vartheta,\vartheta\right)\right\rangle.\]
Thus, the range of $\mathcal L$ is the orthogonal complement of
\[\left\{\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right):\theta\in\ker\mathcal L\right\}\]
with respect to $\langle\cdot,\cdot\rangle$. Indeed, one inclusion is an immediate consequence of the symmetry property and the other inclusion follows from the facts that we already know that $\mathcal L$, being a compact perturbation of the identity, is Fredholm with index zero and that $\mathcal L$ gains no additional kernel when extended to functions $\theta$ of class $H^1$.
\subsection{Transversality condition}
Assuming that the kernel is spanned by the function $\theta(s,z)=\beta^{-(k\nu)^2,\lambda}(s)\cos(k\nu z)$, we have to investigate whether $\mathcal L_\lambda\theta$ is not in the range of $\mathcal L$, which is equivalent to
\[\left\langle\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right),\mathcal L_\lambda\theta\right\rangle\neq 0\]
by the preceding considerations. Differentiating \eqref{eq:L2} and \eqref{eq:L1} with respect to $\lambda$, for general $\theta$ it holds
\begin{align*}
\mathcal L_{\lambda,1}\theta&=-\partial_\lambda\left(\frac{d}{c(\lambda)}\right)\mathcal S\theta-\sigma^{-1}\partial_\lambda\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\partial_z^{-2}\mathcal S\theta\\
&\phantom{=\;}-\sigma^{-1}c_\lambda(\lambda)\partial_z^{-2}\mathcal P\mathcal S\partial_s(\mathcal A_\phi\theta+V[\mathcal S\theta])-\sigma^{-1}c(\lambda)\partial_z^{-2}\mathcal P\mathcal S\partial_s\mathcal A_{\phi\lambda}\theta\\
\mathcal L_{\lambda,2}\theta&=-\mathcal A_{\phi\lambda}\theta,
\end{align*}
where $\mathcal A_{\phi\lambda}\theta$ is the unique solution of
\begin{align*}
\Delta_d(\mathcal I\mathcal A_{\phi\lambda}\theta)&=-d^2|y|^2\left(d^2|y|^2\gamma''(d^2|y|^2\mathcal I\psi^\lambda)+(FF')''(d^2|y|^2\mathcal I\psi^\lambda)\right)\partial_\lambda\mathcal I\psi^\lambda\mathcal I\theta&\text{in }\Omega_0^\mathcal I,\\
\mathcal I\mathcal A_{\phi\lambda}\theta&=0&\text{on }|y|=1.
\end{align*}
Thus, we have
\begin{align*}
&\left\langle\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right),\mathcal L_\lambda\theta\right\rangle\\
&=\frac{2\pi^2\sigma}{d^2c(\lambda)}\Bigg\langle\mathcal S\theta,\partial_\lambda\left(\frac{d}{c(\lambda)}\right)\mathcal S\theta_{zz}+\sigma^{-1}\partial_\lambda\left(\frac{\sigma}{dc(\lambda)}+2c(\lambda)+\frac{F(m(\lambda))^2}{d^2c(\lambda)}+(d^2\gamma+FF')(m(\lambda))\right)\mathcal S\theta\\
&\omit\hfill$\displaystyle+\sigma^{-1}c_\lambda(\lambda)\mathcal P\mathcal S\partial_s(\mathcal A_\phi\theta+V[\mathcal S\theta])+\sigma^{-1}c(\lambda)\mathcal P\mathcal S\partial_s\mathcal A_{\phi\lambda}\theta\Bigg\rangle_{L^2([0,L])}$\\
&\phantom{=\;}-\langle\mathcal I\theta,\Delta_d(-\mathcal I\mathcal A_{\phi\lambda}\theta)\rangle_{L^2(\tilde\Omega_0^\mathcal I)}+\frac{2\pi^2}{d^2}\langle\mathcal S\theta,\mathcal S\partial_s(-\mathcal A_{\phi\lambda}\theta)\rangle_{L^2([0,L])}\\
&=\frac{2\pi^2}{d^2}\left\langle\mathcal S\theta,\partial_\lambda\left(\frac{\sigma d}{c(\lambda)^2}\right)\mathcal S\theta_{zz}+\partial_\lambda\left(\frac{\sigma}{dc(\lambda)^2}+2+\frac{F(m(\lambda))^2}{d^2c(\lambda)^2}+\frac{(d^2\gamma+FF')(m(\lambda))}{c(\lambda)}\right)\mathcal S\theta\right\rangle_{L^2([0,L])}\\
&\phantom{=\;}-\left\langle\mathcal I\theta,d^2|y|^2\left(d^2|y|^2\gamma''(d^2|y|^2\mathcal I\psi^\lambda)+(FF')''(d^2|y|^2\mathcal I\psi^\lambda)\right)\partial_\lambda\mathcal I\psi^\lambda\mathcal I\theta\right\rangle_{L^2(\tilde\Omega_0^\mathcal I)}
\end{align*}
whenever $\mathcal L_1\theta=0$. Now let $\theta(s,z)=\beta^{-(k\nu)^2,\lambda}(s)\cos(k\nu z)$ and notice that $f=\partial_\lambda\beta^{-(k\nu)^2,\lambda}$ solves
\begin{align*}
(\mathcal I f)_{y_iy_i}+\left(d^4|y|^2\gamma'(d^2|y|^2\psi^\lambda)+d^2(FF')'(d^2|y|^2\psi^\lambda)-(k\nu)^2d^2\right)\mathcal I f&\\
=-d^4|y|^2\left(d^2|y|^2\gamma''(d^2|y|^2\mathcal I\psi^\lambda)+(FF')''(d^2|y|^2\mathcal I\psi^\lambda)\right)\mathcal I\beta^{-(k\nu)^2,\lambda}\partial_\lambda\mathcal I\psi^\lambda&&\text{in }|y|<1,\\
\mathcal I f=0&&\text{on }|y|=1.
\end{align*}
Therefore,
\begin{align*}
&\frac{d^2}{L\pi^2}\left\langle\left(\frac{2\pi^2\sigma}{d^2c(\lambda)}\mathcal S\theta,\theta\right),\mathcal L_\lambda\theta\right\rangle\\
&=\frac{1}{2\pi^2}\int_{|y|<1}\mathcal I\beta^{-(k\nu)^2,\lambda}\Big((\mathcal I\partial_\lambda\beta^{-(k\nu)^2,\lambda})_{y_iy_i}\\
&\phantom{=\frac{1}{2\pi^2}\int_{|y|<1}\mathcal I\beta^{-(k\nu)^2,\lambda}\Big(\;}+\left(d^4|y|^2\gamma'(d^2|y|^2\psi^\lambda)+d^2(FF')'(d^2|y|^2\psi^\lambda)-(k\nu)^2d^2\right)\mathcal I\partial_\lambda\beta^{-(k\nu)^2,\lambda}\Big)\,dy\\
&\phantom{=\;}+\partial_\lambda\left(\frac{\sigma}{dc(\lambda)^2}(1-(k\nu)^2d^2)+2+\frac{F(m(\lambda))^2}{d^2c(\lambda)^2}+\frac{(d^2\gamma+FF')(m(\lambda))}{c(\lambda)}\right)\\
&=\partial_\lambda\beta_s^{-(k\nu)^2,\lambda}(1)+\partial_\lambda\left(\frac{\sigma}{dc(\lambda)^2}(1-(k\nu)^2d^2)+2+\frac{F(m(\lambda))^2}{d^2c(\lambda)^2}+\frac{(d^2\gamma+FF')(m(\lambda))}{c(\lambda)}\right)
\end{align*}
after integrating by parts. Thus, we have proved:
\begin{lemma}\label{lma:transversality_condition}
Given $\lambda\in\mathbb R$ with $c(\lambda)\ne 0$ and assuming that the kernel of $\mathcal L(\lambda)$ is one-dimensional spanned by $\theta(s,z)=\beta^{-(k\nu)^2,\lambda}(s)\cos(k\nu z)$ for some $k\ge 1$, the transversality condition
\[\mathcal L_\lambda(\lambda)\theta\notin\im\mathcal L(\lambda)\]
is equivalent to
\[\mathcal D_\lambda(-(k\nu)^2,\lambda)\neq 0,\]
with $\mathcal D$ given in \eqref{eq:d(k,lambda)}.
\end{lemma}
\subsection{Result on local bifurcation}
We summarise our result of this section using the following local bifurcation theorem by Crandall--Rabinowitz \cite[Thm. I.5.1]{Kielhoefer}.
\begin{theorem}\label{thm:CrandallRabinowitz}
Let $X$ be a Banach space, $U\subset\mathbb R\times X$ open, and $\mathcal F\colon U\to X$ have the property $\mathcal F(\cdot,0)=0$. Assume that there exists $\lambda_0\in\mathbb R$ such that $\mathcal F$ is of class $C^2$ in an open neighbourhood of $(\lambda_0,0)$, and suppose that $\mathcal F_x(\lambda_0,0)$ is a Fredholm operator with index zero and one-dimensional kernel spanned by $x_0\in X$, and that the transversality condition $\mathcal F_{\lambda x}(\lambda_0,0)x_0\notin\im \mathcal F_x(\lambda_0,0)$ holds. Then there exists $\varepsilon>0$ and a $C^1$-curve $(-\varepsilon,\varepsilon)\ni t\mapsto(\lambda^t,x^t)$ with $(\lambda^0,x^0)=(\lambda_0,0)$ and $x^t\neq 0$ for $t\neq 0$, and $\mathcal F(\lambda^t,x^t)=0$. Moreover, all solutions of $\mathcal F(\lambda,x)=0$ in a neighbourhood of $(\lambda_0,0)$ are on this curve or are trivial. Furthermore, the curve admits the asymptotic expansion $x^t=tx_0+o(t)$.
\end{theorem}
Applied to our problem, we obtain the following result.
\begin{theorem}\label{thm:LocalBifurcation}
Assume \eqref{ass:SL-spectrum} and that there exists $\lambda_0\in\mathbb R$ with $c(\lambda_0)\ne 0$ such that the dispersion relation
\[\mathcal D(-(k\nu)^2,\lambda_0)=0,\]
with $\mathcal D$ given by \eqref{eq:d(k,lambda)}, has exactly one solution $k_0\in\mathbb N$ and assume that the transversality condition
\[\mathcal D_\lambda(-(k_0\nu)^2,\lambda_0)\neq 0\]
holds. Then there exists $\varepsilon>0$ and a $C^1$-curve $(-\varepsilon,\varepsilon)\ni t\mapsto(\lambda^t,\eta^t,\phi^t)$ with $(\lambda^0,\eta^0,\phi^0)=(\lambda_0,0,0)$, $\eta^t\neq 0$ for $t\neq 0$, and $\mathcal F(\lambda^t,\eta^t,\phi^t)=0$. Moreover, all solutions of $\mathcal F(\lambda,\eta,\phi)=0$ in a neighbourhood of $(\lambda_0,0,0)$ are on this curve or are trivial. Furthermore, the curve admits the asymptotic expansion $(\eta^t,\phi^t)=t\mathcal T(\lambda_0)\theta+o(t)$, where
\begin{align*}
\theta(s,z)&=\beta^{-(k_0\nu)^2,\lambda_0}(s)\cos(k_0\nu z),\\
[\mathcal T(\lambda_0)\theta](x,y)&=\left(-\frac{d}{c(\lambda_0)},\beta^{-(k_0\nu)^2,\lambda_0}(s)-\frac{2\psi^{\lambda_0}(s)+s\psi_s^{\lambda_0}(s)}{c(\lambda_0)}\right)\cos(k_0\nu z).
\end{align*}
\end{theorem}
\begin{proof}
It is straightforward to apply Theorem \ref{thm:CrandallRabinowitz} in view of Lemmas \ref{lma:M_prop}, \ref{lma:kernel}, and \ref{lma:transversality_condition}, noticing that $F_{(\eta,\phi)}(\lambda_0,0,0)$ coincides with $\mathcal L(\lambda_0)$ up to the isomorphism $\mathcal T(\lambda_0)$. Moreover, the asymptotic expansion tells us that $\eta(t)\neq 0$ after possibly shrinking $\varepsilon$.
\end{proof}
\section{Conditions for local bifurcation} \label{sec:conditions}
\subsection{Spectral properties}
In view of the defining equation \eqref{eq:tildebeta} for $\tilde\beta^{\mu,\lambda}$ and the dispersion relation $\mathcal D(\mu,\lambda)=0$ and writing $\varphi=\tilde\beta^{\mu,\lambda}$, we study the eigenvalue problem
\begin{subequations}\label{eigenvalue_problem}
\begin{align}
-d^{-2}s^{-3}(s^3\varphi')'+q^\lambda\varphi&=\mu\varphi&\text{in }(0,1),\\%\label{eigenvalue_problem_ODE}\\
-g(\lambda)\varphi'(1)-h(\lambda)\varphi(1)&=\mu\varphi(1),&\label{eigenvalue_problem_BC}
\end{align}
\end{subequations}
which is a singular Sturm-Liouville problem on $(0,1)$. Here and in the following, we denote
\begin{align*}
q^\lambda(s)&\coloneqq-d^2s^2\gamma'(d^2s^2\psi^\lambda(s))-(FF')'(d^2s^2\psi^\lambda(s)),\\
g(\lambda)&\coloneqq\sigma^{-1}d^{-1}c(\lambda)^2>0,\\
h(\lambda)&\coloneqq d^{-2}+\sigma^{-1}d^{-1}c(\lambda)\left(2c(\lambda)+d^{-2}F(m(\lambda))^2+(d^2\gamma+FF')(m(\lambda))\right).
\end{align*}
Notice that we left out the condition $\tilde\beta^{\mu,\lambda}_s(0)=0$ in view of Lemma \ref{regularity_lemma} below. We first introduce the operators $T$ and $\tau$ via
\[D(T)=D(\tau)=\{\varphi\in L^2_{s^3}(0,1):\varphi,s^3\varphi'\in\text{AC}_{\text{loc}}(0,1],s^{-3}(s^3\varphi')'\in L^2_{s^3}(0,1)\}\]
and
\[T\varphi=-d^{-2}s^{-3}(s^3\varphi')',\quad\tau\varphi=T\varphi+q^\lambda\varphi,\quad\varphi\in D(\tau).\]
We collect some important properties of $T$, $\tau$, and $D(T)$:
\begin{lemma}\label{Ttau_lemma}
The following holds:
\begin{enumerate}[label=(\roman*)]
\item The operators $T$ and $\tau$ are of limit point type at $0$ (and of regular type at $1$).
\item For any $\varphi,\chi\in D(T)$ we have \[\lim_{s\to 0}\left(s^3\varphi'(s)\overline\chi(s)-\varphi(s)s^3\overline\chi'(s)\right)=0.\] In particular, \[\lim_{s\to 0}s^3\varphi(s)=\lim_{s\to 0}s^3\varphi'(s)=0.\]
\end{enumerate}
\end{lemma}
\begin{proof}
It is easy to see that $T$ is of limit point type at $0$, since $\varphi(s)=s^{-2}\notin L^2_{s^3}(0,1)$ solves $T\varphi=0$. Since $q^\lambda\in L^\infty(0,1)$, $\tau$ is also of limit point type at $0$ according to \cite[Corollary 7.4.1]{Zettl}. Thus, (i) is proved. As for (ii), the first statement is an immediate consequence of $T$ being of limit point type at $0$; see \cite[Lemmas 10.2.3, 10.4.1(b)]{Zettl}. Plugging in $\chi(s)=1$ and then $\chi(s)=s$ (which both belong to $D(T)$) yields the second statement.
\end{proof}
As a consequence the following result holds; in particular, this explains why we could leave out $\phi'(0)=0$ in \eqref{eigenvalue_problem}
\begin{lemma}\label{regularity_lemma}
Let $q,f\in C^{0,\alpha}([0,1])$ (or, equivalently, $\mathcal I q,\mathcal I f\in C^{0,\alpha}(\overline{B_1(0)})$) and $\varphi\in D(T)$ satisfy \[T\varphi=q\varphi+f.\] Then, $\mathcal I\varphi\in C^{2,\alpha}(\overline{B_1(0)})$ and solves \[\Delta_4\mathcal I\varphi=\mathcal I q\mathcal I\varphi+\mathcal I f.\] Obviously, the converse also holds. Moreover, in this case $\varphi\in C^{2,\alpha}([0,1])$ and $\varphi'(0)=0$.
\end{lemma}
\begin{proof}
Clearly, $\mathcal I\varphi$ has weak derivatives on $B_1(0)\setminus\{0\}$; in particular, $\nabla_4\mathcal I\varphi=\varphi'e_s$ a.e. First, we claim that this also holds on $B_1(0)$. To this end, we first note that $\mathcal I\varphi\in L^2(B_1(0))$ due to $\varphi\in L^2_{s^3}(0,1)$. Now fix $v\in C_c^\infty(B_1(0);\mathbb R^4)$ and let $\varepsilon>0$. We have to pass to the limit $\varepsilon\to 0$ in the identity \[\int_{\varepsilon\le|y|\le1}\nabla_4\mathcal I\varphi\cdot v\,dy=-\int_{\varepsilon\le|y|\le1}\mathcal I\varphi\nabla_4\cdot v\,dy-\int_{|y|=\varepsilon}\mathcal I\varphi v\cdot e_s\,dS_y;\] note that the surface integral is well-defined since $\varphi\in\text{AC}_{\text{loc}}(0,1]$. Passing to the limit in the volume integrals is easy, as $|\nabla_4\mathcal I\varphi\cdot v|\le|\varphi'||v|$, $|\mathcal I\varphi\nabla_4\cdot v|\le|\varphi||\nabla_4 v|$, and $s^3\varphi,s^3\varphi'\in L^\infty(0,1)$ due to Lemma \ref{Ttau_lemma}(ii). Also because of Lemma \ref{Ttau_lemma}(ii) the surface integral vanishes in the limit, since its modulus can be estimated by $C\varepsilon^3|\varphi(\varepsilon)|$, where $C>0$ only depends on $\|v\|_\infty$.
The next step is to show that $\mathcal I\varphi$ solves $\Delta_4\mathcal I\varphi=\mathcal I q\mathcal I\varphi+\mathcal I f$ on $B_1(0)$ in the weak sense. Clearly, we infer from the preceding considerations that $\mathcal I\varphi\in W^{1,1}(B_1(0))$. For fixed $v\in C_c^\infty(B_1(0))$ it holds that \[-\int_{B_1(0)}\nabla_4\mathcal I\varphi\cdot\nabla_4 v\,dy=-\int\int_0^1\varphi'v_ss^3\,ds\,d\Omega=\int\int_0^1(q\varphi+f)vs^3\,ds\,d\Omega=\int_{B_1(0)}(\mathcal I q\mathcal I\varphi+\mathcal I f)v\,dy,\] where $\int\cdots d\Omega$ denotes the integration with respect to the three angles in spherical coordinates of $\mathbb R^4$. It is very important to notice that here no boundary terms at $s=0$ appear although $v$ does not have to vanish there. This is due to the fact that $\lim_{s\to 0}s^3\varphi'(s)=0$ (see Lemma \ref{Ttau_lemma}(ii)), so the weak form \[-\int_0^1\varphi'w's^3\,ds=\int_0^1(q\varphi+f)ws^3\,ds\] also applies for smooth functions $w$ on $[0,1]$ having support at $s=0$ (but not at $s=1$).
Finally, we infer from elliptic regularity that $\mathcal I\varphi\in C^{2,\alpha}(\overline{B_1(0)})$. Indeed, since $\Delta_4\mathcal I\varphi=\mathcal I q\mathcal I\varphi+\mathcal I f\in L^2(B_1(0))$, we have $\mathcal I\varphi\in H^2(B_1(0))\subset L^p(B_1(0))$, $1\le p<\infty$. Thus, $\Delta_4\mathcal I\varphi\in L^p(B_1(0))$ and $\mathcal I\varphi\in W^{2,p}(B_1(0))\subset C^{0,\alpha}(\overline{B_1(0)})$ for $p$ large. Hence, $\Delta_4\mathcal I\varphi\in C^{0,\alpha}(\overline{B_1(0)})$ and therefore $\mathcal I\varphi\in C^{2,\alpha}(\overline{B_1(0)})$. The remaining statements clearly hold true.
\end{proof}
To introduce a functional-analytic setting when also taking the boundary condition \eqref{eigenvalue_problem_BC} into account, we let $H=L^2_{s^3}(0,1)\times\mathbb C$. In the following, we write elements $u\in H$ as $u=(\varphi,b)$. Equipped with the indefinite inner product \[[u_1,u_2]=\langle d^2\varphi_1,\varphi_2\rangle_{L^2_{s^3}}-g(\lambda)^{-1}b_1\overline{b_2},\] $H$ becomes a Pontryagin $\pi_1$-space. Furthermore, we introduce the operator $K$ given by \[D(K)=\{u\in H:\varphi\in D(\tau),b=\varphi(1)\}\] and \[Ku=\left(\tau\varphi,-g(\lambda)\varphi'(1)-h(\lambda)\varphi(1)\right),\quad u\in D(K),\] which is clearly densely defined. Observe that the eigenvalues (-functions) of $K$ are exactly the eigenvalues (-functions) of \eqref{eigenvalue_problem}. We have the following.
\begin{lemma}
$K$ is self-adjoint.
\end{lemma}
\begin{proof}
We first prove that $K$ is symmetric. To this end, for $u_1,u_2\in H$, $x\in (0,1)$ let \[[u_1,u_2]_x\coloneqq\langle d^2\varphi_1,\varphi_2\rangle_{L^2_{s^3}(x,1)}-g(\lambda)^{-1}b_1\overline{b_2}.\] Now if $u_1,u_2\in D(K)$ we have, after integrating by parts,
\[[Ku_1,u_2]_x-[u_1,Ku_2]_x=x^3\varphi_1'(x)\overline{\varphi_2}(x)-\varphi_1(x)x^3\overline{\varphi_2}'(x).\] Clearly, $K$ is symmetric if and only if the first expression converges to $0$ as $x\to 0$ (for any $u_1,u_2\in D(K)$). But the second expression converges to $0$ due to Lemma \ref{Ttau_lemma}(ii).
To see that $K$ is even self-adjoint, we first note that obviously $H$ admits the fundamental decomposition $H=(L^2_{s^3}(0,1)\times\{0\})\dot+(\{0\}\times\mathbb C)$ into a positive and negative subspace. Associated to this decomposition is the fundamental symmetry \[J=\begin{pmatrix}\mathrm{id}&0\\0&-1\end{pmatrix}\] and the Hilbert inner product $\langle u_1,u_2\rangle_J=[Ju_1,u_2]=\langle d^2\varphi_1,\varphi_2\rangle_{L^2_{s^3}}+g(\lambda)^{-1}b_1\overline{b_2}$. The operator $JK$ is self-adjoint with respect to $\langle\cdot,\cdot\rangle_J$, since now the assumptions of \cite[Theorem 1]{Hinton} are satisfied. In particular, denoting the $J$-adjoint by an upper index $\langle*\rangle$, we have \[D(K)=D(JK)=D\left((JK)^{\langle*\rangle}\right)=D\left(K^{\langle*\rangle}J^{\langle*\rangle}\right)=D\left(K^{\langle*\rangle}J\right)=D(JK^*)=D(K^*),\] as $J^{\langle*\rangle}=J$ and $JK^{\langle*\rangle}J=K^*$ (cf. \cite[Lemma VI.2.1]{Bognar}). Since $K$ is already known to be symmetric, the proof is complete.
\end{proof}
Now we can prove the following important result.
\begin{proposition}
The spectrum of $K$ is purely discrete and consists of only (geometrically) simple eigenvalues.
\end{proposition}
\begin{proof}
Following the proof of \cite[Theorem 2]{Hinton} using the $J$-norm $\|u\|_J=\sqrt{\langle u,u\rangle_J}$, we see that the essential spectra of $K$ and $\tau$ coincide. Notice that the criterion \cite[Theorem XIII.7.1]{DunfordSchwartz} applied there is purely topological and does not make use of an additional structure from an (definite or indefinite) inner product. To see that the essential spectrum of $\tau$ is empty, we can apply a criterion of \cite{Friedrichs}; see also \cite{Hinton}. Indeed, $q^\lambda$ is obviously bounded from below on $(0,1)$ and moreover \[\lim_{s\to 0}\left(q^\lambda(s)+\frac{1}{4d^2s^6\left(\int_s^1\sigma^{-3}\,d\sigma\right)^2}\right)=\lim_{s\to 0}d^{-2}s^{-2}(s^2-1)^{-2}=\infty.\] Finally, it is a priori clear that each eigenvalue of $K$ cannot have (geometric) multiplicity larger than two; the case of multiplicity two is excluded by the fact that $\tau$ is of limit point type at $0$.
\end{proof}
In fact, we can say more about the location of the eigenvalues of $K$. To this end, the following Lemma turns out to be useful.
\begin{lemma}\label{lma:Ku,u}
For any $u\in D(K)$ we have
\[[Ku,u]=\|\varphi'\|_{L^2_{s^3}}^2+\int_0^1 d^2s^3q^\lambda|\varphi|^2\,ds+\frac{h(\lambda)}{g(\lambda)}|\varphi(1)|^2.\]
\end{lemma}
\begin{proof}
The only critical point is to ensure that no boundary terms at $0$ appear after an integration by parts, which again follows from Lemma \ref{Ttau_lemma}(ii).
\end{proof}
\begin{proposition}
$K$ has no or exactly two nonreal eigenvalues, and in the latter case they are the complex conjugate of each other. Moreover, the (real part of the) spectrum of $K$ is bounded from below.
\end{proposition}
\begin{proof}
The first assertion is clear since $H$ is a $\pi_1$-space and $K$ is self-adjoint; cf. \cite{IKL}. To prove the second statement, we use a perturbation argument. First notice that $q^\lambda$ does not affect the domain of the associated operator. Now let $K_0$ be the operator in the case $\gamma=F=0$, which yields $q^\lambda=0$ and $h(\lambda)>0$. By Lemma \ref{lma:Ku,u} we have $[K_0u,u]>0$ if $u\neq 0$. Thus, there exists exactly one negative eigenvector of $K_0$; cf. again \cite{IKL}. Therefore, $K_0$ has exactly one negative eigenvalue $\mu_0$ and its other eigenvalues are positive. With the same proof as in \cite[Lemma 3]{Wahlen06} we conclude that for some constant $C>0$ the estimate
\[\|(K_0-\mu I)^{-1}\|_J\le\frac{C}{|\mu-\mu_0|},\quad\mu\in(-\infty,\mu_0),\]
for the resolvent holds. If $\gamma$ and $F$ are arbitrary, we define the perturbation $A$ via $D(A)=\{u\in H:\varphi\in D(T),b=\varphi(1)\}$ and \[Au=\left(q^\lambda\varphi,-\sigma d^{-1}c(\lambda)\left(d^{-2}F(m(\lambda))^2+(d^2\gamma+FF')(m(\lambda))\right)\varphi(1)\right),\quad u\in D(A).\]
Clearly, $A$ is densely defined and bounded, and we have $K=K_0+A$. Now consider a real $\mu<\mu_0-C\|A\|_J$. Because of \[K-\mu I=\left(I+A(K_0-\mu I)^{-1}\right)(K_0-\mu I)\] and \[\left\|A(K_0-\mu I)^{-1}\right\|_J\le\|A\|_J\cdot\frac{C}{|\mu-\mu_0|}<1,\] the resolvent operator $K-\mu I$ is invertible in view of the Neumann series. This completes the proof.
\end{proof}
Under a certain condition we can infer even more properties of the spectrum of $K$, as we see in what follows.
\begin{proposition}
Assume that
\begin{align}\label{ass:realspec_algsimple}
h(\lambda)>\|q^\lambda_-\|_\infty
\end{align}
where $q^\lambda_-$ denotes the negative part of $q^\lambda$. Then the operator $K$ has only real eigenvalues, $K$ has exactly one eigenvalue $\mu<-\|q^\lambda_-\|_\infty$, and all its other eigenvalues satisfy $\mu>-\|q^\lambda_-\|_\infty$. Moreover, all eigenvalues are algebraically simple.
\end{proposition}
\begin{proof}
Let $\mu$ be an eigenvalue of $K$ and $u=(\varphi,\varphi(1))$ an associated eigenvector. Due to Lemma \ref{lma:Ku,u} we can calculate
\begin{align*}
\mu[u,u]&=[Ku,u]=\|\varphi'\|_{L^2_{s^3}}^2+\int_0^1 d^2s^3q^\lambda|\varphi|^2\,ds+\frac{h(\lambda)}{g(\lambda)}|\varphi(1)|^2\ge -d^2\|q^\lambda_-\|_\infty\|\varphi\|_{L^2_{s^3}}^2+\frac{h(\lambda)}{g(\lambda)}|\varphi(1)|^2\\
&=-\|q^\lambda_-\|_\infty[u,u]+\frac{h(\lambda)-\|q^\lambda_-\|_\infty}{g(\lambda)}|\varphi(1)|^2.
\end{align*}
By assumption and since $\varphi(1)\neq 0$ (otherwise, also $\varphi'(1)=0$ and thus $\varphi\equiv 0$), it follows that
\[(\mu+\|q^\lambda_-\|_\infty)[u,u]>0.\] Hence, $u$ cannot be neutral and $\mu$ has to be real. Since, additionally, by \cite{IKL} -- noting that $H$ is a $\pi_1$-space -- there exists exactly one nonpositive eigenvector of $K$, the first assertion follows immediately. The second statement is a direct consequence of the fact that all eigenvalues are real and no eigenvectors are neutral.
\end{proof}
\begin{remark}
If \eqref{ass:realspec_algsimple} holds, then the assumptions of the next lemmas are satisfied. Moreover, we will discuss \eqref{ass:realspec_algsimple} later when looking at specific examples. Physically speaking, \eqref{ass:realspec_algsimple} is satisfied if the wave speed of the trivial solution at the surface is large compared to the angular components of the velocity and the vorticity (which depend on $\lambda$); more precisely, if
\begin{align*}
|c(\lambda)|&\notin[c_-,c_+],\\
c_\pm&\coloneqq\frac14\left(d\omega^\vartheta-(u^\vartheta)^2\pm\sqrt{(d\omega^\vartheta-(u^\vartheta)^2)^2+8\sigma d(d^2\|\gamma'\|_\infty+\|(FF')'\|_\infty-d^{-2})}\right)
\end{align*}
(where the condition is regarded to be vacuous if $c_\pm$ are not real). In particular, if $\gamma$ and $FF'$ are bounded, this condition is satisfied if \enquote{$c(\lambda)$ is sufficiently large} or, provided additionally $F$ is bounded, if simply \enquote{$|c(\lambda)|$ is sufficiently large}.
\end{remark}
\subsection{Examples}
We now turn to a more detailed investigation of the conditions for local bifurcation for specific examples of $\gamma$ and $F$.
\subsubsection{No vorticity, no swirl}
As a first example, we consider the case without vorticity and swirl, that is, $\gamma=F=0$. By \eqref{eq:trivial_integral_equation} and \eqref{eq:psi^lambda}, the trivial solutions are given by
\[\psi^\lambda(s)=\lambda.\]
Thus,
\[c(\lambda)=2\lambda,\]
that is, $c(\lambda)\ne 0$ if and only if $\lambda\ne 0$. Moreover, $\tilde\beta=\tilde\beta^{-(k\nu)^2,\lambda}$ solves
\[\left(\partial_s^2+\frac3s\partial_s-(k\nu)^2d^2\right)\tilde\beta=0\text{ on }(0,1],\quad\tilde\beta_s(0)=0,\quad\tilde\beta(0)=1.\]
The general solution to the ODE is given by
\[\tilde\beta(s)=c_1\frac{I_1(k\nu ds)}{s}+c_2\frac{K_1(k\nu ds)}{s},\quad c_1,c_2\in\mathbb R,\]
where $I_1$ and $K_1$ are modified Bessel functions of the first and second kind. Since $K_1(x)\to\infty$ as $x\to 0$, we necessarily have $c_2=0$. Determining the remaining constant $c_1$ yields
\[\tilde\beta^{-(k\nu)^2,\lambda}(s)=\frac{2I_1(k\nu ds)}{k\nu ds}\]
and
\[\beta^{-(k\nu)^2,\lambda}(s)=\frac{I_1(k\nu ds)}{sI_1(k\nu d)}.\]
Therefore, using $dI_1/dx=I_0-I_1/x$ (cf. \cite{Amos}),
\[\beta_s^{-(k\nu)^2,\lambda}(1)=\mathcal S\left(\frac{k\nu d\left(I_0(kds)-\frac{I_1(k\nu ds)}{k\nu ds}\right)}{sI_1(k\nu d)}-\frac{I_1(k\nu ds)}{s^2I_1(k\nu d)}\right)=\frac{k\nu dI_0(k\nu d)}{I_1(k\nu d)}-2.\]
Thus, we have
\[\mathcal D(-(k\nu)^2,\lambda)=\frac{k\nu dI_0(k\nu d)}{I_1(k\nu d)}+\frac{\sigma}{dc(\lambda)^2}(1-(k\nu)^2d^2).\]
Noticing that necessarily $(k\nu)^2d^2-1>0$ if $\mathcal D(-(k\nu)^2,\lambda)=0$, the dispersion relation $\mathcal D(-(k\nu)^2,\lambda)=0$ can hence be written as
\begin{align}\label{eq:disprel_no_vortswirl}
\frac{\sigma}{c(\lambda)^2}=\frac{k\nu d^2I_0(k\nu d)}{((k\nu)^2d^2-1)I_1(k\nu d)}.
\end{align}
This dispersion relation was also obtained in \cite{VB-M-S}. Clearly, in order find solutions of \eqref{eq:disprel_no_vortswirl}, we can first choose arbitrary $\nu>0$, $k\in\mathbb N$ with $k\nu>1/d$ and then $\lambda$ such that \eqref{eq:disprel_no_vortswirl} holds. This gives exactly two possible choices $\pm\lambda_0$ for $\lambda$, which correspond to \enquote{mirrored} uniform laminar flows. It is important to notice that, given $c(\lambda)\neq 0$, \eqref{eq:disprel_no_vortswirl} is solved by at most one $k\in\mathbb N$; consequently, the kernel of $\mathcal L(\lambda)$ is one-dimensional if this relation is satisfied for some $k\in\mathbb N$ and is trivial if it fails to hold for all $k$. Indeed, \eqref{eq:disprel_no_vortswirl} obviously cannot hold for $k\nu d\le 1$; moreover, the function \[g(x)\coloneqq\frac{xI_0(x)}{(x^2-1)I_1(x)},\quad x>1,\]
is strictly monotone on $(1,\infty)$ since
\[g'(x)=\frac{x(x^2-1)(I_1(x)^2-I_0(x)^2)-2I_0(x)I_1(x)}{(x^2-1)^2I_1(x)^2}<0,\quad x>1,\]
as $dI_0/dx=I_1$ and $I_0\ge I_1>0$ on $(0,\infty)$; see \cite{Amos}.
Furthermore, it is therefore clear that the transversality condition $\mathcal D_\lambda(-(k\nu)^2,\lambda)\ne 0$ always holds in view of $c_\lambda(\lambda)\ne 0$.
Moreover, it is easy to see that \eqref{ass:realspec_algsimple} is always satisfied since here $q^\lambda=0$ and $h(\lambda)=d^{-2}+\sigma^{-1}d^{-1}c(\lambda)^2>0$.
\subsubsection{Constant $\gamma$, no swirl}
Now let us assume that $\gamma\ne 0$ is a constant and $F=0$. By \eqref{eq:trivial_integral_equation} and \eqref{eq:psi^lambda}, the trivial solutions are given by
\[\psi^\lambda(s)=\lambda-\gamma\int_0^{sd}t^{-3}\int_0^t\tau^3\,d\tau\,dt=\lambda-\frac{\gamma d^2}{8}s^2.\]
Thus,
\begin{align}\label{eq:c_gammaconst}
c(\lambda)=2\lambda-\frac{\gamma d^2}{2}
\end{align}
that is, $c(\lambda)\ne 0$ if and only if $\lambda\ne\frac{\gamma d^2}{4}$. Noticing that $\beta^{-(k\nu)^2,\lambda}$ is the same as in the previous example without vorticity, we moreover have
\[\mathcal D(-(k\nu)^2,\lambda)=\frac{k\nu dI_0(kd)}{I_1(k\nu d)}+\frac{\sigma}{dc(\lambda)^2}(1-(k\nu)^2d^2)+\frac{d^2\gamma}{c(\lambda)}.\]
In order to solve the equation $\mathcal D(-(k\nu)^2,\lambda)=0$ for $1/c(\lambda)$, we see that necessarily
\[1-(k\nu)^2d^2=0\]
or
\begin{align}\label{eq:const_vort_cond}
1-(k\nu)^2d^2\le\frac{d^4\gamma^2I_1(k\nu d)}{4\sigma k\nu I_0(k\nu d)};
\end{align}
obviously, the first case can only occur if $1/(\nu d)\in\mathbb N$. We now want to reformulate the second case. Clearly, \eqref{eq:const_vort_cond} holds if $k\nu d\ge 1$. Let us consider $k\nu d<1$ further. The function
\[\chi(x)\coloneqq\frac{I_1(x)}{x(1-x^2)I_0(x)},\quad 0<x<1,\]
is positive and satisfies, using the result
\begin{align}\label{eq:Bessel_ratio_est}
\frac{x}{1+\sqrt{x^2+1}}\le\frac{I_1(x)}{I_0(x)}\le\frac{x}{\sqrt{x^2+4}}
\end{align}
of \cite{Amos},
\begin{align*}
\chi_x(x)&=\frac{(1-x^2)\left(x(1-(I_1(x)/I_0(x))^2)-2I_1(x)/I_0(x)\right)+2x^2I_1(x)/I_0(x)}{x^2(1-x^2)^2}\\
&\ge\frac{(1-x^2)\left(x\left(1-\frac{x^2}{x^2+4}\right)-\frac{2x}{\sqrt{x^2+4}}\right)+\frac{2x^3}{1+\sqrt{x^2+1}}}{x^2(1-x^2)^2}>0,\qquad 0<x<1;
\end{align*}
here, the last inequality follows from the fact that the numerator is positive at $x=1$ and a nonzero root of it, after some algebra, has to satisfy
\[36x^8+116x^6-64x^4-489x^2-224=0,\]
which can obviously not hold true for $x\in(0,1)$ in view of $36+116<224$. Therefore and because of $I_0(0)=1$, $I_1(0)=0$, and $I_1'(0)=1/2$, the function $\chi\colon (0,1)\to (1/2,\infty)$
is strictly monotonically increasing and onto. Hence, \eqref{eq:const_vort_cond} is always satisfied if $4\sigma d^{-5}\gamma^{-2}\le 1/2$. Otherwise, let $x_1\in(0,1)$ such that $\chi(x_1)=4\sigma d^{-5}\gamma^{-2}$ and $x_0\coloneqq x_1/d$. Thus, we have the equivalence
\[1-(k\nu)^2d^2\left\{\begin{matrix}<\\=\end{matrix}\right\}\frac{d^4\gamma^2I_1(k\nu d)}{4\sigma kI_0(k\nu d)}\Longleftrightarrow k\nu\left\{\begin{matrix}>\\=\end{matrix}\right\}x_0.\]
To conclude, solving $\mathcal D(-(k\nu)^2,\lambda)=0$ for $c(\lambda)$ yields
\begin{align}
c(\lambda)&=-\frac{d^2\gamma I_1(1)}{I_0(1)}&\text{if }k\nu=1/d,\label{eq:disprel_const_vort_1}\\
c(\lambda)&=\frac{2\sigma((k\nu)^2d^2-1)}{d\left(d^2\gamma\pm\sqrt{d^4\gamma^2+\frac{4\sigma k\nu((k\nu)^2d^2-1)I_0(k\nu d)}{I_1(k\nu d)}}\right)}&\text{if }8\sigma\le d^5\gamma^2,k\nu\ne 1/d,\nonumber\\
&&\text{or }\text{if }8\sigma>d^5\gamma^2,k\nu\ge x_0,k\nu\ne 1/d,\label{eq:disprel_const_vort_2}
\end{align}
and else, $\mathcal D(-(k\nu)^2,\lambda)$ cannot vanish. Next, we search for solutions of \eqref{eq:disprel_const_vort_1} and \eqref{eq:disprel_const_vort_2}. First notice that in both cases it suffices to find appropriate $k$ and $c(\lambda)$ (and not $k$ and $\lambda$) since $\mathbb R\ni\lambda\mapsto c(\lambda)\in\mathbb R$ is bijective. Both for the first case (for which $1/(\nu d)\in\mathbb N$ is necessary) and for the second case, we can easily first choose an appropriate $k$
and then $c(\lambda)$ via \eqref{eq:disprel_const_vort_1} or \eqref{eq:disprel_const_vort_2}. The more interesting question is whether there can be multiple solutions for $k$ for fixed $\lambda$. Clearly, it suffices to focus on the second case. To this end, let us introduce $x=k\nu d$ and write \eqref{eq:disprel_const_vort_2} as $c(\lambda)=b^\pm(x)$ with
\[b^\pm(x)=\frac{2\sigma}{d^3\gamma}\frac{x^2-1}{1\pm\sqrt{1+\xi\frac{x(x^2-1)I_0(x)}{I_1(x)}}},\]
where
\[\xi\coloneqq\frac{4\sigma}{d^5\gamma^2}.\]
Here, $b^-(1)$ and possibly $b^\pm(0)$ are to be interpreted as the limit of the above expression as $x$ tends to $1$ or $0$; the limit for $x\to 1$ exists since $x=1$ is a simple root of both nominator and denominator, and the limit for $x\to 0$ also as $I_1(0)=0$ and $I_1'(0)=1/2$. Having clarified this, we see that $b^\pm$ is smooth on $(x_1,\infty)$ and continuous on $[x_1,\infty)$ if $\xi>1/2$, smooth on $(0,\infty)$ and continuous on $[0,\infty)$ if $\xi=1/2$, and smooth on $[0,\infty)$ if $\xi<1/2$. Now notice that it obviously suffices to consider $\gamma>0$ in the following without loss of generality.
We have
\begin{align}\label{eq:disp_rel_bpm}
0=(b^\pm(x))^2\mathcal D(-(k\nu)^2,\lambda)=b^\pm(x)^2f(x)+\frac{\sigma}{d}(1-x^2)+d^2\gamma b^\pm(x),
\end{align}
where
\[f(x)\coloneqq\frac{xI_0(x)}{I_1(x)}.\]
Differentiating \eqref{eq:disp_rel_bpm} with respect to $x$ yields
\begin{align}
b^\pm_x(2b^\pm f+d^2\gamma)&=-(b^\pm)^2f_x+\frac{2\sigma x}{d},\label{eq:b_x}\\
b^\pm_{xx}(2b^\pm f+d^2\gamma)&=-4b^\pm b^\pm_xf_x-2(b^\pm_x)^2f-(b^\pm)^2f_{xx}+\frac{2\sigma}{d}.\label{eq:b_xx}
\end{align}
Thus, if $b^\pm_x=0$ at some $x>0$, then
\begin{align*}
b^\pm_{xx}(2b^\pm f+d^2\gamma)=\frac{2\sigma}{d}-(b^\pm)^2f_{xx}=\frac{2\sigma}{d}\frac{f_x-xf_{xx}}{f_x}.
\end{align*}
Here, we notice that $f_x>0$ for $x>0$ because of
\[f_x(x)=x\left(1-\frac{I_0(x)/I_1(x)}{I_1(x)/I_2(x)}\right)\ge x\left(1-\frac{1+\sqrt{x^2+1}}{1+\sqrt{x^2+9}}\right)>0\]
due to \cite{Amos}. Moreover,
\begin{align}\label{eq:b_bracket_sign}
2b^\pm(x)f(x)+d^2\gamma=\pm d^2\gamma\sqrt{1+\xi\frac{x(x^2-1)I_0(x)}{I_1(x)}}\gtrless 0.
\end{align}
Furthermore, we have
\begin{align}\label{eq:fx-xfxx}
f_x(x)-xf_{xx}(x)=-\frac{2x^2I_0(x)^3}{I_1(x)^3}+\frac{4xI_0(x)^2}{I_1(x)^2}+\frac{2x^2I_0(x)}{I_1(x)}-2x>0.
\end{align}
Instead of presenting a lengthy, not very instructive proof of this inequality we provide a plot of the left-hand side (multiplied by a suitable positive function) in Figure \ref{fig:besselplot} in order to convince the reader of the validity of \eqref{eq:fx-xfxx}.
\begin{figure}[h!]
\centering
\includegraphics{besselplot}
\caption{Demonstration of the validity of \eqref{eq:fx-xfxx}.}
\label{fig:besselplot}
\end{figure}
Thus, putting everything together, $b^\pm_{xx}\gtrless 0$ provided $b^\pm_x=0$. In particular, $b^\pm$ can have at most one critical point on $(0,\infty)$, which, if it exists, has to be a local minimum (maximum). Since moreover $b^\pm$ tends to $\pm\infty$ as $x\to\infty$ by \eqref{eq:Bessel_ratio_est}, we conclude that the monotonicity properties of $b^\pm$ can be characterised by its behaviour near $0$ if $\xi\le1/2$ or near $x_1$ if $\xi>1/2$.
The easy case is $\xi>1/2$. Since
\[b^+(x_1)=b^-(x_1)=\frac{2\sigma(x_1^2-1)}{d^3\gamma},\quad\lim_{\substack{x\to x_1\\x>x_1}}b^\pm_x(x)=\pm\infty\]
due to $\chi_x(x_1)\ne 0$, we conclude that $b^\pm$ is strictly monotonically increasing (decreasing) on $[x_1,\infty)$ and $b^+((x_1,\infty))\cap b^-((x_1,\infty))=\emptyset$.
If $\xi=1/2$, we can argue similarly. Still we have $b^+(0)=b^-(0)=-2\sigma d^{-3}\gamma^{-1}$, but $b^\pm_x$ remains bounded as $x\to 0$. Indeed, from the Taylor expansion
\[1+\frac{x(x^2-1)I_0(x)}{2I_1(x)}=\frac78x^2+\mathcal O(x^4),\quad x\to 0,\]
we infer that
\[\lim_{\substack{x\to 0\\x>0}}b^\pm_x(x)=\pm\frac{2\sigma}{d^3\gamma}\cdot\frac{7/4}{2\sqrt{7/8}}\gtrless0.\]
Therefore, the same conclusions hold as before, namely, $b^\pm$ is strictly monotonically increasing (decreasing) on $[0,\infty)$ and $b^+((0,\infty))\cap b^-((0,\infty))=\emptyset$.
Let us now turn to the case $\xi<1/2$ and take a look at $x=0$. By \eqref{eq:b_xx}, \eqref{eq:b_bracket_sign}, and $b^\pm_x(0)=0$ because of evenness, we see that $b^\pm_{xx}(0)$ has the same sign as $\pm(2\sigma/d-(b^\pm(0))^2f_{xx}(0))$, or vanishes if and only if $2\sigma/d=(b^\pm(0))^2f_{xx}(0)$. Now
\[\frac{2\sigma}{d}-(b^\pm(0))^2f_{xx}(0)=\frac{2\sigma}{d}-\left(\frac{2\sigma}{d^3\gamma}\cdot\frac{-1}{1\pm\sqrt{1-2\xi}}\right)^2\cdot\frac12=\frac{\sigma}{4d\xi}(9\xi-1\pm\sqrt{1-2\xi})\eqqcolon\frac{\sigma}{4d\xi}g^\pm(\xi).\]
First, because of
\[g^+(0)=0,\quad g^+(1/2)=7/2>0,\quad g^+_{\xi\xi}(\xi)=-\frac{1}{(1-2\xi)^{3/2}}<0,\]
$g^+$ is positive on $(0,1/2)$. Therefore, for any $\xi<1/2$, $b^+$ is strictly monotonically increasing on $[0,\infty)$. Second, we have
\[g^-(0)=-2<0,\quad g^-(1/2)=7/2>0,\quad g^-_\xi(\xi)=9+\frac{1}{\sqrt{1-2\xi}}>0,\quad g^-(16/81)=0,\]
and thus
\[g^-(\xi)\begin{cases}<0,&0\le\xi<16/81,\\=0,&\xi=16/81,\\>0,&16/81<\xi<1/2.\end{cases}\]
Hence, $b^-$ is strictly monotonically decreasing on $[0,\infty)$ if $\xi>16/81$ and has exactly one local extremum (which is in fact a global maximum) if $\xi<16/81$. We moreover want to prove that $\max b^-<\min b^+$, that is, $\max b^-<b^+(0)$. To this end, first notice that $b^-<0$ on $[0,\infty)$ since both the nominator and denominator in the definition of $b^-$ have a simple root at $x=1$ and thus $b^-$ cannot have a zero. By \eqref{eq:b_x} we therefore have
\[\max b^-\le -\inf_{x>0}\sqrt{\frac{2\sigma x}{df_x(x)}}=-\sqrt{\frac{2\sigma}{d}}\inf_{x>0}\frac{1}{\sqrt{1-I_0(x)I_2(x)/I_1(x)^2}}\le-\sqrt{\frac{2\sigma}{d}}.\]
Hence,
\[b^+(0)=-\frac{2\sigma}{d^3\gamma(1+\sqrt{1-2\xi})}=-\sqrt{\frac{\sigma}{d}}\cdot\frac{\sqrt{\xi}}{1+\sqrt{1-2\xi}}>-\frac14\sqrt{\frac{\sigma}{d}}>-\sqrt{\frac{2\sigma}{d}}\ge\max b^-,\]
since $\xi\mapsto\sqrt\xi/(1+\sqrt{1-2\xi})$ is strictly monotonically increasing on $[0,16/81]$.
Let us now consider $\xi=16/81$. Differentiating \eqref{eq:b_xx} twice more, evaluating at $0$, and using $b^-_x(0)=b^-_{xx}(0)=b^-_{xxx}(0)=0$ yields
\[b^-_{xxxx}(0)(2b^-(0)f(0)+d^2\gamma)=-b^-(0)^2f_{xxxx}(0)=\frac14b^-(0)^2>0.\]
In particular, $b^-_{xxxx}(0)<0$; hence, $b^-$ is strictly monotonically decreasing.
To summarise, for fixed $\lambda$ we have therefore proved the following, provided $1/(\nu d)\notin\mathbb N$; below in Figure \ref{fig:plots_b} the respective cases are visualised:
\begin{itemize}
\item If $\xi\ge 16/81$:
\begin{itemize}
\item The dispersion relation $\mathcal D(-(k\nu)^2,\lambda)=0$ can have at most one root $k\in\mathbb N$.
\item If $16/81\le\xi<1/2$ and
\[-\frac{2\sigma}{d^3\gamma(1-\sqrt{1-2\xi})}<c(\lambda)<-\frac{2\sigma}{d^3\gamma(1+\sqrt{1-2\xi})},\]
the dispersion relation has no root.
\end{itemize}
\item If $\xi<16/81$:
\begin{itemize}
\item If
\[c(\lambda)>-\frac{2\sigma}{d^3\gamma(1+\sqrt{1-2\xi})}\quad\text{or}\quad c(\lambda)=\max b^-\quad\text{or}\quad c(\lambda)\le-\frac{2\sigma}{d^3\gamma(1-\sqrt{1-2\xi})},\]
the dispersion relation has at most one root.
\item If
\[\max b^-<c(\lambda)\le-\frac{2\sigma}{d^3\gamma(1+\sqrt{1-2\xi})},\]
the dispersion relation has no root.
\item If
\[-\frac{2\sigma}{d^3\gamma(1-\sqrt{1-2\xi})}<c(\lambda)<\max b^-,\]
the dispersion relation has at most two roots.
\end{itemize}
\end{itemize}
\begin{figure}[h!]
\centering
\subfigure[$\xi\ge1/2$.]{\includegraphics[width=0.33\columnwidth]{b_06.pdf}}\subfigure[$16/81\le\xi<1/2$.]{\includegraphics[width=0.33\columnwidth]{b_04.pdf}}\subfigure[$0<\xi<16/81$.]{\includegraphics[width=0.33\columnwidth]{b_005.pdf}}
\caption{Qualitative behaviour of $b^\pm$ for different $\xi$ (in the case $\gamma>0$). Here, $y_0\coloneqq2\sigma d^{-3}\gamma^{-1}(x_1^2-1)$ and $y_\pm\coloneqq-2\sigma d^{-3}\gamma^{-1}/(1\pm\sqrt{1-2\xi})$.}
\label{fig:plots_b}
\end{figure}
If $1/(\nu d)\in\mathbb N$ and
\[c(\lambda)=-\frac{d^2\gamma I_1(1)}{I_0(1)},\]
there is the additional root $k=1/(\nu d)$.
If, however, $\gamma<0$, these statements remain true after reversing all inequalities in the conditions for $c(\lambda)$ and changing $\max b^-$ to $\min b^-$.
Next, let us turn to the transversality condition, fix $\lambda$, and assume that $\mathcal D(-(k\nu)^2,\lambda)=0$ has exactly one solution $k\in\mathbb N$. Since $c_\lambda(\lambda)\ne 0$, it holds that $\mathcal D_\lambda(-(k\nu)^2,\lambda)\ne 0$ if and only if $\xi\le1/2$ or $k\nu d>x_1$ otherwise.
Finally, we have a look at \eqref{ass:realspec_algsimple}. Here, $q^\lambda=0$ and $h(\lambda)=d^{-2}+2\sigma^{-1}d^{-1}\lambda(4\lambda-\gamma d^2)$ by \eqref{eq:c_gammaconst}. Therefore, $h(\lambda)>0$ for all $\lambda\in\mathbb R$ if $\gamma^2<8\sigma d^{-5}\Leftrightarrow\xi>\frac12$, and in the case $\xi\le\frac12$, $h(\lambda)>0$ if and only if $\lambda\notin[\lambda_-,\lambda_+]$ where
\[\lambda_\pm\coloneqq\frac{|\gamma|d^2(\sgn\gamma\pm\sqrt{1-2\xi})}{8}.\]
\section{Global bifurcation}\label{sec:GlobalBifurcation}
The theory for local bifurcation having set up, we now turn to global bifurcation, which is of course the main motivation of our formulation \enquote{identity plus compact}. To this end, we first state the global bifurcation theorem by Rabinowitz.
\begin{theorem}\label{thm:Rabinowitz}
Let $X$ be a Banach space, $U\subset\mathbb R\times X$ open, and $\mathcal F\in C(U;X)$. Assume that $\mathcal F$ admits the form $\mathcal F(\lambda,x)=x+f(\lambda,x)$ with $f$ compact, and that $\mathcal F_x(\cdot,0)\in C(\mathbb R;L(X,X))$. Moreover, suppose that $\mathcal F(\lambda_0,0)=0$ and that $\mathcal F_x(\lambda,0)$ has an odd crossing number at $\lambda=\lambda_0$. Let $S$ denote the closure of the set of nontrivial solutions of $\mathcal F(\lambda,x)=0$ in $\mathbb R\times X$ and $\mathcal C$ denote the connected component of $S$ to which $(\lambda_0,0)$ belongs. Then one of the following alternatives occurs:
\begin{enumerate}[label=(\roman*)]
\item $\mathcal C$ is unbounded;
\item $\mathcal C$ contains a point $(\lambda_1,0)$ with $\lambda_1\neq\lambda_0$;
\item $\mathcal C$ contains a point on the boundary of $U$.
\end{enumerate}
\end{theorem}
The proof of this theorem in the case $U=X$ can be found in \cite[Theorem II.3.3]{Kielhoefer} and is practically identical to the proof for general $U$.
Now we can prove the following result.
\begin{theorem}\label{thm:GlobalBifurcation}
Assume \eqref{ass:SL-spectrum} and that there exists $\lambda_0\neq 0$ such that the dispersion relation
\[\mathcal D(-(k\nu)^2,\lambda_0)=0,\]
with $\mathcal D$ given by \eqref{eq:d(k,lambda)}, has exactly one solution $k_0\in\mathbb N$ and assume that the transversality condition
\[\mathcal D_\lambda(-(k_0\nu)^2,\lambda_0)\neq 0\]
holds. Let $S$ denote the closure of the set of nontrivial solutions of $\mathcal F(\lambda,\eta,\phi)=0$ in $\mathbb R\times X$ and $\mathcal C$ denote the connected component of $S$ to which $(\lambda_0,0,0)$ belongs. Then one of the following alternatives occurs:
\begin{enumerate}[label=(\roman*)]
\item $\mathcal C$ is unbounded in the sense that there exists a sequence $(\lambda_n,\eta_n,\phi_n)\in\mathcal C$ such that
\begin{enumerate}[label=(\alph*)]
\item $|\lambda_n|\to\infty$, or
\item $\|\eta_n\|_{C^{2,\alpha}([0,L])}\to\infty$, or
\item $\|r^{2/p}\gamma(\Psi_n)+r^{2/p-2}F(\Psi_n)F'(\Psi_n)\|_{L^p(Z_{\eta_n})}\to\infty$ with $p\coloneqq\frac{5}{2-\alpha}$, where $Z_{\eta_n}$ denotes a $L$-periodic instance of the axially symmetric fluid domain in $\mathbb R^3$ corresponding to $\eta_n$ and $\Psi_n=r^2\left(\left(\phi_n+\frac{d^2}{(d+\eta_n)^2}\psi^{\lambda_n}\right)\circ H[\eta_n]^{-1}\right)$ is the corresponding original Stokes stream function,
\end{enumerate}
as $n\to\infty$;
\item $\mathcal C$ contains a point $(\lambda_1,0,0)$ with $\lambda_1\neq\lambda_0$;
\item $\mathcal C$ contains a sequence $(\lambda_n,\eta_n,\phi_n)$ such that $\eta_n$ converges to some $\eta$ in $C^{2,\beta}_{0,\mathrm{per},\mathrm{e}}(\mathbb R)$ for any $\beta\in(0,\alpha)$ and such that there exists $z\in[0,L]$ with
\[\eta(z)=-d,\]
that is, intersection of the surface profile with the cylinder axis occurs.
\end{enumerate}
\end{theorem}
\begin{proof}
As was already observed in Lemma \ref{lma:M_prop}, our nonlinear operator $\mathcal F$ is of class $C^2$ and admits the form \enquote{identity plus compact} on each $\mathbb R\times\mathcal U_\varepsilon$, $\varepsilon>0$. Moreover, it is well-known that $F_{(\eta,\phi)}(\lambda,\eta,\phi)$ has an odd crossing number at $(\lambda_0,0,0)$ provided $F_{(\eta,\phi)}(\lambda_0,0,0)$ is a Fredholm operator with index zero and one-dimensional kernel, and the transversality condition holds. These properties, in turn, are consequences of the hypotheses of the theorem in view of Lemmas \ref{lma:kernel} and \ref{lma:transversality_condition} since $F_{(\eta,\phi)}(\lambda_0,0,0)$ coincides with $\mathcal L(\lambda_0)$ up to an isomorphism. For each $\varepsilon>0$, we can thus apply Theorem \ref{thm:Rabinowitz} with $U$ chosen to be the interior of $\mathbb R\times\mathcal U_\varepsilon$. Thus, on each $\mathbb R\times\mathcal U_\varepsilon$, $\mathcal C$ coincides with its counterpart obtained from Theorem \ref{thm:Rabinowitz}. Since $\varepsilon>0$ is arbitrary and $\mathbb R\times\mathcal U=\bigcup_{\varepsilon>0}(\mathbb R\times\mathcal U_\varepsilon)$, it is evident that necessarily
\begin{align}\label{eq:alternative_inf}
\inf_{(\lambda,\eta,\phi)\in\mathcal C}\min_\mathbb R(\eta+d)=0
\end{align}
whenever $\mathcal C$ is bounded in $\mathbb R\times X$ and (ii) fails to hold.
Let us investigate alternative (i) further. In order to show that it can be as stated above, we show that, in view of alternative (i) of Theorem \ref{thm:Rabinowitz}, $\mathcal C$ is bounded in $\mathbb R\times X$ if (i)(a)--(c) and \eqref{eq:alternative_inf} fail to hold. Indeed, along $\mathcal C$ we have $\phi=\mathcal A(\lambda,\eta,\phi)$ and, since \eqref{eq:alternative_inf} does not hold, $\eta+d\ge\varepsilon$ uniformly for some $\varepsilon>0$. Thus,
\begin{align*}
&\|\phi\|_{C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0})}+\|\mathcal I\phi\|_{H^1_\mathrm{per}(\Omega_0^\mathcal I)}\le\|\mathcal I\phi\|_{C^{0,\alpha}_\mathrm{per}(\overline{\Omega_0^\mathcal I})}+\|\mathcal I\phi\|_{H^1_\mathrm{per}(\Omega_0^\mathcal I)}\le C\|\mathcal I\phi\|_{W^{2,p}(\tilde\Omega_0^\mathcal I)}\\
&\le C\Bigg(\|\eta\|_{C^{2,\alpha}([0,L])},\varepsilon^{-1},\Bigg\|\gamma\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)\\
&\phantom{\le\;c\Bigg(\|\eta\|_{C^{2,\alpha}([0,L])},}+\frac{1}{(d+\eta)^2|y|^2}(FF')\left((d+\eta)^2|y|^2\left(\mathcal I\phi+\frac{d^2}{(d+\eta)^2}\mathcal I\psi^\lambda\right)\right)+L^\eta\frac{d^2\mathcal I\psi^\lambda}{(d+\eta)^2}\Bigg\|_{L^p(\tilde\Omega_0^\mathcal I)}\Bigg)\\
&\le C\Bigg(\|\eta\|_{C^{2,\alpha}([0,L])},\varepsilon^{-1},|\lambda|,\Bigg\|s^{3/p}\Bigg[\gamma\left((d+\eta)^2s^2\left(\phi+\frac{d^2}{(d+\eta)^2}\psi^\lambda\right)\right)\\
&\omit\hfill$\displaystyle+\frac{1}{(d+\eta)^2s^2}(FF')\left((d+\eta)^2s^2\left(\phi+\frac{d^2}{(d+\eta)^2}\psi^\lambda\right)\right)\Bigg]\Bigg\|_{L^p(\tilde\Omega_0)}\Bigg)$\\
&\le C\Bigg(\|\eta\|_{C^{2,\alpha}([0,L])},\varepsilon^{-1},|\lambda|,\Bigg\|\Bigg(s^{3/p}\Bigg[\gamma\left((d+\eta)^2s^2\left(\phi+\frac{d^2}{(d+\eta)^2}\psi^\lambda\right)\right)\\
&\omit\hfill$\displaystyle+\frac{1}{(d+\eta)^2s^2}(FF')\left((d+\eta)^2s^2\left(\phi+\frac{d^2}{(d+\eta)^2}\psi^\lambda\right)\right)\Bigg]\Bigg)\circ H[\eta]^{-1}\Bigg\|_{L^p(\tilde\Omega_\eta)}\Bigg)$\\
&\le C\left(\|\eta\|_{C^{2,\alpha}([0,L])},\varepsilon^{-1},|\lambda|,\left\|r^{3/p}\left[\gamma(\Psi)+\frac{1}{r^2}(FF')(\Psi)\right]\right\|_{L^p(\tilde\Omega_\eta)}\right)\\
&\le C\left(\|\eta\|_{C^{2,\alpha}([0,L])},\varepsilon^{-1},|\lambda|,\left\|r^{2/p}\gamma(\Psi)+r^{2/p-2}(FF')(\Psi)\right\|_{L^p(Z_\eta)}\right)
\end{align*}
after using Sobolev's embedding, the Calderón--Zygmund inequality (see \cite[Chapter 9]{GilbargTrudinger}; notice that on the right-hand side the term $\|\mathcal I\phi\|_{L^p(\tilde\Omega_0^\mathcal I)}$ can be left out because of unique solvability of the Dirichlet problem associated to $L^\eta$), and changes of variables via $H[\eta]$ and via cylindrical coordinates in $\mathbb R^5$ and $\mathbb R^3$, and where $\tilde\Omega_\eta$ denotes a periodic instance of $\Omega_{\eta}=H[\eta](\Omega_0)$ and $\Psi$, $Z_\eta$ are analogously defined as in the statement of (c); here, the constant $C>0$ can change in each step.
Finally, we turn to alternative (iii). If \eqref{eq:alternative_inf} holds, but not (i)(b), then clearly we find a sequence as described in (iii) due to the compact embedding of $C_{0,\mathrm{per},\mathrm{e}}^{2,\alpha}(\mathbb R)$ in $C_{0,\mathrm{per},\mathrm{e}}^{2,\beta}(\mathbb R)$.
\end{proof}
\begin{remark}
Alternative (i)(c) says that the angular component of the vorticity, in general given by $\omega^\vartheta=\vec{\omega}\cdot\vec{e}_\vartheta=-r\gamma(\Psi)-(FF')(\Psi)/r$, satisfies $\|r^{2/p-1}\omega^\vartheta_n\|_{L^p(Z_{\eta_n})}\to\infty$ as $n\to\infty$.
\end{remark}
We also have the following.
\begin{proposition}
In Theorem \ref{thm:GlobalBifurcation} the alternative (i)(b) can be replaced by
\begin{enumerate}[leftmargin=40pt]
\item[(i)(b')]
\begin{enumerate}[label=(\greek*)]
\item $\|\eta_n\|_{C^{1,\alpha}([0,L])}\to\infty$, or
\item $\||\vec{u}_n|^2\|_{C^{0,\alpha}(S_n)}\to\infty$ (the square of the velocity [the kinetic energy density] is unbounded in $C^{0,\alpha}$ at the free surface $S_n$), or
\item $|Q(\lambda_n,\eta_n,\phi_n)|\to\infty$ (the Bernoulli constant is unbounded).
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
This follows easily from the Bernoulli equation
\[Q(\lambda,\eta,\phi)=\frac12|\vec{u}|^2-\sigma\left(\frac{\eta_{zz}}{(1+\eta_z^2)^{3/2}}-\frac{1}{(d+\eta)\sqrt{1+\eta_z^2}}\right)\]
at the free surface.
\end{proof}
\small{\textbf{Acknowledgements.}
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no 678698).
A.E., supported in 2020 by the Kristine Bonnevie scholarship 2020 of the Faculty of Mathematics and Natural Sciences, University of Oslo, during his research stay at Lund University, wishes to thank Erik Wahl\'en and the Centre of Mathematical Sciences, Lund University for hosting him. A.E. was partially supported by the DFG under Germany's Excellence Strategy – MATH$^+$: The Berlin Mathematics Research Center (EXC-2046/1 – project ID: 390685689) via the project AA1-12$^*$}
\bibliographystyle{siam}
|
2,877,628,090,253 | arxiv |
\section{Framework of the Burnett spectral method}
\label{sec:general}
Burnett polynomials are introduced in \cite{Burnett1936} to study
high-order approximation to the distribution function for a slightly
non-uniform gas. Here we adopt a normalized form and write Burnett
polynomials as
\begin{displaymath}
p_{lmn}(\bv) = \sqrt{\frac{2^{1-l} \pi^{3/2} n!}{\Gamma(n+l+3/2)}}
L_n^{(l+1/2)} \left( \frac{|\bv|^2}{2} \right) |\bv|^l
Y_l^m \left( \frac{\bv}{|\bv|} \right), \qquad
l,n \in \bbN, \quad m = -l,\cdots,l,
\end{displaymath}
where we have used Laguerre polynomials
\begin{displaymath}
L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \exp(x)}{n!}
\frac{\mathrm{d}^n}{\mathrm{d}x^n}
\left[ x^{n+\alpha} \exp(-x) \right],
\end{displaymath}
and spherical harmonics
\begin{displaymath}
Y_l^m(\bn) = \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}
P_l^m(\cos \theta) \exp(\mathrm{i} m \phi), \qquad
\bn = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)^T
\end{displaymath}
with $P_l^m$ being the associate Legendre polynomial:
\begin{displaymath}
P_l^m(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2}
\frac{\mathrm{d}^{l+m}}{\mathrm{d}x^{l+m}} \left[ (x^2-1)^l \right].
\end{displaymath}
By the orthogonality of Laguerre polynomials and spherical harmonics,
one can find that
\begin{displaymath}
\int_{\bbR^3} \overline{p_{l_1 m_1 n_1}(\bv)} p_{l_2 m_2 n_2}(\bv)
\mathcal{M}(\bv) \dd\bv =
\delta_{l_1 l_2} \delta_{m_1 m_2} \delta_{n_1 n_2}.
\end{displaymath}
Now we introduce the basis function $\varphi_{lmn}(\bv)$ as
\begin{equation}
\varphi_{lmn}(\bv) = p_{lmn}(\bv)\mathcal{M}(\bv).
\end{equation}
For a given distribution function $f \in \mathcal{S}$, we assume that
it has the expansion
\begin{equation} \label{eq:exp_f}
f(\bv) = \sum_{lmn} \tilde{f}_{lmn} \varphi_{lmn}(\bv),
\end{equation}
where the sum is interpreted as
\begin{displaymath}
\sum_{lmn} = \sum_{l=0}^{+\infty} \sum_{m=-l}^l \sum_{n=0}^{+\infty}.
\end{displaymath}
Suppose the corresponding collision term $\mQ[f,f]$ also has the
expansion
\begin{displaymath}
\mQ[f,f](\bv) = \sum_{lmn} \tilde{Q}_{lmn} \varphi_{lmn}(\bv).
\end{displaymath}
By the orthogonality of the Burnett polynomials and the bilinearity of
the operator $\mQ[\cdot,\cdot]$, one can find that
\begin{displaymath}
\tilde{Q}_{lmn} = \sum_{l_1 m_1 n_1} \sum_{l_2 m_2 n_2}
A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}
\tilde{f}_{l_1 m_1 n_1} \tilde{f}_{l_2 m_2 n_2},
\end{displaymath}
where
\begin{equation} \label{eq:A}
A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2} = \int_{\bbR^3}
\overline{p_{lmn}(\bv)}
\mQ[\varphi_{l_1 m_1 n_1}, \varphi_{l_2 m_2 n_2}](\bv) \dd \bv.
\end{equation}
Based on this expansion, it is obvious that \eqref{eq:Boltzmann} is
equivalent to the following ODE system:
\begin{equation}
\label{eq:Boltzmann_ODE}
\begin{aligned}
& \frac{\mathrm{d} \tilde{F}_{lmn}(t)}{\mathrm{d}t} =
\sum_{l_1 m_1 n_1} \sum_{l_2 m_2 n_2}
A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}
\tilde{F}_{l_1 m_1 n_1}(t) \tilde{F}_{l_2 m_2 n_2}(t), \\
& \tilde{F}_{lmn}(0) = \tilde{f}_{lmn}^0 :=
\int_{\bbR^3} \overline{p_{lmn}(\bv)} f^0(\bv) \dd \bv,
\end{aligned}
\end{equation}
where $\tilde{F}_{lmn}(t)$ are the coefficients in the Burnett
series expansion of $F(t)$.
To develop the spectral method, one needs to truncate the Burnett
series to restrict the computation to a finite number of coefficients.
A common choice is to choose a positive integer $M$ and require that
the degree of the polynomial, $l + 2n$, to be less than or equal to
$M$. Thus the spectral method for the homogeneous Boltzmann equation
\eqref{eq:Boltzmann} is
\begin{subequations} \label{eq:ODE}
\begin{align}
\label{eq:ODE1}
& \frac{\mathrm{d} \tilde{F}_{lmn}(t)}{\mathrm{d}t} =
\sum_{\substack{l_1 m_1 n_1\\[2pt] l_1 + 2n_1 \leqslant M}}
\sum_{\substack{l_2 m_2 n_2\\[2pt] l_2 + 2n_2 \leqslant M}}
A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}
\tilde{F}_{l_1 m_1 n_1}(t) \tilde{F}_{l_2 m_2 n_2}(t), \\
& \tilde{F}_{lmn}(0) = \tilde{f}_{lmn}^0, \qquad l + 2n \leqslant M.
\end{align}
\end{subequations}
These ordinary differential equations can be solved by Runge-Kutta
methods. Naively, the computational cost appears to be
$O(N^3)=O(M^9)$, where $N=(M+1)(M+2)(M+3)/6$ is the total number of
$\tilde{F}_{lmn}$, $l+2n\leq M$. It is worth pointing out that there
is a prefactor $1 / 6^3$ of $O(M^9)$ when we count the number of
coefficients $A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}$, and this prefactor
will be directly brought into the computational cost of the collision
term.
However, the actual computational cost can be reduced to $O(M^8)$ due
to the following sparsity of the coefficients $A_{lmn}^{l_1 m_1 n_2,
l_2 m_2 n_2}$:
\begin{theorem} \label{thm:sparsity}
The coefficient $A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}$ is zero if $m
\neq m_1 + m_2$.
\end{theorem}
By taking into account such sparsity, we can find that the number of
nonzero coefficients $A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}$ is $O(M^8)$
with a prefactor $1/297$. This indicates that the evaluation of the
collision can be efficient for not too large $M$. Interestingly, for
some special collision kernel $B(\cdot,\cdot)$, the computational cost
can be further reduced due to the following extra sparsity of the
coefficients $A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}$:
\begin{theorem} \label{thm:sparsityMaxwell}
If the kernel $B(g,\chi)=\sigma(\chi)$ is independent of $g$, the
coefficient $A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}$ is zero if
$l_1+2n_1+l_2+2n_2 \neq l+2n$.
\end{theorem}
A well-known type of collision kernel satisfying the above condition
is the Maxwell molecules, for which the force between a pair of
molecules is always repulsive and proportional to the fifth power of
their distance. The sparsity stated in the above theorem allows us to
reduce the number of nonzero coefficients to $O(M^7)$. By a numerical
test, we find that the prefactor of $O(M^7)$ is around $1 / (2.2\times
10^{3})$. Moreover, the following result also helps save computational
resources:
\begin{theorem} \label{thm:real}
All coefficients $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ are real.
\end{theorem}
The proof of all the above theorems will be provided in Section
\ref{sec:proof}. They help us save both memory and computational time.
For general collision kernel, by Theorem \ref{thm:sparsity}, we do not
need to store the zero coefficients, and the constraint $m = m_1 +
m_2$ reduces the order of time complexity by $1$. For Maxwell
molecules, Theorem \ref{thm:sparsity} and Theorem
\ref{thm:sparsityMaxwell} reduce the order of time complexity by $2$.
Theorem \ref{thm:real} does not reduce the order, but by realizing
that all the coefficients are real, one can reduce the storage
requirement by a half, and the algorithm can also be made faster by
avoiding some operations between complex numbers. Actually, we can
further reduce the computational cost by using the fact that the
distribution functions are real: since
\begin{displaymath}
\varphi_{lmn}(\bv) = (-1)^m \overline{\varphi_{lmn}(\bv)},
\end{displaymath}
the coefficients in \eqref{eq:exp_f} must satisfy $\tilde{f}_{lmn} =
(-1)^m \overline{\tilde{f}_{l,-m,n}}$ to ensure that $f(\bv)$ is real;
therefore, when solving the ordinary differential equations
\eqref{eq:ODE}, we only need to take into account the case $m
\geqslant 0$, which cuts down the computational cost by a half. The
small prefactor of the computational complexity also indicates the low
computational cost is acceptable if $M$ is not too large.
However, the time complexity $O(M^8)$ (or $O(M^7)$ for Maxwell
molecules) still gives huge computational cost when $M$ is large,
especially when solving spatially
inhomogeneous problems. To make the computation even cheaper, we adopt
the idea in \cite{Cai2015, QuadraticCol} which replaces the right-hand
side of \eqref{eq:ODE1} by $\tilde{Q}_{lmn}^{*}(t)$, defined as
\begin{equation} \label{eq:Q_star}
\tilde{Q}_{lmn}^{*}(t) = \left\{
\begin{array}{ll}
\displaystyle \sum \limits_{\substack{l_1 m_1 n_1\\[2pt] l_1 + 2n_1 \leqslant M_0}}
\sum\limits_{\substack{l_2 m_2 n_2\\[2pt] l_2 + 2n_2 \leqslant M_0}}
A_{lmn}^{l_1 m_1 n_2, l_2 m_2 n_2}
\tilde{F}_{l_1 m_1 n_1}(t) \tilde{F}_{l_2 m_2 n_2}(t),
& \text{if } l+2n \leqslant M_0,\\[28pt]
-\mu_{M_0}\tilde{F}_{lmn}(t), & \text{otherwise}.
\end{array}
\right.
\end{equation}
In practice, one can set $M_0$ to be much less than $M$. Thus the
quadratic form is only applied to the first few coefficients whose
associate polynomials have degree less than or equal to $M_0$. When
$l+2n > M_0$, similar to the BGK-type models, we let the coefficient
decay to zero exponentially at a constant rate $\mu_{M_0}$. Thereby we
get the new model
\begin{equation} \label{eq:ODE_final}
\frac{\mathrm{d} \tilde{F}_{lmn}(t)}{\mathrm{d}t} =
\tilde{Q}_{lmn}^{*}(t).
\end{equation}
As in \cite{QuadraticCol, Cai2015}, we choose the decay rate
$\mu_{M_0}$ to be the spectral radius of the linearized collision
operator $\mathcal{L}_{M_0}: \mathcal{S}_{M_0}\rightarrow
\mathcal{S}_{M_0}$ defined as
\begin{equation}
\label{eq:linear_Boltzmann}
\mathcal{L}_{M_0}[f](\bv) =
\sum_{\substack{l_1 m_1 n_1\\[2pt] l_1 + 2n_1 \leqslant M_0}}
\sum_{\substack{n_2 \leqslant (M_0 - l_1)/2}}
\left(A_{l0n_1}^{l0n_2, 000} + A_{l0n_1}^{000, l0n_2} \right)
\tilde{f}_{l_1 m_1 n_2}\psi_{l_1m_1n_1}(\bv),
\end{equation}
where $\mathcal{S}_{M_0} = \text{span}\{\psi_{lmn}(\bv) : l + 2n
\leqslant M_0 \} \cap \mathcal{S}$. We refer the readers to
\cite{Cai2015} for more details.
By now, we have obtained the ordinary differential equations to
approximate the homogeneous Boltzmann equation \eqref{eq:Boltzmann}
under the framework of Burnett polynomials. In order to complete this
algorithm, we still need to find the values of the coefficients
$A_{lmn}^{l_1m_1n_1,l_2m_2n_2}$, which will be detailed in the
following section. Moreover, the implementation of the algorithm will
be discussed deeply to obtain optimal efficiency.
\section{Implementation of the algorithm}
\label{sec:impl}
To implement the algorithm, we first need to find the values of the
coefficients $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$. A formula of these
coefficients has been given in \cite{Kumar}, which reads
\begin{equation}\label{eq:A_lmn}
A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2} =
\sum_{l_3 m_3 n_3} \sum_{l_4 m_4 n_4} \sum_{n_4'}
\overline{\left( \begin{array}{c|c}
n_3 l_3 m_3 & nlm \\ n_4' l_4 m_4 & 000
\end{array} \right)} \left( \begin{array}{c|c}
n_3 l_3 m_3 & n_1 l_1 m_1 \\ n_4 l_4 m_4 & n_2 l_2 m_2
\end{array} \right) V_{n_4 n_4'}^l,
\end{equation}
where
\begin{equation}\label{eq:def_Vnn}
\begin{split}
V_{n n'}^l &= \frac{1}{8\sqrt{2} \pi^{5/2}}
\sqrt{\frac{n!n'!}{\Gamma(n+l+3/2) \Gamma(n'+l+3/2)}}
\int_0^{+\infty} \int_0^{\pi} B(g,\chi)
\left( \frac{g^2}{4} \right)^{l+1} \times {} \\
& \qquad L_n^{(l+1/2)} \left( \frac{g^2}{4} \right)
L_{n'}^{(l+1/2)} \left( \frac{g^2}{4} \right)
[(2l+1)^2 P_l(\cos \chi) - 1]
\exp \left( -\frac{g^2}{4} \right) \dd\chi \dd g,
\end{split}
\end{equation}
and the notation $(\cdot \mid \cdot)$ denotes Talmi coefficients for
equal mass molecules \cite{Talmi1952}. Due to the sparsity of the
Talmi coefficients, the computational cost for evaluating all
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ required in \eqref{eq:ODE} is
$O(M_0^{14})$. Thus, computing the Talmi coefficients is not an
easy task. The reference \cite{Bakri1967} provides a possible
implementation, but the formula involves Wigner 3-$j$ and 9-$j$
symbols, which are also difficult to obtain. Below we are going to
propose another method to compute these coefficients
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ based on the work
\cite{QuadraticCol}. The method also has computational cost
$O(M_0^{14})$, but is much easier to implement.
\subsection{Computation of coefficients
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$}
In \cite{QuadraticCol}, we have calculated the expansion coefficients
of the quadratic collision term $\mQ[f,f](\bv)$ under the framework of
Hermite spectral method. Since both Hermite polynomials and Burnett
polynomials are orthogonal polynomials associated with the same weight
function, we can express the Burnett polynomials by linear
combinations of the Hermite polynomials, and then the coefficients
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ naturally become a linear
combination of the corresponding coefficients in the Hermite spectral
method. Since the expressions for the coefficients in the Hermite
spectral method have been worked out explicitly in
\cite{QuadraticCol}, we do not need to bother using the complicated
symbols in the quantum theory to find the values of $A_{lmn}^{l_1 m_1
n_1, l_2 m_2 n_2}$.
Mathematically, the above framework can be formulated as below. In
\cite{QuadraticCol}, Hermite polynomial $H^{k_1 k_2 k_3}(\bv)$ is
defined as
\begin{equation}
\label{eq:basis}
H^{k_1 k_2 k_3}(\bv) = \frac{(-1)^n}{\mathcal{M}(\bv)}
\frac{\partial^{k_1+k_2+k_3}}{\partial v_1^{k_1} \partial
v_2^{k_2} \partial v_3^{k_2}} \mathcal{M}(\bv), \quad \forall k_1, k_2, k_3 \in \mathbb{N},
\end{equation}
where $\mathcal{M}(\bv)$ is given in \eqref{eq:Maxwellian}. We would
like to express the Burnett polynomials as
\begin{equation}
\label{eq:exp_he_bur}
p_{lmn}(\bv) =
\sum_{(k_1,k_2,k_3) \in I_{l + 2n}}\frac{1}{k_1!k_2!k_3!}C_{lmn}^{k_1k_2k_3}H^{k_1k_2k_3}(\bv),
\end{equation}
where $I_{l+2n}$ is the index set
\begin{equation}
\label{eq:indexset}
I_{l+2n} = \{(k_1, k_2, k_3) \in \bbN^3 \mid k_1+k_2+k_3 = l+2n\},
\end{equation}
and the coefficients $C_{lmn}^{k_1k_2k_3}$ can be calculated as
\begin{equation} \label{eq:C}
C_{lmn}^{k_1k_2k_3} =
\int_{\bbR^3}p_{lmn}(\bv)H^{k_1k_2k_3}(\bv)\mM(\bv) \dd \bv
\end{equation}
based on the orthogonality of Hermite polynomials
\begin{equation}
\label{eq:Her_orth}
\int_{\bbR^3}H^{k_1k_2k_3}(\bv)H^{l_1l_2l_3}(\bv)\mM(\bv) \dd \bv
= \delta_{k_1l_1}\delta_{k_2l_2}\delta_{k_3l_3}k_1!k_2!k_3!.
\end{equation}
Note that when $l + 2n \neq k_1 + k_2 + k_3$, i.e. the degrees of
$H^{k_1 k_2 k_3}$ and $p_{lmn}$ are not equal, the coefficient
$C_{lmn}^{k_1 k_2 k_3}$ defined by \eqref{eq:C} is zero due to the
orthogonality of both polynomials. Once $C_{lmn}^{k_1k_2k_3}$ is
obtained, we just need to substitute \eqref{eq:exp_he_bur} to the
definition of $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ \eqref{eq:A}, which
results in the following formula for these coefficients:
\begin{equation}
\label{eq:A_detail}
\begin{aligned}
A_{lmn}^{l_1m_1n_1, l_2m_2n_2}
&= \sum_{k \in I_{l + 2n}}\sum_{i \in I_{l_1 +2n_1}} \sum_{j \in I_{l_2 + 2n_2}}
\overline{C_{lmn}^{k_1k_2k_3}} C_{l_1m_1n_1}^{i_1i_2i_3}
C_{l_2m_2n_2}^{j_1j_2j_3} \times{} \\
& \qquad \frac{1}{i_1! i_2! i_3! j_1! j_2! j_3! k_1! k_2! k_3!}
\int_{\bbR^3}H^{k_1k_2k_3}(\bv)
\mQ\left[H^{i_1i_2i_3}\mM,H^{j_1j_2j_3}\mM\right](\bv)\dd \bv.
\end{aligned}
\end{equation}
The second line of \eqref{eq:A_detail} has already been evaluated in
\cite[Theorem 1 \& 2]{QuadraticCol} (denoted as
$A_{k_1k_2k_3}^{i_1i_2i_3,j_1j_2j_3}$ therein). Thus we will focus only on
the computation of the coefficients $C_{lmn}^{k_1k_2k_3}$ below.
Define
\begin{equation}
\label{eq:S}
S_{-1} = \frac{1}{2}(v_1 - \mathrm{i} v_2), \quad S_0 = v_3, \quad S_1 =
-\frac{1}{2}(v_1 + \mathrm{i} v_2),
\end{equation}
and
\begin{equation}
\label{eq:gamma}
\gamma_{lm}^{\mu} = \sqrt{\frac{[l + (2\delta_{1, \mu}-1) m + \delta_{1,
\mu}][l - (2\delta_{-1, \mu} - 1)m + \delta_{-1, \mu}]}{(2l
-1)(2l+1)}},
\end{equation}
the recursive formula of the basis functions \cite{Cai2018} is
\begin{equation}
\label{eq:recursive}
\begin{aligned}
S_{\mu}\psi_{lmn}(\bv) &= \frac{1}{2^{|\mu|}}
\left[\sqrt{2(n+l)+3}\gamma_{l+1, m}^{\mu}\psi_{l+1, m+\mu, n}(\bv) -
\sqrt{2n}\gamma_{l+1, m}^{\mu}\psi_{l+1, m+\mu, n-1}(\bv) \right. \\
&\left. + (-1)^{\mu}\sqrt{2(n+l)+1} \gamma_{-l,m}^{\mu}\psi_{l-1, m+\mu, n}(\bv)
-(-1)^{\mu}\sqrt{2(n+1)}\gamma_{-l,m}^{\mu}\psi_{l-1, m+\mu,n+1}(\bv) \right],
\end{aligned}
\end{equation}
where we set $\psi_{lmn}(\bv) = 0$ if $|m| > l$ or either of $l, n$ is
negative. Equations of $C_{lmn}^{k_1 k_2 k_3}$ can be obtained by
multiplying \eqref{eq:recursive} with $H^{k_1k_2k_3}(\bv)$ and taking
integration with respect to $\bv$ on both sides. The integral of the
right-hand side can be written straightforwardly as expressions of
$C_{lmn}^{k_1 k_2 k_3}$, while for the left-hand side, we need to use
the recursion formula of Hermite polynomials
\begin{equation}
\label{eq:recursive_h}
v_s H^{k_1k_2k_3}(\bv) = H^{k_1+\delta_{1s}, k_2+\delta_{2s},k_3+\delta_{3s}}(\bv)
+ k_sH^{k_1-\delta_{1s}, k_2-\delta_{2s},k_3-\delta_{3s}}(\bv), \quad s = 1,2,3.
\end{equation}
Since both the Hermite and Burnett polynomials are orthogonal
polynomials, integrals including the product of polynomials of
different degrees all vanish. Therefore when applying the above
operations, we choose $k_1, k_2, k_3$ such that $k_1 + k_2 + k_3 =
l+2n+1$, and the resulting equations for $\mu = -1,0,1$ are
respectively
\begin{equation}
\label{eq:recursive_c}
\begin{aligned}
a_{l,m+1,n}^{(-1)}C_{l+1, m, n}^{k_1 k_2 k_3} + b_{l,m+1,n}^{(-1)}C_{l-1, m,
n+1}^{k_1 k_2 k_3} &= \frac{1}{2}k_1C_{l,m+1,n}^{k_1-1,k_2,k_3} -
\frac{\mathrm{i}}{2} k_2 C_{l,m+1,n}^{k_1,k_2-1,k_3}, \\
a_{l,m,n}^{(0)}C_{l+1, m, n}^{k_1 k_2 k_3} + b_{l,m,n}^{(0)}C_{l-1, m,
n+1}^{k_1 k_2 k_3} &= k_3C_{l,m,n}^{k_1,k_2,k_3-1}, \\
a_{l,m-1,n}^{(1)}C_{l+1, m, n}^{k_1 k_2 k_3} + b_{l,m-1,n}^{(1)}C_{l-1, m,
n+1}^{k_1 k_2 k_3} &= -\frac{1}{2}k_1C_{l,m-1,n}^{k_1-1,k_2,k_3} -
\frac{\mathrm{i}}{2} k_2C_{l,m-1,n}^{k_1,k_2-1,k_3},
\end{aligned}
\end{equation}
where
\begin{equation}
\label{eq:coe_a}
a_{lmn}^{(\mu)} = \frac{1}{2^{|\mu|}}\sqrt{(2(n+l)
+3)}\gamma_{l+1,n}^{\mu},
\quad b_{lmn}^{(\mu)} = \frac{(-1)^{\mu+1}}{2^{|\mu|}}\sqrt{2(n+1)
}\gamma_{-l,n}^{\mu}, \quad \mu = -1, 0, 1,
\end{equation}
and we have exchanged the left-hand side and the right-hand side since
we would like to solve the coefficients $C_{l+1, m, n}^{k_1 k_2 k_3}$
and $C_{l-1, m, n+1}^{k_1 k_2 k_3}$ from the above equations. This is
possible since on the left-hand sides of \eqref{eq:coe_a}, the sum of
the superscripts $k_1 + k_2 + k_3$ is always greater than the same sum
on the right-hand sides. Hence we can solve all the coefficients
$C_{lmn}^{k_1 k_2 k_3}$ by the order of $k_1 + k_2 + k_3$, so that the
right-hand sides of \eqref{eq:recursive_c} are always known. To start
the computation, we need the ``initial condition'' $C_{000}^{000}=1$,
and the ``boundary conditions'' $C_{lmn}^{k_1k_2k_3} = 0$ if $|m| > l$
or either of $l, n$ is negative. It is not difficult to see that the
computational cost for each coefficient is $O(1)$, and thus the time
complexity for computing all the coefficients $C_{lmn}^{k_1k_2k_3}$ with
$l+2n = k_1 + k_2 + k_3 \leqslant M_0$ is $O(M_0^5)$.
Now we come back to \eqref{eq:A_detail}. From Theorem
\ref{thm:sparsity}, it is known that the total number of nonzero
$A_{lmn}^{l_1m_1n_1, l_2m_2n_2}$ is $O(M_0^8)$, and the computational
cost of each summation symbol on the right-hand side of
\eqref{eq:A_detail} is at most $O(M_0^2)$ from the definition of the
index set $I_{M_0}$. Therefore, based on the knowledge of the second
line of \eqref{eq:A_detail}, the total computational cost for all the
coefficients $A_{lmn}^{l_1m_1n_1, l_2m_2n_2}$ is $O(M_0^{14})$. In
fact, to get the second line of \eqref{eq:A_detail}, the computational
cost is only $O(M_0^{12})$ as stated in \cite{QuadraticCol}. Thus the
overall complexity for finding $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ is
$O(M_0^{14})$. Here we emphasize again that such a computational cost
is only for the precomputation, which needs to be done only once.
Finally, we would like to comment that the computational cost for
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ can be further reduced to one
eighth by using the symmetry of Burnett polynomials
\begin{equation}
\label{eq:odevity}
p_{lmn}(v_1, v_2, -v_3) = (-1)^{l+m}p_{lmn}(v_1, v_2, v_3),
\end{equation}
which means the coefficient $C_{lmn}^{k_1k_2k_3}$ is nonzero only if
$(l+m) - k_3$ is even. By now, we have been able to make the whole
algorithm work, and the rest of this section will be devoted to our
detailed implementation of \eqref{eq:Q_star}, including the design of
the data structure and the detailed steps of our fast algorithm.
\subsection{Data structure: storage of the coefficients}
The optimal data structure to store the coefficients $\tilde{f}_{lmn}$
and $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ should require minimum
``jumps'' in the memory, which means the order of data usage should
match the storage of the data as much as possible. In what follows, we
are going to show by illustration how the data are arranged to achieve
optimal continuity.
\subsubsection{Storage of the coefficients $\tilde{f}_{lmn}$}
Suppose we need to store the coefficients $\tilde{f}_{lmn}$ for all
$l+2n \leqslant M$. We store all the coefficients in a one-dimensional
continuous array. This array can be viewed as the concatenation of
$2M+1$ sections, and each section contains all the coefficients for a
given $m$. Inside each section, the coefficients are ordered as shown
in Figure \ref{fig:f}, where the value of $l+2n$ (the degree of the
corresponding Burnett polynomial) is increasing, and when $l+2n$ is a
constant, the value of $l$ is increasing.
\begin{figure}[!ht]
\centering
\subfigure[Storage pattern of $\tilde{f}_{lmn}$]{
\includegraphics[width=.7\textwidth]{storage-1}
}\qquad\quad
\subfigure[Example: $M=2$]{
\includegraphics[width=.17\textwidth]{storage_ex-1.pdf}
}
\caption{Storage pattern of $\tilde{f}_{lmn}$ showing the two-level
structure of the array. The left column of (a) shows that the first-level
decomposition of the array (one section for each $m$), and the right
column of (a) shows how the elements are stored in the second-level
structure. (b) presents an example with $M=2$.}
\label{fig:f}
\end{figure}
By this storage scheme, for any given $m$, the coefficients associated
with the polynomials of degree less than or equal to $M_0$ are
continuously stored, which makes it easier to perform the
matrix-vector multiplication in the numerical algorithm and achieve
good cache hit ratio.
\subsubsection{Storage of the coefficients $A_{lmn}^{l_1 m_1 n_1, l_2 m_2
n_2}$}
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{storage-2.pdf}
\caption{Storage pattern of $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$. The
left column shows the three-dimensional view of the coefficients for
given $m$, $m_1$ and $m_2$. The right column gives the general
two-level structure of the array.}
\label{fig:A}
\end{figure}
Similar to the storage of $\tilde{f}_{lmn}$, all the coefficients
$A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ are also stored in a continuous
array, which can again be considered as the concatenation of a number
of sections. Each section contains all the coefficients for given $m$,
$m_1$ and $m_2$. Noting that the range of $m$ is from $0$ to $M_0$
while the range of $m_1$ and $m_2$ is from $-M_0$ to $M_0$, we can
find that the total number of sections is $(3M_0 + 2)(M_0 + 1)/2$.
Once $m$ is given, the two indices $l$ and $n$ can be viewed as a
one-dimensional index $ln$ by our ordering rule in the storage pattern
of $\tilde{f}_{lmn}$. Similarly, $l_1 n_1$ and $l_2 n_2$ can also be
regarded as one-dimensional indices. Thus, once $m$, $m_1$ and $m_2$
are given, the coefficients $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ can
be considered as a three-dimensional array, whose three indices are
$ln$, $l_1 n_1$ and $l_2 n_2$ (see the left column of Figure
\ref{fig:A}). Its storage is a simple flattening the three-dimensional
array and is illustrated in the right column of Figure \ref{fig:A}.
\subsection{Details of the algorithm}
Based on the above data structure, the computation of
$\tilde{Q}_{lmn}^*$ can be implemented very efficiently. The general
procedure is as follows:
\renewcommand{\thealgorithm}{ }
\begin{algorithm}[ht]
\label{alg:Q}
\caption{Algorithm to Calculate $\tilde{Q}_{lmn}^*$}
\begin{algorithmic}[1]
\begin{minipage}{0.8\textwidth}
\For {$m$ from $0$ to $M$}
\If{$m > M_0$}
\ForAll{$l,n$ satisfying $l \geqslant m$, $l+2n \leqslant M$}
\State $\tilde{Q}_{lmn}^* \gets -\mu_{M_0} \tilde{f}_{lmn}$
\EndFor
\Else
\ForAll{$l,n$ satisfying $l \geqslant m$, $M_0 < l+2n \leqslant M$}
\State $\tilde{Q}_{lmn}^* \gets -\mu_{M_0} \tilde{f}_{lmn}$
\EndFor
\For {$m_1$ from $m-M_0$ to $M_0$}
\State $m_2 \gets m - m_1$
\ForAll{$l,n$ satisfying $l \geqslant m$, $l+2n \leqslant M_0$}
\State \rule[-.3\baselineskip]{0pt}{1.6\baselineskip}
$\displaystyle \tilde{Q}_{lmn}^* \gets \sum_{l_1+2n_1 \leqslant
M_0} \sum_{l_2+2n_2 \leqslant M_0} A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}
\tilde{f}_{l_1 m_1 n_1} \tilde{f}_{l_2 m_2 n_2}$
\EndFor
\EndFor
\EndIf
\EndFor
\ForAll{$m = -M, \cdots, -1$ and $l,n$ satisfying $l \geqslant |m|$, $l+2n \leqslant M$}
\State $\tilde{Q}_{lmn}^* \gets (-1)^m \overline{\tilde{Q}_{l,-m,n}^*}$
\EndFor
\end{minipage}
\end{algorithmic}
\end{algorithm}
It is worth noting that line 10 can be implemented by two
matrix-vector multiplications:
\begin{align}
\label{eq:tilde_g}
1: & \qquad \tilde{g}_{lmn}^{l_1 m_1 n_1} = \sum_{l_2 + 2n_2
\leqslant M_0} A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}
\tilde{f}_{l_2 m_2 n_2}, \\
\label{eq:tilde_Q}
2: & \qquad \tilde{Q}_{lmn}^* = \sum_{l_1 + 2n_1 \leqslant M_0}
\tilde{g}_{lmn}^{l_1 m_1 n_1} \tilde{f}_{l_1 m_1 n_1}.
\end{align}
Using the storage pattern illustrated in Figure \ref{fig:f} and Figure
\ref{fig:A}, the matrix entries and the vector components involved in
the above operations are automatically continuously stored. The
details are illustrated in Figure \ref{fig:algo}. The left column of
Figure \ref{fig:algo} provides the color coding of the vectors. Each
vertical strip denotes the data structure represented on the right
column of Figure \ref{fig:f}. Since the coefficients with $l + 2n
\leqslant M_0$ and the coefficients with $l + 2n > M_0$ are treated
differently, we distinguish these two parts by shading with slanted
lines. The middle column shows the computation of
$\tilde{g}_{lmn}^{l_1 m_1 n_1}$ (equation \eqref{eq:tilde_g}) for
given $m$ and $m_1$, which is in fact just one matrix-vector
multiplication based on our data structure. The color coding of the
matrix $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}$ is the same as Figure
\ref{fig:A}. The right column gives the computation of
$\tilde{Q}_{lmn}^*$, which contains a matrix-vector multiplication (for
degree less than or equal to $M_0$, equation \eqref{eq:tilde_Q}) and a
vector scaling (for degree greater than $M_0$). The matrix
$\tilde{g}_{lmn}^{l_1 m_1 n_1}$ is a reshaping of the vector in the
middle column. By comparing the matrix form and the vector form of $A$
and $\tilde{g}$, one can observe our data structure automatically
corresponds to the row-major order of these matrices, which makes it
easy to use optimized BLAS libraries such as ATLAS \cite{Whaley2001}
to achieve high numerical efficiency.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{algorithm-1}
\caption{Illustration of the algorithm for given $m$, $m_1$ and $m_2$.}
\label{fig:algo}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
This work aims to model and simulate the binary collision between gas
molecules under the framework of the Burnett polynomials. The special
sparsity of the coefficients is fully utilized, and we have proposed a
method to compute the coefficients in the spectral expansion with high
accuracy based on the work \cite{QuadraticCol}. Moreover, the data
structure and the implementation of the algorithm are carefully
designed to achieve high numerical efficiency.
In order to provide further flexibility, especially when taking into
account the spatial inhomogeneity, we employ the modelling technique
used in \cite{Cai2015,QuadraticCol}, where the quadratic form is
preserved only for the first few moments. It is validated again that
the method is efficient in capturing the evolution of lower-order
moments. The implementation of the spatially inhomogeneous Boltzmann
equation is in progress.
\section*{Acknowledgements} We would like to thank Prof. Ruo Li at
Peking University, China for the valuable suggestions to this research
project. Zhenning Cai is supported by National University of Singapore
Startup Fund under Grant No. R-146-000-241-133. Yanli Wang is
supported by the National Natural Scientific Foundation of China
(Grant No. 11501042) and Chinese Postdoctoral Science Foundation of
China (2018M631233).
\section{Numerical examples} \label{sec:numerical}
In this section, we will show some results of our numerical simulation. In all
the numerical experiments, we consider the inverse-power-law model, for which
the repulsive force between two molecules is proportional to $r^{-\eta}$, with
$r$ and $\eta$ being, respectively, the distance between the two molecules and
a given positive constant. The details about this model can be found in
\cite{Bird}. In all the tests, we use the classical fourth-order Runge-Kutta
method to the equations \eqref{eq:ODE} numerically for some given $M_0$ and
$M$, and the time step is chosen as $\Delta t = 0.01$.
For visualization purposes, we define integration operators $\mathcal{I}_1:
L^1(\mathbb{R}^3) \rightarrow L^1(\mathbb{R})$ and $\mathcal{I}_2:
L^1(\mathbb{R}^3) \rightarrow L^1(\mathbb{R}^2)$ by
\begin{displaymath}
(\mathcal{I}_1 f)(v_1) = \int_{\mathbb{R}} \int_{\mathbb{R}}
f(\bv) \,\mathrm{d}v_2 \,\mathrm{d}v_3, \qquad \forall f\in L^1(\mathbb{R}^3),
\end{displaymath}
and
\begin{displaymath}
(\mathcal{I}_2 f)(v_1, v_2) = \int_{\mathbb{R}} f(\bv) \,\mathrm{d}v_3,
\qquad \forall f \in L^1(\mathbb{R}^3).
\end{displaymath}
These 1D and 2D functions are actually marginal distribution functions (MDFs).
We will only show the plots for these MDFs due to the difficulty in plotting
three-dimensional functions.
Besides, we are also interested in the evolution of the moments of the
distribution function, especially the heat flux $q_i$ and the stress tensor
$\sigma_{ij}$. For a given distribution function $f \in \mathcal{S}$, they are
defined as
\begin{equation}
\label{eq:heatflux_stress}
q_i = \frac{1}{2}\int_{R^3}|\bv|^2 v_i f(\bv) \dd \bv, \qquad
\sigma_{ij} = \int_{\bbR^3}\left(
v_i v_j - \frac{1}{3}\delta_{ij}|\bv|^2
\right) f(\bv) \dd \bv, \qquad i, j = 1, 2, 3.
\end{equation}
The relations between these moments and the coefficients are
\begin{equation}
\label{eq:moment_q_sigma}
\begin{aligned}
& q_1 = \sqrt{5}{\rm Re}(\tilde{f}_{111}), \quad
q_2 = -\sqrt{5}{\rm Im}(\tilde{f}_{111}), \quad
q_3 = -\sqrt{5/2} \tilde{f}_{101}, \\
& \sigma_{11} = \sqrt{2}{\rm Re}(\tilde{f}_{220}) - \tilde{f}_{200}/\sqrt{3}, \quad
\sigma_{12} = -\sqrt{2}{\rm Im}(\tilde{f}_{220}), \quad
\sigma_{13} = -\sqrt{2}{\rm Re}(\tilde{f}_{210}), \\
& \sigma_{22} = -\sqrt{2}{\rm Re}(\tilde{f}_{220}) - \tilde{f}_{200}/\sqrt{3}, \quad
\sigma_{23} = \sqrt{2}{\rm Im}(\tilde{f}_{210}), \quad
\sigma_{33} = 2\tilde{f}_{200}/\sqrt{3}.
\end{aligned}
\end{equation}
\subsection{BKW (Bobylev-Krook-Wu) solution}
In this example, we study the Maxwell gas whose the power index $\eta$
equals $5$. In this case, the kernel $B(g,\chi)$ turns out to be
independent of $g$ (therefore denoted by $B(\chi)$ below), and it is
given in \cite{Bobylev1984, krook1977exact} that the spatially
homogeneous Boltzmann equation \eqref{eq:Boltzmann} admits an exact
solution $F(t) = f^{[\tau(t)]}$, where
\begin{align*}
& \tau(t) = 1 - \frac{2}{5}\exp(-\lambda t),
\qquad \lambda = \frac{\pi}{2} \int_0^{\pi} B(\chi) \sin^2 \chi \dd \chi, \\
& f^{[\tau]}(\bv) = (2\pi \tau)^{-3/2} \exp \left( -\frac{|\bv|^2}{2\tau} \right)
\left[ 1 + \frac{1-\tau}{\tau}
\left( \frac{|\bv|^2}{2\tau} - \frac{3}{2} \right) \right].
\end{align*}
The initial MDFs are plotted in Figure \ref{fig:ex1_init}, in which the contour
lines for exact functions and their numerical approximation are hardly
distinguishable, with the number $M = 20$.
\begin{figure}[!ht]
\centering
\subfigure[Initial MDF $\mathcal{I}_1 f^0$\label{fig:ex1_init_1d}]{%
\includegraphics[width=0.28\textwidth, clip]{ex1_1d_t=0.pdf}
}\quad
\subfigure[Contours of $\mathcal{I}_2 f^0$\label{fig:ex1_init_2d_contour}]{%
\includegraphics[width=0.28\textwidth, clip]{ex1_2d_contour_t=0.pdf}
}\quad
\subfigure[Initial MDF $\mathcal{I}_2 f^0$\label{fig:ex1_init_2d}]{%
\includegraphics[width=0.32\textwidth, clip]{ex1_2d_t=0.pdf}
}
\caption{Initial marginal distribution functions. In (a), the red line
corresponds to the exact solution, while black dashed line
corresponds to $M = 20$ respectively. In (b), the blue solid lines
correspond to the exact solution, and the red dashed lines
correspond to the numerical approximation $M = 20$. Figure (c) shows
only the numerical approximation with $M = 20$.}
\label{fig:ex1_init}
\end{figure}
In Figure \ref{fig:ex1_1d}, the marginal distribution functions
$\mathcal{I}_1F(t)$ at $t = 0.2$, $0.4$ and $0.6$ are shown. Here, $M_0$ is set
as $5$ and $20$. The marginal distribution functions $\mathcal{I}_2F(t)$ are
plotted in Figures \ref{fig:ex1_2d_M0=5_20} and \ref{fig:ex1_2d_M0=20_20},
respectively for $M_0=5$ and $20$. For $M_0 = 5$, the numerical solution
provides a reasonable approximation, but still has noticeable deviations, while
for $M_0 = 20$, the two solutions match perfectly in all cases.
\begin{figure}[!ht]
\centering
\subfigure[$t=0.2$]{%
\includegraphics[width=.30\textwidth]{ex1_1d_t=02.pdf}
}\hfill
\subfigure[$t=0.4$]{%
\includegraphics[width=.30\textwidth]{ex1_1d_t=04.pdf}
}\hfill
\subfigure[$t=0.6$]{%
\includegraphics[width=.30\textwidth]{ex1_1d_t=06.pdf}
}
\caption{Marginal distribution functions $\mathcal{I}_1 F(t)$ at different times.}
\label{fig:ex1_1d}
\end{figure}
\begin{figure}[!ht]
\centering
\subfigure[$t=0.2$]{%
\includegraphics[width=.3\textwidth]{ex1_2d_contour_t=02_M0=5_20.pdf}
} \hfill
\subfigure[$t=0.4$]{%
\includegraphics[width=.3\textwidth]{ex1_2d_contour_t=04_M0=5_20.pdf}
} \hfill
\subfigure[$t=0.6$]{%
\includegraphics[width=.3\textwidth]{ex1_2d_contour_t=06_M0=5_20.pdf}
}
\caption{Comparison of numerical results using $M_0 = 5$ and the exact
solution. The blue contours and the red dashed contours are
respectively the results for $M_0 = 5$ and the exact solution.}
\label{fig:ex1_2d_M0=5_20}
\end{figure}
\begin{figure}[!ht]
\centering
\subfigure[$t=0.2$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex1_2d_contour_t=02_M0=20_20.pdf}
\end{overpic}
} \quad
\subfigure[$t=0.4$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex1_2d_contour_t=04_M0=20_20.pdf}
\end{overpic}
} \quad
\subfigure[$t=0.6$]{%
\begin{overpic}
[width= .3\textwidth, clip]{ex1_2d_contour_t=06_M0=20_20.pdf}
\end{overpic}
}
\caption{Comparison of numerical results using $M_0 = 20$ and the
exact solution. The blue contours and the red dashed contours are
respectively the results for $M_0 = 20$ and the exact solution.}
\label{fig:ex1_2d_M0=20_20}
\end{figure}
Now we consider the time evolution of the coefficients. By expanding the exact
solution into Burnett series, we get the exact solution for the coefficients:
\begin{equation}
\label{eq:exact_ex1}
\tilde{F}_{lmn}(t) = \left\{\begin{array}{ll}
\sqrt{\frac{2\Gamma(n + 3/2)}{\sqrt{\pi} n!}}(1 - n)( 1- \tau(t))^n, &
l = m = 0, \quad n\in \bbN, \\[13pt]
0, & \text{otherwise}.
\end{array} \right.
\end{equation}
Due to the symmetry of the distribution function, the coefficients
$\tilde{F}_{lmn}$ are nonzero for any $t$ only when both of $l$ and $m$ are
zero. From \eqref{eq:exact_ex1}, we see that $\tilde{F}_{000} = 1$ and
$\tilde{F}_{001} = 0$ for any $t$. Hence we will focus on the coefficients
$\tilde{F}_{00n}, n = 2,\cdots,5$. For Maxwell molecules, the discrete kernel
$A_{lmn}^{l_1m_1n_1, l_2m_2n_2}$ is nonzero only when $l + 2n = l_1 + 2 n_1 +
l_2 + 2n_2$. Therefore, for any $M \geqslant M_0 \geqslant 10$, the numerical
results for these coefficients are exactly the same (regardless of round-off
errors), and we just show the results for $M_0 = M =20$ here. Figure
\ref{fig:ex1_moments} gives the comparison between the numerical solution and
the exact solution for these coefficients. In all plots, the two lines almost
coincide with each other.
\begin{figure}[!ht]
\centering
\subfigure[$\tilde{F}_{002}(t)$]{%
\includegraphics[width=.4\textwidth]{ex1_n=2.pdf}
} \hspace{20pt}
\subfigure[$\tilde{F}_{003}(t)$]{%
\includegraphics[width=.4\textwidth]{ex1_n=3.pdf}
} \\
\subfigure[$\tilde{F}_{004}(t)$]{%
\includegraphics[width=.4\textwidth]{ex1_n=4.pdf}
} \hspace{20pt}
\subfigure[$\tilde{F}_{005}(t)$]{%
\includegraphics[width=.4\textwidth]{ex1_n=5.pdf}
}
\caption{The evolution of the coefficients. The red lines correspond to the
reference solution, and the blue dashed lines correspond to the numerical
solution.}
\label{fig:ex1_moments}
\end{figure}
\subsection{Quadruple-Gaussian initial data}
In this example, we perform the numerical test for the hard potential
case where the power index $\eta$ equals $10$. The initial distribution
function is
\begin{equation}
\begin{aligned}
f^0(\bv) = \frac{1}{4\pi^{3/2}} \left[ \exp \Big( -\frac{(v_1 +
u)^2 + v_2^2 + v_3^2 }{2\theta}\Big) +
\exp \Big( -\frac{(v_1 - u)^2 + v_2^2 + v_3^2}{2\theta} \Big)\right. \\
\left. + \exp \Big( -\frac{v_1^2 + (v_2+u)^2 + v_3^2}{2\theta}
\Big) + \exp \Big( -\frac{v_1^2 + (v_2- u)^2 + v_3^2}{2\theta}
\Big) \right],
\end{aligned}
\end{equation}
where $u = \sqrt{2}$ and $\theta = 1/3$. In all our numerical tests,
we use $M = 40$, which gives a good approximation of the initial
distribution function (see Figure \ref{fig:ex2_init}).
\begin{figure}[!ht]
\centering
\subfigure[Initial MDF $\mathcal{I}_1 f^0$\label{fig:ex2_init_1d}]{%
\begin{overpic}[width=0.28\textwidth, clip]{ex2_1d_t=0.pdf}
\end{overpic}
}\quad
\subfigure[Contours of $\mathcal{I}_2 f^0$\label{fig:ex2_init_2d_contour}]{%
\begin{overpic}
[width=0.3\textwidth, clip]{ex2_2d_contour_t=0.pdf}
\end{overpic}
}\quad
\subfigure[Initial MDF $\mathcal{I}_2 f^0$\label{fig:ex2_init_2d}]{%
\begin{overpic}
[width=0.32\textwidth, clip]{ex2_2d_t=0.pdf}
\end{overpic}
}
\caption{Initial marginal distribution functions. In (a), the red
line corresponds to the exact solution, while black dashed line
corresponds to $M = 40$ respectively. In (b), the blue solid lines
correspond to the exact solution, and the red dashed lines
correspond to the numerical approximation $M = 40$. Figure (c)
shows only the numerical approximation $M=40$.}
\label{fig:ex2_init}
\end{figure}
For this example, we set the numerical result with $M_0=20$ as the
reference solution. The numerical results for $M_0=5$, $10$, $15$ are
given respectively in Figure \ref{fig:ex2_2d_M0=5_40},
\ref{fig:ex2_2d_M0=10_40}, and \ref{fig:ex2_2d_M0=15_40}. For each
$M_0$, the marginal distribution functions $\mathcal{I}_2F(t)$ at
$t=0.1$, $0.2$ and $0.3$ are shown.
Due to the high nonequilibrium of this example, when $M_0 = 5$ and
$10$, the ``size'' of the quadratic part in the collision term is too
small to describe the evolution of the distribution function, while
the numerical results for $M_0 = 15$ and $M_0 = 20$ agree well with
each other (except the central area for $t=0.1$, where the
distribution function is very flat). This indicates the observation of
numerical convergence, meaning that $M_0=15$ is sufficient to describe
the evolution of the distribution function. This example is a harder
version of the bi-Gaussian initial data used in \cite{QuadraticCol}
for Hermite basis functions, and therefore requires more degrees of
freedom to give satisfactory numerical results. However, as will be
shown later, by using Burnett basis functions, the computational cost
for $M_0 = 20$ is even smaller than the computational cost for
$M_0 = 15$ using Hermite basis functions, even if the results are
essentially identical.
\begin{figure}[!ht]
\centering
\subfigure[$t=0.1$]{%
\includegraphics[width=.3\textwidth]{ex2_2d_contour_t=01_M0=5_40.pdf}
} \hfill
\subfigure[$t=0.2$]{%
\includegraphics[width=.3\textwidth]{ex2_2d_contour_t=02_M0=5_40.pdf}
} \hfill
\subfigure[$t=0.3$]{%
\includegraphics[width=.3\textwidth]{ex2_2d_contour_t=03_M0=5_40.pdf}
}
\caption{Comparison of numerical results using $M_0 = 5$ and
$M_0 = 20$. The blue contours and the red dashed contours are
respectively the results for $M_0 = 5$ and $M_0 = 20$.}
\label{fig:ex2_2d_M0=5_40}
\end{figure}
\begin{figure}[!ht]
\centering
\subfigure[$t=0.1$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex2_2d_contour_t=01_M0=10_40.pdf}
\end{overpic}
} \quad
\subfigure[$t=0.2$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex2_2d_contour_t=02_M0=10_40.pdf}
\end{overpic}
} \quad
\subfigure[$t=0.3$]{%
\begin{overpic}
[width= .3\textwidth, clip]{ex2_2d_contour_t=03_M0=10_40.pdf}
\end{overpic}
}
\caption{Comparison of numerical results using $M_0 = 10$ and
$M_0 = 20$. The blue contours and the red dashed contours are
respectively the results for $M_0 = 10$ and $M_0 = 20$.}
\label{fig:ex2_2d_M0=10_40}
\end{figure}
\begin{figure}[!ht]
\centering
\subfigure[$t=0.1$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex2_2d_contour_t=01_M0=15_40.pdf}
\end{overpic}
} \hfill
\subfigure[$t=0.2$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex2_2d_contour_t=02_M0=15_40.pdf}
\end{overpic}
} \hfill
\subfigure[$t=0.3$]{%
\begin{overpic}
[width=.3\textwidth, clip]{ex2_2d_contour_t=03_M0=15_40.pdf}
\end{overpic}
}
\caption{Comparison of numerical results using $M_0 = 15$ and
$M_0 = 20$. The blue contours and the red dashed contours are
respectively the results for $M_0 = 15$ and $M_0 = 20$.}
\label{fig:ex2_2d_M0=15_40}
\end{figure}
Now we consider the evolution of the moments. In this example, the
stress tensor and heat flux satisfy
$\sigma_{11} = \sigma_{22} = -0.5\sigma_{33}$ and
$q_i = 0, i = 1,2,3$. Therefore, we focus only on the evolution of
$\sigma_{11}$, which is plotted in Figure \ref{fig:ex2_sigma11}. It
can be seen that the four tests give almost identical results. Even
for $M_0 = 5$ and $10$, while the distribution functions are not
approximated very well, the evolution of the stress tensor is very
accurate.
\begin{figure}[!ht]
\centering
\includegraphics[width=.4\textwidth]{ex2_sigma11.pdf}
\caption{Evolution of $\sigma_{11}(t)$. Four lines are on top of each other.}
\label{fig:ex2_sigma11}
\end{figure}
\subsection{Discontinuous initial data}
In this example, we reconsider the problem with the same discontinuous initial
condition as in \cite{QuadraticCol}:
\begin{displaymath}
f^0(\bv) = \left\{ \begin{array}{ll}
\dfrac{\sqrt[4]{2} (2-\sqrt{2})}{\pi^{3/2}}
\exp \left( -\dfrac{|\bv|^2}{\sqrt{2}} \right), & \text{if } v_1 > 0, \\[10pt]
\dfrac{\sqrt[4]{2} (2-\sqrt{2})}{4\pi^{3/2}}
\exp \left( -\dfrac{|\bv|^2}{2\sqrt{2}} \right), & \text{if } v_1 < 0.
\end{array} \right.
\end{displaymath}
In \cite{QuadraticCol}, the authors used Hermite spectral method to do the
computation up to $M_0 = 15$, which still shows significant difference in the
numerical results compared with $M_0 = 10$. In this paper, we are going to
confirm the reliability of the results obtained with $M_0 = 15$. As in
\cite{QuadraticCol}, we only focus on the evolution of the moments. The
numerical results for the hard potential $\eta = 10$ and soft potential $\eta =
3.1$ with different choices of $M_0$ and $M$ are shown in Figure
\ref{fig:ex3_moments}. For $\eta = 3.1$, the horizontal axes are the scaled
time $t_s = t / \tau$ with $\tau \approx 2.03942$ as in \cite{QuadraticCol}, so
that the two models have the same mean relaxation time near equilibrium.
Since for the homogeneous Boltzmann equation, the behaviors of stress tensor
and heat flux are the same for any $M \geqslant M_0 \geqslant 3$, we let $M =
M_0 = 5$, $10$, $15$ and $20$, and the results are plotted in Figure
\ref{fig:ex3_moments}. The numerical results for $M_0 = 5$, $10$ and $15$ are
exactly the same as \cite{QuadraticCol}. However, due to the significant
enhancement of computational efficiency, we can get the results for $M_0 = 20$,
and the results are almost the same as those for $M_0 = 15$, which indicates
that they should be very close to the exact solution. The whole pictures of
$\sigma_{11}$ and $\sigma_{22}$ show much clearer converging trend of the
numerical solutions with increasing $M_0$, compared with the numerical results
in \cite{QuadraticCol} where the profiles for $M_0 = 20$ was not present. For
the heat flux $q_1$, not surprisingly, the four results are hardly
distinguishable.
\begin{figure}[!ht]
\centering
\subfigure[$\sigma_{11}(t)$ ($\eta = 10$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta10_sigma11.pdf}
} \quad
\subfigure[$\sigma_{11}(t)$ ($\eta = 3.1$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta31_sigma11.pdf}
} \\
\subfigure[$\sigma_{22}(t)$ ($\eta = 10$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta10_sigma22.pdf}
} \quad
\subfigure[$\sigma_{22}(t)$ ($\eta = 3.1$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta31_sigma22.pdf}
} \\
\subfigure[$q_1(t)$ ($\eta = 10$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta10_q1.pdf}
}\quad
\subfigure[$q_1(t)$ ($\eta = 3.1$)]{%
\includegraphics[width=.4\textwidth]{ex3_eta31_q1.pdf}
}
\caption{Evolution of the stress and the heat flux. The left column
shows the results for $\eta = 10$, and the right column shows the
results for $\eta = 3.1$. In the right column, the horizontal axes
are the scaled time.}
\label{fig:ex3_moments}
\end{figure}
Finally, the computational time for one evaluation of the quadratic collision
term under the framework of Burnett series and Hermite series
\cite{QuadraticCol} is plotted in Figure \ref{fig:ex3_time_comparison}. Here,
the number $M$ is fixed as $M = 20$, and $M_0$ increases from $5$ to $20$. It
is clear that the computational cost is greatly reduced by using Burnett basis
functions, especially when $M$ is large.
\begin{figure}[!ht]
\centering
\includegraphics[width=.4\textwidth]{ex3_time_1.pdf}
\caption{Comparison of the computational time for one evaluation of the
collision operator using the method in this paper and that in
\cite{QuadraticCol}. $M$ is fixed as $M=20$ and $M_0$ changes from $5$ to
$20$. The $x$-axis is $M_0$ and the $y$-axis is the logarithm of the
computational time.}
\label{fig:ex3_time_comparison}
\end{figure}
\section{Introduction}
The Boltzmann equation possesses its unambiguous significance in the rarefied
gas dynamics. Using a velocity distribution function $f \in L^1(\mathbb{R}^3)$
to describe the statistical behavior of gas molecules, the Boltzmann equation
incorporates the transport and the collision of particles into a single
equation, which accurately models the gas flow from transitional to free
molecular regimes. By the molecular chaos assumption, the collision between
molecules gives the rate of change for the distribution function as follows:
\begin{equation} \label{eq:quad_col}
\mQ[f,f](\bv) = \int_{\mathbb{R}^3}
\int_{\bn \perp \bg} \int_0^{\pi} B(|\bg|,\chi)
[f(\bv_1') f(\bv') - f(\bv_1) f(\bv)]
\dd\chi \dd\bn \dd\bv_1,
\end{equation}
where $\bn$ is a unit vector and
\begin{align*}
\bg &= \bv - \bv_1, \\
\bv' &= \cos^2(\chi/2) \bv + \sin^2(\chi/2) \bv_1
- |\bg| \cos(\chi/2) \sin(\chi/2) \bn, \\
\bv_1' &= \cos^2(\chi/2) \bv_1 + \sin^2(\chi/2) \bv
+ |\bg| \cos(\chi/2) \sin(\chi/2) \bn.
\end{align*}
The collisional kernel $B(\cdot,\cdot)$ is a nonnegative function involving the
differential cross-section of the collision dynamics. Such a high-dimensional
integral form introduces great difficulty to the numerical simulation of the
Boltzmann equation, and people have been using the stochastic method introduced
by Bird \cite{Bird1963, Bird}, known as direct simulation of Monte Carlo
(DSMC), to solve the Boltzmann equation. Due to the fast development of super
computers, in the past decade, a number of deterministic methods have been
proposed to discretize the integral collision term to avoid numerical
oscillations. The most promising method seems to be the Fourier spectral method
\cite{Pareschi1996, Mouhot2006, Hu2017}, including a variety of its variations
such as the conservative version \cite{Gamba2009}, the positivity preserving
version \cite{Pareschi2000}, the steady-state preserving version
\cite{filbet2015steady} and the entropic version \cite{Cai2019}, where the
technique of fast Fourier transform can be applied to accelerate the
computation. These methods has been applied to spatially inhomogeneous problems
in \cite{Wu2013, Wu2014, Dimarco2018}. Other methods include the fast discrete
velocity method \cite{Mouhot2013} and the discontinuous Galerkin method
\cite{Alekseenko2014}.
Another type of spectral method based on global orthogonal polynomials is also
being studied recently \cite{Cai2015,Gamba2018,QuadraticCol}. In this paper, we
follow the work \cite{Cai2015, Gamba2018} and adopt the spectral method based
on Burnett polynomials \cite{Burnett1936}, which has been applied to the
linearized Boltzmann equation \cite{Cai2015, Cai2018}, and shows great
potential to achieve higher numerical efficiency. To focus on the collision
term, we consider only the spatially homogeneous Boltzmann equation, meaning
that the distribution function is uniform in space, and thus we can use a map
$F: \mathbb{R}_+ \rightarrow L^1(\mathbb{R}^3)$ to describe the evolution of
the distribution function:
\begin{equation} \label{eq:Boltzmann}
\begin{aligned}
& \frac{\mathrm{d}F(t)}{\mathrm{d}t} = \mQ[F(t),F(t)],
\qquad \forall t\in (0,+\infty), \\
& F(0) = f^0 \in L^1(\mathbb{R}^3).
\end{aligned}
\end{equation}
It is well-known that the Boltzmann equation preserves the conservation of
mass, momentum and energy:
\begin{equation} \label{eq:conservation}
\int_{\bbR^3} \begin{pmatrix} 1 \\ \bv \\ |\bv|^2 \end{pmatrix} \mQ[f,f](\bv)
\dd \bv = 0, \qquad \forall f \in L^1(\mathbb{R}^3).
\end{equation}
Thus we can choose appropriate nondimensionalization such that the initial
value $f^0$ in \eqref{eq:Boltzmann} belongs to the following set:
\begin{displaymath}
\mathcal{S} = \left\{ f \in L^1(\bbR^3) : \int_{\bbR^3}
\begin{pmatrix} 1 \\ \bv \\ |\bv|^2 \end{pmatrix} f(\bv) \dd \bv =
\begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \right\},
\end{displaymath}
and by \eqref{eq:conservation}, for all $t > 0$, we always have $F(t) \in
\mathcal{S}$. According to Boltzmann's H-theorem, the steady-state solution of
\eqref{eq:Boltzmann} is the Maxwellian
\begin{equation}
\label{eq:Maxwellian}
\mathcal{M}(\bv) = \frac{1}{(\sqrt{2\pi})^3}
\exp \left( -\frac{|\bv|^2}{2} \right).
\end{equation}
The Burnett polynomials, which will be used in our discretization, are
orthogonal polynomials associated with the weight function $\mathcal{M}(\bv)$.
Therefore our numerical method can represent this steady-state solution
exactly.
In \cite{QuadraticCol, Hu2018}, a similar method using Hermite polynomials,
which are also orthogonal polynomials associated with the weight function
$\mathcal{M}(\bv)$, is studied. In principle, the spectral methods using
Burnett and Hermite polynomials are essentially equivalent, especially when the
series is truncated up to the same degree. The Hermite spectral method was
introduced long ago by Grad \cite{Grad1949} as the moment method. As mentioned
in \cite[pp. 283]{Grad1958}, the Hermite spectral method is frequently
advantageous due to ``the symmetries inherent in the invariant Cartesian
tensor''. Such an advantage has been utilized in \cite{QuadraticCol}, where the
explicit expressions of all the coefficients in the discretization are
formulated using these symmetries. However, the superiority of the Burnett
polynomials introduced in \cite{Burnett1936} is the fact that they are
eigenfunctions of the linearized collision integral for Maxwell molecules
\cite{Chang1952}. Even for non-Maxwell molecules, as will be shown in this
paper, the coefficients also possess some sparsity due to the rotational
invariance of the collision operator. This will result in a considerably faster
algorithm in the computation, which makes the spectral method with Burnett
polynomials preferable in the simulation.
The same basis functions have been used in \cite{Gamba2018}, where the authors
employed numerical integration to find all the coefficients involved in the
discretization of \eqref{eq:Boltzmann}, but the sparsity in the coefficients
was not utilized in the computation. In this paper, we are going to focus on
the detailed implementation of the algorithm, including a much more accurate
way to compute the coefficients, a detailed analysis of the computational cost,
and the design of the data structure to achieve high computational efficiency.
Meanwhile, we also emphasize the modelling technique introduced in
\cite{QuadraticCol} which allows flexible balancing between computational cost
and modelling error.
The rest of this paper is organized as follows. In Section \ref{sec:general},
we present the framework of the Burnett spectral method to solve the
homogeneous Boltzmann equation. In Section \ref{sec:impl}, the detailed
implementation of the algorithm is introduced. We first give an efficient
method to compute the coefficients in the Burnett spectral expansion, and then
discuss the design of the data structure and the implementation of the
algorithm in detail. Some numerical experiments verifying the efficiency of
the Burnett spectral method are carried out in Section \ref{sec:numerical}.
In Section \ref{sec:proof}, we list the proof of the theorems in Section
\ref{sec:general}. Some concluding remarks are made in Section
\ref{sec:conclusion}.
\section{Proof of theorems}\label{sec:proof}
In this section, we prove the three theorems in Section \ref{sec:general}.
Firstly, we introduce two lemmas as following:
\begin{lemma}\label{lem:rotation}
Let $\mathbf{R}$ be an $3\times 3$ orthogonal matrix. Define the
rotation operator $\mathcal{R}$ by
\begin{displaymath}
(\mathcal{R}f)(\bv) = f(\mathbf{R} \bv), \qquad
\forall f: \bbR^3 \rightarrow \mathbb{C}.
\end{displaymath}
Then when $\mathcal{Q}[f,g]$ is well-defined for some functions $f$
and $g$, we have $\mathcal{Q}[f,g](\mathbf{R}\bv) =
\mathcal{Q}[\mathcal{R}f, \mathcal{R}g](\bv)$.
\end{lemma}
\begin{lemma}\label{lem:Talmi}
Talmi coefficient $\Talmi{l_1m_1n_1}{l_2m_2n_2}{l_3m_3n_3}{l_4m_4n_4}$
is zero if
\begin{equation}
l_1+2n_1+l_2+2n_2 \neq l_3+2n_3+l_4+2n_4.
\end{equation}
\end{lemma}
The first lemma is a well-known result and we are not going to prove it in this
paper. The proof of the second lemma can be find in \cite[page 135-137]{Kumar}.
Now we start to prove the theorems.
\begin{proof}[Proof of Theorem \ref{thm:sparsity}]
For any $\eta \in \bbR$, we define the rotation matrix
\begin{displaymath}
\mathbf{R}_{\eta} = \begin{pmatrix}
\cos\eta & -\sin\eta & 0 \\ \sin\eta & \cos\eta & 0 \\ 0 & 0 & 1
\end{pmatrix}.
\end{displaymath}
Using spherical coordinates $\bv = (r \sin\theta \cos\phi, r
\sin\theta \sin\phi, r \cos\theta)^T$, one can see that
\begin{displaymath}
\mathbf{R}_{\eta} \bv = (r \sin\theta \cos(\phi + \eta), r
\sin\theta \sin(\phi + \eta), r \cos\theta)^T.
\end{displaymath}
Therefore
\begin{displaymath}
p_{lmn}(\mathbf{R}_{\eta}\bv) =
\mathrm{e}^{\mathrm{i} m \eta} p_{lmn}(\bv), \qquad
\varphi_{lmn}(\mathbf{R}_{\eta}\bv) =
\mathrm{e}^{\mathrm{i} m \eta} \varphi_{lmn}(\bv).
\end{displaymath}
Let $\mathcal{R}_{\eta}$ be the rotation operator such that
$(\mathcal{R}_{\eta} f)(\bv) = f(\mathbf{R}_{\eta} \bv)$. Now we can
rewrite \eqref{eq:A} as
\begin{equation} \label{eq:A_equality}
\begin{split}
A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2} &= \int_{\bbR^3}
\overline{p_{lmn}(\mathbf{R}_{\eta} \bv)}
\mQ[\varphi_{l_1 m_1 n_1}, \varphi_{l_2 m_2 n_2}](\mathbf{R}_{\eta} \bv)
\dd \bv \\
&= \int_{\bbR^3} \mathrm{e}^{-\mathrm{i} m \eta} \overline{p_{lmn}(\bv)}
\mQ[\mathcal{R}_{\eta} \varphi_{l_1 m_1 n_1},
\mathcal{R}_{\eta} \varphi_{l_2 m_2 n_2}](\bv) \dd \bv \\
&= \mathrm{e}^{\mathrm{i} (m_1+m_2-m) \eta}
\int_{\bbR^3} \overline{p_{lmn}(\bv)}
\mQ[\varphi_{l_1 m_1 n_1}, \varphi_{l_2 m_2 n_2}](\bv) \dd \bv \\
&= \mathrm{e}^{\mathrm{i} (m_1+m_2-m) \eta}
A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2}.
\end{split}
\end{equation}
Here we have used the rotational invariance and bilinearity of the
collision operator $\mQ[\cdot, \cdot]$. Note that
\eqref{eq:A_equality} holds for any $\eta$. If $m \neq m_1 + m_2$,
this shows that $A_{lmn}^{l_1 m_1 n_1, l_2 m_2 n_2} = 0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:sparsityMaxwell}]
Since $B(g,\chi)=\sigma(\chi)$ is independent of $g$, in
\eqref{eq:def_Vnn}, the integrals with respect to $g$ and $\chi$
can be split:
\begin{equation}
\begin{split}
V_{n n'}^l &= \frac{1}{16\sqrt{2} \pi^{5/2}}
\sqrt{\frac{n!n'!}{\Gamma(n+l+3/2) \Gamma(n'+l+3/2)}}
\int_0^{\pi} \sigma(\chi) [(2l+1)^2 P_l(\cos \chi) - 1] \dd\chi
\times{} \\
& \qquad \int_0^{+\infty}
\left( \frac{g^2}{4} \right)^{l+1}
L_n^{(l+1/2)} \left( \frac{g^2}{4} \right)
L_{n'}^{(l+1/2)} \left( \frac{g^2}{4} \right)
\exp\left( -\frac{g^2}{4} \right) \dd g,
\end{split}
\end{equation}
which vanishes if $n\neq n'$, due to the orthogonality of Laguerre
polynomials. Hence, using Lemma \ref{lem:Talmi}, we can obtain that
the summands in \eqref{eq:A_lmn} do not vanish only if
\begin{equation*}
n_4=n_4',\quad
l_3+2n_3+l_4+2n_4'=l+2n,\quad
l_3+2n_3+l_4+2n_4=l_1+2n_1+l_2+2n_2.
\end{equation*}
Direct simplification yields the conclusion in the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:real}]
Set $\mathbf{R}=\diag\{1,-1,1\}$, then using the same approach as that in
the proof of Theorem \ref{thm:sparsity}, one can directly prove this
theorem.
\end{proof}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.